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\section{Introduction}
Modern remote sensing techniques allow recording enormous amounts of spatial
data, including geographical, natural resources, land use, and environmental remote sensing images, which need efficient (preferably real-time) processing~\cite{Rossi94,Foster08}. Such processing involves the reconstruction of missing data that often occur due to different reasons, such as equipment malfunctions and gaps in the coverage of the targeted area that appear as a result of restricted satellite paths or bad weather conditions~\cite{Jun04,Lehman04,Albert12,Bechle13,Emili11}. However, such massive data sets can not be efficiently handled by standard geostatistical methods, such as kriging~\cite{Wack03}. The main drawbacks of such methods are high computational complexity, difficulties to automatize the algorithms to work without subjective user inputs (selection of variogram and search radius for interpolation) and often the necessity of some data pre-processing (e.g., lognormal transformation) if the data does not comply with the Gaussianity requirement~\cite{Dig07}. Thus, there is a need to develop new spatial prediction techniques that overcome these shortcomings~\cite{Cressie08,Hartman08}.
Recently, we have proposed efficient spatial classification methods based on models inspired from statistical physics, in particular discrete spin Ising, Potts, and clock models, employing a heuristic ``energy matching'' principle~\cite{zuk09a,zuk09b}.
In the present study we extend the idea of using spin models for spatial prediction purposes and introduce a new method that is based on a continuous spin planar rotator model. Even though the above mentioned nonparametric discrete-spin-based classification models have been shown to be rather competitive, their performance was based on the assumption of the existence of relevant correlations at some unknown parameters and the prediction results were discrete values even for continuous data. As we empirically demonstrate, the modified planar rotator model {\it inherently} displays flexible short-range spatial correlations that vary significantly over the model's parameter space and could be used for spatial interpolation of continuous geostatistical data.
\section{Model and simulations}
\subsection{Standard planar rotator model}
The Hamiltonian of the standard two-dimensional planar rotator model with nearest-neighbor interactions on a square lattice is defined as
\begin{equation}
\label{Hamiltonian}
H=-J\sum_{\langle i,j \rangle}{\bm s}_{i}{\bm s}_{j}=-J\sum_{\langle i,j \rangle}\cos(\phi_{i}-\phi_j),
\end{equation}
where ${\bm s}_i=(\cos\phi_i,\sin\phi_i)$ is a continuous spin on $i$-th lattice site, represented by a two-dimensional unit vector, $\phi_i \in [0,2\pi]$ is an angle associated with the spin ${\bm s}_i$, $J$ is an exchange interaction parameter and $\langle i,j \rangle$ denotes the sum over nearest neighbors. The Mermin-Wagner theorem~\cite{merm66} prevents any long-range ordering at finite temperatures in such a model, which has, however, been intensively studied in connection with quasi long-range ordering (the so called Kosterlitz-Thouless phase) that appears at low temperatures~\cite{kost73,kost74}. The phase transition is produced by the unbinding of vortex-antivortex pairs at the Kosterlitz-Thouless critical temperature $T_{\rm KT}$, below which all spins are almost aligned even though true long-range order is destroyed by spin fluctuations. The quasi long-range ordering for $T<T_{\rm KT}$ is characterized by the power-law decaying correlation function
\begin{equation}
\label{cor_fun}
C(h;\eta)= \langle {\bm s}(h){\bm s}(0) \rangle = \Big(\frac{h}{L}\Big)^{-\eta(T)},
\end{equation}
where $h$ is the spin separation distance, $L$ is the linear lattice size, and $\eta(T)=T/2\pi J$ is the temperature-dependent exponent.
\subsection{Modified planar rotator model}
Even though the algebraic correlations with the tunable exponent $\eta(T)$ could be relevant for modeling of some spatial processes, there are some deficiencies that prevent the straightforward use of the standard planar rotator model for such purposes. First of all, one needs to define a proper mapping between the spin values and the geostatistical values. Geostatistical data are often positively correlated reflecting a degree of spatial continuity, which means that neighbors with similar values are more likely (i.e., have lower energy) than those with different values. Apparently, this condition is not fulfilled in the standard planar rotator model, described by the Hamiltonian~(\ref{Hamiltonian}); the latter, for example, assigns equal energies to a pair of neighbors whose angles differ by $\theta$ as to a pair whose angles differ by $2\pi+\theta$. This degeneracy could be fixed by defining the energy functional so that it monotonically increases with the turn angle between the neighboring spins on the entire interval $[0,2\pi]$. The latter can be achieved by modifying the Hamiltonian~(\ref{Hamiltonian}) to take the following form of the modified planar rotator (MPR) model
\begin{equation}
\label{Hamiltonian_mod}
H'=-J\sum_{\langle i,j \rangle}\cos[q(\phi_{i}-\phi_j)],
\end{equation}
where $0 < q \leq 1/2$ is a modification factor. In the following we will use a fixed value of $q = 1/2$. The difference between the nature of spatial correlations in the standard and modified models is apparent from snapshots of spin configurations (turn angles) in the low-temperature region, shown in Fig.~\ref{fig:snaps}. While the presence of the vortex-antivortex pairs shows up in Fig.~\ref{fig:p1} in the form of sharp boundaries between the domains of similarly oriented spins, the variation of the spin values in the MPR model, shown in Fig.~\ref{fig:p1_2} is rather smooth, as it is typical for geostatistical data.
\begin{figure}[t]
\centering
\subfigure{\includegraphics[scale=0.5,clip]{snap_t00001_mcs1000_pbc_q1}\label{fig:p1}}\hspace*{5mm}
\subfigure{\includegraphics[scale=0.5,clip]{snap_t00001_mcs1000_pbc_q1_2}\label{fig:p1_2}}
\caption{Snapshots of spin configurations in the low-temperature regime ($T=10^{-4}$) for (a) standard and (b) modified planar rotator model, with periodic boundary conditions.}\label{fig:snaps}
\end{figure}
\subsection{Monte Carlo simulations}
We use Monte Carlo (MC) simulations with the Metropolis update rule and vectorized checkerboard algorithm. The results are presented for a square lattice with the size $L \times L$. We chose $L=128$ as a compromise value between the computational resources needed for MC simulations and calculations of the empirical variograms (see below) on one side and the effort to secure ergodic conditions on the other side. In typical simulations of magnetic systems one would like to suppress boundary effects by employing periodic boundary conditions (see the snapshots in Fig.~\ref{fig:snaps}). However, these are not appropriate for simulation of geostatistical data, since normally there is no reason to assume the data on the opposite boundaries are correlated. Free boundary conditions are not appropriate either since the spatial process is not necessarily confined within the domain boundaries. Instead, we consider it reasonable to apply boundary conditions that assume the smooth continuation of the spatial process beyond the borders. This assumption is implemented by inserting additional nodes that decorate the lattice and requiring that these additional points have the same values as their nearest neighbors inside the lattice. Therefore, if ${\bm s}_{i,j}$ is a spin in the $i$-th row and $j$-th column of the lattice, $i,j=1,\dots L$, then ${\bm s}_{i,L+1}={\bm s}_{i,L}$, ${\bm s}_{L+1,j}={\bm s}_{L,j}$, ${\bm s}_{i,0}={\bm s}_{i,1}$ and ${\bm s}_{0,j}={\bm s}_{1,j}$, where the rows and columns with indices $0$ and $L+1$ respectively are formed by auxiliary nodes around the lattice added to deal with the boundary effects.
\subsection{Modeling spatial variability}
The MC simulations produce spatially correlated spin realizations ${\bm s}_{i}$, which can be represented by their turn angles $\phi_i$. It turns out that there is great flexibility in the spatial correlations as we move in the temperature and simulation time parameter space. Therefore, in order to model the spatial variability (the local variogram) of the simulated spin values we use a very flexible Mat\'{e}rn covariance model. In addition to the trivial parameters of the standard models (Gaussian, exponential, spherical, etc.), controlling the variance, $\sigma^{2}$, and the characteristic covariance length, $\xi$, the Mat\'{e}rn model involves one more parameter, $\nu$, controlling the smoothness of the spatial process. Therefore, it can be considered appropriate for those geostatistical applications in which controlling the smoothness is important. Furthermore, the Mat\'{e}rn model includes the Gaussian and the exponential models as special cases for $\nu \rightarrow \infty $ and $\nu=0.5$, respectively, as well as several other bounded models~\cite{Minasny05}. Due to its flexibility, it has been used in various areas including pedology~\cite{Minasny05,Minasny07,Marchant07}, hydrology~\cite{Rodri74,Pardo09}, topography~\cite{Handcock93}, health modeling~\cite{kamm03}, meteorology~\cite{Handcock94} and environmental modeling~\cite{fuentes02}. The Mat\'{e}rn correlation function has the general form
\begin{equation}
\label{mate_cf} C(h;{\bm \theta'})=\frac{{2}^{1-\nu}}{\Gamma(\nu)}\Big(\frac{h}{\xi}\Big)^{\nu}K_{\nu}\Big(\frac{h}{\xi}\Big),
\end{equation}
where ${\bm \theta'}=(\xi,\nu)$ and $K_{\nu}$ is the modified Bessel function of order $\nu$. Then the corresponding variogram function $\gamma(h;{\bm \theta})$, which is typically used in geostatistics, is related to the correlation function as
\begin{equation}
\label{mate_var} \gamma(h;{\bm \theta})=\sigma_n^{2}+\sigma^{2}\left[1-C(h;{\bm \theta'}) \right],
\end{equation}
where $\sigma^{2}$ is the random field variance, $\sigma_n^{2}$ is the nugget variance that corresponds to uncorrelated fluctuations and ${\bm \theta}=(\sigma_n,\sigma,\xi,\nu)$ is a vector of the complete model parameters.
Having generated realizations of spin values from MC simulations, we can assess their spatial variability by calculating the experimental (i.e., sample-based) variogram as
\begin{equation}
\label{emp_var} \hat{\gamma}(h)=\frac{1}{2N(h)} \sum_{i=1}^{N(h)}\left [\phi(x_i) - \phi(x_i+h) \right]^2,
\end{equation}
where $N(h)$ is the number of pairs of spins separated by the distance $h$.
There are several methods of fitting the model to the experimental variogram. In the present study we used the weighted least squares estimator (hereafter WLS) that was reported to give the best overall results~\cite{zimm91}. The WLS estimator is based on minimizing the weighted sum of squared residuals (objective function), which is defined as follows:
\begin{equation}
\label{wls} O({\bm \theta})= \sum_{k=1}^{k_{\max}}
\frac{N(h_k)}{[\gamma(h_k;{\bm \theta})]^2}[\hat{\gamma}(h_k)-\gamma(h_k;{\bm \theta})]^2,
\end{equation}
where $\hat{\gamma}(h_k)$ is the experimental and $\gamma(h_k;{\bm \theta})$ the model variogram. The summation runs over the $k=1,\ldots,k_{\max}$ lags containing $N(h_{k})$ pairs of points, where $h(k_{\max})$ is less than one half of the maximum lag in the sample.
\section{Results and discussion}
In order to empirically study spatial correlations in the realizations of spin values generated by the modified planar rotator model we run MC simulations at various temperatures $T$ (in units $J/k_B$, where $k_B$ is Boltzmann constant). Each simulation starts from an initial configuration of spatially uncorrelated random spin angles uniformly distributed in the interval $[0,2\pi]$ and the snapshots are collected after $\tau$ MC sweeps. Then, using Eq.~(\ref{emp_var}) the experimental variogram is calculated and subsequently fitted to the appropriate model by the WLS method defined by Eq.~(\ref{wls}). The results obtained at relatively higher temperatures are demonstrated in Fig.~\ref{fig:T001}, for $T=10^{-2}$ and $\tau=10$, 100, 1000 and 10000. Soon after the start of the simulation one can observe the development of spatial correlations both in the snapshots in the form of nucleation of small spin domains with similar values as well as in the small-scale behavior of the experimental variograms. The fitted parameters $\hat{\xi}=1.62$ and $\hat{\nu}=0.62$ for $\tau=10$ in Fig.~\ref{fig:t001_mcs10} indicate that the realization is quite rough with a rather small characteristic length. As the relaxation proceeds the variance decreases and both the characteristic length and the smoothness parameter increase, as shown in Figs.~\ref{fig:t001_mcs100} and~\ref{fig:t001_mcs1000}, for $\tau=100$ and 1000, respectively. Nevertheless, as equilibrium is approached the realizations become rougher again, whereas the characteristic length further increases and saturates only in equilibrium to a temperature-dependent value. The estimated characteristic length $\hat{\xi}=56.51$ for the equilibrium configuration at $\tau=10000$ in Fig.~\ref{fig:t001_mcs10000} implies that the samples on the $128 \times 128$ lattice suffer from lack of ergodicity, which is also reflected in the trend displayed by the corresponding experimental variogram.
\begin{figure}[t!]
\subfigure{\includegraphics[scale=0.54,clip]{comb_t001_mcs10}\label{fig:t001_mcs10}}
\subfigure{\includegraphics[scale=0.54,clip]{comb_t001_mcs100}\label{fig:t001_mcs100}}
\subfigure{\includegraphics[scale=0.54,clip]{comb_t001_mcs1000}\label{fig:t001_mcs1000}}
\subfigure{\includegraphics[scale=0.54,clip]{comb_t001_mcs10000}\label{fig:t001_mcs10000}}
\caption{\label{fig:T001}Experimental and fitted variograms of spin realizations ---shown in the inset--- obtained at temperature $T=10^{-2}$ and at various simulation times $\tau$ equal to (a) 10, (b) 100, (c) $10^3$ and (d) $10^4$ MC sweeps. Mate($\hat{\bm \theta}$) denotes the Mat\'{e}rn model for the inferred parameter values $\hat{\bm \theta}=(\hat{\sigma}_n,\hat{\sigma},\hat{\xi},\hat{\nu})$.}
\end{figure}
\begin{figure}[t!]
\subfigure{\includegraphics[scale=0.54,clip]{comb_t00001_mcs10}\label{fig:t00001_mcs10}}
\subfigure{\includegraphics[scale=0.54,clip]{comb_t00001_mcs100}\label{fig:t00001_mcs100}}
\subfigure{\includegraphics[scale=0.54,clip]{comb_t00001_mcs1000}\label{fig:t00001_mcs1000}}
\subfigure{\includegraphics[scale=0.54,clip]{comb_t00001_mcs10000}\label{fig:t00001_mcs10000}}
\caption{\label{fig:T00001}The same as in Fig.~\ref{fig:T001} for $T=10^{-4}$. Expo($\hat{\bm \vartheta}$) and Gaus($\hat{\bm \vartheta}$) denote respectively the exponential and Gaussian models for the inferred parameter values $\hat{\bm \vartheta}=(\hat{\sigma}_n,\hat{\sigma},\hat{\xi})$. Note that in Fig.~\ref{fig:t00001_mcs10} the fitted Mat\'{e}rn and exponential models coincide as do in Fig.~\ref{fig:t00001_mcs10000} the Mat\'{e}rn and the Gaussian models.}
\end{figure}
As the temperature is decreased the trend of building up correlations in the equilibration process is further enhanced. Since thermal fluctuations are gradually suppressed the realizations become smoother. This tendency is apparent by visual comparison of the snapshots obtained at $T=10^{-4}$, shown in Fig.~\ref{fig:T00001} with those generated in the same stages of the relaxation at $T=10^{-2}$, shown in Fig.~\ref{fig:T001}. The difference between the inferred values of the smoothness parameter for the respective cases becomes evident particularly at later stages ($\tau=10^3$ and $10^4$). As mentioned above, the exponential covariance model with quite rough (non-differentiable) realizations and the Gaussian model with very smooth (infinitely differentiable) realizations can be obtained from the Mat\'{e}rn model as special cases for $\nu=0.5$ and $\nu \rightarrow \infty$, respectively. To emphasize the flexibility of the correlations developed in the relaxation process we also fit the relatively rough data for $\tau=10$ (Fig.~\ref{fig:t00001_mcs10}) to the exponential and the very smooth data for $\tau=10^4$ (Fig.~\ref{fig:t00001_mcs10000}) to the Gaussian models. One can see an excellent collapse of both curves with the best fits to the Mat\'{e}rn model. In spite of the very small value of $\hat{\xi}$ inferred in the Mat\'{e}rn model for $\tau=10^4$ (see the explanation below), the characteristic length is expected to increase with decreasing temperature and increasing simulation time. This expectation can be justified by the fact that the ground state of the MPR model (the equilibrium state at $T=0$ corresponding to the lowest energy), is the ferromagnetic state with all the spins aligned in the same direction. Thus, for $T \rightarrow 0$ one can expect that the correlation (characteristic) length will span the entire lattice and eventually go to infinity for $L \rightarrow \infty$.
Next, we comment on the big discrepancy between the characteristic length parameters $\hat{\xi}$, estimated for the Mat\'{e}rn and Gaussian models, for $T=10^{-4}$ and $\tau=10^4$. In fact, in the Mat\'{e}rn model the parameter $\xi$ has been found to be highly negatively correlated with the smoothness parameter $\nu$~\cite{Minasny07,zuk09c,hris11}. This correlation is also apparent by looking at the objective function of the WLS fit, defined by Eq.~(\ref{wls}), which is shown in Fig.~\ref{obj_fun} in the $\xi-\nu$ parameter space, for the data presented in Fig.~\ref{fig:t00001_mcs10000}. It demonstrates that the samples with small $\hat{\xi}$ and large $\hat{\nu}$ values are quite ``close'' to the samples with large $\hat{\xi}$ and small $\hat{\nu}$ values. Put differently, a small difference between samples from the same population (particularly in non-ergodic samples) can lead to completely different parameter estimates. Thus, one should be careful when performing parameter inference from the empirical variogram. Recently, we have presented a model-independent definition of the correlation length borrowed from statistical field theory and proposed to use a so-called ergodicity index to compare coarse-grained measures corresponding to both trivial (standard) and non-trivial, e.g. Mat\'{e}rn, covariance models with different parameters~\cite{hris11}.
\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.54,clip]{obj_f_t00001_mcs10000}
\end{center}
\caption{\label{obj_fun}The objective function $O$ given by \eqref{wls} over the $\xi-\nu$ parameter space. $O$ represents the fit of the data obtained at $T=10^{-4}$ and $\tau=10000$ (see Fig.~\ref{fig:t00001_mcs10000}) to the Mat\'{e}rn model with $\hat{\sigma}_n=0.001$ and $\hat{\sigma}=0.15$.}
\end{figure}
\section{Summary and outlook}
We have modified the standard planar rotator spin model from statistical physics to display a flexible type of short-range spatial correlations relevant in geostatistical modeling. In particular, the empirical study of spin configurations produced by Monte Carlo simulations in the nonequilibrium regime at various temperatures and stages shows that the smoothness and correlation range vary greatly versus the temperature and the simulation time. This behavior implies that the model has good potential for the simulation and prediction of spatial processes in geophysical and environmental applications. In particular, one can use the model to perform conditional simulations of gridded data, such as remote sensing images. Conditional simulations honor the sample values and reconstruct the variability of missing data, based on simulation parameters inferred from the available samples. Owing to the fact that the model does not show undesirable critical slowing down, the relaxation process is rather fast; furthermore, the short-range nature of the interactions between spin variables allows vectorization of the algorithm. Consequently, the proposed method is significantly more efficient than the conventional geostatistical approaches, and thus applicable to huge datasets, such as satellite and radar images. The implementation details are now under investigation.
\ack
This work was supported by the Scientific Grant Agency of Ministry of Education of Slovak Republic (Grant No. 1/0331/15). We also acknowledge support for a short visit by Prof. \v{Z}ukovi\v{c} at TUC from the Hellenic Ministry of Education - Department of Inter-University Relations, the State Scholarships Foundation of Greece and the Slovak Republic's Ministry of Education through the Bilateral Programme of Educational Exchanges between Greece and Slovakia.
\section*{References}
|
{
"timestamp": "2015-04-14T02:15:31",
"yymm": "1504",
"arxiv_id": "1504.03211",
"language": "en",
"url": "https://arxiv.org/abs/1504.03211"
}
|
\subsection{What {\bfseries{\slshape{FEpX}}}\, can do.}
\label{sec:candu}
{\bfseries{\slshape{FEpX}}}\, is useful for simulating the mechanical behavior of polycrystalline solids at the level of aggregates of grains.
The aggregates may be comprised of grains of a single phase or of multiple phases.
Grains are discretized with finite elements so any sub-volume of an element is a sub-volume of an individual crystal.
The local behaviors associated with the material with any element correspondingly are those of a crystal.
In particular, the behaviors include:
\begin{itemize}
\item{nonlinear kinematics capable of handling motions with both large strains and large rotations;}
\item{anisotropic elasticity based on cubic or hexagonal crystal symmetry; }
\item{anisotropic plasticity based on rate-dependent slip on a restricted number of systems for cubic or hexagonal symmetry; }
\item{evolution of state variables for crystal lattice orientation and slip system strengths;}
\end{itemize}
To accommodate these behaviors the finite element formulation has incorporated a number a numerical features, such as:
\begin{itemize}
\item{higher-order, isoparametric elements with quadrature for integrating over the volume;}
\item{implicit update of the stress in integrations over time;}
\item{monotonic and cyclic loading under quasi-static conditions;}
\end{itemize}
Depending on the goals of the simulation, aggregates might number in grains from only a few to tens of thousands (or more).
The grains can be discretized at a level appropriate for the intent of a simulation.
The number of grains together with the level of discretization within grains set the computational burden for a simulation.
To accommodate combinations with high burden, the {\bfseries{\slshape{FEpX}}}\, code employs scalable parallel methods and executes on clusters.
{\bfseries{\slshape{FEpX}}}\, has been developed to use meshes constructed by instantiation tools for virtual polycrystals and to output data in
an archivable format for subsequent use with visualization tools or other interpretation tools.
With these capabilities {\bfseries{\slshape{FEpX}}}\, is well-suited, for example, to model the mechanical behavior of polycrystals that exhibit inhomogeneous deformations within and among the crystals, to investigate the heterogeneity of stress within a polycrystal, or to examine the roles of neighbors on the behaviors of individual grains. When teamed with appropriate instantiation methods, {\bfseries{\slshape{FEpX}}}\, can be used effectively to model yielding and flow of alloys with complicated phase/grain topologies and morphologies.
\subsection{What {\bfseries{\slshape{FEpX}}}\, cannot do.}
\label{sec:nocandu}
The {\bfseries{\slshape{FEpX}}}\, framework does not encompass many aspects of the behaviors observed in real materials. Some of its principal limitations include:
\begin{itemize}
\item{plastic flow occurs by slip -- no other mechanisms, such as twinning and creep, are modeled; }
\item{deformations are ductile -- no fracture models are included;}
\item{loading is quasi-static -- no inertial effects are modeled; }
\item{loading is mechanical (isothermal) -- coupling with heat transfer (or other physical processes) is not considered;}
\item{boundary conditions are simple -- neither friction models nor changing contact conditions are included.}
\end{itemize}
With these limitations, {\bfseries{\slshape{FEpX}}}\, is not well-suited for modeling applications with complex loading conditions, such as many metal forming and joining processes, or for modeling applications involving fragmentation failure of a dynamically loaded body.
\subsection{What {\bfseries{\slshape{FEpX}}}\, has been used to study.}
\label{sec:our_studies}
A variety of interesting problems arise at physical length scales in which a sample volume encompasses an aggregate of grains. We typically think of an aggregate containing on the order of $10^3-10^6$ grains, but the simulation framework embodied in {\bfseries{\slshape{FEpX}}}\, is appropriate for single-grain or multi-grain samples ($1-10^2$ grains), as well. Some of the applications of {\bfseries{\slshape{FEpX}}}\, published in the open literature are listed below. Users of {\bfseries{\slshape{FEpX}}}\, are encouraged to examine articles in areas of interest to obtain information beyond the scope of this article resulting from the collective experiences of others in applying {\bfseries{\slshape{FEpX}}}.
\begin{itemize}
\item{Grain interactions with attention focused on bulk texture evolution.
Articles published in this area
are: \cite{bea_daw_mat_koc_95,mar_daw_jen_95,sar_daw_96a,sar_daw_96b,mik_daw_98,leb_daw_ker_wen_03,mir_daw_lef_07}.}
\item{Deformation heterogeneity within the grains comprising an aggregate with focus on intra-grain misorientation distributions. Articles published in this area
are: \cite{mik_daw_99,bar_daw_01b,bar_daw_02,que_daw_dri_12}.}
\item{Inter- and intra-grain stress/elastic strain distributions, especially including comparisons to neutron and x-ray diffraction experiments.
Articles published in this area
are: \cite{daw_boy_mac_rog_00,daw_boy_mac_rog_01,daw_boy_rog_05b,mil_par_daw_han_08,rit_daw_mar_10,mar_daw_gar_12}.}
\item{The elasto-plastic transition occurring during loading of polycrystalline solids, with focus on the redirection of stress at the grain level.
Articles published in this area are: \cite{bar_daw_mil_99,han_daw_05b,won_daw_10,sch_won_daw_mil_13}}.
\item{Cyclic loading with interest in evolution of stress and its implications for fatigue failure.
Articles published in this area are: \cite{tur_log_daw_mil_03,won_daw_11}.}
\item{Evolution of dislocation density and associated peak broadening.
Articles published in this area are: \cite{daw_boy_rog_05,won_par_mil_daw_13}}.
\item{Virtual polycrystal instantiation issues, including sensitivity of the stress and deformation to discretization. Articles published in this area are: \cite{log_tur_mil_rog_daw_04,rit_daw_09, que_daw_bar_11}.}
\end{itemize}
Much of the original work in utilizing polycrystal plasticity constitutive models within finite element simulations was focused on bulk texture evolution in macroscopic scale deformation
processes, such as metal forming operations (rolling, extrusion, and sheet forming) and geological processes (particularly mantle convection). In such cases, the relative sizes of finite elements and grains were reversed in comparison to those of the intended applications of the {\bfseries{\slshape{FEpX}}}\, framework described in this article. Consequently, the mechanical properties within an element were derived from an average over an ensemble of crystals once an averaging hypothesis ({\it e.g.} isostress or isostrain) was imposed.
Examples of this type of application are: \cite{mat_daw_89,mat_daw_koc_90,bea_mat_daw_joh_93,bea_daw_mat_koc_kor_94,kum_daw_95,daw_wen_00,daw_mac_wu_03,daw_boy_rog_05b}. While relevant from a historical perspective in the development of {\bfseries{\slshape{FEpX}}}, the
{\bfseries{\slshape{FEpX}}}\, framework described here does not include evaluating properties within an element on the basis an average over a population of grains. Rather, properties within an element are those of a single orientation, consistent with an element spanning only a part of any given grain.
\subsection{Virtual Polycrystal Generated by Regular Tessellation }
\label{sec:dodecahedramesh}
This example was provided by Andrew Poshadel and was developed as part of his research on yielding of dual phase alloys~\cite{poshadel_phd}.
The example demonstrates the application of {\bfseries{\slshape{FEpX}}}\, to a virtual polycrystal with two phases that was built using
dodecahedral grains. The two phases are the austenitic and ferritic phases of a dual steel (LDX-2101), referred to symbolically as the $\gamma$ and $\alpha$ phases, respectively. The stock material has a microstructure consistent with having been rolled or extruded.
\subsubsection{Defining the virtual polycrystal}
Following the summary of the input data given in Section~\ref{sec:input_data}, the required input data to execute {\bfseries{\slshape{FEpX}}}\, can be organized into five categories:
\begin{enumerate}
\item{{\bf Phase Attributes:} The $\alpha$ and $\beta$ phases both have cubic crystal structure -- FCC for the $\gamma$ phase and BCC for the $\alpha$ phase. The single crystal, cubic, elastic moduli for the two phases are listed in Table~\ref{tab:example1_elasticmoduli}. Based on experimental data that indicates the plastic behaviors of the two phases are comparable, the same plasticity parameters were assigned to both phases for the purpose of this example. These parameters are listed in Table~\ref{tab:example1_plasticparms}. The slip systems are different for the two, however, with the FCC $\gamma$ phase using the $\{111\} \, <110>$ systems
and the BCC $\alpha$ phase using the $ \{110\} \, <111>$ systems (See Table~\ref{tab:slip_systems}).
This information is provided to {\bfseries{\slshape{FEpX}}}\, in the *.matl input file.}
\begin{table}[h]
\centering
\begin{tabular}{||c|c|c|c|c||}
\hline
\hline
Phase & Type & $C_{11}$ (GPa) & $C_{12}$ (GPa) & $C_{44}$ (GPa) \\
\hline
$\gamma$ & FCC & 204.6 & 137.7 & 126.2 \\
$\alpha$ & BCC & 236.9 & 140.3 & 116.0 \\
\hline
\hline
\end{tabular}
\caption{LDX 2101 elastic moduli used in the single crystal constitutive equations for the two-phase virtual polycrystal.
Values listed conform to the convention given in Equation~\ref{eq:cubic_stiffness}. }
\label{tab:example1_elasticmoduli}
\end{table}
\begin{table}[h]
\centering
\begin{tabular}{||c|c|c|c|c|c|c|c|c||}
\hline
\hline
Phase & $\dot{\gamma}_0$ (${{\rm s}^{-1}}$) & $m$ & $h_{0}$ (MPa) & $g_0$ (MPa)& $n^\prime$ & $g_1$ (MPa) & $\dot{\gamma_s}$ (${{\rm s}^{-1}}$) & $m^\prime$\\
\hline
$\gamma$ & 1.0 & 0.02 & 391.9 & 237.0 & 1 & 335.0 & $5.0\times10^{10}$ & 0 \\
$\alpha$ & 1.0 & 0.02 & 391.9 & 237.0 & 1 & 335.0 & $5.0\times10^{10}$ & 0 \\
\hline
\hline
\end{tabular}
\caption{LDX 2101 slip parameters used in the single crystal constitutive equations for the two-phase virtual polycrystal.
Values listed conform to the convention given in Equations~\ref{eq:ss_kinetics}, \ref{eq:ss_strength_evolution}, and
\ref{eq:sat_ss_strength}.}
\label{tab:example1_plasticparms}
\end{table}
\item{{\bf Mesh definition:} A finite element mesh underlying the virtual polycrystal was instantiated using a custom {\bfseries{\slshape{MATLAB}}}\, script. It consists of a regular, rectangular layout of elements. These elements can be grouped to form regular dodecahedral grains. The resulting mesh, shown in Figure~\ref{fig:example1_mesh}, has 117,504 10-node tetrahedral elements and 173,829 nodal points.
The corresponding arrays for the nodal point coordinates and element connectivities are provided to {\bfseries{\slshape{FEpX}}}\, in the *.mesh input file. }
\begin{figure}[h!]
\centering
\includegraphics*[width=10cm]{Images/LDX_example_mesh.jpg}
\caption{Finite element mesh for the dual phase steel virtual sample. }
\label{fig:example1_mesh}
\end{figure}
\item{{\bf Phase and grain definition:} The finite elements must be assigned to phases and grains. The first step was to define the spatial distributions of the two phases in one plane using a regular layout of dodecahedral grains.
The second step was to extrude (repeat) the planar layout in the direction perpendicular to the plane to create the cube-shaped polycrystal with a microstructure similar to the stock material. In this example, elements are one of the two phases ($\gamma$ or $\alpha$). Within subdomains of a single phase, there may be one or more crystals. For this dual phase steel, the subdomains of both phases have multiple crystals, although the $\gamma$ phase subdomains generally had fewer crystals than the subdomains of the $\beta$ phase subdomains. The grain definitions follow from the assignment of lattice orientations to the elements. Contiguous elements with the same orientation constitute a grain. Lattice orientations were chosen randomly from measured orientation distributions and assigned to element to create the desired grain arrangement within the phases.
Each grain also was assigned the same initial slip system strength, which in turn was assigned to every element of the grain.
Figure~\ref{fig:example1_phase-and-grain} shows the phase and grain assignments associated with the mesh shown in Figure~\ref{fig:example1_mesh}.
This information is provided in the *.grains and *.kocks files. }
\begin{figure}[h!]
\centering
\subfigure[Phases]
{
\includegraphics*[width=8cm]{Images/LDX_example_phases.png}
}
\subfigure[Grains]
{
\includegraphics*[width=8cm]{Images/LDX_example_angs.png}
}
\caption{Phase and grain distributions for the dual phase steel virtual sample. For the phase distribution, blue indicates the BCC ferritic ($\alpha$) phase; red indicates the FCC austenitic ($\gamma$) phase. For the grain distribution, grains are indicated by domains of uniform color.}
\label{fig:example1_phase-and-grain}
\end{figure}
\item{{\bf Vertex files:} standard definitions of the single crystal yield surface vertices were used for both the FCC and BCC phases.}
\end{enumerate}
\subsubsection{Controlling the loading:}
\begin{enumerate}
\item{{\bf Boundary conditions:} The boundary conditions are intended to simulate the loading applied in a tension test.
The virtual polycrystal is constrained on the bottom and stretched in the z direction by an imposed axial velocity on the top.
Two adjoining lateral surfaces are traction-free while a symmetry condition is applied on the other two. Rigid body translations and rotations have been suppressed by the application of the symmetry conditions. This information is given in the *.bcs file. }
\item{{\bf Target loads:} Simple load control is applied to extend the sample. Several z-direction target loads along a monotonically increasing path (no unloading episodes) are specified to provide points for writing output data. This information is given in the *.loads file.}
\end{enumerate}
\subsubsection{Specifying options:}
\begin{enumerate}
\item{{\bf Load controls:} The ``control by load'' mode is used to control the loading history. }
\item{{\bf Convergence criteria:} The default parameters have been used for both the velocity field and crystal stress iterative procedures. The Newton-Raphson procedure is invoked for the velocity solutions.}
\end{enumerate}
\subsubsection{Selected Simulation Results}
Simulation results are available for postprocessing at the points in the loading designated by the target loads.
These results were aggregated and written to a .vtk file for plotting with {\bfseries{\slshape{Paraview}}}.
Figure~\ref{fig:example1_stresses} shows the distribution of axial component of the stress and the effective plastic strain at the last target load of 590 N.
At this load, the nominal axial strains was approximately 10\%.
The stress shows spatial variations due to the anisotropy of the crystal properties and the
interactions among the grains. There is an increase in the variability of the axial component of the stress as the stress level is increased due in part to the change in the principal directions of the crystal stresses as the stresses move toward a vertex of the single crystal yield surface during the elastic-plastic transition. The effective stress (not show here) exhibits less variability.
Figure~\ref{fig:example1_hardness} shows the evolution of strength over the course of the loading. The deformation of the mesh has been exaggerated by a factor of 4 to facilitate visualizing the heterogeneity of the deformation.
A correlation between the plastic deformation and the strain hardening is evident.
\begin{figure}[h!]
\centering
\subfigure[Axial stress component. ]
{
\includegraphics*[width=8cm]{Images/ldx_example_axialstress_3.png}
}
\subfigure[Effective plastic strain. ]
{
\includegraphics*[width=8cm]{Images/ldx_example_plasticstrain_3.png}
}
\caption{Axial stress and plastic strain distributions at nominal load of 590N shown on the deformed mesh.}
\label{fig:example1_stresses}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics*[width=6cm]{Images/ldx_example_hardness_3.png}
\caption{Slip system strength at 590N shown on a exaggerated (x4) deformation field.}
\label{fig:example1_hardness}
\end{figure}
A great deal more information is available in the simulation output. For example, lattice strains in crystals on designated crystallographic
fibers was collected and averaged for comparisons to experiments in which lattice strains were measured by neutron diffraction during
{\it in situ} loading.
\subsection{Voronoi Tessellated Virtual Polycrystal Generated with {\bfseries{\slshape{Neper}}}}
\label{sec:voronoimesh}
This example was provided by Amanda Mitch and was developed as part of her research on reduced-order representation of
crystal stress distributions for use in a methodology for quantifying residual stress distributions in engineering components~\cite{mitch_ms}.
The example demonstrates the application of {\bfseries{\slshape{FEpX}}}\, to a virtual polycrystal that was built using
a Voronoi tessellation to define the grains. The material is single phase and the single crystal properties are consistent with a FCC crystal type, but do not
represent any particular metal or alloy.
The input data, following the organization given in Section~\ref{sec:input_data}, is summarized below.
\subsubsection{Defining the virtual polycrystal}
\begin{enumerate}
\item{{\bf Phase Attributes:}
The material is single phase with a cubic (FCC) crystal structure. The single crystal, cubic, elastic moduli are listed in Table~\ref{tab:example2_elasticmoduli}. Plasticity parameters are generic, being similar to copper alloy. These parameters are listed in Table~\ref{tab:example2_plasticparms}. The slip systems are the customary primary systems for FCC crystals: the $\{111\} \, <110>$ systems.
This information is provided to {\bfseries{\slshape{FEpX}}}\, in the *.matl input file.
\begin{table}[h]
\centering
\begin{tabular}{||c|c|c|c|c||}
\hline
\hline
Phase & Type & $C_{11}$ (GPa) & $C_{12}$ (GPa) & $C_{44}$ (GPa) \\
\hline
$\alpha$& FCC & 245. & 155. & 62.5 \\
\hline
\hline
\end{tabular}
\caption{Elastic moduli used in the single crystal constitutive equations for the Voronoi-based virtual polycrystal.}
\label{tab:example2_elasticmoduli}
\end{table}
\begin{table}[h]
\centering
\begin{tabular}{||c|c|c|c|c|c|c|c|c||}
\hline
\hline
Phase & $\dot{\gamma}_0$ (${{\rm s}^{-1}}$) & $m$ & $h_{0}$ (MPa) & $g_0$ (MPa)& $n^\prime$ & $g_1$ (MPa) & $\dot{\gamma_s}$ (${{\rm s}^{-1}}$) & $m^\prime$\\
\hline
$\alpha$ & 1.0 & 0.05 & 200. & 210.& 1 & 330.& $5.0\times10^{10}$ & $5.0\times10^{-3}$ \\
\hline
\hline
\end{tabular}
\caption{Slip parameters used in the single crystal constitutive equations for the Voronoi-based virtual polycrystal.}
\label{tab:example2_plasticparms}
\end{table}
}
\item{{\bf Mesh definition:}
A virtual polycrystal was instantiated using the {\bfseries{\slshape{Neper}}}\, code. {\bfseries{\slshape{Neper}}}\, builds a Voronoi construction of the domain to define grains and then discretizes the grains into finite elements. The resulting mesh, shown in Figure~\ref{fig:example2_mesh_grains}, has 96,758 10-node tetrahedral elements and 134,362 nodal points.
The corresponding arrays for the nodal point coordinates and element connectivities are provided to {\bfseries{\slshape{FEpX}}}\, in the *.mesh input file.
\begin{figure}[h!]
\centering
\subfigure[Mesh]
{
\includegraphics*[width=6cm]{Images/harmonic_app2_mesh.png}
}
\subfigure[Grains]
{
\includegraphics*[width=6cm]{Images/harmonic_app2_grains.png}
}
\caption{Mesh and grains for the {\bfseries{\slshape{Neper}}}-built polycrystal. }
\label{fig:example2_mesh_grains}
\end{figure}
}
\item{{\bf Phase and grain definition:}
All grains are the same phase. The grain designations for the finite elements follow from
the Voronoi tessellation and are shown in Figure~\ref{fig:example2_mesh_grains}.
This information is provided in the *.grain and *.kocks files.
}
\item{{\bf Vertex files:}
a standard definition for the vertices of a FCC single crystal yield surface was used.}
\end{enumerate}
\subsubsection{Controlling the loading:}
\begin{enumerate}
\item{{\bf Boundary conditions:} The boundary conditions are intended to simulate the triaxial loading of the sample such that the
stress components remain in constant proportions given by.:
\begin{equation}
[\sigma]
=
\sigma_1 \left[
\begin{array}{c c c }
1.0 & 0 & 0 \\
0 & -0.625 & 0 \\
0 & 0 & -0.375
\end{array}
\right]
\label{eq:triaxial_stress_state}
\end{equation}
Normal velocity components of different magnitudes are applied to three adjacent surfaces while the opposing surfaces are fixed in place.
The surface velocities are adjusted to achieve tractions consistent with the target ratios for triaxial stress state. }
\item{{\bf Target loads:} Two target loads along a monotonically increasing path (no unloading episodes) are specified to provide points for writing output data. At the first target load, the response is essentially elastic; the second target load is sufficient to induce plastic strains on the order of 3\%. The target load information is given in the *.loads file. For triaxial loading, three normal forces are specified for each target load consistent with the
desired stress state specified in Equation~\ref{eq:triaxial_stress_state}.}
\end{enumerate}
\subsubsection{Post-Processing:}
\begin{enumerate}
\item{{\bf Lightup:} Fiber-based quantities are computed for 6 fibers: (100) and (111) crystal planes in the $[100], [010] {\rm and} [001]$ sample directions.}
\end{enumerate}
\subsubsection{Specifying options:}
\begin{enumerate}
\item{{\bf Load controls:} The ``Triax-CSR'' mode is used to control the loading history. Using this option, the magnitude of the velocities are adjusted to impose a specified loading rate (increase in the applied forces). The relative values of the imposed surface velocities are adjusted to maintain the specified state of triaxial stress. }
\item{{\bf Convergence criteria:} The default parameters have been used for both the velocity field and crystal stress iterative procedures. The Newton-Raphson procedure is invoked for the velocity solutions.}
\item{{\bf Lightup:} The option to compute fiber-based quantities is specified.}
\end{enumerate}
\subsubsection{Selected Simulation Results}
Stress distributions for the polycrystal are shown in Figure~\ref{fig:example2_stresscomps} at the second target load. In these images, the normal components of the stress are plotted over the deformed mesh. The total deformation is not large, so the change in shape from the initial configuration shown in Figure~\ref{fig:example2_mesh_grains} is difficult to discern. The differences in overall shade between the three images
reflects the triaxial stress condition intentionally imposed on the polycrystal. There are variations over the polycrystal for all of the components
stemming from the elastic and plastic anisotropy.
\begin{figure}[h!]
\centering
\subfigure[ xx component]
{
\includegraphics*[width=6cm]{Images/harmonic_app2_stress-xx.png}
}
\subfigure[yy component]
{
\includegraphics*[width=6cm]{Images/harmonic_app2_stress-yy.png}
}
\subfigure[zz component]
{
\includegraphics*[width=6cm]{Images/harmonic_app2_stress-zz.png}
}
\caption{Normal stress component distributions at $\sigma_{11} = 225$MPa}
\label{fig:example2_stresscomps}
\end{figure}
In Figure~\ref{fig:example2_straincomps} the normal components of the elastic strain are depicted. A noticeable contrast to the stress distributions
is evident. For the elastic strains, the net effect of the stress triaxiality and grain interactions is to produce distributions that span approximately the
same range in strain for all the normal strain components. Unlike the stress, it is difficult to discern the triaxiality of the stress from differences in the lattice lattice (elastic) strain distributions.
Figure~\ref{fig:example2_plasticstrain} shows the effective plastic strain at the second target load. Again, the distribution is not uniform: some elements display several percentage plastic strain while other have almost no plastic strain.
Details are available in \cite{mitch_ms}.
\begin{figure}[h!]
\centering
\subfigure[ xx component]
{
\includegraphics*[width=6cm]{Images/harmonic_app2_strain-xx.png}
}
\subfigure[yy component]
{
\includegraphics*[width=6cm]{Images/harmonic_app2_strain-yy.png}
}
\subfigure[zz component]
{
\includegraphics*[width=6cm]{Images/harmonic_app2_strain-zz.png}
}
\caption{Normal lattice (elastic) strain component distributions at $\sigma_{1} = 225$MPa}
\label{fig:example2_straincomps}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics*[width=10cm]{Images/harmonic_app2_plasticstrain.png}
\caption{Effective plastic strain distribution at $\sigma_{1} = 225$MPa}
\label{fig:example2_plasticstrain}
\end{figure}
The distribution of stresses over a polycrystal depends strongly on the levels of anisotropy in the elastic and plastic behaviors of the constituent crystals. The stress state must vary spatially to satisfy compatibility and equilibrium
if the properties vary. Often these level are different, which is evident by observing the stress distributions as the polycrystal is loaded through the elastic-plastic transition. At lower loads, the behavior is essentially purely elastic and the distribution is controlled by the elastic moduli. At high loads, the behavior is dominated by the single crystal yield surface. This is illustrated by the relative changes in average elastic strains along the selected fibers given
in Table~\ref{tab:example2_fiberstrains}. One can readily observe that between the target loads for $\sigma_{1} = 200$MPa and $\sigma_{1} = 225$MPa
there is not the same proportional increase in strains, as would be expected in the response remained linear and superposition could be applied.
The adjustment in stress that accompanies yielding implies that the principal axes of the strain rotate as the stress moves toward a vertex of the single crystal yield surface.
\begin{table}[h]
\centering
\begin{tabular}{||c|c|c|c|c|c|c||}
\hline
\hline
$\sigma_{1}$ (MPa) & $(100)||[100]$ & $(111)||[100]$ & $(100)||[010]$ & $(111)||[010]$ & $(100)||[001]$ & $(111)||[001]$ \\
\hline
200 & 0.00211 & 0.00196 & -0.00115 & -0.00117 &-0.00075 & -0.00078\\
225 & 0.00221 & 0.00244 & -0.00105 & -0.00160 & -0.00044 & -0.00113 \\
\hline
\hline
\end{tabular}
\caption{Average lattice strains along selected fibers.}
\label{tab:example2_fiberstrains}
\end{table}
\pagebreak
\subsection{Voxel-Based Virtual Polycrystal Generated from 3-D Serial Section Maps}
\label{sec:voxelmesh}
This example was provided by Donald Boyce and was developed as part of an ONR-sponsored project on strength and ductility in titanium alloys. The example demonstrates the application of {\bfseries{\slshape{FEpX}}}\, to a virtual polycrystal that was built by mapping
voxel data to a regular mesh to define the grains. The titanium alloy being modeled is Ti-6Al-4V, which is two-phase at
room temperature. However, since the volume fraction of the BCC phase is only about 7\%, this analysis is performed assuming a single (HCP) phase.
\subsubsection{Defining the virtual polycrystal}
\begin{enumerate}
\item{Phase Attributes:
The principal phase for this titanium alloy has hexagonal symmetry (HCP), and is designated as the $\alpha$ phase. The single crystal, hexagonal, elastic moduli for it are listed in Table~\ref{tab:example3_elasticmoduli}.
Input to the code does not include $C_{33}$. Rather, it is computed internally to assure that the constraint to decouple the
deviatoric and volumetric responses is satisfied. The plasticity parameters were estimated from fitting stress-strain data for the alloy and are listed in Table~\ref{tab:example3_plasticparms}. This type of alloy exhibits very little strain hardening; the choice of parameters accomplishes this by setting the initial slip system strength to the saturation value. The slip systems include prismatic, basal and pyramidal systems as per Table~\ref{tab:slip_systems}.
This information is provided to {\bfseries{\slshape{FEpX}}}\, in the *.matl input file.
\begin{table}[h]
\centering
\begin{tabular}{||c|c|c|c|c|c||}
\hline
\hline
Phase & Type & $C_{11}$ (GPa) & $C_{12}$ (GPa) & $C_{13}$ (GPa) & $C_{44}$ (GPa) \\
\hline
$\alpha$ & HCP & 161.4 & 91.0 & 69.5 & 46.7 \\
\hline
\hline
\end{tabular}
\caption{Titanium elastic moduli used in the single crystal constitutive equations for the voxel-based virtual polycrystal.}
\label{tab:example3_elasticmoduli}
\end{table}
\begin{table}[h]
\centering
\begin{tabular}{||c|c|c|c|c|c|c|c|c||}
\hline
\hline
Phase & $\dot{\gamma}_0$ (${{\rm s}^{-1}}$) & $m$ & $h_{0}$ (MPa) & $g_0$ (MPa) & $n^\prime$ & $g_1$ (MPa) & $\dot{\gamma_s}$ (${{\rm s}^{-1}}$) & $m^\prime$\\
\hline
$\alpha$ & 1.0 & 0.01 & 1000. &500. & 1 & 500. & $5.0\times10^{10}$ & $0.01$ \\
\hline
\hline
\end{tabular}
\caption{Titanium slip parameters used in the single crystal constitutive equations for the voxel-based virtual polycrystal.}
\label{tab:example3_plasticparms}
\end{table}
\begin{table}[h]
\centering
\begin{tabular}{||c|c|c||}
\hline
\hline
Basal & Prismatic & Pyramidal\\
\hline
1 & 1 & 3 \\
\hline
\hline
\end{tabular}
\caption{Relative strength for the titanium slip system used in the single crystal constitutive equations for the voxel-based virtual polycrystal.}
\label{tab:example3_ss-strengths}
\end{table}
}
\item{{\bf Mesh definition:}
The finite element mesh underlying the virtual polycrystal was instantiated by mapping voxel-based (3D) orientation map onto a regular mesh using a custom {\bfseries{\slshape{MATLAB}}}\, script (available in the {\bfseries{\slshape{OdfPf}}}\, package). The orientation map was obtained from serial section data measured using electron back-scattered diffraction (EBSD). The finite element mesh spans a volume of $20\mu m \times 20 \mu m \times 60 \mu m$ that
coincides with an interior portion of the experimental volume. The element size was chosen to give and resolution comparable to the spatial resolution of the data.
The resulting mesh, shown in Figure~\ref{fig:example3_mesh}, has 144,000 10-node tetrahedral elements and 230,401 nodal points.
The corresponding arrays for the nodal point coordinates and element connectivities are provided to {\bfseries{\slshape{FEpX}}}\, in the *.mesh input file, along with
definition of the six sample surfaces in terms of the mesh elements.
\begin{figure}[h!]
\centering
\subfigure[Finite element mesh]
{
\includegraphics*[width=6cm]{Images/voxel_example_mesh.png}
}
\subfigure[Lattice orientations]
{
\includegraphics*[width=6cm]{Images/voxel_example_latticeorienations_0.png}
}
\caption{Finite element mesh for the mill annealed titanium alloy and grain orientations assigned to the elements.}
\label{fig:example3_mesh}
\end{figure}
}
\item{{\bf Phase and grain definition:}
For every element of the mesh, the voxel that is closest to the element is identified using the distance between the centroid of the element and the centroid of the voxel). The orientation data of the voxel is then used to assign the lattice orientation for the element.
The simulation assumes all grains are of the same HCP phase.
All the grains are assigned the same initial slip system strength, which in turn was assigned to every element of the grain.
Figure~\ref{fig:example3_mesh} also shows the grain assignment associated with the mesh.
While some grains are evident, noisy or missing information in the orientation data makes crisp definition of grains difficult.
The grain assignment and lattice orientation data are given in the *.grain and *.kocks files.
}
\item{{\bf Vertex files:}
Vertices of a HCP single crystal yield surface was used having a topology consistent with
the prescribed ratios of the slip system strengths of $(1:1:3)$ for the basal, prismatic and pyramidal slip systems, respectively.
}
\end{enumerate}
\subsubsection{Controlling the loading:}
\begin{enumerate}
\item{{\bf Boundary conditions:}
Boundary conditions are chosen to mimic a tensile test: there is a fixed velocity applied on the upper surface (positive $y$ face) while the lower surface is fixed from translation in the $y$ direction. The lateral surfaces have
two traction free surfaces (positive $x$ and $z$) and two symmetry surfaces (negative $x$ and $z$).
This information is in the *.bcs file.
}
\item{{\bf Target loads:}
Simple load control is applied to compress the sample. Three y-direction target loads along a monotonically increasing path (no unloading episodes) are specified to provide points for writing output data. The final target load was sufficient to compress the sample by approximately 1\%, overall. This information is given in the *.loads file.
}
\end{enumerate}
\subsubsection{Specifying options (information in the *.options file):}
\begin{enumerate}
\item{{\bf Load controls: } The ``control by load'' mode is used to control the loading history.
}
\item{{\bf Convergence criteria: } Default parameters have been used for both the velocity field and crystal stress iterative procedures. The Newton-Raphson procedure is invoked for the velocity solutions.
}
\end{enumerate}
\subsubsection{Selected Simulation Results}
Figure~\ref{fig:example3_axialstress} shows the axial stress and the axial lattice strain at the final target load.
Here the grain structure is more evident.
Because the load is sufficient to cause wide-spread yielding, higher stresses and strains typically occur in grains whose lattices are at stronger orientations.
Note that the stress distribution is not merely a scaled version of the strain distribution. This is a result of the elastic anisotropy, which implies that the eigenvectors of the stress and strain tensors do not necessarily align.
At first glance, the stress levels depicted in Figure~\ref{fig:example3_axialstress} might appear to exceed stress limits imposed the single crystal yield surface. However, the large grain size relative to the sample size has an effect such that the deformation is more highly constrained.
The consequence is that the mean stress (shown in Figure~\ref{fig:example3_plasticstrainrate}) is larger at many points than would be expected for a case of simple tension.
The plastic straining induced by this highly heterogeneous stress field is very inhomogeneous, as is evident from the distribution of the effective plastic strain shown in
Figure~\ref{fig:example3_plasticstrainrate}. This plot shows how plastic flow interconnects through the polycrystal leaving some domains relatively undeformed.
\begin{figure}[h!]
\centering
\subfigure[Axial stress ]
{
\includegraphics*[width=6cm]{Images/voxel_example_axialstress_3.png}
}
\subfigure[Axial lattice (elastic) strain]
{
\includegraphics*[width=6cm]{Images/voxel_example_axiallatticestrain_3.png}
}
\caption{Axial stress and axial lattice strain distributions at the third target load.}
\label{fig:example3_axialstress}
\end{figure}
\begin{figure}[h!]
\centering
\subfigure[Mean stress]
{
\includegraphics*[width=6cm]{Images/voxel_example_meanstress_3.png}
}
\subfigure[Effective plastic strain]
{
\includegraphics*[width=6cm]{Images/voxel_example_plasticstrain_3.png}
}
\caption{Mean stress and plastic strain distributions at the third target load.}
\label{fig:example3_plasticstrainrate}
\end{figure}
\subsection{Matrix notation for tensorial quantities}
To facilitate the presentation and implementation of the finite element formulation, tensor quantities are written as matrices.
Vectors map directly to one-dimensional column or row matrices. For second order tensors, column vectors
are defined for with a designated ordering to the components. For the Kirchhoff stress and elastic strain, which are symmetric tensors, we use:
\begin{equation}
{\boldsymbol{\tau}} \rightarrow \left\{ \tau \right\} = \left\{ \tau_{11}\,\, \tau_{22}\,\, \tau_{33}\,\, \sqrt{2}\tau_{23}\,\, \sqrt{2}\tau_{31}\,\, \sqrt{2}\tau_{12} \right\}^T
\label{eq:vector_stress}
\end{equation}
\begin{equation}
{\tnsr{e}^e} \rightarrow \left\{ \sf{e}^e \right\} = \left\{ \sf{e}^e _{11}\,\, \sf{e}^e _{22}\,\, \sf{e}^e _{33}\,\, \sqrt{2}\sf{e}^e _{23}\,\, \sqrt{2}\sf{e}^e _{31}\,\, \sqrt{2}\sf{e}^e _{12} \right\}^T
\label{eq:vector_strain}
\end{equation}
where the $\sqrt{2}$ factor appears for the shear components in both tensors, which preserves the inner product
relation
\begin{equation}
{\boldsymbol{\tau}} \cdot {\tnsr{e}^e} = \left\{ \tau \right\}^T \left\{ \sf{e}^e \right\}
\label{eq:inner_product}
\end{equation}
For the deviatoric parts of the second order tensors, a five-component form is adopted. For the Kirchhoff stress and
the deformation rate:
\begin{equation}
{\boldsymbol{\tau}}^\prime \rightarrow \left\{ \tau^\prime \right\} = \left\{ \frac{1}{\sqrt{2}}(\tau^\prime_{11}- \tau^\prime_{22})\,\, \sqrt{\frac{3}{2}}\tau^\prime_{33}\,\, \sqrt{2}\tau^\prime_{23}\,\, \sqrt{2}\tau^\prime_{31}\,\, \sqrt{2}\tau^\prime_{12} \right\}^T
\label{eq:vector_dev-stress}
\end{equation}
\begin{equation}
\tnsr{d}^\prime \rightarrow \left\{ \sf{d}^\prime \right\} = \left\{ \frac{1}{\sqrt{2}}(\sf{d}^\prime_{11}- \sf{d}^\prime_{22})\,\, \sqrt{\frac{3}{2}}\sf{d}^\prime_{33}\,\, \sqrt{2}\sf{d}^\prime_{23}\,\, \sqrt{2}\sf{d}^\prime_{31}\,\, \sqrt{2}\sf{d}^\prime_{12} \right\}^T
\label{eq:vector_dev-defrate}
\end{equation}
\begin{equation}
{\tnsr{e}^e}^\prime \rightarrow \matdlateps = \left\{ \frac{1}{\sqrt{2}}(\sf{e}^\prime_{11}- \sf{e}^\prime_{22})\,\, \sqrt{\frac{3}{2}}\sf{e}^\prime_{33}\,\, \sqrt{2}\sf{e}^\prime_{23}\,\, \sqrt{2}\sf{e}^\prime_{31}\,\, \sqrt{2}\sf{e}^\prime_{12} \right\}^T
\label{eq:vector_dev-lateps}
\end{equation}
where the inner product again is preserved:
\begin{equation}
{\boldsymbol{\tau}}^\prime \cdot \tnsr{d}^\prime = \left\{ \tau^\prime \right\}^T \left\{ \sf{d}^\prime \right\}
\label{eq:inner_product_dev}
\end{equation}
Fourth-order tensors, namely the crystal elastic stiffness and compliance tensors, are commonly written as 6x6 matrices and populated according to the crystal symmetries.
Hooke's law written using the matrix form in a crystal coordinate bases for cubic and hexagonal crystal types are:
\begin{equation}
\left\{
\begin{array}{c} \tau_{11} \\ \tau_{22} \\ \tau_{33} \\ \tau_{23} \\ \tau_{13} \\ \tau_{12} \end{array}
\right\}
=
\left[
\begin{array}{c c c c c c}
C_{11} & C_{12} & C_{12} & & & \\
C_{12} & C_{11} & C_{12} & & & \\
C_{12} & C_{12} & C_{11} & & & \\
& & & C_{44} & & \\
& & & & C_{44} & \\
& & & & & C_{44}
\end{array}
\right]
\left\{
\begin{array}{c} e_{11} \\ e_{22} \\ e_{33} \\ 2e_{23} \\ 2e_{13} \\ 2e_{12} \end{array}
\right\} \hspace{0.5cm}{\rm Cubic \, Symmetry}
\label{eq:cubic_stiffness}
\end{equation}
and
\begin{equation}
\left\{
\begin{array}{c} \tau_{11} \\ \tau_{22} \\ \tau_{33} \\ \tau_{23} \\ \tau_{13} \\ \tau_{12} \end{array}
\right\}
=
\left[
\begin{array}{c c c c c c}
C_{11} & C_{12} & C_{13} & & & \\
C_{12} & C_{11} & C_{13} & & & \\
C_{13} & C_{13} & C_{33} & & & \\
& & & C_{44} & & \\
& & & & C_{44} & \\
& & & & & (C_{11}-C_{12})/2
\end{array}
\right]
\left\{
\begin{array}{c} e_{11} \\ e_{22} \\ e_{33} \\ 2e_{23} \\ 2e_{13} \\ 2e_{12} \end{array}
\right\} \hspace{0.5cm}{\rm Hexagonal \, Symmetry}
\label{eq:hexagonal_stiffness}
\end{equation}
where only the nonzero values are displayed.
A cautionary remark is added here regarding a factor of 2 that may appear with $C_{44}$ in other expressions of Hooke's law. {\bfseries{\slshape{FEpX}}}\, expects values for the elastic moduli consistent with Equation~\ref{eq:cubic_stiffness} or \ref{eq:hexagonal_stiffness} even though stiffness or compliance matrices internal to {\bfseries{\slshape{FEpX}}}\, are constructed somewhat differently. A restriction is placed on the moduli for hexagonal crystals vis-\`{a}-vis coupling of the shear and volumetric responses, as described in the next paragraph.
In the kinematic development, the motion is split into volumetric and deviatoric parts according to Equations~\ref{eq:kinematic_decomp_meandefrate} and \ref{eq:kinematic_decomp_devdefrate}. This split is convenient for the
numerical implementation, but limits the generality of the hexagonal behaviors. In particular, the volumetric and
deviatoric responses separate only when $C_{11}+C_{12} = C_{13}+C_{33}$. Thus, only four of the five nonzero moduli
are independent. To guarantee that this constraint is imposed, $(C_{11}, C_{12}, C_{13}, C_{44})$ are read in the input for {\bfseries{\slshape{FEpX}}}\, and $C_{33}$ is computed to satisfy the constraint.
When split into volumetric and deviatoric parts, Equation~\ref{eq:hookes_law} gives:
\begin{equation}
\trace\matkirch = \frac{\kappa}{3} \trace\matlateps
\label{eq:volumetric_hooke}
\end{equation}
and
\begin{equation}
\matdkirch = \matxdelasticity\matdlateps
\label{eq:deviatoric_hooke}
\end{equation}
where the stress and strain vectors are consistent with Equations~\ref{eq:vector_stress} - \ref{eq:vector_dev-defrate}.
Table~\ref{tab:elasticmoduli} lists the values of the $\kappa$ and the nonzero (diagonal) entries of $\matxdelasticity$ in terms of the
moduli presented in Equations~\ref{eq:cubic_stiffness} and \ref{eq:hexagonal_stiffness}.
\begin{table}[h]
\centering
\begin{tabular}{||c|c|c||}
\hline
\hline
Parameter & Cubic &Hexagonal \\
\hline
$\kappa$ & $3(C_{11}+ 2C_{12})$ & $3(C_{11}+ C_{12}+C_{13})$ \\
$c^\prime_{11}$ & $C_{11}-C_{12}$ & $C_{11}-C_{12}$ \\
$c^\prime_{22}$ & $C_{11}-C_{12}$ & $3(C_{33}-C_{13})$ \\
$c^\prime_{33}$ & $C_{44}$ & $C_{44}$ \\
$c^\prime_{44}$ & $C_{44}$ & $C_{44}$ \\
$c^\prime_{55}$ & $C_{44}$ & $C_{11}-C_{12}$ \\
\hline
\hline
\end{tabular}
\caption{Values of the moduli used in the separated form of Hooke's law}
\label{tab:elasticmoduli}
\end{table}
\subsection{Time-discretized elastoplastic relations}
Equations~\ref{eq:kinematic_decomp_meandefrate}, \ref{eq:kinematic_decomp_devdefrate}, \ref{eq:hookes_law}
and \ref{eq:ss_const_law} are now merged into a single equation that relates the Cauchy stress to the
total deformation rate. First, the spatial time-rate change of the elastic strain is approximated with a finite difference expression:
\begin{equation}
\matlatepsdot = \deeteei \biggl( \matlateps - \matlatepsold \biggr)
\label{eq:euler_approx_strain_rate}
\end{equation}
where $\matlateps$ is the elastic strain at the end of the time step and $\matlatepsold$ is the elastic strain at the beginning of the time step. The difference approximation is employed in an implicit algorithm, wherein the equations are solved at the time corresponding to the end of the time step. This time corresponds to the current configuration.
Writing the time rate change of the strain in terms of strains at two times facilitates substitution of Hooke's law -- namely at the end of the time step. The elastic strain at the beginning of the time step is known from the solution for the preceding time step.
For the volumetric part of the motion, combining Equations~\ref{eq:kinematic_decomp_meandefrate} and \ref{eq:hookes_law}
with the difference expression gives:
\begin{equation}
-\pi = \frac{\kappa\Delta t }{ \beta } \trace\matdefrate + \frac{\kappa} {\beta} \trace\matlatepsold
\label{eq:discret_volumetric_ep-law}
\end{equation}
Turning to the deviatoric (shearing) part of the motion, inserting Equation~\ref{eq:euler_approx_strain_rate} into Equation~\ref{eq:kinematic_decomp_devdefrate} gives:
\begin{equation}
\matddefrate = \deeteei \matdlateps + \matlatdefrate + \matpxspinhat \matdlateps - \deeteei \matdlatepsold
\label{eq:discret_deviatoric_ep-law}
\end{equation}
where $\matpxspinhat$ is the matrix form of $\hat{\tnsr{w}}^{p}$:
\begin{equation}
\matpxspinhat =
\left[
\begin{array}{c c c c c}
0 & 0 & -2{\hat w_{12}^p} & -{\hat w_{13}^p} & {\hat w_{23}^p} \\
0 & 0 & 0 & {\sqrt{3}} {\hat w_{13}^p} & {\sqrt{3}} {\hat w_{23}^p} \\
2{\hat w_{12}^p}& 0 & 0 & -{\hat w_{23}^p} & -{\hat w_{13}^p} \\
{\hat w_{13}^p} & -{\sqrt{3}}{\hat w_{13}^p} & {\hat w_{23}^p} & 0 & -{\hat W_{12}^p} \\
-{\hat w_{23}^p} & -{\sqrt{3}}{\hat w_{23}^p} & {\hat w_{13}^p} & {\hat w_{12}^p} & 0
\end{array}
\right]
\end{equation}
The equations for plastic slip
(Equation~\ref{eq:ss_const_law}) for the plastic deformation rate:
\begin{equation}
\matlatdefrate = \matxplasticity \matdkirch
\end{equation}
\begin{equation}
\matxplasticity
=
\sum_{\alpha} \left( \frac{f(\tau^\alpha, g)}{\tau^\alpha} \right)
\matsymschmid \matsymschmid^{\rm T}
\end{equation}
together with Equation~\ref{eq:deviatoric_hooke}
are substituted to render an equation that gives the deviatoric Cauchy stress in terms of the total deviatoric deformation rate and
a matrix, $\mathhh$, that accounts for the spin and the elastic strain at the beginning of the time step:
\begin{equation}
\matdcauchy = \matxep \bigg( \matddefrate- \mathhh \bigg)
\label{eq:discrete_devCauchy}
\end{equation}
where:
\begin{equation}
\matxep^{{ -\hspace{-.1em}1}} = \frac{\beta}{\Delta t} {\matxdelasticity}^{{ -\hspace{-.1em}1}} +\beta \matxplasticity
\label{eq:discrete_ep_stiffness}
\end{equation}
\begin{equation}
\mathhh = \matpxspinhat \matdlateps - \deeteei \matdlatepsold
\label{eq:discrete_spin_correction}
\end{equation}
Equations~\ref{eq:discrete_devCauchy}, \ref{eq:discrete_ep_stiffness} and \ref{eq:discrete_spin_correction} will be used in the weak form of equilibrium to write the stress in terms of the deformation rate.
\subsection{Interpolation functions}
{\bfseries{\slshape{FEpX}}} employs a standard isoparametric mapping framework for discretizing the problem domain and for representing the solution variables.
The mapping of the coordinates of points provided by the elemental interpolation functions, $\Big[ \,\mathsf{N}(\xi, \eta, \zeta) \,\Big]$, and the coordinates of the nodal points, $\left\{ X \right\}$:
\begin{equation}
\left\{ x \right\} = \Big[ \,\mathsf{N}(\xi, \eta, \zeta) \,\Big] \left\{ X \right\}
\label{eq:coord_mapping}
\end{equation}
where $(\xi, \eta, \zeta)$ are local coordinates within an element.
The same mapping functions are used for the solution (trial) functions which, together with the nodal point values of the
velocity, $\Big\{ \mathsf{V} \Big\}$, specify the velocity field over the elemental domains:
\begin{equation}
\matvel = \Big[ \,\mathsf{N}(\xi, \eta, \zeta) \,\Big] \Big\{ \mathsf{V} \Big\}
\label{eq:trial_functions}
\end{equation}
The deformation rate is computed from the spatial derivatives (derivatives with respect to $\vctr{x}$) of the mapping functions and the nodal velocities as:
\begin{equation}
\matdefrate = \Big[ \,\mathsf{B} \,\Big] \Big\{ \mathsf{V} \Big\}
\label{eq:trial_function_derivatives}
\end{equation}
$\Big[ \,\mathsf{B} \,\Big]$ is computed using the derivatives of $\Big[ \,\mathsf{N}(\xi, \eta, \zeta) \,\Big]$ with respect to local coordinates, $(\xi, \eta, \zeta)$, together with the Jacobian of the mapping specified by Equation~\ref{eq:coord_mapping}, following standard finite element procedures for isoparametric elements.
{\bfseries{\slshape{FEpX}}}\, relies principally on a 10-node, tetrahedral, serendipity element, as shown in Figure~\ref{fig:tet_element}. This $C^0$ element provides pure quadratic interpolation of the velocity field.
\begin{figure}[h]
\begin{center}
\includegraphics*[width=10cm]{Images/ten_node_tet.jpg}
\caption{10-node tetrahedral element with quadratic interpolation of the velocity, shown in the parent configuration and bounded by a unit cube.}
\label{fig:tet_element}
\end{center}
\end{figure}
{\bfseries{\slshape{FEpX}}}\, employs a Galerkin methodology for constructing a weighted residual. The weight functions therefore use the same interpolation functions as used for the coordinate map and the trial functions:
\begin{equation}
\Big\{ \psi \Big\} = \Big[ \,\mathsf{N}(\xi, \eta, \zeta) \,\Big] \Big\{ {\it \Psi } \Big\}
\label{eq:weight_functions}
\end{equation}
\subsection{Finite element residual for the velocity field}
\label{sec:fe_formulation}
Equilibrium is enforced by requiring a global weighted residual to vanish:
\begin{equation}
R_u = \int_{\cal{B}} \gtnsr{\psi} \cdot \left( {\rm div} {{\boldsymbol{\sigma}}}^T + {\vctr{\iota}}\right) {\mathrm{d}} {\cal B} = 0
\end{equation}
The residual is manipulated in the customary manner (integration by parts and application of the divergence theorem) to obtain the weak form:
\begin{equation}
R_u =
- \int_{\cal{B}} \; \trace \left( {{\gtnsr{\sigma}}^\prime}^{\rm T} \,{\rm grad } \vctr{\psi} \right) {\mathrm{d}} {\cal{B}}
+ \int_{\cal B} \pi \, {\rm div} \vctr{\psi} {\mathrm{d}} {\cal B}
+ \int_{\partial {\cal B}} \vctr{t} \cdot \vctr{\psi} {\mathrm{d}}\Gamma +
\int_{\cal{B}} \vctr{\iota} \cdot \vctr{\psi} {\mathrm{d}}{\cal B}
\end{equation}
Introduction of the trial and weight functions gives a residual vector for the discretized weak form for each element:
\begin{equation}
\Big\{ R^{\it ele}_u \Big\}
=
\bigg[ \matstiffd + \matstiffv \bigg] \Big\{ \mathsf{V} \Big\}
-
\Big\{ \mathsf{f}^{\it ele}_a \Big\}
-
\Big\{ \mathsf{f}^{\it ele}_d \Big\}
-
\Big\{ \mathsf{f}^{\it ele}_v \Big\}
\end{equation}
where
\begin{equation}
\matstiffd
=
\int_{\cal{B}} \Big[ \,\mathsf{B} \,\Big]^{\rm T} \matcapX^{\rm T} \matxep \matcapX \Big[ \,\mathsf{B} \,\Big] {\mathrm{d}} {\cal B}
\label{eq:k_d}
\end{equation}
\begin{equation}
\matstiffv =
\int_{\cal{B}} \betaoverkappa \Big[ \,\mathsf{B} \,\Big]^{\rm T} \matcapX^{\rm T} \Big\{ \mathsf{\delta} \Big\} \Big\{ \mathsf{\delta} \Big\}^{\rm T} \matcapX \Big[ \,\mathsf{B} \,\Big] {\mathrm{d}} {\cal B}
\end{equation}
\begin{equation}
\Big\{ \mathsf{f}^{\it ele}_a \Big\}
=
\int_{\partial {\cal{B}}} \Big[ \,\mathsf{N}(\xi, \eta, \zeta) \,\Big]^{\rm T} \mattraction {\mathrm{d}} \Gamma
+ \int_{\cal{B}} \Big[ \,\mathsf{N}(\xi, \eta, \zeta) \,\Big]^{\rm T} \matbodyforce {\mathrm{d}} {\cal B}
\end{equation}
\begin{equation}
\Big\{ \mathsf{f}^{\it ele}_v \Big\}
= \int_{\cal B} \Big[ \,\mathsf{B} \,\Big]^{\rm T} \matcapX^{\rm T} \betaoverkappa \Big\{ \mathsf{\delta} \Big\}^{\rm T} \matlatepsold {\mathrm{d}} {\cal B}
\end{equation}
\begin{equation}
\Big\{ \mathsf{f}^{\it ele}_d \Big\}
= \int_{\cal B} \Big[ \,\mathsf{B} \,\Big]^{\rm T} \matcapX^{\rm T} \matxep \mathhh {\mathrm{d}} {\cal B}
\label{eq:f_d}
\end{equation}
The integrals appearing in Equations~\ref{eq:k_d}-\ref{eq:f_d} are evaluated by numerical quadrature.
Assembling the elemental matrices and requiring that the residual vanish for all independent variations in the
weights gives:
\begin{equation}
\bigg[ \Matstiffd + \Matstiffv \bigg] \Big\{ \mathsf{V} \Big\}
=
\Big\{ \mathsf{F}_a \Big\}
+
\Big\{ \mathsf{F}_d \Big\}
+
\Big\{ \mathsf{F}_v \Big\}
\label{eq:global_system}
\end{equation}
The essential boundary conditions are applied prior to solving for the nodal velocities, as described in Section~\ref{sec:fe_boundary-conditions}. The matrix equation given by Equation~\ref{eq:global_system} is nonlinear ($\Matstiffd$ and $\Matstiffv$ depend on $\Big\{ \mathsf{V} \Big\}$). The solution methodology used in {\bfseries{\slshape{FEpX}}}\, is outlined in Section~\ref{sec:fe_nonlinear_methods}.
\subsection{Nonlinear solution algorithm for obtaining the velocity field}
\label{sec:fe_nonlinear_methods}
To solve Equation~\ref{eq:global_system} for the velocity field, an iterative methodology is invoked. This methodology is a hybrid procedure that utilizes a combination of successive approximations (Picard) and Newton-Raphson updates.
To accomplish this, the assembled residual force vector, $\Big\{ R_u \Big\}$, is defined as:
\begin{equation}
\Big\{ R_u \Big\} = \bigg[ \Matstiffd + \Matstiffv \bigg] \Big\{ \mathsf{V} \Big\}
-
\Big\{ \mathsf{F}_a \Big\}
-
\Big\{ \mathsf{F}_d \Big\}
-
\Big\{ \mathsf{F}_v \Big\}
\label{eq:global_residual}
\end{equation}
The goal of the iterative process is to drive the residual to zero through a series of corrections, $\Big\{ \Delta\mathsf{U} \Big\}$, to an estimate of the velocity field. Denoting the estimate of the velocity on the $i^{\rm th}$ iteration as $\Big\{ \mathsf{V} \Big\}^{i}$ and the estimate on the next iteration as $ \Big\{ \mathsf{V} \Big\}^{i+1}$, the iteration procedure is written simply as:
\begin{equation}
\Big\{ \mathsf{V} \Big\}^{i+1} = \Big\{ \mathsf{V} \Big\}^{i} + \Big\{ \Delta\mathsf{U} \Big\}^{i+1}
\label{eq:iterative_velocity}
\end{equation}
where $\Big\{ \Delta\mathsf{U} \Big\}^{i+1}$ is determined from the solution of
\begin{equation}
\bigg[ \Matstiffd^{{\rm type}} + \Matstiffv \bigg] \Big\{ \Delta\mathsf{U} \Big\}^{i+1} = -{\Big\{ R_u \Big\}}^{i}
\label{eq:velocity_iterations}
\end{equation}
Here, $\Matstiffd^{{\rm type}}$ refers to either a tangent modulus, $\Matstiffd^{{\rm tan}}$ or a secant modulus, $\Matstiffd^{{\rm sec}}$, as specified by the hybrid procedure.
Convergence is based on changes in the norm of $\Big\{ \mathsf{V} \Big\}$ becoming small.
\subsection{Time marching and boundary conditions}
\label{sec:fe_boundary-conditions}
The intent of the {\bfseries{\slshape{FEpX}}}\, framework and the derivative finite element code is to simulate the deformation of virtual polycrystals over time. To this end, time histories are approximated
by solving for the velocity field at a series of discrete times. Simply stated, {\bfseries{\slshape{FEpX}}}\, computes the velocity field at the end of time interval using Equation~\ref{eq:global_residual} knowing the velocity field and state at beginning of the time interval.
The geometry and state variables (lattice orientations and slip system strengths) are updated concurrently with
the velocity field at the end of the time step.
The time marching method is documented in ~\cite{mar_daw_98b,man_daw_lee_92}.
Over the course of a deformation history, the applied boundary conditions often change. This may imply that either
natural or essential boundary conditions in Equations~\ref{eq:surface_traction} and \ref{eq:surface_velocity} are functions of time. Presently, {\bfseries{\slshape{FEpX}}}\, allows the user to change the values of imposed velocities or forces at nodes over the course of a deformation, but does not allow the user to change the type of boundary condition. That is, a nodal point with imposed velocity (essential boundary condition) will have an essential boundary condition throughout a simulation, but the value of the velocity that is imposed may change with time. The same applies for nodes with natural boundary conditions -- the force may change with time, but the condition at that node will remain a natural boundary condition throughout the simulation.
Given this limitation, the boundary condition options within {\bfseries{\slshape{FEpX}}}\, accommodate a number of possibilities. For example,
it is anticipated that a common use of {\bfseries{\slshape{FEpX}}}\, will be to simulate the response of virtual polycrystals being subjected to boundary conditions that replicate mechanical tests performed using a load frame.
The control of mechanical tests can be designed to provide
programmed force histories, programmed displacement histories, or combinations of these. {\bfseries{\slshape{FEpX}}}\, offers the
capabilities to impose boundary conditions to mimic mechanical test histories. Some of the possibilities include:
\begin{itemize}
\item specified crosshead/actuator velocity,
\item specified load history,
\item unloading episodes, and
\item uniaxial and biaxial loading modes.
\end{itemize}
\section{Introduction}
\label{chapter:purpose}
\input{Purpose_Scope}\pagebreak[4]
\section{Capabilities of {\bfseries{\slshape{FEpX}}}}
\label{chapter:capabilities}
\input{Capabilities}\pagebreak[4]
\section{Nomenclature}
\label{chapter:nomenclature}
\input{Nomenclature}\pagebreak[4]
\section{Governing Equations}
\label{chapter:governing_equations}
\input{Governing_Equations}\pagebreak[4]
\section{Finite Element Implementation}
\label{chapter:fem_implementation}
\input{FE_Implementation}\pagebreak[4]
\section{Input and Output Data}
\label{chapter:IO}
\input{IO_Data}\pagebreak[4]
\section{Example Problems}
\label{chapter:examples}
\input{Example_Problems}\pagebreak[4]
\section{Acknowledgments}
Support was provided by the US Office of Naval Research (ONR) under contract N00014-12-1-0075. The authors wish to thank Andrew Poshadel and Amand Mitch for the example problems they provided. Thanks also to Andrew Poshadel, Matthew Kasemer and Robert Carson for their comments on the manuscript.
\newpage
\bibliographystyle{unsrt}
\subsection{Kinematics and balance laws}
\label{sec:strongforms}
The motion of the virtual polycrystal is assumed to be a smooth mapping of all points within
the domain, $\cal{B}$. The domain in this context refers to the union of all crystal volumes that collectively
define the virtual polycrystal. The internal crystal interfaces are smooth surfaces that remain
coherent throughout the motion. Under the motion, the current coordinates may be written as
a function of reference coordinates for every point in the domain:
\begin{equation}
\vctr{x} = \vctr{\chi}(\vctr{X})
\label{eq:mapping}
\end{equation}
The deformation gradient is defined locally as:
\begin{equation}
\tnsr{f} = \frac{\partial\vctr{x}}{\partial\vctr{X}}
\label{eq:defgrad}
\end{equation}
While the mapping, $\chi$, is smooth, the deformation gradient will be so only within the interior of elements.
Discontinuities can arise across element boundaries, whether the boundaries lie within crystals or on the interface between crystals.
The time-rate-change of the deformation gradient is given by:
\begin{equation}
\dot{\tnsr{f}} = \tnsr{l} \cdot \tnsr{f}
\end{equation}
where the velocity gradient, $\tnsr{l}$, is computed from the velocity field, $\vctr{v}(\vctr{x})$, as:
\begin{equation}
\tnsr{l} = \frac{\partial\vctr{v}}{\partial\vctr{x}}
\label{eq:velocitygradient}
\end{equation}
Again, while the velocity is smooth everywhere, its gradient may have discontinuities across element boundaries.
The velocity gradient is decomposed into its symmetric and skew parts:
\begin{equation}
\tnsr{l} = \tnsr{d} +\tnsr{w}
\label{eq:velgrad_decomp}
\end{equation}
where $\tnsr{d}$ is the deformation rate (symmetric part) and $\tnsr{w}$ is the spin (skew part).
The motion of the polycrystal is driven by the stress.
The Cauchy stress, ${\boldsymbol{\sigma}}(\vctr{x})$, is a field variable defined over the polycrystal domain.
Under the loading assumptions, inertia in the balance of linear momentum is neglected,
giving static equilibrium in the local form as:
\begin{equation}
{\rm div} {{\boldsymbol{\sigma}}}^T + \vctr{\iota} \,=\, \boldsymbol{0}
\label{eq:equilibrium}
\end{equation}
where $\vctr{\iota}$ is the body force vector. Body forces are neglected in the current implementation of {\bfseries{\slshape{FEpX}}}.
This equation applies to the interior of the crystals. Across crystal interfaces, continuity of the
traction is needed. Applying the Cauchy formula, this condition may be written
for two contacting crystals, $i$ and $j$, as:
\begin{equation}
\vctr{t}(\vctr{x})^i \,=\, \ \vctr{t}(\vctr{x})^j
\label{eq:crystal_tract_vec}
\end{equation}
where the tractions are related to the stress by means of the Cauchy formula:
\begin{equation}
\vctr{t} \,=\, {\vctr{\nu}(\vctr{x})} \cdot {\boldsymbol{\sigma}}
\label{eq:cauchyformula}
\end{equation}
The Cauchy stress may be split into deviatoric and spherical parts:
\begin{equation}
{\boldsymbol{\sigma}} = {\gtnsr{\sigma}}^\prime - \pi \tnsr{ I}
\label{eq:deviator}
\end{equation}
which is central to the material response as only the deviatoric part drives plastic flow.
\subsection{Constitutive equations}
\label{sec:constitutive}
The material behavior is quantified with a set of constitutive equations, here written at the level of the single crystal.
The behavior includes both elastic (recoverable strains upon removal of the stress) and plastic (non-recoverable strains upon removal of the stress) responses.
These can occur concurrently which requires coupling of the motions for the two via a kinematic decomposition. The decomposition is not strictly derivable from the mapping, but rather involve assumptions regarding the behavior. Consequently, it is part of constitutive model. The elastic response is limited to linear behavior following Hooke's law for anisotropic behavior. The plastic response is nonlinear and rate-dependent (viscoplastic). It is assumed to be isochoric and independent of the mean stress. The set of equations are summarized in the following subsections, starting with the kinematic decomposition. The equations for the elastic and plastic responses follow discussion of the kinematic decomposition and are broken into two parts: fixed state relations and evolution relations.
\subsubsection{Elastoplastic kinematic decomposition}
The deformation at a material point\footnote{In the context of {\bfseries{\slshape{FEpX}}}, a material point is a volume of material that is small in comparison to the finite element in which it resides (and thus, small in comparison to an individual crystal), yet large enough to fully reflect the crystal structure and the deformation processes. For slip, this means that the dimensions of the volume are much larger than a Burger's vector.}
is a combination of the elastic and plastic parts. In addition, a rotation occurs as part of the complete motion.
These are shown schematically in Figure~\ref{fig:kinematic_decomposition}.
\begin{figure}[h]
\begin{center}
\includegraphics*[width=8cm]{Images/decomposition.jpg}
\caption{Kinematic decomposition for motion by a combination of plastic slip, rotation and elastic straining. }
\label{fig:kinematic_decomposition}
\end{center}
\end{figure}
The decomposition that describes this motion consists of breaking the deformation gradient into a sequence of three parts: a plastic part, a rotation and an elastic part, given as:
\begin{equation}
\tnsr{f} = \tnsr{f}^e \tnsr{f}^\star \tnsr{f}^p = \tnsr{v}^e \tnsr{r}^\star{\tnsr{f}}^p
\label{eq:kinematic_decomp}
\end{equation}
Each part of the decomposition brings the material point to a new configuration, starting with reference coordinates, $\vctr{X}$, and finishing at the current coordinates, $\vctr{x}$.
The elastic part is a pure stretch which, by assuming small elastic strains, can be approximated with:
\begin{equation}
\tnsr{v}^e = \tnsr{I} + {\tnsr{e}^e}
\label{eq:small_elastic_stretch}
\end{equation}
where
\begin{equation}
|| {\tnsr{e}^e} || << 1
\end{equation}
The plastic part involves both stretch and rotation as a consequence of being a linear combination
of slip modes, each of which is simple shear. The distinct rotation in $\tnsr{f}^\star$ (or equivalently, $\tnsr{r}^\star$)
is the rotation beyond that included in ${\tnsr{f}}^p$ that is needed for consistency with the overall mapping
given by Equation~\ref{eq:mapping}.
The primitive solution field variable of {\bfseries{\slshape{FEpX}}}\, is the velocity, owing principally to the code's legacy of being a tool for modeling
viscoplastic flow. To cast the kinematic decomposition in rate form, the velocity gradient first is written in terms of the deformation gradient and its time-rate-of-change:
\begin{equation}
\tnsr{l} = \dot\xdefgrad \tnsr{f}^\invrs
\label{eq:kinematic_decomp_rate}
\end{equation}
where the velocity gradient is subsequently decomposed into the deformation rate and spin, as per Equation~\ref{eq:velgrad_decomp}. The deviatoric deformation rate is obtained by subtracting the volumetric part from the total:
\begin{equation}
\tnsr{d}^\prime = \tnsr{d} - \frac{1}{3}\trace{\tnsr{d}}
\label{eq:deviatoric_defrate}
\end{equation}
Substituting Equation~\ref{eq:kinematic_decomp} into Equation~\ref{eq:kinematic_decomp_rate} and separating the deformation rate into its volumetric and deviatioric parts with Equation~\ref{eq:deviatoric_defrate} gives:
\begin{equation}
\trace ( \tnsr{d}) = \trace( \dot{\tnsr{e}}^e )
\label{eq:kinematic_decomp_meandefrate}
\end{equation}
and
\begin{equation}
\tnsr{d}^\prime = \dot{{\tnsr{e}}^e}^\prime + \hat{\tnsr{d}^p}^\prime +
{\tnsr{e}^e}^\prime \hat{\tnsr{w}}^{p} - \hat{\tnsr{w}}^{p} {\tnsr{e}^e}^\prime
\label{eq:kinematic_decomp_devdefrate}
\end{equation}
in which the small elastic strain approximation from Equation~\ref{eq:small_elastic_stretch} has been invoked.
The spin becomes:
\begin{equation}
\tnsr{w} = \hat{\tnsr{w}}^{p} + {\tnsr{e}^e}^\prime \hat{\tnsr{d}^p}^\prime - \hat{\tnsr{d}^p}^\prime {\tnsr{e}^e}^\prime
\label{eq:kinematic_decomp_spin}
\end{equation}
where, again, small elastic strains are assumed.
In Equations~\ref{eq:kinematic_decomp_spin} and \ref{eq:kinematic_decomp_devdefrate},
the hat over $\tnsr{w}^{p}$ and $ {\tnsr{d}^{p}}^\prime$ indicates mapping to configuration $\cal{\hat B}$ using $\tnsr{r}^\star$, as indicated in Figure~\ref{fig:kinematic_decomposition}:
\begin{equation}
\hat{\tnsr{w}}^{p} = \tnsr{r}^\star \tnsr{w}^{p} {\tnsr{r}^\star}^{\rm T}
\end{equation}
\begin{equation}
\hat{\tnsr{d}^p}^\prime = \tnsr{r}^\star {\tnsr{d}^{p}}^\prime {\tnsr{r}^\star}^{\rm T}
\end{equation}
The deformations associated with elastic and plastic parts of the kinematic decomposition are intimately connected to the
crystallographic lattice. The orientation of the lattice relative to a set of global base vectors at a designated material point is given by
the associated quaternion, $\vctr{q}$, which is parameterized by the components: $(q_0, \vec{q})$.
It is convenient to use other representations for orientation as well, depending on the task at hand.
Two frequently used representations are the Rodrigues vector:
\begin{equation}
\vctr{r} = \vctr{n} \tan \frac{\phi}{2} = \vec{q}/q_0
\label{eq:rod_vect}
\end{equation}
and the rotation tensor:
\begin{equation}
\tnsr{R} = {\frac{1}{1+\vctr{r} \cdot \vctr{r}}}\big({{\tnsr{I}(1-\vctr{r} \cdot \vctr{r})}+{2(\vctr{r} \otimes \vctr{r} + \tnsr{I}\times \vctr{r})}}\big)
\label{eq:rotation_tensor}
\end{equation}
Properties that are dependent on lattice orientation are indicated as a function of $\vctr{q}$. Note also that
changes in the lattice orientation that accompany the motion are embedded in $\tnsr{r}^\star$, so the evolution rate of
$\vctr{q}$ is defined in terms of the rate of change of $\tnsr{r}^\star$.
With the kinematic decomposition specified, the Kirchhoff stress is written based on
the material point volume in the $\hat{\cal B}$ configuration as:
\begin{equation}
{\boldsymbol{\tau}} = \beta {\boldsymbol{\sigma}}
\hspace{0.5 cm} {\rm where} \hspace{0.5cm}
\beta = {\rm det} (\tnsr{v}^e)
\label{eq:kirchhoff_defn}
\end{equation}
\subsubsection{Fixed state constitutive relations }
The kinematic decomposition must be accompanied by equations relating the elastic and plastic deformations to the stress.
For the elastic deformations, this relation is simply Hooke's law, written using the $\hat{\cal B}$ configuration as a reference volume:
\begin{equation}
{\boldsymbol{\tau}} = \boldsymbol{\cal{C}}(\quaternion) {\tnsr{e}^e}
\label{eq:hookes_law}
\end{equation}
Here, the anisotropic behavior stemming from the crystal symmetry is indicated by the
orientation dependence of the elastic stiffness.
The structure of the elastic stiffness (occurrence of zero or repeated components in $\boldsymbol{\cal{C}}(\quaternion)$) reflects the application of symmetry conditions to the fully anisotropic version of Hooke's law. These are more easily presented using a vector representation of
the stress and strain tensors, as is done in Section~\ref{chapter:fem_implementation}.
For the plastic flow, the relation is a combination of several equations that describe crystallographic slip on a limited number of slip systems (commonly called restricted slip). First are equations for the kinematic decomposition written in terms of slip using
the Schmid tensor's symmetric and skew parts:
\begin{equation}
\hat{\tnsr{l}}^p
=\hat{\tnsr{d}^p}^\prime + \hat{\tnsr{w}}^{p}
\label{eq:plastic_velgrad}
\end{equation}
where
\begin{equation}
\hat{\tnsr{d}^p}^\prime = \sum_{\alpha} \dot{\gamma}^\alpha \hat{\tnsr{p}}^\alpha
\hspace{0.5 cm} {\rm and} \hspace{0.5cm}
\hat{\tnsr{w}}^{p} = {\dot{\tnsr{r}}}^\star {\tnsr{r}^\star}^\transp + \sum_{\alpha} \dot{\gamma}^\alpha \hat{\tnsr{q}}^\alpha
\label{eq:ss_superposition}
\end{equation}
and
\begin{equation}
\hat{\tnsr{p}}^\alpha = \hat{\tnsr{p}}^\alpha(\vctr{q}) = \mbox{\rm sym} \, ({\hat{\vctr{s}}}^{\alpha}\otimes {\hat{\vctr{m}}}^{\alpha})
\end{equation}
\begin{equation}
\hat{\tnsr{q}}^\alpha = \hat{\tnsr{q}}^\alpha(\vctr{q}) = \mbox{\rm skw} \, ({\hat{\vctr{s}}}^{\alpha}\otimes {\hat{\vctr{m}}}^{\alpha})
\end{equation}
The slip systems commonly assumed for face-centered cubic and hexagonal close-packed crystal types
are shown in Figures~\ref{fig:fcc_ss} and \ref{fig:hcp_ss}, respectively.
\begin{figure}[h]
\centering
\subfigure[FCC slip systems ]
{
\includegraphics*[width=5cm]{Images/fcc_ss.jpg}
}
\vspace{1cm}
\subfigure[BCC slip systems]
{
\includegraphics*[width=6.6cm]{Images/bcc_ss.jpg}
}
\caption{Primary slip systems for face-centered cubic (FCC) and body-centered cubic (BCC) crystal types.}
\label{fig:fcc_ss}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics*[width=12cm]{Images/hcp_ss.jpg}
\caption{Primary slip systems for hexagonal close-packed (HCP) crystal types.}
\label{fig:hcp_ss}
\end{center}
\end{figure}
\begin{table}[h]
\centering
\begin{tabular}{||c|c|c|c|c||}
\hline
\hline
Crystal Type & Name & Number & $\vctr{m}$ & $\vctr{s}$ \\
\hline
FCC & octahedral & 12 & $\{111\} $& $<110> $\\
BCC & - & 12& $\{110\} $& $<111> $\\
HCP & basal & 6 & $\{0001\} $& $<11\bar{2}0> $\\
- & prismatic & 6 & $\{10\bar{1}0\} $& $<1 1 \bar{2}0> $\\
- & pyramidal & 6 & $\{10\bar{1}1\} $& $<1 1 \bar{2}3> $\\
\hline
\hline
\end{tabular}
\caption{Slip system vectors for (unstressed) FCC, BCC and HCP crystal types given a coordinate system attached to the lattice orientation.
}
\label{tab:slip_systems}
\end{table}
Next is an equation that defines the kinetics of slip, which introduces the rate dependence of plastic flow
using a power law expression between the resolved shear stress and the slip system shearing rate:
\begin{equation}
\dot{\gamma}^\alpha = f(\tau^\alpha , g^\alpha) = \dot{\gamma}_0 \left( {\frac{ \arrowvert \tau^\alpha \arrowvert }{g^\alpha}}\right)^{\frac{1}{m} }{\rm sgn}(\tau^\alpha)
\label{eq:ss_kinetics}
\end{equation}
The resolved shear stress is scaled by the slip system strength, $g^\alpha$, which in general may be different
for each slip system. However, in {\bfseries{\slshape{FEpX}}}\, the slip system strengths are the same for each family of slip systems
within a grain. Thus, the slip system strengths are all the same within each of the finite elements that
discretize a grain for either FCC and BCC crystal types.
For HCP crystals, the basal, prismatic and pyramidal strengths within each of the finite elements discretizing a grain can have different values, but all the systems
of a given type have the same value. The values of the strength evolve with deformation according to
the evolution equations, as discussed in Section~\ref{sec:state_evolution}. The `sgn' term forces the shearing
to be in the same direction as the shear stress. Finally, the resolved shear stress is the projection of the
crystal stress tensor onto the slip plane and into the slip direction, which is readily computed with the
symmetric part of the Schmid tensor:
\begin{equation}
\tau^\alpha=\trace( \hat{\tnsr{p}}^\alpha {\boldsymbol{\tau}}^\prime)
\label{eq:rss_projection}
\end{equation}
The equations for slip are combined in a single, nonlinear relation as:
\begin{equation}
\hat{\tnsr{d}^p}^\prime = \boldsymbol{\cal{M}}(\vctr{q},\dot{\gamma}^\alpha) {\boldsymbol{\tau}}^\prime
\label{eq:ss_const_law}
\end{equation}
Combining Equations~\ref{eq:kinematic_decomp_meandefrate}, \ref{eq:kinematic_decomp_devdefrate}, \ref{eq:hookes_law} and \ref{eq:ss_const_law} into a single equation that relates the Cauchy stress to the
total deformation rate requires an additional step to discretize the elastic strain rate. This step is
introduced in Section~\ref{chapter:fem_implementation}.
\subsubsection{State evolution equations}
\label{sec:state_evolution}
There are two state variables at every material point that are updated as a deformation progresses,
the lattice orientation and the slip system strength (also called hardnesses)\footnote{One could argue that the elastic strain also is a state variable, as it quantifies the shape of the unit cell. However, it is updated as an integral part of solving for the velocity field, rather than separate from solving for the velocity field as are summarized in this section for the lattice orientations and slip system hardnesses. See Section~\ref{chapter:fem_implementation}.}. The rate of lattice re-orientation follows directly from
the Equations~\ref{eq:plastic_velgrad} and \ref{eq:ss_superposition}, assuming that the slip system shearing rates are known.
Written in terms of the Rodrigues vector:
\begin{equation}
\dot\vctr{r} = \frac{1}{2} \vctr{\omega} + (\vctr{\omega} \cdot \vctr{r}) \vctr{r} + \vctr{\omega} \times \vctr{r}
\end{equation}
where
\begin{equation}
\vctr{\omega} = {\rm vect}\left( \hat{\tnsr{w}}^{p} - \sum_{\alpha} \dot{\gamma}^\alpha \hat{\tnsr{q}}^\alpha \right)
\end{equation}
Evolution of the slip system strengths is governed by an additional, empirical relationship which follows
as modified Voce form:
\begin{equation}
\dot{g^\alpha} = h_{0} {\left(\frac{g_{s}(\dot{\gamma}) - g^\alpha} {g_{s}(\dot{\gamma}) - g_{0}}\right)}^{n^\prime} {\dot{\gamma}}
\label{eq:ss_strength_evolution}
\end{equation}
Here, a saturation strength appears that is assumed to depend on a net local plastic strain rate
computed from the sum of the magnitudes of the slip system shearing rate:
\begin{equation}
g_{s}(\dot{\gamma}) = g_1 \left( \frac{\dot{\gamma}}{\dot{\gamma_s}} \right)^{m^\prime}
\hspace{0.5 cm} {\rm and} \hspace{0.5cm}
\dot{\gamma} = \sum_{\alpha} \arrowvert \dot{\gamma}^\alpha \arrowvert
\label{eq:sat_ss_strength}
\end{equation}
This equation is used to update the strength of the slip systems, family-by-family, in each element of the
finite element mesh used to discretize a polycrystal.
\subsection{Boundary conditions}
Consistent with solid mechanics theoretical framework, the boundary condition applied to a
surface of a virtual polycrystal may be either an imposed velocity or an imposed traction.
For tractions, this is stated simply as:
\begin{equation}
\vctr{t}(\vctr{x}) = \bar{\vctr{t} }
\label{eq:surface_traction}
\end{equation}
while for the velocity condition, it is:
\begin{equation}
\vctr{v}(\vctr{x}) = \bar{\vctr{v}}
\label{eq:surface_velocity}
\end{equation}
where the overbar indicates a known quantity.
\subsection{{\bfseries{\slshape{FEpX}}}\, input data }\label{sec:input_data}
The body being loaded and deformed in a {\bfseries{\slshape{FEpX}}}\, simulation is a virtual polycrystal -- a set of grains (each grain being a single crystal) that forms a fully-dense solid. The finite element mesh needed to build a virtual polycrystal must be created in advance, which can be done with {\bfseries{\slshape{Neper}}}\, or other mesh generation codes.
The input data is organized as follows:
\begin{itemize}
\item {\bf Setting up a job --} a script to execute {\bfseries{\slshape{FEpX}}}\, together with a file which lists the input files described below.
\item {\bf Defining a virtual polycrystal --} a set of files: a file that specifies the single crystal elastic and plastic material properties for each phase; a file that defines the finite element mesh (nodal coordinates, element connectivity, and surface elements); a file that designates the phase and grain numbers for each element; a file that provides the starting lattice orientation for each grain; and, one or more files that define the vertices of the single crystal yield surface for each phase.
\item {\bf Controlling the deformation --} a set of two files: a file designates the type of boundary condition (essential or natural) for each degree of freedom of all the nodes and the corresponding velocity or force; and, a file specifies target loads or displacements, depending on the loading type.
\item {\bf Postprocessing for diffraction data --} a file that provides information for averaging strains and stresses over crystallographic fibers.
\item {\bf Prescribing optional input --} a file that specifies the choices for loading protocols, post-processing options, various solution method options, and associated convergence limits.
\end{itemize}
\subsection{{\bfseries{\slshape{FEpX}}}\, output data}
\label{sec:output_data}
In the present implementation, the {\bfseries{\slshape{FEpX}}}\, code writes files for solution variables at times designated in the input data. The solution variables written to files are:
\begin{itemize}
\item {\bf Geometry:} the nodal point coordinates and the nodal point velocities.
\item {\bf Deformation:} the effective deformation rate, the effective plastic deformation rate, the slip system shearing rates, the effective strain, and the effective plastic strain.
\item {\bf State Variables:} the slip system strengths and the lattice orientations.
\item {\bf Stress:} the stress tensor. Note that the components of the Cauchy stress tensor are written in the global frame
in the following order: $ \sigma_{11}, \sigma_{12} , \sigma_{13}, \sigma_{22}, \sigma_{23}, \sigma_{33} $.
\item {\bf Elastic strain:} the elastic strain tensor. As with the stress, the components of the elastic strain tensor are written in the global frame
in the following order: $ \epsilon_{11}, \epsilon_{12}, \epsilon_{13}, \epsilon_{22}, \epsilon_{23}, \epsilon_{33}$.
\item {\bf Fiber averages:} mean and standard deviation values of the stress, elastic strain, and slip system activities taken over designated crystallographic fibers.
\item {\bf Monitors:} the resultant force on each external surface.
\end{itemize}
When executed on a parallel architecture, {\bfseries{\slshape{FEpX}}}\, writes output files on every node which are written to the master node at the end of the job. This leaves the results spread over many output files and in a format that is inconvenient for postprocessing. An {\bfseries{\slshape{OdfPf}}}\, script is available to concatenate all of the output files into a single data structure that is readily used within postprocessing tools.
\subsection{Exporting Input and Output Data }
\label{sec:hdf5_io}
Exporting input and output data is done for two reasons. One is so that the results may be archived in a data management system appropriate for the project of interest. The other is to interface with visualization codes or other specialty post-processing software, such as forward projectors or virtual instruments. The {\bfseries{\slshape{OdfPf}}}\, package includes scripts for these purposes:
\begin{itemize}
\item The HDF-5 file format is a commonly used standard for writing archivable files. A matlab script is available that prepares an HDF-5 file with the {\bfseries{\slshape{FEpX}}}\, standard input and output data. This script can be modified to include other information (such as postprocessing results) if desired.
\item To facilitate visualization of the results, matlab scripts are available to export files that can be read in by {\bfseries{\slshape{Paraview}}}\, or {\bfseries{\slshape{VisIt}}}. The applications of {\bfseries{\slshape{FEpX}}}\, shown in Section~\ref{chapter:examples} were plotted with {\bfseries{\slshape{Paraview}}}.
\end{itemize}
\subsection{{\bfseries{\slshape{OdfPf}}}\, capabilities}
\label{sec:odfpf}
{\bfseries{\slshape{OdfPf}}}\, figures prominently in the use of {\bfseries{\slshape{FEpX}}}\, to simulate the behavior of virtual polycrystals, both in the instantiation of virtual polycrystals and in the comparison of simulation results to diffraction data. {\bfseries{\slshape{OdfPf}}}\, is a function set is a collection of {\bfseries{\slshape{MATLAB}}}\, functions which operate on ODF's (orientation distribution functions) and PF's (pole figures). It handles plotting of the ODF using Rodrigues parameters, plotting of pole figures and inverse pole figures, evaluation of pole figures inverse pole figures from ODF's, and it provides many tools for computing ODF's from pole figures.
Archival publications are available that cover various aspects of its use or the use of a Rodrigues parameterization of orientation space in quantitative texture analyses. Relevant articles include: \cite{kum_daw_00,bar_boy_daw_02, bernierodf_etal_2006,schurenUncert_2011}.
\subsection{Variables used in theoretical description}
\label{sec:nomen-theory}
\begin{tabular}[b]{lll}
\nomenclature{$\beta$}{determinant of the elastic stretch, $\tnsr{v}^e$ }\\
\nomenclature{$\dot{\gamma}^\alpha$}{shearing rate of the $\alpha$ slip system }\\
\nomenclature{$\vctr{\nu}$}{surface normal vector}\\
\nomenclature{$\vctr{\iota}$}{body force vector}\\
\nomenclature{$\pi$}{mean stress (${\rm tr}({\boldsymbol{\sigma}})/3$) }\\
\nomenclature{$\phi$}{rotation angle associated with $\vctr{r}$}\\
\nomenclature{${\boldsymbol{\sigma}}$}{Cauchy stress }\\
\nomenclature{${\boldsymbol{\tau}}$}{Kirchhoff stress }\\
\nomenclature{$\tau^\alpha$}{resolved shear stress on the $\alpha$ slip system }\\
\nomenclature{$\vctr{\omega}$}{spin vector associated with $\vctr{v}$ }\\
\nomenclature{$\vctr{\chi}$}{mapping function of motion}\\
\nomenclature{$\cal{B}$}{domain of the polycrystal}\\
\nomenclature{${\partial {\cal B}}$}{surface of the polycrystal}\\
\nomenclature{$\boldsymbol{\cal{C}}$}{elasticity (stiffness) tensor}\\
\nomenclature{$\tnsr{d}$}{deformation rate (symmetric part of $\tnsr{l}$)}\\
\nomenclature{$\hat{\tnsr{d}^p}^\prime$}{plastic deformation rate (symmetric part of $\hat{\tnsr{l}}^p$)}\\
\nomenclature{${\tnsr{e}^e}$}{elastic strain}\\
\nomenclature{$\tnsr{f}$}{deformation gradient}\\
\nomenclature{$\tnsr{f}^e$}{elastic part of the deformation gradient}\\
\nomenclature{$\tnsr{f}^\star$}{rotational part of the deformation gradient associated with the lattice rotation}\\
\nomenclature{$\tnsr{f}^p$}{plastic part of the deformation gradient}\\
\nomenclature{$g^\alpha$}{ strength of the $\alpha$ slip system}\\
\nomenclature{$\tnsr{l}$}{velocity gradient}\\
\nomenclature{$\hat{\tnsr{l}}^p$}{plastic velocity gradient}\\
\nomenclature{$\vctr{m}^\alpha$}{normal to the slip plane for the $\alpha$ slip system}\\
\nomenclature{$\boldsymbol{\cal{M}}$}{plasticity (stiffness) tensor}\\
\nomenclature{$\vctr{n}$}{axis vector associated with $\vctr{r}$}\\
\nomenclature{$\tnsr{p}^\alpha$}{symmetric part of the Schmid tensor for the $\alpha$ slip system }\\
\nomenclature{$\tnsr{q}^\alpha$}{skew part of the Schmid tensor for the $\alpha$ slip system}\\
\nomenclature{$\vctr{q}$}{quaternion representation of lattice orientation $(q_0, \vec{q})$}\\
\nomenclature{$\vctr{r}$}{Rodrigues vector for the orientation of the crystallographic lattice}\\
\nomenclature{$\tnsr{r}^\star$}{rotational part of the deformation gradient associated with the lattice rotation ($=\tnsr{f}^\star $)}\\
\nomenclature{$\tnsr{R}$}{rotation operator corresponding to $\vctr{r}$}\\
\nomenclature{$\vctr{s}^\alpha$}{slip direction for the $\alpha$ slip system}\\
\nomenclature{$\vctr{t}$}{traction vector}\\
\nomenclature{$\bar{\vctr{t}}$}{imposed traction vector on the surface}\\
\nomenclature{$\Delta t$}{time step}\\
\nomenclature{$\vctr{v}$}{velocity vector of a point in the current configuration}\\
\nomenclature{$\bar{\vctr{v}}$}{ imposed velocity vector on the surface}\\
\nomenclature{$\tnsr{v}^e$}{elastic stretch}\\
\nomenclature{$\tnsr{w}$}{spin (skew part of $\tnsr{l}$)}\\
\nomenclature{$\hat{\tnsr{w}}^{p}$}{plastic spin (skew part of $\hat{\tnsr{l}}^p$)}\\
\nomenclature{$\vctr{x}$}{position vector of a point in the current configuration}\\
\nomenclature{$\vctr{X}$}{position vector of a point in the reference configuration}
\end{tabular}
\bigskip
\subsection{Parameters appearing in the constitutive models}
\label{sec:nomen-parameters}
\begin{tabular}[b]{lll}
\nomenclature{$\dot{\gamma}_0$}{fixed-state strain rate scaling coefficient }\\
\nomenclature{$\dot{\gamma_s}$}{saturation strength strain rate scaling coefficient}\\
\nomenclature{$\kappa$}{elastic bulk modulus}\\
\nomenclature{$c_{ij}$}{components of the elastic stiffness}\\
\nomenclature{$g_{0}$}{initial slip system strength}\\
\nomenclature{$g_{1}$}{reference value of saturation strength}\\
\nomenclature{$h_{0}$}{strength hardening rate coefficient }\\
\nomenclature{$m$}{fixed-state strain rate sensitivity}\\
\nomenclature{$m^\prime$}{saturation strength rate scaling exponent}\\
\nomenclature{$n^\prime$}{power on modified Voce hardening term }
\end{tabular}\bigskip
\subsection{Variables used in implementation description}
\label{sec:nomen-implementation}
\begin{tabular}[b]{lll}
\nomenclature{$ \left\{ \delta \right\}$}{matrix trace operator} \\
\nomenclature{$\Big\{ \psi \Big\}$}{matrix form of the weights}\\
\nomenclature{$ \Big\{ {\it \Psi } \Big\}$}{nodal point weights vector }\\
\nomenclature{$\left\{ \sigma \right\}$}{vector matrix form of Cauchy stress}\\
\nomenclature{$\left\{ \tau \right\}$}{vector matrix form of Kirchhoff stress}\\
\nomenclature{$ \Big[ \,\mathsf{B} \,\Big] $}{spatial derivatives of the interpolation functions, $[ N ]$ }\\
\nomenclature{$\matxelasticity$}{matrix form of the elastic stiffness}\\
\nomenclature{$\left\{ \sf{d} \right\}$}{vector matrix form of deformation rate}\\
\nomenclature{$\left\{ \sf{e}^e \right\}$}{vector matrix form of elastic strain}\\
\nomenclature{$\Big\{ \mathsf{f}^e_a \Big\} , \Big\{ \mathsf{F}_a \Big\}$}{elemental and global surface traction and body force matrices}\\
\nomenclature{$\Big\{ \mathsf{f}^e_v \Big\} , \Big\{ \mathsf{F}_d \Big\}$}{elemental and global initial elastic strain matrices}\\
\nomenclature{$\Big\{ \mathsf{f}^e_d \Big\} , \Big\{ \mathsf{F}_v \Big\}$}{elemental and global spin correction stiffness matrices}\\
\nomenclature{$\mathhh$}{vector matrix form of the spin correction and initial elastic strain terms }\\
\nomenclature{$\matstiffd, \Matstiffd$}{elemental and global deviatoric stiffness matrices}\\
\nomenclature{$\matstiffv, \Matstiffv$}{elemental and global volumetric stiffness matrices}\\
\nomenclature{$\matxplasticity$}{matrix form of the plastic stiffness}\\
\nomenclature{$[ N ]$}{interpolation functions for interpolation of the velocity distribution}\\
\nomenclature{$\matsymschmid$}{vector matrix form of the symmetric part of the Schmid tensor}\\
\nomenclature{$R_u$}{equilibrium weighted residual }\\
\nomenclature{$\Big\{ R_u \Big\}$}{elemental residual vector}\\
\nomenclature{$\matpxspinhat $}{matrix form of the plastic spin }\\
\nomenclature{$\matvel $}{matrix form of the velocity}\\
\nomenclature{$ \Big\{ \mathsf{V} \Big\} $}{nodal point velocity vector }\\
\nomenclature{$\left\{ x \right\}$}{matrix form of the position}\\
\nomenclature{$\matcapX$}{coefficient matrix of factors necessary to deliver correct inner product from matrix multiplications.}
\end{tabular}
\subsection{Purpose }
\label{sec:purpose}
The purpose of this article is to lay out a complete system of equations for modeling the anisotropic, elasto-viscoplastic response of polycrystalline solids comprised of aggregates of grains and to summarize a finite element formulation that may be employed to compute the motion and stress in polycrystals governed by the system of equations under imposed loadings. The governing equations together with associated solution methodologies define a modeling framework, referred to as {\bfseries{\slshape{FEpX}}}, that is focused at a physical length scale of an ensemble of grains. There is an associated finite element code, also named {\bfseries{\slshape{FEpX}}}, that follows the framework. A major motivation for archiving this article is to provide a thorough and accessible reference that researchers who utilize the code can readily cite. However, the article
stands independently in providing a complete summary of a crystal-scale model for the elasto-viscoplastic response of polycrystalline aggregates and a finite element formulation that enables solving the model equations over motions that entail large deformations.
The content provided here regarding the governing equations and finite element framework draws primarily from the following published articles:
\cite{daw_mar_98p,mar_daw_98a,mar_daw_98b,bar_daw_mil_99}.
The present article is not intended to serve as a primer for
computational crystal plasticity, so background knowledge of solid mechanics, including crystal plasticity,
and nonlinear finite element methods is assumed. Rather, it strives to encapsulate the full set of equations, assumptions, and solution approximations necessary to document simulation results with sufficient detail to
facilitate those results being reproduced by others.
\subsection{Scope}
\label{sec:scope}
The scope of this article is limited to the theory and methods that define the {\bfseries{\slshape{FEpX}}}\, framework, plus a general overview of
the data flow within the framework and the interfaces with tools to instantiate virtual polycrystals and to
visualize simulation results.
Consequently, there are sections of the article devoted
to these topics, as listed in the Table of Contents. Also provided are representative examples to illustrate application
of the derivative finite element code to modeling of single and dual phase metallic alloys.
No detailed information is included on the specific formatting used for problem definition, code execution, or exported simulation results.
That information is contained in separate documentation associated with the code itself.
\subsection{Complementary modeling tools}
\label{sec:complementarysoftwar}
The role of {\bfseries{\slshape{FEpX}}}\, in the modeling of polycrystals is to solve the boundary value problem associated with the elastoplastic
response of a polycrystalline solid arising from applied mechanical loading. Separate tools are needed to instantiate a virtual polycrystal and
to discretize it with finite elements. {\bfseries{\slshape{FEpX}}}\, accepts the finite element mesh generated by custom {\bfseries{\slshape{MATLAB}}}\, scripts (available in the {\bfseries{\slshape{OdfPf}}}\, package) and by the {\bfseries{\slshape{Neper}}}\, program~\cite{que_daw_bar_11}. Separate packages for visualizations also are needed. Export scripts are available for writing files that can be imported by {\bfseries{\slshape{Paraview}}}\cite{paraview}, and {\bfseries{\slshape{VisIt}}}\cite{HPV:VisIt}.
|
{
"timestamp": "2015-04-14T02:17:45",
"yymm": "1504",
"arxiv_id": "1504.03296",
"language": "en",
"url": "https://arxiv.org/abs/1504.03296"
}
|
\section{Introduction}
\label{sec:intro}
The two-loop sunrise integral with non-vanishing internal masses is the first and simplest integral in quantum field theory, which
cannot be expressed in terms of multiple polylogarithms.
It has already received considerable attention in the
literature \cite{Broadhurst:1993mw,Berends:1993ee,Bauberger:1994nk,Bauberger:1994by,Bauberger:1994hx,Caffo:1998du,Laporta:2004rb,Groote:2005ay,Groote:2012pa,Bailey:2008ib,MullerStach:2011ru,Adams:2013nia,Bloch:2013tra,Remiddi:2013joa,Adams:2014vja,Caffo:2002ch,Pozzorini:2005ff,Caffo:2008aw}.
However, the question which generalisations of multiple polylogarithms appear in the evaluation of the two-loop sunrise
integral is still open.
A first and partial answer has been given for the two-loop sunrise integral in two space-time dimensions \cite{Bloch:2013tra,Adams:2014vja}.
In two space-time dimensions the two-loop sunrise integral is finite and does not require regularisation.
Furthermore only one graph polynomial contributes in the Feynman parameter representation.
It turns out that in two space-time dimensions the two-loop sunrise integral is given
as a product of a period with a sum of elliptic dilogarithms.
This has been shown for the equal mass case in \cite{Bloch:2013tra} and for the general unequal mass case in \cite{Adams:2014vja}.
The arguments of the elliptic dilogarithms have a nice geometric interpretation as the images
in the Jacobi uniformisation of the intersection points of the variety defined by
the graph polynomial with the integration region.
In this paper we study the two-loop sunrise integral around four space-time dimensions.
Our motivations are as follows:
First of all, four space-time dimensions is the physical value
and it is the sunrise integral in four space-time dimensions which enters
precision calculations in high-energy particle physics.
Secondly, we hope to learn more about elliptic generalisations of polylogarithms.
In particular we are interested in generalisations with higher weight or depth.
Various elliptic generalisations of (multiple) polylogarithms have been discussed in the
mathematical literature \cite{Beilinson:1994,Levin:1997,Levin:2007,Brown:2011,Wildeshaus,Bloch:2013tra,Adams:2014vja}.
The definitions of the generalised functions differ among the articles in the mathematical literature and depend on the postulated
mathematical properties one would like to keep in going from the (multiple) polylogarithms towards the elliptic case.
We are interested to find out, which definition nature has chosen in quantum field theory.
Unlike in two space-time dimensions, the sunrise integral in four space-time dimensions is divergent and requires regularisation.
Dimensional regularisation is the method of choice.
In $D=4-2\varepsilon$ space-time dimensions, the $\varepsilon$-expansion of the two-loop sunrise integral starts at $1/\varepsilon^2$, corresponding to two
ultraviolet divergences.
The terms proportional to $1/\varepsilon^2$ and $1/\varepsilon$ are
very well known and our main interest in this article are the terms proportional to $\varepsilon^0$.
Using dimensional-shift relations we can relate the ${\mathcal O}(\varepsilon^0)$-term around four space-time dimensions
to the ${\mathcal O}(\varepsilon^0)$-term and the ${\mathcal O}(\varepsilon^1)$-term
of the sunrise integral around two space-time dimensions.
In the equal mass case the coefficient of the ${\mathcal O}(\varepsilon^1)$-term in this relation
vanishes and the ${\mathcal O}(\varepsilon^0)$-term around four space-time dimensions
is related to the ${\mathcal O}(\varepsilon^0)$-term around two space-time dimensions alone.
In this paper we treat the more general case of unequal masses and discuss as a specialisation the case of equal masses.
The ${\mathcal O}(\varepsilon^0)$-term of the sunrise integral around two space-time dimensions
is already known \cite{Bloch:2013tra,Adams:2014vja}.
In this paper we compute the ${\mathcal O}(\varepsilon^1)$-term of the sunrise integral around two space-time dimensions
and express it in terms of generalisations of Clausen and Glaisher functions towards the elliptic case.
We briefly recall that the Clausen and Glaisher functions are related to the real and imaginary parts of the classical polylogarithms.
Generalisations of multiple polylogarithms towards the elliptic case start
to make their appearance in physics \cite{Bloch:2013tra,Adams:2014vja,Bloch:2014qca,Broedel:2014vla}.
The ${\mathcal O}(\varepsilon^1)$-term of the sunrise integral around two space-time dimensions
gives us information on elliptic generalisations of multiple polylogarithms of depth greater than one.
In addition we observe that this term is not of uniform weight, it contains terms of weight three and four.
We discuss in detail the occurrence of the weight four terms.
This paper is organised as follows:
In section~\ref{sec:definition} we introduce the two-loop sunrise integral.
Section~\ref{sec:variables} is devoted to variables associated to this integral.
Our principal method to calculate the integral is based on differential equations, and the differential
equations obeyed by the two-loop sunrise integral are discussed in section~\ref{sec:dgl}.
The solution of any differential equation requires boundary values, and the boundary values for the
two-loop sunrise integral are presented in section~\ref{sec:boundary}.
In section~\ref{sec:Clausen} we introduce generalisations of the Clausen and Glaisher functions to the
elliptic setting. These functions will be needed to express our final result for the two-loop sunrise integral.
In section~\ref{sec:result_S1} we give the result for the ${\mathcal O}(\varepsilon^1)$-part
of the sunrise integral around two space-time dimensions.
Using dimensional-shift identities
we can relate the terms up to the finite part in the $\varepsilon$-expansion of the two-loop sunrise integral
around four space-time dimensions to the ${\mathcal O}(\varepsilon^0)$-part and the ${\mathcal O}(\varepsilon^1)$-part
of the sunrise integral around two space-time dimensions.
This is done in section~\ref{sec:shift}.
Section~\ref{sec:equal_mass_case} discusses the two-loop sunrise integral around
four space-time dimensions for the special case, where all masses are equal.
Section~\ref{sec:weights} is devoted to a detailed discussion of the transcendental weights occurring in our result
for the ${\mathcal O}(\varepsilon^1)$-part
of the sunrise integral around two space-time dimensions.
Finally, section~\ref{sec:conclusions} contains our conclusions.
In an appendix we included explicit formulae, which are too long to be presented in the main text of this paper.
\section{Definition of the sunrise integral}
\label{sec:definition}
The two-loop integral corresponding to the sunrise graph with arbitrary masses is given
in $D$-dimensional Minkowski space by
\begin{eqnarray}
\label{def_sunrise}
\lefteqn{
S_{\nu_1 \nu_2 \nu_3}\left( D, p^2, m_1^2, m_2^2, m_3^2, \mu^2 \right)
=
} & &
\\
& &
\left(\mu^2\right)^{\nu-D}
\int \frac{d^Dk_1}{i \pi^{\frac{D}{2}}} \frac{d^Dk_2}{i \pi^{\frac{D}{2}}}
\frac{1}{\left(-k_1^2+m_1^2\right)^{\nu_1}\left(-k_2^2+m_2^2\right)^{\nu_2}\left(-\left(p-k_1-k_2\right)^2+m_3^2\right)^{\nu_3}}.
\nonumber
\end{eqnarray}
In eq.~(\ref{def_sunrise}) the three internal masses are denoted by $m_1$, $m_2$ and $m_3$.
Without loss of generality we assume that the masses are ordered as
\begin{eqnarray}
\label{ordering_masses}
0 < m_1 \le m_2 \le m_3.
\end{eqnarray}
The arbitrary scale $\mu$ is introduced to keep the integral dimensionless.
We denote by $\nu=\nu_1+\nu_2+\nu_3$ the sum of the exponents of the propagators.
The quantity $p^2$ denotes the momentum squared (with respect to the Minkowski metric)
and we will write
\begin{eqnarray}
t & = & p^2.
\end{eqnarray}
We perform our calculation in the region defined by
\begin{eqnarray}
-t & \ge & 0,
\end{eqnarray}
and in the vicinity of the equal mass point.
Our result can be continued analytically to all other regions of interest by
Feynman's $i0$-prescription, where we substitute $t \rightarrow t + i0$.
The symbol $+i0$ denotes an infinitesimal positive imaginary part.
Where it is not essential we will suppress the dependence on the masses $m_i$ and the scale $\mu$ and simply write
$S_{\nu_1 \nu_2 \nu_3}( D, t)$ instead of $S_{\nu_1 \nu_2 \nu_3}( D, t, m_1^2, m_2^2, m_3^2, \mu^2)$.
In terms of Feynman parameters the two-loop integral is given by
\begin{eqnarray}
\label{def_Feynman_integral}
S_{\nu_1 \nu_2 \nu_3}\left( D, t\right)
& = &
\frac{\Gamma\left(\nu-D\right)}{\Gamma\left(\nu_1\right)\Gamma\left(\nu_2\right)\Gamma\left(\nu_3\right)}
\left(\mu^2\right)^{\nu-D}
\int\limits_{\sigma} x_1^{\nu_1-1} x_2^{\nu_2-1} x_3^{\nu_3-1} \frac{{\cal U}^{\nu-\frac{3}{2}D}}{{\cal F}^{\nu-D}} \omega
\end{eqnarray}
with the two Feynman graph polynomials
\begin{eqnarray}
{\cal U} & = & x_1 x_2 + x_2 x_3 + x_3 x_1,
\nonumber \\
{\cal F} & = & - x_1 x_2 x_3 t
+ \left( x_1 m_1^2 + x_2 m_2^2 + x_3 m_3^2 \right) {\cal U}.
\end{eqnarray}
The differential two-form $\omega$ is given by
\begin{eqnarray}
\omega & = & x_1 dx_2 \wedge dx_3 - x_2 dx_1 \wedge dx_3 + x_3 dx_1 \wedge dx_2.
\end{eqnarray}
The integration is over
\begin{eqnarray}
\sigma & = & \left\{ \left[ x_1 : x_2 : x_3 \right] \in {\mathbb P}^2 | x_i \ge 0, i=1,2,3 \right\}.
\end{eqnarray}
In order to facilitate a comparison with results in the literature we remark that
the definition of the sunrise integral in eq.~(\ref{def_sunrise}) is in Minkowski space.
In a space with Euclidean signature one defines
the two-loop sunrise integral as
\begin{eqnarray}
\lefteqn{
S_{\nu_1 \nu_2 \nu_3}^{\mathrm{eucl}}\left( D, P^2, m_1^2, m_2^2, m_3^2, \mu^2 \right)
=
} & &
\nonumber \\
& &
\left(\mu^2\right)^{\nu-D}
\int \frac{d^DK_1}{\pi^{\frac{D}{2}}} \frac{d^DK_2}{\pi^{\frac{D}{2}}}
\frac{1}{\left(K_1^2+m_1^2\right)^{\nu_1}\left(K_2^2+m_2^2\right)^{\nu_2}\left(\left(P-K_1-K_2\right)^2+m_3^2\right)^{\nu_3}}.
\end{eqnarray}
The momenta in Euclidean space are denoted by capital letters, while the ones in Minkowski space are denoted by lower case letters.
We have with $P^2=-p^2$ the relation
\begin{eqnarray}
S_{\nu_1 \nu_2 \nu_3}^{\mathrm{eucl}}\left( D, P^2, m_1^2, m_2^2, m_3^2, \mu^2 \right)
& = &
S_{\nu_1 \nu_2 \nu_3}\left( D, p^2, m_1^2, m_2^2, m_3^2, \mu^2 \right).
\end{eqnarray}
We are primarily concerned with the integral $S_{111}(4-2\varepsilon,t)$.
This integral has a Laurent expansion in $\varepsilon$, starting at $\varepsilon^{-2}$:
\begin{eqnarray}
\label{expansion_4D}
S_{111}\left( 4-2\varepsilon, t\right)
& = &
e^{-2 \gamma \varepsilon} \left[ \frac{1}{\varepsilon^2} S_{111}^{(-2)}(4,t) + \frac{1}{\varepsilon} S_{111}^{(-1)}(4,t) + S_{111}^{(0)}(4,t)
+ {\cal O}\left(\varepsilon\right) \right].
\end{eqnarray}
The first two terms $S_{111}^{(-2)}(4,t)$ and $S_{111}^{(-1)}(4,t)$ are rather simple, while the term $S_{111}^{(0)}(4,t)$
is in the focus of this paper.
With the help of dimensional-shift relations we relate the integral $S_{111}(4-2\varepsilon,t)$ to the integral
$S_{111}(2-2\varepsilon,t)$, which has an expansion in $\varepsilon$ starting at $\varepsilon^0$:
\begin{eqnarray}
\label{expansion_2D}
S_{111}\left( 2-2\varepsilon, t\right)
& = &
e^{-2 \gamma \varepsilon} \left[ S_{111}^{(0)}(2,t) + \varepsilon S_{111}^{(1)}(2,t)
+ {\cal O}\left(\varepsilon^2\right) \right].
\end{eqnarray}
The term $S_{111}^{(0)}(2,t)$ has been given in \cite{Adams:2014vja} in terms of elliptic dilogarithms,
the term $S_{111}^{(1)}(2,t)$ will be given in this paper.
In using dimensional-shift relations we will encounter simpler integrals, obtained by pinching internal propagators.
For the sunrise integral any integral obtained by pinching propagators is a product of tadpole integrals.
The tadpole integral is given by
\begin{eqnarray}
\label{def_tadpole}
T_{\nu}\left( D, m^2, \mu^2 \right)
& = &
\left(\mu^2\right)^{\nu-\frac{D}{2}}
\int \frac{d^Dk}{i \pi^{\frac{D}{2}}}
\frac{1}{\left(-k^2+m^2\right)^{\nu}}
\;\; = \;\;
\frac{\Gamma\left(\nu-\frac{D}{2}\right)}{\Gamma\left(\nu\right)}
\left( \frac{m^2}{\mu^2} \right)^{\frac{D}{2}-\nu}.
\end{eqnarray}
\section{Variables related to the sunrise integral}
\label{sec:variables}
The sunrise integral defined in eq.~(\ref{def_sunrise}) depends for a given space-time dimension $D$
on the variables $t$, $m_1^2$, $m_2^2$, $m_3^2$ and $\mu^2$.
It is clear from the definition that the sunrise integral will not change the value under a simultaneous
rescaling of all five quantities.
This implies that the integral depends only on the four dimensionless ratios $t/\mu^2$,
$m_1^2/\mu^2$, $m_2^2/\mu^2$ and $m_3^2/\mu^2$.
It will be convenient to view the non-trivial part of the integral as a function of five new variables
\begin{eqnarray}
\label{set_new_variables}
q, \;\; w_1, \;\; w_2, \;\; w_3 \;\; \mbox{and} \;\; \frac{m_1^2 m_2^2 m_3^2}{\mu^6}.
\end{eqnarray}
The variables $w_1$, $w_2$ and $w_3$ satisfy
\begin{eqnarray}
w_1 w_2 w_3 & = & 1,
\end{eqnarray}
therefore there are again only four independent variables.
The variables $q$, $w_1$, $w_2$ and $w_3$ are closely related
to the elliptic curve defined by ${\mathcal F}=0$.
Before we discuss this change of variables we first introduce some convenient abbreviations.
Let us start by denoting the pseudo-thresholds by
\begin{eqnarray}
\label{def_pseudo_thresholds}
\mu_1 = m_1+m_2-m_3,
\;\;\;
\mu_2 = m_1-m_2+m_3,
\;\;\;
\mu_3 = -m_1+m_2+m_3,
\end{eqnarray}
and the threshold by
\begin{eqnarray}
\label{def_thresholds}
\mu_4 = m_1+m_2+m_3.
\end{eqnarray}
In the vicinity of the equal mass point we can assume $m_3 < m_1+m_2$, or equivalently $\mu_1>0$.
With this assumption and with the ordering of eq.~(\ref{ordering_masses}) we have
\begin{eqnarray}
0 < \mu_1 \le \mu_2 \le \mu_3 < \mu_4.
\end{eqnarray}
We further set
\begin{eqnarray}
\label{def_D}
D & = &
\left( t - \mu_1^2 \right)
\left( t - \mu_2^2 \right)
\left( t - \mu_3^2 \right)
\left( t - \mu_4^2 \right).
\end{eqnarray}
The variable $D$ will appear in the algebraic prefactor of our result.
It will be convenient to introduce
the monomial symmetric polynomials $M_{\lambda_1 \lambda_2 \lambda_3}$ in the variables $m_1^2$, $m_2^2$ and $m_3^2$.
These are defined by
\begin{eqnarray}
M_{\lambda_1 \lambda_2 \lambda_3} & = &
\sum\limits_{\sigma} \left( m_1^2 \right)^{\sigma\left(\lambda_1\right)} \left( m_2^2 \right)^{\sigma\left(\lambda_2\right)} \left( m_3^2 \right)^{\sigma\left(\lambda_3\right)},
\end{eqnarray}
where the sum is over all distinct permutations of $\left(\lambda_1,\lambda_2,\lambda_3\right)$. A few examples are
\begin{eqnarray}
M_{100}
& = &
m_1^2 + m_2^2 + m_3^2,
\nonumber \\
M_{111}
& = &
m_1^2 m_2^2 m_3^2,
\nonumber \\
M_{210}
& = &
m_1^4 m_2^2 + m_2^4 m_3^2 + m_3^4 m_1^2 + m_2^4 m_1^2 + m_3^4 m_2^2 + m_1^4 m_3^2.
\end{eqnarray}
In addition, we introduce the abbreviations
\begin{eqnarray}
\label{def_delta}
\Delta = \mu_1 \mu_2 \mu_3 \mu_4,
\;\;\;\;\;\;
\delta_1 = -m_1^2 + m_2^2 + m_3^2,
\;\;\;\;\;\;
\delta_2 = m_1^2 - m_2^2 + m_3^2,
\;\;\;\;\;\;
\delta_3 = m_1^2 + m_2^2 - m_3^2
\end{eqnarray}
and
\begin{eqnarray}
\label{def_v_variables}
v_1 = \frac{1+i\frac{\sqrt{\Delta}}{\delta_1}}{1-i\frac{\sqrt{\Delta}}{\delta_1}},
\;\;\;\;\;\;
v_2 = \frac{1+i\frac{\sqrt{\Delta}}{\delta_2}}{1-i\frac{\sqrt{\Delta}}{\delta_2}},
\;\;\;\;\;\;
v_3 = \frac{1+i\frac{\sqrt{\Delta}}{\delta_3}}{1-i\frac{\sqrt{\Delta}}{\delta_3}}.
\end{eqnarray}
In the vicinity of the equal mass point we have
\begin{eqnarray}
\Delta > 0,
\;\;\;\;\;\;
\delta_i > 0, & & i \in \{1,2,3\}.
\end{eqnarray}
The variables $v_1$, $v_2$ and $v_3$ are
then
complex numbers of unit norm.
In addition they satisfy
\begin{eqnarray}
v_1 v_2 v_3 & = & 1.
\end{eqnarray}
We now describe in detail the change of variables from the set $t, m_1^2, m_2^2, m_3^2, \mu^2$ to the set specified in eq.~(\ref{set_new_variables}).
The equation
\begin{eqnarray}
{\mathcal F} & = & 0
\end{eqnarray}
is a polynomial equation of degree three in the Feynman parameters $x_1$, $x_2$ and $x_3$.
The cubic curve intersects the domain of integration in the three points
\begin{eqnarray}
\label{intersection_F_sigma}
P_1 = \left[1:0:0\right],
\;\;\;
P_2 = \left[0:1:0\right],
\;\;\;
P_3 = \left[0:0:1\right].
\end{eqnarray}
The cubic curve together with a choice of a point $O \in \{ P_1, P_2, P_3 \}$ as origin defines an elliptic curve.
All three choices will lead to the same Weierstrass normal form, given by
\begin{eqnarray}
y^2
& = &
4 \left(x-e_1\right)\left(x-e_2\right)\left(x-e_3\right),
\;\;\;\;\;\;
\mbox{with}
\;\;\;
e_1+e_2+e_3=0,
\end{eqnarray}
and
\begin{eqnarray}
\label{def_roots}
e_1
& = &
\frac{1}{24 \mu^4} \left( -t^2 + 2 M_{100} t + \Delta + 3 \sqrt{D} \right),
\nonumber \\
e_2
& = &
\frac{1}{24 \mu^4} \left( -t^2 + 2 M_{100} t + \Delta - 3 \sqrt{D} \right),
\nonumber \\
e_3
& = &
\frac{1}{24 \mu^4} \left( 2 t^2 - 4 M_{100} t - 2 \Delta \right).
\end{eqnarray}
The modulus $k$ and the complementary modulus $k'$ of the elliptic curve are given by
\begin{eqnarray}
\label{def_modulus}
k = \sqrt{\frac{e_3-e_2}{e_1-e_2}},
& &
k' = \sqrt{1-k^2} = \sqrt{\frac{e_1-e_3}{e_1-e_2}}.
\end{eqnarray}
The periods of the elliptic curve can be taken as
\begin{eqnarray}
\label{def_periods}
\psi_1 =
2 \int\limits_{e_2}^{e_3} \frac{dx}{y}
=
\frac{4 \mu^2}{D^{\frac{1}{4}}} K\left(k\right),
& &
\psi_2 =
2 \int\limits_{e_1}^{e_3} \frac{dx}{y}
=
\frac{4 i \mu^2}{D^{\frac{1}{4}}} K\left(k'\right),
\\
\phi_1 =
\frac{8\mu^4}{D^{\frac{1}{2}}} \int\limits_{e_2}^{e_3} \frac{\left(x-e_2\right) dx}{y}
=
\frac{4 \mu^2}{D^{\frac{1}{4}}} \left( K\left(k\right)- E\left(k\right) \right),
& &
\phi_2 =
\frac{8\mu^4}{D^{\frac{1}{2}}} \int\limits_{e_1}^{e_3} \frac{\left(x-e_2\right) dx}{y}
=
\frac{4 i \mu^2}{D^{\frac{1}{4}}} E\left(k'\right).
\nonumber
\end{eqnarray}
$K(x)$ and $E(x)$ denote the complete elliptic integral of the first kind and second kind, respectively:
\begin{eqnarray}
K(x)
=
\int\limits_0^1 \frac{dt}{\sqrt{\left(1-t^2\right)\left(1-x^2t^2\right)}},
& &
E(x)
=
\int\limits_0^1 dt \sqrt{\frac{1-x^2t^2}{1-t^2}}.
\end{eqnarray}
The Legendre relation for the periods reads
\begin{eqnarray}
\psi_1 \phi_2 - \psi_2 \phi_1
& = &
\frac{8 \pi i \mu^4}{D^{\frac{1}{2}}}.
\end{eqnarray}
The Wronskian is given by
\begin{eqnarray}
\label{def_Wronski}
W & = &
\psi_1 \frac{d}{dt} \psi_2 - \psi_2 \frac{d}{dt} \psi_1
=
-
4 \pi i \mu^4
\;
\frac{\left( 3 t^2 - 2 M_{100} t + \Delta \right)}{t\left( t - \mu_1^2 \right)\left( t - \mu_2^2 \right)\left( t - \mu_3^2 \right)\left( t - \mu_4^2 \right)}.
\end{eqnarray}
We denote the ratio of the two periods $\psi_2$ and $\psi_1$ by
\begin{eqnarray}
\tau
& = &
\frac{\psi_2}{\psi_1}
\end{eqnarray}
and the nome by
\begin{eqnarray}
\label{def_nome}
q & = & e^{i\pi \tau}.
\end{eqnarray}
The nome $q$ will be one of our new variables, and we can think of the variable $q$ as replacing
the variable $t$.
A useful relation for expressing $q$ in terms of $t$ (or vice versa) is given by
\begin{eqnarray}
\label{basic_relation_power_series}
\frac{t}{\left( \mu_1^2 - t \right) \left( \mu_2^2 - t \right) \left( \mu_3^2 - t\right) \left( \mu_4^2 -t \right)}
& = &
- \frac{1}{m_1^2 m_2^2 m_3^2}
\frac{\eta\left(\frac{\tau}{2}\right)^{24}\eta\left(2\tau\right)^{24}}{\eta\left(\tau\right)^{48}}.
\end{eqnarray}
In this equation, Dedekind's $\eta$-function, defined by
\begin{eqnarray}
\eta\left(\tau\right)
& = &
e^{\frac{\pi i \tau}{12}} \prod\limits_{n=1}^\infty \left( 1- e^{2 \pi i n \tau} \right)
=
q^{\frac{1}{12}} \prod\limits_{n=1}^\infty \left( 1 - q^{2n} \right),
\end{eqnarray}
appears. The left-hand side of eq.~(\ref{basic_relation_power_series})
has a power series in $t$, the right-hand side
of eq.~(\ref{basic_relation_power_series}) has a power series in $q$.
Eq.~(\ref{basic_relation_power_series}) can be used to express $t$ as a power series in $q$,
or vice versa.
We have
\begin{eqnarray}
t =
- q \frac{\Delta^2}{m_1^2 m_2^2 m_3^2}
+ {\cal O}\left(q^2\right),
& &
q =
- t \frac{m_1^2 m_2^2 m_3^2}{\Delta^2}
+ {\cal O}\left(t^2\right),
\end{eqnarray}
therefore $t \rightarrow 0$ implies $q \rightarrow 0$ and vice versa.
It remains to define the variables $w_1$, $w_2$ and $w_3$. These are given by
\begin{eqnarray}
\label{def_arguments_w_i}
w_i
= e^{i \beta_i},
\;\;\;\;
\beta_i
= \pi \frac{F\left(u_i,k\right)}{K\left(k\right)},
\;\;\;\;
u_i
= \sqrt{\frac{e_1-e_2}{x_{j,k}-e_2}},
\;\;\;\;
x_{j,k}
= e_3 + \frac{m_j^2 m_k^2}{\mu^4}.
\end{eqnarray}
In the definition of $u_i$ we used the convention that $(i,j,k)$ is a permutation of $(1,2,3)$.
In the definition of $\beta_i$ the incomplete elliptic integral of the first kind appears,
defined by
\begin{eqnarray}
F\left(z,x\right)
& = &
\int\limits_0^z \frac{dt}{\sqrt{\left(1-t^2\right)\left(1-x^2t^2\right)}}.
\end{eqnarray}
In the case $q=0$ (or equivalently $t=0$) we have
\begin{eqnarray}
\lim\limits_{q \rightarrow 0} w_j & = & v_j,
\;\;\;\;\;\;\;\;\; j \in \{ 1,2,3 \},
\end{eqnarray}
while in the equal mass case $m_1=m_2=m_3$ we have
\begin{eqnarray}
\left. w_j \right|_{m_1=m_2=m_3}
& = & e^{\frac{2\pi i}{3}},
\;\;\;\;\;\;\;\;\; j \in \{ 1,2,3 \}
\end{eqnarray}
for all $q$ (or equivalently all $t$).
There is a simple geometric interpretation for the variables $w_1$, $w_2$ and $w_3$:
We recall that the points $P_1$, $P_2$ and $P_3$
are the intersection points of ${\mathcal F}=0$ with the integration region $\sigma$.
One of these points is chosen as origin of the elliptic curve.
The set
\begin{eqnarray}
\left\{ w_1, w_2, w_3, w_1^{-1}, w_2^{-1}, w_3^{-1} \right\}
\end{eqnarray}
is obtained from the images in the Jacobi uniformisation of the two points not chosen as origin
by considering all three choices of origins.
More details can be found in \cite{Adams:2014vja}.
\section{Differential equations}
\label{sec:dgl}
In this section we present the (fourth-order) differential equation for the sunrise integral $S_{111}(D,t)$ in $D$ space-time dimensions
with arbitrary masses.
From this equation we deduce a differential equation for the ${\mathcal O}(\varepsilon^1)$-piece $S_{111}^{(1)}(2,t)$.
The latter differential equation is then solved up to quadrature.
In two space-time dimensions the integral $S_{111}(2,t)=S_{111}^{(0)}(2,t)$ satisfies
a second-order differential equation \cite{MullerStach:2011ru}:
\begin{eqnarray}
\label{second_order_dgl}
\left[ p_2 \frac{d^2}{d t^2} + p_1 \frac{d}{dt} + p_0 \right] S_{111}^{(0)}\left(2,t\right) & = & \mu^2 p_3,
\end{eqnarray}
The coefficients $p_2$, $p_1$ and $p_0$ as well as $p_3$ are collected in appendix~\ref{appendix:coeff}.
The left hand side defines a Picard-Fuchs operator
\begin{eqnarray}
\label{def_L2}
L^{(0)}_2\left(2\right) & = &
p_2 \frac{d^2}{d t^2} + p_1 \frac{d}{dt} + p_0.
\end{eqnarray}
The periods $\psi_1$ and $\psi_2$ are solutions of the homogeneous differential equation \cite{Adams:2013nia}
\begin{eqnarray}
L^{(0)}_2\left(2\right) \psi_i & = & 0,
\;\;\;\;\;\;\;\;\;
i \in \{ 1,2 \}.
\end{eqnarray}
We will encounter several differential operators.
We will use the notation
\begin{eqnarray}
L^{(j)}_{r,i}\left(D\right),
\end{eqnarray}
where $r$ denotes the order of the differential operator and $D$ denotes the associated space-time dimension.
In the case where $D$ is an integer and a superscript $j$ is present, this superscript
denotes the order in the $\varepsilon$-expansion to which this operator belongs.
Finally, $i$ is a label to distinguish differential operators with identical $r$, $D$, $j$.
In $D$ dimensions the integral $S_{111}(D,t)$ satisfies a fourth-order differential equation.
\begin{eqnarray}
\label{diff_eq_D_dim}
\left[
P_4 \frac{d^4}{dt^4}
+
P_3 \frac{d^3}{dt^3}
+
P_2 \frac{d^2}{dt^2}
+
P_1 \frac{d}{dt}
+
P_0
\right] S_{111}\left(D,t\right)
& = &
\mu^2
\left[
c_{12} T_{12}
+
c_{13} T_{13}
+
c_{23} T_{23}
\right].
\end{eqnarray}
Here we used the abbreviation
\begin{eqnarray}
T_{ij} & = &
T_1\left(D,m_i^2,\mu^2\right) T_1\left(D,m_j^2,\mu^2\right).
\end{eqnarray}
The explicit expressions for the coefficients $P_4$, $P_3$, $P_2$, $P_1$, $P_0$ and $c_{12}$, $c_{13}$, $c_{23}$ are rather long and given
in appendix~\ref{appendix:coeff}.
There are several possibilities to obtain the fourth-order differential equation:
It can either be derived by using the relations given in \cite{Caffo:1998du},
by using the program ``Reduze'' \cite{Studerus:2009ye,vonManteuffel:2012np} or by the algorithm given in \cite{MullerStach:2012mp}.
Eq.~(\ref{diff_eq_D_dim}) defines a fourth-order Picard-Fuchs operator
\begin{eqnarray}
L_4\left(D\right)
& = &
P_4 \frac{d^4}{dt^4}
+
P_3 \frac{d^3}{dt^3}
+
P_2 \frac{d^2}{dt^2}
+
P_1 \frac{d}{dt}
+
P_0.
\end{eqnarray}
The Picard-Fuchs operator $L_4(D)$ has a polynomial
dependence on the number of space-time dimensions $D$.
Around $D=2-2\varepsilon$ we can write
\begin{eqnarray}
L_4\left(2-2\varepsilon\right)
& = &
\sum\limits_{j=0}^5 \varepsilon^j \; L^{(j)}_4\left(2\right).
\end{eqnarray}
Of particular relevance for the ${\mathcal O}(\varepsilon^1)$-part $S_{111}^{(1)}(2,t)$
will be the operators $L^{(0)}_4(2)$ and $L^{(1)}_4(2)$.
The operator $L^{(0)}_4(2)$ factorises
\begin{eqnarray}
L^{(0)}_4\left(2\right) & = &
L^{(0)}_{1,a}\left(2\right)
\;\;
L^{(0)}_{1,b}\left(2\right)
\;\;
L^{(0)}_{2}\left(2\right).
\end{eqnarray}
The differential operator $L^{(0)}_{2}(2)$ is the one we already encountered in eq.~(\ref{def_L2}).
The two other factors $L^{(0)}_{1,a}(2)$ and $L^{(0)}_{1,b}(2)$ are first-order differential operators:
\begin{eqnarray}
L^{(0)}_{1,a}\left(2\right)
& = &
8 t^2
\left[
\frac{\left(5t^2 + 2M_{100} t + 7 \Delta \right)}{\left( 15 t^2 - 2 M_{100} t - 3 \Delta \right) \left(3t^2-2 M_{100}t+\Delta\right)} \frac{d}{dt}
\right. \nonumber \\
& & \left.
- \frac{60 t^3 - 12 M_{100} t^2 - \left(60 M_{200} - 88 M_{110} \right) t - 12 M_{100} \Delta}{\left( 15 t^2 - 2 M_{100} t - 3 \Delta \right) \left(3t^2-2 M_{100}t+\Delta\right)^2}
\right],
\nonumber \\
L^{(0)}_{1,b}\left(2\right)
& = &
\left( 15 t^2 - 2 M_{100} t - 3 \Delta \right) \frac{d}{dt} - \left( 30 t - 2 M_{100} \right).
\end{eqnarray}
Solutions to the homogeneous equations
\begin{eqnarray}
L^{(0)}_{1,a}\left(2\right) \; \psi_{a}\left(t\right) = 0,
& &
L^{(0)}_{1,b}\left(2\right) \; \psi_{b}\left(t\right) = 0
\end{eqnarray}
are
\begin{eqnarray}
\psi_{a}\left(t\right) & = &
\left( 3 t^2 - 2 M_{100} t + \Delta \right)
\left( 5 t^2 + 2 M_{100} t + 7 \Delta \right),
\nonumber \\
\psi_{b}\left(t\right) & = & 15 t^2 - 2 M_{100} t - 3 \Delta.
\end{eqnarray}
Substituting the $\varepsilon$-expansion of $S_{111}(2-2\varepsilon,t)$ given in eq.~(\ref{expansion_2D})
into the $D$-dimensional differential equation~(\ref{diff_eq_D_dim}) gives a coupled system of differential
equations for $S_{111}^{(j)}(2,t)$, where the differential equation for $S_{111}^{(j)}(2,t)$ will involve
the lower order integrals $S_{111}^{(i)}(2,t)$ with $i<j$.
This system can be solved order by order in $\varepsilon$.
At
order $\varepsilon^0$
one finds
\begin{eqnarray}
\label{dgl_S_111_0}
L^{(0)}_{1,a}\left(2\right)
\;\;
L^{(0)}_{1,b}\left(2\right)
\;\;
L^{(0)}_{2}\left(2\right)
\;\;
S_{111}^{(0)}(2,t)
& = &
- 32 \mu^2 t^2 \left( 15 t^2 + 14 M_{100} t + 77 \Delta \right).
\end{eqnarray}
From eq.~(\ref{second_order_dgl}) we know already that
\begin{eqnarray}
L^{(0)}_{2}\left(2\right)
\;\;
S_{111}^{(0)}(2,t)
& = &
\mu^2 p_3(t),
\end{eqnarray}
and eq.~(\ref{dgl_S_111_0}) reduces to
\begin{eqnarray}
\mu^2
\;\;
L^{(0)}_{1,a}\left(2\right)
\;\;
L^{(0)}_{1,b}\left(2\right)
\;\;
p_3(t)
& = &
- 32 \mu^2 t^2 \left( 15 t^2 + 14 M_{100} t + 77 \Delta \right),
\end{eqnarray}
which is easily verified.
\\
\\
At order $\varepsilon^1$ we have
\begin{eqnarray}
\label{dgl_S_111_1}
L^{(0)}_{1,a}\left(2\right)
\;\;
L^{(0)}_{1,b}\left(2\right)
\;\;
L^{(0)}_{2}\left(2\right)
\;\;
S_{111}^{(1)}(2,t)
& = &
I_1\left(t\right),
\end{eqnarray}
with
\begin{eqnarray}
\label{def_I_t}
I_1\left(t\right)
& = &
- L_4^{(1)}(2) \;\; S_{111}^{(0)}(2,t)
\nonumber \\
& &
- \mu^2
\left\{
912 t^4 + 1344 M_{100} t^3 + \left( 9088 M_{110} - 4416 M_{200} \right) t^2
+ 512 M_{100} \Delta t
+ 112 \Delta^2
\right. \nonumber \\
& & \left.
+ d\left(t,m_1^2,m_2^2,m_3^2\right) \ln \frac{m_1^2}{\mu^2}
+ d\left(t,m_2^2,m_3^2,m_1^2\right) \ln \frac{m_2^2}{\mu^2}
+ d\left(t,m_3^2,m_1^2,m_2^2\right) \ln \frac{m_3^2}{\mu^2}
\right\}
\end{eqnarray}
and
\begin{eqnarray}
\lefteqn{
d\left(t,m_1^2,m_2^2,m_3^2\right)
=
-320 t^4
- \left( 352 m_1^2 + 272 m_2^2 + 272 m_3^2 \right) t^3
} & & \nonumber \\
& &
+ \left( 1440 m_1^4 + 1744 m_2^4 + 1744 m_3^4 - 3120 m_1^2 m_2^2 - 3120 m_1^2 m_3^2 - 3616 m_2^2 m_3^2 \right) t^2
\nonumber \\
& &
+ \left( -544 m_1^6 + 272 m_2^6 + 272 m_3^6 + 1296 m_1^4 m_2^2 + 1296 m_1^4 m_3^2 - 1024 m_1^2 m_2^4 - 1024 m_1^2 m_3^4
\right. \nonumber \\
& & \left. - 272 m_2^4 m_3^2 - 272 m_2^2 m_3^4 \right) t
\nonumber \\
& &
+ \left( 224 m_1^4 - 112 m_2^4 - 112 m_3^4 - 112 m_1^2 m_2^2 - 112 m_1^2 m_3^2 + 224 m_2^2 m_3^2 \right) \Delta.
\end{eqnarray}
Eq.~(\ref{dgl_S_111_1}) is a fourth-order differential equation for the integral $S_{111}^{(1)}(2,t)$.
The fourth-order differential operator in eq.~(\ref{dgl_S_111_1}) factorises
into two first-order differential operators
and a second-order differential operator.
The inhomogeneous term $I_1(t)$ has a Taylor expansion in $t$, starting with $t^2$.
Eq.~(\ref{dgl_S_111_1}) is easily solved for $L^{(0)}_{2}(2) \; S_{111}^{(1)}(2,t)$:
\begin{eqnarray}
\label{L2_S_111_1}
L^{(0)}_{2}\left(2\right) \;\; S_{111}^{(1)}(2,t)
& = &
I_2\left(t\right),
\end{eqnarray}
with
\begin{eqnarray}
\lefteqn{
I_2\left(t\right)
= }
& & \\
& &
C_1 \psi_b(t)
+ C_2 \psi_b(t) \int\limits_0^t \frac{\psi_a(t_1) dt_1}{p_{1,b}(t_1) \psi_b(t_1)}
+ \psi_b(t) \int\limits_0^t \frac{\psi_a(t_1) dt_1}{p_{1,b}(t_1) \psi_b(t_1)}
\int\limits_0^{t_1} \frac{I_1\left(t_2\right) dt_2}{p_{1,a}(t_2) \psi_a(t_2)},
\nonumber
\end{eqnarray}
where $C_1$ and $C_2$ are two integration constants to be determined from the boundary conditions.
In eq.~(\ref{L2_S_111_1}) we used the notation that
\begin{eqnarray}
L^{(0)}_{1,a}\left(2\right)
=
p_{1,a} \frac{d}{dt} + p_{0,a},
& &
L^{(0)}_{1,b}\left(2\right)
=
p_{1,b} \frac{d}{dt} + p_{0,b}.
\end{eqnarray}
For the homogeneous solutions we have
\begin{eqnarray}
C_1 \psi_b(t) & = &
C_1 \left( 15 t^2 - 2 M_{100} t - 3 \Delta \right),
\nonumber \\
C_2 \psi_b(t) \int\limits_0^t \frac{\psi_a(t_1) dt_1}{p_{1,b}(t_1) \psi_b(t_1)}
& = &
\frac{C_2}{3} \left(3 t^2 + 6 M_{100} t - 7 \Delta \right) t.
\end{eqnarray}
The differential equation~(\ref{L2_S_111_1}) is now of the same type as eq.~(\ref{second_order_dgl}),
only the inhomogeneous term differs.
It can be solved with the same methods as used for eq.~(\ref{second_order_dgl}) by changing variables from $t$
to the nome $q$. This change of variables is described in detail in \cite{Adams:2014vja}.
One finds
\begin{eqnarray}
\label{solution_S1_quadrature}
S_{111}^{(1)}(2,t)
& = &
C_3 \psi_1 + C_4 \psi_2
-
\frac{\psi_1}{\pi}
\int\limits_0^q \frac{dq_1}{q_1}
\int\limits_0^{q_1} \frac{dq_2}{q_2}
\;
\frac{I_2\left(q_2\right) \psi_1\left(q_2\right)^3}{\pi p_2\left(q_2\right) W\left(q_2\right)^2}.
\end{eqnarray}
$C_3$ and $C_4$ are two further integration constants, which need to be determined from boundary conditions.
Eq.~(\ref{solution_S1_quadrature}) gives the solution for the integral $S_{111}^{(1)}(2,t)$ up to quadrature.
We still need to determine the integration constants. In addition we would like
to express the integral $S_{111}^{(1)}(2,t)$ in terms of elliptic generalisations of multiple polylogarithms.
Let us also mention that eq.~(\ref{solution_S1_quadrature}) can be used to obtain the $q$-expansion (or equivalently the $t$-expansion) of
$S_{111}^{(1)}(2,t)$ to high orders in $q$ (or $t$).
\section{Boundary values}
\label{sec:boundary}
In this section we give the boundary values at $t=0$ for the first two coefficients of the $\varepsilon$-expansion of $S_{111}(2-2\varepsilon,t)$
(i.e. $S_{111}^{(0)}(2,0)$ and $S_{111}^{(1)}(2,0)$)
and for the first three coefficients of the $\varepsilon$-expansion of $S_{111}(4-2\varepsilon,t)$
(i.e. $S_{111}^{(-2)}(4,0)$, $S_{111}^{(-1)}(4,0)$ and $S_{111}^{(0)}(4,0)$)
We will use the boundary values $S_{111}^{(1)}(2,0)$ and $S_{111}^{(0)}(4,0)$ together with regularity conditions at $t=0$ to fix the integration
constants $C_1$, $C_2$, $C_3$ and $C_4$ of the previous section.
Let us start with the boundary value at $t=0$ of the sunrise integral in $D=2-2\varepsilon$ dimensions.
We have
\begin{eqnarray}
S_{111}\left( 2-2\varepsilon, 0\right)
& = &
\Gamma\left(1+2\varepsilon\right)
\left(\mu^2\right)^{1+2\varepsilon}
\int\limits_{\sigma} \frac{\omega}{\left( x_1 m_1^2 + x_2 m_2^2 + x_3 m_3^2 \right)^{1+2\varepsilon} {\cal U}^{1-\varepsilon}}.
\end{eqnarray}
By a change of variables we can relate this integral
to the one-loop three-point function
in $4+2\varepsilon$ space-time dimensions (please note the sign of the $\varepsilon$-part)
with massless internal lines
and three external masses.
The change of variables can be found in \cite{Adams:2013nia} and the result of the one-loop three point function can be taken from
\cite{Bern:1994kr,Lu:1992ny}.
One obtains for the $\varepsilon$-expansion of $S_{111}( 2-2\varepsilon, 0)$
\begin{eqnarray}
\sum\limits_{j=0}^\infty \varepsilon^j S_{111}^{(j)}\left(2,0\right)
& = &
e^{2 \gamma \varepsilon} \Gamma\left(1+2\varepsilon\right)
\left( \frac{\sqrt{\Delta}}{\mu^2} \right)^{-1-2\varepsilon}
\left[ \frac{1}{2\varepsilon^2} \frac{\Gamma\left(1+\varepsilon\right)^2}{\Gamma\left(1+2\varepsilon\right)}
\left( f_1+f_2+f_3\right)
- \frac{\pi}{\varepsilon} \right],
\end{eqnarray}
with
\begin{eqnarray}
f_j
& = &
\frac{1}{i}
\left[
\left(-v_j\right)^{-\varepsilon} \; {}_2F_1\left(-2\varepsilon,-\varepsilon;1-\varepsilon; v_j \right)
-
\left(-v_j^{-1}\right)^{-\varepsilon} \; {}_2F_1\left(-2\varepsilon,-\varepsilon;1-\varepsilon; v_j^{-1} \right)
\right].
\end{eqnarray}
The expansion of the hypergeometric function reads
\begin{eqnarray}
{}_2F_1\left(-2\varepsilon,-\varepsilon;1-\varepsilon; x \right)
& = &
1 + 2 \varepsilon^2 \mathrm{Li}_2\left(x\right)
+ \varepsilon^3 \left[ 2 \mathrm{Li}_3\left(x\right) - 4 \mathrm{Li}_{2,1}\left(x,1\right) \right]
+ {\mathcal O}\left(\varepsilon^4\right).
\end{eqnarray}
We obtain for $S_{111}^{(0)}(2,0)$ and $S_{111}^{(1)}(2,0)$
\begin{eqnarray}
S_{111}^{(0)}\left(2,0\right)
& = &
\frac{2\mu^2}{\sqrt{\Delta}}
\sum\limits_{j=1}^3
\frac{1}{2i}
\left[ \mathrm{Li}_2\left(v_j\right) - \mathrm{Li}_2\left(v_j^{-1}\right) \right],
\nonumber \\
S_{111}^{(1)}\left(2,0\right)
& = &
\frac{2\mu^2}{\sqrt{\Delta}}
\sum\limits_{j=1}^3
\frac{1}{2i}
\left\{
- 2 \mathrm{Li}_{2,1}\left(v_j,1\right) - \mathrm{Li}_3\left(v_j\right)
+ 2 \mathrm{Li}_{2,1}\left(v_j^{-1},1\right) + \mathrm{Li}_3\left(v_j^{-1}\right)
\right. \nonumber \\
& & \left.
- 2 \ln\left(\frac{\sqrt{\Delta}}{\mu^2}\right) \left[ \mathrm{Li}_2\left(v_j\right) - \mathrm{Li}_2\left(v_j^{-1}\right) \right]
\right\}.
\end{eqnarray}
The definition of the multiple polylogarithms is given in eq.~(\ref{def_multiple_polylogs}).
Let us now turn to the values of the coefficients of the Laurent expansion of $S_{111}(4-2\varepsilon,t)$ at $t=0$.
At $t=0$ we find
\begin{eqnarray}
\label{boundary_t_0}
\lefteqn{
S_{111}^{(-2)}(4,0)
=
- \frac{M_{100}}{2 \mu^2},
} & &
\nonumber \\
\lefteqn{
S_{111}^{(-1)}(4,0)
=
- \frac{3 M_{100}}{2 \mu^2}
+ \frac{m_1^2}{\mu^2} \ln\left(\frac{m_1^2}{\mu^2}\right)
+ \frac{m_2^2}{\mu^2} \ln\left(\frac{m_2^2}{\mu^2}\right)
+ \frac{m_3^2}{\mu^2} \ln\left(\frac{m_3^2}{\mu^2}\right),
} & &
\nonumber \\
\lefteqn{
S_{111}^{(0)}(4,0)
=
\frac{\Delta}{2\mu^4} S_{111}^{(0)}(2,0)
- \frac{7 M_{100}}{2\mu^2}
+ 3
\left[ \frac{m_1^2}{\mu^2} \ln\left(\frac{m_1^2}{\mu^2}\right)
+ \frac{m_2^2}{\mu^2} \ln\left(\frac{m_2^2}{\mu^2}\right)
+ \frac{m_3^2}{\mu^2} \ln\left(\frac{m_3^2}{\mu^2}\right)
\right]
} & &
\nonumber \\
& &
- \frac{1}{2}
\left[ \frac{m_1^2}{\mu^2} \ln^2\left(\frac{m_1^2}{\mu^2}\right)
+ \frac{m_2^2}{\mu^2} \ln^2\left(\frac{m_2^2}{\mu^2}\right)
+ \frac{m_3^2}{\mu^2} \ln^2\left(\frac{m_3^2}{\mu^2}\right)
\right]
\nonumber \\
& &
- \frac{1}{2\mu^2}
\left[
\left( m_1^2 + m_2^2 - m_3^2 \right) \ln\left(\frac{m_1^2}{\mu^2}\right) \ln\left(\frac{m_2^2}{\mu^2}\right)
+ \left( m_1^2 - m_2^2 + m_3^2 \right) \ln\left(\frac{m_1^2}{\mu^2}\right) \ln\left(\frac{m_3^2}{\mu^2}\right)
\right. \nonumber \\
& & \left.
+ \left( -m_1^2 + m_2^2 + m_3^2 \right) \ln\left(\frac{m_2^2}{\mu^2}\right) \ln\left(\frac{m_3^2}{\mu^2}\right)
\right]
- \frac{M_{100}}{2 \mu^2} \zeta_2.
\end{eqnarray}
The expression for $S_{111}^{(0)}(4,0)$ is obtained from the expression given in \cite{Caffo:1998du}
by noting that the function
\begin{eqnarray}
L & = &
\mathrm{Li}_2\left(- \frac{m_2}{m_1} t_3 \right)
+ \mathrm{Li}_2\left(- \frac{m_1}{m_2} t_3 \right)
+ \zeta_2
+ \frac{1}{2}\ln^2 t_3
\nonumber \\
& &
+ \frac{1}{2} \left[
\ln\left(t_3+\frac{m_2}{m_1}\right)
- \ln\left(t_3+\frac{m_1}{m_2}\right)
+ \frac{3}{4} \ln\frac{m_1^2}{m_2^2}
\right]
\ln\frac{m_1^2}{m_2^2},
\nonumber \\
t_3 & = &
\frac{1}{2m_1m_2} \left( m_3^2 - m_1^2 - m_2^2 + \sqrt{-\Delta} \right)
\end{eqnarray}
can be written in a more symmetrical way as
\begin{eqnarray}
L & = &
- \frac{1}{2}
\left[
\mathrm{Li}_2\left( v_1 \right) + \mathrm{Li}_2\left( v_2 \right) + \mathrm{Li}_2\left( v_3 \right)
- \mathrm{Li}_2\left( v_1^{-1} \right) - \mathrm{Li}_2\left( v_2^{-1} \right) - \mathrm{Li}_2\left( v_3^{-1} \right)
\right].
\end{eqnarray}
The constants of integration $C_1$, $C_2$, $C_3$ and $C_4$ are determined as follows:
In section~\ref{sec:shift} we discuss dimensional-shift relations.
These relations allow us to relate the integral $S_{111}(4-2\varepsilon,t)$ to the integral $S_{111}(2-2\varepsilon,t)$, and we can determine
the integration constants from the following four conditions:
\begin{enumerate}
\item The requirement that $S_{111}^{(0)}(4,t)$ is regular at $t=0$, i.e. there is no pole at $t=0$ for $S_{111}^{(0)}(4,t)$.
\item The boundary value $S_{111}^{(0)}(4,0)$.
\item The requirement that $S_{111}^{(1)}(2,t)$ is regular at $t=0$, i.e. there is no logarithmic singularity at $t=0$ for $S_{111}^{(1)}(2,t)$.
\item The boundary value $S_{111}^{(1)}(2,0)$.
\end{enumerate}
The explicit expressions for $C_1$ and $C_2$ are given in appendix~\ref{section_integration_constants}.
Condition $3$ implies $C_4=0$.
With
\begin{eqnarray}
\psi_1\left(t=0\right) & = & \frac{2 \pi \mu^2}{\sqrt{\Delta}}
\end{eqnarray}
one finds for $C_3$
\begin{eqnarray}
C_3
& = &
\frac{\sqrt{\Delta}}{2\pi \mu^2} S_{111}^{(1)}\left(2,0\right).
\end{eqnarray}
\section{Generalisations of the Clausen and Glaisher functions}
\label{sec:Clausen}
In this section we introduce elliptic generalisations of the Clausen and Glaisher functions.
These generalisations will show up in our final result.
The classical polylogarithms are defined by
\begin{eqnarray}
\mathrm{Li}_n\left(x\right) & = & \sum\limits_{j=1}^\infty \; \frac{x^j}{j^n},
\end{eqnarray}
and the multiple polylogarithms by
\begin{eqnarray}
\label{def_multiple_polylogs}
\mathrm{Li}_{n_1,n_2,...,n_k}\left(x_1,x_2,...,x_k\right)
& = &
\sum\limits_{j_1=1}^\infty \sum\limits_{j_2=1}^{j_1-1} ... \sum\limits_{j_k=1}^{j_{k-1}-1}
\frac{x_1^{j_1}}{j_1^{n_1}} \frac{x_2^{j_2}}{j_2^{n_2}} ... \frac{x_k^{j_k}}{j_k^{n_k}}.
\end{eqnarray}
The sum representation gives rise to a quasi-shuffle product and one has for example
\begin{eqnarray}
\mathrm{Li}_{n_1}\left(x_1\right) \mathrm{Li}_{n_2}\left(x_2\right)
& = &
\mathrm{Li}_{n_1,n_2}\left(x_1,x_2\right)
+
\mathrm{Li}_{n_2,n_1}\left(x_2,x_1\right)
+
\mathrm{Li}_{n_1+n_2}\left(x_1 \cdot x_2\right).
\end{eqnarray}
We recall that the Clausen functions are given by
\begin{eqnarray}
\mathrm{Cl}_n(\varphi) & = &
\left\{ \begin{array}{rl}
\frac{1}{2i} \left[ \mathrm{Li}_n\left( e^{i \varphi} \right)
-\mathrm{Li}_n\left( e^{-i \varphi} \right)
\right],
& \mbox{$n$ even,} \\
& \\
\frac{1}{2} \left[ \mathrm{Li}_n\left( e^{i \varphi} \right)
+\mathrm{Li}_n\left( e^{-i \varphi} \right)
\right],
& \mbox{$n$ odd}, \\
\end{array} \right.
\end{eqnarray}
while the Glaisher functions are given by
\begin{eqnarray}
\mathrm{Gl}_n(\varphi) & = &
\left\{ \begin{array}{rl}
\frac{1}{2} \left[ \mathrm{Li}_n\left( e^{i \varphi} \right)
+\mathrm{Li}_n\left( e^{-i \varphi} \right)
\right]
& \mbox{$n$ even,} \\
& \\
\frac{1}{2i} \left[ \mathrm{Li}_n\left( e^{i \varphi} \right)
-\mathrm{Li}_n\left( e^{-i \varphi} \right)
\right],
& \mbox{$n$ odd.} \\
\end{array} \right.
\end{eqnarray}
The Clausen and Glaisher functions correspond to the real and imaginary part of $\mathrm{Li}_n(e^{i\varphi})$,
and the actual assignment depends on whether $n$ is even or odd.
Let us now consider the elliptic setting.
In \cite{Adams:2014vja} we considered
the following generalisation depending on three variables $x$, $y$, $q$ and two (integer) indices $n$, $m$:
\begin{eqnarray}
\mathrm{ELi}_{n;m}\left(x;y;q\right) & = &
\sum\limits_{j=1}^\infty \sum\limits_{k=1}^\infty \; \frac{x^j}{j^n} \frac{y^k}{k^m} q^{j k}.
\end{eqnarray}
We define the weight of $\mathrm{ELi}_{n;m}(x;y;q)$ to be $w=n+m$.
The definition is symmetric under the exchange of the pair $(x,n)$ with $(y,m)$.
The two summations are coupled through the variable $q$.
In the special case $q=1$ the two summations decouple and we obtain a product of classical polylogarithms:
\begin{eqnarray}
\mathrm{ELi}_{n;m}\left(x;y;1\right) & = &
\mathrm{Li}_{n}\left(x\right) \mathrm{Li}_{m}\left(y\right).
\end{eqnarray}
In addition we introduce the following linear combinations
\begin{eqnarray}
\label{def_classical_E}
\lefteqn{
\mathrm{E}_{n;m}\left(x;y;q\right)
= } & & \\
& = &
\left\{ \begin{array}{ll}
\frac{1}{i}
\left[
\frac{1}{2} \mathrm{Li}_n\left( x \right) - \frac{1}{2} \mathrm{Li}_n\left( x^{-1} \right)
+ \mathrm{ELi}_{n;m}\left(x;y;q\right) - \mathrm{ELi}_{n;m}\left(x^{-1};y^{-1};q\right)
\right],
& \mbox{$n+m$ even,} \\
& \\
\frac{1}{2} \mathrm{Li}_n\left( x \right) + \frac{1}{2} \mathrm{Li}_n\left( x^{-1} \right)
+ \mathrm{ELi}_{n;m}\left(x;y;q\right) + \mathrm{ELi}_{n;m}\left(x^{-1};y^{-1};q\right),
& \mbox{$n+m$ odd.} \\
\end{array}
\right.
\nonumber
\end{eqnarray}
The special case $(n,m)=(2,0)$ appeared already in \cite{Adams:2014vja}.
Eq.~(\ref{def_classical_E}) gives the generalisation to arbitrary indices $(n,m)$.
In general, the functions $\mathrm{E}_{n;m}(x;y;q)$ are not symmetric under the exchange of the pair $(x,n)$ with $(y,m)$,
nor do they have for $m\neq 0$ a uniform weight.
The functions $\mathrm{E}_{n;m}(x;y;q)$ can be thought of as elliptic generalisations of the Clausen and Glaisher functions.
In particular we have
\begin{eqnarray}
\label{examples_elliptic_generalisations}
\lim\limits_{q \rightarrow 0} \mathrm{E}_{1;0}\left(e^{i \varphi}; y; q \right)
& = &
\mathrm{Cl}_1\left(\varphi\right),
\nonumber \\
\lim\limits_{q \rightarrow 0} \mathrm{E}_{2;0}\left(e^{i \varphi}; y; q \right)
& = &
\mathrm{Cl}_2\left(\varphi\right),
\nonumber \\
\lim\limits_{q \rightarrow 0} \mathrm{E}_{3;1}\left(e^{i \varphi}; y; q \right)
& = &
\mathrm{Gl}_3\left(\varphi\right).
\end{eqnarray}
We now turn to the multi-variable case.
In order to keep the notation simple in this case, we introduce a prefactor $c_n$ and a sign $s_n$, both depending on an index $n$ by
\begin{eqnarray}
c_n = \frac{1}{2} \left[ \left(1+i\right) + \left(1-i\right)\left(-1\right)^n\right] =
\left\{ \begin{array}{rl}
1, & \mbox{$n$ even}, \\
i, & \mbox{$n$ odd}, \\
\end{array} \right.
& &
s_n = (-1)^n =
\left\{ \begin{array}{rl}
1, & \mbox{$n$ even}, \\
-1, & \mbox{$n$ odd}. \\
\end{array} \right.
\end{eqnarray}
With the help of these definitions we can write the definition of the functions $\mathrm{E}_{n;m}(x;y;q)$
uniformly as
\begin{eqnarray}
\lefteqn{
\mathrm{E}_{n;m}\left(x;y;q\right)
= } & &
\\
& = &
\frac{c_{n+m}}{i}
\left[
\left( \frac{1}{2} \mathrm{Li}_n\left( x \right) + \mathrm{ELi}_{n;m}\left(x;y;q\right) \right)
- s_{n+m}
\left( \frac{1}{2} \mathrm{Li}_n\left( x^{-1} \right) + \mathrm{ELi}_{n;m}\left(x^{-1};y^{-1};q\right)
\right)
\right].
\nonumber
\end{eqnarray}
It turns out, that in addition to the functions defined above we only need a single depth two elliptic object
in order to express all results of this paper.
This function depends on five variables $x_1$, $x_2$, $y_1$, $y_2$, $q$ and five (integer) indices $n_1$, $n_2$, $m_1$, $m_2$ and $o$.
This new function is defined as follows:
\begin{eqnarray}
\lefteqn{
\mathrm{E}_{n_1,n_2;m_1,m_2;2o}\left(x_1,x_2;y_1,y_2;q\right)
=
} & & \\
& &
\frac{c_{n_1+m_1}}{i} \frac{c_{n_2+m_2}}{i}
\left\{
\left[ \mathrm{ELi}_{n_1+o; m_1+o}\left(x_1;y_1;q\right) - s_{n_1+m_1} \mathrm{ELi}_{n_1+o; m_1+o}\left(x_1^{-1};y_1^{-1};q\right) \right]
\right. \nonumber \\
& & \left.
\times
\frac{1}{2}
\left[ \mathrm{Li}_{n_2}\left(x_2\right) - s_{n_2+m_2} \mathrm{Li}_{n_2}\left(x_2^{-1}\right) \right]
\right. \nonumber \\
& & \left.
+
\sum\limits_{j_1=1}^\infty
\sum\limits_{k_1=1}^\infty
\sum\limits_{j_2=1}^\infty
\sum\limits_{k_2=1}^\infty
\frac{\left( x_1^{j_1} y_1^{k_1} - s_{n_1+m_1} x_1^{-j_1} y_1^{-k_1} \right)}{j_1^{n_1} k_1^{m_1}}
\frac{\left( x_2^{j_2} y_2^{k_2} - s_{n_2+m_2} x_2^{-j_2} y_2^{-k_2} \right)}{j_2^{n_2} k_2^{m_2}}
\frac{q^{j_1 k_1 + j_2 k_2}}{\left(j_1 k_1 + j_2 k_2\right)^o}
\right\}.
\nonumber
\end{eqnarray}
Please note that this definition is asymmetric in the quadruplets $(n_1,m_1,x_1,y_1)$ and $(n_2,m_2,x_2,y_2)$.
Let us briefly discuss the weights of the individual pieces. The first term, consisting of products of the form
\begin{eqnarray}
\mathrm{ELi}_{n_1+o; m_1+o}\left(x_1;y_1;q\right) \mathrm{Li}_{n_2}\left(x_2\right)
\end{eqnarray}
is of weight $w_1=n_1+n_2+m_1+2o$, while the quadruple sum of the form
\begin{eqnarray}
\sum\limits_{j_1=1}^\infty
\sum\limits_{k_1=1}^\infty
\sum\limits_{j_2=1}^\infty
\sum\limits_{k_2=1}^\infty
\frac{ x_1^{j_1} y_1^{k_1} }{j_1^{n_1} k_1^{m_1}}
\frac{ x_2^{j_2} y_2^{k_2} }{j_2^{n_2} k_2^{m_2}}
\frac{q^{j_1 k_1 + j_2 k_2}}{\left(j_1 k_1 + j_2 k_2\right)^o}
\end{eqnarray}
is of weight $w_2=n_1+n_2+m_1+m_2+2o$.
For $m_2=0$ the two weights coincide.
For $o > 0$ we can express this function as an $o$-fold iterated integral over $q$ (the variables $x_1$, $x_2$, $y_1$ and $y_2$
are treated as constants in the integration):
\begin{eqnarray}
\label{E_depth_two_integral_repr}
\lefteqn{
\mathrm{E}_{n_1,n_2;m_1,m_2;2o}\left(x_1,x_2;y_1,y_2;q\right)
=
} & & \\
& &
\int\limits_0^q \frac{dq_1}{q_1} \int\limits_0^{q_1} \frac{dq_2}{q_2} ... \int\limits_0^{q_{o-1}} \frac{dq_o}{q_o}
\left[ \mathrm{E}_{n_1; m_1}\left(x_1;y_1;q_o\right) - \mathrm{E}_{n_1; m_1}\left(x_1;y_1;0\right) \right]
\mathrm{E}_{n_2;m_2}\left(x_2;y_2;q_o\right).
\nonumber
\end{eqnarray}
In eq.~(\ref{E_depth_two_integral_repr}) the asymmetry with respect to the quadruplets $(n_1,m_1,x_1,y_1)$ and $(n_2,m_2,x_2,y_2)$
manifests itself by the fact, that the constant part with respect to the Taylor expansion in $q$ is subtracted out
from $\mathrm{E}_{n_1; m_1}(x_1;y_1;q)$, but not from $\mathrm{E}_{n_2;m_2}(x_2;y_2;q)$.
This subtraction ensures that the integrand has a Taylor expansion starting at $q^1$.
Therefore the integral is well defined at the lower integration boundary $q=0$ and no regularisation is needed.
Let us write an explicit example. At depth two there is only one combination of indices, which
will appear in the results of this paper.
The indices are given by $(n_1,n_2)=(0,1)$, $(m_1,m_2)=(-2,0)$ and $2o=4$.
We have
\begin{eqnarray}
\lefteqn{
\mathrm{E}_{0,1;-2,0;4}\left(x_1,x_2;y_1,y_2;-q\right)
=
} & & \\
& &
\frac{1}{i}
\left\{
\left[ \mathrm{ELi}_{2; 0}\left(x_1;y_1;-q\right) - \mathrm{ELi}_{2; 0}\left(x_1^{-1};y_1^{-1};-q\right) \right]
\times
\frac{1}{2}
\left[ \mathrm{Li}_{1}\left(x_2\right) + \mathrm{Li}_{1}\left(x_2^{-1}\right) \right]
\right. \nonumber \\
& & \left.
+
\sum\limits_{j_1=1}^\infty
\sum\limits_{k_1=1}^\infty
\sum\limits_{j_2=1}^\infty
\sum\limits_{k_2=1}^\infty
\frac{k_1^2}{j_2 \left(j_1 k_1 + j_2 k_2\right)^2}
\left( x_1^{j_1} y_1^{k_1} - x_1^{-j_1} y_1^{-k_1} \right)
\left( x_2^{j_2} y_2^{k_2} + x_2^{-j_2} y_2^{-k_2} \right)
\left(-q\right)^{j_1 k_1 + j_2 k_2}
\right\}.
\nonumber
\end{eqnarray}
The corresponding integral representation reads
\begin{eqnarray}
\lefteqn{
\mathrm{E}_{0,1;-2,0;4}\left(x_1,x_2;y_1,y_2;-q\right)
=
} & & \\
& &
\int\limits_0^{q} \frac{dq_1}{q_1} \int\limits_0^{q_1} \frac{dq_2}{q_2}
\left[ \mathrm{E}_{0; -2}\left(x_1;y_1;-q_2\right) - \mathrm{E}_{0; -2}\left(x_1;y_1;0\right) \right]
\mathrm{E}_{1; 0}\left(x_2;y_2;-q_2\right).
\nonumber
\end{eqnarray}
\section{The result for $S_{111}^{(1)}(2,t)$}
\label{sec:result_S1}
In this section we present the ${\mathcal O}(\varepsilon^1)$-term $S_{111}^{(1)}(2,t)$ of the sunrise integral around two space-time
dimensions in terms of elliptic generalisations of the Clausen and Glaisher functions discussed in the previous section.
Let us start by recalling that the ${\mathcal O}(\varepsilon^0)$-term $S_{111}^{(0)}(2,t)$ of the $\varepsilon$-expansion of
$S_{111}(2-2\varepsilon,t)$ is given by
\begin{eqnarray}
S_{111}^{(0)}\left(2,t\right)
& = &
\frac{\psi_1}{\pi} E^{(0)},
\end{eqnarray}
with
\begin{eqnarray}
\label{res_E_0}
E^{(0)}
& = &
\mathrm{E}_{2;0}\left(w_1;-1;-q\right)
+
\mathrm{E}_{2;0}\left(w_2;-1;-q\right)
+
\mathrm{E}_{2;0}\left(w_3;-1;-q\right).
\end{eqnarray}
This motivates the following ansatz for $S_{111}^{(1)}(2,t)$:
\begin{eqnarray}
S_{111}^{(1)}\left(2,t\right)
& = &
\frac{\psi_1}{\pi} E^{(1)}.
\end{eqnarray}
We find that $E^{(1)}$ is given by
\begin{eqnarray}
\label{res_E_1}
\lefteqn{
E^{(1)}
=
} & &
\nonumber \\
& &
\left\{
- \frac{2}{3} \sum \limits_{j=1}^3 \ln\left( \frac{m_j^2}{\mu^2} \right)
- 6 \mathrm{E}_{1,0}\left(-1;1;-q\right)
+ \sum\limits_{j=1}^3 \left[
\mathrm{E}_{1;0}\left(w_j;1;-q\right)
- \frac{1}{3} \mathrm{E}_{1;0}\left(w_j;-1;-q\right)
\right]
\right\} E^{(0)}
\nonumber \\
& &
- 2
\sum\limits_{j=1}^3
\frac{1}{2i}
\left\{
\mathrm{Li}_{2,1}\left(w_j,1\right) - \mathrm{Li}_{2,1}\left(w_j^{-1},1\right)
+ \mathrm{Li}_3\left(w_j\right) - \mathrm{Li}_3\left(w_j^{-1}\right)
\right. \nonumber \\
& & \left.
+ 3 \ln\left(2\right) \left[
\mathrm{Li}_2\left( w_j \right)
-\mathrm{Li}_2\left( w_j^{-1} \right)
\right]
\right\}
\nonumber \\
& &
+
\sum\limits_{j=1}^3
\left[
4 \mathrm{E}_{0,1;-2,0;4}\left( w_j,w_j; -1,-1; -q \right)
-
6 \mathrm{E}_{0,1;-2,0;4}\left( w_j,w_j; 1,-1; -q \right)
\right. \nonumber \\
& & \left.
+
6 \mathrm{E}_{0,1;-2,0;4}\left( w_j,-1; -1,1; -q \right)
\right]
\nonumber \\
& &
+
\sum\limits_{j_1=1}^3
\sum\limits_{j_2=1}^3
\left[
2 \mathrm{E}_{0,1;-2,0;4}\left( w_{j_1},w_{j_2}; 1,-1; -q \right)
- \mathrm{E}_{0,1;-2,0;4}\left( w_{j_1},w_{j_2}; -1,-1; -q \right)
\right. \nonumber \\
& & \left.
- \mathrm{E}_{0,1;-2,0;4}\left( w_{j_1},w_{j_2}; -1,1; -q \right)
\right]
\nonumber \\
& &
+
\sum\limits_{j=1}^3 \mathrm{E}_{3;1}\left(w_j;-1;-q\right).
\end{eqnarray}
Eq.~(\ref{res_E_1}) is one of the main results of this paper.
A few remarks are in order:
Eq.~(\ref{res_E_1}) gives the result for $E^{(1)}$ entirely in the variables $q$, $w_1$, $w_2$, $w_3$
and $m_1^2 m_2^2 m_3^2 / \mu^6$.
Although not as elegant as the result for $E^{(0)}$ in eq.~(\ref{res_E_0}), it is still far from trivial
that $E^{(1)}$ can be expressed in a few lines.
As arguments of the (elliptic) multiple polylogarithms only the variables $w_j^{\pm 1}$ and the values $\pm 1$ occur.
The first line gives all terms proportional to the ${\mathcal O}(\varepsilon^0)$ result $E^{(0)}$.
These terms are all of weight one, multiplying $E^{(0)}$, which is of weight two, yielding the total weight three.
The second and third line contain terms, which depend on the variables $w_j$, but not explicitly on $q$.
These are ordinary multiple polylogarithms.
Terms like these are expected,
since $S_{111}^{(1)}(2,t)$ depends not only on the graph polynomial ${\mathcal F}$, but also
on the graph polynomial ${\mathcal U}$.
The ordinary multiple polylogarithms are all of weight three.
The next four lines contain functions of depth two.
Surprisingly, we only encounter the function $\mathrm{E}_{0,1;-2,0;4}$, which is of weight $1-2+4=3$.
The last line gives probably the most interesting part:
It contains the elliptic polylogarithm $\mathrm{E}_{3;1}$,
an elliptic generalisation of the Glaisher function $\mathrm{Gl}_3$ according to eq.~(\ref{examples_elliptic_generalisations}).
The elliptic polylogarithm $\mathrm{E}_{3;1}$ is not homogeneous in the weight,
having parts of weight $3$ and parts of weight $4$.
The occurrence of $\mathrm{E}_{3;1}$ is discussed in more detail in section \ref{sec:weights}.
Furthermore, the result shows an explicit $\ln(2)$. These $\ln(2)$-terms are spurious and cancel with
$\ln(2)$-terms from $\mathrm{E}_{1;0}(-1;1;-q)$ and
$\mathrm{E}_{0,1;-2,0;4}( w_j,-1; -1,1; -q )$.
\section{The result for the sunrise integral around four space-time dimensions}
\label{sec:shift}
In this section we discuss dimensional-shift relations, which relate the integrals $S_{111}(4-2\varepsilon,t)$ and
$S_{111}(2-2\varepsilon,t)$.
This allows us to express the finite part $S_{111}^{(0)}(4,t)$ of the sunrise integral around four space-time dimensions
in terms of the already calculated integrals $S_{111}^{(0)}(2,t)$ and $S_{111}^{(1)}(2,t)$.
The divergent parts $S_{111}^{(-2)}(4,t)$ and $S_{111}^{(-1)}(4,t)$ are rather simple and listed for completeness.
There are several relations, which relate integrals in $D$ space-time dimensions
to integrals in $(D+2)$ space-time dimensions.
The Tarasov relations read \cite{Tarasov:1996br,Tarasov:1997kx}
\begin{eqnarray}
S_{\nu_1 \nu_2 \nu_3}\left( D, t\right)
& = &
\nu_1 \nu_2 S_{(\nu_1+1) (\nu_2+1) \nu_3}\left( D+2, t\right)
+
\nu_2 \nu_3 S_{\nu_1 (\nu_2+1) (\nu_3+1)}\left( D+2, t\right)
\nonumber \\
& &
+
\nu_1 \nu_3 S_{(\nu_1+1) \nu_2 (\nu_3+1)}\left( D+2, t\right).
\end{eqnarray}
From eq.~(\ref{def_Feynman_integral}) we find
\begin{eqnarray}
\mu^2 \frac{d}{dt}
S_{\nu_1 \nu_2 \nu_3}\left( D, t\right)
& = &
\nu_1 \nu_2 \nu_3
S_{(\nu_1+1) (\nu_2+1) (\nu_3+1)}\left( D+2, t\right).
\end{eqnarray}
For a fixed space-time dimension we can always reduce any integral to a linear combination of sunrise master integrals
and simpler integrals.
Inverting the relations, which relate integrals in $D$ dimensions to integrals in $(D+2)$ dimensions, we can express
$S_{111}(4-2\varepsilon,t)$ in terms of four sunrise master integrals for $D=2-2\varepsilon$ and simpler tadpole integrals.
A possible basis of sunrise master integrals is given by
\begin{eqnarray}
\label{derivative_basis}
S_{111}\left(D,t\right),
\;\;\;\;\;\;
\mu^2 \frac{d}{dt} S_{111}\left(D,t\right),
\;\;\;\;\;\;
\mu^4 \frac{d^2}{dt^2} S_{111}\left(D,t\right),
\;\;\;\;\;\;
\mu^6 \frac{d^3}{dt^3} S_{111}\left(D,t\right).
\end{eqnarray}
In this basis one finds for $S_{111}^{(0)}(4,t)$
\begin{eqnarray}
\label{S4_dimensional_shift}
S_{111}^{(0)}(4,t)
& = &
\frac{1}{\mu^4}
L_3^{(-1)}\left(2\right) \;\; S_{111}^{(1)}(2,t)
+ \frac{1}{\mu^4} L_3^{(0)}\left(2\right) \;\; S_{111}^{(0)}(2,t)
+
R^{(0)},
\end{eqnarray}
where the remainder function $R^{(0)}$ contains the contributions from the tadpoles.
The differential operator $L_3^{(-1)}(2)$ factorises again:
\begin{eqnarray}
L_3^{(-1)}\left(2\right)
& = &
L^{(-1)}_{1}\left(2\right)
\;\;
L_2^{(0)}\left(2\right),
\end{eqnarray}
with $L^{(-1)}_{1}(2)$ given by
\begin{eqnarray}
\label{def_L_1_m1}
L^{(-1)}_{1}\left(2\right)
& = &
\frac{-33t^3+13 M_{100}t^2 + \left(22 M_{110}-15 M_{200} \right)t - 3 M_{100} \Delta}
{24 t \left(3t^2-2 M_{100}t+\Delta\right) \left(5t^2 + 2M_{100} t + 7 \Delta \right)}
\frac{d}{dt}
\nonumber \\
& &
+
\frac{69 t^2 - 22 M_{100} t + 9 M_{200} -10 M_{110}}{24 t \left(3t^2-2 M_{100}t+\Delta\right) \left(5t^2 + 2M_{100} t + 7 \Delta \right)}.
\end{eqnarray}
The explicit expressions for the differential operator $L_3^{(0)}(2)$ and the remainder function $R^{(0)}$ are rather long and not
provided here.
We have used eq.~(\ref{S4_dimensional_shift}) internally for the determination of the integration constants $C_1$ and $C_2$.
In order to present our final result on $S_{111}^{(0)}(4,t)$ it is better to use an alternative basis of sunrise master integrals
given by
\begin{eqnarray}
S_{111}\left(D,t\right),
\;\;\;\;\;\;
\mu^2 \frac{\partial}{\partial m_1^2} S_{111}\left(D,t\right),
\;\;\;\;\;\;
\mu^2 \frac{\partial}{\partial m_2^2} S_{111}\left(D,t\right),
\;\;\;\;\;\;
\mu^2 \frac{\partial}{\partial m_3^2} S_{111}\left(D,t\right).
\end{eqnarray}
This is the basis of sunrise master integrals used in \cite{Caffo:1998du}.
We recall that
\begin{eqnarray}
\mu^2 \frac{\partial}{\partial t}
S_{111}\left( D, t\right)
& = &
S_{222}\left( D+2, t\right),
\nonumber \\
\mu^2 \frac{\partial}{\partial m_1^2}
S_{111}\left( D, t\right)
& = &
- S_{211}\left( D, t\right),
\nonumber \\
\mu^2 \frac{\partial}{\partial m_2^2}
S_{111}\left( D, t\right)
& = &
- S_{121}\left( D, t\right),
\nonumber \\
\mu^2 \frac{\partial}{\partial m_3^2}
S_{111}\left( D, t\right)
& = &
- S_{112}\left( D, t\right).
\end{eqnarray}
Let us define for $k \in \{-1,0\}$ two partial differential operators
\begin{eqnarray}
\tilde{L}_3^{(k)}(2) & = &
C^{(k)}_0
+
\sum\limits_{i=1}^3 C^{(k)}_i m_i^2 \frac{\partial}{\partial m_i^2}.
\end{eqnarray}
For $k=-1$ the coefficients are given by
\begin{eqnarray}
C^{(-1)}_i & = &
\frac{\left(t-m_i^2\right)}{6 \mu^4 t} \left[ \left( 2 m_i^2 - m_j^2 - m_k^2 \right) t
- 3 m_j^4 - 3 m_k^4 - m_i^2 m_j^2 - m_i^2 m_k^2 + 8 m_j^2 m_k^2 \right],
\nonumber \\
C^{(-1)}_0 & = &
\frac{1}{4} \left( C^{(-1)}_1 + C^{(-1)}_2 + C^{(-1)}_3 \right),
\end{eqnarray}
while for $k=0$ we have
\begin{eqnarray}
C^{(0)}_0 & = &
\frac{1}{12 \mu^4 t}
\left[
- M_{100} t^2 + \left( -26M_{200}+36M_{110} \right)t + M_{300} + 8 M_{210} - 78 M_{111}
\right]
\nonumber \\
C^{(0)}_i & = &
\frac{\left(t-m_i^2\right)}{12 \mu^4 t} \left[ -t ^2 + \left( 12 m_i^2 - 3 m_j^2 - 3 m_k^2 \right) t
- m_i^4 - 16 m_j^4 - 16 m_k^4 - 3 m_i^2 m_j^2 - 3 m_i^2 m_k^2
\right. \nonumber \\
& & \left.
+ 66 m_j^2 m_k^2 \right].
\end{eqnarray}
Please note that all coefficients of $\tilde{L}_3^{(-1)}(2)$ vanish in the equal mass case.
For $S_{111}^{(0)}\left(4,t\right)$ we find
\begin{eqnarray}
\label{res_S4}
\lefteqn{
S_{111}^{(0)}\left(4,t\right)
=
\tilde{L}_3^{(-1)}(2) S_{111}^{(1)}\left(2,t\right)
+
\tilde{L}_3^{(0)}(2) S_{111}^{(0)}\left(2,t\right)
+ \frac{13}{8} \frac{t}{\mu^2} - 3 \frac{M_{100}}{\mu^2} - \frac{1}{2} \zeta_2 \frac{M_{100}}{\mu^2}
} & &
\\
& &
- \sum\limits_{(i,j,k) \in {\mathbb Z}_3}
\left[
\frac{\left(4m_i^2 + m_j^2 + m_k^2\right) t + m_i^2\left(m_j^2+m_k^2\right) - 2 m_j^2 m_k^2}{12 \mu^2 t} \ln^2\left(\frac{m_i^2}{\mu^2}\right)
\right. \nonumber \\
& &
\left.
- \frac{\left(m_i^2 - 2m_j^2 - 2m_k^2\right) t + m_i^2\left(m_j^2+m_k^2\right) - 2 m_j^2 m_k^2}{6 \mu^2 t} \ln\left(\frac{m_j^2}{\mu^2}\right) \ln\left(\frac{m_k^2}{\mu^2}\right)
\right. \nonumber \\
& &
\left.
+ \frac{2 t^2 - \left(24m_i^2 + 6m_j^2 + 6m_k^2\right) t + 2m_i^4 - m_j^4 - m_k^4 - 6m_i^2\left(m_j^2+m_k^2\right) + 12 m_j^2 m_k^2}{12 \mu^2 t} \ln\left(\frac{m_i^2}{\mu^2}\right)
\right].
\nonumber
\end{eqnarray}
The notation $(i,j,k) \in {\mathbb Z}_3$ stands
for a sum over the three cyclic permutations of $(1,2,3)$.
Eq.~(\ref{res_S4}) expresses the finite part $S_{111}^{(0)}(4,t)$ of the sunrise integral in $4-2\varepsilon$ dimensions
in terms of $S_{111}^{(1)}(2,t)$ and $S_{111}^{(0)}(2,t)$, derivatives thereof and simpler terms.
The results for $S_{111}^{(1)}(2,t)$ and $S_{111}^{(0)}(2,t)$ have been given in section~\ref{sec:result_S1}.
Please note that $\tilde{L}_3^{(-1)}(2)$ vanishes in the equal mass case.
Therefore it follows that in the equal mass case the ${\mathcal O}(\varepsilon)$-part $S_{111}^{(1)}(2,t)$ does not affect
$S_{111}^{(0)}(4,t)$.
However, in the unequal mass case the ${\mathcal O}(\varepsilon)$-part $S_{111}^{(1)}(2,t)$ is required
for $S_{111}^{(0)}(4,t)$.
Eq.~(\ref{res_S4}) together with eq.~(\ref{res_E_1}) is the main result of this paper.
For completeness we list also the divergent terms of $S_{111}(4-2\varepsilon,t)$. These read
\begin{eqnarray}
S_{111}^{(-2)}(4,t)
& = &
- \frac{M_{100}}{2 \mu^2},
\nonumber \\
S_{111}^{(-1)}(4,t)
& = &
\frac{t}{4 \mu^2}
- \frac{3 M_{100}}{2 \mu^2}
+ \sum\limits_{i=1}^3 \frac{m_i^2}{\mu^2} \ln\left(\frac{m_i^2}{\mu^2}\right).
\end{eqnarray}
\section{The equal mass case}
\label{sec:equal_mass_case}
This section is devoted to the equal mass case $m_1=m_2=m_3=m$.
In the equal mass case the result for the sunrise integral around four space-time dimensions is significantly
simpler.
We have already seen that the contribution from $S_{111}^{(1)}(2,t)$ in
eq.~(\ref{res_S4}) drops out in the equal mass case.
Furthermore we have now a second-order differential equation for all $D$ \cite{Laporta:2004rb}:
\begin{eqnarray}
\lefteqn{
\left\{
2 t \left(t-9m^2\right) \left(t-m^2\right) \frac{d^2}{dt^2}
+ \left[ 3 \left(4-D\right) t^2 + 10 \left(D-6\right) t m^2 + 9 D m^4 \right] \frac{d}{dt}
\right.
} & & \nonumber \\
& &
\left.
+ \left(D-3\right) \left[ \left(D-4\right) t + \left(D+4\right) m^2 \right]
\right\}
S_{111}\left(D,t\right)
=
- 3 \left(D-2\right)^2 \mu^2 \left[ T_1\left(D\right) \right]^2,
\end{eqnarray}
with $T_1(D)=T_1(D,m^2,\mu^2)$.
A convenient basis of sunrise master integrals is therefore
\begin{eqnarray}
S_{111}\left(D,t\right),
\;\;\;\;\;\;
\mu^2 \frac{d}{dt} S_{111}\left(D,t\right).
\end{eqnarray}
In terms of this basis the dimensional-shift relation simplifies and we have
\begin{eqnarray}
S_{111}\left(4-2\varepsilon,t\right)
& = &
\frac{1}{6 \left(1-2\varepsilon\right)\left(1-3\varepsilon\right)\left(2-3\varepsilon\right)}
\left\{
\frac{\left(t+3m^2\right)\left(t-m^2\right)\left(t-9m^2\right)}{\mu^4} \frac{d}{dt} S_{111}\left(2-2\varepsilon,t\right)
\right. \nonumber \\
& & \left.
+ \left[ \frac{\left(t-m^2\right)\left(t-9m^2\right)}{\mu^4}
+ \varepsilon \frac{\left( t^2 + 22 m^2 t - 87 m^4 \right)}{\mu^4} \right] S_{111}\left(2-2\varepsilon,t\right)
\right. \nonumber \\
& & \left.
+ 3 \left(1-\varepsilon\right)^2 \left[ - \frac{6 \mu^2}{m^2} + \varepsilon \frac{\mu^2 \left( t + 21 m^2 \right)}{m^4} \right] \left[ T_1\left(4-2\varepsilon\right) \right]^2
\right\}.
\end{eqnarray}
For the coefficients of the $\varepsilon$-expansion we have now
\begin{eqnarray}
S_{111}^{(-2)}(4,t)
& = &
- \frac{3 m^2}{2 \mu^2},
\\
S_{111}^{(-1)}(4,t)
& = &
\frac{t}{4 \mu^2}
- \frac{9 m^2}{2 \mu^2}
+ \frac{3 m^2}{\mu^2} \ln\left(\frac{m^2}{\mu^2}\right),
\nonumber \\
S_{111}^{(0)}(4,t)
& = &
\frac{13 t}{8 \mu^2}
- \frac{9 m^2}{\mu^2}
- \frac{3 m^2}{2 \mu^2} \zeta_2
- \left( \frac{t}{2\mu^2} - \frac{9 m^2}{\mu^2} \right) \ln\left(\frac{m^2}{\mu^2}\right)
- \frac{3 m^2}{\mu^2} \left(\ln\left(\frac{m^2}{\mu^2}\right) \right)^2
\nonumber \\
& &
+ \frac{\left(t-m^2\right)\left(t-9m^2\right)}{12 \mu^4} S_{111}^{(0)}\left(2,t\right)
+ \frac{\left(t+3m^2\right)\left(t-m^2\right)\left(t-9m^2\right)}{12 \mu^4} \frac{d}{dt} S_{111}^{(0)}\left(2,t\right),
\nonumber
\end{eqnarray}
again showing the absence of $S_{111}^{(1)}(2,t)$ in the expression for $S_{111}^{(0)}(4,t)$.
\section{Weights}
\label{sec:weights}
In this section we discuss in more detail the transcendental weights associated with individual terms and in particular the
occurrence of the functions $\mathrm{E}_{3;1}$.
The building blocks of all our expressions are the multiple polylogarithms of the form
\begin{eqnarray}
\label{example_polylogs}
\mathrm{Li}_{n_1,n_2,...,n_k}\left(x_1,x_2,...,x_k\right)
& = &
\sum\limits_{j_1=1}^\infty \sum\limits_{j_2=1}^{j_1-1} ... \sum\limits_{j_k=1}^{j_{k-1}-1}
\frac{x_1^{j_1}}{j_1^{n_1}} \frac{x_2^{j_2}}{j_2^{n_2}} ... \frac{x_k^{j_k}}{j_k^{n_k}},
\end{eqnarray}
the generalisation of the classical polylogarithms to the elliptic case
\begin{eqnarray}
\label{example_elliptic_polylogs}
\mathrm{ELi}_{n_1;m_1}\left(x_1;y_1;q\right) & = &
\sum\limits_{j_1=1}^\infty \sum\limits_{k_1=1}^\infty \; \frac{x_1^{j_1}}{j_1^{n_1}} \frac{y_1^{k_1}}{k_1^{m_1}} q^{j_1 k_1},
\end{eqnarray}
and quadruple sums of the form
\begin{eqnarray}
\label{example_quadruple_sum}
\sum\limits_{j_1=1}^\infty
\sum\limits_{k_1=1}^\infty
\sum\limits_{j_2=1}^\infty
\sum\limits_{k_2=1}^\infty
\frac{ x_1^{j_1} y_1^{k_1} }{j_1^{n_1} k_1^{m_1}}
\frac{ x_2^{j_2} y_2^{k_2} }{j_2^{n_2} k_2^{m_2}}
\frac{q^{j_1 k_1 + j_2 k_2}}{\left(j_1 k_1 + j_2 k_2\right)^o}.
\end{eqnarray}
For $x_1=x_2=...x_k=1$, $y_1=y_2=1$ and $q=1$ the summand is in all cases a homogeneous function of the summation variables.
We defined the transcendental weight as the negative of the degree of homogeneity with respect to the summation variables in this
case.
Thus eq.~(\ref{example_polylogs}) is of weight $w_1=n_1+n_2+...+n_k$,
eq.~(\ref{example_elliptic_polylogs}) is of weight $w_2=n_1+m_1$, and
eq.~(\ref{example_quadruple_sum}) is of weight $w_3=n_1+n_2+m_1+m_2+2 o$.
With this weight counting, the function
\begin{eqnarray}
\mathrm{ELi}_{3;1}\left(x_1;y_1;q\right)
\end{eqnarray}
is of weight $4$ and we would like to discuss the occurrence of this function in the result for $S_{111}^{(1)}(2,t)$.
It turns out, that the occurrence of this function can be related to the fact that
\begin{eqnarray}
\ln q
\end{eqnarray}
should be counted as weight $2$.
The latter fact can be seen as follows: We have for example
\begin{eqnarray}
\int\limits_0^x \mathrm{ELi}_{n;m}\left(x';y;q\right) d \ln x'
& = &
\mathrm{ELi}_{n+1;m}\left(x;y;q\right),
\end{eqnarray}
but
\begin{eqnarray}
\int\limits_0^q \mathrm{ELi}_{n;m}\left(x;y;q'\right) d \ln q'
& = &
\mathrm{ELi}_{n+1;m+1}\left(x;y;q\right).
\end{eqnarray}
Thus an integration with respect to $d\ln q = dq / q$ increases the weight by $2$.
The final result for $S_{111}^{(1)}(2,t)$ does not contain any $\ln q$-terms, however a convenient method
for calculating $S_{111}^{(1)}(2,t)$ splits the integral into two parts, each part containing $\ln q$-terms.
The $\ln q$-terms cancel in the sum, but leave the functions $\mathrm{ELi}_{3;1}$ as remainders.
In order to understand how this happens, it is sufficient to consider the equal mass case $m_1=m_2=m_3=m$ for
$S_{111}^{(1)}(2,t)$.
In the equal mass case the differential equations for $S_{111}^{(1)}(2,t)$ (and $S_{111}^{(0)}(2,t)$) simplify and we find
with
\begin{eqnarray}
\lefteqn{
L^{(0)}_{2,\mathrm{equal}}
=
p_{2,\mathrm{equal}} \frac{d^2}{dt^2}
+
p_{1,\mathrm{equal}} \frac{d}{dt}
+
p_{0,\mathrm{equal}},
} & &
\\
& &
p_{2,\mathrm{equal}} \; = \; t \left(t-m^2\right) \left(t-9m^2\right),
\;\;\;\;\;\;
p_{1,\mathrm{equal}} \; = \; 3 t^2 - 20 t m^2 + 9 m^4,
\;\;\;\;\;\;
p_{0,\mathrm{equal}} \; = \; t- 3 m^2,
\nonumber
\end{eqnarray}
the following differential equations
\begin{eqnarray}
L^{(0)}_{2,\mathrm{equal}} S_{111}^{(0)}\left(2,t\right)
& = &
- 6 \mu^2,
\\
L^{(0)}_{2,\mathrm{equal}} S_{111}^{(1)}\left(2,t\right)
& = &
12 \mu^2 \ln\left(\frac{m^2}{\mu^2}\right)
+
\left[
\left(-3t^2 + 10 t m^2 + 9 m^4 \right) \frac{d}{dt} -3t + 5m^2
\right] S_{111}^{(0)}\left(2,t\right).
\nonumber
\end{eqnarray}
In order to find $S_{111}^{(1)}(2,t)$, let us make the ansatz
that $S_{111}^{(1)}(2,t)$ consists of a part proportional to $S_{111}^{(0)}(2,t)$ and a remainder $\tilde{S}_{111}^{(1)}(2,t)$:
\begin{eqnarray}
S_{111}^{(1)}\left(2,t\right)
& = &
\tilde{S}_{111}^{(1)}\left(2,t\right)
+
F_1(t) S_{111}^{(0)}\left(2,t\right).
\end{eqnarray}
The differential equation for $\tilde{S}_{111}^{(1)}(2,t)$ is then
\begin{eqnarray}
\label{diff_eq_S1_equal_mass_case}
L^{(0)}_{2,\mathrm{equal}} \tilde{S}_{111}^{(1)}\left(2,t\right)
& = &
12 \mu^2 \ln\left(\frac{m^2}{\mu^2}\right)
+
6 \mu^2 F_1\left(t\right)
-
\left[
\left(2 p_{2,\mathrm{equal}} \frac{d F_1(t)}{dt} +3t^2 - 10 t m^2 - 9 m^4 \right) \frac{d}{dt}
\right.
\nonumber \\
& &
\left.
+ p_{2,\mathrm{equal}} \frac{d^2F_1(t)}{dt^2}
+ p_{1,\mathrm{equal}} \frac{dF_1(t)}{dt}
+ 3t - 5m^2
\right] S_{111}^{(0)}\left(2,t\right).
\end{eqnarray}
We are free to choose $F_1(t)$
(different choices for $F_1(t)$ will lead to different expressions for $\tilde{S}_{111}^{(1)}(2,t)$)
and a convenient choice for $F_1(t)$ will be the one, which eliminates
$d S_{111}^{(0)}(2,t)/dt$ on the right-hand side of eq.~(\ref{diff_eq_S1_equal_mass_case}).
This amounts to the choice
\begin{eqnarray}
\frac{d}{dt} F_1(t)
& = &
\frac{-3t^2 + 10 t m^2 + 9 m^4}{2 t \left(t-m^2\right)\left(t-9m^2\right)}
\;\; = \;\;
\frac{1}{2t} - \frac{1}{t-m^2} - \frac{1}{t-9m^2},
\end{eqnarray}
and hence
\begin{eqnarray}
F_1(t)
& = &
- \ln\left( \frac{m \left(t-m^2\right)\left(t-9m^2\right)}{3 \mu^4 \sqrt{t}} \right).
\end{eqnarray}
It turns out that this choice does not only eliminate the terms proportional to the derivative of
$S_{111}^{(0)}(2,t)$, but also the terms proportional to $S_{111}^{(0)}(2,t)$ itself
and the differential equation for $\tilde{S}_{111}^{(1)}(2,t)$ reads
\begin{eqnarray}
\label{simplified_dgl}
L^{(0)}_{2,\mathrm{equal}} \tilde{S}_{111}^{(1)}\left(2,t\right)
& = &
\mu^2 \tilde{I}_{2,\mathrm{equal}}\left(t\right),
\;\;\;\;\;\;\;\;\;
\tilde{I}_{2,\mathrm{equal}}\left(t\right)
\;\; = \;\;
6 \ln\left(\frac{3 m^3 \sqrt{t}}{\left(t-m^2\right)\left(t-9m^2\right)} \right).
\end{eqnarray}
The inhomogeneous term of this differential equation is significantly simpler
than the generic inhomogeneous term of eq.~(\ref{diff_eq_S1_equal_mass_case}).
But please note that $F_1(t)$ and $\tilde{I}_{2,\mathrm{equal}}(t)$ contain both terms
proportional to $\ln(t)$ (which can be related to $\ln(-q)$-terms).
In order to determine $\tilde{S}_{111}^{(1)}(2,t)$ from eq.~(\ref{simplified_dgl})
one needs first the homogeneous solutions. These are spanned by $\psi_1$ and $\psi_2$, defined
in eq.~(\ref{def_periods}).
For the case at hand it will be convenient to use instead of the basis $\{\psi_1,\psi_2\}$
the basis given by
\begin{eqnarray}
\psi_1,
& &
\psi_1 \ln\left(-q\right).
\end{eqnarray}
We then write the full solution as
\begin{eqnarray}
\tilde{S}_{111}^{(1)}\left(2,t\right)
& = &
c_1 \psi_1
+ c_2 \psi_1 \ln\left(-q\right)
+ \tilde{S}_{111,\mathrm{special}}^{(1)}\left(2,t\right)
\end{eqnarray}
with $\tilde{S}_{111,\mathrm{special}}^{(1)}(2,0)=0$.
From the boundary values one finds now
\begin{eqnarray}
c_1
& = &
\frac{3}{2 \pi i}
\left\{
\mathrm{Li}_3\left(r_3\right) - \mathrm{Li}_3\left(r_3^{-1}\right)
- 2 \left[ \mathrm{Li}_{21}\left(r_3,1\right) + \mathrm{Li}_3\left(r_3\right) - \mathrm{Li}_{21}\left(r_3^{-1},1\right) - \mathrm{Li}_3\left(r_3^{-1}\right) \right]
\right. \nonumber \\
& & \left.
- \ln\left(3\right) \left( \mathrm{Li}_2\left(r_3\right) - \mathrm{Li}_2\left(r_3^{-1}\right) \right)
\right\},
\nonumber \\
c_2
& = &
- \frac{1}{2} \frac{3}{2 \pi i} \left[ \mathrm{Li}_2\left(r_3\right) - \mathrm{Li}_2\left(r_3^{-1}\right) \right].
\end{eqnarray}
In the following we will denote by
\begin{eqnarray}
r_p & = & \exp\left(\frac{2\pi i}{p}\right)
\end{eqnarray}
the $p$-th root of unity.
For $\tilde{S}_{111,\mathrm{special}}^{(1)}(2,t)$ one finds
along the lines leading to eq.~(\ref{solution_S1_quadrature})
\begin{eqnarray}
\tilde{S}_{111,\mathrm{special}}^{(1)}\left(2,t\right)
=
-
\frac{\psi_1}{\pi}
\frac{1}{2i}
\sum\limits_{j=1}^\infty
\sum\limits_{k=1}^\infty
\int\limits_0^q \frac{dq_1}{q_1}
\int\limits_0^{q_1} \frac{dq_2}{q_2}
\left(r_3^j - r_3^{-j} \right)
k^2 \left(-1\right)^k
\left(-q_2\right)^{jk}
\tilde{I}_{2,\mathrm{equal}}(q_2).
\end{eqnarray}
In order to perform the integration we need the $q$-expansion of $\tilde{I}_{2,\mathrm{equal}}$.
The $q$-expansion of $\tilde{I}_{2,\mathrm{equal}}$ is given by
\begin{eqnarray}
\lefteqn{
\tilde{I}_{2,\mathrm{equal}}\left(q\right)
=
3 \ln\left(-q\right)
} \\
& &
+ 12
\sum\limits_{j=1}^\infty \sum\limits_{k=1}^\infty
\frac{1}{j}
\left[ 6 r_2^j + r_3^j + r_3^{2 j} - 6 r_4^j - 6 r_4^{3 j} + 3 r_6^j + 3 r_6^{5 j}
- 2 r_{12}^j - 2 r_{12}^{5 j} - 2 r_{12}^{7 j} - 2 r_{12}^{11 j} \right] q^{j k}.
\nonumber
\end{eqnarray}
Of particular relevance for the weight $4$ part is the term $3 \ln(-q)$.
We have
\begin{eqnarray}
\label{special_solution_equal_mass_case}
\lefteqn{
-
3 \frac{\psi_1}{\pi}
\frac{1}{2i}
\sum\limits_{j=1}^\infty
\sum\limits_{k=1}^\infty
\int\limits_0^q \frac{dq_1}{q_1}
\int\limits_0^{q_1} \frac{dq_2}{q_2}
\left(r_3^j - r_3^{-j} \right)
k^2 \left(-1\right)^k
\left(-q_2\right)^{jk}
\ln\left(-q_2\right)
= } & &
\nonumber \\
& &
-
\frac{3}{2i}
\frac{\psi_1}{\pi}
\ln\left(-q\right)
\left(
\mathrm{ELi}_{2;0}\left(r_3;-1;-q\right)
-
\mathrm{ELi}_{2;0}\left(r_3^{-1};-1;-q\right)
\right)
\nonumber \\
& &
+
\frac{3}{i} \frac{\psi_1}{\pi}
\left(
\mathrm{ELi}_{3;1}\left(r_3;-1;-q\right)
-
\mathrm{ELi}_{3;1}\left(r_3^{-1};-1;-q\right)
\right).
\end{eqnarray}
Counting the weight of $\ln(-q)$ as two,
we notice that in eq.~(\ref{special_solution_equal_mass_case}) all terms are of weight $4$.
The two terms on the right-hand side of eq.~(\ref{special_solution_equal_mass_case})
combine nicely with parts of the boundary terms to give
\begin{eqnarray}
\label{partial_combination}
-
\frac{3}{2}
\frac{\psi_1}{\pi}
\ln\left(-q\right)
\mathrm{E}_{2;0}\left(r_3;-1;-q\right)
+
3 \frac{\psi_1}{\pi}
\mathrm{E}_{3;1}\left(r_3;-1;-q\right).
\end{eqnarray}
We note that eq.~(\ref{partial_combination}) contains terms of weight $3$ and $4$.
The term proportional to $\ln(-q)$ in eq.~(\ref{partial_combination}) cancels
the corresponding logarithmic singularity of $\ln(-q)$ (or equivalently $\ln(t)$)
in $F_1(t) S_{111}^{(0)}(2,t)$.
The second term proportional $\mathrm{E}_{3;1}$ explains the occurrence of $\mathrm{E}_{3;1}$ in the final
result for $S_{111}^{(1)}(2,t)$ in eq.~(\ref{res_E_1}).
\section{Conclusions}
\label{sec:conclusions}
In this paper we presented the result for the ${\mathcal O}(\varepsilon^1)$-part
of the sunrise integral around two space-time dimensions.
The result is expressed in terms of generalisations of the Clausen and Glaisher functions towards
the elliptic case.
The ${\mathcal O}(\varepsilon^1)$-part gives us information on elliptic generalisations of multiple polylogarithms
of depth greater than one.
It is worth noting that the ${\mathcal O}(\varepsilon^1)$-part
of the sunrise integral around two space-time dimensions contains terms of weight three and four.
It is not of uniform weight.
We discussed in detail the occurrence of the weight four terms.
Using dimensional-shift relations we expressed the finite part of
the sunrise integral around four space-time dimensions
in terms
of the ${\mathcal O}(\varepsilon^0)$-part and the ${\mathcal O}(\varepsilon^1)$-part
of the sunrise integral around two space-time dimensions.
\subsection*{Acknowledgements}
C.B. thanks Humboldt University for hospitality and support.
\begin{appendix}
\section{The coefficients of the differential equation}
\label{appendix:coeff}
We first recall from \cite{MullerStach:2011ru} the coefficients of the second-order differential equation
for $S_{111}(2,t)$. This differential equation reads
\begin{eqnarray}
\left[ p_2 \frac{d^2}{d t^2} + p_1 \frac{d}{dt} + p_0 \right] S_{111}\left(2,t\right) & = & \mu^2 p_3,
\end{eqnarray}
where $p_0$, $p_1$ $p_2$ and $p_3$ are given by
\begin{eqnarray}
\label{def_p0}
p_2 & = &
t \left( t - \mu_1^2 \right)
\left( t - \mu_2^2 \right)
\left( t - \mu_3^2 \right)
\left( t - \mu_4^2 \right)
\left( 3 t^2 - 2 M_{100} t + \Delta \right),
\nonumber \\
p_1 & = &
9 t^6
- 32 M_{100} t^5
+ \left( 37 M_{200} + 70 M_{110} \right) t^4
- \left( 8 M_{300} + 56 M_{210} + 144 M_{111} \right) t^3
\\
& &
- \left( 13 M_{400} - 36 M_{310} + 46 M_{220} - 124 M_{211} \right) t^2
\nonumber \\
& &
- \left( -8 M_{500} + 24 M_{410} - 16 M_{320} - 96 M_{311} + 144 M_{221} \right) t
\nonumber \\
& &
- \left( M_{600} - 6 M_{510} + 15 M_{420} - 20 M_{330} + 18 M_{411} - 12 M_{321} - 6 M_{222} \right),
\nonumber \\
p_0 & = &
3 t^5
- 7 M_{100} t^4
+ \left( 2 M_{200} + 16 M_{110} \right) t^3
+ \left( 6 M_{300} - 14 M_{210} \right) t^2
\nonumber \\
& &
- \left( 5 M_{400} - 8 M_{310} + 6 M_{220} - 8 M_{211} \right) t
+ \left( M_{500} - 3 M_{410} + 2 M_{320} + 8 M_{311} - 10 M_{221} \right).
\nonumber
\end{eqnarray}
The polynomial $p_3$ appearing in the inhomogeneous part is given by
\begin{eqnarray}
p_3 & = &
-2 \left( 3 t^2 - 2 M_{100} t + \Delta \right)^2
\\
& &
+ 2 c\left(t,m_1,m_2,m_3\right) \ln \frac{m_1^2}{\mu^2}
+ 2 c\left(t,m_2,m_3,m_1\right) \ln \frac{m_2^2}{\mu^2}
+ 2 c\left(t,m_3,m_1,m_2\right) \ln \frac{m_3^2}{\mu^2},
\nonumber
\end{eqnarray}
with
\begin{eqnarray}
\lefteqn{
c\left(t,m_1,m_2,m_3\right) = } & &
\nonumber \\
& &
\left( -2 m_1^2 + m_2^2 + m_3^2 \right) t^3
+ \left( 6 m_1^4 - 3 m_2^4 - 3 m_3^4 - 7 m_1^2 m_2^2 - 7 m_1^2 m_3^2 + 14 m_2^2 m_3^2 \right) t^2
\nonumber \\
& &
+ \left( -6 m_1^6 + 3 m_2^6 + 3 m_3^6 + 11 m_1^4 m_2^2 + 11 m_1^4 m_3^2 - 8 m_1^2 m_2^4 - 8 m_1^2 m_3^4 - 3 m_2^4 m_3^2 - 3 m_2^2 m_3^4 \right) t
\nonumber \\
& &
+ \left( 2 m_1^8 - m_2^8 - m_3^8 - 5 m_1^6 m_2^2 - 5 m_1^6 m_3^2 + m_1^2 m_2^6 + m_1^2 m_3^6 + 4 m_2^6 m_3^2 + 4 m_2^2 m_3^6
\right. \nonumber \\
& & \left.
+ 3 m_1^4 m_2^4 + 3 m_1^4 m_3^4 - 6 m_2^4 m_3^4
+ 2 m_1^4 m_2^2 m_3^2 - m_1^2 m_2^4 m_3^2 - m_1^2 m_2^2 m_3^4 \right).
\end{eqnarray}
In $D$ dimensions the integral $S_{111}(D,t)$ satisfies a fourth-order differential equation:
\begin{eqnarray}
\left[
P_4 \frac{d^4}{dt^4}
+
P_3 \frac{d^3}{dt^3}
+
P_2 \frac{d^2}{dt^2}
+
P_1 \frac{d}{dt}
+
P_0
\right] S_{111}\left(D,t\right)
=
\mu^2
\left[
c_{12} T_{12}
+
c_{13} T_{13}
+
c_{23} T_{23}
\right].
\end{eqnarray}
The coefficients $P_j$ (with $0 \le j \le 4$) read
\begin{eqnarray}
\lefteqn{
P_4
=
8 t^3
\left( t - \mu_1^2 \right) \left( t - \mu_2^2 \right) \left( t - \mu_3^2 \right) \left( t - \mu_4^2 \right)
\left[ \left(7-D\right) t^2 - 2 \left(D-3\right) M_{100} t + \left(13-3D\right) \Delta \right],
} & &
\nonumber \\
\lefteqn{
P_3
=
4 t^2
\left\{
\left( 5 D^2 - 71 D + 252 \right) t^6
- \left( 2 D^2 - 82 D + 500 \right) M_{100} t^5
\right.
} & & \nonumber \\
& & \left.
- \left( 33 D^2 - 343 D + 520 \right) M_{200} t^4
- \left( 14 D^2 - 130 D - 144 \right) M_{110} t^4
\right. \nonumber \\
& & \left.
+ \left( 52 D^2 - 772 D + 2040 \right) M_{300} t^3
- \left( 20 D^2 - 100 D + 312 \right) M_{210} t^3
\right. \nonumber \\
& & \left.
- \left( 152 D^2 -2552 D + 9744 \right) M_{111} t^3
- \left( 13 D^2 - 503 D + 1780 \right) M_{400} t^2
\right. \nonumber \\
& & \left.
+ \left( 36 D^2 - 684 D + 2192 \right) M_{310} t^2
+ \left( 124 D^2 - 1332 D + 4592 \right) M_{211} t^2
\right. \nonumber \\
& & \left.
- \left( 46 D^2 - 362 D + 824 \right) M_{220} t^2
- \left( 18 D^2 + 46 D - 508 \right) M_{500} t
\right. \nonumber \\
& & \left.
+ \left( 54 D^2 + 138 D - 1524 \right) M_{410} t
- \left( 120 D^2 + 904 D - 6288 \right) M_{311} t
\right. \nonumber \\
& & \left.
+ \left( 132 D^2 + 1532 D - 9528 \right) M_{221} t
- \left( 36 D^2 + 92 D - 1016 \right) M_{320} t
\right. \nonumber \\
& & \left.
- 3 D \left( 3 D - 13 \right) \Delta^3
\right\},
\nonumber \\
\lefteqn{
P_2
=
2 t
\left\{
- \left( 9 D^3 - 183 D^2 + 1224 D - 2688 \right) t^6
- \left( 8 D^3 - 62 D^2 - 326 D + 2360 \right) M_{100} t^5
\right.
} & &
\nonumber \\
& & \left.
+ \left( 51 D^3 - 999 D^2 + 5718 D - 9360 \right) M_{200} t^4
- \left( 6 D^3 - 78 D^2 + 828 D - 3696 \right) M_{110} t^4
\right. \nonumber \\
& & \left.
- \left( 24 D^3 - 852 D^2 + 6948 D - 15360 \right) M_{300} t^3
+ \left( 56 D^3 - 852 D^2 + 4516 D - 8928 \right) M_{210} t^3
\right. \nonumber \\
& & \left.
+ \left( 192 D^3 - 4392 D^2 + 31560 D - 71856 \right) M_{111} t^3
- \left( 19 D^3 - 73 D^2 - 1636 D + 6640 \right) M_{400} t^2
\right. \nonumber \\
& & \left.
+ \left( 12 D^3 + 28 D^2 - 2304 D + 8528 \right) M_{310} t^2
+ \left( 20 D^3 + 772 D^2 - 9536 D + 27248 \right) M_{211} t^2
\right. \nonumber \\
& & \left.
+ \left( 14 D^3 - 202 D^2 + 1336 D - 3776 \right) M_{220} t^2
- \left( 114 D^2 - 414 D - 312 \right) M_{500} t
\right. \nonumber \\
& & \left.
+ \left( 342 D^2 - 1242 D - 936 \right) M_{410} t
+ \left( 144 D^3 - 2472 D^2 + 7608 D + 1872 \right) M_{311} t
\right. \nonumber \\
& & \left.
- \left( 288 D^3 - 4260 D^2 + 12732 D + 1872 \right) M_{221} t
- \left( 228 D^2 - 828 D - 624 \right) M_{320} t
\right. \nonumber \\
& & \left.
- 3 D \left(D-2\right) \left(3D-13\right) \Delta^3
\right\},
\nonumber \\
\lefteqn{
P_1
=
\left( 7 D^4 - 175 D^3 + 1610 D^2 - 6440 D + 9408 \right) t^6
} & &
\nonumber \\
& &
+ \left( 14 D^4 - 270 D^3 + 1708 D^2 - 3792 D + 1440 \right) M_{100} t^5
\nonumber \\
& &
- \left( 27 D^4 - 767 D^3 + 7182 D^2 - 27352 D + 36480 \right) M_{200} t^4
\nonumber \\
& &
+ \left( 38 D^4 - 782 D^3 + 6172 D^2 - 22384 D + 31296 \right) M_{110} t^4
\nonumber \\
& &
- \left( 12 D^4 + 52 D^3 - 2712 D^2 + 16688 D - 29760 \right) M_{300} t^3
\nonumber \\
& &
- \left( 20 D^4 - 532 D^3 + 4888 D^2 - 19376 D + 28608 \right) M_{210} t^3
\nonumber \\
& &
- \left( 56 D^4 - 2168 D^3 + 24688 D^2 - 112576 D + 179136 \right) M_{111} t^3
\nonumber \\
& &
+ \left( 17 D^4 - 273 D^3 + 1342 D^2 - 1416 D - 2880 \right) M_{400} t^2
\nonumber \\
& &
- \left( 20 D^4 - 180 D^3 + 184 D^2 + 2592 D - 7488 \right) M_{310} t^2
\nonumber \\
& &
- \left( 140 D^4 - 2220 D^3 + 11656 D^2 - 19680 D - 1728 \right) M_{211} t^2
\nonumber \\
& &
+ \left( 6 D^4 + 186 D^3 - 2316 D^2 + 8016 D - 9216 \right) M_{220} t^2
\nonumber \\
& &
- \left( 2 D^4 - 34 D^3 + 292 D^2 - 1088 D + 1248 \right) M_{500} t
\nonumber \\
& &
+ \left( 6 D^4 - 102 D^3 + 876 D^2 - 3264 D + 3744 \right) M_{410} t
\nonumber \\
& &
+ \left( 104 D^4 - 1128 D^3 + 2992 D^2 + 1440 D - 7488 \right) M_{311} t
\nonumber \\
& &
- \left( 220 D^4 - 2460 D^3 + 7736 D^2 - 3648 D - 7488 \right) M_{221} t
\nonumber \\
& &
- \left( 4 D^4 - 68 D^3 + 584 D^2 - 2176 D + 2496 \right) M_{320} t
\nonumber \\
& &
- D \left(D-2\right) \left(D-4\right) \left(3D-13\right) \Delta^3,
\nonumber \\
\lefteqn{
P_0 =
\left(D-3\right) \left(D-4\right)
\left\{
- \left( D^3 - 21 D^2 + 146 D - 336 \right) t^5
- \left( 3 D^3 - 51 D^2 + 258 D - 360 \right) M_{100} t^4
\right.
} & &
\nonumber \\
& & \left.
+ \left( 2 D^3 - 70 D^2 + 568 D - 1280 \right) M_{200} t^3
- \left( 12 D^3 - 204 D^2 + 1144 D - 2192 \right) M_{110} t^3
\right. \nonumber \\
& & \left.
+ \left( 6 D^3 - 66 D^2 + 120 D + 240 \right) M_{300} t^2
- \left( 6 D^3 - 50 D^2 + 40 D + 336 \right) M_{210} t^2
\right. \nonumber \\
& & \left.
- \left( 28 D^3 - 300 D^2 + 584 D + 1392 \right) M_{111} t^2
- \left( D^3 - 33 D^2 + 182 D - 240 \right) M_{400} t
\right. \nonumber \\
& & \left.
+ \left( 12 D^3 - 148 D^2 + 560 D - 624 \right) M_{310} t
- \left( 12 D^3 - 84 D^2 + 48 D + 144 \right) M_{211} t
\right. \nonumber \\
& & \left.
- \left( 22 D^3 - 230 D^2 + 756 D - 768 \right) M_{220} t
\right. \nonumber \\
& & \left.
+ \left(D-2\right) \left(3D-13\right)
\left[ \left(D-4\right) M_{300} - \left(D-4\right) M_{210} + \left(2D-16\right) M_{111} \right]
\Delta
\right\}.
\end{eqnarray}
In the inhomogeneous term we have
\begin{eqnarray}
\lefteqn{
c_{12} =
\left( D-2 \right)^2 \left(D-4\right)
\left\{
\left( D^2 - 13 D + 42 \right) t^4
\right. } & & \nonumber \\
& & \left.
+ D^2 \left( 2 m_1^2 + 2 m_2^2 + 4 m_3^2 \right) t^3
- D \left( 24 m_1^2 + 24 m_2^2 + 40 m_3^2 \right) t^3
+ \left( 62 m_1^2 + 62 m_2^2 + 76 m_3^2 \right) t^3
\right. \nonumber \\
& & \left.
+ D^2 \left(-4 m_1^4 -4 m_2^4 + 2 m_3^4 + 10 m_1^2 m_3^2 + 10 m_2^2 m_3^2 + 8 m_1^2 m_2^2 \right) t^2
\right. \nonumber \\
& & \left.
+ D \left( 62 m_1^4 + 62 m_2^4 + 22 m_3^4 - 132 m_1^2 m_3^2 - 132 m_2^2 m_3^2 - 108 m_1^2 m_2^2 \right) t^2
\right. \nonumber \\
& & \left.
+ \left(-198 m_1^4 -198 m_2^4 -180 m_3^4 + 450 m_1^2 m_3^2 + 450 m_2^2 m_3^2 + 348 m_1^2 m_2^2 \right) t^2
\right. \nonumber \\
& & \left.
+ D^2 \left( -2 m_1^6 -2 m_2^6 -4 m_3^6 + 2 m_1^4 m_2^2 + 2 m_1^2 m_2^4 -8 m_1^4 m_3^2 -8 m_2^4 m_3^2 + 14 m_1^2 m_3^4 + 14 m_2^2 m_3^4
\right. \right. \nonumber \\
& & \left. \left.
+ 16 m_1^2 m_2^2 m_3^2 \right) t
\right. \nonumber \\
& & \left.
+ 32 D m_3^2 \left( 3 m_1^4 +3 m_2^4 + 2 m_3^4 -5 m_1^2 m_3^2 -5 m_2^2 m_3^2 -4 m_1^2 m_2^2 \right) t
\right. \nonumber \\
& & \left.
+ \left( 42 m_1^6 + 42 m_2^6 -180 m_3^6 -42 m_1^4 m_2^2 -42 m_1^2 m_2^4 -288 m_1^4 m_3^2 -288 m_2^4 m_3^2 + 426 m_1^2 m_3^4
\right. \right. \nonumber \\
& & \left. \left.
+ 426 m_2^2 m_3^4
+ 192 m_1^2 m_2^2 m_3^2 \right) t
\right. \nonumber \\
& & \left.
+ \left( 3 D -13 \right)
\left[
D \left( -m_1^4-m_2^4 + m_3^4 + 2 m_1^2 m_2^2 \right) + 4 m_1^4 + 4 m_2^4 - 6 m_3^4 + 2 m_1^2 m_3^2 + 2 m_2^2 m_3^2
\right. \right. \nonumber \\
& & \left. \left.
- 8 m_1^2 m_2^2
\right]
\Delta
\right\}.
\end{eqnarray}
$c_{13}$ and $c_{23}$ are obtained by permutation of the masses.
\section{The integration constants}
\label{section_integration_constants}
The explicit expressions for the two integration constants $C_1$ and $C_2$ appearing in eq.~(\ref{L2_S_111_1}) are given by
\begin{eqnarray}
C_1
& = &
- \frac{4}{3} \left( M_{300} - M_{210} + 5 M_{111} \right) S_{111}^{(0)}\left(2,0\right)
+ \mu^2 \left[
\frac{2}{3} \Delta
\right. \nonumber \\
& &
+ \frac{2}{3} \left( 4 m_1^4 - m_2^4 - m_3^4 - 3 m_1^2 m_2^2 - 3 m_1^2 m_3^2 + 2 m_2^2 m_3^2 \right) \ln\left(\frac{m_1^2}{\mu^2}\right)
\nonumber \\
& &
+ \frac{2}{3} \left( 4 m_2^4 - m_3^4 - m_1^4 - 3 m_2^2 m_3^2 - 3 m_2^2 m_1^2 + 2 m_3^2 m_1^2 \right) \ln\left(\frac{m_2^2}{\mu^2}\right)
\nonumber \\
& &
+ \frac{2}{3} \left( 4 m_3^4 - m_1^4 - m_2^4 - 3 m_3^2 m_1^2 - 3 m_3^2 m_2^2 + 2 m_1^2 m_2^2 \right) \ln\left(\frac{m_3^2}{\mu^2}\right)
\nonumber \\
& &
- \frac{1}{3} \left( 2 m_1^4 - m_2^4 - m_3^4 - m_1^2 m_2^2 - m_1^2 m_3^2 + 2 m_2^2 m_3^2 \right) \ln^2\left(\frac{m_1^2}{\mu^2}\right)
\nonumber \\
& &
- \frac{1}{3} \left( 2 m_2^4 - m_3^4 - m_1^4 - m_2^2 m_3^2 - m_2^2 m_1^2 + 2 m_3^2 m_1^2 \right) \ln^2\left(\frac{m_2^2}{\mu^2}\right)
\nonumber \\
& &
- \frac{1}{3} \left( 2 m_3^4 - m_1^4 - m_2^4 - m_3^2 m_1^2 - m_3^2 m_2^2 + 2 m_1^2 m_2^2 \right) \ln^2\left(\frac{m_3^2}{\mu^2}\right)
\nonumber \\
& &
- \frac{2}{3} \left( m_1^4 + m_2^4 - 2 m_3^4 + m_1^2 m_3^2 + m_2^2 m_3^2 - 2 m_1^2 m_2^2 \right) \ln\left(\frac{m_1^2}{\mu^2}\right) \ln\left(\frac{m_2^2}{\mu^2}\right)
\nonumber \\
& &
- \frac{2}{3} \left( m_2^4 + m_3^4 - 2 m_1^4 + m_2^2 m_1^2 + m_3^2 m_1^2 - 2 m_2^2 m_3^2 \right) \ln\left(\frac{m_2^2}{\mu^2}\right) \ln\left(\frac{m_3^2}{\mu^2}\right)
\nonumber \\
& & \left.
- \frac{2}{3} \left( m_3^4 + m_1^4 - 2 m_2^4 + m_3^2 m_2^2 + m_1^2 m_2^2 - 2 m_3^2 m_1^2 \right) \ln\left(\frac{m_3^2}{\mu^2}\right) \ln\left(\frac{m_1^2}{\mu^2}\right)
\right],
\end{eqnarray}
\begin{eqnarray}
\lefteqn{
C_2 =
\frac{4}{7 \Delta^2} \left( 7 M_{600} - 44 M_{510} + 113 M_{420} - 77 M_{411} + 121 M_{321} - 152 M_{330} - 258 M_{222} \right) S_{111}^{(0)}\left(2,0\right)
} & &
\nonumber \\
& &
+ \mu^2 \left\{
\frac{1}{7 \Delta} \left( 121 M_{300} - 121 M_{210} + 270 M_{111} \right)
\right. \nonumber \\
& &
+ \left(2 m_1^2 - m_2^2 - m_3^2 \right) \ln^2\left(\frac{m_1^2}{\mu^2} \right)
+ \left(2 m_2^2 - m_3^2 - m_1^2 \right) \ln^2\left(\frac{m_2^2}{\mu^2} \right)
+ \left(2 m_3^2 - m_1^2 - m_2^2 \right) \ln^2\left(\frac{m_3^2}{\mu^2} \right)
\nonumber \\
& &
+ 2 \left( m_1^2 + m_2^2 - 2 m_3^2 \right) \ln\left(\frac{m_1^2}{\mu^2} \right) \ln\left(\frac{m_2^2}{\mu^2} \right)
+ 2 \left( m_2^2 + m_3^2 - 2 m_1^2 \right) \ln\left(\frac{m_2^2}{\mu^2} \right) \ln\left(\frac{m_3^2}{\mu^2} \right)
\nonumber \\
& &
+ 2 \left( m_3^2 + m_1^2 - 2 m_2^2 \right) \ln\left(\frac{m_3^2}{\mu^2} \right) \ln\left(\frac{m_1^2}{\mu^2} \right)
\nonumber \\
& &
+ \frac{2}{7 \Delta^2}
\left[
32 m_1^{10}
+ 55 \left(m_2^2+m_3^2 \right) m_1^8
- \left( 355 m_2^4 + 355 m_3^4 - 400 m_2^2 m_3^2 \right) m_1^6
\right. \nonumber \\
& & \left.
+ 5 \left(m_2^2+m_3^2\right) \left( 83 m_2^4 + 83 m_3^4 - 124 m_2^2 m_3^2 \right) m_1^4
- \left( m_2^2 - m_3^2 \right)^2 \left( 145 m_2^4 + 145 m_3^4 + 546 m_2^2 m_3^2 \right) m_1^2
\right. \nonumber \\
& & \left.
-2 \left( m_2^2 - m_3^2 \right)^4 \left( m_2^2 + m_3^2 \right)
\right] \ln\left(\frac{m_1^2}{\mu^2} \right)
\nonumber \\
& &
+ \frac{2}{7 \Delta^2}
\left[
32 m_2^{10}
+ 55 \left(m_3^2+m_1^2 \right) m_2^8
- \left( 355 m_3^4 + 355 m_1^4 - 400 m_3^2 m_1^2 \right) m_2^6
\right. \nonumber \\
& & \left.
+ 5 \left(m_3^2+m_1^2\right) \left( 83 m_3^4 + 83 m_1^4 - 124 m_3^2 m_1^2 \right) m_2^4
- \left( m_3^2 - m_1^2 \right)^2 \left( 145 m_3^4 + 145 m_1^4 + 546 m_3^2 m_1^2 \right) m_2^2
\right. \nonumber \\
& & \left.
-2 \left( m_3^2 - m_1^2 \right)^4 \left( m_3^2 + m_1^2 \right)
\right] \ln\left(\frac{m_2^2}{\mu^2} \right)
\nonumber \\
& &
+ \frac{2}{7 \Delta^2}
\left[
32 m_3^{10}
+ 55 \left(m_1^2+m_2^2 \right) m_3^8
- \left( 355 m_1^4 + 355 m_2^4 - 400 m_1^2 m_2^2 \right) m_3^6
\right. \nonumber \\
& & \left.
+ 5 \left(m_1^2+m_2^2\right) \left( 83 m_1^4 + 83 m_2^4 - 124 m_1^2 m_2^2 \right) m_3^4
- \left( m_1^2 - m_2^2 \right)^2 \left( 145 m_1^4 + 145 m_2^4 + 546 m_1^2 m_2^2 \right) m_3^2
\right. \nonumber \\
& & \left. \left.
-2 \left( m_1^2 - m_2^2 \right)^4 \left( m_1^2 + m_2^2 \right)
\right] \ln\left(\frac{m_3^2}{\mu^2} \right)
\right\}.
\end{eqnarray}
\end{appendix}
|
{
"timestamp": "2015-04-14T02:16:34",
"yymm": "1504",
"arxiv_id": "1504.03255",
"language": "en",
"url": "https://arxiv.org/abs/1504.03255"
}
|
\section{INTRODUCTION}
The standard physical model of cosmology is based on the solution of
general relativity describing a spatially homogeneous and isotropic
spacetime, known as the Friedmann-Lema\^{i}tre-Robertson-Walker
(FLRW) solution. It is assumed that the geometry of our Universe is
smooth on large scales. One of the major tasks in modern cosmology
is to precisely determine the parameters which characterize the
postulated model by fitting the observational data. The cornerstone
of observational evidence that supports the FLRW model is the
existence of highly isotropic cosmic microwave background radiation
(CMBR). It could be inferred that the spacetime should be exactly
FLRW when the background radiation appears to be exactly isotropic
to a given family of observers~\cite{Ehlers68}. Therefore,we can
prove the Universe to be FLRW just from our own observations of the
CMBR by taking the Copernican principle into consideration.
Moreover, this result could be extended to the case of an almost
isotropic background radiation, which hints at an almost FLRW
spacetime~\cite{Stoeger95}. Although this simple solution of
Einstein field equation provides an excellent description for the
universe on large scales, it also makes clear that we need to
understand the departures from a spatially homogeneous model when
interpreting observational data. Indeed, departures from perfect
homogeneity change the distance-redshift relation. However, in
practice, cosmological observations are usually fitted just using
relationships derived from homogeneous models.
The fact that matter is not continuously distributed can imprint
most cosmological observations probing quantities related to light
propagation(as discussed in detail in Ref.~\cite{Clarkson12}), in
particular regarding the propagation of light with narrow beams,
such as the redshift, the angular diameter distance, the luminosity
distance, and the image distortion. The importance of quantifying
the effects of inhomogeneities on light propagation was first
pointed out by Zel'dovich~\cite{Zeldovich64} and
Kantowski~\cite{Kantowski69}. They designed an ``empty beam"
approximation by arguing that photons should mostly propagate in
vacuum. Later, this was generalized by Dyer and Roeder as the
``partially filled beam" approach~\cite{Dyer72,Dyer73}. More
generally, the early work of Ref.~\cite{Zeldovich64} stimulated many
studies on this issue~\cite{Dashevskii66,
Bertotti66,Gunn67a,Gunn67b,Refsdal70,Weinberg76,Dyer81,Linder88,Fang89,Wux90,Tomita98,Rose01,Kibble05}.
In this framework, the proportion of clumped matter with respect to
the homogeneous fluid is characterized by the clumpiness or
smoothness parameter. In addition, they arrived at an equation for
the angular diameter distance which, via the Etherington relation,
connects to the observable luminosity distance. We refer to it here
as the Zel'dovich-Kantowski-Dyer-Roeder (ZKDR) luminosity distance.
Since the 1960s, a rich literature has formed which concerns the
ZKDR approach and its cosmological implications. Phenomenologies and
investigations involving many different physical aspects were
performed, such as analytical or approximate
expressions~\cite{Kantowski98,Kantowski00,Demianski03}, critical
redshift for the angular diameter distance~\cite{Sereno01},
gravitational lensing~\cite{Covone05,Giovi01}, and accelerated
expanding Universe models driven by particle
creation~\cite{Campos04}. Recently, some quantitative analysis from
such compact radio sources as standard
rulers~\cite{Alcaniz04,Santos08a}, and such type Ia supernovae (SNe
Ia) or gamma-ray bursts (GRBs) as standard
candles~\cite{Santos08b,Busti11,Busti12,Breto13,Yang13,Lima14} were
also performed. To be specific, in Ref.~\cite{Santos08b},
constraints on the dark energy and smoothness parameter from the
so-called gold SN Ia sample released by the High-z Supernova
team~\cite{Riess07} and the first year results of the Supernova
Legacy Survey (SNLS), which is a planned five-year
project~\cite{Astier05}, were examined. The results suggested that
SNe Ia data alone was incapable of constraining the smoothness
parameter although the gold SN Ia provided a little more stringent
constraint since this sample extended to appreciably higher
redshifts. Later, Busti {\it et al.}~\cite{Busti12} performed an
updated investigation where the statistical analysis was based on
the 557 SNe Ia Union2 compilation data~\cite{Amanullah10} and 59
Hymnium GRBs~\cite{Hao10}, and almost the same conclusion was
achieved. More recently, this issue was also studied by using
Union2.1 SN Ia~\cite{Suzuki12} plus nine long GRBs in $1.55\leq
z\leq3.57$~\cite{Tsutsui12} and the constrained value of the
smoothness parameter indicated a clumped Universe~\cite{Breto13}. On
the other hand, as concluded in their work, this result may be an
indication that the ZKDR approximation is not a precise form of
describing the effects of clumpiness in the expanding Universe.
However, in these previous analysis, all distances of SNe Ia and
GRBs applied to test the inhomogeneity of the Universe were derived
from a global fit in the context of standard dark energy scenarios
where the clumpiness has vanished, i.e., the flat $\Lambda$ cold
dark matter ($\Lambda$CDM) or $w$CDM model. That is, the light-curve
fitting parameters accounting for the distance estimation in SNe Ia
observations (e.g., $\alpha$ and $\beta$ in the most widely used
SALT2 training method~\cite{Guy07}) are left as free parameters (on
the same weight as cosmological parameters) and are determined by
fitting the distances of SNe Ia, which is a linear combination of
light-curve fitting parameters and observed quantities, to the
model-predicted ones in the context of the standard $\Lambda$CDM or
$w$CDM scenario. Therefore, HDs constructed in this way are somewhat
model dependent. Moreover, cosmological implications on nonstandard
dark energy scenarios or a Universe with homogeneity taken into
consideration derived from these HDs are model
biased~\cite{Conley11}. It has been shown that this kind of bias
cannot be neglected and may be significant in the era of precision
cosmology~\cite{Zheng12,Zheng14}. Certainly, this kind of bias also
hides in the GRB cosmology where luminosity relations being
responsible for distance estimation of GRB are calibrated with the
model-dependent HDs of low-redshift SNe Ia~\cite{Liang08,Liang10}.
In this paper, we first reconstruct Hubble diagrams for the latest
SNe Ia and for long GRB observations by calibrating the light-curve
fitting parameters and luminosity relations, respectively, in the
context of an inhomogeneous Universe with the cosmological constant.
These Hubble diagrams can lead to unbiased tests for the matter
density parameter $\Omega_m$ as well as the clumpiness parameter
$\eta$. For the joint light-curve analysis of the SDSS-II and the
SNLS (JLA SN Ia) in the range of $0.01\leq
z\leq1.23$~\cite{Betoule14}, the constraints are
$\Omega_m=0.29^{+0.07}_{-0.05}$ and $\eta=0.76^{+0.24}_{-0.65}$,
slightly indicating a clumped Universe. For the long GRBs in the
range of $1.48\leq z\leq8.20$~\cite{Fayin11}, the best fits are
$\Omega_m=0.42\pm0.06$ and $\eta=1.00^{+0.00}_{-0.12}$, strongly
supporting a homogeneous Universe. For the combination of these two
probes, the constraints are $\Omega_m=0.34\pm0.02$ and
$\eta=1.00^{+0.00}_{-0.02}$, also favoring a universe full of FLRW
fluid with a very high confidence level. We suggest that the matter
density parameter $\Omega_m$ is mainly determined by the SNe Ia
observations while the clumpiness parameter $\eta$ is primarily
constrained from the observed GRB events. Moreover, it is also shown
that larger scales are explored, the test more strongly implies a
homogeneous Universe. These reasonable results may be an indication
that the ZKDR approximation remains to be a precise description for
the luminosity distance-redshift relation in a locally inhomogeneous
Universe with the cosmological constant.
\section{\bf THE ZKDR LUMINOSITY DISTANCE}
For most cosmological models, angular or apparent size distance,
which is proportional to the square root of the cross-sectional area
$A(z)$, is related to the luminosity distance by
$d_\mathrm{A}(z)=d_\mathrm{L}(z)/(1+z)^2$. In the model only
including dark matter and dark energy, the luminosity distance
$d_\mathrm{L}(z)$, which accounts for a partially depleted mass
density in the observing beam but neglects lensing by external
masses, is obtained by integrating the second-order differential
equation for $A(z)$ of an observing beam from the source at redshift
$z$ to the observer at $z=0$~\cite{Kantowski98,Kantowski01}:
\begin{equation}\label{eq1}
(1+z)^2E(z)\frac{d}{dz}\bigg[(1+z)^2E(z)\frac{d}{dz}\sqrt{A(z)}\bigg]+\frac{3}{2}\eta\Omega_m(1+z)^5\sqrt{A(z)}=0,
\end{equation}
where $E(z)$ is the reduced Hubble parameter at redshift $z$
\begin{equation}\label{eq2}
E(z)=\frac{H(z)}{H_0}=(1+z)\sqrt{1+\Omega_mz+\Omega_\Lambda[(1+z)^{-2}-1]},
\end{equation}
and the phenomenological parameter $\eta=1-\rho_{\mathrm{cl}}/\rho$
is the so-called clumpiness or smoothness parameter which quantifies
the amount of matter in clumps relative to the amount of matter
uniformly distributed. The required boundary conditions for
Eq.~(\ref{eq1}) are
\begin{equation}\label{eq3}
\sqrt{A}\mid_{z=0}=0,~~~~~~~~~~~~\frac{d\sqrt{A}}{dz}\mid_{z=0}=-\sqrt{\delta\Omega}\frac{c}{H_0},
\end{equation}
where $\delta\Omega$ is the solid angle of the beam. By using an
approximate change of variables
\begin{eqnarray}\label{eq4}
h(A,z)&\equiv&(1+z)\sqrt{\frac{A}{\delta\Omega}},\\
\zeta(z)&=&\frac{\Omega_m}{1-\Omega_m}(1+z)^3+1,
\end{eqnarray}
Eq.~(\ref{eq1}) can be transformed into a hypergeometric equation
\begin{equation}\label{eq5}
(1-\zeta)\zeta\frac{d^2h}{d\zeta^2}+\big(\frac{1}{2}-\frac{7}{6}\zeta\big)\frac{dh}{d\zeta}+\frac{\nu(\nu+1)}{36}=0.
\end{equation}
The resulting luminosity distance is then given by
\begin{equation}\label{eq7}
d_L(z)=(1+z)h(\zeta(0)).
\end{equation}
Expressed in terms of hypergeometric functions, Eq.~(\ref{eq7})
becomes
\begin{eqnarray}\label{eq8}
d_L(z;\Omega_m,\nu)=&\frac{c}{H_0}&\frac{2(1+z)}{\Omega_m^{1/3}(1+2\nu)}[1+\Omega_mz(3+3z+z^2)]^{\nu/6}\nonumber\\
&\times&\bigg\{~_2F_1\bigg(-\frac{\nu}{6},\frac{3-\nu}{6};\frac{5-2\nu}{6};\frac{1-\Omega_m}{1+\Omega_mz(3+3z+z^2)}\bigg)\nonumber\\
&\times&_2F_1\bigg(\frac{1+\nu}{6},\frac{4+\nu}{6};\frac{7+2\nu}{6};1-\Omega_m\bigg)\nonumber\\
&-&[1+\Omega_mz(3+3z+z^2)]^{-(1+2\nu)/6}~_2F_1\bigg(-\frac{\nu}{6},\frac{3-\nu}{6};\frac{5-2\nu}{6};1-\Omega_m\bigg)\nonumber\\
&\times&_2F_1\bigg(\frac{1+\nu}{6},\frac{4+\nu}{6};\frac{7+2\nu}{6};\frac{1-\Omega_m}{1+\Omega_mz(3+3z+z^2)}\bigg)\bigg\}.
\end{eqnarray}
The parameter $\nu$ presented in Eqs.~(\ref{eq5}) and~\ref{eq8}
corresponds to the clumpiness parameter $\eta$ by
\begin{equation}\label{eq9}
\eta=\frac{1}{6}(3+\nu)(2-\nu).
\end{equation}
The range for $\nu$ is $0\leq\nu\leq2$, where $\nu=0(\eta=1)$ is
related to a FLRW fluid, while $\nu=2(\eta=0)$ to a totally clumped
case.
Actually, the ZKDR approach has been criticized by several authors
(e.g., a few detailed comments gathered in Ref.~\cite{Breto13}).
However, so far, confrontations of the ZKDR luminosity distance with
observations have not led to conclusive results in the sense of
totally excluding this model. Moreover, we should keep in mind that
most previous tests in this field were somewhat dependent on the
standard dark energy model (the flat $\Lambda$CDM or $w$CDM).
Therefore, it is necessary to clarify the validity and the scope of
the ZKDR luminosity distance in describing the Universe in a
model-unbiased way. Here, we follow the simplest treatment, where
$\eta$ is assumed to be a constant.
\section{\bf SAMPLES AND RESULTS}
We carry out analysis by using the latest observations of standard
candles, including the joint light-curve analysis of the SDSS-II and
SNLS supernova samples~\cite{Betoule14}--which is referred to as JLA
SN Ia in the literature--and the long gamma-ray bursts reported in
Ref.~\cite{Fayin11}. Descriptions for the samples, methodology, and
results are presented in this section.
\subsection{\bf Type Ia supernovae}
The cosmic acceleration was discovered 16 years ago by measuring
accurate distances to distant SNe
Ia~\cite{Riess98,Schmidt98,Perlmutter99}. The reason for the
acceleration remains uncertain and a large experimental effort in
observational cosmology has been driven to reveal the mechanism of
this ostensibly counterintuitive phenomenon. By precisely mapping
the distance-redshift relation up to redshift $z\approx1$, SNe Ia
remain, at this stage, the most promising probe of the late-time
history of the Universe. Because of the variability of the large
spectra features, distance estimation for SNe Ia is based on the
empirical observation that these events form a homogeneous class
whose remaining variability is reasonably well captured by two
parameters~\cite{Tripp98}. One of them characterizes the stretching
of the light curve ($X_1$ in what follows), and the other describes
the color at maximum brightness ($\mathcal{C}$ in what follows).
With the assumption that SNe Ia at all redshifts with the identical
color, shape and galactic environment have, on average, the same
intrinsic luminosity, the distance estimator (distance modulus:
$\mu=5\log\big[\frac{d_\mathrm{L}}{\mathrm{Mpc}}\big]+25$) used in
most cosmological analysis is quantified by a linear model,
\begin{equation}\label{eq10}
\mu_\mathrm{B}(\alpha, \beta; M)=m_\mathrm{B}^*-M+\alpha\times
X_1-\beta\times\mathcal{C},
\end{equation}
where $m_\mathrm{B}^*$ is the observed peak magnitude in the
rest-frame $B$ band, and $\alpha$ and $\beta$ are nuisance
parameters which characterize the stretch-luminosity and
color-luminosity relationships, corresponding to the well-known
broader-brighter and bluer-brighter relationships, respectively. The
value of $M$ is another nuisance parameter representing the absolute
magnitude of a fiducial SNe Ia. In general, $\alpha$ and $\beta$ are
left as free parameters (on the same weight as cosmological
parameters) that are determined in the global fit in the context of
standard dark energy scenario to construct the Hubble diagram for
SNe Ia. It should be noted that cosmological implications derived
from this Hubble diagram for other nonstandard models, which are
different from the standard $\Lambda$CDM (or $w$CDM) scenario used
to carry out the global fit, are model biased.
In order to achieve model-unbiased constraints on the clumpiness of
the Universe, we should fit the light-curve fitting parameters
($\alpha$ and $\beta$) and the model parameters ($\Omega_m$ and
$\nu$) simultaneously to construct a Hubble diagram of SNe Ia in an
inhomogeneity-allowed scenario by confronting the distances
estimated from SNe Ia observations via Eq.~(\ref{eq10}) with the
ones predicted from the ZKDR luminosity distance model,
\begin{equation}\label{eq11}
\mu_{\mathrm{mod}}(z;\mathbf{\theta_1},\mu_0)=5\log_{10}[D_\mathrm{L}(z;\mathbf{\theta_1})]+\mu_0.
\end{equation}
Here $D_\mathrm{L}$ is the Hubble-constant free luminosity distance,
$\mathbf{\theta_1}$ represents the model parameter vector
$(\Omega_m,\nu)$ and $\mu_0=5\log_{10}[c/H_0]+25$. For the latest
JLA SN Ia, the standard $\chi^2$ function is given by
\begin{equation}\label{eq12}
\chi^2(\mu_0,M;\mathbf{\theta_1},\mathbf{\theta_2})=\sum_{i=1}^{740}\frac{[\mu_{\mathrm{mod}}(z_i;
\mathbf{\theta_1},\mu_0)-\mu_{\mathrm{B},i}(\mathbf{\theta_2};M)]^2}{\sigma_{\mu,i}^2},
\end{equation}
where $\mathbf{\theta_2}$ denotes the vector of light-curve fitting
parameters $(\alpha,\beta)$ and $\sigma_{\mu,i}$ is the error on the
distance modulus for the $i$th SNe Ia. It should be noted that we
take only the statistical uncertainties into account and they are
also dependent on the light-curve fitting parameters. In order to
marginalize over the nuisance parameters, $H_0$ and $M$, we expand
the $\chi^2$ function with respect to $\widetilde{\mu_0}=\mu_0+M$
as~\cite{Pietro03,Nesseris05,Perivolaropoulos05}
\begin{equation}\label{eq13}
\chi^2(\mathbf{\theta_1},\mathbf{\theta_2};\widetilde{\mu_0})=A-2\widetilde{\mu_0}B+\widetilde{\mu_0}^2C,
\end{equation}
where
\begin{eqnarray}\label{eq14}
A(\mathbf{\theta_1},\mathbf{\theta_2})&=&\sum_{i=1}^{740}\frac{[\mu_{\mathrm{mod}}(z_i;
\mathbf{\theta_1},\mu_0=0)-\mu_{\mathrm{B},i}(\mathbf{\theta_2};M=0)]^2}{\sigma_{\mu,i}^2}\;,\\
B(\mathbf{\theta_1},\mathbf{\theta_2})&=&\sum_{i=1}^{740}\frac{[\mu_{\mathrm{mod}}(z_i;
\mathbf{\theta_1},\mu_0=0)-\mu_{\mathrm{B},i}(\mathbf{\theta_2};M=0)]}{\sigma_{\mu,i}^2}\;,\\
C(\mathbf{\theta_2})&=&\sum_{i=1}^{740}\frac{1}{\sigma_{\mu,i}^2}\;.
\end{eqnarray}
Equation~(\ref{eq13}) has a minimum at $\widetilde{\mu_0}=B/C$, and
it is
\begin{equation}\label{eq15}
\widetilde{\chi}^2(\mathbf{\theta_1},\mathbf{\theta_2})=A-\frac{B^2}{C}.
\end{equation}
Therefore, we can minimize
$\widetilde{\chi}^2(\mathbf{\theta_1},\mathbf{\theta_2})$ to get rid
of the dependence on nuisance parameters.
The constraint on the light-curve fitting parameters vector is
presented in Fig.~\ref{Fig1}. The best fit value is $(\alpha,~
\beta)=(0.13,~3.17)$, which is marginally compatible with the result
estimated in the flat $\Lambda$CDM at a 1$\sigma$ confidence level.
By applying a minimization of $\widetilde{\chi}^2$, we can get an
estimation for $\widetilde{\mu_0}$ which is a combination of $H_0$
and $M$. Here, we break the degeneracy by fixing
$H_0=70~\mathrm{km~s}^{-1}~\mathrm{Mpc}^{-1}$ and obtain $M=-19.08$.
With the constraint on ($\alpha,~\beta$) and estimation of $M$, an
indicative Hubble diagram in the framework of the ZKDR luminosity
distance model is constructed and shown in Fig.~\ref{Fig2}.
Moreover, results for confidence regions constrained in the
($\Omega_m,~\nu$) plane are presented in Fig.~\ref{Fig3} and
Tab.~\ref{Tab2}. We suggest that the clumpiness parameter $\eta$ is
poorly constrained, being bounded on the interval
$0.16\leq\eta\leq1.00$ within a 1$\sigma$ confidence level. However,
a tighter constraint is obtained for the matter density parameter
$\Omega_m$, being restricted on the interval
$0.25\leq\Omega_m\leq0.37$(1$\sigma$). These are very similar to
what was obtained in previous analyses~\cite{Santos08a,Busti12}, but
quite different from the results included in Ref.~\cite{Breto13}.
That is, our unbiased tests slightly indicate an inhomogeneity and
the standard FLRW cosmology is consistent with SNe Ia observations
within a 1$\sigma$ confidence level.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.60\textwidth, height=0.45\textwidth]{figsnlc.eps}
\caption{\label{Fig1} Constraints on the light-curve fitting
parameters, $\alpha$ and $\beta$, from the global fit in the context
of a clumpy Universe. The triangle and star represent the best fits
when the ZKDR approximation and the standard $\Lambda$CDM framework
are considered, respectively.}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.60\textwidth, height=0.45\textwidth]{fighd.eps}
\caption{\label{Fig2} Hubble diagram of the standard candles
constructed from the global fit in the context of a clumpy Universe.
The distance modulus redshift relation of the best-fit ZKDR
approximation for a fixed
$H_0=70~\mathrm{km}~\mathrm{s}^{-1}~\mathrm{Mpc}^{-1}$ is shown as
the solid line.}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.60\textwidth, height=0.45\textwidth]{figsnfiled.eps}
\caption{\label{Fig3} Confidence regions in the ($\Omega_m$, $\nu$)
plane for the model with a ZKDR luminosity distance constrained from
the JLA SN Ia.}
\end{figure}
\subsection{\bf Long gamma-ray bursts}
Gamma-ray bursts (GRBs), which are the most intensive explosions
observed in the Universe and thus are visible across much larger
distances than SNe Ia, are deemed as a potential probe to explore
the Universe at higher redshift, a redshift of at least $6$ and up
to even $z=10$~\cite{Lamb2000, Bromm2002, Lin2004, Krimm2006}.
Specifically, relations between the luminosity/energy and the
measurable properties of the prompt gamma-ray emission imply that
GRBs may be appropriate candidates for cosmological standard
candles. In the past few years, several empirical luminosity
relations have been statistically inferred from observations. For
instance, several two-variable relations: the relation between
spectral lag and luminosity
($\tau_{\mathrm{lag}}-L$)~\cite{Norris2000}, the relation between
variability and luminosity ($V-L$)~\cite{Fenimore2000,
Reichart2001}, the relation between peak spectral energy and
luminosity ($E_\mathrm{peak}-L$)~\cite{Schaefer2003, Yonetoku2004},
the relation between peak spectral energy and collimation-corrected
energy ($E_\mathrm{peak}-E_\gamma$)~\cite{Ghirlanda2004}, the
relation between the minimum raising time in the GRB light curve and
luminosity ($\tau_{\mathrm{RT}}-L$)~\cite{Schaefer2007}, and the
relation between peak spectral energy and isotropic energy
($E_{\mathrm{peak}}-E_\mathrm{\gamma,iso}$)~\cite{Amati2002}--have
been successfully deduced from observations. Meanwhile, a few
multivariable relations have also been obtained, such as the
connection between $E_{\mathrm{iso}}$, $E_\mathrm{peak}$, and the
break time of the optical afterglow light curves
($t_\mathrm{b}$)~\cite{Liang2005}, the correlation between the
luminosity, $E_\mathrm{peak}$, and the rest-frame ``high-signal"
time scale ($T_{0.45}$)~\cite{Firmani2006}. Moreover, these
luminosity relations have been proposed to calibrate GRBs as
distance indicators (see, e.g.,
Refs.~\cite{Ghirlanda2006a,Schaefer2007} for reviews).
In particular, in Refs.~\cite{Busti12,Breto13}, distances of GRBs
used to constrain the clumpiness of the Universe are obtained by
calibrating their luminosity relations with low-redshift SNe
Ia~\cite{Liang08,Hao10,Tsutsui12}. However, it is necessary to make
clear that distances of SNe Ia quoted to calibrate luminosity
relations are estimated from a global fit in the frame of a standard
dark energy model. In other words, the distances of GRBs given in
Refs.~ \cite{Liang08,Hao10,Tsutsui12} are still somewhat dependent
on the standard dark energy model and thus subsequent tests for the
inhomogeneity of the Universe derived from them are model biased. In
this work, we construct the Hubble diagram of 116 long
GRBs~\cite{Fayin11} in the framework of an inhomogeneous Universe by
calibrating their luminosity/energy relations in the global fit
where the context of the ZKDR luminosity distance model is
considered. This Hubble diagram can then lead to an unbiased
examination of the clumpiness of the Universe. In
Ref.~\cite{Fayin11}, six luminosity correlations
($\tau_{\mathrm{lag}}-L$, $V-L$, $E_{\mathrm{peak}}-L$,
$E_{\mathrm{peak}}-E_{\gamma}$, $\tau_{\mathrm{RT}}-L$,
$E_{\mathrm{peak}}-E_{\gamma, \mathrm{iso}}$) have been derived from
the latest observations of 116 long GRBs. In their work, it was also
found that the intrinsic scatter of the $V-L$ correlation was too
large to infer an inherent correlation between these two quantities
using the currently observed GRB events. What is more, the
luminosity correlations $E_{\mathrm{peak}}-E_{\gamma}$ and
$E_{\mathrm{peak}}-E_{\gamma, \mathrm{iso}}$ mirror almost the same
physics, we should include one of them to avoid strong correlation
among the luminosity correlations. Therefore, we choose the
$E_{\mathrm{peak}}-E_{\gamma}$ correlation, which has a smaller
intrinsic scatter, and then use the rest four correlations for the
following analysis. The same as previous works that derived
cosmological implications from GRBs, we use only the subsample at
$z>1.4$ for the complimentary redshift range to the SN Ia.
The remaining four luminosity correlations involved in this paper
are
\begin{align}
\label{eq:GRB-lag-L}
\log \frac{L}{1 \; \mathrm{erg} \; \mathrm{s}^{-1}}
&= a_1+b_1 \log
\left[
\frac{\tau_{\mathrm{lag}}(1+z)^{-1}}{0.1\;\mathrm{s}}
\right]
,
\end{align}
\begin{align}
\label{eq:GRB-E_peak-L}
\log \frac{L}{1 \; \mathrm{erg} \; \mathrm{s}^{-1}}
&= a_2+b_2 \log
\left[
\frac{E_{\mathrm{peak}}(1+z)}{300\;\mathrm{keV}}
\right]
,
\end{align}
\begin{align}
\label{eq:GRB-E_peak-E_gamma}
\log \frac{E_{\gamma}}{1\;\mathrm{erg}}
&= a_3+b_3 \log
\left[
\frac{E_{\mathrm{peak}}(1+z)}{300\;\mathrm{keV}}
\right]
,
\end{align}
\begin{align}
\label{eq:GRB-tau_RT-L}
\log \frac{L}{1 \; \mathrm{erg} \; \mathrm{s}^{-1}}
&= a_4+b_4 \log
\left[
\frac{\tau_{\mathrm{RT}}(1+z)^{-1}}{0.1\;\mathrm{s}}
\right]
,
\end{align}
where $a$ and $b$ are the intercept and the slope of the relation,
respectively. In these correlations, the isotropic peak luminosity
$L$ is given by
\begin{equation}\label{eq:GRB-L-P_bolo}
L=4 \pi d_L^2 P_{\mathrm{bolo}},
\end{equation}
where $P_{\mathrm{bolo}}$ is the bolometric flux of gamma rays in
the burst. The isotropic energy released in a burst is
\begin{equation}\label{eq:GRB-E_iso-S_bolo}
E_{\gamma,\mathrm{iso}}=4 \pi d_L^2 S_{\mathrm{bolo}} (1+z)^{-1},
\end{equation}
where $S_{\mathrm{bolo}}$ is the bolometric fluence of gamma rays in
the burst at redshift $z$. The total collimation-corrected energy
can be calculated by
\begin{equation}\label{eq:GRB-E_gamma-S_bolo}
E_{\gamma}= E_{\gamma,\mathrm{iso}} (1-\cos\theta_{\mathrm{jet}}),
\end{equation}
where $\theta_{\mathrm{jet}}$ is the opening angle of the jet.
In order to completely avoid any circularity and obtain
model-unbiased constraints on the clumpiness of the Universe from
GRBs~\cite{Schaefer2003,Schaefer2007}, we separately calibrate each
luminosity relation, Eqs.~\ref{eq:GRB-lag-L}-\ref{eq:GRB-tau_RT-L},
by carrying out a similar simultaneous global fitting route
presented in the above subsection. Results are shown in
Tab.~\ref{Tab1}. Here, $\sigma_{\mathrm{int}}$ is the systematic
error and it can be estimated by finding the value such that an
$\chi^2$ fit to each relation calibration curve produces a value of
reduced $\chi^2$ of unity~\cite{Schaefer2007}. This quantity
accounts the extra scatter of the luminosity relations. In this
global fitting route, we marginalize the nuisance parameter Hubble
constant by fixing $H_0=70~\mathrm{km~s}^{-1}~\mathrm{Mpc}^{-1}$.
Following the method about uncertainty calculation and distance
estimation from calibrated luminosity
relations~\cite{Schaefer2007,Liang08}, as shown in Fig.~\ref{Fig2},
we construct a Hubble diagram of GRBs in the context of the ZKDR
luminosity distance scenario. In addition, results concerning the
constraints on model parameters are presented in Fig.~\ref{Fig4} and
Tab.~\ref{Tab2}. It is suggested that a Universe composed only by
homogeneously distributed matter is strongly favored by GRB
observations. This is greatly different from what was obtained in
previous works~\cite{Busti12,Breto13}.
Finally, we perform a joint analysis from the combination of JLA SN
Ia and long GRBs. Results are displayed in Fig.~\ref{Fig5} and
Tab.~\ref{Tab2}. Within a 1$\sigma$ confidence level, the matter
density parameter is restricted in the interval
$0.32\leq\Omega_m\leq0.36$ and the smoothness parameter is bounded
in the interval $0.98\leq\eta\leq1.00$. It is shown that the
constraint on the matter density parameter is mainly dependent on
SNe Ia observations while the estimation of the smoothness parameter
is basically determined by the long GRBs. The fact that high
redshift GRBs prefer a homogeneous Universe with a great
significance of probability can be understood as follows: they
explore much larger scales of the Universe and should contribute to
diminishing the corresponding space parameter. That is, since the
Universe is more homogeneous on larger scales (a higher redshift),
higher value of the smoothness parameter $\eta$ is favored. In
addition, it should be noted that, although large redshift GRBs are
very important for the tests of the clumpiness parameter, there are
only four GRBs at redshift larger than $5$.
\begin{table}[!h]
\begin{tabular}{|c|c|c|c|c|}
\hline
~Luminosity relation ~&~~$a(1\sigma)$~~&~~$b(1\sigma)$ ~~&~~$\sigma_{\mathrm{int}}$~~&~~$N(\mathrm{z}_{\mathrm{GRB}}>1.4)$~~\\
\hline
$\tau_{\mathrm{lag}}-L$ &~~ $52.60\pm0.04$~~&~~$-0.76\pm0.06$~~&~~0.12~~&~~26~~\\
\hline
$E_{\mathrm{peak}}-L$ &~~ $52.10\pm0.04$~~&~~$1.38\pm0.12$~~&~~0.16~~&~~62~~\\
\hline
$E_{\mathrm{peak}}-E_\gamma$ &~~ $50.36\pm0.07$~~&~~$1.56\pm0.20$~~&~~0.01~~&~~12~~\\
\hline
$\tau_{\mathrm{RT}}-L$ &~~ $52.95\pm0.05$~~&~~$-1.03\pm0.13$~~&~~0.16~~&~~36~~\\
\hline
\end{tabular}
\tabcolsep 0pt \caption{\label{Tab1} Summary of the constraints on
luminosity relations of GRBs from the global fit in the context of a
clumpy Universe.} \vspace*{5pt}
\end{table}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.60\textwidth, height=0.45\textwidth]{figgrbfiled.eps}
\caption{\label{Fig4} Confidence regions in the ($\Omega_m$, $\nu$)
plane for the model with a ZKDR luminosity distance constrained from
the long GRBs.}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.60\textwidth, height=0.45\textwidth]{figtotfiled.eps}
\caption{\label{Fig5} Confidence regions in the ($\Omega_m$, $\nu$)
plane for the model with a ZKDR luminosity distance constrained from
the combination of JLA SN Ia and long GRBs.}
\end{figure}
\begin{table}[!h]
\begin{tabular}{|c|c|c|c|}
\hline
~Sample ~&~~$\Omega_m(1\sigma)$~~&~~$\nu(1\sigma)$ ~~&~~$\eta(1\sigma)$~~\\
\hline
~~JLA SN Ia~~&~~ $0.25\leq\Omega_m\leq0.37$~~&~~$0.00\leq\nu\leq1.80$~~&~~$0.16\leq\eta\leq1.00$~~\\
\hline
~~Long GRBs~~&~~ $0.38\leq\Omega_m\leq0.49$~~&~~$0.00\leq\nu\leq0.48$~~&~~$0.88\leq\eta\leq1.00$~~\\
\hline
~~Joint analysis~~&~~ $0.32\leq\Omega_m\leq0.36$~~&~~$0.00\leq\nu\leq0.12$~~&~~$0.98\leq\eta\leq1.00$~~\\
\hline
\end{tabular}
\tabcolsep 0pt \caption{\label{Tab2} Summary of the unbiased
constraints on model parameters in the ZKDR luminosity distance from
observations of standard candles. } \vspace*{5pt}
\end{table}
\section{CONCLUSIONS AND DISCUSSIONS}
In the era of precision cosmology, where one aims at determining the
cosmological parameters at the percent level, distance estimations
for standard candles and rulers with increasing accuracy are
expected to provide powerful constraints on dark energy or other
fundamental dynamical parameters. However, it is necessary to be
aware of the physical hypothesis underlying these probes when we
proceed with such a program. As far as we know, the Universe is
effectively inhomogeneous at least in the small-scale domain.
Furthermore, notice that even the large-scale homogeneity also has
been challenged~\cite{Uzan08}. In this topic, the method based on
the ZKDR luminosity distance is a simple alternative and is usually
applied to quantitatively assessing the influences of the clumpiness
on the light propagation. In the past few years, there has been a
rich literature concerning the constraints on the smoothness
parameter from observations of standard
candles~\cite{Kantowski01,Santos08b,Busti12,Breto13,Lima14}.
However, we should keep in mind that distances of SNe Ia applied to
test the inhomogeneity were estimated from the global fit in the
context of a standard homogeneous dark energy model, i.e., the flat
$\Lambda$CDM or $w$CDM model. Therefore, in these previous analyses,
constraints on the smoothness parameter from the distance modulus of
SNe Ia were somewhat model biased. Meanwhile, results obtained from
GRBs suffered the same problem since the distances of them were
determined by calibrating luminosity relations with low-redshift SNe
Ia.
In this paper, we first construct Hubble diagrams for SNe Ia and
GRBs by calibrating the light-curve fitting parameters and
luminosity relations, respectively, in the global fit where the
context of the ZKDR luminosity distance model is taken into account.
And then, these Hubble diagrams can lead to unbiased tests for the
inhomogeneity of the Universe. For the JLA SN Ia, as shown in
Fig.~\ref{Fig3}, constraint on the smoothness parameter is not
stringent and slightly implies a locally inhomogeneous background,
while the matter density parameter is well constrained, being
bounded in the interval $0.25\leq\Omega_m\leq0.37$(1$\sigma$). For
the long GRBs, as shown in Fig.~\ref{Fig4}, the Universe with matter
uniformly distributed is favored with a high confidence level. This
is completely different from what was obtained in
Ref.~\cite{Breto13}. Finally, we perform a joint analysis which
provides good constraints on both model parameters. At a 1$\sigma$
confidence level, the intervals are $0.32\leq\Omega_m\leq0.36$ and
$0.98\leq\eta\leq1.00$. It is suggested that the constraint on the
matter density parameter is mainly based on the observations of
low-redshift SNe Ia, while the test for the clumpiness parameter is
primarily determined from the observations of high-redshift GRBs.
Just as expected, the investigation on the inhomogeneity was very
sensitive to the scales explored by the observations, i.e., the
Universe should be more homogeneous on larger scales. These also may
be an indication that the ZKDR approximation remains to be a precise
description for the luminosity distance-redshift relation in a
locally inhomogeneous Universe with the cosmological constant.
Frankly, it should be pointed out that constraints on the model
parameters from low-redshift SNe Ia and high-redshift GRBs are
somewhat inconsistent. This inconsistency may imply that the
assumption with the smoothness parameter $\eta$ being a constant is
not accurate enough to fit the practical observations. That is, the
smoothness parameter $\eta$ might evolve with cosmic time (or
redshift). Moreover, the intrinsic scatters in GRB observations may
also lead to this tension. Therefore, in the near future, a more
precise and larger sample of high-redshift GRB data (even some other
distance measurements with new methods, e.g., extremely luminous
active galactic nuclei readily observed over a range of distances
from $\sim10~\mathrm{Mpc}$ to $z>7$~\cite{Watson2011, Wang2013,
Yoshii2014}) and a plausible extension of the ZKDR approach are
expected to perform more accurate tests for the inhomogeneity and
contribution of matter in the Universe.
\section*{Acknowledgments}
We are grateful to the anonymous referee for his or her helpful
comments. This work was supported by the Ministry of Science and
Technology National Basic Science Program (Project 973) under Grants
No. 2012CB821804 and No. 2014CB845806, the Strategic Priority
Research Program ``The Emergence of Cosmological Structure" of the
Chinese Academy of Sciences (No. XDB09000000), the National Natural
Science Foundation of China under Grants No. 11373014 and No.
11073005, the China Postdoc Grant No. 2014T70043, and the
Fundamental Research Funds from the Central Universities and
Scientific Research Foundation of Beijing Normal University.
|
{
"timestamp": "2015-04-28T02:12:32",
"yymm": "1504",
"arxiv_id": "1504.03482",
"language": "en",
"url": "https://arxiv.org/abs/1504.03482"
}
|
\section*{Introduction}
\label{intro}
Since the publication of their results by the PAMELA collaboration~\citep{Adriani2009,Adriani2013}, the positron fraction \textit{i.e.} the flux of cosmic-ray positrons divided by the flux of electrons and positrons, has attracted a lot of interest. Indeed, PAMELA observed a raise of this quantity together with the cosmic-ray energy between 10 and 200~GeV which has been confirmed by AMS-02~\citep{Aguilar2013}.
The AMS-02 experiment should have the ability to measure the positron fraction at even higher energies. What ever is the correct explanation for this rise, the positron fraction must either saturate or decline. In the latter case, how abrupt a decline might we expect? The AMS-02 collaboration is prone to explain that a sharply falling positron fraction would be a smoking gun for Dark Matter (see for instance the AMS-02 press conference of September 2014\footnote{\url{http://goo.gl/sf71o6}} or Manuela Vecchi's presentation at SUGAR 2015). In this work we studied whether this affirmation was motivated by any solid scientific argument.
Because stars do not contain anti-matter, positrons, like anti-protons or anti-deuterons, that we find in cosmic rays, are expected to be produced as {\it secondary} particles by cosmic ray nuclei while they propagate and interact in the interstellar medium (ISM). It is now clear that the increase observed in the positron fraction cannot be explained by the simplest models of secondary production. Various alternatives have been proposed, such as a modification of the propagation model~\citep{Katz2009,Blum2013}, or primary positron production scenarios, with pulsars~(\textit{e.g.}, \citealp{Grasso09,Hooper09,Delahaye2010,Blasi11,Linden13}) or Dark Matter annihilation~(\textit{e.g.}, \citealp{Delahaye2008,Arkani-Hamed09,Cholis09,Cirelli09}) as sources. As of today, it is not possible to conclude which explanation is the correct one because they all suffer from theoretical uncertainties and because the current data cannot lift degeneracies. Finding a way to discriminate among these explanations has been looked for by variuos authors (see for instance \citealp{Ioka2010,Kawanaka2010,Pato2010,Mauro2014}); here we want to test more specifically the possibility of a sharp drop of the positron fraction. An original aspect of our work is to also convolve our results with the cosmic-ray production parameter space for pulsars allowed by theory. We investigate the following question: what constraints could we put on Dark Matter annihilation and primary pulsar scenarios if the next AMS-02 data release were to show a sharply dropping positron fraction?
According the AMS-02 collaboration, a sharp drop can only be explained if the positron excess originates from the annihilation of Dark Matter particles with a mass of several hundred GeV. However, we show here that such a feature would be highly constraining in terms of Dark Matter scenarios.
In fact pulsar models could lead to a sharp fall of the positron fraction at the cost of some parameter tuning.
In this proceedings we summarize the results of our earlier work~\citep{Delahaye2014}; readers interested in the details of the method are advised to consult that earlier reference.
\section*{Positron flux morphology}
\label{sec-1}
If it is quite straightforward that the positron flux coming from Dark Matter cannot reach energies higher than the mass of the Dark Matter particle (or even half this quantity in case of a decaying Dark Matter); if the injection has a rather sharp shape like in the case of annihilation into a electron-positron pair, one can expects the flux after propagation to be quite sharp too. However, the morphology of the positron flux due to a bursting source spatially located, like a pulsar, can be less intuitive. Figure~\ref{fig-1} recalls results from~\cite{Delahaye2010} and compares the flux of positron coming from a single source located 500 pc from us at various times after the injection (left panel) or 500~kyr old located a different distances from us (right panel). We make here the hypothesis that all the cosmic-rays are released at once, for instance at the beginning of the Sedov phase in the case of a pulsar interacting with a supervova remnant. The spectrum of the cosmic-rays at injection in the interstellar medium is a power-law in energy with an exponential cut-off : $\propto E^{-\sigma} \exp\left(E/E_c\right)$.
\begin{figure*}
\centering
\includegraphics[width=\columnwidth,clip]{fig_electron_single_source_age_effect_500pc}
\includegraphics[width=\columnwidth,clip]{fig_electron_single_source_dist_effect_5kyr}
\caption{Impact of the distance (left panel) and of the age (right panel) of a pulsar on the positron flux received at the Earth. In order to show only the effects of the propagation, the injection energy cut-off has been set to the very high value of 100~TeV for all the cases. Continuous, dashed and dotted lines correspond to an injection respectively of $\sigma=$1, 1.5 and 2.
The fluxes displayed here are corrected by a factor $E^{1+\sigma}$ to ease the comparison.
It clearly appears that distance has little impact on the shape of the flux at high energies. One should also note that the flux coming from old pulsars drops more sharply, whatever the distance.}
\label{fig-1}
\end{figure*}
From Figure~\ref{fig-1}, one can see that older sources can give a sharper drop at high energy and that the distance affects the low energy part of the spectrum but not so much the high energy one. This means that a sharp fall of the positron fraction could be due to a relatively old pulsar within around 2~kpc from the Sun but not necessarily extremely close.
\section*{A sharply falling positron fraction}
\label{sec-2}
Let us now consider the hypothesis advertised by the AMS-02 collaboration of a sharply falling positron fraction. We have considered two cases with a sudden drop at 350~GeV and 600~GeV down to the level expected from secondaries (computed as in~\cite{Delahaye2009}). The first case with a drop at 350~GeV is for the discussion's sake only since AMS-02 has now published data up to 500~GeV. Considering that the low energy part could be explained by far away pulsars, we tried to fit the feature of the sharp fall only (the eight last bins that appear darker on Figure~\ref{fig-2}) either with a Dark Matter component or with a single bursting pulsar for which we have considered three possible values of the coefficient $\sigma$: 1.0, 1.5 and 2.0.
\begin{figure*}
\centering
\includegraphics[width=0.9\columnwidth,clip]{fig_flux_max_low0329}
\includegraphics[width=0.9\columnwidth,clip]{fig_flux_max_high0329}
\caption{Best fit fluxes for the max parameter set. Upper panel for a positron drop at 350~GeV, lower panel at 600~GeV. Data up to 350~GeV is from AMS-02~\citep{Aguilar2013}, above this energy, the bins are mock data. The lines correspond respectively to Dark Matter annihilating into $e^+e^-$ or $\mu^+\mu^-$ or to a pulsar with injection spectrum parameter of 1, 1.5 or 2.
Note that for the pulsar cases, a smooth distribution of far away pulsars, with the same injection spectrum (but a lower cut-off) has been added to reproduce the data at intermediate energies (10 to 150 GeV).}
\label{fig-2}
\end{figure*}
As what can see from Figure~\ref{fig-2}, a Dark Matter annihilating into a muon pair does not reproduce well a sharp fall of the positron fraction if the fall happens at too low energy but gives better results if the fraction saturates a little before falling. When Dark Matter annihilates into an electron-positron pair, the fall is too sharp and it is hard to reproduce a fraction that increases slowly and then drops. However, all the pulsars cases can give a good fit.
Note that in all Dark Matter cases, the annihilation cross-sections (or boost factors) required to fit the data are very high. This is already known for quite some time and raises a large number of issues concerning consistency of such a results with other observations such as anti-protons~\citep{Donato2009}, $\gamma$-rays~\citep{Cirelli:2009dv}, synchrotron emission~\citep{Linden:2011au} etc.
The question is hence, what are the criteria a pulsar would have to fulfil to reproduce a sharply falling positron fraction. Using a fast semi-analytical method for the propagation of cosmic-ray, we have been able to scan a large parameter space. The results of this scan are displayed on Figure~\ref{fig-3}, which shows what ages and distances of the pulsar can accomodate the data, for different injection power-law $\sigma$ (columns) and different propagation parameter sets (lines). The dark area represents paramteres giving a good fit ($\chi^2$/dof $\leq 1$) whereas lighter colors correspond to $2\sigma$ contours. Purple and yellow dots are sources from respectively the ATNF~\cite{Manchester2005} and Green~\cite{Green2009} catalogues.
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth,clip]{fig_scan_novuple0329}
\caption{Energy cut-off at injection (top) and total energy going to cosmic-ray (bottom) for an injection spectrum $\sigma=1$. The left-hand panels correspond to the min case and a drop at 350~GeV, whereas the right-hand panels are for a drop at 600~GeV and the max propagation parameters.}
\label{fig-3}
\end{figure}
One can see that for all values of the power-law parameter $\sigma$ it is possible to reproduce a sharply falling positron fraction but that $\sigma=2$ gives a good fit only for a relatively small parameter space, that can be considered as fine tuning. The intuition we had from Figure~\ref{fig-1} is confirmed, the fit procedure prefers a given age but does not care very much about the distance of the source to the Sun. Also, what is important to note is that there are actually a couple of existing sources that are with the correct age and distance parameters.
One cannot conclude anything without also performing an analysis of the energetics involved. For each point of our scan we have computed the amount of energy the source would have to inject into cosmic-rays to give the correct flux at the Earth today. This estimates cannot be extremely precise and depends on the assumptions made for the lowest and highest energies of the cosmic-rays considered, however the values we find amount to around 5 to 10\% of the progenitor supernova ejecta energy. The further the source, the higher the fraction of energy that has to go into cosmic-rays. Our parameter scan favours a relatively old (a few hundred kyr old) close-by source (within $\sim 1$\,kpc), capable of supplying at least ${\cal E}_{\rm tot}\sim 10^{47-48}\,$erg into electrons and positrons, accelerated with a hard spectrum. The discussion concerning the production of such cosmic-rays by a pulsar cannot fit in these short proceedings but the interested reader can refer to \citep{Delahaye2014}.
AMS-02 has also published some limits on the anisotropy of the positron ratio (positron flux divided by negative electron flux). Let us first stress that it is surprising to choose to work with this quantity. Indeed, if there were only a single source of electrons and positrons in the sky, even though most cosmic-rays would come from the same direction, this quantity would be equal to zero since both electrons and positrons would have the same anisotropy. Since electrons are dominating over positrons this quantity is only 20\% smaller than the individual fluxes anisotropies but if the fraction were to increase, this could be problematic. Anyway, we have also computed the anisotropy one would get from a single pulsar responsible for the positron fraction and never found a value that was excluded by the data. An estimate of the positron flux anisotropy $A$ can be read from Figure~\ref{fig-3} (dashed lines).
Most of the parameter space for pulsars compatible with a sharply falling positron fraction is hence compatible with everything we know about cosmic-rays and pulsars.
\section*{Conclusion}
Though the idea that a sharply falling positron fraction would be a proof of Dark Matter is advertised widely by the AMS-02 collaboration, we show here that this is of course not the case. In the contrary, pulsars could explain such a feature in a much more natural way than Dark Matter. However this does not mean that the question is not interesting and one could learn a lot about pulsars if indeed such an unlikely feature were to be observed in the near future.
More precisely, if we really were to observe a sharply falling positron fraction, this would actually teach us that only few pulsars can accelerate electron and positron cosmic-rays. We would then have to understand what are the conditions that make that pulsars can inject cosmic-rays or not. Also, depending of the energy at which the fall is taking place, we could actually determine the age of the pulsar responsible for the feature and look for it in the catalogues and in the sky, allowing to observe more closely the few candidate sources. Finally, by giving us some indication on the injection, this may also help understand what is the precise mechanism that allows these pulsars to inject electron-positron pairs in the interstellar medium.
A sharply falling positron fraction is quite unlikely but if it is observed it would be exciting. Not for the reason that this would prove anything about Dark Matter but because this could potentially teach us a lot about pulsars and cosmic-ray acceleration mechanisms.
\vspace{1cm}
The slides of this presentation are available online~\url{http://www.fysik.su.se/~tdela/SUGAR_2015.html}
\linebreak This work was supported in part by ERC project 267117 (Dark Matters) hosted by Universit\'e Pierre et Marie Curie—Paris 6. K.K. acknowledges financial support from PNHE and ILP.
|
{
"timestamp": "2015-04-15T02:00:43",
"yymm": "1504",
"arxiv_id": "1504.03336",
"language": "en",
"url": "https://arxiv.org/abs/1504.03336"
}
|
\section{\label{}}
\\
\\
{\it Generalized Fluctuation Theorem} :
We prove the following inequality below:
\begin{eqnarray}
\mathrm{I} _{x\rightarrow y}\geq -\mathrm{E} [\Theta _{\nu }^{t}]. \label{jsu_transfer}
\end{eqnarray}
Here, we define the following generalized forms of the entropy production in terms of a conditional distribution $\nu $:
\begin{eqnarray}
\hspace{-0.5cm} \Theta _{\nu }^t\equiv \log \frac{\pi (x^{t+1}|y^{t+1})}{\nu (x^t|x^{t+1},y^{t})}, \ \hspace{-0.05cm} \widehat{\Theta } _{\nu }^t\equiv \Theta _{\nu }^t\hspace{-0.04cm} +\log \frac{p_s(x^t|y^t)}{p_s(x^{t+1}|y^{t+1})}. \label{generalized_entropy_production}
\end{eqnarray}
We can regard $\nu $ as representing physical quantities computed in the neural system on the basis of $x^t, y^t, y^{t+1}$ and adjusted through learning (see supplemental materials). First, we have the apparent identity
\begin{eqnarray}
&\lefteqn{\exp \left [\log p_s(x^t|y^{t})+\log \pi (x^{t+1}|y^{t+1})-\widehat{\Theta } _{\nu }^t\right ]} &\nonumber \\
&&=\exp \left [\log p_s(x^{t+1}|y^{t+1})+\log \nu (x^t|x^{t+1},y^t)\right ]. \label{apparent_identity}
\end{eqnarray}
Multiplying both sides by $p_s(y^{t+1},y^{t})$ and summing them over relevant random variables, we obtain a generalized form of the fluctuation theorem:
\begin{eqnarray}
\mathrm{E} \left [e^{-I_{tr}^{t+1}-\widehat{\Theta } _{\nu }^t}\right ]=1, \ \ I_{tr}^{t+1}\equiv \log \frac{p_s(y^{t+1}|x^{t},y^{t})}{p_s(y^{t+1}|y^{t})} .\label{equality_transfer}
\end{eqnarray}
Applying Jensen's inequality ($\exp \mathrm{E} [F(Z)]<\mathrm{E} [\exp F(Z)]$, which applies to any random variable $Z$ and any function $F$) to Eq.(\ref{equality_transfer}) gives
\begin{eqnarray}
\mathrm{E} \left [-I_{tr}^{t+1}-\widehat{\Theta } _{\nu }^t\right ]\leq 0. \label{after_Jensen}
\end{eqnarray}
Noting that $\mathrm{E} [\widehat{\Theta } _{\nu }^t]=\mathrm{E} [\Theta _{\nu }^t]$ and $\mathrm{I} _{x\rightarrow y}=\mathrm{E} [I _{tr}^{t+1}]$, we have the inequality in Eq.(\ref{jsu_transfer}).
It is found that, for fixed $\pi $, the right-hand side of Eq.(\ref{jsu_transfer}) is maximal if and only if
\begin{eqnarray}
\nu (x^t|x^{t+1},y^t)=p_s(x^t|x^{t+1},y^{t}). \label{equality_without}
\end{eqnarray}
Furthermore, we can prove that the equality in Eq.(\ref{jsu_transfer}) holds if and only if, in addition to Eq.(\ref{equality_without}), the mutual information takes the maximal value for the fixed $\pi $ and hence satisfies $\mathrm{I} [x^t;y^t]=\mathrm{H} [y^t]$, under suitable conditions (see supplemental materials).
If the neural network has sufficient capacity, it is expected that there is some optimal $\pi $ that maximizes both $\mathrm{I} _{x\rightarrow y}$ and $\mathrm{I} [x^t;y^t]$, simultaneously. In this case, the above analysis implies that the optimal $\pi $ is obtained by maximizing $-\mathrm{E}[\Theta _{\nu }^t]$ with respect to $\pi $ and $\nu $. In conclusion, we find that, for a neural network with a large capacity, the maximization of $-\mathrm{E}[\Theta _{\nu }^t]$ leads to the maximization of $\mathrm{I} _{x\rightarrow y}$ and $\mathrm{I} [x^t;y^t]$. Because the maximization of $\mathrm{I} [x^t;y^t]$ served as the definition of Infomax in previous studies \cite{Linsker:1988vn,Bell:1995vn,Bell:1997ve}, the maximization of $-\mathrm{E}[\Theta _{\nu }^t]$ provides a generalized Infomax.
\\
\\
{\it Structures of Neural Networks} : To maximize $-\mathrm{E}[\Theta _{\nu }^t]$, the neural network must be able to adjust $\pi $ and $\nu $ to optimal conditional distributions through learning. For this purpose, in the remainder of this Letter, we parameterize $\pi (x^t|y^t)$ as
\begin{eqnarray}
\hspace{-0.7cm} \pi (x^{t}|y^{t})&=&\prod _{\ell =1}^{L}\prod _{i=N_{\ell -1}+1}^{N_{\ell }}\hspace{-0.3cm} \pi _i(x_i^{t}|y^{t}, \{ x_k^{t}\} _{k=1}^{N_{\ell -1}}), \label{conditional_form} \\
\pi _i(x_i^{t}&=&1|y^{t},\{ x_k^{t}\} _{k=1}^{N_{\ell -1}})=g(e _i^t), \nonumber \\
e _i^t&=&\sum _{1\leq j\leq M_{\ell }}\rho _{ij}\{ g(\xi _{(\ell ),j}^{t})-\frac{1}{2} \} -h _0,\nonumber \\
\xi _{(\ell ),j}^{t}&=&\sum _{1\leq k\leq d}v_{jk}^{(\ell )}y_k^t+\sum _{1\leq k\leq N_{\ell -1}}w_{jk}^{(\ell )}x_k^t-h _j^{(\ell )}.\label{transition_model}
\end{eqnarray}
Here, $g(e_i^t)$ is the logistic function $(1+\tanh (e_i^t))/2$. Equation(\ref{transition_model}) describes the situation in which each neuron computes its own transition probability, $\pi _{i}$, through the intermediate units $\xi _{(\ell ),j}^t$, with the adjustable parameters $\rho _{ij}, v_{jk}^{(\ell )}, w_{jk}^{(\ell )}$ and $h _j^{(\ell )}$ and the constant parameter $h_0$. These parameters represent the synaptic strengths and intrinsic properties of the neurons and intermediate units. Note that we assume a layered structure of the system, as illustrated in Fig.2, in which neuron $x_i^t$ in layer $\ell $ receives an input $g(e_i^t)$ from the neurons in layers $1$ through $\ell -1$ and the environment through the $\ell $-th intermediate layer. It is known that an arbitrary continuous mapping of $\{y_{k}^t\} _{k=1}^d$ and $\{ x_k^t\} _{k=1}^{N_{\ell -1}}$ to $e _i^{t}$ can be approximated by the last two lines in Eq.(\ref{transition_model}) to arbitrary precision if the number of the intermediate units, $M_{\ell }$, is sufficiently large \cite{Funahashi:1989wi}. Thus, any conditional probability of the form given in Eq.(\ref{conditional_form}) can be represented in terms of $g(e _i^t)$, as in Eq.(\ref{transition_model}) . Increasing the number of layers of the neural network increases its capability to represent various conditional probabilities. We believe that the capability to represent a wide variety of conditional probabilities will allow for realization of the optimal $\pi $, and therefore such capability is necessary for our purposes. We model $\nu $ in the same way as $\pi $. Note that $\nu $ is not used for the realization of neural states. We consider the situation in which only the values of $\log \nu (x^t|x^{t+1},y^t)$ are calculated through some biological mechanisms based on the realized states, $x^t$, $x^{t+1}$ and $y^t$ (see supplemental materials for details regarding $\nu $). \\
\\
\begin{figure}
\includegraphics[width=85mm]{fig2.eps}
\caption{(Color online). An illustration of the layered neural networks for the modeling of $\pi $ ($\nu $ is modeled in a similar manner).}
\end{figure}
{\it A Simple Model of Animals Learning to Explore Environments} : We have shown that neural systems can maximize the transfer entropy and mutual information through a learning mechanism based on a generalized fluctuation theorem. In order to characterize the present learning mechanism, we must clarify the role of the maximization of the transfer entropy in biological contexts, while that of the mutual information has been investigated in previous studies \cite{Linsker:1988vn,Bell:1995vn,Bell:1997ve}. In the following sections, we show that the maximization of the transfer entropy can be understood as a mechanism for the active exploration by an animal of its environment. In order to clearly demonstrate this effect in biological contexts, we introduce a learning problem in which an animal seeks to obtain rewards (e.g., food, water, etc.) through active exploration. \\
\\
Concretely, an animal with a neural system represented by the state $x^t$ moves around in a two-dimensional grid. At each position in the grid, a value of a reward associated with that position is defined (Fig.3(a)). Specifically, in each timestep, the animal takes either one step or zero steps, with the number and direction determined by the values of the specialized neurons, as shown in Fig.3(b). The state of the environment, $y^t$, is specified by the position of the animal and the status of reward configuration in the grid. At each timestep, the animal ``receives'' the reward $r^{t}=r(y^{t})$ at its present position. As shown in Fig.3(a), at most positions in the grid, the reward takes a negative value fixed throughout the simulation. Such a negative reward is interpreted as a punishment. The size of the punishment is minimal in the center of the grid and increases in each direction moving away from the center. At eight (fixed) positions in the outer region of the grid, there are positive rewards. The value of each is initially $R$. If such a positive reward is visited by the animal, the reward at the position is 0 for the subsequent 100 timesteps and then reset to $R$. The animal receives inputs from the environment as twelve real variables $\{ y_k^t\} \ (1\leq k\leq 12)$. The inputs $\{ y_k^t\} $ consist of the coordinate values of the animal's position in the grid $(k=1,2)$, the presence or absence of a reward $R$ at the animal's current position ($y_{3}^t=1\ \mathrm{or} \ 0$) and the values of the rewards at all positions within one step of the current position $(4\leq k\leq 12)$, as shown in Fig.3(c). This set of values allows the animal to predict the immediate consequence of its movement. Initially, the model parameters that determine $\pi $ are set in such a way that the animal primarily attempts to avoid negative rewards, mimicking the innate behavior of real animals (see supplemental materials). With this model, it is very natural to consider maximization of the average reward, $\mathrm{E} [r^t]$, by adjusting the animal's behavior represented by $\pi$, because animals must do so for survival. This maximization problem is called a {\it reinforcement learning problem}. However, in general, it is known that algorithms that simply maximize $\mathrm{E} [r^t]$ do not reach an optimal outcome in most realistic situations because there is a lack of new experiences, unless some mechanisms for active exploration are included \cite{Sutton:1998:IRL:551283,wiering2012reinforcement}. In the present case, in order to obtain the rewards $R$ to realize a larger $\mathrm{E} [r^t]$, the animal must possess a mechanism that allows it to explore the outer region and tolerate the punishment incurred there. In the following, we show that maximization of the transfer entropy in addition to the average reward provides this mechanism.
\\
\\
\begin{figure}
\includegraphics[width=85mm]{fig3.eps}
\caption{(Color). A simple model of an animal exploring in a two-dimensional grid: (a) the configuration of reward in the grid, (b) animal's movement according to the values of the specialized neurons, (c) surrounding reward values as input variables.}
\end{figure}
We consider the following learning problem:
\begin{eqnarray}
\max \ \left (\mathrm{I} _{x\rightarrow y}+\beta \mathrm{E} [r^t]\right ), \label{learning_problem}
\end{eqnarray}
where $\beta \geq 0$. First, we theoretically analyze the optimal $\pi $ for the above problem. Since the neural control over the environment is deterministic in the above model; that is, for given $x^t$ and $y^t$, we have $\mu (y^{t+1}|x^t,y^t)=1\ \mathrm{or} \ 0$, the optimization of $\pi (x^{t}|y^t)$ reduces to that of $\alpha (y^{t+1}|y^t)\equiv \sum _{x^{t}}\mu (y^{t+1}|x^t,y^{t})\pi (x^t|y^t)$. As we know from the basic theory of reinforcement learning, it is helpful for analysis of the maximization problem treated here to consider the following functions of $y\in \mathcal{Y}$:
\begin{eqnarray}
\hspace{-0.6cm} V _{r, \alpha }^{(\gamma )}(y) \hspace{-0.02cm} &=&\hspace{-0.02cm} \mathrm{E} \hspace{-0.04cm} \left [\sum _{s=1}^{\infty }\gamma ^sr^{t+s}\Big{|} y^t=y\right ]\hspace{-0.06cm} -\hspace{-0.02cm} \mathrm{E} \hspace{-0.04cm} \left [\sum _{s=1}^{\infty }\gamma ^sr^{t+s}\right ]\hspace{-0.04cm} , \nonumber \\
\hspace{-0.6cm} V _{I, \alpha }^{(\gamma )}(y) \hspace{-0.02cm} &=&\hspace{-0.02cm} \mathrm{E} \hspace{-0.04cm} \left [\sum _{s=1}^{\infty }\gamma ^sI_{tr}^{t+s}\Big{|} y^t=y\right ]\hspace{-0.06cm} -\hspace{-0.02cm} \mathrm{E} \hspace{-0.04cm} \left [\sum _{s=1}^{\infty }\gamma ^sI_{tr}^{t+s}\right ]\hspace{-0.04cm} , \label{excess_function}
\end{eqnarray}
where $\gamma $ is a parameter satisfying $0\leq \gamma <1$. The above quantities with $\gamma \rightarrow 1$ represent the average amounts of ``excess reward'' and ``excess information'', obtained from the initial state $y$ until the system has relaxed into the steady state. This is analogous to the definition of the ``excess heat'' in steady-state thermodynamics \cite{Oono:1998uj,Sasa:2006gg}. With these limits, we can prove that the learning problem, Eq.(\ref{learning_problem}), has a unique optimal distribution $\alpha ^*$ of the following form (see supplemental materials):
\begin{eqnarray}
\hspace{-0.6cm} \alpha ^*\hspace{-0.04cm} (y^{t+1}|y^t)\hspace{-0.05cm} \propto \hspace{-0.05cm} \exp [\beta \{ r^{t+1}\hspace{-0.17cm}+\hspace{-0.06cm} V_{r,\alpha ^{*}}^{(1)}(y^{t+1})\hspace{-0.03cm} \} \hspace{-0.06cm} +\hspace{-0.06cm} V_{I,\alpha ^{*}}^{(1)}(y^{t+1})]. \label{Bellman_optimal_solution}
\end{eqnarray}
Inspecting Eq.(\ref{Bellman_optimal_solution}), we understand that the animal shows the following three types of behaviors determined by the value of $\beta $. In the case with finite $\beta (>0)$, the animal moves with high probability in a direction for which large future reward is expected, and with small (but non-zero) probability in a direction for which small future reward is expected. It is known that such exploratory behavior, with (infrequent) excursions in directions with low expected payoff, is necessary for neural systems to find larger rewards \cite{Sutton:1998:IRL:551283,wiering2012reinforcement}. Contrastingly, in the case with $\beta \rightarrow \infty $, the optimal behavior is deterministic, and exploration is stifled. In the case with $\beta =0$, the animal is completely insensitive to the values of reward. Hence, we see behavior that is a compromise between the drive to explore and the drive to acquire large rewards, represented by $\mathrm{I} _{x\rightarrow y}$ and $\mathrm{E} [r^t]$, respectively.
\\
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{\it Numerical Simulations} : In order to confirm the theoretical results obtained in the above, we carried out simulations in which we maximized $\mathrm{E} [\beta r^t-\Theta _{\nu }^t]$ by applying a stochastic gradient algorithm to the model depicted in Fig.3 (see supplemental materials for the algorithms and discussion of its biological counterparts). It is expected that this maximization will result in the maximization in Eq.(\ref{learning_problem}). We first examine the case with $\beta =\infty $, i.e., that in which the animal attempts to maximize $\mathrm{E} [r^t]$ ($R=600$). In this case, we observe that the environmental entropy, $\mathrm{H} [y^t]$, decreases monotonically and that $\mathrm{E} [r^t]$ becomes fixed at zero (Fig.4(b),(c)). This indicates that the animal has learned only to avoid the outer areas and remains for all times at the origin. Hence, the learning has essentially failed. By contrast, setting $\beta =0.1$ and $R=0$, we observe that $-\mathrm{E} [\Theta _{\nu }^t]$ increases monotonically in Fig.4(a). We also observe that $\mathrm{H} [y^t]$ increases in a similar manner to $-\mathrm{E} [\Theta _{\nu }^t]$ and that $\mathrm{I} [x^t;y^t]$ almost realizes the maximal value, and satisfies $\mathrm{I} [x^t;y^t]=\mathrm{H} [y^t]$ (Fig.4(b)). Hence, we have confirmed that the maximization of $-\mathrm{E} [\Theta _{\nu }^t]$ leads to exploration and maximization of $\mathrm{I} [x^t;y^t]$, as theoretically predicted above. Finally, with $\beta =0.1$ and $R=600$, we find that the animal is able to increase $\mathrm{E} [r^t]$ through the exploration (Fig.4(c)).
\begin{figure}
\includegraphics[width=85mm]{fig4.eps}
\caption{(Color). The values of (a) $-\mathrm{E} [\Theta _{\nu }^t]$ (the red line is out of range), (b) $\mathrm{H} [y^t]$ and $\mathrm{I} [x^t;y^t]$, and (c) $\mathrm{E} [r^t]$, during the course of learning. Neural networks with $L=4$, $N_1=30$, $N_2=60$, $N_3=62$, $N_4=64$, $M_1=M_2=120$, and $M_3=M_4=60$ were simulated with $\beta=\infty ,0.1$, and $R=0,600$.}
\end{figure}
\\
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{\it Conclusion} : We have shown on the basis of theoretical and numerical analysis that assuming that the learning process exhibited by neural systems is based on a principle described by a generalized fluctuation theorem, this system will learn an effective form of exploring behavior (by maximizing the transfer entropy, $\mathrm{I} _{x\rightarrow y}$) and acquiring information about its environment (by maximizing the mutual information, $\mathrm{I} [x^t;y^t]$). Although informational quantities other than the transfer entropy have been considered as mechanisms for the exploration \cite{azar2012dynamic,peters2010relative,Still:2012dn,Ay:2012tu,Bialek:2001wv}, it has not been elucidated how those quantities are maximized in neural systems. We believe that use of the transfer entropy as a mechanism for exploration is more plausible, because the present learning mechanism can be utilized for it. Although the demonstration is limited to the case of Markovian environmental dynamics and neural networks without memory, this work will be generalized to more complex systems in the near future using the foundation laid by the present work.\\
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This work was supported by JST CREST from MEXT.
\newpage
\clearpage
\newpage
\begin{widetext}
\subsection{Supplemental Materials}
{\it Proof of The Maximization of The Mutual Information, $\mathrm{I} [x^t;y^t]$} : In this section, we prove that the maximization of the mutual information, $\mathrm{I} [x^t;y^t]$, and Eq.(\ref{equality_without}) are equivalent to equality in Eq.(\ref{jsu_transfer}), under suitable conditions. First, we note that we can replace $\nu (x^t|x^{t+1},y^t)$ by $\nu (x^t|x^{t+1},y^t,y^{t+1})$ in Eqs.(\ref{jsu_transfer}), (\ref{generalized_entropy_production}), (\ref{apparent_identity}), (\ref{equality_transfer}) and (\ref{after_Jensen}). In this case, equality in Eq.(\ref{jsu_transfer}) follows from equality in the Jensen inequality ($\exp F(Z)= \mathrm{E} [\exp F(Z)]$ with probability 1):
\begin{eqnarray}
e^{-I_{tr}^{t+1}-\widehat{\Theta } _{\nu }^t}=\mathrm{E} [e^{-I_{tr}^{t+1}-\widehat{\Theta } _{\nu }^t}]=1.
\end{eqnarray}
This implies $-I_{tr}^{t+1}-\widehat{\Theta } _{\nu }^t=0$ with probability 1. By rearranging terms, we have
\begin{eqnarray}
\nu (x^t|x^{t+1},y^t,y^{t+1})&=&p_s(x^{t}|y^{t+1},y^{t},x^{t+1}). \label{equality_with}
\end{eqnarray}
Hence, by including $y^{t+1}$ as a conditioning variable of $\nu $, we can easily obtain the equality. However, by reducing the number of conditioning variables, we can also obtain the maximization of the mutual information, $\mathrm{I} [x^t;y^t]$, as we noted in the main text. We prove this in the following.
\\
\\
First, we obtain an explicit expression of the optimal $\nu (x^t|x^{t+1},y^t)$ in Eq.(\ref{equality_without}) from the following inequality:
\begin{eqnarray}
\mathrm{E} \left [\log \frac{\nu (x^t|y^t,x^{t+1})}{p_s(x^t|y^t,x^{t+1})}\right ]=\sum _{y^t,x^{t+1}}p_s(y^t, x^{t+1})\sum _{x^t}p_s(x^t|y^t,x^{t+1})\log \frac{\nu (x^t|y^t,x^{t+1})}{p_s(x^t|y^t,x^{t+1})}\leq 0.
\end{eqnarray}
The above inequality is derived from the inequality $F-1\geq \log F$ for positive real $F$, and thus, the optimality condition in Eq.(\ref{equality_without}) is obtained from the equality, $F-1=\log F\leftrightarrow F=1$ with probability 1:
\begin{eqnarray}
\nu (x^t|x^{t+1},y^t)&=&p_s(x^{t}|y^{t},x^{t+1}).\label{equality_without_again}
\end{eqnarray}
Then, in order to analyze the equality condition of Eq.(\ref{jsu_transfer}) for $\nu (x^t|x^{t+1},y^t)$, we calculate the difference $\Delta $ between the values of $-\mathrm{E} [\Theta _{\nu }^t]$ as calculated with Eqs. (\ref{equality_without_again}) and (\ref{equality_with}), writing
\begin{eqnarray}
\Delta =\mathrm{E} \left [\log \frac{p_s(x^t|y^{t},y^{t+1},x^{t+1})}{p_s(x^t|y^{t},x^{t+1})} \right ]=\mathrm{I} [y^{t+1};x^{t}|x^{t+1},y^{t}].
\end{eqnarray}
Because $-\mathrm{E} [\Theta _{\nu }^t]=\mathrm{I} _{x\rightarrow y}$ in the case with Eq.(\ref{equality_with}), we obtain
\begin{eqnarray}
\mathrm{I} _{x\rightarrow y}+\mathrm{E} [\Theta _{\nu }^{t}]=\mathrm{I} [y^{t+1};x^t|x^{t+1},y^{t}], \label{error_from_equality}
\end{eqnarray}
in the case with Eq.(\ref{equality_without_again}).
\\
\\
Next, we show that the condition
\begin{eqnarray}
\mathrm{I} [y^{t+1};x^t|x^{t+1},y^{t}]=0, \label{control_condition_equality}
\end{eqnarray}
is equivalent to the maximization of the mutual information,
\begin{eqnarray}
\mathrm{I} [x^t;y^t]=\mathrm{H} [y^t], \label{maximal_mutual_information}
\end{eqnarray}
on the assumption that the environmental state space $\mathcal{Y} $ is not too finely partitioned in comparison with the precision of neural control over the environment. Precisely, we assume that there is no coarse-grained partition $\mathcal{Y^{\prime }}$ of $\mathcal{Y}$ such that the neural control has the same precision on the two partitions $\mathcal{Y} $ and $\mathcal{Y} ^{\prime }$ of the environmental state space. We also assume that $p_s(y^{t+1}|y^t)\neq 1$ for all $y^t$ and $y^{t+1}$, which holds for most sets of values of the model parameters in a general model.
A coarse-grained partition $\mathcal{Y} ^{\prime }$ is a set of subcollections of $\mathcal{Y} $ such that $y^{\prime }\cap y^{\prime \prime }=\emptyset $ for any $y^{\prime }\neq y ^{\prime \prime }\in \mathcal{Y} ^{\prime }$ and $\cup _{y^{\prime }\in \mathcal{Y} ^{\prime }} y^{\prime }=\mathcal{Y} $.
For any such coarse-grained partition $\mathcal{Y} ^{\prime }$, we require
\begin{eqnarray}
\hspace{-0.8cm} \mathrm{I} [y^{t+1,\prime };x^t|y^t]\neq \mathrm{I} [y^{t+1};x^t|y^t],\ y^{t+1}\in y^{t+1,\prime }\in \mathcal{Y} ^{\prime }, \label{rough_partition}
\end{eqnarray}
where we have defined the random variable $y^{t+1,\prime }$, which takes values in $\mathcal{Y} ^{\prime }$ with
$y^{t+1}\in y^{t+1,\prime }$.
Under this assumption, we first show that the conditional mutual information,
\begin{eqnarray}
\mathrm{I} [y^{t+1};x^{t+1}|y^{t}]=\sum _{x^{t+1},y^{t+1},y^t}p_s(y^t,y^{t+1},x^{t+1})\log \frac{p_s(y^{t+1}|x^{t+1},y^t)}{p_s(y^{t+1}|y^t)},
\end{eqnarray}
must be maximal. Note that the conditional mutual information takes its maximal value and hence satisfies
\begin{eqnarray}
\mathrm{I} [x^{t+1};y^{t+1}|y^t]=\mathrm{H} [y^{t+1}|y^t] \label{maximality_conditional_mi}
\end{eqnarray}
if and only if $p_s(y^{t+1}|x^{t+1},y^t)=1$ with probability 1.
Thus, to obtain the desired result, we show that $p_s(y^{t+1}|x^{t+1},y^t)>0$ for multiple $y^{t+1}$ with some $y^t=y_0$ and $x^{t+1}=x_0$ contradicts Eq.(\ref{control_condition_equality}). \\
\\
First, we define the set
\begin{eqnarray}
\overline{y} =\{ y^{t+1}\in \mathcal{Y} |p_s(y^{t+1}|x^{t+1}=x_0,y^t=y_0)>0\} ,
\end{eqnarray}
and a coarse-grained partition of $\mathcal{Y} $ as
\begin{eqnarray}
\mathcal{Y} ^{\prime }=\{ \{ y\} \} _{y\in {\mathcal{Y} \setminus \overline{y} }}\cup \{ \overline{y} \} .
\end{eqnarray}
Here, $\mathcal{Y} \setminus \overline{y}$ is the relative complement of $\overline{y}$ in $\mathcal{Y}$, which consists of all the elements of $\mathcal{Y}$ that are not contained in $\overline{y}$.
The assumption in Eq.(\ref{rough_partition}) requires
\begin{eqnarray}
\mathrm{I} [y^{t+1};x^t|y^t]-\mathrm{I} [y^{t+1,\prime };x^t|y^t]&=&\mathrm{I} [y^{t+1},y^{t+1,\prime };x^t|y^t]-\mathrm{I} [y^{t+1,\prime };x^t|y^t] \nonumber \\
&=&\mathrm{I} [y^{t+1};x^t|y^{t+1,\prime },y^t]\nonumber \\
&>&0, \label{positive_control_assumption}
\end{eqnarray}
where the first equality holds because $y^{t+1}$ uniquely determines $y^{t+1,\prime }$ and thus the additional inclusion of $y^{t+1,\prime }$ in the first term does not affect the value of the conditional mutual information. Also, Eq.(\ref{control_condition_equality}) implies
\begin{eqnarray}
\mathrm{I} [y^{t+1};x^t|y^t,x^{t+1}]=\mathrm{I} [y^{t+1};x^t|y^{t+1,\prime },y^t,x^{t+1}]=0.
\end{eqnarray}
Now, recall that the inclusion of additional conditioning variables (in this case, $y^{t+1,\prime }$) always reduces the value of the conditional mutual information. The right-hand side of the above equation can be written as
\begin{eqnarray}
&&\mathrm{E} \left [\log \frac{p_s(y^{t+1}|x^t,y^{t+1,\prime },y^t,x^{t+1})}{p_s(y^{t+1}|y^{t+1,\prime },y^t,x^{t+1})} \right ]\nonumber \\
&=&\mathrm{E} \left [\log \left \{ \frac{p_s(x^{t+1}|y^{t+1})p_s(y^{t+1}|y^{t+1,\prime },y^t,x^t)}{\sum _{\widetilde{y} ^{t+1}\in \mathcal{Y} }p_s(x^{t+1}|\widetilde{y} ^{t+1})p_s(\widetilde{y} ^{t+1}|y^{t+1,\prime },y^t,x^t)} \frac{\sum _{\widehat{y} ^{t+1}\in \mathcal{Y} }p_s(x^{t+1}|\widehat{y} ^{t+1})p_s(\widehat{y} ^{t+1}|y^{t+1,\prime },y^t)}{p_s(x^{t+1}|y^{t+1})p_s(y^{t+1}|y^{t+1,\prime },y^t)} \right \} \right ] \nonumber \\
&=&0, \label{contradiction_equality}
\end{eqnarray}
with the dummy variables $\widetilde{y} ^{t+1}$ and $\widehat{y} ^{t+1}$ having the same (conditional) distributions as $y^{t+1}$.
The above equality requires that the argument of the logarithm be 1 with probability 1, since $F-1\geq \log F$, $F-1= \log F\leftrightarrow F=1$ and
\begin{eqnarray}
&&-\mathrm{E} \left [\log \frac{p_s(y^{t+1}|x^t,y^{t+1,\prime },y^t,x^{t+1})}{p_s(y^{t+1}|y^{t+1,\prime },y^t,x^{t+1})} \right ]\nonumber \\
&=&\sum _{x^t,y^{t+1,\prime },y^t,x^{t+1}}p_s(x^t,y^{t+1,\prime },y^t,x^{t+1})\sum _{y^{t+1}}p_s(y^{t+1}|x^t,y^{t+1,\prime },y^t,x^{t+1})\log \frac{p_s(y^{t+1}|y^{t+1,\prime },y^t,x^{t+1})}{p_s(y^{t+1}|x^t,y^{t+1,\prime },y^t,x^{t+1})} \nonumber \\
&\leq &\sum _{x^t,y^{t+1,\prime },y^t,x^{t+1}}p_s(x^t,y^{t+1,\prime },y^t,x^{t+1})\sum _{y^{t+1}}p_s(y^{t+1}|x^t,y^{t+1,\prime },y^t,x^{t+1})\left (\frac{p_s(y^{t+1}|y^{t+1,\prime },y^t,x^{t+1})}{p_s(y^{t+1}|x^t,y^{t+1,\prime },y^t,x^{t+1})}
-1\right ) \nonumber \\
&=&0.
\end{eqnarray}
Hence, noting that $p_s(x^{t+1}=x_0|y^{t+1})>0$ for $y^{t+1}\in \overline{y} $, we have
\begin{eqnarray}
\frac{p_s(y^{t+1}|y^{t+1,\prime }=\overline{y} ,y^{t},x^t)}{p_s(y^{t+1}|y^{t+1,\prime }=\overline{y} ,y^{t})} =\frac{\sum _{\widehat{y} ^{t+1}\in \mathcal{Y} }p_s(x^{t+1}=x_0|\widehat{y} ^{t+1})p_s(\widehat{y} ^{t+1}|y^{t+1,\prime }=\overline{y} ,y^t)}{\sum _{\widetilde{y} ^{t+1}\in \mathcal{Y} }p_s(x^{t+1}=x_0|\widetilde{y} ^{t+1})p_s(\widetilde{y} ^{t+1}|y^{t+1,\prime }=\overline{y} ,y^t,x^t)}, \ \ \ \forall y^{t+1}\in \overline{y} . \label{first_equation_for_contradiction}
\end{eqnarray}
Furthermore, Eq.(\ref{first_equation_for_contradiction}) with Eq.(\ref{positive_control_assumption}) implies
\begin{eqnarray}
\frac{p_s(y^{t+1}|y^{t+1,\prime }=\overline{y} ,y^t,x^t)}{p_s(y^{t+1}|y^{t+1,\prime }=\overline{y} ,y^{t})}=c\neq 1, \label{cneq1}
\end{eqnarray}
for all $y^{t+1}\in \overline{y} $ and some $y^{t}$ and $x^t$, noting
\begin{eqnarray}
\frac{p_s(y^{t+1}|y^{t+1,\prime },y^t,x^t)}{p_s(y^{t+1}|y^{t+1,\prime },y^{t})}=1, \ \ \ \forall y^{t+1,\prime }\in \mathcal{Y} ^{\prime } \setminus \{ \overline{y} \} . \label{always1}
\end{eqnarray}
Here, note that $c=1$ in Eq.(\ref{cneq1}) with Eq.(\ref{always1}) implies $\mathrm{I} [y^{t+1};x^t|y^{t+1,\prime },y^t]=0$, violating the assumption in Eq.(\ref{positive_control_assumption}).
However, this implies
\begin{eqnarray}
1=\sum _{y^{t+1}\in \overline{y} }p_s(y^{t+1}|y^{t+1,\prime }=\overline{y} ,y^t,x^t)=c\sum _{y^{t+1}\in \overline{y} }p_s(y^{t+1}|y^{t+1,\prime }=\overline{y} ,y^{t})=c\neq 1.
\end{eqnarray}
This contradiction completes the proof of the maximality of the conditional mutual information, Eq.(\ref{maximality_conditional_mi}). \\
\\
Next, we show the equivalence of the maximality of the conditional mutual information and the maximality of the mutual information. As we have discussed, Eq.(\ref{maximality_conditional_mi}) implies
\begin{eqnarray}
p_s(y^{t+1}|x^{t+1},y^{t})=\frac{p_s(y^t|y^{t+1})p_s(y^{t+1}|x^{t+1})}{\sum _{\widetilde{y} ^{t+1}\in \mathcal{Y} }p_s(y^t|\widetilde{y} ^{t+1})p_s(\widetilde{y} ^{t+1}|x^{t+1})}=1, \label{remove_cond}
\end{eqnarray}
with probability 1. Then, the assumption $p_s(y^{t+1}|y^{t})\neq 1$ implies that $p_s(y^t|y^{t+1})$ is positive for multiple $y^{t+1}$. Thus, the condition Eq.(\ref{remove_cond}) implies $p_s(y^{t+1}|x^{t+1})=1$ with probability 1, or equivalently, that the mutual information, $\mathrm{I} [x^{t};y^{t}]$, is maximal and hence satisfies
\begin{eqnarray}
\mathrm{I} [x^t;y^t]=\mathrm{H} [y^{t}]. \label{max_mi}
\end{eqnarray}
Conversely, the equality in Eq.(\ref{max_mi})
implies
\begin{eqnarray}
\mathrm{I} [y^{t+1};x^t|x^{t+1},y^t]&=&\mathrm{H} [y^{t+1}|x^{t+1},y^t]-\mathrm{H} [y^{t+1}|x^t,x^{t+1},y^t]\nonumber \\
&\leq &\mathrm{H} [y^{t+1}|x^{t+1}]\nonumber \\
&=&\mathrm{H} [y^{t}]-\mathrm{I} [x^t;y^t] \nonumber \\
&=&0.
\end{eqnarray}
Thus we have recovered the condition Eq.(\ref{control_condition_equality}). This completes the proof that equality in Eq.(\ref{jsu_transfer}) is equivalent to Eq.(\ref{equality_without}) and Eq.(\ref{maximal_mutual_information}).\\
\\
In the above proof, we note that different definitions of $\nu $ also lead to the maximization of the mutual information $\mathrm{I} [x^t;y^t]$ in different manners, although we do not note this point in the main text for the simplicity of presentation. Concretely, we consider the model described by the causal network in Fig.5 by splitting $x^t$ into $x_{(1)}^t$ and $x_{(2)}^t$, and define a generalized entropy production as
\begin{eqnarray}
\Theta _{\nu }^t=\log \frac{\pi (x_{(1)}^{t+1}|y^{t+1})}{\nu (x_{(1)}^t|x_{(2)}^t,y^{t+1})} .
\end{eqnarray}
Then, in the same manner as above, we can prove that the equality, $-\mathrm{E} [\Theta _{\nu }^t]=\mathrm{I} _{x_{(1)}\rightarrow y}$, implies the maximization of the mutual information, $\mathrm{I} [x_{(2)}^t;y^{t,\prime }]=\mathrm{H} [y^{t,\prime }]$, in some coarse-grained partition $\mathcal{Y} ^{\prime }$ that satisfies
\begin{eqnarray}
\mathrm{I} [x_{(1)}^t;y^{t,\prime }|y^{t+1}]=\mathrm{I} [x_{(1)}^t;y^{t}|y^{t+1}].
\end{eqnarray}
Further results in this direction will be investigated in the future reports.
\begin{figure}
\includegraphics[width=85mm]{fig5.eps}
\caption{Representation of the dynamics with an additional neural network $z^t$ as a causal network.}
\end{figure}
\\
\\
{\it Modeling of $\nu $ with a Neural Network} : We compute $\nu$ in the same way as $\pi $, explicitly writing
\begin{eqnarray}
\nu (x^{t}|x^{t+1},y^t)&=&\prod _{\ell =1}^L\prod _{i=N_{\ell -1}+1}^{N_{\ell }}\nu _i(x_i^{t}|\{x_k^t\} _{k=1}^{N_{\ell -1}},x^{t+1},y^t), \nonumber \\
\nu _i(x_i^{t}&=&1|\{x_k^t\} _{k=1}^{N_{\ell -1}},x^{t+1}, y^t)=g(f _i^{t+1}), \nonumber \\
f _i^{t+1}&=&\sum _{1\leq j\leq M_{\ell }}\kappa _{ij}\{ g(\eta _{(\ell ),j}^{t+1})-\frac{1}{2} \} -m_0,\nonumber \\
\eta _{(\ell ),j}^{t+1}=\hspace{-0.15cm} \sum _{1\leq k\leq N}z_{jk}^{(\ell )}&x_k^{t+1}&+\hspace{-0.25cm} \sum _{1\leq k\leq N_{\ell -1}}\hspace{-0.25cm} u_{jk}^{(\ell )}x_k^{t}+\hspace{-0.15cm} \sum _{1\leq k\leq d}q_{jk}^{(\ell )}y_{k}^t-m_j^{(\ell )}.\ \ \label{backward_transition_model}
\end{eqnarray}
Here, the neurons in the $\ell $-th layer receive inputs from $\{ x_k^t\} _{1\leq k\leq N_{\ell -1}}$, $x^{t+1}$ and $y^t$ through the intermediate units, $\eta _{(\ell ),j}^{t+1}$, with the adjustable parameters $\kappa _{ij}$, $z_{jk}^{(\ell )}$, $u_{jk}^{(\ell )}$, $q_{jk} ^{(\ell )}$ and $m_j^{(\ell )}$ and the constant parameter $m_0$. This computation may seem strange, because here the neurons receive inputs from the future states. However, this is not problematic, because the goal of the computation is not to realize the states of the neural network but to calculate the value of $\log \nu (x^t|x^{t+1},y^t)$. Consider the following situation for this computation, for example. The intermediate units in the $\ell $-th layer receive inputs at time $t+1$ from $x^{t+1}$ and also from $\{ x_k^t\} _{1\leq k\leq N_{\ell -1}}$ and $y^t$ through some time-delay mechanisms. These intermediate units send outputs $\{ g(\eta _{(\ell ),j}^{t+1})\} _{1\leq j\leq M_{\ell }}$ to the neurons in the $\ell $-th layer. At this time, the $i$-th neuron in this layer possesses memory of its own state at time $t$, $x_i^t$, through some mechanism. Then, the $i$-th neuron can compute the value of $\nu _i(x_i^{t}|\{x_k^t\} _{k=1}^{N_{\ell -1}},x^{t+1}, y^t)$ as a function of $x_i^t$ and $\{ g(\eta _{(\ell ),j}^{t+1})\} _{1\leq j\leq M_{\ell }}$. The value of $\log \nu $ is the sum of these values of $\nu _i$ over the neurons in the neural network. \\
\\
{\it Proofs of the Relations Used in the Theoretical Analysis of the Reinforcement Learning Problem} : In this section, our goal is to prove Eq.(\ref{Bellman_optimal_solution}). First, we define the following functions called ``value functions'' in the field of reinforcement learning:
\begin{eqnarray}
\widehat{V} _{r, \alpha }^{(\gamma )}(y)&=&\mathrm{E} \left [\sum _{s=1}^{\infty }\gamma ^sr^{t+s}\Big{|} y^t=y\right ], \nonumber \\
\widehat{V} _{I, \alpha }^{(\gamma )}(y)&=&\mathrm{E} \left [\sum _{s=1}^{\infty }\gamma ^sI_{tr}^{t+s}\Big{|} y^t=y\right ], \nonumber \\
\widehat{V} _{\alpha }^{(\gamma )}(y)&=&\widehat{V} _{I, \alpha }^{(\gamma )}(y)+\beta \widehat{V} _{r, \alpha }^{(\gamma )}(y).
\label{value_functions}
\end{eqnarray}
Then, we can write the learning problem Eq.(\ref{learning_problem}) in terms of the value functions as
\begin{eqnarray}
\mathrm{I} _{x\rightarrow y}+\beta \mathrm{E} [r^t]=\lim _{\gamma \rightarrow 1}(1-\gamma ) \widehat{V} _{\alpha }^{(\gamma )}(y),\ \ \ \ \forall y\in \mathcal{Y} . \label{learning_problem_rewritten}
\end{eqnarray}
By definition, the value function satisfies the following recursive relation called the ``Bellman equation'':
\begin{eqnarray}
\hspace{-0.5cm} \widehat{V} _{\alpha }^{(\gamma )}(y^{t})&=&\sum _{y^{t+1}\in \mathcal{Y} }\gamma \alpha (y^{t+1}|y^t)\nonumber \\
\times &\{ &\beta r(y^{t+1})-\log \alpha (y^{t+1}|y^t)+\widehat{V} _{\alpha }^{(\gamma )}(y^{t+1})\} . \label{Bellman_equation}
\end{eqnarray}
Next, we show that for fixed $\gamma $, it is known that an optimal control $\alpha ^*(y^{t+1}|y^t)$ maximizes the value function at all $y \in \mathcal{Y} $, in comparison with suboptimal controls. Explicitly, for any control $\alpha $, the following inequality holds:
\begin{eqnarray}
\widehat{V} ^{(\gamma )}_{\alpha ^{*}}(y)\geq \widehat{V} ^{(\gamma )}_{\alpha }(y),\ \ \ \ \forall y\in \mathcal{Y} .\label{maximality_same_time}
\end{eqnarray}
\\
\\
In order to prove Eq.(\ref{maximality_same_time}), we consider the following operator called a backup operator, operating on functions of the environmental state $y^t$:
\begin{eqnarray}
B\phi (y^t)=\max _{\alpha (y^{t+1}|y^t)}\sum _{y^{t+1}\in \mathcal{Y} }\alpha (y^{t+1}|y^t)\{ \gamma r(y^{t+1})-\gamma \log \alpha (y^{t+1}|y^t)+\gamma \phi (y^{t+1})\} .
\end{eqnarray}
We first show that this operation results in contraction in the space of functions of environmental states $y^t$ with respect to max norm:
\begin{eqnarray}
\parallel \phi \parallel _{\infty }=\max _{y^t\in \mathcal{Y} }|\phi (y^t)|.
\end{eqnarray}
For two functions $\phi _1$ and $\phi _2$, a fixed $\alpha (y^{t+1}|y^t)$, and an operator $B_{\alpha }$ defined as
\begin{eqnarray}
B_{\alpha }\phi (y^t)=\sum _{y^{t+1}\in \mathcal{Y} }\alpha (y^{t+1}|y^t)\{ \gamma r(y^{t+1})-\gamma \log \alpha (y^{t+1}|y^t)+\gamma \phi (y^{t+1})\} ,
\end{eqnarray}
we have
\begin{eqnarray}
\parallel B_{\alpha }\phi _1-B_{\alpha }\phi _2 \parallel _{\infty }&=&\parallel \sum _{y^{t+1}}\alpha (y^{t+1}|y^t)\gamma \{ \phi _1(y^{t+1})-\phi _2(y^{t+1})\} \parallel _{\infty } \nonumber \\
&\leq &\max _{y^{t}\in \mathcal{Y} }\left \{ \sum _{y^{t+1}}|\alpha (y^{t+1}|y^t)|\right \} \parallel \gamma (\phi _1-\phi _2)\parallel _{\infty } \nonumber \\
&=&\gamma \parallel \phi _1-\phi _2\parallel _{\infty }.
\end{eqnarray}
Then, with the distribution $\alpha ^{(i),*}(y^{t+1}|y^{t})$ maximizing $B_{\alpha }\phi _i(y^t)$ $(i=1,2)$, we have
\begin{eqnarray}
\parallel B\phi _1-B\phi _2\parallel _{\infty }&=&\parallel B_{\alpha ^{(1),*}}\phi _1-B_{\alpha ^{(2),*}}\phi _2\parallel _{\infty }\nonumber \\
&\leq &\max _i\parallel B_{\alpha ^{(i),*}}\phi _1-B_{\alpha ^{(i),*}}\phi _2\parallel _{\infty }\nonumber \\
&\leq &\gamma \parallel \phi _1-\phi _2\parallel _{\infty }.
\end{eqnarray}
This proves that the backup operation yields a contraction of the space of functions on the environmental state space $\mathcal{Y} $, and that there is a unique fixed point of this operation in this space of functions. Because the backup operation always increases the values of any value function at any point in $\mathcal{Y} $, we have Eq.(\ref{maximality_same_time}).
Hence, when we consider the maximality condition of $\widehat{V} _{\alpha }^{(\gamma )}(y)$ with respect to $\alpha (y^{t+1}|y^t)$, it is sufficient to consider the stationarity condition of $B\widehat{V} _{\alpha }^{(\gamma )}(y^t)$ by differentiating it with respect to $\alpha (y^{t+1}|y^t)$ and simply putting the derivative of $\widehat{V} _{\alpha }^{(\gamma )}(y^{t+1})$ to be zero. Solving the stationarity condition with the Lagrange multiplier corresponding to $\sum _{y^{t+1}}\alpha (y^{t+1}|y^t)=1$, we obtain
\begin{eqnarray}
\alpha ^*(y^{t+1}|y^t)&\propto &\exp [\beta \{ r(y^{t+1})+\widehat{V} _{r,\alpha ^{*}}^{(\gamma )}(y^{t+1})\} +\widehat{V} _{I,\alpha ^{*}}^{(\gamma )}(y^{t+1})]. \label{optimal_solution_gamma}
\end{eqnarray}
The optimal condition for the learning problem, the maximization of Eq.(\ref{learning_problem_rewritten}), is obtained by taking the limit $\gamma \rightarrow 1$ in Eq.(\ref{optimal_solution_gamma}). In order to avoid divergence, we need to replace the value functions in Eq.(\ref{optimal_solution_gamma}) with the functions defined in Eq.(\ref{excess_function}) that represent the ``excess reward'' and ``excess information''.
\\
\\
{\it Derivation and Biological Plausibility of the Learning Rule} : In the gradient ascent method used for the simulation, we update each parameter $\theta \in \{ \rho _{ij}, v_{jk}^{(\ell )}, w_{jk}^{(\ell )}, h _j^{(\ell )}, \kappa _{ij}, z_{jk}^{(\ell )}, u_{jk}^{(\ell )}, q_{jk} ^{(\ell )}, m_j^{(\ell )}\} $ as follows:
\begin{eqnarray}
\theta ^{t+1}&=&\theta ^t+\epsilon \left \{ \tau (\beta r^{t+1}-\Theta _{\nu }^t)\psi _{\theta }^t-\frac{\partial }{\partial \theta } \Theta _{\nu }^t\right \} \nonumber \\
\psi _{\theta }^t&=&\frac{1}{\tau } \frac{\partial }{\partial \theta } \log \pi (x^{t+1}|y^{t+1})+\left (1-\frac{1}{\tau }\right )\psi _{\theta }^{t-1}. \label{learning rule}
\end{eqnarray}
Here, the constant $\tau $ is a positive real number that is large compared with the mixing time of the dynamics. We set $\epsilon $ to such a small value that the change in the model parameters in each update does not affect the stationarity on a time scale of $\tau $. Then, in the above learning rule, the expectation value of the change in the parameter $\theta $ in each update is equal to the gradient of $\mathrm{E} [\beta r^t-\Theta _{\nu }^t] $ with respect to $\theta $ as we show below. Thus, we can regard the learning rule as a stochastic gradient ascent algorithm to maximize $\mathrm{E} [\beta r^t-\Theta _{\nu }^t] $.
\\
\\
In the gradient ascent method, we must calculate the gradient of the following quantity with respect to $\theta $:
\begin{eqnarray*}
\mathrm{E} [\beta r^t-\Theta _{\nu }^t]=\sum _{x^{t},x^{t+1},y^t,y^{t+1}}p_s(y^t)\pi (x^t|y^{t})\mu (y^{t+1}|y^t,x^t)\pi (x^{t+1}|y^{t+1})\{ \beta r(y^{t+1})-\Theta _{\nu }^{t}\} .
\end{eqnarray*}
In this calculation, we find that differentiation of the stationary distribution $p_s(y^t)$ is apparently intractable, while differentiation of the other components is easily carried out. We note, however, that we do not need to differentiate the stationary distribution explicitly, assuming that the stationary distribution is a smooth function of any model parameter $\theta $. In this case, small changes in $p_s(y^{t-\tau })$ for $\tau \gg 1$ vanish at $t$ and $t-1$, and thus terms including the derivatives of $p_s(y^{t-\tau })$ are negligible (see also \cite{Baxter99directgradient-based}). Thus, we can compute the gradient as follows:
\begin{eqnarray}
&&\frac{\partial }{\partial \theta } \mathrm{E} [\beta r(y^{t+1})-\Theta _{\nu }^{t}]\nonumber \\
&=&\lim _{\tau \rightarrow \infty }\sum _{y^{t-\tau }, x^{t-\tau },\cdots ,y^{t+1},x^{t+1}}p_s(y^{t-\tau })\frac{\partial }{\partial \theta } \left [ \pi (x^{t-\tau }|y^{t-\tau })\prod _{s=0}^{\tau }\mu (y^{t-s+1}|x^{t-s},y^{t-s})\pi (x^{t-s+1}|y^{t-s+1}) \{ \beta r(y^{t+1})-\Theta _{\nu }^{t}\} \right ] \nonumber \\
&=&\lim _{\tau \rightarrow \infty }\sum _{y^{t-\tau }, x^{t-\tau },\cdots ,y^{t+1},x^{t+1}}p_s(y^{t-\tau }, x^{t-\tau },\cdots ,y^{t+1},x^{t+1})\sum _{s=0}^{\tau +1}\frac{\frac{\partial }{\partial \theta } \pi (x^{t-s+1}|y^{t-s+1})}{\pi (x^{t-s+1}|y^{t-s+1})} \{ \beta r(y^{t+1})-\Theta _{\nu }^{t}\} \nonumber \\
&\ &+\lim _{\tau \rightarrow \infty }\sum _{y^{\tau }, x^{t-\tau },\cdots ,y^{t+1},x^{t+1}}p_s(y^{t-\tau }, x^{t-\tau },\cdots ,y^{t+1},x^{t+1})\frac{\partial }{\partial \theta } \{ \beta r(y^{t+1})-\Theta _{\nu }^{t}\} \nonumber \\
&=&\lim _{\tau \rightarrow \infty }\mathrm{E} \left [\{ \beta r(y^{t+1})-\Theta _{\nu }^t \} \sum _{s=0}^{\tau +1}\frac{\partial }{\partial \theta } \log \pi (x^{t-s+1}|y^{t-s+1})\right ]-\mathrm{E} \left [\frac{\partial }{\partial \theta } \Theta _{\nu }^{t}\right ]. \label{derivative_batch}
\end{eqnarray}
Note that the third equality follows from $\frac{\partial }{\partial \theta } r(y^{t+1})=0$. In order to decompose the expectation values into time-stepwise quantities, we introduce the auxiliary variable $\psi _{\theta }^t$, defined through
\begin{eqnarray}
\psi _{\theta }^t=\frac{1}{\tau } \frac{\partial }{\partial \theta } \log \pi (x^{t+1}|y^{t+1})+(1-\frac{1}{\tau } )\psi _{\theta }^{t-1},\ \ \ \mathrm{and} \ \ \ \psi _{\theta }^t=0 \ \ (t\leq 0). \label{online_process}
\end{eqnarray}
Then, we have
\begin{eqnarray*}
\psi _{\theta }^t=\frac{1}{\tau } \sum _{s=0}^{\infty }(1-\frac{1}{\tau } )^s\frac{\partial }{\partial \theta } \log \pi (x^{t-s+1}|y^{t-s+1}).
\end{eqnarray*}
If the process under consideration is stationary, $\psi _{\theta }^t$ approaches the long-time average of $\frac{\partial }{\partial \theta } \log \pi (x^{t+1}|y^{t+1})$ as $\tau \rightarrow \infty $ and $t/\tau \rightarrow \infty $. Similarly, assuming that the correlation of $\beta r(y^{t+1})-\Theta _{\nu }^t$ with $\frac{\partial }{\partial \theta } \log \pi (x^{t-\tau +1}|y^{t-\tau +1})$ is small for $\tau \gg 1$ and that $T\gg \tau $, we have
\begin{eqnarray}
\mathrm{E} \left [\{ \beta r(y^{t+1})-\Theta _{\nu }^{t}\} \sum _{s=0}^{\tau +1}\frac{\partial }{\partial \theta } \log \pi (x^{t-s+1}|y^{t-s+1})\right ]&\approx &\frac{\tau }{T} \sum _{t=1}^{T}(\beta r(y^{t+1})-\Theta _{\nu }^t)\psi _{\theta }^t.\label{online_correlation}
\end{eqnarray}
Then, applying a well-known argument in stochastic approximation theory \citep{robbins1951stochastic}, we obtain the learning rule given in Eq.(\ref{learning rule}) as a stepwise approximation of the gradient in Eq.(\ref{derivative_batch}). \\
\\
Finally, we derive the exact form of the learning rule with respect to several $\theta $ and present its interpretation. Note that Eq.(\ref{learning rule}) is composed of $\log \pi (x^{t+1}|y^{t+1})$, $\log \nu (x^t|x^{t+1},y^t)$ and their derivatives with respect to $\theta $. First, we show that these components are easily calculated in a neuron-wise manner. Note that $\Theta _{\nu }^t$, $\log \pi (x^{t+1}|y^{t+1})$ and $\log \nu (x^t|x^{t+1},y^t)$ are decomposed as
\begin{eqnarray}
\Theta _{\nu }^t&=&\log \pi (x^{t+1}|y^{t+1})-\log \nu (x^t|x^{t+1},y^t) \nonumber \\
&=&\sum _{\ell =1}^L\sum _{i=N_{\ell -1}+1}^{N_{\ell }} \log \pi _i(x_i^{t+1}|y^{t+1}, \{ x_k^{t+1}\} _{k=1}^{N_{\ell -1}})-\sum _{\ell =1}^L\sum _{i=N_{\ell -1}+1}^{N_{\ell }}\log \nu _i(x_i^t|\{ x_k^t\} _{k=1}^{N_{\ell -1}}, x^{t+1}, y^t) \nonumber \\
&=&\sum _i\log \chi (e_i^{t+1}, x_i^{t+1})-\sum _i\log \chi (f_i^{t}, x_i^{t}),
\end{eqnarray}
where
\begin{eqnarray}
\chi (a,b)&=&\left \{
\begin{array}{ccc}
g(a)&,\ & \mathrm{if\ \ } b=1, \nonumber \\
-g(a)&,\ & \mathrm{if\ \ } b=0. \nonumber
\end{array}
\right. \\
\end{eqnarray}
Then, the derivatives of $\Theta _{\nu }^t$, $\log \pi (x^{t+1}|y^{t+1})$ and $\log \nu (x^t|x^{t+1},y^t)$ (with respect to $v_{jk}^{(\ell )},\rho _{ij},z_{jk}^{(\ell )},\kappa _{ij}$, for example) are calculated as follows.
First, denoting the derivative of $\chi (a,b)$ with respect to $a$ as $\chi _a(a,b)$,
\begin{eqnarray}
\frac{\partial }{\partial v_{jk}^{(\ell )}} \log \pi (x^{t+1}|y^{t+1})&=&\sum _{i=N_{\ell -1}+1}^{N_{\ell }} \frac{\partial }{\partial v_{jk}^{(\ell )}}\log \pi _i(x_i^{t+1}|y^{t+1},\{ x_k^{t+1}\} _{k=1}^{N_{\ell -1}} ) \nonumber \\
&=&\sum _{i=N_{\ell -1}+1}^{N_{\ell }}\frac{\chi _a(e_i^{t+1},x_i^{t+1})}{\chi (e_i^{t+1},x_i^{t+1})} \rho _{ij}g^{\prime }(\xi _{(\ell ),j}^{t+1})y_k^{t+1}.
\end{eqnarray}
\begin{eqnarray}
\frac{\partial }{\partial \rho _{ij}} \log \pi (x^{t+1}|y^{t+1})&=&\frac{\partial }{\partial \rho _{ij}}\log \pi _i(x_i^{t+1}|y^{t+1},\{ x_k^{t+1}\} _{k=1}^{N_{\ell -1}} ) \nonumber \\
&=&\frac{\chi _a(e_i^{t+1},x_i^{t+1})}{\chi (e_i^{t+1},x_i^{t+1})} g(\xi _{(\ell ,j)}^{t+1}).
\end{eqnarray}
\begin{eqnarray}
\frac{\partial }{\partial z_{jk}^{(\ell )}} \log \nu (x^t|x^{t+1},y^t)&=&\sum _{i=N_{\ell -1}+1}^{N_{\ell }} \frac{\partial }{\partial z_{jk}^{(\ell )}}\log \nu _i(x_i^{t}|\{ x_k^{t}\} _{k=1}^{N_{\ell -1}},x^{t+1},y^{t}) \nonumber \\
&=&\sum _{i=N_{\ell -1}+1}^{N_{\ell }}\frac{\chi _a(f_i^{t+1},x_i^{t})}{\chi (f_i^{t+1},x_i^{t})} \kappa _{ij}^{(\ell )}g^{\prime }(\eta _{(\ell ),j}^{t+1})x_k^{t+1}.
\end{eqnarray}
\begin{eqnarray}
\frac{\partial }{\partial \kappa _{ij}} \log \nu (x^t|x^{t+1},y^t)&=&\frac{\partial }{\partial \kappa _{ij}}\log \nu _i(x_i^{t+1}|\{ x_k^{t}\} _{k=1}^{N_{\ell -1}},x^{t+1},y^{t}) \nonumber \\
&=&\frac{\chi _a(f_i^{t+1},x_i^{t})}{\chi (f_i^{t+1},x_i^{t})} g(\eta _{(\ell ,j)}^{t+1}).
\end{eqnarray}
It should be noted that calculations of the derivatives involve quantities only for related neurons and intermediate units. For example, the derivative with respect to $v_{jk}^{(\ell )}$ used only information regarding $y_k^{t+1}$, $\xi _{(\ell ),k}^{t+1}$ and $\{ \mu _i^{t+1}, x_i^{t+1}, \rho _{ij}\} $ of the $i$-th neuron to which the $j$-th intermediate unit is connected. Thus, we can regard the change in the synaptic strength $v_{jk}^{(\ell )}$ as being determined by the local interactions at the synapse on the $j$-th intermediate unit. Continuing with this line of argument, we can obtain even more realistic forms of learning rules for actual neural systems. However, we do not go into detail here, because the argument becomes quite complicated and is beyond the scope of the current study.
\\
\\
{\it Initial Values of Model Parameters and Values of Learning Parameters Used in the Simulation } : In the numerical simulation of our model of learning, we used initial values of the model parameters that results in behavior in which the animal primarily attempts to avoid negative reward, mimicking innate behavior of real animals. We set the values of the model parameters involved in the inputs to the movement-related neurons as shown in Fig.6. A neuron controlling motion in one of four directions receives connections with relatively strong positive weights, $\rho _0$, from a specialized intermediate unit (for example, from $\xi _{(4),1}^t$ to $x_N^t$). The intermediate unit receives connections from the environmental variables that take the values of the rewards within one step of the animal's position, $y_k^t$ ($4\leq k\leq 12$), with the weight-values $v_0$, $-v_0$ and $0$, as illustrated in Fig.6. These initial values of the weight parameters make the neurons controlling motion take a value of 1 when relative amounts of the reward in the corresponding direction are large. We chose the other weight parameters with small random values in accordance with the following:
$\rho _{ij}\sim [-0.05:0.05]\ (i\leq N_2)$; $v_{ij}^{(\ell )}\sim [-0.05:0.05]\ (\ell =1,2)$; $\rho _{ij}=0$ if $N_2< i\leq N_4=N$ and $(i, j)\neq (N,1),(N-1,2),(N-2,1),(N-3,2)$; $v_{ij}^{(\ell )}=0$ ($\ell =3,4$ except the red and blue synaptic weights in Fig.6); $w_{ij}^{(\ell )}\sim [-0.05:0.05]\ (\ell =1,2)$; $w_{ij}^{(\ell )}=0\ (\ell =3,4)$; $h_0=\log 20$; $h_j^{(\ell )}=0$; $\kappa _{ij}\sim [-0.05:0.05]$; $z_{jk}^{(\ell )}\sim [-0.05:0.05]$; $u_{jk}^{(\ell )}\sim [-0.05:0.05]$; $q_{jk}^{(\ell )}\sim [-0.05:0.05]$; $m_j^{(\ell )}=0$; $m_0=0$.\\
In the updates of the model parameters according to Eq.(\ref{learning rule}), we used the following (fixed) values of learning parameters: $\epsilon =3.0\times 10^{-5}$; $\tau =50$.
\begin{figure}
\includegraphics[width=85mm]{fig6.eps}
\caption{(Color online). Initial values of model parameters for synaptic weights to the movement-related neurons.}
\end{figure}
\end{widetext}
|
{
"timestamp": "2015-04-14T02:13:37",
"yymm": "1504",
"arxiv_id": "1504.03132",
"language": "en",
"url": "https://arxiv.org/abs/1504.03132"
}
|
\section{Introduction}
In many different areas of science, physical limitations make it impossible
to measure the sign (phase in the complex case) of a signal but obtaining amplitudes is
relatively easy. Well known examples are X-ray crystallography, astronomy, or
diffraction imaging \cite{mill90,fienup93,qui10}.
The problem of retrieving a signal up to a global sign (phase in the complex case)
from intensity measurements is often referred to as \emph{phase
retrieval}. More formally, let $\reals_\sim^n$ be the set of equivalence classes
$[\vecx]=\{\vecx\}\cup\{-\vecx\}$ with $\vecx\in\reals^n$. Phase retrieval is the problem of recovering
$[\vecx]\in\reals_\sim^n$ from $m$ phaseless measurements of the
form\footnote{For a vector $\vecu\in\reals^k$, we define the element-wise
absolute value operation as $|\vecu|=\tp{(|u_1|,\dots,|u_k|)}$.}
$\vecy=|\matA\vecx| \in \reals^m$
with measurement matrix $\matA\in\reals^{m\times n}$.
It is by no means clear how large $m$ has to be to allow for recovery of
$[\vecx]\in\reals^n_\sim$ from $m$ phaseless measurements. Thus
from the very beginning, there have been a number of works regarding
recovery conditions for this problem in the context of specific
applications \cite{bruck79}. More recently, this question has been studied in
more abstract terms, asking for the smallest number $m$ of phaseless measurements that
is required to make the mapping
$[\vecx]\mapsto |\matA\vecx|$ injective without imposing structural
assumptions on $\matA$. In \cite{balan06}, the authors showed that
at least $2n-1$ such measurements are necessary and generically sufficient to
guarantee injectivity.
Furthermore, it was shown that semidefinite programming can be used to
recover $[\vecx]$ if $\rmatA$ is random with i.i.d. Gaussian entries or with
i.i.d. rows that are uniformly distributed on a sphere, as long as $m \geq
c_0 n$ for a sufficiently large constant $c_0$ \cite{cand12a}. Other
phase retrieval methods for which theoretical performance guarantees are
available can be found, e.g., in \cite{cand13a,wald12,netr12,cand14a}.
Recently, there has been also interest in \emph{sparse phase retrieval},
where the number $s$ of nonzero coefficients of the vector $\vecx$ is much
smaller than $n$. This a-priori knowledge about $\vecx$
can be used to reduce the number of measurements significantly. For instance,
$\mathcal{O}(s \log(n/s))$ measurements were shown to be sufficient for
stable sparse phase retrieval \cite{eldar14}. If the rows of the measurement
matrix $\matA$ are a generic choice of vectors in $\reals^n$, injectivity of
the mapping $[\vecx]\mapsto |\matA\vecx|$ is guaranteed provided that
$m\geq 2s$ \cite{akta14}.
\emph{Contributions:}
Following the approach introduced for compressed sensing \cite{wuve10} and signal separation \cite{stribo13}
problems, we formulate phase retrieval as an analog source coding problem.
Assuming that the unknown vector $\rvecx$ is random with a certain distribution,
we derive asymptotic recovery results for $[\rvecx]$. Our results hold for Lebesgue almost all
(a.a.) measurement matrices $\matA$. However, our results are in terms of probability of error (with respect to the distribution
of $\rvecx$)
and hence
do not provide worst-case guarantees.
Specifically, we study the asymptotic setting $n \rightarrow \infty$ where the vector $\vecx$ is a realization of a
random process; for each $n$, we let $m=\lfloor Rn \rfloor$ for a parameter
$R$, which we denote \emph{compression rate}.
In Theorem \ref{thm1} we show that we can recover $[\rvecx]$ from $m$
phaseless measurements with arbitrarily small probability of error for a.a.
measurement matrices $\matA$, provided that $n$ is sufficiently large and the
compression rate $R$ is larger than the (lower) Minkowski dimension
compression rate (see Definition \ref{dfndimrate}) of $\rvecx$. It is
remarkable that the obtained result is identical to the corresponding result
in compressive sensing \cite{wuve10} where $\vecy=\matA\vecx$, so that we can
conclude that \emph{in terms of achievability results, phaseless linear
measurements are ``as good'' as linear measurements with full phase
information:} Ignoring the sign of $m$ measurements only leaves us with an
ambiguity with respect to an overall sign factor of $\vecx$.
\emph{Notation:}
Roman letters $\matA,\matB,\ldots$ and $\veca,\vecb,\ldots$
designate deterministic matrices and vectors, respectively.
Boldface letters $\rmatA,\rmatB,\ldots$ and $\rveca,\rvecb,\ldots$ denote random matrices and
random vectors, respectively.
For the distribution of a random matrix $\rmatA$ and a random vector $\rveca$, we write $\mu_\rmatA$ and $\mu_\rveca$, respectively.
The $i$th component of the vector $\vecu$ (random vector $\rvecu$) is $\vecuc_i$ ($\rvecuc_i$).
The superscript $\tp{}$ stands for transposition. For a matrix $\matA$, $\tr(\matA)$ denotes its trace.
The identity matrix of suitable size is denoted by $\matI$.
For a vector $\vecu$, we write $\|\vecu\|=\sqrt{\tp{\vecu}\vecu}$ for its Euclidean norm.
For the Euclidean space $(\reals^k,\|\cdot\|)$, we denote the open ball of
radius $r$ centered at $\vecu\in \reals^k$ by $\setB_k(\vecu,r)$, $V(k,r)$
stands for its volume.
The Borel sigma algebra on $\reals$ is denoted by $\colB_\reals$.
We write $\reals_\geq$ for the set of nonnegative real numbers with Borel sigma algebra $\colB_{\reals_\geq}$.
For $\vecu,\vecv\in\reals^k$, $\vecu\equivalent \vecv$ means that either $\vecu=\vecv$ or $\vecu=-\vecv$ and we write for the corresponding equivalence classes $[\vecu]=\{\vecu\}\cup\{-\vecu\}$.
For a set $\setS\subseteq\reals^k$, $\setS_\sim=\{[\vecu] \mid \vecu\in\setS\}$.
The indicator function on a set $\setU$ is denoted by $\ind{\setU}$.
\section{Main Results}
We start by formulating phase retrieval as a source coding problem.
\begin{dfn} (Source vector)\label{dfnsource}
Let $(\rvecxc_i)_{i\in\naturals}$ be a stochastic process on
$(\reals^\naturals,\colB_\reals^{\otimes\naturals})$. Then, for
$n\in\naturals$, the source vector $\rvecx$ of length $n$ is given by
$\rvecx=\tp{(\rvecxc_1,\dots,\rvecxc_n)}\in\reals^n$.
\end{dfn}
\begin{dfn} (Code, achievable rate) \label{dfncode}
For $\rvecx$ as in Definition \ref{dfnsource} and $\varepsilon >0$, an
$(n,m)$ code consists of
\begin{enumerate}[(i)]
\item measurements $|\matA \cdot|:\reals^n\to \reals^m_\geq$;
\item a decoder $g: \reals^m_\geq\to \reals^n$ that is measurable with
respect to $\colB_{\reals_\geq}^{\otimes m}$ and
$\colB_\reals^{\otimes n}$.
\end{enumerate}
We call $R$ with $0< R\leq 1$ an $\varepsilon$-achievable rate if there
exists an $N(\varepsilon)\in\naturals$ and a sequence of $(n,\lfloor
Rn\rfloor)$ codes with decoders $g$ such that
\begin{align*}
\matP[g(|\matA \rvecx|)\not\equivalent \rvecx ] \leq \varepsilon
\end{align*}
for all $n\geq N(\varepsilon)$.
\end{dfn}
Next, we introduce the Minkowski dimension compression rate for source
vectors.
\begin{dfn} (Minkowski dimension)\label{dfndim}
Let $\setU$ be a nonempty bounded set in $\reals^n$. The lower Minkowski
dimension of $\setU$ is defined as
\begin{align*}
\underline{\dim}_\mathrm{B}(\setU)=\liminf_{\rho\to 0} \frac{\log N_\setU(\rho)}{\log \frac{1}{\rho}}
\end{align*}
and the upper Minkowski dimension of $\setU$ is defined as
\begin{align*}
\overline{\dim}_\mathrm{B}(\setU)=\limsup_{\rho\to 0} \frac{\log N_\setU(\rho)}{\log \frac{1}{\rho}}
\end{align*}
where $N_\setU(\rho)$ is the covering number of $\setU$ given by
\begin{align*}
N_\setU(\rho)&=\min\Big\{k \in\naturals\mid \setU\subseteq \bigcup_{i\in\{1,\dots,k\}} \setB_n(\vecu_i,\rho),\ \vecu_i\in \reals^n\Big\}.
\end{align*}
If $\underline{\dim}_\mathrm{B}(\setU)=\overline{\dim}_\mathrm{B}(\setU)$, we
write $\dim_\mathrm{B}(\setU)$.
\end{dfn}
\begin{dfn}(Minkowski dimension compression rate)\label{dfndimrate}
For $\rvecx$ from Definition \ref{dfnsource} and $\varepsilon >0$, we define
the lower Minkowski dimension compression rate as
\begin{align}
\underline{R}_\mathrm{B}(\varepsilon)&=\limsup_{n\to\infty} \underline{a}_n(\varepsilon),\quad\text{where} \nonumber \\
\underline{a}_n(\varepsilon)&=\inf\Big\{\frac{\underline{\dim}_\mathrm{B}(\setU)}{n} \;\Big\vert\; \setU \subset\reals^n,\ \matP[\rvecx\in\setU]\ \geq 1-\varepsilon\Big\}.\nonumber
\end{align}
and the upper Minkowski dimension compression rate as
\begin{align}
\overline{R}_\mathrm{B}(\varepsilon)&=\limsup_{n\to\infty} \overline{a}_n(\varepsilon),\quad\text{where} \nonumber \\
\overline{a}_n(\varepsilon)&=\inf\Big\{\frac{\overline{\dim}_\mathrm{B}(\setU)}{n} \;\Big\vert\; \setU \subset\reals^n,\ \matP[\rvecx\in\setU]\ \geq 1-\varepsilon\Big\}.\nonumber
\end{align}
The sets $\setU$ in the definitions for $\underline{a}_n(\varepsilon)$ and
$\overline{a}_n(\varepsilon)$ are assumed to be nonempty and bounded.
\end{dfn}
\begin{exa}\label{exa1}
The source vector $\rvecx$ from Definition \ref{dfnsource} has a
mixed discrete-continuous distribution if for each $n\in \naturals$ the
random variables $\rvecxc_i$, $i\in\{1,\dots,n\}$, are independent and
distributed according to
\begin{align*}
\mu_{\rvecxc_i}=(1-\lambda)\mu_{{d}}+\lambda\mu_{{c}},\quad i\in\{1,\dots, n\}
\end{align*}
where $0\leq \lambda\leq 1$ is the mixing parameter, $\mu_{{c}}$ is a
distribution on $(\reals,\setB_\reals)$, absolutely continuous with respect
to Lebesgue measure, and $\mu_{{d}}$ is a discrete distribution. Then,
\cite[Th. 15]{wuve10}
\begin{align}
\underline{R}_\mathrm{B}(\varepsilon)=\overline{R}_\mathrm{B}(\varepsilon)= \lambda,\quad 0<\varepsilon <1.\nonumber
\end{align}
\end{exa}
The following result states that every rate $R>\underline{R}_\mathrm{B}(\varepsilon)$ is $\varepsilon$-achievable for Lebesgue a.a. matrices $\matA$.
\begin{thm}\label{thm1}
Let $0<\varepsilon < 1$ and $\rvecx$ as in Definition \ref{dfnsource}.
Then, for Lebesgue a.a. matrices
$\matA\in\reals^{m\times n}$ with $m=\lfloor Rn\rfloor$, $R$ is an
$\varepsilon$-achievable rate provided that $R>\underline{R}_\mathrm{B}(\varepsilon)$.
\end{thm}
\begin{proof}
Since $R>\underline{R}_\mathrm{B}(\varepsilon)$ and $m=\lfloor Rn\rfloor$,
Definition \ref{dfndimrate} implies that there exists a sequence of nonempty
bounded sets $\setU_n\subseteq \reals^n$ and an $N(\varepsilon)\in\naturals$
such that
\begin{align}
\underline{\dim}_\mathrm{B}(\setU) &< m\label{eq:nk}\\
\matP\big[\rvecx\in\setU\big]&\geq 1-\varepsilon \label{eq:ek}
\end{align}
for all $\setU=\setU_n$ with $n\geq N(\varepsilon)$. In the remainder of the
proof we assume that $n$ is sufficiently large for \eqref{eq:nk} and
\eqref{eq:ek} to hold.
The claim now follows from Proposition \ref{pro1} below.
\end{proof}
\begin{prp}\label{pro1}
Let $\varepsilon \geq 0$, $\rvecx\in\reals^n$ a random vector, and
$\setU\subseteq\reals^n$ a nonempty bounded set with
$\matP[\rvecx\in\setU]\geq 1-\varepsilon$. Then, for Lebesgue
a.a. matrices $\matA\in\reals^{m\times n}$, there exists a decoder $g$ with
$\matP[g(|\matA \rvecx|)\not\equivalent \rvecx ] \leq \varepsilon$ provided
that $\underline{\dim}_\mathrm{B}(\setU) < m$.
\end{prp}
\begin{proof}
See Section \ref{proofthm1}.
\end{proof}
\begin{rem}
By \cite[Sec. 3.2, Properties (i)--(iii)]{fa90}, the lower Minkowski
dimension of any bounded nonempty subset in $\reals^n$ containing only
vectors
with no more than $s$ nonzero entries is at most $s$.
Therefore, Proposition \ref{pro1} implies that any $s$-sparse random vector
$\rvecx\in\reals^n$ can be recovered with arbitrarily small probability of
error (by increasing the size of the set $\setU$ in Proposition \ref{pro1}),
provided that $m>s$. This result holds for an arbitrary distribution of
$\rvecx$ and a.a. matrices $\matA\in\reals^{m\times n}$. The best known
recovery threshold for deterministic $s$-sparse vectors is $m\geq 2s$
\cite{akta14}.
\end{rem}
\begin{rem}\label{remmc}
It is worth noting that formally phase retrieval can be formulated as a
matrix completion problem with measurements
$\vecyc_i^2=\tr(\veca_i\tp{\veca_i}\vecx\tp{\vecx})$ using rank-one
measurement matrices $\matA_i=\veca_i\tp{\veca_i}$, $i=1,\dots,m$. However,
compared to the rank-one measurement matrices used in the matrix completion
problem \cite{cazh14,ristbo15}, the matrices $\veca_i\tp{\veca_i}$ are
symmetric. This complicates the proof of Proposition \ref{pro1}
significantly and forces us to develop a novel
concentration of measure result (Lemma \ref{lemcomindep}). On the other hand, in phase retrieval we are interested in
recovering symmetric rank-one matrices $\vecx\tp{\vecx}$ (which is equivalent
to the recovery of $[\vecx]$), whereas matrix completion deals with the
recovery of arbitrary low-rank matrices.
\end{rem}
In the mixed discrete-continuous case we can strengthen the result of Theorem \ref{thm1} through the following lemma.
\begin{lem}
Let $0<\varepsilon < 1$ and $\rvecx$ be distributed according to the mixed
discrete-continuous distribution in Example \ref{exa1} with mixing parameter
$\lambda$. Then, for Lebesgue a.a. matrices $\matA\in\reals^{m\times n}$ with
$m=\lfloor Rn\rfloor$, $R$ is $\varepsilon$-achievable provided that
$R>\lambda$. Moreover, $R\geq \lambda$ is also a necessary condition for $R$
being $\varepsilon$-achievable.
\end{lem}
\begin{proof} Achievability: Follows from Theorem \ref{thm1} and Example \ref{exa1}.
Converse: Suppose that a rate $R<\lambda$ is $\varepsilon$-achievable for some
$\varepsilon$ with $0<\varepsilon<1$. This implies that there exists a set $\setK\subseteq\reals^n$
and a matrix $\matA\in\reals^{m\times n}$ with $m=\lfloor Rn\rfloor$ such that
\begin{enumerate}[(a)]
\item
$\Pr[\rvecx\in\setK]\geq 1-\varepsilon$;
\item
$|\matA \cdot|$ is one-to-one on $\setK_\sim$
\end{enumerate}
for $n$ sufficiently large. From $(b)$ it follows that there can be at most
one equivalence class $[\vecu]\in \setK_\sim$ with $\matA \vecu=\matA
(-\vecu)= 0$
(if there was more than one such
equivalence class then the mapping $|\matA \cdot|$ would not be one-to-one on
$\setK_\sim$).
Suppose first that there is no equivalence class $[\vecu]=\{\vecu,-\vecu\}\in \setK_\sim$ with $\matA \vecu=\matA (-\vecu)= 0$ and $\vecu\neq\veczero$. Then, (b) implies that $\matA$ is one-to-one on $\setK$ which, together with (a) and $R<\lambda$, leads to a contradiction to the converse part of \cite[Thm. 6]{wuve10}.
Now suppose that there is an equivalence class $[\vecu]=\{\vecu,-\vecu\}\in
\setK_\sim$ with $\matA \vecu=\matA (-\vecu)= 0$ and $\vecu\neq\veczero$. Let
$\tilde R$ be such that $R < \tilde R < \lambda$ and set $\tilde m=\lfloor
\tilde R n\rfloor$. Then, $\tilde m > m$ for $n$ sufficiently large. Let
$\tilde\matA=\tp{(\tp{\matA},\vecu,\veczero,\dots,\veczero)}\in\reals^{\tilde
m\times n}$. Then, (b) implies that $\tilde\matA$ is one-to-one on $\setK$
which, together with (a) and $\tilde R<\lambda$, leads to a contradiction to
the converse part of \cite[Thm. 6]{wuve10}.
\end{proof}
\section{Proof of Proposition \ref{pro1}}\label{proofthm1}
Let
\begin{align*}
\setF(\vecy)
&=\mleft\{\vecu\in\reals^n \big|\vecu\in\setU, |\matA \vecu|=\vecy\mright\}\\
&\phantom{=}\cup\mleft\{\vecu \in\reals^n\big|-\vecu\in\setU, |\matA \vecu|=\vecy\mright\},\quad \vecy\in\reals^m_\geq.
\end{align*}
For a vector $\vecu\in\setF(\vecy)\setminus\{\veczero\}$, let $\bar \vecuc$
denote the first nonzero component of $\vecu$. We then define the reduced set
\begin{align*}
\bar\setF(\vecy)=\mleft\{\vecu\in\setF(\vecy)\setminus\{\veczero\}\big| \bar\vecuc=|\bar\vecuc|\mright\}\cup(\setF(\vecy)\cap\{\veczero\}),\quad \!\!\vecy\in\reals^m_\geq.
\end{align*}
We define the decoder $g: \reals^m_\geq\to \reals^n$ by
\begin{align*}
g(\vecy)=
\begin{cases}
\vecu,& \text{if}\ \bar\setF(\vecy) = \{\vecu\}\\
\vece, & \text{else}
\end{cases}
\end{align*}
where $\vece$ is some fixed vector in the complement of $\setU$ (used
to declare a decoding error). Then, we have
\begin{align}
&\matP\big[g(|\matA\rvecx|)\not\equivalent\rvecx\big]\nonumber\\
&=\matP\big[g(|\matA\rvecx|)\not\equivalent\rvecx,\rvecx\in\setU\big]+\matP\big[g(|\matA\rvecx|)\not\equivalent\rvecx,\rvecx\notin\setU\big]\nonumber\\
&\leq\matP\big[g(|\matA\rvecx|)\not\equivalent\rvecx,\rvecx\in\setU\big]+\varepsilon\nonumber\\
&=\matP\mleft[\exists\vecu\in\setU \big| \vecu\not\equivalent\rvecx, |\matA\vecu|=|\matA\rvecx|, \rvecx\in\setU\mright]+\varepsilon \label{eq:errorbound1}
\end{align}
where \eqref{eq:errorbound1} follows from the definition of the decoder. Fix an arbitrary $r>0$. Suppose that we can
show that
\begin{align}\label{eq:havetoshow}
P(\vecx) &= \matP\big[\exists \vecu\in\setU\ \text{with}\ \vecu\not\equivalent\vecx, |\rmatA\vecu|=|\rmatA\vecx|\big]=0,\quad \vecx\in\setU
\end{align}
where $\rmatA\in\reals^{m\times n}$ has independent rows that are uniformly
distributed on $\setB_n(\veczero,r)$. Then,
\begin{align}
&\int\limits_{\setA(r)}\matP\mleft[\exists\vecu\in\setU \big| \vecu\not\equivalent\rvecx, |\matA\vecu|=|\matA\rvecx|, \rvecx\in\setU\mright]\operatorname{d}\!\mu_\rmatA \nonumber\\
&=\int\limits_{\setU}
\matP\big[\exists \vecu\in\setU\ \text{with}\ \vecu\not\equivalent\vecx, |\rmatA\vecu|=|\rmatA\vecx|\big]
\operatorname{d}\!\mu_\rvecx\nonumber\\
&=0\label{eq:argumentFubini}
\end{align}
where we used Fubini's Theorem and set $\setA(r)=\setB_n(\veczero,r)\times\dots\times \setB_n(\veczero,r)$.
Since $r$ is
arbitrary, \eqref{eq:argumentFubini} implies that
\begin{align}\label{eq:zeroaa}
\matP\mleft[\exists\vecu\in\setU \big| \vecu\not\equivalent\rvecx, |\matA\vecu|=|\matA\rvecx|, \rvecx\in\setU\mright]=0
\end{align}
for Lebesgue a.a. matrices $\matA$. Hence, combining
\eqref{eq:errorbound1} and \eqref{eq:zeroaa} proves the Proposition provided
that we can show that \eqref{eq:havetoshow} holds, which is done in Section
\ref{proofhavetoshow}
\section{Proof of \eqref{eq:havetoshow}}\label{proofhavetoshow}
Suppose first that $\vecx=\veczero$. Then, $P(\vecx)=0$ if and only if
\begin{align}\label{eq:casexzero}
\matP\big[\exists \vecu\in\setU\setminus\{\veczero\}\ \text{with}\ \rmatA\vecu=\veczero]=0.
\end{align}
Since $\underline{\dim}_\mathrm{B}\big(\setU\big)<m$, \eqref{eq:casexzero} follows from \cite[Prop. 1]{stribo13}.
Therefore, we can assume in what follows that $\vecx\neq\veczero$.
We can upper-bound $P(\vecx)\leq P_1(\vecx)+P_2(\vecx)$
with
\begin{align*}
P_i(\vecx)=\matP\big[\exists \vecu\in\setU_i(\vecx)\ \text{with}\ |\rmatA\vecu|=|\rmatA\vecx|\big],\quad i\in\{1,2\}
\end{align*}
where we defined
\begin{align}
\setU_1(\vecx)&=\{\vecu\in\setU \mid \rank (\vecx,\vecu) =2\}\nonumber\\%\label{eq:U1}\\
\setU_2(\vecx)&=\{\vecu\in\setU \mid \rank (\vecx,\vecu) =1\}\setminus \{\vecu\in\setU| \vecu\equivalent\vecx\}.\nonumbe
\end{align}
We have to show that $P_i(\vecx)=0$ for $i\in\{1,2\}$.
First, we establish $P_2(\vecx)=0$. We have (recall that $\vecx\neq \veczero$)
\begin{align}
&P_2(\vecx)\nonumber\\
&=
\matP\big[\exists \vecu\in\setU\ \text{with}\ \rank (\vecx,\vecu) =1, \vecu\not\sim\vecx, |\rmatA\vecu|=|\rmatA\vecx|]\nonumber\\
&=\matP\big[\rmatA\vecx=\veczero]\nonumber\\
&=0\nonumber
\end{align}
where we used \cite[Prop. 1]{stribo13} together with $\underline{\dim}_\mathrm{B}\big(\{\vecx\}\big)=0$ in the last step. It remains to show that $P_1(\vecx)=0$.
To this end, we first present an auxiliary lemma.
\begin{lem}\label{lemprobtri}
Let $r>0$, $\emptyset \neq \setS\subseteq \setB_n(\veczero,L)$, $\rho
>0$, $\vecx\in \setB_n(\veczero,L)$, and $\rmatA\in\reals^{m\times n}$ with
independent rows that are uniformly distributed on $\setB_n(\veczero,r)$.
Then, there exist $\vecs_l(\rho)\in \setS$, $l=1,\dots,N_\setS(\rho)$ with $N_\setS(\rho)$ being the
covering number of $\setS$, such that
\begin{align}\label{eq:probtri}
&\matP\big[\exists \vecu\in\setS\ \text{with}\ \big\||\rmatA\vecu|-|\rmatA\vecx|\big\|\leq\rho\big]\nonumber\\
&\leq
\sum_{l=1}^{N_\setS(\rho)}\matP\mleft[\big||\tp{\rveca}\vecs_l(\rho)|^2 - |\tp{\rveca}\vecx|^2\big|\leq 2Lr(2r+1)\rho\mright]^m
\end{align}
where $\rveca$ is uniformly distributed on $\setB_n(\veczero,r)$.
\end{lem}
\begin{proof}
Let $\setS \subseteq \bigcup_{l\in\{1,\dots,N_\setS(\rho)\}}
\setB_n(\vecv_l(\rho),\rho)$, $\vecv_l(\rho)\in
\reals^n$, be a \emph{minimal} covering of $\setS$ according to the
definition of the covering number, cf. Definition \ref{dfndim}. Then, there
exist $\vecs_l(\rho)\in
\setS\cap\setB_n(\vecv_l(\rho),\rho)$ for all
$l=1,\dots,N(\rho)$. Hence, the balls
$\setB_n(\vecs_l(\rho),2\rho)$ cover the set $\setS$ and have
centers in $\setS$. We can upper bound the lhs in \eqref{eq:probtri} by
\begin{align}
&\matP\big[\exists \vecu\in\setS\ \text{with}\ \big\||\rmatA\vecu|-|\rmatA\vecx|\big\|\leq\rho\big]\nonumber\\
&\!\leq \!\!\sum_{l=1}^{N_\setS(\rho)}\!\! \matP\big[\exists \vecu\in\setS\cap\setB_n(\vecs_l(\rho),2\rho)\ \text{with}\ \big\||\rmatA\vecu|-|\rmatA\vecx|\big\|\leq\rho\big]\nonumber\\
&\!\leq \!\!\sum_{l=1}^{N_\setS(\rho)}\!\! \matP\big[\exists \vecu\in\setS\cap\setB_n(\vecs_l(\rho),2\rho)\ \text{with}\
\big||\tp{\rveca}\vecu| - |\tp{\rveca}\vecx|\big|\leq\rho\big]^m\label{eq:Pr2}
\end{align}
where $\eqref{eq:Pr2}$ follows from the fact that the rows of $\rmatA$ are
independent and uniformly distributed on $\setB_n(\veczero,r)$. Using the
triangle inequality we obtain
\begin{align}\label{eq:tri1}
\big||\tp{\rveca}\vecs_l(\rho)| - |\tp{\rveca}\vecx|\big|
&\leq
\big||\tp{\rveca}\vecx| - |\tp{\rveca}\vecu|\big|+\big||\tp{\rveca}\vecu| - |\tp{\rveca}\vecs_l(\rho)|\big|.
\end{align}
The second term on the rhs
of \eqref{eq:tri1} can be further upper bounded by
\begin{align}
\big||\tp{\rveca}\vecu| - |\tp{\rveca}\vecs_l(\rho)|\big|
&\leq \big|\tp{\rveca}(\vecu -\vecs_l(\rho))\big|\nonumber\\
&\leq \|\rveca\|\|\vecu -\vecs_l(\rho)\|\nonumber\\
&\leq 2r\rho\label{eq:tri2}
\end{align}
where \eqref{eq:tri2} follows from $\vecu\in
\setB_n(\vecs_l(\rho),2\rho)$ and
$\rveca\in \setB_n(\veczero,r)$.
Combining \eqref{eq:tri1} and \eqref{eq:tri2} gives
\begin{align}\label{eq:tri3}
\big||\tp{\rveca}\vecx| - |\tp{\rveca}\vecu|\big|\geq
\big||\tp{\rveca}\vecs_l(\rho)| - |\tp{\rveca}\vecx|\big|
-2r\rho.
\end{align}
Using \eqref{eq:tri3} in \eqref{eq:Pr2} yields
\begin{align}
&\matP\big[\exists \vecu\in\setS\ \text{with}\ \big\||\rmatA\vecu|-|\rmatA\vecx|\big\|\leq\rho\big]\nonumber\\
&\leq \sum_{l=1}^{N_\setS(\rho)}\matP\mleft[\big||\tp{\rveca}\vecs_l(\rho)| - |\tp{\rveca}\vecx|\big|\leq (2r+1)\rho\mright]^m\nonumber\\
&\leq
\sum_{l=1}^{N_\setS(\rho)}\matP\mleft[\big||\tp{\rveca}\vecs_l(\rho)|^2 - |\tp{\rveca}\vecx|^2\big|\leq 2Lr(2r+1)\rho\mright]^m
\label{eq:Pr3}
\end{align}
where $\eqref{eq:Pr3}$ follows from $\big||\tp{\rveca}\vecs_l(\rho)|^2
- |\tp{\rveca}\vecx|^2\big|=\big|(|\tp{\rveca}\vecs_l(\rho)| +
|\tp{\rveca}\vecx|)(|\tp{\rveca}\vecs_l(\rho)| -
|\tp{\rveca}\vecx|)\big|\leq 2Lr\big||\tp{\rveca}\vecs_l(\rho)| -
|\tp{\rveca}\vecx|\big|$.
\end{proof}
We now continue with the proof of $P_1(\vecx)=0$. Since $\setU$ is a bounded set, there exists an $L\in\reals$ such that
\begin{align}\label{eq:boundedU}
\|\vecu\|\leq L,\quad \vecu\in\setU.
\end{align}
We define the sets $\setT_{j}(\vecx)$ by
\begin{align*}
\setT_j(\vecx)&=\mleft\{\vecu\in\setU_1(\vecx)\Big| \sqrt{\|\vecu\|^2\|\vecx\|^2-|\tp{\vecu}\vecx|^2}>\frac{1}{j}\mright\},\quad j\in\naturals.
\end{align*}
Since
\begin{align*
P_1(\vecx)\leq \sum_{j\in\naturals}\matP\big[\exists \vecu\in\setT_{j}(\vecx)\ \text{with}\ |\rmatA\vecu|=|\rmatA\vecx|\big]
\end{align*}
it is sufficient to prove that
\begin{align
P_1^{(j)}(\vecx)=\matP\big[\exists \vecu\in\setT_{j}(\vecx)\ \text{with}\ |\rmatA\vecu|=|\rmatA\vecx|\big]&=0\nonumber
\end{align}
for all $j\in\naturals$.
Suppose, by contradiction, that there exists a $j\in\naturals$ such that $P_1^{(j)}(\vecx)>0$. Then,
\begin{align}
\liminf_{\rho\to 0} \frac{\log P_1^{(j)}(\vecx)}{\log \frac{1}{\rho}}
&=0.\label{eq:contr}
\end{align}
Furthermore, $\setT_{j}(\vecx)
\neq \emptyset$ and by \cite[Sec. 3.2, Property (ii)]{fa90} (recall that
$\setT_{j}(\vecx)\subseteq \setU_1(\vecx)\subseteq\setU$ ) we get
\begin{align}\label{eq:dimT}
\underline{\dim}_\mathrm{B}\big(\setT_{j}(\vecx)\big)<m.
\end{align}
We have
\begin{align}
&
\liminf_{\rho\to 0} \frac{\log P_1^{(j)}(\vecx)}{\log \frac{1}{\rho}} \nonumber\\
&
=\liminf_{\rho\to 0} \frac{\log\matP\big[\exists \vecu\in\setT_{j}(\vecx)\ \text{with}\ |\rmatA\vecu|=|\rmatA\vecx|\big]}{\log \frac{1}{\rho}}\nonumber\\
&
\leq
\liminf_{\rho\to 0}\nonumber\\
&
\ \ \ \frac{\log
\Big(\!
\sum_{l=1}^{N_{\!\setT_{j}(\vecx)}(\rho)}\!\!\matP\mleft[\big||\tp{\rveca}\vecs^{(j)}_{l}(\rho , \vecx)|^2 \! - \!|\tp{\rveca}\vecx|^2\big|\leq\tilde\rho\mright]^m\Big)}{\log \frac{1}{\rho}} \label{eq:Pr4b}\\
&\leq
\liminf_{\rho\to 0}
\frac{\log
\Big(
{\tilde\rho}^m\sum_{l=1}^{N_{\!\setT_{j}(\vecx)}(\rho)}{f\big(\tilde\rho,r,\vecs^{(j)}_{l}(\rho , \vecx),\vecx\big)}^m \Big)}{\log \frac{1}{\rho}} \label{eq:Pr4c}\\
&\leq
\liminf_{\rho\to 0}
\frac{\log \big({\tilde\rho}^m N_{\setT_{j}(\vecx)}(\rho) {\tilde f(\tilde\rho,r,L,j)}^m\big)}{\log \frac{1}{\rho}}\label{eq:Pr4d}\\
&=\underline{\dim}_\mathrm{B}(\setT_{j}(\vecx))-m + m \lim_{\rho\to 0}
\frac{\log \tilde f(\tilde\rho,r,L,j)}{\log \frac{1}{\rho}}\nonumber\\
&=\underline{\dim}_\mathrm{B}(\setT_{j}(\vecx))-m\nonumber\\
&<0\label{eq:Pr4e}
\end{align}
where in
\eqref{eq:Pr4b} we applied Lemma \ref{lemprobtri} with
$\setS=\setT_{j}(\vecx)$ and set $\tilde\rho=2Lr(2r+1)\rho$,
\eqref{eq:Pr4c} follows from Lemma \ref{lemcomindep} below with
$\vecu=\vecs^{(j)}_{l}(\rho , \vecx)$, $\vecv=\vecx$, and $\delta=\tilde\rho$
where $f$ is defined in \eqref{eq:fR}, in
\eqref{eq:Pr4d} we used that
\begin{align}
&f\big(\tilde\rho,r,\vecs^{(j)}_{l}(\rho , \vecx),\vecx\big)\nonumber\\
&\leq
\tilde f(\tilde\rho,r,L,j)\nonumber\\
&=\frac{2(2r)^{n-2}j}{V(n,r)}\Big(1+\log\Big(2+\frac{8r^2L^2}{\tilde\rho}\Big)\Big),\quad l=1,\dots, N_{\!\setT_{j}(\vecx)}(\rho)\nonumber
\end{align}
which follows from \eqref{eq:boundedU} and the fact that $\vecs^{(j)}_{l}(\rho , \vecx) \in
\setT_{j}(\vecx)$, $l=1,\dots, N_{\setT_{j}(\vecx)}(\rho)$, and
in
\eqref{eq:Pr4e} we applied \eqref{eq:dimT}.
But \eqref{eq:Pr4e} is a contradiction to \eqref{eq:contr}. Therefore, $P_1^{(j)}(\vecx)=0$ for all $j\in\naturals$,
which implies in turn that $P_1(\vecx)=0$ and concludes the proof of
\eqref{eq:havetoshow}.
\vspace*{0mm}
\section{Concentration of measure result}
\begin{lem}\label{lemcomindep}
Let $r>0$, $\rveca$ be uniformly distributed on $\setB_n(\veczero,r)$,
$\matC=\vecu\tp{\vecu}-\vecv\tp{\vecv}$ with linearly independent vectors
$\vecu,\vecv\in\reals^n$, and $\delta>0$. Then
\begin{align}\label{eq:bound}
\matP\big[|\tp{\rveca}\matC\rveca|\leq\delta\big] \leq \delta f(\delta,r,\vecu,\vecv)
\end{align}
with
\begin{align}
&f(\delta,r,\vecu,\vecv)=\nonumber\\
&\frac{2(2r)^{n-2}\Big(1+\log\Big(2+\frac{2r^2\big(\|\vecu+\vecv\|\|\vecu-\vecv\|-\big|\|\vecu\|^2-\|\vecv\|^2\big|\big)}{\delta}\Big)\Big)}{\sqrt{\|\vecu\|^2\|\vecv\|^2-|\tp{\vecu}\vecv|^2}V(n,r)}
\label{eq:fR}
\end{align}
\end{lem}
\begin{proof} We have
\begin{align}
&\matP\big[|\tp{\rveca}\matC\rveca|\leq\delta\big]\nonumber\\
&=\frac{1}{V(n,r)}\int\limits_{\setB_n(\veczero,r)} \ind{\mleft\{\veca\in\reals^n \big| |\tp{\veca}\matC\veca|<\delta\mright\}} \operatorname{d}\!\veca\nonumber\\%\label{eq:com1a}\\
&=\frac{1}{V(n,r)}\int\limits_{\setB_n(\veczero,r)} \ind{\mleft\{\veca\in\reals^n \big| |\tp{\veca}\matW\matR
\matJ\tp{\matR}\tp{\matW}\veca|<\delta\mright\}} \operatorname{d}\!\veca\label{eq:com1b}\\
&=\frac{1}{V(n,r)}\int\limits_{\setB_n(\veczero,r)} \ind{\mleft\{\vecb\in\reals^n \big| |\tp{\vecc}\matR
\matJ\tp{\matR}\vecc |<\delta\mright\}} \operatorname{d}\!\vecb\label{eq:com1c}\\
&\leq \frac{(2r)^{n-2}}{V(n,r)}\int\limits_{\setB_2(\veczero,r)} \ind{\mleft\{\vecc\in\reals^2 \big| |\tp{\vecc}\matR
\matJ\tp{\matR}\vecc |<\delta\mright\}} \operatorname{d}\!\vecc \label{eq:com1}
\end{align}
where \eqref{eq:com1b} follows from Lemma \ref{lemQR} with $\matR$ and
$\matJ$ defined in \eqref{eq:JR} and $\matW$ defined in \eqref{eq:W} and
\eqref{eq:com1c} follows from changing variables to $\veca = \bar\matW\vecb$
with $\bar\matW=(\matW, \matZ)\in\reals^{n\times n}$ where
$\matZ\in\reals^{n\times (n-2)}$ is chosen in such a way that
$\bar\matW\tp{\bar\matW}=\matI$ and $\vecc=\tp{(\veccc_1,\veccc_2)}$ with
$\veccc_1=\vecbc_1$ and $\veccc_2=\vecbc_2$.
The bound \eqref{eq:detRJR} on the determinant of the matrix
$\matR\matJ\tp{\matR}$ implies that one eigenvalue of $\matR\matJ\tp{\matR}$,
say $\lambda_1$, is positive and the other eigenvalue of
$\matR\matJ\tp{\matR}$, say $-\lambda_2$, is negative.
We can assume
without loss of generality that $\lambda_1 \geq \lambda_2$. Using the
eigendecomposition
$\matR\matJ\tp{\matR}=\matU\diag(\lambda_1,-\lambda_2) \tp{\matU}$,
where $\matU\in\reals^{2\times 2}$ with $\matU\tp{\matU}=\matI$, and changing
variables to $\vecc=\matU\vecd$, we can further upper bound \eqref{eq:com1}
by
\begin{align}
&\frac{(2r)^{n-2}}{V(n,r)}
\int\limits_{\setB_2(\veczero,r)}
\ind{\mleft\{\vecc\in\reals^2 \big| |\tp{\vecc}\matR\matJ\tp{\matR}\vecc |<\delta\mright\}} \operatorname{d}\!\vecc\nonumber\\
&=
\frac{(2r)^{n-2}}{V(n,r)}
\int\limits_{\setB_2(\veczero,r)}
\ind{\mleft\{\vecd\in\reals^2 \big| \mleft|\lambda_1\vecdc_1^2- \lambda_2\vecdc_2^2\mright|<\delta\mright\}} \operatorname{d}\!\vecd \nonumber\\
&=
\frac{(2r)^{n-2}}{\sqrt{\lambda_1\lambda_2}V(n,r)}\int\limits_{\reals^2}
\ind{\mleft\{\vect\in\reals^2 \big|\frac{\vectc_1^2}{\lambda_1}+\frac{\vectc_2^2}{\lambda_2}\leq r^2\mright\}}\nonumber\\
&\phantom{=\frac{(2r)^{n-2}}{\sqrt{\lambda_1\lambda_2}V(n,r)}\int\limits_{\reals^2}}
\times\ind{\mleft\{\vect\in\reals^2 \big| \mleft|\vectc_1^2- \vectc_2^2\mright|<\delta \mright\}}
\operatorname{d}\!\vect\label{eq:com2a}\\
&\leq
\frac{(2r)^{n-2}}{\sqrt{\lambda_1\lambda_2}V(n,r)}\int\limits_{\reals^2}
\ind{\mleft\{\vect\in\reals^2 \big|\vectc_1^2\leq \lambda_1r^2, \vectc_2^2\leq \lambda_2r^2\mright\}}\nonumber\\
&\phantom{=\frac{(2r)^{n-2}}{\sqrt{\lambda_1\lambda_2}V(n,r)}\int\limits_{\reals^2}}
\times\ind{\mleft\{\vect\in\reals^2 \big| \mleft|\vectc_1^2- \vectc_2^2\mright|<\delta \mright\}}
\operatorname{d}\!\vect\label{eq:com3a}
\end{align}
where in \eqref{eq:com2a} we changed variables to $\vect=\diag(\sqrt{\lambda_1},\sqrt{\lambda_2})\vecd$.
The integral in \eqref{eq:com3a} measures the area that is inside the rectangle $\{\vect\mid \vectc_1^2\leq \lambda_1r^2, \vectc_2^2\leq \lambda_2r^2\}$ and the two hyperbolas $\{\vect\mid\vectc_1^2- \vectc_2^2=\pm\delta\}$ (see Figure \ref{picture}). The bound \eqref{eq:bound} can then be established by performing the following to steps:
\begin{enumerate}
\item deriving an upper bound on the integral in \eqref{eq:com3a}.
\item finding an expression of the eigenvalues of $\matR\matJ\tp{\matR}$ in terms of the vectors $\vecu$ and $\vecv$,
\end{enumerate}
which will be done next.
We have
\pgfplotsset{every axis/.append style={
axis x line=middle,
axis y line=middle,
axis line style={<->},
xlabel={$t_1$},
ylabel={$t_2$},
}}
\tikzset{>=stealth}
\begin{figure}
\begin{center}
\begin{tikzpicture}
\begin{axis}[
xmin=-5,xmax=5,
ymin=-5,ymax=5]
\addplot [red,thick,domain=-2:2] ({cosh(x)}, {sinh(x)});
\addplot [red,thick,domain=-2:2] ({-cosh(x)}, {sinh(x)});
\addplot [red,thick,domain=-2:2] ({sinh(x)},{cosh(x)});
\addplot [red,thick,domain=-2:2] ({sinh(x)},{-cosh(x)});
\addplot[red,dashed] expression {x};
\addplot[red,dashed] expression {-x};
\addplot [blue,thick,domain=-4:4] ({x}, {2});
\addplot [blue,thick,domain=-4:4] ({x}, {-2});
\addplot [blue,thick,domain=-2:2] ({-4}, {x});
\addplot [blue,thick,domain=-2:2] ({4}, {x});
\end{axis}
\end{tikzpicture}
\caption{Intersection of the rectangle
$\{\vect\mid \vectc_1^2\leq \lambda_1r^2, \vectc_2^2\leq \lambda_2r^2\}$
with the two hyperbolas
$\{\vect\mid\vectc_1^2- \vectc_2^2=\pm\delta\}$ for
$\delta=1$, $\lambda_1=16/r^2$, and $\lambda_2=4/r^2$.}\label{picture}
\end{center}
\end{figure}
\begin{align}
&\frac{(2r)^{n-2}}{\sqrt{\lambda_1\lambda_2}V(n,r)}\int\limits_{\reals^2}
\ind{\mleft\{\vect\in\reals^2 \big|\vectc_1^2\leq \lambda_1r^2, \vectc_2^2\leq \lambda_2r^2\mright\}}\nonumber\\
&\phantom{=\frac{(2r)^{n-2}}{\sqrt{\lambda_1\lambda_2}V(n,r)}\int\limits_{\reals^2}}
\times\ind{\mleft\{\vect\in\reals^2 \big| \mleft|\vectc_1^2- \vectc_2^2\mright|<\delta \mright\}}
\operatorname{d}\!\vect\nonumber\\
&\leq
\frac{(2r)^{n-2}}{\sqrt{\lambda_1\lambda_2}V(n,r)}\int\limits_{\reals^2}
\ind{\mleft\{\vect\in\reals^2 \big|\vectc_1^2+\vectc_2^2\leq \delta+2\lambda_2r^2\mright\}}\nonumber\\
&\phantom{=\frac{(2r)^{n-2}}{\sqrt{\lambda_1\lambda_2}V(n,r)}\int\limits_{\reals^2}}
\times\ind{\mleft\{\vect\in\reals^2 \big| \mleft|\vectc_1^2- \vectc_2^2\mright|<\delta \mright\}}
\operatorname{d}\!\vect
\label{eq:com2b}\\
&=
\frac{(2r)^{n-2}}{\sqrt{\lambda_1\lambda_2}V(n,r)}\int\limits_{\opR^2}
\ind{\mleft\{\vecz\in\opR^2 \big|\veczc_1^2+\veczc_2^2\leq \delta+2\lambda_2r^2\mright\}} \nonumber\\
&\phantom{=\frac{(2r)^{n-2}}{\sqrt{\lambda_1\lambda_2}V(n,r)}\int\limits_{\reals^2}}
\times
\ind{\mleft\{\vecz\in\opR^2 \big| \mleft|\veczc_1\veczc_2\mright|<\frac{\delta}{2} \mright\}}
\operatorname{d}\!\vecz
\label{eq:com2c}\\
&\leq
\frac{(2r)^{n-2}}{\sqrt{\lambda_1\lambda_2}V(n,r)}\int\limits_{\opR^2}
\ind{\mleft\{\vecz\in\opR^2 \big|\veczc_1^2\leq \delta+2\lambda_2r^2,\, \veczc_2^2\leq \delta+2\lambda_2r^2\mright\}}\nonumber\\
&\phantom{=\frac{(2r)^{n-2}}{\sqrt{\lambda_1\lambda_2}V(n,r)}\int\limits_{\reals^2}}
\times
\ind{\mleft\{\vecz\in\opR^2 \big| \mleft|\veczc_1\veczc_2\mright|<\frac{\delta}{2} \mright\}}
\operatorname{d}\!\vecz
\nonumber\\
&=
\frac{4(2r)^{n-2}}{\sqrt{\lambda_1\lambda_2}V(n,r)}\int\limits_{\opR_\geq^2}
\ind{\mleft\{\vecz\in\opR^2 \big|\veczc_1\leq \sqrt{\delta+2\lambda_2r^2}\mright\}}\nonumber\\
&\phantom{=\frac{4(2r)^{n-2}}{\sqrt{\lambda_1\lambda_2}V(n,r)}\int\limits_{\opR_\geq^2}}
\times
\ind{\mleft\{\vecz\in\opR^2 \big|\veczc_2\leq \min\big(\sqrt{\delta+2\lambda_2r^2},\frac{\delta}{2\veczc_1}\big)\mright\}}
\operatorname{d}\!\vecz
\nonumber\\
&\leq
\frac{4(2r)^{n-2}}{\sqrt{\lambda_1\lambda_2}V(n,r)}\int\limits_{\opR_\geq^2}
\ind{\mleft\{\vecz\in\opR^2 \big|\veczc_1\leq \frac{\delta}{2\sqrt{\delta+2\lambda_2r^2}}\mright\}}\nonumber\\
&\phantom{=\frac{4(2r)^{n-2}}{\sqrt{\lambda_1\lambda_2}V(n,r)}\int\limits_{\opR_\geq^2}}
\times
\ind{\mleft\{\vecz\in\opR^2 \big|\veczc_2\leq \sqrt{\delta+2\lambda_2r^2}\mright\}}
\operatorname{d}\!\vecz\nonumber\\
&\phantom{=}
+\frac{4(2r)^{n-2}}{\sqrt{\lambda_1\lambda_2}V(n,r)}\int\limits_{\opR_\geq^2}
\ind{\mleft\{\vecz\in\opR^2 \big| \frac{\delta}{2\sqrt{\delta+2\lambda_2r^2}}< \veczc_1\leq\sqrt{\delta+2\lambda_2r^2}\mright\}}\nonumber\\
&\phantom{=\frac{4(2r)^{n-2}}{\sqrt{\lambda_1\lambda_2}V(n,r)}\int\limits_{\opR_\geq^2}}
\times
\ind{\mleft\{\vecz\in\opR^2 \big|\veczc_2\leq \frac{\delta}{2\veczc_1}\mright\}}
\operatorname{d}\!\vecz
\nonumber\\
&=\frac{2\delta(2r)^{n-2}}{\sqrt{\lambda_1\lambda_2}V(n,r)}\Big(1+\log\Big(2+\frac{4\lambda_2r^2}{\delta}\Big) \Big)
\label{eq:com3R}
\end{align}
where in \eqref{eq:com2b} we used that $\vect_2^2 \leq \lambda_2r^2$ and
$|\vectc_1^2- \vectc_2^2|<\delta$ imply $\vectc_1^2 + \vectc_2^2 \leq
\delta+2\lambda_2r^2$, and in \eqref{eq:com2c} we applied the orthogonal transformation
$z_1=(1/\sqrt{2})(t_1+t_2)$, $z_2=(1/\sqrt{2})(t_1-t_2)$.
Combining \eqref{eq:com1} with \eqref{eq:com3R} and using the expressions
\eqref{eq:detRJR} and \eqref{eq:lminRJR} gives \eqref{eq:fR}.
\end{proof}
\section{Properties of certain rank two matrices}
\begin{lem}\label{lemQR}
Let $\vecu,\vecv\in\reals^n$ be linearly independent and
$\matC=\vecu\tp{\vecu}-\vecv\tp{\vecv}$. Then,
\begin{align}\label{eq:CQR}
\matC=
\matW
\matR
\matJ
\tp{\matR}
\tp{\matW}
\end{align}
with
\begin{align}\label{eq:JR}
\matJ=
\begin{pmatrix}
1&0\\
0&-1
\end{pmatrix},&&
\matR=
\begin{pmatrix}
\|\vecu\|&\frac{\tp{\vecu}\vecv}{\|\vecu\|}\\
0&\|\vecv-\frac{\tp{\vecu}\vecv}{\|\vecu\|^2}\vecu\|
\end{pmatrix}
\end{align}
and
\begin{align}\label{eq:W}
\matW=\begin{pmatrix}
\frac{\veca}{\|\veca\|},\frac{\vecb}{\|\vecb\|}
\end{pmatrix}
\end{align}
where the orthonormal vectors $\veca/\|\veca\|$ and $\vecb/\|\vecb\|$ are
defined by
\begin{align}
\veca&=\vecu\label{eq:veca}\\
\vecb&=\vecv-\frac{\tp{\vecu}\vecv}{\|\vecu\|^2}\veca.\label{eq:vecb}
\end{align}
Moreover,
\begin{align}
\det(\matR\matJ\tp{\matR})
&=|\tp{\vecu}\vecv|^2-\|\vecu\|^2\|\vecv\|^2<0\label{eq:detRJR}\\
\tr(\matR\matJ\tp{\matR}) &= \|\vecu\|^2-\|\vecv\|^2\label{eq:trRJR}\\
\sigma_2(\matR\matJ\tp{\matR})
&=\frac{1}{2}\|\vecu+\vecv\|\|\vecu-\vecv\| - \frac{1}{2}\big|\|\vecu\|^2-\|\vecv\|^2\big|
\label{eq:lminRJR}
\end{align}
where $\sigma_1(\matR\matJ\tp{\matR})\geq\sigma_2(\matR\matJ\tp{\matR})$ are the singular values of $\matR\matJ\tp{\matR}$.
\end{lem}
\begin{proof}
We can rewrite $\matC
\matA\matJ\tp{\matA}$ with
$\matA=(\vecu,\vecv)$.
Hence, to prove \eqref{eq:CQR}, it is sufficient to show that $\matA=\matW\matR$.
Using the definitions of the vectors $\veca$ and $\vecb$ in \eqref{eq:veca}
and \eqref{eq:vecb}, we can rewrite
\begin{align*}
\matA
&=
\begin{pmatrix}
\veca,\frac{\tp{\vecu}\vecv}{\|\vecu\|^2}\veca+\vecb
\end{pmatrix}\\
&=
\begin{pmatrix}
\veca,\vecb
\end{pmatrix}
\begin{pmatrix}
1& \frac{\tp{\vecu}\vecv}{\|\vecu\|^2}\\
0&1
\end{pmatrix}\\
&=
\begin{pmatrix}
\frac{\veca}{\|\veca\|},\frac{\vecb}{\|\vecb\|}
\end{pmatrix}
\begin{pmatrix}
\|\vecu\|& \frac{\tp{\vecu}\vecv}{\|\vecu\|}\\
0&\|\vecv-\frac{\tp{\vecu}\vecv}{\|\vecu\|^2}\vecu\|
\end{pmatrix}\\
&=\matW\matR
\end{align*}
which proves \eqref{eq:CQR}.
The explicit form of the determinant in \eqref{eq:detRJR} follows from the
fact that
\begin{align}
\det(\matR\matJ\tp{\matR})
&=\det(\matR)\det(\matJ)\det(\tp{\matR})\nonumber\\
&=-|\det(\matR)|^2\nonumber\\
&=-\|\vecu\|^2\mleft\|\vecv-\frac{\tp{\vecu}\vecv}{\|\vecu\|^2}\vecu\mright\|^2\nonumber\\
&=|\tp{\vecu}\vecv|^2-\tp{\vecu}\vecu\tp{\vecv}\vecv\nonumber\\
&<0\label{eq:cs}
\end{align}
where \eqref{eq:cs} follows from the Cauchy-Schwarz inequality \cite[Sec.
0.6.3]{hojo90} and $\vecu$ and $\vecv$ being linearly
independent. The expression for the trace \eqref{eq:trRJR} follows from
$\tr(\matR\matJ\tp{\matR})=\tr(\matC)$. Finally, \eqref{eq:lminRJR} follows
from
\begin{align}
\sigma_2(\matR\matJ\tp{\matR})
&=\frac{1}{2}\big(\sigma_1(\matR\matJ\tp{\matR})+\sigma_2(\matR\matJ\tp{\matR})\big)\nonumber\\
&\phantom{=}-\frac{1}{2}\big(\sigma_1(\matR\matJ\tp{\matR})-\sigma_2(\matR\matJ\tp{\matR})\big)\nonumber\\
&=\frac{1}{2}\sqrt{\tr(\matR\matJ\tp{\matR})^2-4\det(\matR\matJ\tp{\matR})}-\frac{1}{2}|\tr(\matR\matJ\tp{\matR})|\nonumber\\
&=\frac{1}{2}\|\vecu+\vecv\|\|\vecu-\vecv\|- \frac{1}{2}\big|\|\vecu\|^2-\|\vecv\|^2\big|.\nonumber
\end{align}
\end{proof}
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2015-04-23T02:09:37",
"yymm": "1504",
"arxiv_id": "1504.03024",
"language": "en",
"url": "https://arxiv.org/abs/1504.03024"
}
|
\section{Introduction}
\label{intro}
The accurate determination of the transition probability of transitions in heavy
alkali earth systems is an important step in the research program to measure Atomic Parity Violation
(APV) in such systems \cite{Portela,Wansbeek3,Willmann,Portela2,Sahoo1,Sahoo2,Sahoo3,Geetha,Roberts}. In the research reported here,
a single trapped Ba$^+$ ion has been investigated and the lifetime of its 5d$^2$D$_{5/2}$ state has been measured.
This provides essential input for testing atomic structure and, in particular, the atomic wavefunctions of the involved states
at percent level accuracy. Such measurements are highly sensitive to variations of parameters that determine the experiment's
performance during long periods (i.e. several hours) and which may cause systematic uncertainties. In particular, such effects
may arise from interactions of the ion with background gas.
There are two main reasons for choosing single trapped Ba$^+$ ion in UHV to perform precise lifetime measurements.
Firstly, Barium~(Ba) is a heavy alkaline earth metal. The Ba$^+$ ion has a rather simple electronic configuration. Precise
measurements provide for accurate tests of the atomic wavefunctions. Secondly, systematic errors
due to collisions with other particles (such as different species) are highly suppressed.
The lifetime of the metastable 5d$^2$D$_{5/2}$ state in Ba$^+$ has been measured earlier in different experiments
\cite{Nagourney,Madej,Plumelle,Royen,Gurell,Auchter,Yu}. Calculations are presently performed by different independent theory groups
\cite{Sahoo1,Sahoo2,Sahoo4,Dzuba,Iskrenova,Guet,Guet1}. All the measurements to date as well as calculated values for the lifetime of
5d$^2$D$_{5/2}$ state in Ba$^+$ have been compiled in Table 1.
\begin{table}[h]
\centering
\caption{Calculations and measurements of the lifetime of the 5d$^2$D$_{5/2}$ state in Ba$^+$(see also Fig. 6).
Note, some of the values have been reported without error bars.}
\label{tab:1}
\hspace*{-2cm}
\begin{tabular}{llllll}
\hspace{1.0cm}Theory & \hspace{-4.0cm}Experiments \\
\begin{tabular}{llllll}
\hline\noalign{\smallskip}
Value[s] & Year & Reference & Value[s] & Year & Reference \\
\noalign{\smallskip}\hline\noalign{\smallskip}
29.8(3) & 2012 & \hspace*{0.45cm}\cite{Sahoo1,Sahoo2} & 31.2(0.9) & 2014 & \hspace*{0.45cm}\cite{Auchter} \\
30.3(4) & 2008 & \hspace*{0.45cm}\cite{Iskrenova} & 32.0(2.9) & 2007 & \hspace*{0.45cm}\cite{Royen} \\
30.8 & 2007 & \hspace*{0.45cm}\cite{Guet1} & 32.3 & 1997 & \hspace*{0.45cm}\cite{Yu} \\
31.6 & 2007 & \hspace*{0.45cm}\cite{Gurell} & 34.5(3.5) & 1990 & \hspace*{0.45cm}\cite{Madej} \\
30.3 & 2001 & \hspace*{0.45cm}\cite{Dzuba} & 32(5) & 1986 & \hspace*{0.45cm}\cite{Nagourney} \\
37.2 & 1991 & \hspace*{0.45cm}\cite{Guet} & 47(16) & 1980 & \hspace*{0.45cm}\cite{Plumelle} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
\end{tabular}
\end{tabular}
\end{table}
\vspace*{-1.50cm}
\section{Experimental setup}
\label{sec:1}
The trap for Ba$^+$ in this experiment is a hyperbolic Paul trap \cite{Paul}. It consists of a ring electrode and two end caps
made of copper. The electrodes are mounted on a Macor holder. The chosen geometry results in a harmonic pseudopotential at the
center of the trap when AC voltages are applied between the ring and the two endcaps. The latter are grounded. The operating RF
frequency for the trap is $\Omega_{RF}$ = 5.44~MHz.
The trap with its Macor holder is mounted on Oxygen Free High Conductivity (OFHC) copper base plate. In order to trap ions,
there is a Ba oven (0.9 mm diameter $\times$ 40 mm length resistively heated stainless steel tube) which contains a mixture of
BaCO$_{3}$ and Zr. This oven produces a flux of order of $10^6$ thermal Ba atoms/s.
\begin{figure}[h]
\begin{minipage}[t]{0.50\linewidth}
\includegraphics[width=\textwidth]{fig1}
\caption{Hyperbolic Paul trap of Ba$^+$ ion. On top, images of 3, 2 and 1 ion are given.}
\label{fig:minipage1}
\end{minipage}
\quad
\begin{minipage}[t]{0.50\linewidth}
\includegraphics[width=\textwidth]{fig2}
\caption{Level scheme of Ba$^+$ ion. The lowest $^2$S$_{1/2}$, $^2$P$_{1/2}$ and $^2$D$_{3/2}$ electronic
states form a closed three level system. }
\label{fig:minipage2}
\end{minipage}
\end{figure}
A laser at wavelength 413~nm is used to produce Ba$^+$ ions in the trap by
two-photon photoionisation. We use laser light at $\lambda_{1} = 493$~nm (frequency doubled from a Coherent MBR-110 Ti:Sa
laser) for driving the 6s$^2$S$_{1/2}$-6p$^2$P$_{1/2}$ cooling transition and laser light at $\lambda_{2} = 649$~nm (produced from
Coherent CR-699 ring dye laser) for the 6p$^2$P$_{1/2}$-5d$^2$D$_{3/2}$
repump transition (see Fig. 2). In the experiments reported here, the power of $\lambda_{1}$ is between 6~\textmu{}W and 50~\textmu{}W
and that of $\lambda_{2}$ is between 6~\textmu{}W and 45~\textmu{}W. The Gaussian radius of the laser beams is about 60~\textmu{}m
at the position of the ion for all the measurements. Fluorescence from the 6s$^2$S$_{1/2}$-6p$^2$P$_{1/2}$ transition
in the Ba$^+$ ion is detected with a photomultiplier tube (PMT) and an EMCCD camera. Fig. 1 shows our hyperbolic
Paul trap together with the image of ions that are trapped and localized at the potential minimum of the trap.
\vspace*{-0.50cm}
\section{Electron shelving technique}
\label{sec:2}
Ba$^+$ ions have a closed three-level system. One of the excited states, the 5d$^2$D$_{5/2}$
state, is long-lived (see Fig. 2). Simultaneous laser radiation at $\lambda_{1}$ and $\lambda_{2}$ is therefore needed
to cool the ion in the center of the trap. When the ion is exposed to the light of two laser beams at wavelengths
$\lambda_{1}$ and $\lambda_{2}$ (see Fig. 2), there is a closed cycle of 6s$^2$S$_{1/2}$-6p$^2$P$_{1/2}$-5d$^2$D$_{3/2}$
transitions. Observing the fluorescence from the 6p$^2$P$_{1/2}$-6s$^2$S$_{1/2}$ transition implies that the ion is ``not shelved''
in the 5d$^2$D$_{5/2}$ state. The electron shelving technique is employed in our experiment to determine the lifetime
of the 5d$^2$D$_{5/2}$ state. With an additional fiber-coupled high power LED (M455F1) at $\lambda_{3} = 455$~nm wavelength the
ion can be ``shelved'' to the 5d$^2$D$_{5/2}$ state via excitation to the 6p$^2$P$_{3/2}$ state and this
state's subsequent decay. The direct observation of ``quantum jumps'' in a single Ba$^+$ ion between the 5d$^2$D$_{5/2}$
and 6s$^2$S$_{1/2}$ states has been first demonstrated by Nagourney et al. \cite{Nagourney}.
The decay of the 6p$^2$P$_{3/2}$ state is the start of a shelving period which ends with a quantum jump from the
5d$^2$D$_{5/2}$ state to 6s$^2$S$_{1/2}$ state. Fig.~3 displays the highest PMT count rate (2200cnts/s) when the ion is not
shelved and the lowest count rate (600cnts/s) as background when it is shelved to the metastable 5d$^2$D$_{5/2}$ state.
The ``on/off'' and ``off/on'' transitions in the fluorescence signal corresponds to the start and end of one single interval, when the
ion was in the 5d$^2$D$_{5/2}$ state.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{fig3}
\caption{Quantum Jumps observed in single Ba$^+$ ion. Left : PMT count rate as a function of time.
Right: EMCCD image of the ion in the unshelved state(top) and shelved state(bottom).}
\end{figure}
\vspace*{-0.50cm}
\section{Measurements}
\label{sec:3}
In order to measure the lifetime $\tau_{D_{5/2}}$, a total of 5046 individual shelved periods
have been recorded in 71 data samples and analysed. They were taken under in part significantly different conditions to
enable observing and correcting for systematic errors \cite{Portela3}. Fig.~4 represents one example of the analysed
samples. It shows an exponential decay. Such a decay function is fitted to each data set using a binned
log-likelihood method. The lifetime $\tau_{D_{5/2}}$ is obtained for each data sample from the corresponding fit parameters.
We note that experimental situations can be created where ion heating results in longer measured durations of
individual dark periods than the actual dwell time of the ion in the D$_{5/2}$ state. This can be seen in the slow
recovery of the fluorescence light.
\begin{figure}[h]
\begin{minipage}[t]{0.48\linewidth}
\includegraphics[width=\textwidth]{fig4}
\caption{One sample of the lifetime measurements in single Ba$^+$ ion with 96 shelved periods.}
\label{fig:minipage3}
\end{minipage}
\quad
\begin{minipage}[t]{0.49\linewidth}
\includegraphics[width=\textwidth]{fig5}
\caption{Lifetime of the 5d$^2$D$_{5/2}$ state versus residual gas pressure in a single Ba$^+$ ion.
68$\%$ and 95$\%$ confidence intervals are given.}
\label{fig:minipage4}
\end{minipage}
\end{figure}
Collisions with background gas can reduce the lifetime of the metastable state.
In order to extrapolate the absolute value for the lifetime to zero pressure, the
lifetime $\tau_{D_{5/2}}$ was measured at different background pressures. Fig.~5 displays
the results for a selection of 1600 out of 5046 shelved periods. The uncertainty for the lifetime in
each value corresponds to the statistical error from fitting an exponential decay
to the data. A range of pressures between $2.5 \times 10^{-10}$ and $8.7 \times 10^{-10}$ mbar was
explored by changing the temperature of the vacuum chamber in the range from 289
K to 296 K and by adjusting the pumping speed of the ion pump. For the small
change in temperature needed here, changes in the collision cross-sections between the ion and
the residual gas atoms can be neglected. A linear function is fitted to the data.
The lifetime of the 5d$^2$D$_{5/2}$ state is found to be $\tau_{D_{5/2}} = 26.4(1.7)$~s.
3446 shelved periods are used to check for systematics, such as potentially arising from laser intensities,
laser frequency detunings, rf voltages for trap and effects from the operating conditions of the ion pump.
No significant effects have been observed.
\begin{figure}[h]
\centering
\includegraphics[width=0.6\textwidth]{fig6}
\caption{Lifetime of the 5d$^2$D$_{5/2}$ state in a single Ba$^+$ ion versus time within the last four decades.
Squares represent measurements and triangles represent calculated values. Note, the result Guet2 \cite{Guet1}
differs from Guet1 \cite{Guet} by an omitted term.}
\end{figure}
\vspace*{-1.30cm}
\section{Conclusions}
\label{sec:4}
In summary, the lifetime of the metastable 5d$^2$D$_{5/2}$ has been measured for a single Ba$^+$ ion.
The measured value is preliminary because cross checks for systematics are still ongoing.
Our result agrees within 2$\sigma$ with the most recent theoretical value $\tau_{D_{5/2}} = 29.8(3)$~s \cite{Sahoo2}
and with the latest independent experimental value of $\tau_{D_{5/2}} = 31.2(9)$~s \cite{Auchter}. Fig.~6 displays the time
evolution of the measurements and the theory values for the lifetime of the 5d$^2$D$_{5/2}$ state in a Ba$^+$ ion.
\vspace*{-0.50cm}
\begin{acknowledgements}
We thank Leo Huismann, Oliver B\"{o}ll, and Otto Dermois for their technical assistance. We acknowledge the
financial support from FOM Programme 114 (TRI$\mu$P/AGOR) and FOM programme 125 (Broken Mirrors and Drifting Constants).
\end{acknowledgements}
\vspace{-0.50cm}
|
{
"timestamp": "2015-04-14T02:09:46",
"yymm": "1504",
"arxiv_id": "1504.03023",
"language": "en",
"url": "https://arxiv.org/abs/1504.03023"
}
|
\section{Introduction}
\label{SecIntro}
Orthogonal polynomials with respect to a differential operator were introduced in \cite{ApLoMa02} as a generalization of the notion of orthogonal polynomials. Analytic and algebraic properties of these classes of polynomials have been considered for some classes of first order differential operators in \cite{BePiMaUr11, PiBeUr10}, for a Jacobi differential operator in \cite{BorPij12}, and for differential operators of arbitrary order with polynomials coefficients in \cite{Bor12}. In this paper, we consider orthogonal polynomials with respect to a Laguerre or Hermite operator and a positive Borel measure $\mu $ with unbounded support on $\mathds{R}$.
We denote by $\mathcal{L}_L$ the Laguerre and by $\mathcal{L}_H$ the Hermite differential operators on the linear space $\mathds{P}$ of all polynomials, i.e. for all $ f \in \mathds{P}$ and $\alpha>-1$
\begin{eqnarray}
\label{Lag_DO}
\mathcal{L}_L[f] & = & xf^{\prime \prime} + (1+\alpha -x) f^{\prime}= x^{-\alpha}\,e^{x} \left(x^{\alpha+1}\,e^{-x}\, f^{\prime}\right)^{\prime}, \\
\label{Her_DO}
\mathcal{L}_H[f] & = &\frac{1}{2} f^{\prime \prime} -xf^{\prime}=\frac{1}{2} e^{x^2} \left(e^{-x^2}\, f^{\prime}\right)^{\prime}.
\end{eqnarray}
Each one of these second order differential operators has a system of monic polynomials which are eigenfunctions of the operator and orthogonal with respect to a measure. Let $\{L^{\alpha}_n\}_{n=0}^{\infty}$ be the monic Laguerre polynomials with $\alpha > -1$ and $\{H_n\}_{n=0}^{\infty}$ the monic Hermite polynomials, then
$$
\begin{array}{cccc}
\langle L^{\alpha}_n, L^{\alpha}_m \rangle_{L} & = & \displaystyle \int L^{\alpha}_n(x) L^{\alpha}_m(x) dw_{L}^{\alpha}(x) & \left\{ \begin{array}{ccc}
=0 & \mbox{if} & n \neq m, \\
\neq 0 & \mbox{if} & n=m, \\
\end{array}\right. \\
\langle H_n, H_m \rangle_{H} & = & \displaystyle \int H_n(x) H_m(x) dw_{H}(x) & \left\{ \begin{array}{ccc}
=0 & \mbox{if} & n \neq m, \\
\neq 0 & \mbox{if} & n=m, \\
\end{array}\right.
\end{array}$$
where $dw_{L}^{\alpha}(x)= x^{\alpha}\,e^{-x}dx$ and $dw_{H}(x)= e^{-x^2}dx$. In addition,
\begin{equation}\label{eigen_poly}
\mathcal{L}_L[L^{\alpha}_n]=-n L^{\alpha}_n \quad \mbox{and} \quad \mathcal{L}_H[H_n]=-n H_n.
\end{equation}
To unify the approach, we will denote by $\mathcal{L}$ the Laguerre or Hermite differential operator ($\mathcal{L}_L$ or $\mathcal{L}_H$) in the sequel, by $dw$ the Laguerre or Hermite measure ($dw_{L}^{\alpha}$ or $dw_{H}$), by $L_n$ the $n$th Laguerre or Hermite monic orthogonal polynomial ($L^{\alpha}_n$ or $H_n$) and by $\Delta$ the set $\mathds{R}_+$ or $\mathds{R}$, respectively. We will refer to one or the other depending on the case we are solving.
Let $\mu$ be a finite positive Borel measure, supported on $\Delta$ and $\{P_n\}_{n=0}^{\infty}$ the corresponding system of monic orthogonal polynomials, i.e.
\begin{equation}\label{OrthPoly_mu}
\langle P_n,P_k \rangle_{\mu}=\int P_n(x) P_k(x) d\mu(x) \left\{ \begin{array}{ll}
\not =0 & \mbox{ if } n=k, \\
= 0 & \mbox{ if } n \neq k. \\
\end{array}\right.
\end{equation}
We say that $Q_n$ is the $n$th monic orthogonal polynomial with respect to the pair $(\mathcal{L}, \mu)$ if $Q_n$ has degree $n$
and
\begin{equation}\label{OrthDiff_01}
\int \, \mathcal{L}[Q_n] (x) \; x^k d\mu(x)=0 \quad
\mbox{for all} \quad 0 \leq k \leq n-1,
\end{equation}
or, equivalently,
\begin{equation}\label{OrthDiff_02}
\mathcal{L}[Q_n]=\lambda_n \,P_n,
\end{equation}
where $\lambda_n=-n$.
It was shown in \cite[\S2]{BorPij12} that it is not always possible to guarantee the existence of a system of polynomials $\{Q_n\}_{n\in \mathds{Z}_+}$ orthogonal with respect to the pair $(\mathcal{L}^{(\alpha,\beta)},\mu)$, where $\mathcal{L}^{(\alpha,\beta)}$ is the Jacobi differential operator and $\mu$ an arbitrary positive finite Borel measure. As will be shown later (cf. Propositions \ref{(Cor1)LH} and \ref{Theo_EU}), a similar situation occurs for the case of Laguerre and Hermite operators. Let $m \in \mathds{N}$ be fixed, a fundamental role in the existence of infinite sequences of polynomials $\{Q_n\}_{n>m}$ orthogonal with respect to the pair $(\mathcal{L},\mu)$ is played by the class $\mathcal{P}_m(\Delta)$ defined as the family of finite positive Borel measures $\mu$ supported on $\Delta$ for which there exist a polynomial $\rho$ of degree $m$, such that $\displaystyle \mu= \left(\rho \right)^{-1} w$.
If $\mu\in \mathcal{P}_m(\Delta)$ it is not difficult to see that if $n>m$, then
\begin{eqnarray}\label{PnLn}
P_n(z) &=& \sum_{k=0}^{m} b_{n,n-k} \; L_{n-k}(z),\quad b_{n,n-k}=\frac{1}{\tau_{n-k}} \int P_n(x) L_{n-k}(x) dw(x), \\
\label{Lag-Herm_Norm2}
\tau_n &=&\|L_n\|_{w}^2=\int L_n^2(x) dw(x)=\left\{\begin{array}{rl}
n! \; \Gamma(n+\alpha+1) &, \mu\in\mathcal{P}_m(\mathds{R}_+), \\
n! \sqrt{\pi}2^{-n}&, \mu\in\mathcal{P}_m(\mathds{R}),
\end{array} \right.
\end{eqnarray}
and from \eqref{OrthDiff_02} we obtain that the monic polynomial of degree $n$, for $n>m$ defined by the formula
\begin{eqnarray}\label{Qhat_Def}
\widehat{Q}_n(z) &=& \sum_{k=0}^{m} \frac{\lambda_n}{\lambda_{n-k}}b_{n,n-k} \; L_{n-k}(z),
\end{eqnarray}
is orthogonal with respect to $(\mathcal{L},\mu)$.
Notice that from the equivalence between relations \eqref{OrthDiff_01} and \eqref{OrthDiff_02}, the polynomial $\widehat{Q}_n+c, c\in \mathds{C}$, is orthogonal with respect to $(\mathcal{L},\mu)$ so that we do not have a unique monic orthogonal polynomial of degree $n$. We had a similar situation when we studied the orthogonality with respect to a Jacobi operator. A natural way to define a unique sequence would be to consider a sequence of complex numbers $\{\zeta_n\}_{n=m+1}^{\infty}$, and define the sequence $\{Q_n\}_{n=m+1}^{\infty}$ satisfying \eqref{OrthDiff_01}, as the polynomial solution of the initial value problem
\begin{equation}\label{IVP_nLH}\left\{\begin{array}{rcl}
\mathcal{L}[y] & = & \lambda_n \,P_n, \quad n > m,\\
y(\zeta_n) &=& 0.
\end{array}\right.
\end{equation}
We say that $\{Q_n\}_{n=m+1}^{\infty}$ is the sequence of monic orthogonal polynomials with respect to the pair $(\mathcal{L}, \mu)$ such that $Q_n(\zeta_n)=0$.
Notice that the initial value problem \eqref{IVP_nLH} has the unique polynomial solution
\begin{equation}\label{(13)LH}
y(z)=Q_n(z)= \widehat{Q}_n(z) - \widehat{Q}_n(\zeta_n).
\end{equation}
In this paper, we study some analytic and algebraic properties of the sequence of orthogonal polynomials with respect to a Laguerre or Hermite differential operator. In order to study the asymptotic properties of the sequence of polynomials we shall normalize them with an adequate parameter.
Let $x_{n}$ be the modulus of the largest zero of the $n$th orthogonal polynomial with respect to $\mu$ (or $w$), from \cite[Lemma 11 with $\lambda=2$]{Rakh81} for the Hermite case and \cite[Coroll. (p. 191) with $\gamma=1$]{Rakh81} for the Laguerre case,
we get
\begin{eqnarray}\label{asympzero}
\displaystyle \lim_{n\to\infty}c_n^{-1}x_n=1,
\end{eqnarray}
where $c_n$ is usually called Mhaskar-Rakhmanov-Saff constant, here with the closed expression
\begin{equation}\label{LargestZ-LH}
c_{n}= \left\{ \begin{array}{rllcl}
4n & , & \mu \in \mathcal{P}_m(\mathds{R}_+) & \mbox{or}& w(x)=x^{\alpha}e^{-x},x>0, \\
\sqrt{2n} & , & \mu \in \mathcal{P}_m(\mathds{R})& \mbox{or}& w(x)= e^{-x^2},x\in \mathds{R}.
\end{array}
\right.
\end{equation}
Throughout this paper we denote the functions $\varphi(z)=z+\sqrt{z^2-1}$ and $\psi(z)=2z-1+2\sqrt{z(z-1)}$, where the branch of each root is selected from the condition $ \varphi(\infty)=\infty$ and $ \psi(\infty)=\infty $, respectively. Let $\Delta_c$ be the interval $[0,1]$ in the Laguerre case and $[-1,1]$ in the Hermite case. Let $\mathfrak{P}_n(z)=c_n^{-n}P_n(c_nz)$ be the normalized monic orthogonal polynomials with respect to a measure $\mu\in\mathcal{P}_m(\Delta)$.
To each generic polynomial $q_{n}$, let $\displaystyle \mu_n={n^{-1}}\sum_{q_{n}(\omega)=0}\delta_{\omega}$ be the normalized root counting measure, where $\delta_{\omega}$ is the Dirac measure with mass $1$ at the point $\omega$. From \cite[Ths. 4 \& 4']{Rakh81} we find that the limit distribution $\nu_w$ of the zero counting measure of the normalized Laguerre and Hermite polynomials is
$$
d\nu_w(t)=\begin{cases}
\displaystyle {2}{\pi^{-1}}\sqrt{\frac{1-t}{t}}dt,\quad t\in[0,1]& \text{Laguerre case,}\\
\displaystyle{2}{\pi^{-1}}\sqrt{1-t^2}dt,\quad t\in[-1,1] & \text{Hermite case.}\\
\end{cases}
$$
From \cite[Chs. III \& IV]{safftotik97} we have that
\begin{eqnarray}
\label{Pnrootc} \displaystyle \lim_{n \rightarrow
\infty}\left|\mathfrak{P}_n(z)\right|^{\frac{1}{n}}&=&
\begin{cases}
\frac{1}{e }\,\left|\psi(z)\right|\, e^{2\Re [1/\varphi(z)]} & \mu \in \mathcal{P}_m(\mathds{R}_+),\\
\frac{1}{2 \sqrt{e}} \,\left|\varphi(z)\right| \, e^{\Re{[z/\varphi(z)]}} & \mu \in \mathcal{P}_m(\mathds{R}),
\end{cases}
\end{eqnarray}
uniformly on compact subsets $K \subset \mathds{C} \setminus \Delta_c$.
We are interested in asymptotic properties of the normalized monic orthogonal polynomials with respect to a pair $(\mathcal{L},\mu)$ defined by
\begin{equation}\label{(13)LHnor}
\mathfrak{Q}_n(z)=\mathfrak{\widehat{Q}}_n(z)-\mathfrak{\widehat{Q}}_n(\zeta_n),
\end{equation}
where $\mathfrak{\widehat{Q}}_n(z)= c_n^{-n}\;\widehat{Q}_n(c_nz)$. For these polynomials we prove the followings results
\begin{theorem}\label{countingmeasureconv}
Let $\mu \in \mathcal{P}_m(\Delta)$, where $m \in \mathds{N}$. Then:
\begin{itemize}
\item [a)] If $\nu_n,\sigma_n$ denote the root counting measure of ${\widehat{\mathfrak{Q}}}_n$ and ${\widehat{\mathfrak{Q}}}^{\prime}_n$ respectively then $\displaystyle \nu_n \;\;{\stackrel{\rm *}{\longrightarrow}}\;\; \nu_{w}$ and $\displaystyle \sigma_n \;\;{\stackrel{\rm *}{\longrightarrow}}\;\; \nu_{w}$
in the weak star sense.
\item [b)] The set of accumulation points of the zeros of $\left \{\mathfrak{\widehat{Q}}_n\right \}_{n=m+1}^{\infty}$ is $\Delta_c$.
\end{itemize}
\end{theorem}
\begin{theorem}\label{Th2}
Let $m \in \mathds{N}$, $\mu \in \mathcal{P}_m(\Delta)$. Then, for every compact subset $K$ of $\mathds{C} \setminus \Delta_c$ we have uniformly
\begin{eqnarray}\label{StrQ}
\lim_{n \to \infty}\frac{\mathfrak{P}_n(z)}{\widehat{\mathfrak{Q}}_n(z)}&=&\begin{cases}
\displaystyle 1 \quad & \mu \in \mathcal{P}_m(\mathds{R}_+)\\
\displaystyle 1 & \mu \in \mathcal{P}_m(\mathds{R})\\
\end{cases}
\\
\label{nrootcon}\displaystyle \lim_{n\to \infty}\left|\widehat{\mathfrak{Q}}_{n}(z)\right|^{\frac{1}{n}}\,&=&
\begin{cases}
\frac{1}{e }\,\left|\psi(z)\right|\, e^{2\Re [1/\varphi(z)]} & \mu \in \mathcal{P}_m(\mathds{R}_+),\\
\frac{1}{2 \sqrt{e}} \,\left|\varphi(z)\right| \, e^{\Re{[z/\varphi(z)]}} & \mu \in \mathcal{P}_m(\mathds{R}).\\
\end{cases}
\end{eqnarray}
\end{theorem}
The following result shows that the set of accumulation points of the zeros of the sequence of normalized polynomials, orthogonal with respect to $(\mathcal{L},\mu)$ is contained in a curve.
\begin{theorem}\label{zeroloc}
Let $m \in \mathds{N} $ and $\mu \in \mathcal{P}_m(\Delta)$. If $\{\zeta_n\}_{n=m+1}^{\infty}$ is a sequence of complex numbers with limit $\zeta \in \mathds{C} \setminus \Delta_c$. Then:
\begin{itemize}
\item [a)] The accumulation points of zeros of the sequence $\{\mathfrak{Q}_{n}\}_{n=m+1}^{\infty}$ such that $\mathfrak{Q}_{n}(\zeta_n)=0$ are located on the set $E=\mathcal{E}(\zeta) \bigcup \Delta_c$, where $\mathcal{E}(\zeta)$ is the curve
\begin{equation}
\mathcal{E}(\zeta):= \displaystyle\{z\in \mathds{C}: \Psi(z)=\Psi(\zeta)\},
\end{equation}
$\Psi(z)=\left|\psi(z)\right|e^{2\Re [1/\varphi(z)]}$ for $\mu \in \mathcal{P}_m(\mathds{R}_+)$, and $\displaystyle \Psi(z)=\left|\varphi(z)\right|e^{\Re{[z/\varphi(z)]}}$ for $\mu \in \mathcal{P}_m(\mathds{R})$.
\item [b)] If $\displaystyle \mathfrak{d}(\zeta)=\inf_{x\in\Delta_c}|\zeta-x|>2$ then $E=\mathcal{E}(\zeta)$ and for $n$ sufficiently large are simple.
\end{itemize}
\end{theorem}
The relative asymptotic behavior between the sequences of polynomials $\{\mathfrak{Q}_{n}\}_{n>m}$ and $\{\mathfrak{P}_{n}\}_{n>m}$ reads as
\begin{theorem}\label{RelativeAsymptoticLH}
Let $\{\zeta_n\}_{n>m}$ be a sequence of complex numbers with limit $\zeta \in \mathds{C} \setminus \Delta_c$, $m \in \mathds{N}$, $\mu \in \mathcal{P}_m(\Delta)$ and $\{\mathfrak{Q}_n\}_{n>m}$ be the sequence of normalized monic orthogonal polynomials with respect to the pair $(\mathcal{L},\mu)$ such that $\mathfrak{Q}_n(\zeta_n)=0$, then:
\begin{enumerate}
\item Uniformly on compact subsets of $\Omega=\{z \in \mathds{C} : |\Psi(z)| > |\Psi(\zeta)| \}$,
\begin{equation}
\label{AsintComp1}
\frac{\mathfrak{Q}_{n}(z)}{\mathfrak{P}_n(z)} \;\; {\mathop{\mbox{\Large $\rightrightarrows$}}_{n\to\infty}}\;\; 1.
\end{equation}
\item Uniformly on compact subsets of $\Omega=\{z \in \mathds{C} : |\Psi(z)| < |\Psi(\zeta)| \} \setminus \Delta_c$
\begin{equation}
\label{AsintComp2}
\frac{\mathfrak{Q}_{n}(z)}{\mathfrak{P}_n(\zeta_n)} \;\; {\mathop{\mbox{\Large $\rightrightarrows$}}_{n\to\infty}}\;\; -1,
\end{equation}
where $\Psi$ is as defined in Theorem \ref{zeroloc}.
If $\displaystyle \mathfrak{d}(\zeta)>2$ then \eqref{AsintComp2} holds for $\Omega=\{z \in \mathds{C} : |\Psi(z)| < |\Psi(\zeta)| \}$.
\end{enumerate}
\end{theorem}
The paper continues as follows. Section \ref{Sec_EU} is dedicated to the study of existence, uniqueness and some results concerning the properties of the zeros of orthogonal polynomials with respect to the Laguerre or Hermite operators. In Sections \ref{ZeroLocAsymtpBeh} and \ref{withouthat} we study the asymptotic behavior of the polynomials $\widehat{\mathfrak{Q}}_{n}$ and $\mathfrak{Q}_{n}$ respectively. Finally, in Section \ref{Sec_FluidLH} we show a fluid dynamics model for the zeros of these polynomials.
\section{The polynomial ${Q}_{n}$}
\label{Sec_EU}
First of all, we are interested in discussing systems of polynomials such that for some $m\in \mathds{N}$, for all $n>m$, they are solutions of \eqref{OrthDiff_02}. In order to classify those measures $\mu$ for which the existence of such sequences of orthogonal polynomials with respect to $(\mathcal{L},\mu)$ can be guaranteed, we prove a preliminary lemma.
\begin{lemma} \label{Theo_EUverII} Let $\mu$ be a finite positive Borel measure with support contained on $\mathds{R}$ and let $n\in \mathds{N}$ be fixed. Then, the differential equation \eqref{OrthDiff_02} has a
monic polynomial solution $Q_n$ of degree $n$, which is unique up to an additive constant, if and only if
\begin{equation}\label{(11)verII}
\int P_n(x)dw(x)=0, \; \mbox{where $P_n$ is as \eqref{OrthPoly_mu}.}
\end{equation}
\end{lemma}
\begin{proof} Suppose that there exists a polynomial $Q_n$ of degree $n$, such that $\mathcal{L}[Q_n]=-n \,P_n$. Then, integrating \eqref{Lag_DO} or \eqref{Her_DO} with respect to the Laguerre measure on $\mathds{R}_+$ or Hermite measure on $\mathds{R}$ respectively we have \eqref{(11)verII}.
Conversely, suppose that $P_n$ satisfies \eqref{(11)verII}. Let $Q_n$ be the polynomial of degree $n$ defined by $\displaystyle Q_n(z) = L_n(z) + \sum_{k=0}^{n-1}a_{n,k} L_k(z),$ where $a_{n,0}$ is an arbitrary constant and
$\displaystyle a_{n,k}= \frac{\lambda_n}{\lambda_{k}\tau_{k}}\int P_n(x) L_{k}(x) dw(x),$ $k=1,\ldots ,n-1.$ From the linearity of $\mathcal{L}[\cdot]$ and \eqref{eigen_poly} we get that $\mathcal{L}[Q_n]=-n \,P_n$.\end{proof}
From the preceding lemma, as in \cite[Coroll. 2.2]{BorPij12}, we obtain
\begin{proposition} \label{(Cor1)LH} Let $w$ be the Laguerre or Hermite measure and $\mu$ a finite positive Borel measure on $\Delta$, such that $d\mu(x)=r(x) d w(x)$ with $r \in {L}^2(w)$. Then, $m$ is the smallest natural number such that for each $n>m$ there exists a monic polynomial $Q_n$ of degree $n$, unique up to an additive constant and orthogonal with respect to $(\mathcal{L}, \mu)$ if and only if $r^{-1}$ is a polynomial of degree $m$.
\end{proposition}
\begin{proof} Suppose that $m$ is the smallest natural number such that for each $n>m$ there exists a monic polynomial $Q_n$ of degree $n$, unique up to an additive constant and orthogonal with respect to $(\mathcal{L}, \mu)$. According to Lemma \ref{Theo_EUverII}
$$\int L_n(x)\frac{d\mu(x)}{r(x)}=\int L_n(x)dw(x)
\left\{ \begin{array}{cc}
=0 & \mbox{if } n>m,\\
\neq 0 & \mbox{if } n=m.
\end{array}
\right.$$
But this is equivalent to saying that $\displaystyle
\frac{1}{r(x)}= \sum_{k=0}^{m} c_{k} L_k(x) $ with $c_m \neq 0$. The converse is straightforward.
\end{proof}
It is possible to give another characterization, in terms of the quasi orthogonality concept, for the existence of a system of polynomials such that for all $n>m$, for some $m\in \mathds{N}$, they are solutions of \eqref{OrthDiff_02}.
\begin{proposition}\label{Theo_EU}
Let $\mu$ be a finite positive Borel measure on $\mathds{R}$ and $\{P_n\}_{n=0}^{\infty}$ the sequence of monic orthogonal polynomials with respect to $\mu$. Then, $m$ is the smallest natural number such that for each $n>m$ there exists, except for an additive constant, a unique monic polynomial $Q_n$, orthogonal with respect to the pair $(\mathcal{L},\mu)$, if and only if for all $n > m$
$$
\int P_n(x)\, x^k dw(x)=0, \quad \mbox{for } k=0,1, \dots, n-m,
$$
i.e. the polynomial $P_n$ is quasi-orthogonal of order $n-m+1$ with respect to the measure $w$.
\end{proposition}
\begin{proof}
Assume that $m$ is the smallest natural number such that for each $n>m$ there exists a monic polynomial $Q_n$ of degree $n$, unique up to an additive constant and orthogonal with respect to $(\mathcal{L}, \mu)$. From Lemma \ref{Theo_EUverII} we have that \eqref{(11)verII} holds for $n>m$. From the three term recurrence relation for $\{P_n\}_{n=0}^{\infty}$
\begin{eqnarray}\label{P_n_3trr}
xP_n(x) & = & P_{n+1}(x)+ \beta_n P_n(x)+\alpha_n^2 P_{n-1}(x),\; n \geq 1,\\
\nonumber & & P_0(x)=1, \; P_{-1}(x)=0, \; \alpha_n,\beta_n \in \mathds{R} \mbox{ and } \alpha_n \neq 0,\\
\mbox{thus }\; \int P_n(x) x^k dw(x) &=& 0 \quad \mbox{for all } \; 0 \leq k<n -m,\label{EU_Cond_3}
\end{eqnarray}
which implies that the polynomial $P_n$ is quasi-orthogonal of order $n-m+1$ with respect to the measure $w$ (Laguerre or Hermite).
Conversely, assume that $m$ is the smallest natural number such that for $n > m$, the polynomial $P_n$ is quasi-orthogonal of order $n-m+1$ with respect to the measure $dw$. Then we have that
$$P_n(x)=L_n(x)+\sum_{k=1}^{m}d_{n-k}L_{n-k}(x),$$
which implies that for all integers $n > m$ the polynomials $P_n$ satisfy the condition \eqref{(11)verII}. From Lemma \ref{Theo_EUverII} we have that there exists a monic polynomial $Q_n$ of degree $n$, unique up to an additive constant and orthogonal with respect to $(\mathcal{L}, \mu)$, for all $n>m$.
\end{proof}
From the above proposition, we deduce in particular that the differential equation \eqref{OrthDiff_02} has, except for an additive constant, a unique monic polynomial solution $Q_n$ of degree $n$ \emph{for all the natural numbers} only if $P_n=L_n$ and $d\mu=dw$. Hence $Q_n=L_n$, the polynomial eigenfunctions of $\mathcal{L}$, whose properties are well known.
Let us continue by noting that the polynomials $Q_n$ and $\displaystyle \widehat{Q}_n$ (see \eqref{Qhat_Def} and \eqref{(13)LH}) are primitives of the same polynomial $Q_n^{\prime}$ (or $\widehat{Q}_n^{\prime}$) and
\begin{equation}\label{SeudoOrtogo_04}
\int \, \widehat{Q}_n(x) \, x^k \, dw(x)=0, \quad k=0,1,\ldots,n-m-1.
\end{equation}
Applying classical arguments \cite{Sho37}, it is not difficult to prove the following result, which will be used in the sequel.
\begin{proposition}\label{Zero_Loc01} The polynomial $ \widehat{Q}_n$ defined by \eqref{Qhat_Def} for all $n>m$, has at least $(n-m)$ zeros and $(n-m-1)$ critical points of odd multiplicity on $\Delta$.
\end{proposition}
For $m=2$ we denote by $\mathcal{\widetilde{P}}_2[\mathds{R}]$ the class of measures of the form $\displaystyle d\mu=\frac{e^{-x^2}}{x^2+x_1^2}dx,x_1\neq 0$ in the Hermite case. The following proposition shows some results concerning the zeros of $\widehat{Q}_n$ and $\widehat{Q}^{\prime}_n$ for measures $\mu \in \mathcal{P}_1[\mathds{R}_+]$ or $ \mu \in \mathcal{\widetilde{P}}_2[\mathds{R}]$.
\begin{proposition} \label{Zero_Loc02}
Assume that $\mu \in \mathcal{P}_1[\mathds{R}_+]$ or $ \mu \in \mathcal{\widetilde{P}}_2[\mathds{R}]$, then the zeros of $\widehat{Q}_n$ and $\widehat{Q}^{\prime}_n$ are real and simple. The critical points of $Q_n$ interlace the zeros of $P_n$.
\end{proposition}
\begin{proof}\
\noindent 1.- \emph{Laguerre case.}
If $m=1$ and $\mu \in \mathcal{P}_1[\mathds{R}_+]$ from Proposition \ref{Zero_Loc01} the polynomial $\widehat{Q}_n$ has at least $(n-1)$ real zeros of odd multiplicity on $\mathds{R}_+$. But, $\widehat{Q}_n$ is a polynomial with real coefficients and degree $n$, consequently the zeros of $\widehat{Q}_n$ are real and simple. As ${Q}^{'}_n = \widehat{Q}^{'}_n $, from Rolle's theorem all the critical points of ${Q}_n$ are real, simple, and $(n-2)$ of them are contained on $\mathds{R}_+^*=]0,\infty[$.
Denote $\displaystyle G(x)= x^{\alpha+1}\,e^{-x}\, Q_n^{\prime}(x)$, with $\alpha \in ]-1,\infty[$. Notice that $G$ is a real--valued, continuous and differentiable function on $\mathds{R}_+^*$. Suppose that there exists $x \in \mathds{R}_+^*$ such that $G(x)=0$. As $G(0)=0$ from Rolle's Theorem there exists $x^{\prime} \in \mathds{R}_+^*$ such that $G^{\prime}(x^{\prime})=0$. But, $\displaystyle G^{\prime}(x)=x^{\alpha}\,e^{-x} \mathcal{L}_L[Q_n]=\lambda_n x^{\alpha}\,e^{-x}\, P_n(x)$ and all the critical points of $G$ are contained on $\mathds{R}_+^*$. Hence all the critical points of ${Q}_n$ belong to $\mathds{R}_+^*$.
\medskip\noindent 2.- \emph{Hermite case.}
Consider now $ \mu \in \mathcal{\widetilde{P}}_2[\mathds{R}]$, that is, $m=2$ and $\displaystyle d\mu(x)=\frac{e^{-x^2}}{x^2+x_1^2}dx$, $x_1\neq 0$. Using the relations \eqref{Qhat_Def} and \cite[5.6.1]{Szg75} we have that for $k>1$
\begin{eqnarray}
\label{HtL}\widehat{Q}_{2k}(z)&=&L^{-1/2}_{k}(z^2)+\frac{k}{k-1}\frac{\langle P_{2k},H_{2k-2},\rangle_H}{\langle H_{2k-2},H_{2k-2},\rangle_H} L^{-1/2}_{k-1}(z^2),\\
\nonumber \widehat{Q}_{2k+1}(z)&=&zL^{1/2}_{k}(z^2)+\frac{2k+1}{2k-1}\frac{\langle P_{2k+1},H_{2k-1},\rangle_H}{\langle H_{2k-1},H_{2k-1},\rangle_H}zL^{1/2}_{k-1}(z^2).
\end{eqnarray}
As $L^{-1/2}_{n}(z^2), zL^{1/2}_{n}(z^2)$ are the $2n$ and $2n+1$ monic orthogonal polynomials of degree $2n$ and $2n+1$ respectively with respect to the measure $d\mu(x)=e^{-x^2}dx$, from \eqref{HtL} and \cite[Th. 3.3.4]{Szg75} we have that the zeros of $\widehat{Q}_{n} , n>2$ are real.
The statement that critical points of $Q_n$ interlace the zeros of $P_n$ follows by applying Rolle's theorem to the functions $\displaystyle G(x)= x^{\alpha+1}\,e^{-x}\, Q_n^{\prime}(x)$ and $\displaystyle G(x)= e^{-x^{2}}\, Q_n^{\prime}(x)$, for both the Laguerre and Hermite cases.
\end{proof}
We conjecture that Proposition \ref{Zero_Loc02} is still valid for any measure in the class $\mathcal{P}_m(\Delta)$, $m>1$, for the Laguerre case or $m>2$, $m$ even, for the Hermite case.
Finally, we find asymptotic bounds for the coefficients $b_{n,n-k}$ that define the polynomial $\widehat{Q}_{n}$.
\begin{proposition}\label{Lem_Coeff}
Let $m \in \mathds{N}$ and $\mu \in \mathcal{P}_m(\Delta)$. Then for $n$ large enough, there exist constants $C^{L}_{\rho}$ and $C^{H}_{\rho}$ such that
$$
|b_{n,n-k}|= \frac{\left|\langle P_n, L_{n-k} \rangle_w \right|}{\|L_{n-k}\|_{w}^2} <
\left\{ \begin{array}{cc}
C^{L}_{\rho}\,n^k & \text{Laguerre case,} \\ & \\
C^{H}_{\rho}\, \sqrt{n^{k}} & \text{Hermite case,}
\end{array}
\right.
$$
for $k=1,\ldots,m$.
\end{proposition}
\begin{proof} Let $\displaystyle\rho(x)=\sum_{j=1}^{m}\rho_jx^j$ and $\displaystyle \rho_+=\max_{0 \leq j \leq m}{|\rho_j|}$. From the Cauchy--Schwarz inequality we have
\begin{eqnarray}
|b_{n,n-k}| & \leq & \frac{\|P_n \|_{\mu}}{\|L_{n-k}\|_{w}^2} \sqrt{\langle \rho L_{n-k}, L_{n-k} \rangle_w} \leq \frac{\|\rho L_{n-m} \|_{\mu}}{|\rho_m|\,\|L_{n-k}\|_{w}^2} \sqrt{\langle \rho L_{n-k}, L_{n-k} \rangle_w} \nonumber \\ \label{coeff0}
& \leq & \frac{\rho_+}{|\rho_m|\,\|L_{n-k}\|_{w}^2} \sqrt{\sum_{j=0}^{m}\left|\langle x^j, L_{n-m}^2 \rangle_w\right|} \sqrt{\sum_{j=0}^{m}\left|\langle x^j, L_{n-k}^2 \rangle_w\right|}
\end{eqnarray}
We analyze separately the Laguerre and Hermite cases. Without loss of generality we can assume that $n>2m$.
\medskip\noindent\emph{$\bullet$ Laguerre case $(L_n=L_n^{\alpha},\; \Delta=\mathds{R}_{+} \; \mbox{ and } \;dw(x)=x^{\alpha}e^{-x}dx)$.} From \cite[(III.4.9) and (I.2.9)]{Rus05} we have the connection formula
\begin{equation*}\label{Connect01}
L_{n-k}^{\alpha}(z)= \sum_{\nu=k}^{k+j} \binom{j}{\nu-k}\frac{(n-k)!}{(n-\nu)!} L_{n-\nu}^{\alpha+j}(z),
\end{equation*}
then from \eqref{Lag-Herm_Norm2} and the orthogonality
\begin{eqnarray*}
\langle x^j, \left( L_{n-k}^{\alpha}\right)^2 \rangle_L &=& \sum_{\nu=k}^{k+j} \binom{j}{\nu-k}\frac{(n-k)!}{(n-\nu)!} \int \,\left(L_{n-\nu}^{\alpha+j}(x)\right)^2 x^{\alpha+j} e^{-x}dx, \\
&=& \sum_{\nu=k}^{k+j} \binom{j}{\nu-k} (n-k)! \Gamma(n-\nu+j+\alpha+1) ,\\
& \leq & 2^j (n-k)! \Gamma(n-k+j+\alpha+1) ,\\ \mbox{and }\;
\sum_{j=0}^{m} \langle x^j, L_{n-k}^2 \rangle_w & \leq & (n-k)! \sum_{j=0}^{m} 2^j \Gamma(n-k+j+\alpha+1), \\
& \leq & (2^{m+1}-1) (n-k)! \Gamma(n-k+m+\alpha+1).
\end{eqnarray*}
Hence, from \eqref{coeff0}, \eqref{Lag-Herm_Norm2} and $n$ large enough
\begin{eqnarray*}
\nonumber |b_{n,n-k}| & \leq & \frac{\rho_+ (2^{m+1}-1)}{|\rho_m|} \sqrt{\frac{(n-m)! \Gamma(n+\alpha+1) \Gamma(n+m-k+\alpha+1)}{(n-k)! \Gamma^2(n-k+\alpha+1)}},
\\
& \leq & \frac{\rho_+ (2^{m+1}-1)}{|\rho_m|} \sqrt{\frac{(n+\alpha)^{k+m}}{(n-m)^{m-k}}} \leq \frac{\rho_+ 2^m(2^{m+1}-1)}{|\rho_m|}\; n^k.
\end{eqnarray*}
\medskip\noindent\emph{$\bullet$ Hermite case $(L_n=H_n,\; \Delta=\mathds{R} \; \mbox{ and } \;dw(x)=e^{-x^2}dx)$. } By the symmetry property of the Hermite polynomials, if $\nu$ is an odd number
$$
\int x^{\nu}\,H_{n-k}^{2}(x)dw(x)=0.
$$
Hence, from \eqref{coeff0}
$$
|b_{n,n-k}| \leq \frac{\rho_+}{|\rho_m|\,\|H_{n-k}\|_{w}^2} \sqrt{\sum_{j=0}^{\left\lfloor\frac{m}{2}\right\rfloor}\|x^j\,H_{n-m}\|_{w}^{2}} \sqrt{\sum_{j=0}^{\left\lfloor\frac{m}{2}\right\rfloor} \|x^j\,H_{n-k}\|_{w}^{2}},
$$
where for all $x \in \mathds{R}$, the symbol $\left\lfloor x\right\rfloor$ denote the largest integer less than or equal to $x$. As it is well known (cf. \cite[(5.5.6) and (5.5.8)]{Szg75}), the Hermite polynomials satisfy the recurrence relation
$z H_{n}(z)= H_{n+1}(z)+ \frac{n}{2} H_{n-1}(z),
$ from which we get by induction on $j$
\begin{equation}\label{RecuReiterate}
z^j H_{n}(z)= \sum_{\nu=0}^{j} \sigma_{j,\nu}(n) H_{n+j-2\nu}(z),
\end{equation}
where $\sigma_{j,\nu}(n)$ is a polynomial in $n$ of degree equal to $\nu$ and leading coefficient $2^{-\nu}\binom{j}{v}$ (i.e. $\sigma_{j,\nu}(n)= 2^{-\nu}\binom{j}{v} n^{\nu}+ \cdots $). Hence, from \eqref{Lag-Herm_Norm2}, for $n$ large enough
\begin{eqnarray*}
\|x^j H_{n-k}\|_{w}^2 &=& \sum_{\nu=0}^{j} \sigma_{j,\nu}^2(n-k) \|H_{n-k+j-2\nu}\|_{w}^2, \\
& \leq & \frac{\sqrt{\pi}\;(n-k-j)!}{2^{n-k+j}}\left(\sum_{\nu=0}^{j} 2^{2\nu} \sigma_{j,\nu}^2(n-k) (n-k+j)^{2j-2\nu} \right), \\ & \leq & \frac{2 \sqrt{\pi}\;(n-k-j)! (n-k)^{2j}}{2^{n-k}}\binom{2j}{j},
\end{eqnarray*} with $j=0,1, \ldots, \left\lfloor\frac{m}{2}\right\rfloor$, therefore
\begin{eqnarray*}
|b_{n,n-k}| & \leq & \frac{\rho_+ \;2^{n-k}}{\sqrt{\pi}|\rho_m|\,(n-k)!} \sqrt{\sum_{j=0}^{\left\lfloor\frac{m}{2}\right\rfloor}\|x^j\,H_{n-m}\|_{w}^{2}} \sqrt{\sum_{j=0}^{\left\lfloor\frac{m}{2}\right\rfloor} \|x^j\,H_{n-k}\|_{w}^{2}}\\
& \leq & \frac{2 m! \rho_+ }{|\rho_m|} \sqrt{\sum_{j=0}^{\left\lfloor\frac{m}{2}\right\rfloor}(n-m)^{ 2j} \frac{\left(n-m-j\right)!}{(n-k)!}} \sqrt{\sum_{j=0}^{\left\lfloor\frac{m}{2}\right\rfloor} (n-k)^{ 2j} \frac{\left(n-k-j\right)!}{{(n-k)!}}} \\
& \leq & \frac{2 m! \rho_+ }{|\rho_m|} \sqrt{\sum_{j=0}^{\left\lfloor\frac{m}{2}\right\rfloor} \frac{(n-m)^{ 2j}}{(n-m-j)^{m+j-k}}} \sqrt{\sum_{j=0}^{\left\lfloor\frac{m}{2}\right\rfloor} \frac{(n-k)^{ 2j}}{(n-m-j)^j}} \\
& \leq & \frac{2 m! \rho_+ }{|\rho_m|} \sqrt{ 8 m (n-k)^{-\left\lfloor\frac{m}{2}\right\rfloor}} \sqrt{2 m (n-k)^{\left\lfloor\frac{m}{2}\right\rfloor}} \, n^k=
\frac{8 m (m)! \rho_+ }{|\rho_m|} \;n^k.
\end{eqnarray*}\end{proof}
\section{The polynomial $\mathfrak{\widehat{Q}}_{n}$}\label{ZeroLocAsymtpBeh}
In this section we prove asymptotic properties of the normalized monic orthogonal polynomials with respect to a Laguerre or Hermite differential operator. We recall that as in Section \ref{SecIntro}, $\Delta_c$ denotes the interval $[0,1]$ in the Laguerre case and $[-1,1]$ in the Hermite case, and the sequence of real numbers $\{c_n\}_{n=1}^{\infty}$ is given by \eqref{LargestZ-LH}. Set $\mathfrak{L}_{n,\nu}(z)= c_n^{-\nu}\;L_{\nu}(c_nz); \mathfrak{L}_{n}(z)\equiv\mathfrak{L}_{n,n}(z)$ and $\mathfrak{P}_{n,\nu}(z)= c_n^{-\nu}\;P_{\nu}(c_nz); \mathfrak{P}_{n}(z)\equiv\mathfrak{P}_{n,n}(z)$.
We prove now some preliminary lemmas.
\begin{lemma}\label{loc}
Let $m \in \mathds{N}$, $\mu \in \mathcal{P}_m(\Delta)$ and $\zeta$ such that $\widehat{\mathfrak{Q}}_n(\zeta)=0$. Then for all $n$ sufficiently large
$d_c(\zeta)< 2\varpi_c,$ where
$$\varpi_c=\left\{ \begin{array}{ll}
1+ 2^{-1}\,C^{L}_{\rho} & \mbox{Laguerre case},\\
1+ \sqrt{2}\,C^{H}_{\rho} & \mbox{Hermite case},
\end{array}
\right.$$
$\displaystyle d_c(z)= \min_{x \in \Delta_c}|z-x|$, and $C^{L}_{\rho}$ and $C^{H}_{\rho}$ are the same constants of Proposition \ref{Lem_Coeff}.
\end{lemma}
\begin{proof}
For each fixed $n>m$, we have that
$$ x_n^{-n} \widehat{Q}_n(x_nz)= \sum_{k=0}^{m} \frac{ \lambda_n\,b_{n,n-k}}{x_n^k\lambda_{n-k}} \; x_n^{-n+k}L_{n-k}(x_nz),$$
where $x_n$ is the zero of the largest modulus of $L_n$. It follows that the smallest interval containing the zeros of $\{x_n^{-k}L_{k}(x_nz)\}_{k=0}^{n}$ is $\Delta_{c}$. Hence, if $\zeta$ is such that $\widehat{Q}_n(x_n\zeta)=0$, from \cite[Coroll. 1]{Sch98}, Proposition \ref{Lem_Coeff}, \eqref{LargestZ-LH}, and \eqref{asympzero} we have,
\begin{eqnarray}\label{preloc}
d_c(\zeta) &\leq & 1+\max_{1\leq k \leq m}\left|\frac{ \lambda_n\,b_{n,n-k}}{x_n^k\lambda_{n-k}}\right|<1+2\max_{1\leq k \leq m}\left|\frac{ b_{n,n-k}}{x_n^k}\right|\leq \varpi_c,
\end{eqnarray}
where
$$
\varpi_c=\left\{ \begin{array}{ll}
1+ 2^{-1}\,C^{L}_{\rho} & \mbox{Laguerre case},\\
1+ \sqrt{2}\,C^{H}_{\rho} & \mbox{Hermite case}.
\end{array}
\right.
$$
Notice that $\displaystyle \mathfrak{\widehat{Q}}_n\left(\frac{x_n}{c_n}z\right)= c_{n}^{-n}\widehat{Q}_n(x_nz)$; therefore, if $\zeta$ is such that $\widehat{Q}_n(x_n\zeta)=0$ then $\displaystyle\zeta^*=\frac{x_n}{c_n}\zeta$ is such that $\mathfrak{\widehat{Q}}_n(\zeta^*)=0$. From \eqref{asympzero} and \eqref{LargestZ-LH} we have that for $n$ large, $\displaystyle\left| \frac{x_n}{c_n}\right|<2$. Using now \eqref{preloc} we obtain the lemma.
\end{proof}
If $\{\Pi_n\}_{n=0}^{\infty}$ is a sequence of orthogonal polynomials with respect to either the measures $\mu$ or $w$ we denote by $\{\mathfrak{\Pi}_n\}_{n=0}^{\infty}$ the sequence of monic normalized polynomials, that is,
\begin{equation}\label{genricPL}
\mathfrak{\Pi}_n(z)=c_n^{-n}\Pi_n(c_n z) \; \mbox{ and }\; \mathfrak{\Pi}_{n,\nu}(z)=c_n^{-\nu}\Pi_{\nu}(c_n z).
\end{equation}
From the interlacing property of the zeros of consecutive orthogonal polynomials, if $K$ is a compact subset of $\mathds{C}\setminus \Delta_c$ it follows that there exist a constant $ M_*$ such that for $n$ large enough
\begin{equation}
\label{Lacotger} \left|\frac{ \mathfrak{\Pi}_{n,n-k}(z)}{\mathfrak{\Pi}_{n}(z)} \right| < M_k\leq M_*, \quad k=1,\ldots ,m,
\end{equation}
uniformly on $z \in K$, where $\displaystyle M_k=2\sup_{\substack{ z \in K \\ x \in \Delta_c}} |z-x|^{-k}$, $M_* = \max\{M_1,\ldots, M_m\}$.
The following lemma is needed to study the modulus of the sequence $\displaystyle \left\{ \frac{\mathfrak{P}_{n} }{\mathfrak{L}_{n} }\right\}_{n=0}^{\infty}$.
\begin{lemma}\label{h}
Suppose that $m \in \mathds{N}$ is fixed, and $K\subset \mathbb{C}\setminus \Delta_c$ a compact subset. Then, for $n$ sufficiently large
\begin{eqnarray}
\label{h0} \displaystyle \left |\left(\frac{c_{n+m}}{c_n}\right)^{n} \frac{\mathfrak{\Pi}_n(z)}{\mathfrak{\Pi}_n(\frac{c_{n+m}}{c_n}z)} \right |&< &3^{\displaystyle\frac{2m}{d}},\quad n>n_0, \forall z\in K,
\end{eqnarray}
where $\displaystyle d=\inf_{\substack{ z \in K \\ x \in \Delta_c}}|z-x|$ and $\mathfrak{\Pi}_n$ as in \eqref{genricPL}.
\end{lemma}
\begin{proof}Let us define the monic polynomial $\displaystyle \mathfrak{\Pi}^*_{n}(z)=\left(\frac{c_n}{c_{n+m}}\right)^{n}\mathfrak{\Pi}_n\left(\frac{c_{n+m}}{c_n}z\right)$. We have that \eqref{h0} is equivalent to proving that
$$\displaystyle \left |\frac{\mathfrak{\Pi}_n(z)}{\mathfrak{\Pi}^*_{n}(z)} \right |\leq 3^{\displaystyle\frac{2m}{d}},\quad n>n_0, \forall z\in K.$$
If $\{z^{*}_{k,n}\}_{k=1}^{n}$, $\{z_{k,n}\}_{k=1}^{n}$ denotes the zeros of the polynomials $\mathfrak{\Pi}^*_{n}$, $\mathfrak{\Pi}_{n}$ respectively, we have the relation $\displaystyle z^{*}_{k,n}=\frac{c_n}{c_{n+m}}z_{k,n}, k=1,\ldots ,n$. If we denote $\displaystyle k_n=\frac{c_n}{c_{n+m}}$, we have, for all $n$ sufficiently large
\begin{eqnarray}
\displaystyle \left |\frac{\mathfrak{\Pi}_n(z)}{\mathfrak{\Pi}^*_{n}(z)} \right |&\leq & \left |\prod_{k=1}^{n}\left(1+\frac{(k_n-1)z_{k,n}}{z-k_nz_{k,n}}\right) \right |\leq \prod_{k=1}^{n} \left(1+|k_n-1|\left|\frac{z_{k,n}}{z-k_nz_{k,n}}\right|\right) \\
\nonumber &\leq &\prod_{k=1}^{n} \left(1+ \frac{2|k_n-1|}{d} \right)\leq \left(1+ \frac{2|k_n-1|}{d} \right)^{n}< 3^{\displaystyle\frac{2n|k_n-1| }{d}}\leq 3^{\displaystyle\frac{2m}{d}},
\end{eqnarray}
where $\displaystyle d=\inf_{\substack{ z \in K \\ x \in \Delta_c}}|z-x|$.
\end{proof}
We prove now that the modulus of the sequence $\displaystyle \left\{ \frac{\mathfrak{P}_{n} }{\mathfrak{L}_{n} }\right\}_{n=0}^{\infty}$ is uniformly bounded from above and below in the interior of $\mathbb{C}\setminus \Delta_c$.
\begin{lemma}\label{LPright}
Let $\mu \in \mathcal{P}_m(\Delta)$, where $m \in \mathds{N}$ and $K\subset \mathbb{C}\setminus \Delta_c$ a compact subset. Then, for all $n$ sufficiently large there exists a constant $C^*$ such that
$$\displaystyle \left | \frac{\mathfrak{P}_{n}(z)}{\mathfrak{L}_{n}(z)}\right |\leq C^*,\quad n>n_0, \forall z\in K.$$
\end{lemma}
\begin{proof}
From Relation \eqref{PnLn} we deduce that
$\displaystyle
\frac{\mathfrak{P}_n(z)}{\mathfrak{L}_n(z)}=1 + \sum_{k=1}^{m}\frac{b_{n,n-k}}{c_n^k} \frac{\mathfrak{L}_{n,n-k}(z)}{\mathfrak{L}_n(z)}.
$ Hence, from Proposition \ref{Lem_Coeff}, and Lemma \ref{h} we deduce that for $n$ large enough
\begin{eqnarray}\label{RP}
\left|\frac{\mathfrak{P}_n(z)}{\mathfrak{L}_n(z)}\right|\leq 1 + \sum_{k=1}^{m} C_{\rho}\left|\frac{\mathfrak{L}_{n,n-k}(z)}{\mathfrak{L}_n(z)}\right|,
\end{eqnarray}
Using \eqref{RP} and \eqref{Lacotger} we deduce the lemma.
\end{proof}
\begin{lemma}\label{LPleft}
Let $\mu \in \mathcal{P}_m(\Delta)$, where $m \in \mathds{N}$ and $K\subset \mathbb{C}\setminus \Delta_c$ is a compact subset. Then, for all $n$ sufficiently large there exists a constant $C$ such that
$$\displaystyle C\leq \left | \frac{ \mathfrak{P}_{n}(z)}{\mathfrak{L}_{n}(z)}\right |,\quad n>n_0, \forall z\in K.$$
\end{lemma}
\begin{proof} We have that $\displaystyle \rho(z)L_n( z) =\sum_{k=0}^{m} \mathfrak{b}_{n,n-k} P_{n+m-k}(z),$ where $\displaystyle
\mathfrak{b}_{n,n-k}=\frac{\int L_{n}(x)P_{n+m-k}(x)\rho(x) d\mu(x)}{\|P_{n+m-k}(x)\|_{\mu}^2},$ or equivalently,
\begin{eqnarray}\label{LgPg}
\frac{\rho(c_{n+m}z)}{c^m_{n+m}}\left(\frac{c_n}{c_{n+m}}\right)^{n}\frac{\mathfrak{L}_n(\frac{c_{n+m}}{c_n} z)}{\mathfrak{L}_n(z)}\frac{ \mathfrak{L}_{n}(z)}{\mathfrak{P}_{n+m}(z)}=\sum_{k=0}^{m}\frac{\mathfrak{b}_{n,n-k}}{c_{n+m}^{k}}\frac{\mathfrak{P}_{n+m,n+m-k}(z)}{\mathfrak{P}_{n+m}(z)}.
\end{eqnarray}
From the Cauchy Schwartz inequality we have that
$$|\mathfrak{b}_{n,n-k}|\leq \frac{\left(\int L_n^2(x)dw(x)\right)^{1/2}(\int P_{n+m-k}^2(x)dw(x))^{1/2}}{\|P_{n+m-k}\|_{\mu}^2}=\frac{\|L_{n}\|_{w}\|P_{n+m-k}\|_{w}}{\|P_{n+m-k}\|_{\mu}^2}.$$
Using an infinite--finite range inequality for the case in which $w$ is a Laguerre weight, cf. \cite{safftotik97}, we have that there exists a constant $k_L$ such that for all $n$ large enough
$$
\frac{k_L}{n^m}\int_{0}^{\infty} L^2_n(x) dw(x)\leq \frac{k_{0,L}}{(4n)^m}\int_{0}^{\infty} P^2_n(x) dw(x) \leq \frac{1}{\rho_+(4n)^m}\int_{0}^{4n} P^2_n(x) dw(x) \leq \int_{0}^{\infty} P_{n}^2(x)d\mu(x),
$$
where $\displaystyle \rho_+=\max_{0 \leq j \leq m}{|\rho_j|}$. Analogously, for the case of an Hermite weight, for all $n$ large enough, we have that there exists a constant $k_H$ such that
$$
\frac{k_H}{n^{m/2}}\int_{-\infty}^{\infty} L^2_n(x) dw(x) \leq \frac{k_{0,H}}{(2n)^{m/2}}\int_{-\infty}^{\infty} P^2_n(x) dw(x) \leq \frac{1}{\rho_+(2n)^{m/2}}\int_{-\sqrt{2n}}^{\sqrt{2n}} P^2_n(x) dw(x)\leq\int_{-\infty}^{\infty} P_{n}^2(x)d\mu(x),
$$
Hence, for all $n$ large enough
\begin{eqnarray}\label{PnmLnw}
\displaystyle \|P_n\|^2_{\mu} &\geq & k_Ln^{-m}\|L_{n}\|^2_{w},\quad \mbox{Laguerre case,}\\
\nonumber \displaystyle \|P_n\|^2_{\mu} &\geq & k_Hn^{-m/2}\|L_{n}\|^2_{w},\quad \mbox{Hermite case.}
\end{eqnarray}
From \eqref{PnLn} and Proposition \ref{Lem_Coeff} we deduce that for $n$ large enough, there exists a constant $k_1$ such that
\begin{equation}\label{PnwLnw}
\|P_n\|_{w}\leq k_1\|L_n\|_{w}.
\end{equation}
Inequalities \eqref{PnmLnw} and \eqref{PnwLnw} give us that there exists a constant $M^*$ such that for all $n$ large enough
\begin{equation}\label{bgerm}
\frac{|\mathfrak{b}_{n,n-k}|}{c_{n+m}^k}\leq M^*, \quad 1\leq k\leq m.
\end{equation}
From \eqref{Lacotger} it follows that there exists a constant $ M_*$ such that for all $z \in K$
\begin{equation}\label{Pacot}
\left|\frac{ \mathfrak{P}_{n+m,n+m-k}(z)}{\mathfrak{P}_{n+m}(z)} \right| < M_*, \quad k=1,\ldots ,m.
\end{equation}
Using Lemma \ref{h}, \eqref{LgPg}, \eqref{bgerm} and \eqref{Pacot} we obtain
\begin{equation}\label{PreLgPgnormal}
\displaystyle \left |\frac{\rho(c_{n+m}z)}{c^m_{n+m}}\right |\left | \frac{ \mathfrak{L}_{n}(z)}{\mathfrak{P}_{n+m}(z)}\right |\leq 3^{\displaystyle\frac{2m}{d}}\left(1+m\,M^{*}\,M_*\right),
\end{equation}
with $d$ as in Lemma \ref{h}. Hence, from \eqref{Lacotger}, \eqref{Pacot}, \eqref{PreLgPgnormal} and Lemma \ref{h} we obtain that for all $n$ sufficiently large there exists $M>0$ such that
\begin{equation}\label{acotuniff}
\displaystyle \left |\frac{\rho(c_{n+m}z)}{c^m_{n+m}}\right |\left | \frac{\mathfrak{L}_{n}(z)}{\mathfrak{P}_{n}(z)}\right |\leq M, \quad \forall z \in K.
\end{equation}
Let us denote by $\{z_{k}\}_{k=1}^{m}$ the roots of the polynomial $\rho$, and $\displaystyle d^*=\inf_{z\in K}|z|$. Let us choose $\varepsilon$ so that for $n$ large enough $\displaystyle \left|\frac{z_{k}}{c_{n+m}}\right|<\varepsilon<d^*, k=1,\ldots ,m$. Hence,
\begin{equation}\label{acotuniff1}
\left(d^*-\varepsilon\right)^m \leq \prod_{k=1}^{m}\left(\left|z\right |-\left|\frac{z_k}{c_{n+m}}\right|\right)\leq \prod_{k=1}^{m}\left |\left(z-\frac{z_k}{c_{n+m}}\right)\right |= \left |\frac{\rho(c_{n+m}z)}{c^m_{n+m}}\right |.
\end{equation}
Therefore, from \eqref{acotuniff} and \eqref{acotuniff1}, for all $n$ large enough we have that
\begin{equation*}
\displaystyle \left | \frac{\mathfrak{L}_{n}(z)}{\mathfrak{P}_{n}(z)}\right |\leq \frac{M}{\left(d^*-\varepsilon\right)^m}, \quad \forall z \in K,
\end{equation*}
and this proves the lemma.
\end{proof}
\begin{lemma}\label{QLAcotac}
Let $\mu \in \mathcal{P}_m(\Delta)$, where $m \in \mathds{N}$ and $K\subset \mathbb{C}\setminus \Delta_c$ is a compact subset. Then,
$$\displaystyle\left | \frac{\mathfrak{\widehat{Q}}_n(z)}{\mathfrak{L}_{n}(z)}- \frac{\mathfrak{P}_n(z)}{\mathfrak{L}_{n}(z)}\right | \rightrightarrows 0, \quad \forall z\in K.$$
\end{lemma}
\begin{proof}
For each fixed $n>m$, we have that
\begin{equation}\label{QLacot0}
\frac{\widehat{\mathfrak{Q}}_n(z)-\mathfrak{P}_{n}(z)}{\mathfrak{L}_{n}(z)}= \sum_{k=0}^{m} \left(\frac{\lambda_n}{\lambda_{n-k}}-1\right)\frac{b_{n,n-k}}{c_n^k} \; \frac{\mathfrak{L}_{n,n-k}(z)}{\mathfrak{L}_{n}(z)}.
\end{equation}
As $\lambda_n=-n$ in the Laguerre case and $\lambda_n=-2n$ in the Hermite case, then for each $k$ fixed, $k=1,\ldots,m$,
\begin{equation}\label{eigenvalueslim}
\lim_{n \to \infty}\frac{ \lambda_n}{\lambda_{n-k}}= 1.
\end{equation}
From \eqref{Lacotger}, \eqref{QLacot0}, \eqref{eigenvalueslim} and Proposition \ref{Lem_Coeff} we deduce the lemma.
\end{proof}
\begin{proof}\emph{[Theorem \ref{countingmeasureconv}]}
$a)$ From \cite[(5.1.14),(5.5.10)]{Szg75} we have that
$\displaystyle \mathfrak{L}^{\prime}_{n,n-k}=(n-k)\mathfrak{\widetilde{L}}_{n,n-1-k}$, where
$$ \mathfrak{\widetilde{L}}_{n,n-1-k}=\begin{cases}
\displaystyle c^{-(n-1-k)}_nL_{n-1-k}^{\alpha+1}(c_nz),& \quad \text{Laguerre case},\\
\displaystyle c^{-(n-1-k)}_nH_{ n-1-k}(c_nz), & \quad \text{Hermite case}.\\
\end{cases}$$
Let us define
\begin{eqnarray*}
d\widetilde{w}(x)&=&\begin{cases}
\displaystyle dw_{L}^{\alpha+1}(x),& \text{Laguerre case},\\
\displaystyle dw_{H}(x), & \text{Hermite case}.\\
\end{cases}\\
dw_n(x)&=&\begin{cases}
\displaystyle c_n^{-1}dw_{L}^{\alpha}(c_nx),& \text{Laguerre case},\\
\displaystyle c_n^{-1}dw_{H}(c_nx), & \text{Hermite case}.\\
\end{cases}\\
d\widetilde{w}_n(x)&=&\begin{cases}
\displaystyle c_n^{-1}dw_{L}^{\alpha+1}(c_nx),& \text{Laguerre case},\\
\displaystyle c_n^{-1}dw_{H}(c_nx), & \text{Hermite case}.\\
\end{cases}
\end{eqnarray*}
Notice that $\{\mathfrak{L}_{n,n-k}\}_{k=0}^{n}$ and $\{\mathfrak{\widetilde{L}}_{n,n-k}\}_{k=0}^{n}$ are monic orthogonal polynomials with respect to $w_n, \widetilde{w}_n$ respectively, hence, from \cite[(11)]{GoRak86}, we have that the sequences $\{\mathfrak{L}_{n,n-k}\}_{n=0}^{\infty}$ and $\{\mathfrak{\widetilde{L}}_{n,n-k}\}_{n=0}^{\infty}$ for every $k=0,\ldots ,m$ satisfy that
\begin{eqnarray}\label{GoRak}
\lim_{n\to\infty}\|w_n\mathfrak{L}_{n,n-k}\|_{L^{2}(\Delta)}^{1/n}=e^{-F_w}, \lim_{n\to\infty}\|\widetilde{w}_n\mathfrak{\widetilde{L}}_{n,n-k}\|_{L^{2}(\Delta)}^{1/n}=e^{-F_w},
\end{eqnarray}
where $F_w$ is the modified Robin constant for the weights $w,\widetilde{w}$ (or the extremal constant according to the terminology of \cite{GoRak86}) and $\|.\|_{L^{2}(\Delta)}$ denotes the $L^2$--norm with the Lebesgue measure with support on $\Delta$.
From \cite[Ths. 1 \& 2]{markett80} we have that
\begin{eqnarray}\label{nik}
\|w_n\mathfrak{L}_{n,n-k}\|_{L^{\infty}(\Delta)} &\leq & K_{1}n^{\beta}\|w_n \mathfrak{L}_{n,n-k}\|_{L^{2}(\Delta)},\\
\nonumber \|\widetilde{w}_n \mathfrak{\widetilde{L}}_{n,n-k}\|_{L^{\infty}(\Delta)} &\leq & K_{2}n^{\beta}\|\widetilde{w}_n \mathfrak{\widetilde{L}}_{n,n-k}\|_{L^{2}(\Delta)},
\end{eqnarray}
where $K_{1}, K_{2}$ are constants that do not depend on $n$, $\beta=1/2$ for the Laguerre case, and $\beta=1/4$ for the Hermite case. Using \eqref{GoRak}, \eqref{nik}, and \cite[(11)]{GoRak86} we obtain that
\begin{eqnarray}\label{GoRakinfty}
\lim_{n\to\infty}\|w_n\mathfrak{L}_{n,n-k}\|_{L^{\infty}(\Delta)}^{1/n}=e^{-F_w}, \lim_{n\to\infty}\|\widetilde{w}_n\mathfrak{\widetilde{L}}_{n,n-k}\|_{L^{\infty}(\Delta)}^{1/n}=e^{-F_w}.
\end{eqnarray}
Then we have
\begin{eqnarray*}
\|w_n{\widehat{\mathfrak{Q}}}_n\|_{L^{\infty}(\Delta)} &\leq & \sum_{k=0}^{m} \left|\frac{ \lambda_n\,b_{n,n-k}}{c_n^k\lambda_{n-k}}\right|\; \|w_n \mathfrak{L}_{n,n-k}\|_{L^{\infty}(\Delta)} \\
&\leq & \left|\frac{(m+1)\lambda_n\,b_{n,n-k^*(n)}}{c_n^{k^*(n)}\lambda_{n-k^*(n)}}\right|\; \|w_n\mathfrak{L}_{n,n-k^*(n)}\|_{L^{\infty}(\Delta)} ,
\end{eqnarray*}
and
\begin{eqnarray*}
\|\widetilde{w}_n{\widehat{\mathfrak{Q}}}^{\prime}_n\|_{L^{\infty}(\Delta)} &\leq & \displaystyle \sum_{k=0}^{m} \left|\frac{(n-k) \lambda_n\,b_{n,n-k}}{c_n^{k}\lambda_{n-k}}\right|\; \|\widetilde{w}_n\mathfrak{\widetilde{L}}_{n,n-1-k}\|_{L^{\infty}(\Delta)}\\
& \leq& \left|\frac{(m+1)(n-k^{**}(n)) \lambda_n\,b_{n,n-k^{**}(n)}}{c_n^{k^{**}(n)}\lambda_{n-k^{**}(n)}}\right|\; \|\widetilde{w}_n\mathfrak{\widetilde{L}}_{n,n-1-k^{**}(n)}\|_{L^{\infty}(\Delta)},
\end{eqnarray*}
where $\|.\|_{L^{\infty}(\Delta)}$ denotes the sup norm and $k^*(n), k^{**}(n)$ denote positive integer numbers such that the following equalities hold
$$\left|\frac{ \lambda_n\,b_{n,n-k^*(n)}}{c_n^{k^*(n)}\lambda_{n-k^*(n)}}\right|\|w_n \mathfrak{L}_{n,n-k}\|_{L^{\infty}(\Delta)}=\max_{k=0,\ldots,m}\left|\frac{ \lambda_n\,b_{n,n-k}}{c_n^k\lambda_{n-k}}\right|\; \|w_n \mathfrak{L}_{n,n-k}\|_{L^{\infty}(\Delta)},$$
\begin{eqnarray*}
& & \left|\frac{(n-k^{**}(n)) \lambda_n\,b_{n,n-k^{**}(n)}}{c_n^{k^{**}(n)}\lambda_{n-k^{**}(n)}}\right|\; \|\widetilde{w}_n\mathfrak{\widetilde{L}}_{n,n-1-k^{**}(n)}\|_{L^{\infty}(\Delta)} \\ & & = \max_{k=0,\ldots,m}\left|\frac{(n-k) \lambda_n\,b_{n,n-k}}{c_n^k\lambda_{n-k}}\right|\; \|\widetilde{w}_n \mathfrak{\widetilde{L}}_{n,n-k}\|_{L^{\infty}(\Delta)}.
\end{eqnarray*}
From these last inequalities and \eqref{GoRakinfty} we deduce that
$$
\lim_{n\to \infty}\left(\|w_n{\widehat{\mathfrak{Q}}}_n\|_{L^{\infty}(\Delta)}\right)^{1/n}=e^{-F_w}, \;
\lim_{n\to \infty}\left(\|\widetilde{w}_n{\widehat{\mathfrak{Q}}}^{\prime}_n\|_{ L^{\infty}(\Delta)}\right)^{1/n}=e^{-F_w},
$$
Therefore, if $\nu_n,\delta_n$ denote the root counting measure of ${\widehat{\mathfrak{Q}}}_n$ and ${\widehat{\mathfrak{Q}}}^{\prime}_n$ respectively, from \cite[Th. 1.1]{MeOrPi} we deduce that $\nu_n\;\;{\stackrel{\rm *}{\longrightarrow}}\;\; \nu_{w}$, $\delta_n\;\;{\stackrel{\rm *}{\longrightarrow}}\;\; \nu_{w}$ in the weak star sense.
$b)$ From Lemma \ref{LPleft}, if $\varepsilon$ is sufficiently small and $K\subset \mathds{C}\setminus\Delta_c$ is a compact subset, for all $n$ sufficiently large we have that, for some positive constant $C$,
\begin{equation*}
C-\varepsilon\leq \left|\frac{\mathfrak{P}_{n}(z)}{\mathfrak{L}_{n}(z)}\right|-\varepsilon\leq \left|\frac{\widehat{\mathfrak{Q}}_n(z)}{\mathfrak{L}_{n}(z)}\right|.
\end{equation*}
From this fact and from Lemma \ref{loc} we deduce that the set of accumulation points is contained on $\Delta_c$ and from $a)$ of the present theorem we deduce that the set of accumulation points of the zeros of ${\widehat{\mathfrak{Q}}}_n$ is $\Delta_c$.
\end{proof}
\begin{proof}\emph{[Theorem \ref{Th2}]} From $b)$ of Theorem \ref{countingmeasureconv} we deduce that for the Laguerre case
\begin{eqnarray*}
\lim_{n\to\infty}\frac{\widehat{Q}_n^{\prime}(c_nz)}{\widehat{Q}_n(c_nz)}=\lim_{n\to\infty}\frac{\widehat{Q}_n^{\prime\prime}(c_nz)}{\widehat{Q}_n^{\prime}(c_nz)}=\frac{1}{2\pi }\int_{0}^{1}\frac{1}{z-t}\sqrt{\frac{1-t}{t}}dt =\frac{1}{2}\left(1-\sqrt{1-1/z}\right),
\end{eqnarray*}
and for the Hermite case
\begin{eqnarray*}
\lim_{n\to\infty} \frac{\widehat{Q}_n^{\prime}(c_nz)}{c_n\widehat{Q}_n(c_nz)}=\lim_{n\to\infty}\frac{\widehat{Q}_n^{\prime\prime}(c_nz)}{c_n\widehat{Q}_n^{\prime}(c_nz)}=\frac{1}{\pi}\int_{-1}^{1}\frac{\sqrt{1-t^2}}{z-t}\,dt=z\left(1-\sqrt{1-1/z^2}\right) ,
\end{eqnarray*}
on compact subsets $K\subset \mathds{C} \setminus \Delta_c$. From \eqref{OrthDiff_02} and the preceding relations we have for the Laguerre case
\begin{eqnarray}
\label{strongL}\frac{P_n(c_nz)}{\widehat{Q}_n(c_nz)}&=&\frac{zc_n}{\lambda_n}\frac{\widehat{Q}_n^{\prime\prime}(c_nz)}{\widehat{Q}^{\prime}_n(c_nz)}\frac{\widehat{Q}_n^{\prime}(c_nz)}{\widehat{Q}_n(c_nz)}+\left(\frac{1+\alpha-c_nz}{\lambda_n}\right)\frac{\widehat{Q}_n^{\prime}(c_nz)}{\widehat{Q}_n(c_nz)},
\end{eqnarray}
and for the Hermite case
\begin{eqnarray}
\label{strongH}\frac{P_n(c_nz)}{\widehat{Q}_n(c_nz)}&=&\frac{1}{2}\frac{1}{\lambda_n}\frac{\widehat{Q}_n^{\prime\prime}(c_nz)}{\widehat{Q}^{\prime}_n(c_nz)}\frac{\widehat{Q}_n^{\prime}(c_nz)}{\widehat{Q}_n(c_nz)}-\left(\frac{c_nz}{\lambda_n}\right)\frac{\widehat{Q}_n^{\prime}(c_nz)}{\widehat{Q}_n(c_nz)}.
\end{eqnarray}
Taking limits in \eqref{strongL} and \eqref{strongH} we obtain \eqref{StrQ}. Relation \eqref{nrootcon} follows from \eqref{Pnrootc} and \eqref{StrQ}.
\end{proof}
\section{The polynomial $\mathfrak{Q}_{n}$}\label{withouthat}
Some basic properties of the zeros of $\mathfrak{Q}_n$ follow directly from \eqref{Lag_DO} and \eqref{Her_DO}. For example, the multiplicity of the zeros of $\mathfrak{Q}_n$ is at most $3$, a zero of multiplicity $3$ is also a zero of $\mathfrak{P}_n$ and a zero of multiplicity $2$ is a critical point of $\widehat{\mathfrak{Q}}_n$. In the next lemma we prove conditions for the boundedness of the zeros of $\mathfrak{Q}_n$ and determine their asymptotic behavior.
\begin{lemma} \label{Th5LH} Let $\mu \in \mathcal{P}_m(\Delta)$, where $m \in \mathds{N}$ and define for $z\in\mathds{C}$, $\displaystyle\mathfrak{D}(z)=\sup_{x\in \Delta_c}|z-x|$. If $\{\zeta_n\}_{n=m+1}^{\infty}$ is a sequence of complex numbers with limit $\zeta \in \mathds{C}$, then for every $\;d>1$ there is a positive number $N_d$, such that $\{z \in \mathds{C} : \mathfrak{Q}_n(z)=0 \} \subset \{z \in \mathds{C} : |z| \leq \mathfrak{D}(\zeta)+d\}$ whenever $n >N_d$.
\end{lemma}
\begin{proof}\
As $\mathfrak{Q}_n(z)=0$ then $\widehat{\mathfrak{Q}}_n(z)=\widehat{\mathfrak{Q}}_n(\zeta_n)$. From Gauss--Lucas theorem (cf. \cite[\S 2.1.3]{She02}), it is known that the critical points of $\widehat{\mathfrak{Q}}_n $ are in the convex hull of its zeros and from $b)$ of Theorem \ref{countingmeasureconv} the zeros of the polynomials $\{\widehat{\mathfrak{Q}}_n\}_{n=m+1}^{\infty}$ accumulate on $\Delta_c$. Hence, from the \emph{bisector theorem} (see \cite[\S 5.5.7]{She02} ) $|z| \leq \mathfrak{D}(\zeta_n)+1$ and the lemma is established.
\end{proof}
We are now ready to prove Theorem \ref{zeroloc}.
\begin{proof}\emph{[Theorem \ref{zeroloc}]} From Lemma \ref{Th5LH} we have that the zeros of $\mathfrak{Q}_{n}$ are located in a compact set. From \eqref{(13)LHnor} the zeros of $\mathfrak{Q}_n$ satisfy
the equation
\begin{equation}\label{RaizCerosLH}
\left|\widehat{\mathfrak{Q}}_{n}(z)\right|^{\frac{1}{n}}
= \left|\widehat{\mathfrak{Q}}_{n}(\zeta_n)\right|^{\frac{1}{n}}\,.
\end{equation}
If $z \in \mathds{C} \setminus \Delta_c$, taking limit when $n \rightarrow \infty$, from Lemma \ref{Th5LH}, and using \eqref{nrootcon} of Theorem \ref{Th2} on both sides of \eqref{RaizCerosLH}, we have that the zeros of the sequence of polynomials $\{\mathfrak{Q}_n\}_{n=m+1}^{\infty}$ cannot accumulate outside the set
$$ \{z\in \mathds{C}: \Psi(z)=\Psi(\zeta)\}\, \bigcup \, \Delta_c.$$
To verify the second statement of the theorem, note that if $z$ is a zero of $\mathfrak{Q}_n$, from \eqref{(13)LHnor} we get
\begin{equation}\label{Cero_condLH}
\prod_{k=1}^{n} \left|\frac{z-\widehat{x}_{n,k}}{\zeta_n-\widehat{x}_{n,k}} \right| = 1,\; \mbox{where $\widehat{x}_{n,k}$ are the zeros of $\widehat{\mathfrak{Q}}_n$.}
\end{equation}
Let $\displaystyle \mathcal{V}_{\varepsilon}(\Delta_c)$ be the $\varepsilon$-neighborhood of $\Delta_c$ defined as $\displaystyle \mathcal{V}_{\varepsilon}(\Delta_c)= \{ z \in \mathds{C} : \mathfrak{d}(z) < \epsilon \}$, as $\displaystyle \lim_{n \to \infty} \zeta_n=\zeta$, then for all $\varepsilon >0$ there is a $N_{\varepsilon}>0$ such that $|\mathfrak{d}(\zeta_n)-\mathfrak{d}(\zeta)|< \varepsilon $ whenever $n > N_{\varepsilon}$.
If $\mathfrak{d}(\zeta)>2$, let us choose $\displaystyle \varepsilon = \varepsilon_{\zeta} = \frac{1}{2} \, \left(\mathfrak{d}(\zeta)-2 \right)$ and suppose that there is a $z_0 \in \mathcal{V}_{\varepsilon_{\zeta}}(\Delta_c)$ such that $\mathfrak{Q}_n(z_0)=0$ for some $n > N_{\varepsilon_{\zeta}}$. Hence
\begin{equation}\label{Cero_cond}
\prod_{k=1}^{n}\left|\frac{z_0-\widehat{x}_{n,k}}{\zeta_n-\widehat{x}_{n,k}} \right| < \left( \frac{2+\varepsilon_{\zeta}}{\mathfrak{d}(\zeta_n)} \right)^{n}< 1,
\end{equation}
which is a contradiction with \eqref{Cero_condLH}, hence $\{z \in \mathds{C} : \mathfrak{Q}_n(z)=0 \} \bigcap \mathcal{V}_{\varepsilon_n}(\Delta_c)= \varnothing$ for all $n >N_{\varepsilon_{\zeta}}$, i.e. the zeros of $\mathfrak{Q}_n$ cannot accumulate on $\displaystyle \mathcal{V}_{\varepsilon_{\zeta}}(\Delta_c).$
From \eqref{(13)LHnor} it is straightforward that a multiple zero of $\mathfrak{Q}_n$ is also a critical point of $\widehat{\mathfrak{Q}}_n$. But, from b) of Theorem \ref{countingmeasureconv} and the Gauss--Lucas theorem the set of accumulation points of $\widehat{\mathfrak{Q}}_n$ is $\Delta_c$, where we have that for $n$ sufficiently large the zeros of ${\mathfrak{Q}}_n$ are simple.
\end{proof}
\begin{proof} \emph{[Theorem \ref{RelativeAsymptoticLH}]}
\noindent 1.- Let us prove first that
\begin{equation}\label{AsintComp}
\frac{\mathfrak{Q}_{n}(z)}{\widehat{\mathfrak{Q}}_n (z)} =1 - \frac{\widehat{\mathfrak{Q}}_n (\zeta_n)}{\widehat{\mathfrak{Q}}_n (z)} \;\; {\mathop{\mbox{\Large $\rightrightarrows$}}_{n\to\infty}}\;\; 1,
\end{equation}
uniformly on compact subsets $K$ of the set $\{z \in \mathds{C} : |\Psi(z)| > |\Psi(\zeta)| \}$.
In order to prove \eqref{AsintComp} it is sufficient to show that
\begin{equation}\label{SufAsint}
\frac{\widehat{\mathfrak{Q}}_n (\zeta_n)}{\widehat{\mathfrak{Q}}_n (z)} \;\; {\mathop{\mbox{\Large $\rightrightarrows$}}_{n\to\infty}}\;\; 0,
\end{equation}
uniformly on $K$.
From \cite{GeAssch90} and Lemmas \ref{LPright}, \ref{LPleft}, we have that for all $n$ large enough it is possible to find constants $c^*,c$ such that
\begin{equation}\label{SemStrongLH}
c^* \leq \left|\frac{\mathfrak{P}_n(z)}{\Psi^n(z)}\right| \leq c,
\end{equation}
uniformly on compact subsets of $\mathds{C} \setminus \Delta_c$.Then we have
$$
\left|\frac{\widehat{\mathfrak{Q}}_n (\zeta_n)}{\widehat{\mathfrak{Q}}_n (z)}\right|= \left|\frac{\widehat{\mathfrak{Q}}_n(\zeta_n)}{\mathfrak{P}_n(\zeta_n)}\right|
\left|\frac{\mathfrak{P}_n(z)}{\widehat{\mathfrak{Q}}_n (z)}\right|\left|\frac{\mathfrak{P}_n(\zeta_n)}{\Psi^n(\zeta_n)}\right| \left|\frac{\Psi^n(z)}{\,\mathfrak{P}_n(z)}\right| \left|\left(\frac{\Psi(\zeta_n)}{\Psi(z)}\right)\right|^n.
$$
From \eqref{StrQ} of Theorem \ref{Th2} and \eqref{SemStrongLH} the first four factors on the right hand side of the previous formula are bounded; meanwhile, the last factor tends to $0$ when $n \to \infty$, and we get \eqref{SufAsint}. Finally, the assertion 1 is straightforward from \eqref{StrQ} of Theorem \ref{Th2}.
\medskip \noindent 2.- For the assertion 2 of the theorem it is sufficient to prove that
\begin{equation}\label{AsintComp-2}
\frac{\mathfrak{Q}_{n}(z)}{\widehat{\mathfrak{Q}}_n (\zeta_n)} =\frac{\widehat{\mathfrak{Q}}_n (z)}{\widehat{\mathfrak{Q}}_n (\zeta_n)} - 1 \;\; {\mathop{\mbox{\Large $\rightrightarrows$}}_{n\to\infty}}\;\; -1,
\end{equation}
uniformly on compact subsets $K$ of the set $\{z \in \mathds{C} : |\Psi(z)| < |\Psi(\zeta)| \}\setminus \Delta_c$. Note that
$$\frac{\widehat{\mathfrak{Q}}_n (z)}{\widehat{\mathfrak{Q}}_n (\zeta_n)} = \frac{\widehat{\mathfrak{Q}}_n (z)}{\mathfrak{P}_n(z)} \frac{\mathfrak{P}_n(\zeta_n)}{\widehat{\mathfrak{Q}}_n(\zeta_n)} \frac{\mathfrak{P}_n(z)} {\Psi^n(z)}
\frac{\Psi^n(\zeta_n)}{\mathfrak{P}_n(\zeta_n)}
\left(\frac{\Psi(z)}{\Psi(\zeta_n)}\right)^n.
$$
Now, the first part of the assertion 2 is straightforward from \eqref{StrQ} of Theorem \ref{Th2} and \eqref{SemStrongLH}.
If $\displaystyle \mathfrak{d}(\zeta)>2$, let $\displaystyle \mathcal{V}_{\varepsilon}(\Delta_c)= \{ z \in \mathds{C} : \mathfrak{d}(z) < \epsilon \}$ be a $\varepsilon$--neighborhood of $\Delta_c$, where $\displaystyle \varepsilon = \varepsilon_{\zeta} = \frac{\mathfrak{d}(\zeta)}{2}-1$. By the same reasoning used to deduce \eqref{Cero_cond} we get that
\begin{equation}\label{Normal1}
\left|\frac{\widehat{\mathfrak{Q}}_n (z)}{\widehat{\mathfrak{Q}}_n (\zeta_n)}\right| < \kappa^{n}, \quad \mbox{for all } z \in \mathcal{V}_{\varepsilon}(\Delta_c), \kappa <1.
\end{equation}
Hence from the first part of the assertion 2 and \eqref{Normal1} we get the second part of the assertion 2.
\end{proof}
\section{A fluid dynamics model}
\label{Sec_FluidLH}
In this section we show a hydrodynamical model for the zeros of the orthogonal polynomials with respect to the pair $(\mathcal{L},\mu)$. In \cite{BorPij12}, we gave a hydrodynamic interpretation for the critical points of orthogonal polynomials with respect to a Jacobi differential operator.
Let us consider a flow of an incompressible fluid in the complex plane, due to a system of $n$ \emph{different} points ($n>1$) fixed at $w_i$, $ 1 \leq i \leq n$. At each point $w_i$ of the system there is defined a complex potential $\mathcal{V}_{i}$, which for the Laguerre case equals to the sum of a \emph{source(sink)} with a fixed strength $\Re[c_i]$ plus a \emph{vortex} with a fixed strength $\Im[c_i]$ plus a \emph{uniform stream} $U_i$ at infinity. Here $c_i$ and $d_i$ are fixed complex numbers which depend on the position of the remaining points $\{w_i\}_{i=1}^{n}$, see \cite[Ch. VIII]{MTho98} for the terminology. We shall call \emph{$n$ system} to the set of the $n$ points fixed at $w_i$ with its respective potential of velocities.
Define the functions
$$\displaystyle f_i(w_1,\ldots, w_n)=\frac{R_n^{\prime \prime}(w_i)}{\displaystyle R_n^{\prime}(w_i)},\quad i=1,\dots ,n \quad \mbox{where} \quad R_n(z)=\prod_{i=1}^{n}(z-w_i).$$
The complex potentials $\mathcal{V}_{L}$ (Laguerre case) or $\mathcal{V}_{H}$ (Hermite case) at any point $z$ (see \cite[Ch. 10]{Dur08}), by the principle of superposition of solutions, are given by
\begin{equation}
\mathcal{V}_{L}(z) = \sum_{i=1}^{n}\mathcal{V}_{L,i} = \sum_{i=1}^{n} \left(-z + (1+\alpha-w_i) \log{(z-w_i)} + (z+w_i \log{(z-w_i)} ) f_i(w_1,\ldots, w_n)\right), \label{(14)LHQn}
\end{equation}
and
\begin{equation}
\label{(14')HQ}\mathcal{V}_{H}(z)= {\sum_{i=1}^{n}\mathcal{V}_{H,i}} = \sum_{i=1}^{n}
\left(-z+ \frac{1}{2}(f_i(w_1,\ldots, w_n)-2w_i)\log{(z-w_i)}\right).
\end{equation}
From a complex potential $\mathcal{V}$, a complex velocity $\mathbf{V}$ can be derived by differentiation ($\mathbf{V}(z)=\frac{d\mathcal{V}}{dz}$). A standard problem associated with the complex velocity is to find the zeros, that correspond to the set of \emph{stagnation points}, i.e. points where the fluid has zero velocity.
We are interested in the problem: Build an $n$ system (location of the points $w_1, \ldots, w_{n}$) such that the stagnation points are at preassigned points with \emph{nice} properties. As it is well known, the zeros of the orthogonal polynomials with respect to a finite positive Borel measures on $\mathds{R}$ have a rich set of \emph{nice} properties (cf. in \cite[Ch. VI]{Szg75}), and will be taken as preassigned stagnation points. Here we consider $\mu\in \mathcal{P}_1[\mathds{R}_+]$ or $\mu\in \mathcal{\widetilde{P}}_2[\mathds{R}]$. In the next paragraph we establish the statement of the problem for both Laguerre and Hermite cases.
\begin{quote}
\noindent\textbf{Problem.} \emph{ Let $\{x_1, \ldots, x_n\}$ be the set of zeros of the $n$th orthogonal polynomial $P_n$ ($n>1$ for the Laguerre case and $n>2$ for the Hermite) with respect to a measure $\mu\in \mathcal{P}_1[\mathds{R}_+]$ or $\mu\in \mathcal{\widetilde{P}}_2[\mathds{R}]$. Suppose a flow is given, with complex potential $\mathcal{V}_{L}$ (Laguerre case) or $\mathcal{V}_{H}$ (Hermite). Build an $n$ system (location of the points $w_1, \ldots, w_{n}$) such that the stagnation points are attained at the points $z=x_i$, with $i=1,2,\ldots,n$.}
\end{quote}
Consider first the Laguerre case. If $x_{k}$ ($k=1,\ldots, n.$) are stagnation points then
\begin{equation}\label{(15)LH}
\frac{\partial \mathcal{V}_{L}}{\partial z}(x_k)= (1+\alpha-x_k){\sum_{i=1}^{n} \frac{1}{x_k-w_i}}+
x_k\sum_{i=1}^{n}\frac{R_n^{\prime \prime}(w_i)}{R_n^{\prime}(w_i)(x_k-w_i)}= 0.
\end{equation}
We are looking for a solution $\displaystyle R_n(z)=\prod_{i=1}^{n}(z-w_i)$, with $w_i\neq w_j\neq x_k, \forall i,j,k$, $i\neq j$, such that \eqref{(15)LH} holds. This assumption implies that the sum in the second term of the left hand side of expression \eqref{(15)LH} is the partial-fraction decomposition of $\displaystyle \frac{R_n^{\prime \prime}}{R_n}$ evaluated at $x=x_k$. Therefore, \eqref{(15)LH} is equivalent to
$$ x_k R_n^{\prime \prime}(x_k)+(1+\alpha-x_k)R_n^{\prime}(x_k)= 0, \quad
k=1,\,2,\, \dots, n.$$
Note that $xR^{\prime\prime}_n(x)+ (1+\alpha-x)R^{\prime}_n(x)$ is a polynomial of degree $n$, with leading coefficient $\lambda_n$ that vanishes at the zeros of $P_n$, i.e.
\begin{equation}\label{(16L)}
xR^{\prime\prime}_n(x)+ (1+\alpha-x)R^{\prime}_n(x)= \lambda_n P_n(x).
\end{equation}
Observe that expression \eqref{(16L)} is equivalent to \eqref{OrthDiff_02}. From Proposition \ref{Zero_Loc02}, the zeros of $\widehat{Q}_n, \widehat{Q}^{\prime}_n$ are real, simple and $Q^{\prime}_n(x_k)\neq 0$. Therefore, $R_n =\widehat{Q}_n $ is a solution. Hence, an answer to our problem yields the $n$ points as the $n$ zeros of the polynomial $\widehat{Q}_n $.
For the Hermite case we have a similar situation. Thus, if $x_{k}$ is a stagnation point, $ \displaystyle \frac{\partial \mathcal{V}_{H}}{\partial z}(x_k)=0$, which gives
\begin{equation}\label{(15')Q}
x_k\sum_{i=1}^{n} {\frac{1}{x_k-w_i}}-\frac{1}{2} \sum_{i=1}^{n}\frac{R_n^{\prime \prime}(w_i)}{R_n^{\prime}(w_i)(x_k-w_i)}= 0, \quad
k=1,\,2,\, \ldots, n.
\end{equation}
Again, we can deduce that the expression \eqref{(15')Q} equals to $\displaystyle \frac{1}{2}R_n^{\prime \prime}(x_k)-x_kR_n^{\prime}(x_k) = 0,$ for $k=1, \ldots, n$.
Note that $\frac{1}{2}R^{\prime\prime}_n(x)-xR^{\prime}_n(x)$ is a polynomial of degree $n$, with leading coefficient $\lambda_n$ that vanishes at the zeros of $P_n$, i.e.
\begin{equation}\label{(16H)}
\frac{1}{2}R^{\prime\prime}_n(x)-xR^{\prime}_n(x)= \lambda_n P_n(x).
\end{equation}
Therefore, the expression \eqref{(16H)} is equivalent to \eqref{OrthDiff_02}. From Proposition \ref{Zero_Loc02}, the zeros of $\widehat{Q}_n, \widehat{Q}^{\prime}_n$ are real and simple and $Q^{\prime}_n(x_k)\neq 0$, which implies that $R_n =\widehat{Q}_n $ is a solution to our problem. As a conclusion,
\begin{quote}
\noindent\textbf{Answer.} \emph{The flow of an incompressible two--dimensional fluid, due to $n$ points with complex potential $\mathcal{V}_{L}$ or $ \mathcal{V}_{H} $, located at the zeros of the $n$th orthogonal polynomial $\widehat{Q}_n$ with respect to $(\mathcal{L}, \mu)$, with $\mu \in \mathcal{P}_1[\mathds{R}_+]$ or $\mu\in \mathcal{\widetilde{P}}_2[\mathds{R}]$ has its $n$ stagnation points at the $n$ zeros of the $n$th orthogonal polynomial $\widehat{Q}_n$.}
\end{quote}
It would be interesting to consider the uniqueness of the solution obtained. In other words, what could be said about the solutions of the form $Q_n(z)= \widehat{Q}_n(z) - \widehat{Q}_n(\zeta_n)$ and to extend this model to more general classes of measures $\mu$. It would be also of interest to decide if these stagnation or equilibrium points are stable.
\begin{ack*}
The authors thank the comments and suggestions made by the referees which helped improve the manuscript.
\end{ack*}
|
{
"timestamp": "2015-04-14T02:17:47",
"yymm": "1504",
"arxiv_id": "1504.03297",
"language": "en",
"url": "https://arxiv.org/abs/1504.03297"
}
|
\section{introduction}
Graphene-based field-effect transistors have been synthesized,\cite{Science-graphene}
but the ratio of high and low resistances (switching ratio) is limited (less than a factor of $10$)
because of the zero-gap band structure of pristine graphene.
A possible solution is to introduce an energy gap into graphene by using, for instance,
bilayer graphene,\cite{PhysRevLett.99.216802,Nat.Mater.7.151}
nanoribbons,\cite{PhysRevLett.98.206805} or chemical derivatives.\cite{Science.323.610}
Recently, Britnell {\it et al.}\cite{Science.335.947,NNANO.8.100,NCOMMS.4.1794}
reported an alternative transistor architecture, a vertical field effect tunneling transistor (FETT),
exhibiting a fairly high ($\sim 50$) switching ratio.
In FETT devices, a graphene$|$barrier$|$graphene trilayer is utilized as the current-carrying channel,
as in Fig.~\ref{fig:struct}(a);
the electric current flows into the channel through one graphene layer and out through the other.
Inside the trilayer, electrons cross the thin barrier via tunneling.
The chemical potentials of the two graphene layers and the tunneling conductance can be tuned by gate voltages.
The bottom graphene layer (closer to gate electrode, $\mathrm{Gr_B}$)
can only partially screen the gate voltage because of
the low density of states (DOS) near the Dirac point,
and so the top graphene layer (away from gate electrode, $\mathrm{Gr_T}$) is also affected by the gate voltage.
In a typical field-effect device, the current-carrying channel
is separated by a thick dielectric from a metallic gate electrode.
A gate voltage applied between the gate electrode and the current-carrying channel
controls the free carrier density in the channel.
The spatial distribution of free carriers can be obtained by solving the electrostatic Poisson equation.
Macroscopic physical quantities of device components are used as parameters for Poisson's equation,
including dielectric constants and electron affinities of dielectrics and work functions of gate electrodes, {\it etc}.
\cite{APL.90.143108}
However, the extrapolation of the macroscopic theory of electrostatics to the nanometer scale is unjustified,
because interfaces often exhibit dielectric screening properties different from the bulk.\cite{Jackson}
The applicability of the macroscopic theory should be examined by studying the materials-specific
dielectric screening properties of interfaces, one of motivations of this work.
The density functional theory (DFT) method fully takes the atomic structure of a device into account,
and microscopic electrostatics quantities such as potential and charge density can be solved self-consistently,
from which microscopic dielectric properties can be deduced.\cite{PhysRevLett.91.267601}
One can extract an effective dielectric constant from microscopic calculations to represent the dielectric screening of the interface,
which is not necessarily a constant but a quantity that depends on the thickness of the interface and other factors.
\begin{figure}[b]
\begin{center}
\includegraphics[width=0.7 \linewidth]{struct4}
\caption{
\label{fig:struct}
(Color Online)
(a) Schematic of a graphene-based field effect tunneling transistor.
The core graphene$|${\it h}-BN$|$graphene structure is separated from the doped Si (which serves as the gate electrode)
by {\it h}-BN{} crystal and $\mathrm{SiO}_x$ slabs (which serve as dielectrics).
The graphene layers closer to and further away from the gate electrode are labelled as $\mathrm{Gr_B}${} and $\mathrm{Gr_T}$, respectively.
(b) The model used in our simulations.
The dielectrics ({\it h}-BN\ and $\mathrm{SiO}_x$) are replaced by vacuum.
The core graphene$|${\it h}-BN$|$graphene structure is embedded betwen a semi-infinite vacuum ($z>z_1$)
and an ideally metallic medium ($z<-z_1$, which replaces the doped Si gate).
}
\end{center}
\end{figure}
The boundary condition is crucial in solving for the electrostatic potential.
In an FET device, for example,
the electrostatic potential at the surface of a metallic gate electrode should be constant,
and the electric field (derivative of the potential) in vacuum
far away from the device should be zero.
This mixed boundary condition is different from the conventional and widely-implemented periodic boundary condition.
Otani {\textit{et al.}{}}\cite{PhysRevB.73.115407} proposed a Green's-function-based
effective screening medium (ESM) approach to solve the electrostatic potential
under several different boundary conditions,
which is promising for simulating the electric-field effect in planar devices.
In this work, graphene-based FETT devices were simulated using a DFT+ESM method.
The computational approach, based on first principles, enables us
to understand interface effects quantitatively, and thus enables computational design of functional systems.
The rest of the paper is organized as follows.
In the next Section, details of the computational method are presented.
The distribution of free carriers and the band structures of graphene layers
under different gate voltages are presented in Sec.~\ref{sec:results}.
The effective dielectric constant of the {\it h}-BN{} thin barrier in FETT is analyzed in Sec.~\ref{sec:discuss}.
Finally, we give a summary on our results in Sec.~\ref{sec:summary}.
\section{simulation method}
\label{sec:method}
The structure of the FETT used in experiments\cite{Science.335.947} is schematically shown in Fig.~\ref{fig:struct}(a).
Doped silicon is used as the gate electrode, which is separated from the current-carrying $\mathrm{Gr_T}$$|${\it h}-BN$|$$\mathrm{Gr_B}$\ trilayer channel
by insulating silicon oxide ($\sim300\,\,\textrm{nm}$) and {\it h}-BN\ thin films ($\sim50\,\,\textrm{nm}$).
The gate voltage is applied between the doped silicon and the trilayer.
In our DFT simulations,
the doped silicon gate electrode was replaced by a semi-infinite ideally metallic medium for $z<-z_1$
[Fig.~\ref{fig:struct}(b); the $z$-direction is perpendicular to the $\mathrm{Gr_T}$$|${\it h}-BN$|$$\mathrm{Gr_B}${} trilayer],
and the dielectrics between the gate electrode and the $\mathrm{Gr_T}$$|${\it h}-BN$|$$\mathrm{Gr_B}${} trilayer were replaced by
vacuum to save computational resources.
Thus the $\mathrm{Gr_T}$$|${\it h}-BN$|$$\mathrm{Gr_B}$\ trilayer is sandwiched by a semi-infinite vacuum medium ($z>z_1$)
and an ideally metallic medium ($z<-z_1$), see Fig.~\ref{fig:struct}(b).
Note that the semi-infinite vacuum and metallic media are not explicitly included in calculations,
instead they are used as boundary conditions for the Hartree potential ($V_H$),
\begin{equation}
\label{eq:boundary}
{\partial V_H \over \partial z}|_{z=z_1} = 0, \qquad V_H|_{z=-z_1} = 0.
\end{equation}
We adopted the ESM method\cite{PhysRevB.73.115407} to solve the Hartree potential.
The purpose to employ this method is twofold:
(i) Long-range Coulomb interactions with periodic images are avoided.
(ii) The non-periodic boundary condition of Eq.~\eqref{eq:boundary} for the Hartree potential can be imposed.
A gate voltage can be simulated by adding extra electrons
or holes to the $\mathrm{Gr_T}$$|${\it h}-BN$|$$\mathrm{Gr_B}$\ trilayer;\cite{PhysRevB.73.115407}
the areal density of free carriers is proportional to the gate voltage.
In experiments the carrier density in graphene can be tuned
by gate voltage up to $\sim 10^{13}\, \mathrm{cm}^{-2}$,
which is equivalent to $5.24\times 10^{-3}$ electrons per primitive unit cell of graphene,
or an electric displacement field of $\Dg = 0.016\,\mathrm{C/m^2}$ in the dielectrics.
The measured interlayer distance between {\it h}-BN\ and graphene layers\cite{Nat.Mater.11.764} is $3.32\,\mathrm{\AA}$, and
the interlayer distance between {\it h}-BN\ layers $3.33\,\mathrm{\AA}$ is taken as half of the lattice constant $c$ of crystalline {\it h}-BN.\cite{BN-book}
In-plane lattice constants for both graphene and {\it h}-BN{} were set to $2.46\,\mathrm{\AA}$ due to the small lattice mismatch between them.
In the $x$-$y$ plane a periodic boundary condition was applied with a dense $155 \times 155$ Monkhorst-Pack\cite{PhysRevB.13.5188} $k$-mesh.
The cutoff energy for wave functions and the Methfessel-Paxton\cite{PhysRevB.40.3616} spreading energy
were taken to be $70\,\mathrm{Ryd}$ and $10^{-3}\,\mathrm{Ryd}$, respectively.
The Perdew-Burke-Ernzerhof parameterization\cite{PhysRevLett.77.3865} of the generalized gradient approximation (GGA)
to the exchange-correlation functional and
Trouiller-Martins norm-conserving pseudopotentials were used.
DFT calculations were performed using the \textsc{Quantum ESPRESSO} package\cite{QE-2009}.
Charge density in the $\mathrm{Gr_B}$$|${\it h}-BN$|$$\mathrm{Gr_T}$\ trilayer was calculated self-consistently for different gate voltages.
In order to illustrate the distribution of free carriers across the trilayer,
it is convenient to integrate the charge density in the $x$-$y$ plane
and define a charge density along $z$-direction $\rho(z)$.
The density of free carriers is defined as charge density difference
for a device under a finite gate voltage $\Vg$ with respect to that under zero gate voltage.
Compared to gate voltages, the electric displacement field $\Dg$ in the dielectrics
is a more convenient quantity, because it is independent of the thickness or the dielectric constant of the dielectrics.
Gate voltages are expressed as $\Dg$ hereafter.
\begin{figure}[b]
\begin{center}
\includegraphics[width=\linewidth]{delta_rho}
\caption{
\label{fig:delta_rho}
(Color Online)
(a1) The distribution of free carriers under $\Dg=\pm 3\times 10^{-3}\,\mathrm{C/m^2}$ and
(a2) the charge density under zero gate voltage of a $\mathrm{Gr_T}$$|$monolayer {\it h}-BN$|$$\mathrm{Gr_B}${} system.
(b) The free carrier distribution in a graphene$|$multilayer {\it h}-BN$|$ graphene FETT
under $\Dg=3\times 10^{-3}\,\mathrm{C/m^2}$
with the positions of the top and (c) the bottom graphene layer aligned, respectively.
}
\end{center}
\end{figure}
\section{results}
\label{sec:results}
A FETT with a monolayer {\it h}-BN{} barrier is used as an example to illustrate the charge and free carrier densities
in such devices..
The charge and free carrier density on each layer can be analyzed using the Bader decomposition,\cite{Bader}
in which boundaries between atoms are defined by zero flux surfaces.
There are three peaks in the charge density curve $\rho(z)$,
corresponding to the $\mathrm{Gr_T}$, {\it h}-BN, and $\mathrm{Gr_B}${} layers, respectively; cf. Fig.~\ref{fig:delta_rho}(a2).
Free carrier densities $\Delta\rho(z)$,
defined as the change in charge density induced by gate voltages,
under $\Dg=\pm 0.003\,\mathrm{C/m^2}$ are shown in Fig.~\ref{fig:delta_rho}(a1),
where it can be seen that they have opposite sign but almost the same amplitude,
in accordance with the electron-hole symmetry of graphene in the vicinity of the Fermi energy.
For the one-dimensional charge density $\rho(z)$, atomic layers are divided by minima of $\rho(z)$,
as denoted by vertical dashed lines in Fig.~\ref{fig:delta_rho}(a2)
(zero flux surfaces shrink to points for the one-dimensional charge density.)
A large portion of the free carriers accumulate on $\mathrm{Gr_B}${} at the side closer to the gate electrode;
thus the free carrier density on $\mathrm{Gr_B}${} is larger than that on $\mathrm{Gr_T}$.
The asymmetric shape of $\Delta\rho(z)$ with respect to the central {\it h}-BN{} layer is a consequence of
the asymmetric environment of the $\mathrm{Gr_B}$$|${\it h}-BN$|$$\mathrm{Gr_T}$\ trilayer,
with the metallic gate electrode on one side and vacuum on the other side.
\begin{figure}[b]
\begin{center}
\includegraphics[width=\linewidth]{bandstructure1}
\caption{
\label{fig:bandstructure}
(a) Band structure of a FETT with five layers of {\it h}-BN{} as barrier at $\Dg=0.016\,\mathrm{C/m^2}$,
and (b) the difference in chemical potential between $\mathrm{Gr_B}${} and $\mathrm{Gr_T}${} as a function of gate voltage
(the dashed line is a guide to the eye).
}
\end{center}
\end{figure}
The {\it h}-BN{} barrier can screen the applied gate voltage by developing an internal electric polarization,
which reduces the effect of the gate voltage and the free carrier density in $\mathrm{Gr_T}$.
We performed calculations on devices with a {\it h}-BN{} barrier thickness of up to eight monolayers.
The free carrier density for these devices as a function of $z$
for $\Dg=3\times 10^{-3}\,\mathrm{C/m^2}$
is shown in Figs.~\ref{fig:delta_rho}(b) and \ref{fig:delta_rho}(c),
shifted to make the positions of the $\mathrm{Gr_T}${} (b) or $\mathrm{Gr_B}${} (c) layers coincide.
The distribution of free carriers on $\mathrm{Gr_B}${} is almost independent of the thickness of the {\it h}-BN{} barrier, cf. Fig.~\ref{fig:delta_rho}(c).
The amplitude of the electric polarization in the {\it h}-BN{} barrier shrinks for thicker barriers (Fig.~\ref{fig:delta_rho}(c)),
and so does the free carrier density in $\mathrm{Gr_T}$(Fig.~\ref{fig:delta_rho}(b)), indicating that thicker {\it h}-BN{} barriers provide stronger screening.
The chemical potential of graphene, defined as the energy difference between the Fermi energy and the charge neutrality point,
can be efficiently tuned by using gate voltages because of the small DOS near the Fermi energy.%
\cite{Science.335.947,NNANO.8.100,NCOMMS.4.1794}
The band structure of a $\mathrm{Gr_T}$$|${\it h}-BN$|$$\mathrm{Gr_B}${} trilayer in the vicinity of the Fermi energy
has contributions only from graphene layers, because
{\it h}-BN{} is a wide gap insulator with a calculated energy band gap of $4.77\,\textrm{eV}$.
The band structure of graphene shows a tiny energy gap of $0.05\,\textrm{eV}$
[Fig.~\ref{fig:bandstructure}(a)] induced by interaction with {\it h}-BN.\cite{PhysRevB.76.073103}
A FETT with a {\it h}-BN{} barrier thicker than one monolayer shows no hybridization between $\mathrm{Gr_B}${} and $\mathrm{Gr_T}${} in its band structure.
As an example, the band structure of a FETT with a five-layer-thick {\it h}-BN{} barrier for
$\Dg=0.016\,\mathrm{C/m^2}$ is shown in Fig.~\ref{fig:bandstructure}(a),
where bands from $\mathrm{Gr_T}${} and $\mathrm{Gr_B}${} shift upward rigidly by $\sim 0.1\,\textrm{eV}$ and $\sim 0.2\,\textrm{eV}$ with respect to the Fermi energy, respectively.
The bands from $\mathrm{Gr_T}${} are always shifted away from the Fermi level by a larger energy than those from $\mathrm{Gr_B}${}
[Fig.~\ref{fig:bandstructure}(b)].
\section{discussion}
\label{sec:discuss}
Macroscopic electrostatic models
have been used to calculate the electrostatic potential and free carrier density in graphene based FETT.
\cite{Science.335.947,JAP.113.136502,APL.101.033503,IEEE.60.268}
In these models, the {\it h}-BN\ barrier between graphene layers is assumed to have a dielectric constant equal to that of the {\it h}-BN\ crystal.
However, interfaces can exhibit significantly different dielectric properties compared to the bulk,\cite{PhysRevLett.91.267601}
and the dielectric properties of a few-layer-thick {\it h}-BN{} barrier sandwiched between two graphene layers need to be revisited.
We seek to find the (effective) dielectric constant $\ensuremath{ \epsilon_{\mathrm{BN}}}$ of a {\it h}-BN{} barrier in a FETT and compare to its bulk value,
in order to investigate if the dielectric constant is still a valid physical concept for thin {\it h}-BN{} films
and to investigate whether $\ensuremath{ \epsilon_{\mathrm{BN}}}$ of a {\it h}-BN{} barrier is modified by the interfaces with graphene layers.
\begin{figure}[b!]
\begin{center}
\includegraphics[width=\linewidth]{dielectric_GrBNGr}
\caption{
\label{fig:dielectric_GrBNGr}
(a) The polarization of {\it h}-BN{} layers in a FETT with a five-layer {\it h}-BN{} thin film
induced by an gate voltage of $\Dg=6.114 \times 10^{-3}\,\mathrm{C/m^2}$.
(b) The calculated average relative dielectric permittivity of {\it h}-BN{} thin films
embedded by graphene layers as a function of the number of {\it h}-BN{} atomic layers,
and the dashed line denotes the relative dielectric permittivity of bulk {\it h}-BN{}.
}
\end{center}
\end{figure}
The dielectric constants of {\it h}-BN{} thin films ($\ensuremath{ \epsilon_{\mathrm{BN}}}$) sandwiched between graphene layers
can be deduced from the electric field inside the {\it h}-BN{} thin film ($\Eint$)
and the electric polarization ($P$) induced by gate voltages,
\begin{equation}
\label{eq:epsilon1}
\ensuremath{ \epsilon_{\mathrm{BN}}} = \frac{\Eint+P/\epsilon_0}{\Eint},
\end{equation}
where $\Eint$ is equal to the slope of the self-consistent Kohn-Sham effective potential in the $z$-direction.
$\Eint$ is also related to the difference between chemical potentials of graphene layers $\Delta\mu$ by
$ e \Eint d =\Delta\mu $, where $d$ is the distance between the two graphene layers.
The polarization can be expressed as the summation of centers of Wannier functions
according to the modern theory of polarization.
\cite{PhysRevB.47.1651,RevModPhys.66.899}
In practice we followed the method proposed in Refs.~\onlinecite{PhysRevB.75.205121,PhysRevB.80.224110}
which extends the Wannier function theory of polarization
\cite{PhysRevB.47.1651,RevModPhys.66.899,PhysRevB.71.144104,PhysRevLett.97.107602}
to metal-insulator heterostructures.
The hybrid Wannier functions
\cite{PhysRevB.56.12847,PhysRevB.64.115202},
which are exponentially localized in the $z$ direction
but Bloch-like in the $x$-$y$ plane,
were constructed using the parallel-transport method.\cite{PhysRevB.56.12847,PhysRevB.64.115202}
The first Brillouin zone was sampled by discrete $k$-points of the type
$\mathbf{k} = \mathbf{k_\perp} + j\,\mathbf{b}$,
where the vectors $\mathbf{k_\perp}$ form a
$ N_\perp \times N_\perp$ uniform mesh in the $x$-$y$ plane,
and $ \mathbf{b} = (0,0,2\pi N_\| /L )$ is along the $z$-direction with $L$ the height of the unit cell
and $N_\|$ the number of $k$-points along the $z$-direction.
The total number of $k$-points is $N_\perp^2 N_\|$.
We used $N_\perp=7$ and $N_\|=3$, which are sufficient to converge the polarization.
The matrices
$ M_{mn}(\mathbf{k} ) = \braket{ u_{m,\mathbf{k}} }{ u_{n,\mathbf{k} + \mathbf{b}}}$
were constructed where $\ket{ u_{m,\mathbf{k}} }$ is periodic under lattice translations and $n$ is the band index;
$\ket{ u_{m,\mathbf{k}} }$ is normalized such that
$\braket{ u_{m,\mathbf{k}} } { u_{m,\mathbf{k}} } =1$.
Singular value decomposition of each $M$ matrix was done utilizing the LAPACK library: $M=U \Sigma V^{\dagger}$
where $U$ and $V$ are complex unitary matrices and $\Sigma$ is a diagonal matrix with diagonal elements very close to 1 for small $\mathbf{b}$.
A new matrix $\tilde{M} = UV^{\dagger}$ was constructed corresponding to each $M$ matrix, and
a global matrix $\Lambda(\mathbf{k}_{\perp}) = \prod_{j=0}^{N_\|-1} \tilde{M}( \mathbf{k_\perp} + j\mathbf{b}) $
was then constructed for each $\mathbf{k_\perp}$ point.
The centers of hybrid Wannier functions were calculated as $z_m = (-L/2\pi) \,\textrm{Im}[ \ln \lambda_m]$,
where the $\lambda_m$ are the eigenvalues of $\Lambda$.
The number of bands considered to construct the hybrid Wannier functions,
i.e., the dimension of the $M$ matrices, is equal to four for each graphene or {\it h}-BN{} atomic layer,
the number of occupied bands in most of the first Brillouin zone
except for the small portion near the $K$-point (see Fig.~\ref{fig:bandstructure}).
This choice does not affect the calculations of the polarization inside {\it h}-BN{} thin films
because the bands near the Fermi energy are contributed by graphene layers.
Four of the resulting hybrid Wannier functions can be assigned to each graphene or {\it h}-BN{} layer
according to the positions of their centers.
Two of them are very close to the atomic plane (within $10^{-3} \,\mathrm{\AA}$) and
the other two are located about $ 0.4 \,\mathrm{\AA} $ above and beneath the atomic plane respectively.
The center of charge for each atomic layer is equal to the average value of the corresponding four Wannier functions.
The dipole moment corresponding to each {\it h}-BN{} layer was calculated using the shift of the center of charge under an electric field,
and the polarization was calculated with the thickness of each {\it h}-BN{} layer set to be $ 3.33\,\,\mathrm{\AA} $.
The calculated polarization of the {\it h}-BN{} layer adjacent to the graphene layers
is almost the same as {\it h}-BN{} layers deeply inside [see Fig.~\ref{fig:dielectric_GrBNGr}(a)],
indicating that interface with graphene layers has little effect on the dielectric properties of {\it h}-BN{} thin films.
As a result, the average dielectric constant for {\it h}-BN{} thin films embedded by graphene layers
is independent of the thickness; see Fig.~\ref{fig:dielectric_GrBNGr}(b).
\begin{figure}[b!]
\begin{center}
\includegraphics[width=\linewidth]{surface_charge}
\caption{
\label{fig:surface_charge}
(Color online)
Interface charge redistribution \rhoif{} in a FETT device with eight-layer {\it h}-BN{} for
$\Dg=0.016\,\mathrm{C/m^2}$,
where \rhoif{} is the total charge redistribution $\Delta\rho$
minus the bulk charge redistribution $\rhob$ at the center of {\it h}-BN;
\rhob{} is denoted by the patterned rectangle in the top panel.
The absolute value of \rhoif{} is plotted on a logarithmic scale in the bottom panel.
The positions of the atomic planes of $\mathrm{Gr_B}$, $\mathrm{Gr_T}${} and the two interface {\it h}-BN{} layers
$\mathrm{BN_1}$ and $\mathrm{BN_8}$ are denoted by vertical dashed lines.
}
\end{center}
\end{figure}
The effect of an interface with graphene on the dielectric properties of {\it h}-BN{}
can be analyzed using the interface charge redistribution, denoted as \rhoif{}
and defined as the difference in charge redistribution
at the interface with respect to that in the bulk (denoted as \rhob).
In practice we used the charge redistribution at the center of the {\it h}-BN{} as the bulk,
as shown in Fig.~\ref{fig:surface_charge}.
Thus one can obtain \rhoif{} by subtracting \rhob{} from the total charge redistribution.
Because {\it h}-BN{} is a wide-band-gap insulator, \rhoif{} should decay exponentially away from the interface.
The strength of the interface effect is determined by the amplitude of \rhoif{} near the interface.
An example the charge redistribution in a FETT device with eight-layer {\it h}-BN{} for $\Dg=0.016\,\mathrm{C/m^2}$
is shown in Fig.~\ref{fig:surface_charge}.
The difference \rhoif{} is large near the graphene layers, decays quickly into {\it h}-BN,
and becomes invisible after crossing the first {\it h}-BN\ atomic layers
($\mathrm{BN_1}$ and $\mathrm{BN_8}$ in Fig.~\ref{fig:surface_charge}).
The amplitude of \rhoif{} is also presented on a logarithmic scale
in the lower panel of Fig.~\ref{fig:surface_charge}.
The decay of \rhoif{} into {\it h}-BN{} is approximately exponential.
Most importantly, the amplitude of \rhoif{} is about $50$ times smaller than that of \rhob{}
between $\mathrm{BN_1}$ and $\mathrm{BN_8}$ in Fig.~\ref{fig:surface_charge}:
\rhoif{} is very small inside the atomic plane of the first {\it h}-BN{} layer.
As a result, any interface effect on the dielectric properties of {\it h}-BN\ is very weak,
which explains why the calculated dielectric constant of {\it h}-BN{} in FETT devices is close to the bulk value.
\begin{figure}[h]
\begin{center}
\includegraphics[width=\linewidth]{shift_6Si111}
\caption{
\label{fig:Si111}
(Color online)
(Upper panel)
The atomic structure of a hydrogen saturated Si(111) thin film with a thickness of about $ 2\,\,\textrm{nm} $.
Si and H atoms are represented by large blue and small pink spheres, respectively.
Solid lines denote the boundary of the unit cell.
(Lower panel)
The shift of the hybrid Wannier functions induced by an electric field
of $ 0.0385\,\mathrm{V/\,\mathrm{\AA}} $ along the $z$-direction.
Red circles ($\circ$) and blue discs ($\bullet$) represent
hybrid Wannier functions with higher and lower polarizability, respectively.
}
\end{center}
\end{figure}
We also compared the {\it h}-BN{} thin films with silicon thin films,
because the latter are known to exhibit lower dielectric permittivity than for the corresponding bulk.
\cite{PhysRevLett.91.267601,PhysRevB.71.144104}
The in-plane lattice constant of Si(111) slabs was chosen to be the experimental lattice constant of {\it fcc}-Si.
Dangling bonds on both surfaces are saturated by hydrogen, as shown in the upper panel of Fig.~\ref{fig:Si111},
and the Si-H bond length is $ 1.50\,\,\mathrm{\AA} $ after structure optimization.
We plotted in Fig.~\ref{fig:Si111} the shift of each hybrid Wannier functions
induced by an electric field of $ 0.0385\,\mathrm{V/\,\mathrm{\AA}} $ along the $z$-direction,
where the electric field was applied using the ESM method using a metal$|$slab$|$metal configuration.
The hybrid Wannier functions can be divided into two categories according to their polarizability.
The hybrid Wannier functions located at canted Si-Si bonds with respect to the $z$-direction
(denoted as $\bullet$ in Fig.~\ref{fig:Si111}) exhibit lower polarizability,
and they show negligible deviations at the surface.
On the other hand, the hybrid Wannier functions located at parallel Si-Si or Si-H bonds
with respect to the $z$-direction (denoted as $\circ$ in Fig.~\ref{fig:Si111}) exhibit higher polarizability.
We also observed that the hybrid Wannier functions of Si-H bonds at the surface show a polarizability 12\% lower
than those corresponding to parallel Si-Si bonds.
The 12\% lower polarizability of the Si(111) surfaces shown in Fig.~\ref{fig:Si111}
is not as severe as reported by previous studies.\cite{PhysRevB.71.144104,JAP.99.054309}
In those studies,
the macroscopic polarization and dielectric constant were obtained after a smoothing procedure.
The purpose of the smoothing procedure is to eliminate the dielectric nonlocality,
but this procedure reduces the dielectric permittivity of surfaces artificially
because the dielectric constant of vacuum is smeared into the surface.
\section{summary}
\label{sec:summary}
The distribution of free carriers and the band structure of graphene layers
in graphene based FETT have been simulated using the DFT+ESM method.
The dielectric properties of {\it h}-BN{} thin films sandwiched between graphene layers in FETT
were investigated using the theory of microscopic permittivity
and found to have a dielectric permittivity close to that of crystalline {\it h}-BN.
The small amplitude of interface charge redistribution inside the atomic plane of the first {\it h}-BN{} layer
proves that the effect of the interface with graphene on the dielectric properties of {\it h}-BN{} is weak.
In this study we have demonstrated the DFT+ESM method as a promising approach
to simulate field-effect devices with a planar structure.
Once the charge density and effective potential of a field-effect device
are self-consistently obtained, the scattering of transport electrons
and electric conductivity can be calculated using scattering theory.
\begin{acknowledgments}
This work was supported by the US Department of Energy (DOE),
Office of Basic Energy Sciences (BES),
under Contract No. DE-FG02-02ER45995.
This research used resources of the National Energy Research Scientific Computing Center.
\end{acknowledgments}
|
{
"timestamp": "2015-04-15T02:03:15",
"yymm": "1504",
"arxiv_id": "1504.03396",
"language": "en",
"url": "https://arxiv.org/abs/1504.03396"
}
|
\section{Introduction}
In this article we study the properties of linear operators when we allow the leading coefficient functions to vanish on the boundary of the domain. For example, the differential equation:
\begin{align}
\label{some elliptic operator}
L u = - \text{div} ( \Phi \nabla u ) = f,
\end{align}
where $f \in L^p(\Omega)$, and $\Phi(x) \in \mathbb{C}^{d\times d}$ has been extensively studied when $\Phi$ is uniformly positive definite on $\bar{\Omega}$. The operator $L$ is called uniformly elliptic. For more on such operators, see \cites{demengel2012functional, gilbarg1977elliptic, han2011elliptic, krylov1996lectures, maz2011sobolev} and the references therein.
Less is known when the uniform positivity assumption on $\Phi$ is relaxed. In \cite[\S 6.6]{gilbarg1977elliptic}, Trudinger and Gilberg partially relax the condition. In particular, they assume $\Phi \in \mathscr{C}^{0,\gamma}(\bar{\Omega})$ for some $\gamma \in (0,1)$, and if $x_0 \in \partial \Omega$ then there exists a suitably chosen $y \in \mathbb{R}^d$ such that $\Phi(x_0) \cdot (x_0-y) \neq 0$. With this restriction, they establish existence and uniqueness of solutions to \eqref{some elliptic operator}.
In \cite{murthy1968boundary}, Murthy and Stampacchia studied the properties of weak solutions to \eqref{some elliptic operator} in the case where there exists a positive function $m$ with $m^{-1} \in L^p(\Omega)$ such that
\begin{align} \label{generic restriction}
{v} \cdot \Phi(x) {v} \geq m(x) |{v}|^2,
\qquad \text{for a.e. } x\in \Omega, \text{ and all } {v} \in \mathbb{R}^d.
\end{align}
Studies on the properties of solutions to $Lu = f$, when $L$ is non-uniformly elliptic, can be found in \cites{trudinger1971non-uniformly, trudinger1981harnack, trudinger1977maximum}, as well as \cite{coffman1976bvp}, and \cite{franchi1998irregular}, where the authors assume restrictions on $\Phi$ that are similar to \eqref{generic restriction}. The results presented here address the Fredholm properties of $L$ in the case when $\Phi = 0$ on $\partial \Omega$, and/or when $v \cdot \Phi(x) v \geq m(x) |v|$ for some positive function $m \in \mathscr{C}^{1}(\bar{\Omega})$ with $m^{-1} \not\in L^p(\Omega)$.
Other examples of a differential equation with vanishing coefficients arise when studying linear stability of solutions to non-linear PDE. The operator
\begin{align}\label{linearized compacton equation}
Lu = (\varphi u)_{xxx} + (\varphi u)_{x} + b u_x,
\end{align}
defined on $L^p(-1, 1)$ where $\varphi(x) = a \cos^2(\tfrac{\pi}{2} x)$ and $a,b \in \mathbb{R}$, arose when studying compactly supported solutions to
\begin{align*}
u_t = (u^2)_{xxx} + (u^2)_x.
\end{align*}
Our results can be used to accomplish two goals. The first is assessing the solvability of the boundary value problem $Lu = f$ where $u = g$ on the boundary. To that end, we analyze the Fredholm properties of $L$. Our second goal is establishing well-posedness (or ill-posedness) of the Cauchy problem $u_t = Lu$, where $u = g$ when $t = 0$. For this goal, we present results on the spectrum of $L$.
The operators studied here are linear differential operators on $L^p(\Omega)$, elliptic or otherwise, that have coefficient functions on the leading order derivative term that vanish on $\partial \Omega$. For the matrix valued function $\Phi : \bar{\Omega} \to \mathbb{C}^{d\times d}$, we only require that at least one eigenvalue vanishes on $\partial \Omega$. The operators shown in \eqref{some elliptic operator} and \eqref{linearized compacton equation} are examples of operators that can be analyzed using the results presented here.
\section{The main results} \label{section:main results}
\subsection{Preliminaries}
Throughout this article we make the following assumptions:
\begin{itemize}
\item The domain $\Omega \subset \mathbb{R}^d$ is open and bounded and $\partial \Omega$ is Lipshitz continuous.
\item The function $\varphi : \bar{\Omega} \to \mathbb{R}$ is such that $\varphi \in \mathscr{C}^k(\bar{\Omega})$ for some positive integer $k$, $\varphi > 0$ on $\Omega \subset \mathbb{R}^d$, and $\ker \varphi = \partial \Omega$.
\item The ambient function space for the differential operator $L$ is $L^p(\Omega)$ where $1 < p < \infty$.
\end{itemize}
We say the scalar valued function $\varphi$ is \emph{simply vanishing on} $\partial \Omega$ if for each $y \in \partial \Omega$ there exists an $a \neq 0$ such that
\begin{align}\label{vanishing_definition intro}
\lim_{x \to y} \frac{\varphi(x)}{\text{dist}(x,\partial \Omega)} = a,
\end{align}
where the limit is taken in $\Omega$. For the matrix valued function $\Phi : \bar{\Omega} \to \mathbb{C}^{d\times d}$ we make restrictions on its eigenfunctions, defined as the functions $\varphi_i$ such that $\Phi(x) v = \varphi_i(x)v$ for some $v \in \mathbb{C}^d$. We say the matrix valued function $\Phi : \bar{\Omega} \to \mathbb{C}^{d\times d}$ is \emph{simply vanishing on} $\partial \Omega$ if $\Phi(x)$ is positive semi-definite in $\bar{\Omega}$ and for each fixed $i$, the eigenfunction $\varphi_i$ is either strictly positive on $\bar{\Omega}$ or simply vanishing on $\partial \Omega$, with at least one $i$ such that $\varphi_i$ is simply vanishing on $\partial \Omega$.
\subsection{Results}
In this section, we summarize the results proved in this article. In the following theorem, $\lfloor a \rfloor$ denotes the integer part of $a$.
\begin{thm}
\label{thm:kinda hardy intro}
Let $\Omega \subset \mathbb{R}^d$ be a bounded open set and let $\varphi \in \mathscr{C}^1(\bar{\Omega})$ be simply vanishing on $\partial \Omega$. Fix $m \in \mathbb{N}$ and $1 < p < \infty$. Assume $k \in \mathbb{N}$ is such that
\begin{align*}
k > \tfrac{d}{p} + (m-1)\big\lfloor \tfrac{d}{p} \big\rfloor,
\end{align*}
and that the boundary $\partial \Omega$ is $\mathscr{C}^k$. If $u \in L^{p}(\Omega)$ and $\varphi^m u \in W^{k+m,p}(\Omega)$ then
\begin{align*}
u \in W^{\kappa,p}(\Omega), \quad \text{where } \kappa \coloneqq k- m\big\lfloor \tfrac{d}{p} \big\rfloor,
\end{align*}
and there exists a $c > 0$, independent of $u$, such that
\begin{align*}
\|u\|_{W^{\kappa, p}(\Omega)} \leq c \|\varphi^{m} u\|_{W^{k+m,p}(\Omega)}.
\end{align*}
\end{thm}
The above result is proven in section~\ref{subsection:domain_vanishing} as Theorem~\ref{thm:kinda hardy}, with the estimate proven in Remark~\ref{rem:implicit bound}.
The following theorem is proven in section~\ref{subsection:compactness} as Theorem~\ref{thm:not closed but closable}.
\begin{thm}\label{thm:not closed but closable intro}
Let $\Omega \subset \mathbb{R}^d$ be an open and bounded set with $\mathscr{C}^{0,1}$ boundary. Let $\varphi \in \mathscr{C}^{1}(\bar{\Omega})$ be simply vanishing on $\partial \Omega$. If $A$ is Fredholm on $L^p(\Omega)$ with domain $W^{k,p}(\Omega)$ for some $k > 0$, then $\varphi^m A$, $m \geq 1$, is not closed on its natural domain,
\begin{align*}
\textnormal{D} (\varphi^m A) = \{ u \in L^p : u \in \textnormal{D} (A), \; \varphi^m Au \in L^{p}(\Omega) \} \equiv \textnormal{D} (A),
\end{align*}
but $\varphi^m A$ is closable.
\end{thm}
The above theorem tells us that even simple operators do not have the desirable property of being closed on their `natural domain'. For example, the operator $L u = \sin(\pi x) u_{xx}$ is not closed on $W^{2,p}(0,1)$ for any $p \in (1,\infty)$ by Theorem~\ref{thm:not closed but closable intro}. We can use Theorem~\ref{thm:kinda hardy intro} to get an estimate on the properties of the domain, as demonstrated in the following example.
\begin{example}\label{exa:example_for_domain}
Let $\Omega = (0,1)$ and consider the operator $L$ acting on $L^p(\Omega)$ defined by $Lu = \varphi u_{xxx}$ where $\varphi$ is simply vanishing. The operator $L$ is of the form $\varphi A$ where $A$ is Fredholm. By Theorem~\ref{thm:not closed but closable intro}, $L$ is not closed on its natural domain, $W^{3,p}(\Omega)$, but is closable. Let $\bar{L}$ denote the closure of $L$, and let $\varphi^{(k)} : L^p(\Omega) \to W^{3-k,p}(\Omega)$ denote the multiplication operator $u \mapsto \varphi^{(k)} u$ where the function $\varphi^{(k)}$ denotes the $k$-th derivative of the function $\varphi$, and as an abuse of notation we set $\varphi = \varphi^{(0)}$. After rewriting $L$ as
\begin{align*}
L u = (\varphi u)_{xxx} - 3 (\varphi^{(1)} u )_{xx} + 3 (\varphi ^{(2)} u)_x - \varphi^{(3)} u,
\end{align*}
we see that
\begin{align*}
\textnormal{D} (\bar{L}) = \textnormal{D} ( A^3 \varphi ) \cap \textnormal{D} ( A^2 \varphi^{(1)} ) \cap \textnormal{D} ( A \varphi^{(2)} ) \cap \textnormal{D} ( \varphi^{(3)} ),
\end{align*}
where $A$ is the derivative operator on $L^p(\Omega)$. Specifically, if $\varphi \in \mathscr{C}^3(\bar{\Omega})$ is simply vanishing, then the fact that $ \textnormal{D} (A^3) = W^{3,p}(\Omega)$ implies $ \textnormal{D} (A^3\varphi) \subset W^{2,p}(\Omega)$ by Theorem~\ref{thm:kinda hardy intro}. By the same theorem, we have $W^{2,p}(\Omega) \subset \textnormal{D} (A^{k} \varphi^{(3-k)})$ for $k = 0,1,2$ so the best we can do is $ \textnormal{D} (\bar{L}) \subset W^{2,p}(\Omega)$.
More concretely, if we set $\varphi(x) = \sin(\pi x)$ and fix $p=2$, then one can construct functions $u \in \textnormal{D} (\bar{L})$ such that $u \not\in W^{3,2}(\Omega)$ but $u \in W^{2,2}(\Omega)$. See \cite[\S 3.1]{jordon2013properties}.
\end{example}
The following theorem speaks about the range of the multiplication operator. It
is proved in section~\ref{subsec:range vanishing} as Theorem~\ref{thm:range closed vanishing}.
\begin{thm}
\label{thm:range closed vanishing intro}
Let $\Omega \subset \mathbb{R}^d$ be open and bounded with $\mathscr{C}^{0,1}$ boundary and assume the function $\varphi \in \mathscr{C}^k(\bar{\Omega})$ is simply vanishing on $\partial \Omega$. Then the range of the operator $u \mapsto \varphi^m u$ is closed in $W^{k,p}(\Omega)$ whenever $k \geq m$ and is not closed when $k < m$.
\end{thm}
If we know the range of the multiplication operator $u \mapsto \varphi^m u$ is closed in $W^{k,p}(\Omega)$ then we necessarily have
\begin{align*}
\|u\|_{L^p(\Omega)} \leq c \|\varphi^m u\|_{W^{k,p}(\Omega)}
\end{align*}
for some constant $c > 0$.
The matrix valued analogs of Theorems~\ref{thm:kinda hardy intro} and \ref{thm:range closed vanishing intro} are proved in section~\ref{subsection:matrix_functions}. One implication is illustrated in the following example.
\begin{example}
Let $\Omega\subset \mathbb{R}^d$ be open and bounded with $\mathscr{C}^{0,1}$ boundary. Let $\Phi \in \mathscr{C}^1(\bar{\Omega}; \mathbb{R}^{d\times d})$ be simply vanishing on $\partial \Omega$. Then by the matrix analog of Theorem~\ref{thm:range closed vanishing intro} (which is Theorem~\ref{thm:range closed vanishing_matrices}) we know that the range of $\Phi^m$, $m \in \mathbb{N}$, is closed in
\begin{align*}
W^{1,2}(\Omega^d) \coloneqq \underbrace{W^{1,2}(\Omega) \times \cdots \times W^{1,2}(\Omega),}_{d \text{ copies}}
\end{align*}
if and only if $m = 1$. This implies $\Phi^m : L^2(\Omega^d) \to W^{1,2}(\Omega^d)$ is semi-Fredholm if and only if $m=1$. Now, it is well known that the weak gradient $\nabla : W^{1,2}(\Omega) \subset L^2(\Omega) \to L^2(\Omega^d)$ and its adjoint, $ \text{div} (\cdot) : L^2(\Omega^d) \to L^2(\Omega)$, are semi-Fredholm. Thus, the non-uniformly elliptic operator,
\begin{align}
Lu = \text{div} ( \Phi^m \nabla u),
\end{align}
is semi-Fredholm on $L^2(\Omega)$ if and only if $m=1$.
\end{example}
\begin{thm}
\label{thm:roots equal spectrum intro}
Let $\Omega \subset \mathbb{R}^d$ be open and bounded with $\mathscr{C}^{0,1}$ boundary, $m,k \in \mathbb{N}$ with $0 < m < k$, and let $A$ be densely defined on $L^p(\Omega)$. Assume the following:
\begin{itemize}
\item The operator $A$ is closed on $L^p(\Omega)$ and $ \textnormal{D} (A) \subset W^{k,p}(\Omega)$.
\item There exists a $u \in \textnormal{D} (A)$ such that $u \notin W^{m,p}_0(\Omega) \cap W^{k,p}(\Omega)$.
\item The resolvent set $\rho(A)$ is non-empty.
\end{itemize}
If $\varphi \in \mathscr{C}^k(\bar{\Omega})$ is simply vanishing then
\begin{align*}
\sigma_{\text{ess}} (A \varphi^m ) = \sigma_{\text{ess}} (\overline{\varphi^m A^*} ) =\mathbb{C}.
\end{align*}
Moreover, if either $A\varphi^m$ or $\overline{\varphi^m A^*}$ is Fredholm then
\begin{align*}
\sigma_{\text{p}} (\overline{\varphi^m A^*} ) = \mathbb{C}.
\end{align*}
\end{thm}
In the above theorem, $ \sigma_{\text{p}} (L)$ and $ \sigma_{\text{ess}} (L)$ denote the point spectrum and essential spectrum of $L$ respectively. The definition of the essential spectrum is given in section~\ref{section:spectrum} and the result is proven as Theorem~\ref{thm:roots equal spectrum}. Its import is demonstrated in the following example.
\begin{example}
This example continues from Example~\ref{exa:example_for_domain}, where $\Omega = (0,1)$, and $L = \varphi u_{xxx}$. For any $u \in W^{3,p}(\Omega)$, there exists $a,b \in \mathbb{R}$ such that $u + ax + b \in W^{1,p}_0(\Omega)$, which implies
\begin{align*}
W^{3,p}(\Omega) = W^{3,p}(\Omega) \cap W^{1,p}_0(\Omega) \oplus \text{span}\{1,x\}.
\end{align*}
Now consider the multiplication operator $\varphi : L^p(\Omega) \to W^{3,p}(\Omega)$ given by $u \mapsto \varphi u$ where $\varphi$ is simply vanishing on $\partial \Omega$. Then we know that the range of $\varphi$ is $W^{3,p}(\Omega) \cap W^{1,p}_0(\Omega)$, which has co-dimension 2 in $W^{3,p}(\Omega)$. Thus, if we let $A$ denote three applications of the weak derivative operator on $L^p(\Omega)$, then we know that $\rho(A)$ is nonempty, and $ \textnormal{D} (A) = W^{3,p}(\Omega) \subset\subset L^p(\Omega)$ by the Rellich-Kondrachov Theorem. Then we see that $A \varphi$ has finite dimensional nullspace and the range has finite co-dimension. This shows $A \varphi$ is Fredholm. Applying Theorem~\ref{thm:roots equal spectrum intro} yields $ \sigma_{\text{p}} (\bar{L}) = \sigma_{\text{p}} (\overline{\varphi A}) = \mathbb{C}$.
More concretely, we have $ \sigma_{\text{p}} (\overline{\sin(\pi x) u_{xxx}} ) = \mathbb{C}$. The same holds if we set $\varphi(x) = \sin^2(\pi x)$, but not necessarily when we set $\varphi(x) = \sin^m(\pi x)$ where $m \in \mathbb{N}$ and $m \geq 3$.
\end{example}
\subsection{Outline of the article}
The article is structured as follows:
\begin{itemize}
\item Section~\ref{section:preliminaries} introduces the notation and basic definitions that are used in this article.
\item Section~\ref{section:basics} goes over some basic properties of closed and Fredholm operators.
\item Section~\ref{section:properties_vanishing} covers the properties of the operators $u \mapsto \varphi u$ and $\mathbf{u} \mapsto \Phi \mathbf{u}$. The domain and range of the operator $u \mapsto \varphi u$ is covered in sections~\ref{subsection:domain_vanishing} and \ref{subsec:range vanishing} respectively. Matrix valued operators are handled in section~\ref{subsection:matrix_functions}
\item We study various properties of differential operators composed with vanishing operators in section~\ref{section:differential_with}. In particular, we focus on the Fredholm properties and spectra of the operators $A\varphi$ and $\varphi A$ where $A$ is a Fredholm differential operator.
\end{itemize}
\section{Notation and definitions}\label{section:preliminaries}
We will review some of the basic definitions and introduce the notation used in this article. We will use capital letters, such as $W$, $X$, $Y$, or $Z$, to denote a Banach space. We use $\mathcal{B}(X, Y)$ to denote the set of bounded linear operators from $X$ to $Y$, and $\mathcal{C}(X,Y)$ to denote the set of closed and densely defined linear operators from $X$ to $Y$. The sets $\mathcal{B}(X)$ and $\mathcal{C}(X)$ denote the sets $\mathcal{B}(X,X)$ and $\mathcal{C}(X,X)$ respectively.
The domain and range of a linear operator $A$ will be denoted by $ \textnormal{D} (A)$ and $ \textnormal{R} (A)$ respectively, and we use $ \textnormal{N} (A)$ to denote the nullspace of $A$. If $A \in \mathcal{C}(X,Y)$, then $ \textnormal{D} (A)$ equipped with the graph norm,
\begin{align*}
\|x\|_{ \textnormal{D} (A)} \coloneqq \|x\|_{X} + \|Ax\|_{Y}, \qquad x \in \textnormal{D} (A),
\end{align*}
is a Banach space, and we call $\|\cdot\|_{ \textnormal{D} (A)}$ the $A$\emph{-norm}. When referring to the composition of two linear operators $A$ and $B$, the subspace
\begin{align*}
\textnormal{D} (AB) = \{ \, x \in \textnormal{D} (B) : Bx \in \textnormal{D} (A) \, \},
\end{align*}
is called the \emph{natural domain} of $AB$.
If $A$ is a linear operator from $X$ to $Y$, any closed operator ${A}_1$ where $ \textnormal{D} (A) \subset \textnormal{D} (A_1)$ and $A = {A}_1$ on $ \textnormal{D} (A)$ is called a {\it closed extension} of $A$. We call $A$ {\it closable} if there exists a closed extension of $A$. We denote $\bar{A}$ as the closure of $A$, and it is the `smallest' closed extension, in the sense that $ \textnormal{D} (\bar{A}) \subset \textnormal{D} (A_1)$ for any operator $A_1$ that is a closed extension of $A$. An operator is closable if every sequence $\{x_n\} \subset \textnormal{D} (A)$ where $x_n \to 0$ in $X$ and $Ax_n \to y$ in $Y$ implies $y = 0$.
A Banach space $Y$ is said to be \emph{continuously embedded} in another Banach space $X$ if there exists an operator $P \in \mathcal{B}(Y,X)$ that is one-to-one. The space $Y$ is said to be \emph{compactly embedded} in $X$ if $P$ is also compact and we write $Y \subset\subset X$ whenever $Y$ is compactly embedded in $X$. For Sobolev spaces, we take $P$ to be the inclusion operator, which we denote as $\iota$.
Most of the analysis takes place in $L^p(\Omega)$ and the Sobolev spaces $W^{k,p}(\Omega)$, where $k \in \mathbb{N}$, $\Omega \subset \mathbb{R}^d$ is an open and bounded set, and, unless stated otherwise, $1 < p < \infty$. The closure of a set $\Omega \subset \mathbb{R}^d$ will be denoted by $\bar{\Omega}$ and the boundary of $\Omega$ will be denoted by $\partial \Omega$. The space $\mathscr{C}^k(\Omega)$ denotes the space of all functions from $\Omega$ to $\mathbb{R}$ that are $k$-times continuous differentiable everywhere in $\Omega$, and $\mathscr{C}^{k}_{0}(\Omega) \subset \mathscr{C}^k(\Omega)$ denotes the subspace of those functions with compact support in $\Omega$. The space $W^{k,p}_0(\Omega)$ denotes the closure of $\mathscr{C}^{\infty}_{0}(\Omega)$ in $W^{k,p}(\Omega)$. If $u$ is weakly differentiable, we let $D^{\alpha}u$ denote the $\alpha$-th weak derivative of $u$, where $\alpha = (\alpha_1,\ldots,\alpha_d) \in \mathbb{Z}^d_+$ is a multi-index, and we let $|\alpha| = \alpha_1 + \cdots +\alpha_d$ denote the order of $\alpha$. We use $\nabla^{(k)} u$ to denote the vector of all weak derivatives of $u$ with order $k$, and set $\nabla u = \nabla^{(1)} u$ to be the gradient of $u$.
We say $\partial \Omega$ is $\mathscr{C}^{k,\gamma}$ if for each point $y \in \partial \Omega$, there exists an $r > 0$ and a $\mathscr{C}^{k,\gamma}$ function $\gamma : \mathbb{R}^{d-1} \to \mathbb{R}$ such that
\begin{align*}
\Omega \cap B(y,r) = \{ x \in B(y,r) : x_d > \gamma(x_1,\ldots, x_{d-1})\},
\end{align*}
where $B(x,r) \coloneqq \{ y \in \mathbb{R}^d : | x - y| < r\}$, and $\mathscr{C}^{k,\gamma}$ is a H\"{o}lder space.
\begin{definition}
Let $\Omega \subset \mathbb{R}^d$ be a bounded and open set. For any $\varphi \in \mathscr{C}^1(\bar{\Omega})$, we will call the multiplication operator $u \mapsto \varphi u$ \emph{vanishing} if $\ker \varphi = \partial \Omega$. As an abuse of notation, we will use $\varphi$ to refer to the multiplication operator $u \mapsto \varphi u$.
\end{definition}
\begin{definition}
Let $\Omega \subset \mathbb{R}^d$ be a bounded and open set. Take $\text{dist}(x,\partial \Omega) \coloneqq \inf_{y \in \partial \Omega} |x - y|$ to be the distance from $x$ to the boundary of $\Omega$. Let $\varphi \in \mathscr{C}^{1}(\bar{\Omega})$. We say $\varphi$ is \emph{simply vanishing on} $\partial \Omega$ if $\ker \varphi = \partial \Omega$, and for each $y \in \partial \Omega$ there exists an $a \neq 0$ such that
\begin{align}\label{vanishing_definition}
\lim_{x \to y} \frac{\varphi(x)}{\text{dist}(x,\partial \Omega)} = a,
\end{align}
where the limit is taken within $\Omega$. The multiplication operator $u \mapsto \varphi u$ is called simply vanishing on $\partial \Omega$ if the function $\varphi$ is simply vanishing on $\partial \Omega$.
\label{def:simply_vanishing}
\end{definition}
\begin{definition}
Let $\Omega \subset \mathbb{R}^d$ be a bounded and open set. We say the function $\varphi \in \mathscr{C}^{m}(\bar{\Omega})$ is \emph{vanishing of order} $m$ on $\partial \Omega$ if $\ker \varphi = \partial \Omega$, $D^{\alpha} \varphi = 0$ on $\partial \Omega$ when $|\alpha| < m$, and $\nabla^{(m)} \varphi \neq 0$ on $\partial \Omega$. The multiplication operator $u \mapsto \varphi u$ is called vanishing of order $m$ on $\partial \Omega$ if the function $\varphi$ is vanishing of order $m$ on $\partial \Omega$.
\label{def:simply_vanishing_equivalent}
\end{definition}
Functions that are vanishing of order 1 are simply vanishing functions. To see why, take $\Omega\subset\mathbb{R}^d$ with $\mathscr{C}^1$ boundary and assume $\varphi \in \mathscr{C}^k(\bar{\Omega})$ is simply vanishing. Fix any $y \in \partial \Omega$ and let $\Omega_n = B(y,n^{-1}) \cap \Omega$. Since $\partial \Omega$ is $\mathscr{C}^1$, there exists a point $x_n \in \Omega_n$ such that $|x_n - y| = \text{dist}(x_n,\partial \Omega)$. Given $\varphi$ is simply vanishing, there exists an $a \neq 0$ such that
\begin{align*}
a = \lim_{n \to\infty} \frac{\varphi(x_n)}{\text{dist}(x_n,\partial \Omega)}
=\lim_{n \to\infty} \frac{|\varphi(x_n) - \varphi(y)|}{|x_n - y|} = |\nabla \varphi(y)|,
\end{align*}
which shows $\nabla \varphi(y) \neq 0$. Since $y \in \partial \Omega$ was arbitrary, we see that $\nabla \varphi \neq 0$ on $\partial \Omega$ whenever $\partial \Omega$ is $\mathscr{C}^1$.
We have a similar definition for matrix-valued functions. Let $\Phi : \bar{\Omega} \to \mathbb{C}^{d \times d}$ be Hermitian for each $x \in \bar{\Omega}$. Then there exists a unitary matrix $\mathbf{U}(x)$ and a real diagonal matrix $\mathbf{D}(x)$ such that
\begin{align}\label{eq:schur's decomposition}
\Phi = \mathbf{U} \mathbf{D} \mathbf{U}^*,
\end{align}
by Schur's decomposition theorem. If $\Phi \in \mathscr{C}^1(\bar{\Omega}; \mathbb{C}^{d\times d})$, we can choose $\mathbf{U}$ and $\mathbf{D}$ in $\mathscr{C}^{1}(\bar{\Omega}; \mathbb{C}^{d\times d})$ so that \eqref{eq:schur's decomposition} holds. Thus, we lose no generality by assuming the operator $\mathbf{D}$ has the form $\mathbf{D} = \text{diag}(\varphi_1, \ldots, \varphi_d)$ for some functions $\varphi_i \in \mathscr{C}^1(\bar{\Omega})$ where $i = 1,\ldots, d$.
\begin{definition}
Let $\Omega \subset \mathbb{R}^d$ be open and bounded, and let $\Phi \in \mathscr{C}^{m}(\bar{\Omega}; \mathbb{C}^{d \times d})$. We say the function $\Phi$ is \emph{vanishing of order} $m$ if $\Phi$ is positive semi-definite on $\bar{\Omega}$, and the matrix $\mathbf{D} = \text{diag}(\varphi_1,\ldots,\varphi_d)$ in its Schur decomposition has the following property: for each $i = 1,\ldots,d$, either $\varphi_i > 0$ on $\bar{\Omega}$ or $\varphi_i$ is vanishing of order $m_i$ on $\partial \Omega$ with $m_i \leq m$, with at least one function $\varphi_i$ that is vanishing of order $m$ on $\partial \Omega$. The multiplication operator $\mathbf{u} \mapsto \Phi \mathbf{u}$ is called vanishing of order $m$ if the function $\Phi$ is vanishing of order $m$.
\label{def:simply vanishing matrices2}
\end{definition}
\subsection{Fredholm and semi-Fredholm operators}\label{subsection:fredholm}
We will utilize Fredholm operator theory when describing the properties of the multiplication operators $u \mapsto \varphi u$ and $\mathbf{u} \mapsto \Phi \mathbf{u}$. In this section, we briefly review the theory of Fredholm operators.
Let $X$ and $Y$ be Banach spaces. An operator $A : X \to Y$ is called \emph{Fredholm} if
\begin{enumerate}[label=(\alph*)]
\item The domain of $A$ is dense in $X$,
\item The operator $A$ is closed on its domain,
\item The nullspace of $A$ is finite dimensional, \label{nullspace_finite_dimensional}
\item The range of $A$ is closed in $Y$,
\item The co-dimension of the range of $A$ is finite dimensional, \label{codimension_of_the_range}
\end{enumerate}
where the co-dimension of a closed subspace $M \subset Y$, denoted $ \textnormal{co-\!} \dim M$, is the dimension of the quotient space $Y /M$. We use $\mathcal{F}(X,Y)$ to denote the set of Fredholm operators from $X$ to $Y$ and write $\mathcal{F}(X)$ in place of $\mathcal{F}(X,X)$. Note that property \ref{codimension_of_the_range} is equivalent to requiring that the nullspace of the adjoint operator $A^*$ be finite dimensional. The \emph{index} of a Fredholm operator $A$ is defined as
\begin{align*}
\text{ind}(A) \coloneqq \dim \textnormal{N} (A) - \textnormal{co-\!} \dim \textnormal{R} (A).
\end{align*}
The set of \emph{semi-Fredholm} operators from $X$ to $Y$, denoted $\mathcal{F}_+(X,Y)$, is the set of operators that satisfy all the properties of Fredholm opertors except possibly property \ref{codimension_of_the_range}. The set of semi-Fredholm operators from $X$ to $X$ will be denoted by $\mathcal{F}_+(X)$. We note that our definition for $\mathcal{F}_+(X,Y)$ is sometimes referred to as the set of \emph{upper semi-Fredholm} operators.
The following characterization of Fredholm operators is useful.
\begin{thm}[\cite{schechter2002principles}, Theorem 7.1]
\label{thm:equivalence_of_fredholm}
Let $X$ and $Y$ be Banach spaces. Then $A \in \mathcal{F}(X,Y)$ if and only if there exists closed subspaces $X_0 \subset X$ and $Y_0 \subset Y$ where $Y_0$ is finite dimensional and $X_0$ has finite co-dimension such that
\begin{align*}
X = X_0 \oplus \textnormal{N} (A), \qquad Y = \textnormal{R} (A) \oplus Y_0.
\end{align*}
Moreover, there exists operators $A_0 \in \mathcal{B}(Y,X)$, $K_1 \in \mathcal{B}(X)$, and $K_2 \in \mathcal{B}(Y)$ where
\begin{itemize
\item The $ \textnormal{N} (A_0) = Y_0$,
\item The $ \textnormal{R} (A_0) = X_0 \cap \textnormal{D} (A)$,
\item $A_0 A = I - K_1$ on $ \textnormal{D} (A)$,
\item $A A_0 = I - K_2$ on $Y$,
\item The $ \textnormal{N} (K_1) = X_0$, while $K_1 = I$ on $ \textnormal{N} (A)$,
\item The $ \textnormal{N} (K_2) = \textnormal{R} (A)$, while $K_2 = I$ on $Y_0$.
\end{itemize}
\end{thm}
Note that $K_1$ and $K_2$ are projection operators and their ranges are finite dimensional. The operator $A_0$ from Theorem~\ref{thm:equivalence_of_fredholm} will be referred to as the \emph{pseudo-inverse} of $A$, since $A A_0 A = A$ and $A_0 A A_0 = A_0$.
An equivalent characterization of semi-Fredholm operators is as follows: if $X$ and $Y$ are Banach spaces and $A \in \mathcal{C}(X,Y)$, then $A$ is not semi-Fredholm if and only if there exists a bounded sequence $\{x_k\} \subset \textnormal{D} (A)$ having no convergent subsequence such that $\{Ax_k\}$ converges. A proof of this equivalence can be found in \cite{schechter1976remarks} or \cite[p. 177]{schechter2002principles}.
\begin{rem}
Suppose $\Omega\subset \mathbb{R}^d$ is open and bounded, $\varphi \in \mathscr{C}^{m}(\bar{\Omega})$ is simply vanishing on $\partial \Omega$ and $\zeta \in \mathscr{C}^{m}(\bar{\Omega})$ is vanishing of order $m$ on $\partial \Omega$. Then the multiplication operator $u \mapsto \varphi^m u$ is semi-Fredholm from $L^p(\Omega)$ to $W^{k,p}(\Omega)$ if and only if the mapping $u \mapsto \zeta u$ is semi-Fredholm. Moreover, $ \textnormal{D} (\varphi^m) = \textnormal{D} (\zeta)$. To see why, use the fact that the multiplication operators $u \mapsto \zeta \varphi^{-m} u$ and $u \mapsto \varphi^{m} \zeta^{-1} u$ are one-to-one and onto $L^p(\Omega)$ and that the composition of a semi-Fredholm operator with a Fredholm operator is semi-Fredholm.
\end{rem}
\section{Basic properties of closed operators}\label{section:basics}
Fredholm operators are closed under composition. That is, if $X,Y$, and $Z$ are Banach spaces, then $A \in \mathcal{F}(X,Y)$ and $B \in \mathcal{F}(Y,Z)$ implies $BA \in \mathcal{F}(X,Z)$ with $\text{ind}(BA) = \text{ind}(B) + \text{ind}(A)$. Moreover, if $B \in \mathcal{C}(Y,Z)$ and $BA \in \mathcal{F}(X,Z)$ then we necessarily have $B \in \mathcal{F}(Y,Z)$. These claims are proved, respectively, in \cite[p. 157]{schechter2002principles} as Theorem 7.3, and \cite[p. 162]{schechter2002principles} as Theorem 7.12. As for semi-Fredholm operators, we have the following.
\begin{lem}[\cite{schechter1976remarks}, Lemma 4]\label{lem:reverse_products_and_semifredholm}
Let $X,Y$, and $Z$ be Banach spaces and let $A \in \mathcal{F}(X,Y)$, and $B \in \mathcal{C}(Y,Z)$. If $BA \in \mathcal{F}_+(X,Z)$, then $B \in \mathcal{F}_+(Y,Z)$.
\end{lem}
One of the theorems that we use throughout this article is the following consequence of the Closed Graph Theorem. A proof of the Closed Graph Theorem can be found in many functional analysis textbooks, such as \cite[p. 62]{schechter2002principles} or \cite[p. 166]{kato1995perturbation}.
\begin{lem}\label{lem:equivalence_of_norms}
Let $X$ be a Banach space and $Y \subset X$. If there exists a norm that converts $Y$ into a Banach space then there exists a $c > 0$ such that $\|y\|_X \leq c \|y\|_Y$ for all $y \in Y$. If $Y = X$ then their norms are equivalent.
\end{lem}
\begin{proof}
Let $\iota$ denote the inclusion map from $Y$ to $X$. It is a closed operator with domain equal to $Y$, so by the Closed Graph Theorem it is bounded. If $Y=X$ then apply the above argument to the inclusion map from $X$ to $Y$.
\end{proof}
The above lemma is useful for showing that special subsets of $L^p(\Omega)$ have certain properties --- such as compactness --- since they can inherit such properties from other Sobolev spaces. One of the most important consequences of compactness is the following theorem.
\begin{thm}\label{thm:compactness equals closed range}
Let $X$ and $Y$ be Banach spaces. If $A \in \mathcal{C}(X,Y)$ with $ \textnormal{D} (A) \subset\subset X$, then $A \in \mathcal{F}_+(X,Y)$.
\end{thm}
\begin{proof}
We prove this by contradiction. Suppose $A$ is not semi-Fredholm. Since $A \in \mathcal{C}(X, Y)$, this implies there exists a bounded sequence $\{x_n \}\subset \textnormal{D} (A)$ having no convergent subsequence, such that $\{Ax_n \}$ is convergent in $Y$. But if $\{x_n\}$ is bounded in $X$ and $\{A x_n\}$ is convergent in $Y$, then $\{x_n\}$ is a bounded sequence in the $A$-norm. Since $ \textnormal{D} (A) \subset\subset X$ there exists a subsequence of $\{x_n\}$ that is convergent in $X$, the desired contradiction.
\end{proof}
\begin{rem}
We know an operator $A \in \mathcal{C}(X,Y)$ has closed range if and only if there exists a $c > 0$ such that
\begin{align*}
\inf_{z \in \textnormal{N} (A)} \|x - z\|_{X} \leq c \|Ax\|_{Y}, \qquad \text{for all } x \in \textnormal{D} (A).
\end{align*}
Theorem~\ref{thm:compactness equals closed range} can be used to quickly establish estimates involving differential operators. We can, for example, establish Poincar\'e inequalities.
It is also well known that $W^{1,p}_{0}(\Omega) \subset\subset L^p(\Omega)$ for any bounded and open set $\Omega \subset \mathbb{R}^d$. Since the weak gradient operator $\nabla$ is closed on $W^{1,p}_0(\Omega)$, we get $ \textnormal{R} (\nabla)$ is closed by Theorem~\ref{thm:compactness equals closed range}. This implies the existence of a $c > 0$ such that
\begin{align*}
\|u\|_{L^p(\Omega)} = \inf_{a \in \textnormal{N} (\nabla)} \|u - a\|_{L^p(\Omega)} \leq c\|\nabla u\|_{L^p(\Omega)},
\end{align*}
for any $u \in W^{1,p}_0(\Omega)$.
\end{rem}
\section{Properties of vanishing operators}\label{section:properties_vanishing}
\subsection{The domain of a vanishing operator}\label{subsection:domain_vanishing}
In this section we establish properties of the domain of the vanishing operator $\varphi : L^p(\Omega) \to W^{k,p}(\Omega)$. In particular, we will establish the embedding of the domain of $\varphi$ in various Sobolev spaces.
If $\varphi \in \mathscr{C}^1(\bar{\Omega})$, then the mapping $u \mapsto \varphi u$ is bounded from $L^p(\Omega)$ to $L^p(\Omega)$. This implies that a natural choice for its domain is $L^p(\Omega)$. Whenever a vanishing operator $\varphi$ is composed with a differential operator $A$ to form $A\varphi$ --- $A$ being an operator that is closed on $W^{k,p}(\Omega)$ --- it makes sense to think of $\varphi$ as a densely defined operator that maps some subset of $L^p(\Omega)$ to the space $W^{k,p}(\Omega)$.
This and subsequent sections rely heavily on Hardy's inequality, so we include the statement for the reader's convenience.
\begin{thm}[\cite{wannebo1990hardy}, Hardy's Inequality]\label{thm:hardys_inequality}
Let $\Omega \subset \mathbb{R}^d$ be a bounded open set with $\mathscr{C}^{0,1}$ boundary, and $\delta(x) = \inf_{y \in \partial \Omega} |x - y|$. Then, for all $u \in \mathscr{C}^{\infty}_{0}(\Omega)$,
\begin{align*}
\| \delta^{-m} u\|_{L^p(\Omega)}
\leq c \| \nabla^{(m)} u \|_{L^{p}(\Omega)},
\end{align*}
where $c>0$ depends on $\Omega$, $p$, $d$, and $m$.
\end{thm}
See \cites{lehrback2014weighted, hajlasz1999pointwise} for recent developments on the assumptions necessary for Hardy's inequality. The interested reader should consult \cite[\S 2.7]{maz2011sobolev} for a treatment of optimal constants for Hardy's inequality.
We begin with basic properties of the domain and range of the multiplication operator $u \mapsto \varphi^m u$.
\begin{lem}\label{lem:statement of range}
Let $\Omega \subset \mathbb{R}^d$ be open and bounded and $\varphi \in \mathscr{C}^1(\bar{\Omega})$ be simply vanishing on $\partial \Omega$. For each $k,m \in \mathbb{Z}_+$, the multiplication operator $\varphi^m : L^p(\Omega) \to W^{k,p}(\Omega)$ defined as $u \mapsto \varphi^m u$ is closed on
\begin{align*}
\textnormal{D} (\varphi^m) = \{ u \in L^p(\Omega) : \varphi^m u \in W^{k,p}(\Omega) \}.
\end{align*}
Moreover, if $\varphi \in \mathscr{C}^m(\bar{\Omega})$ and $m \leq k$ then $ \textnormal{R} (\varphi^m) = W^{k,p}(\Omega) \cap W^{m,p}_{0}(\Omega)$.
\end{lem}
\begin{proof}
The proof is broken into two claims.
\begin{claim}{1}
The multiplication operator $u \mapsto \varphi^m u$ is closed on $ \textnormal{D} (\varphi^m)$.
\end{claim}
We first show it is closable. Suppose $u_n \to 0$ in $L^p(\Omega)$ and $\varphi^m u_n \to y$ in $W^{k,p}(\Omega)$. Then we know that $\varphi^m u_n \to y$ in $L^p(\Omega)$. But since $\varphi$ is bounded and $u_n \to 0$ in $L^p(\Omega)$ we can conclude $\varphi^m u_n \to 0 = y$ in $L^p(\Omega)$. This shows the multiplication operator $\varphi^m$ is closable on its domain. But any closed extension cannot be defined on a set larger than $ \textnormal{D} (\varphi^m)$, implying the domain of any closed extension must be $ \textnormal{D} (\varphi^m)$. This completes the proof of the claim.
\begin{claim}{2}
If $\varphi \in \mathscr{C}^m(\bar{\Omega})$ and $m \leq k$ then $ \textnormal{R} (\varphi^m) = W^{k,p}(\Omega) \cap W^{m,p}_{0}(\Omega)$.
\end{claim}
Take $v \in W^{k,p}(\Omega) \cap W^{m,p}_{0}(\Omega)$. Since $v \in W^{m,p}_0(\Omega)$ we can apply Hardy's inequality (Theorem~\ref{thm:hardys_inequality}) to show that $\varphi^{-m} v \in L^p(\Omega)$. Since this implies $\varphi^{-m} v \in \textnormal{D} (\varphi^m)$ we have $W^{k,p}(\Omega) \cap W^{m,p}_{0}(\Omega) \subset \textnormal{R} (\varphi^m)$.
For the other direction, first note that since $\varphi^m \in \mathscr{C}^m(\bar{\Omega}) \cap W^{m,p}_0(\Omega)$ there exists a sequence $\{\phi_n\} \subset \mathscr{C}^{\infty}_0(\Omega)$ such that $\phi_n \to \varphi^m$ in $W^{m,p}(\Omega)$ and $\{D^{\alpha}\phi_n\}$ is uniformly bounded when $|\alpha| \leq m$\footnote{One such example are the functions $\phi_{n} = \varphi^m \mathbf{1}_{\Omega_{n}} * \eta_{1 / 3n}$, where $\mathbf{1}_{\Omega_n}$ is an indicator function for the set $\Omega_{n}= \{x \in \Omega : \text{dist}(x, \partial \Omega) > 1/n \}$ and $\eta_{\epsilon}$ is a mollifier.}. Let $u \in \textnormal{D} (\varphi^m)$ be arbitrary, and set $v_n = \phi_n - \varphi^m$. Since $D^{\alpha}(\phi_n u)$ is bounded by $cD^{\alpha}(\varphi u)$ for some constant $c$ and $|D^{\alpha}(v_n u)|^p \to 0$ almost everywhere when $|\alpha| \leq m$ we have
\begin{align*}
\lim_{n\to\infty} \|D^{\alpha} (\phi_n u) - D^{\alpha} (\varphi^m u) \|_{L^p(\Omega)}
=\lim_{n\to\infty} \|D^{\alpha} (v_n u ) \|_{L^p(\Omega)} = 0,
\end{align*}
when $|\alpha| \leq m$ by dominated convergence. Noting that $\phi_n u \in W^{m,p}_0(\Omega)$ shows $\varphi^m u \in W^{m,p}_0(\Omega)$ and completes the proof.
\end{proof}
The following lemma establishes the relative compactness of $ \textnormal{D} (\varphi^m)$ in $L^p(\Omega)$ when $\varphi^m$ maps to $W^{k,p}(\Omega)$ for $k > m$. It uses the relative compactness of $W^{m+1,p}(\Omega)$ in $W^{m,p}(\Omega)$. This is implied by the Rellich-Kondrachov Theorem, which establishes that for $1 \leq p < \infty$, $W^{1,p}(\Omega)$ is compactly embedded in $L^p(\Omega)$ whenever $\Omega$ is a bounded domain with Lipshitz continuous boundary. See \cite[p. 168]{adams2003sobolev} Thoerem 6.3 for the full statement and proof of the Rellich-Kondrachov Theorem.
\begin{lem}\label{lem:compactness implies compactness almost}
Let $\Omega \subset \mathbb{R}^d$ be open and bounded with $\mathscr{C}^{0,1}$ boundary and let $\varphi \in \mathscr{C}^{1}(\bar{\Omega})$ be simply vanishing on $\partial \Omega$. If
\begin{align*}
\textnormal{D} (\varphi^m) = \{u \in L^p(\Omega) : \varphi^m u \in W^{m+1,p}(\Omega) \}
\end{align*}
then $ \textnormal{D} (\varphi^m) \subset\subset L^p(\Omega)$.
\end{lem}
\begin{proof}
Since $\Omega$ is bounded and $\partial \Omega$ is $\mathscr{C}^{0,1}$, we can use the Rellich-Kondrachov Theorem to establish $W^{m+1,p}(\Omega) \subset \subset W^{m,p}(\Omega)$. Suppose $\{ u_n \} \subset \textnormal{D} (\varphi^m)$ is such that $\|u_n\|_{ \textnormal{D} (\varphi^m)} \leq 1$ for each $n$. Then $\| \varphi^m u_n\|_{W^{m+1,p}(\Omega)} \leq 1$ for each $n$, so there exists a subsequence that is convergent in $W^{m,p}(\Omega)$. After relabeling the convergent subsequence, we take this to be the entire sequence. Applying Hardy's inequality (Theorem~\ref{thm:hardys_inequality}) yields
\begin{align*}
\lim_{n,k\to \infty} \|u_n - u_k\|_{L^{p}(\Omega)}
\leq \lim_{n,k\to \infty} c \| \varphi^m u_n - \varphi^m u_k\|_{W^{m,p}(\Omega)} = 0,
\end{align*}
completing the proof.
\end{proof}
Next we establish the embedding of $ \textnormal{D} (\varphi)$ in various Sobolev spaces. To do so, we will use the fact that when $\Omega \subset \mathbb{R}^d$ is open and bounded with $\mathscr{C}^k$ boundary, the map
\[
u \mapsto u |_{\partial \Omega}
\]
from $\mathscr{C}^k(\bar{\Omega})$ to $\mathscr{C}^{k}(\partial \Omega)$ can be extended to a continuous surjective linear map from $W^{k,p}(\Omega)$ to $W^{k-1/p,p}(\partial \Omega)$ where $1 < p < \infty$ (see \cite[p. 158]{demengel2012functional} Theorem 3.79).
We would like to highlight the fact that the trace map $T$ on $W^{k,p}(\Omega)$ is defined on a Banach space and has range that is onto the Banach space $W^{k-1/p,p}(\partial \Omega)$. As for the nullspace of $T$, a classical result states that when $\partial \Omega$ is $\mathscr{C}^1$, $Tu = 0$ if and only if $u \in W^{1,p}_0(\Omega)$; (see \cite[p. 259]{evans1998partial} Theorem 2, or \cite[p. 138]{demengel2012functional} Corollary 3.46).
Let $\Omega$ be an open and bounded set with $\mathscr{C}^k$ boundary and let $T$ denote the continuous trace operator from $W^{k,p}(\Omega)$ onto $W^{k-1/p,p}(\partial \Omega)$. We will need a trace-like operator that is one-to-one. To define this new operator, first set $W_0 \coloneqq W^{1,p}_0(\Omega) \cap W^{k,p}(\Omega) = \textnormal{N} (T)$. Clearly $W_0$ is a closed subspace of $W^{k,p}(\Omega)$. Next let $\hat{W}^k$ denote the quotient space $W^{k,p}(\Omega)/W_0$ and define the operator $\hat{T}$ from $\hat{W}^k$ to $W^{k-1/p,p}(\partial \Omega)$ as
\begin{align*}
\hat{T}[u] = Tu, \qquad [u] \in \hat{W}^k.
\end{align*}
The operator $\hat{T}$ is well-defined, linear, and one-to-one. To see that $\hat{T}$ is closed, take $\{[u_n]\} \subset \hat{W}^k$ such that $[u_n] \to [u]$ and $\hat{T}[u_n] \to y$ as $n\to \infty$. Then $\| [u_n] - [u]\|_{\hat{W}^k} \to 0$ implies the existence of a sequence $\{v_n\} \subset W_0$ such that
\begin{align*}
u_n - v_n \to u, \qquad \text{in } W^{k,p}(\Omega).
\end{align*}
Since $\hat{T}[u_n] \to y$, we know that $T u_n = T(u_n - v_n) \to y$ as $n \to \infty$. By the boundedness of $T$ we get $T u = y$. This implies $\hat{T}[u] = y$ and proves that $\hat{T}$ is closed. Applying the Closed Graph Theorem shows that $\hat{T}$ is bounded. Moreover, the fact that $T$ is surjective implies that $\hat{T}$ is surjective as well. This tells us that $\hat{T}^{-1}$ exists and is a bounded linear operator from $W^{k-1/p,p}(\partial \Omega)$ onto $\hat{W}^k$, by the Bounded Inverse Theorem.
We also need the following general Sobolev space theorem. We use $\lfloor a \rfloor$ to denote the integer part of $a$.
\begin{thm}[\cite{adams2003sobolev} p. 85, Sobolev Imbedding]\label{thm:general_inequality}
Let $\Omega \subset \mathbb{R}^d$ be open and bounded with a $\mathscr{C}^{0,1}$ boundary. Assume $u \in W^{k,p}(\Omega)$, and that $kp> d$. Set
\begin{align*}
\kappa = k - \left\lfloor \tfrac{d}{p} \right\rfloor - 1, \qquad
\gamma = \left\{
\begin{array}{ll}
1 - \left( \tfrac{d}{p} - \left\lfloor \tfrac{d}{p} \right\rfloor\right), & \textnormal{if } \tfrac{d}{p} \not \in \mathbb{N} \\
\textnormal{any number in } (0,1), & \textnormal{otherwise.}
\end{array}\right.
\end{align*}
Then there exists a function $u^*$ such that $u^* = u$ a.e. and $u^* \in \mathscr{C}^{\kappa,\gamma}(\bar{\Omega})$.
\end{thm}
We can now establish the following lemma.
\begin{lem}\label{lem:kinda hardy}
Let $\Omega \subset \mathbb{R}^d$ be a bounded open set and let $\varphi \in \mathscr{C}^{1}(\bar{\Omega})$ be simply vanishing on $\partial \Omega$. Assume $k \in \mathbb{N}$ with $kp > {d}$ and that the boundary $\partial \Omega$ is $\mathscr{C}^k$. Then for every $u \in L^{p}(\Omega)$ where $\varphi u \in W^{k+1,p}(\Omega)$ we have
\begin{align*}
\varphi D^{\alpha} u \in W^{1,p}_{0}(\Omega) \cap W^{k,p}(\Omega), \qquad
\text{for any } \alpha \text{ with } |\alpha| = 1.
\end{align*}
\end{lem}
\begin{proof}
The proof is divided into two claims.
\begin{claim}{1}
If $|\alpha| = 1$ then $D^{\alpha} (\varphi u) = u D^{\alpha}\varphi$ on $\partial \Omega$.
\end{claim}
Since $kp > d$ and $\varphi u \in W^{k+1,p}(\Omega)$, we know there exists a $\gamma \in (0,1)$ dependent on $d$ and $p$ such that
\begin{align*}
\varphi u \in \mathscr{C}^{k-\big\lfloor \tfrac{d}{p} \big\rfloor,\gamma}(\bar{\Omega}),
\end{align*}
by Theorem~\ref{thm:general_inequality}. This shows $\varphi u \in \mathscr{C}^{1}(\bar{\Omega})$. Also, since $u = \varphi^{-1} \varphi u$ whenever $\varphi \neq 0$, the fact that $\varphi \in \mathscr{C}^{1}(\bar{\Omega})$ and is nonzero in $\Omega$ implies $u \in \mathscr{C}^1(\Omega)$.
Now, since $\varphi$ is simply vanishing on $\partial \Omega$, we know that for any $y \in \partial \Omega$,
\begin{align*}
\lim_{x \to y} \frac{|x-y|}{|\varphi(x)|}
= \lim_{x \to y} \frac{|x-y|}{|\varphi(x)-\varphi(y)|} = |\nabla \varphi(y)|^{-1},
\end{align*}
where the limit is taken in $\Omega$. Note that $\nabla \varphi \neq 0$ on $\partial \Omega$, so that this is well defined. Next, given $\varphi u \in W^{1,p}(\Omega)$ and $u \in L^p(\Omega)$, we know $\varphi u \in W^{1,p}_{0}(\Omega)$ by Lemma~\ref{lem:statement of range}. We already know $\varphi u$ is continuous on $\bar{\Omega}$, which implies $\varphi u = 0$ on $\partial \Omega$. Thus, for any $y \in \partial \Omega$ we can take any sequence in $\Omega$ that converges to $y$ and obtain
\begin{align}
\lim_{x \to y} |u(x)|
= \lim_{x \to y}
\frac{|\varphi(x) u(x) - \varphi(y)u(y)|}{|x - y|}
\frac{|x-y|}{|\varphi(x)|}
= |\nabla (\varphi u)(y)| |\nabla \varphi(y)|^{-1}.
\label{eq:limit_with_product}
\end{align}
By Leibniz's rule, $D^{\alpha} (\varphi u) = uD^{\alpha} \varphi + \varphi D^{\alpha} u$ when $|\alpha| = 1$, so the above limit implies $D^{\alpha} (\varphi u) = u D^{\alpha}\varphi$ on $\partial \Omega$.
\begin{claim}{2}
The function $\varphi D^{\alpha}u$ is in $W^{1,p}_{0}(\Omega) \cap W^{k,p}(\Omega)$ whenever $|\alpha| = 1$.
\end{claim}
Set ${W}_0 \coloneqq W^{1,p}_{0}(\Omega) \cap W^{k,p}(\Omega)$ and $\hat{W}^k \coloneqq W^{k,p}(\Omega)/W_0$. By assumption, $\varphi u \in W^{k+1,p}(\Omega)$ so $D^{\alpha}(\varphi u) \in W^{k,p}(\Omega)$ whenever $|\alpha| = 1$. This implies the coset $[D^{\alpha}(\varphi u)]$ is in $\hat{W}^k$ and that its trace $\hat{T}[D^{\alpha}(\varphi u)]$ is in $W^{k-1/p,p}(\partial \Omega)$.
By claim 1,
\begin{align*}
u D^{\alpha} \varphi = D^{\alpha}(\varphi u) \qquad \text{on } \partial \Omega,
\end{align*}
which shows that $u D^{\alpha} \varphi |_{\partial \Omega} \in W^{k-1/p,p}(\partial \Omega)$ and that $\hat{T}^{-1} (u D^{\alpha} \varphi |_{\partial \Omega}) \in \hat{W}^k$. The one-to-one nature of $\hat{T}^{-1}$ implies $[D^{\alpha}(\varphi u)] = [uD^{\alpha}\varphi]$. Since these cosets are equal, there exists a function $v \in W_0$ such that
\begin{align*}
u D^{\alpha} \varphi = D^{\alpha} (\varphi u) - v.
\end{align*}
But again, $D^{\alpha} (\varphi u) = uD^{\alpha} \varphi + \varphi D^{\alpha} u$, so it must be the case that $v = \varphi D^{\alpha} u$. We then conclude that $\varphi D^{\alpha} u \in W_0 = W^{1,p}_{0}(\Omega) \cap W^{k,p}(\Omega)$, completing the proof.
\end{proof}
\begin{thm}\label{thm:kinda hardy_pre}
Let $\Omega \subset \mathbb{R}^d$ be a bounded open set and let $\varphi \in \mathscr{C}^{1}(\bar{\Omega})$ be simply vanishing on $\partial \Omega $. Assume $k \in \mathbb{N}$ with $kp > {d}$ and that $\partial \Omega$ is $\mathscr{C}^k$. Then $u \in L^{p}(\Omega)$ and $\varphi u \in W^{k+1,p}(\Omega)$ implies $u \in W^{\kappa,p}(\Omega)$ where $\kappa \coloneqq k- \big\lfloor \tfrac{d}{p} \big\rfloor$.
\end{thm}
\begin{proof}
Fix the multi-index $\alpha$ with $|\alpha| \leq \kappa$. Choose a finite sequence of multi-indices $\{\alpha_n\}_{n\leq |\alpha|}$ each with $|\alpha_n| = 1$ such that $\sum \alpha_n = \alpha$. By assumption, $kp > d$ so we may apply Lemma~\ref{lem:kinda hardy} to $u$ to obtain
\begin{align}\label{consequence of lemma kinda hardy}
\varphi D^{\alpha_1} u \in W^{1,p}_{0}(\Omega) \cap W^{k+1-|\alpha_1|,p}(\Omega).
\end{align}
Applying Hardy's inequality to $\varphi D^{\alpha_1} u$ yields
\begin{align}\label{using hardys after lemma}
D^{\alpha_1} u \in L^{p}(\Omega).
\end{align}
Moreover, given $|\alpha| \leq \kappa = k - \big\lfloor \tfrac{d}{p} \big\rfloor$, we know that
\begin{align}\label{index inequality kinda hardy}
k+1 - |\alpha_1| \geq k+1 - |\alpha| \geq 1 + \big\lfloor \tfrac{d}{p} \big\rfloor > \tfrac{d}{p}.
\end{align}
Since \eqref{consequence of lemma kinda hardy}, \eqref{using hardys after lemma}, and \eqref{index inequality kinda hardy} all hold, we may apply Lemma~\ref{lem:kinda hardy} to $D^{\alpha_1}u$ to obtain $\varphi D^{\alpha_1+\alpha_2} u \in W^{1,p}_{0}(\Omega) \cap W^{k-1,p}(\Omega)$. Another application of Hardy's inequality shows $D^{\alpha_1 + \alpha_2} u \in L^{p}(\Omega)$. We continue inductively applying Lemma~\ref{lem:kinda hardy} and Hardy's inequality at each step to finally show that
\begin{align*}
\varphi D^{\alpha} u \in W^{1,p}_{0}(\Omega) \cap W^{k+1-|\alpha|,p}(\Omega), \quad \text{and} \quad D^{\alpha}u \in L^p(\Omega).
\end{align*}
Since this applies to any multi-index $\alpha$ with $|\alpha| \leq \kappa$, we see that $u \in W^{\kappa,p}(\Omega)$, completing the proof.
\end{proof}
Iteratively applying the above theorem yields the following.
\begin{thm}\label{thm:kinda hardy}
Let $\Omega \subset \mathbb{R}^d$ be a bounded open set and let $\varphi \in \mathscr{C}^1(\bar{\Omega})$ be simply vanishing on $\partial \Omega$. Fix $m \in \mathbb{N}$ and $1 < p < \infty$. Assume $k \in \mathbb{N}$ is such that
\begin{align}\label{the lower bound for k kinda hardy}
k > \tfrac{d}{p} + (m-1)\big\lfloor \tfrac{d}{p} \big\rfloor,
\end{align}
and that $\partial \Omega$ is $\mathscr{C}^k$. If $u \in L^{p}(\Omega)$ and $\varphi^m u \in W^{k+m,p}(\Omega)$ then
\begin{align}\label{statement of kappa kinda hardy}
u \in W^{\kappa,p}(\Omega), \quad \text{where } \kappa \coloneqq k- m\big\lfloor \tfrac{d}{p} \big\rfloor.
\end{align}
\end{thm}
\begin{proof}
For convenience, we define the variables $\kappa_1, \ldots, \kappa_m$ as follows
\begin{align*}
\kappa_j \coloneqq k + m - j - j \big\lfloor \tfrac{d}{p} \big\rfloor, \qquad j = 1, \ldots, m.
\end{align*}
We know $\varphi^{m-1} u \in L^p(\Omega)$, $\varphi^m u \in W^{k+m,p}(\Omega)$, and $k + m - 1 > p^{-1} d$. By Theorem~\ref{thm:kinda hardy_pre}, this implies $\varphi^{m-1} u \in W^{\kappa_1,p}(\Omega)$. If $m > 1$, we see that $\kappa_1 - 1 > p^{-1}d$, and we have $\varphi^{m-2} u \in L^p(\Omega)$ and $\varphi^{m-1} u \in W^{\kappa_1,p}(\Omega)$. Thus we get $\varphi^{m-2} u \in W^{\kappa_2,p}(\Omega)$ by the same theorem. Continuing inductively, we apply Theorem~\ref{thm:kinda hardy_pre} at each step to get $\varphi^{m-j} u \in W^{\kappa_j,p}(\Omega)$ for $m > j$. When $j = m - 1$ we have $u \in L^p(\Omega)$ and $\varphi u \in W^{\kappa_{m-1},p}(\Omega)$. Since
\begin{align*}
\kappa_{m-1} -1 = k - (m-1)\big\lfloor \tfrac{d}{p} \big\rfloor > \tfrac{d}{p},
\end{align*}
we may apply Theorem~\ref{thm:kinda hardy_pre} one more time to get $u \in W^{\kappa_m,p}(\Omega)$, as desired.
\end{proof}
\begin{rem}\label{rem:implicit bound}
There is an implicit estimate accompanying Theorem~\ref{thm:kinda hardy}. Assume $\varphi \in \mathscr{C}^1(\bar{\Omega})$ and consider the multiplication operator $\varphi^m : L^p(\Omega) \to W^{k+m,p}(\Omega)$ where $k$ satisfies \eqref{the lower bound for k kinda hardy}. Theorem~\ref{thm:kinda hardy} tells us that $u \in \textnormal{D} (\varphi^m)$ implies $u \in W^{\kappa,p}(\Omega)$ where $\kappa$ is given by \eqref{statement of kappa kinda hardy}. Thus, $ \textnormal{D} (\varphi^m) \subset W^{\kappa,p}(\Omega)$. By the closedness of $\varphi^m$, $ \textnormal{D} (\varphi^m)$ is a Banach space with the operator norm. We can conclude that, for some constants $c_0, c_1 > 0$ and for all $u \in \textnormal{D} (\varphi^m)$,
\begin{align}
\|u\|_{W^{\kappa,p}(\Omega)} & \leq c_0 \|u\|_{ \textnormal{D} (\varphi^m)}
= c_0 \|u\|_{L^{p}(\Omega)} + c_0 \|\varphi^m u\|_{W^{k+m,p}(\Omega)} \label{first inequality from closed graph} \\
& \leq c_1 \|\varphi^m u\|_{W^{k+m,p}(\Omega)}, \label{second inequality from hardys inequality}
\end{align}
where \eqref{first inequality from closed graph} follows from Lemma~\ref{lem:equivalence_of_norms} applied to the Banach spaces $ \textnormal{D} (\varphi^m)$ and $W^{\kappa, p}(\Omega)$ and \eqref{second inequality from hardys inequality} from Hardy's inequality.
\end{rem}
\subsection{The range of a vanishing operator}\label{subsec:range vanishing}
Having a closed range is a very useful property for linear operators. As we will see in section~\ref{section:spectrum}, it is often necessary for establishing basic properties of the spectrum. Showing the multiplication operator $u \mapsto \varphi u$ has closed range requires keeping track of the multiplicity of the roots of the function $\varphi$. This is formally established in the following result.
\begin{thm}\label{thm:range closed vanishing}
Let $\Omega \subset \mathbb{R}^d$ be open and bounded with $\mathscr{C}^{0,1}$ boundary and assume the function $\varphi \in \mathscr{C}^k(\bar{\Omega})$ is simply vanishing on $\partial \Omega$. Then the range of the operator $u \mapsto \varphi^m u$ is closed in $W^{k,p}(\Omega)$ whenever $k \geq m$ and is not closed when $k < m$.
\end{thm}
\begin{proof}
As usual, we treat $\varphi$ as an operator from some dense subset of $L^p(\Omega)$ to $W^{k,p}(\Omega)$. We start with the following.
\begin{claim}{1}
If $k \geq m$ then the range of $\varphi^m$ is closed in $W^{k,p}(\Omega)$.
\end{claim}
If $k = m$ then $ \textnormal{R} (\varphi^m) = W^{m,p}_{0}(\Omega)$ as discussed in Lemma~\ref{lem:statement of range}, which clearly establishes the closedness of $ \textnormal{R} (\varphi^m)$ in $W^{m,p}(\Omega)$. If $k > m$ then we may apply Lemma~\ref{lem:compactness implies compactness almost} to show $ \textnormal{D} (\varphi^m) \subset\subset L^p(\Omega)$. Invoking Theorem~\ref{thm:compactness equals closed range} proves $\varphi^m$ is semi-Fredholm, which implies $ \textnormal{R} (\varphi^m)$ is closed in $W^{k,p}(\Omega)$.
\begin{claim}{2}
If $k < m$ then the range of $\varphi^m$ is not closed in $W^{k,p}(\Omega)$.
\end{claim}
We prove this claim by contradiction. Suppose $\varphi^m$ has closed range in $W^{k,p}(\Omega)$. This implies $\varphi^m$ is semi-Fredholm from $L^p(\Omega)$ to $W^{k,p}(\Omega)$. Since $\varphi^k$ is Fredholm from $L^p(\Omega)$ to $W^{k,p}_{0}(\Omega)$, and since $\varphi^{m} = \varphi^{m-k} \varphi^k$ is semi-Fredholm from $L^p(\Omega)$ to $W^{k,p}(\Omega)$, Lemma~\ref{lem:reverse_products_and_semifredholm} implies $\varphi^{m-k}$ is semi-Fredholm from $W^{k,p}_{0}(\Omega)$ to $W^{k,p}(\Omega)$. Now, for any function $u \in \mathscr{C}^{\infty}_{0}(\Omega)$ we know that $v = \varphi^{k-m} u \in \mathscr{C}^k_{0}(\Omega)$, so $\varphi^{m-k}v \in \mathscr{C}^{\infty}_{0}(\Omega)$, implying $\varphi^{m-k}$ is onto the subspace $\mathscr{C}^{\infty}_{0}(\Omega)$. This implies
\begin{align*}
\mathscr{C}^{\infty}_{0}(\Omega) \subset \textnormal{R} (\varphi^{m-k}).
\end{align*}
Since $ \textnormal{R} (\varphi^{m-k})$ is closed, we know that $W^{k,p}_{0}(\Omega)$ is a subspace of $ \textnormal{R} (\varphi^{m-k})$. But we also know that $\varphi^{k} \in W^{k,p}_{0}(\Omega)$, so there exists a function $v \in W^{k,p}_{0}(\Omega)$ such that $\varphi^{m-k} v = \varphi^{k}$, which implies $v = \varphi^{2k-m}$. But $\varphi^{2k-m}$ cannot be in $W^{k,p}_{0}(\Omega)$ as Hardy's inequality would then show
\begin{align*}
\|\varphi^{k-m}\|_{L^p(\Omega)} = \|\varphi^{-k} \varphi^{2k-m} \|_{L^p(\Omega)} \leq c \| \varphi^{2k-m} \|_{W^{k,p}(\Omega)} < \infty.
\end{align*}
This is our desired contradiction.
\end{proof}
\begin{example}[The Legendre differential equation]
Set $\Omega = (-1,1)$. Let us analyze the operator $L$ given by
\begin{align*}
Lu(x) = \frac{d}{dx}\left[ (1-x^2) \frac{d}{dx} u(x) \right],
\end{align*}
acting on $L^p(\Omega)$, where as usual we assume $1 < p <\infty$. Let $A$ denote the derivative operator on $L^p(\Omega)$ and $\varphi(x) = (1 - x^2)$. The domain of $A$ is $W^{1,p}(\Omega)$, the nullspace of $A$ is $\text{span}\{ 1\}$, and the range of $A$ is equal to $L^p(\Omega)$. Since $\varphi$ is simply vanishing on $\partial \Omega$, we know the range of the multiplication operator $u \mapsto \varphi u$ is equal to $W^{1,p}_0(\Omega)$ by Lemma~\ref{lem:statement of range}.
If $u \in W^{1,p}(\Omega)$, then $u \in \mathscr{C}^{0,1/p}(\bar{\Omega})$ by Sobolev Imbedding (Theorem~\ref{thm:general_inequality}). Thus, we can find a unique line $l(x)$ such that $u + l = 0$ on $\partial \Omega$, implying $u + l \in W^{1,p}_0(\Omega)$. Since $u \in W^{1,p}(\Omega)$ was arbitrary, this implies
\begin{align*}
W^{1,p}(\Omega) = W^{1,p}_0(\Omega) \oplus \text{span}\{1,x\}.
\end{align*}
If we take $\tilde{A}$ to be the restriction of the derivative operator to $W^{1,p}_0(\Omega)$, then $\dim \textnormal{N} (\tilde{A}) = 0$ and $ \textnormal{co-\!} \dim \textnormal{R} (\tilde{A}) = 1$. Since $A$, $\tilde{A}$, and $\varphi : L^p(\Omega) \to W^{1,p}_0(\Omega)$ are all Fredholm we know that $L = \tilde{A}\varphi A$ is Fredholm, with
\begin{align*}
\text{ind}(L) = \text{ind}(\tilde{A}\varphi A) = \text{ind}(\tilde{A}) + \text{ind}(\varphi) + \text{ind}( A) = -1 + 0 + 1 = 0,
\end{align*}
and $ \textnormal{N} (L) = \text{span}\{1\}$.
In terms of the domain of $L$, we automatically get $ \textnormal{D} (L) \subset \textnormal{D} (A) = W^{1,p}(\Omega)$. The interesting thing to note is that $L$ cannot be semi-Fredholm if $ \textnormal{D} (L) \subseteq W^{2,p}(\Omega)$. To see this, first note that $A$ maps $W^{2,p}(\Omega)$ onto $W^{1,p}(\Omega)$ and that the range of $\varphi : W^{1,p}(\Omega) \to W^{1,p}_0(\Omega)$ is not closed. Since this implies that $\varphi A : W^{2,p}(\Omega) \to W^{1,p}_0(\Omega)$ cannot be semi-Fredholm, we know that $L = \tilde{A}\varphi A$ cannot be semi-Fredholm.
\end{example}
\subsection{Matrix-valued functions} \label{subsection:matrix_functions}
One of our goals is to aid in the analysis of
\begin{align}\label{L_operator2}
Lu = \text{div} ( \Phi \nabla u),
\end{align}
when the matrix-valued function $\Phi : \bar{\Omega} \to \mathbb{C}^{d \times d}$ is positive semi-definite for each $x\in \bar{\Omega}$. With that end in mind, this section focuses on the multiplication operator $\mathbf{u} \mapsto \Phi \mathbf{u}$ where $\Phi \in \mathscr{C}^1(\bar{\Omega};\mathbb{C}^{d\times d})$ and $\mathbf{u}(x) \in \mathbb{C}^d$ for almost every $x \in \Omega$. As we will see shortly, the properties that were established for the multiplication operator $u \mapsto \varphi u$ apply for the multiplication operator $\mathbf{u} \mapsto \Phi \mathbf{u}$ as well.
In order for the operator $L$ defined in \eqref{L_operator2} to be uniformly elliptic, the matrix $\Phi : \bar{\Omega} \to \mathbb{C}^{d\times d}$ must be uniformly positive definite. This section, like the ones before it, focus on the violation of this positivity assumption. Specifically, we assume $\Phi$ is vanishing of order $m$ (recall Definition~\ref{def:simply vanishing matrices2}). Another way to express this is as follows: for each fixed $V \subset\subset \Omega$ we have
\begin{align} \label{rayleigh thingy}
\inf_{x \in V^{}} \inf_{v \in \mathbb{C}^d} {\bar{v} \cdot \Phi(x) v} \geq c_V|v|,
\end{align}
where $c_V > 0$ is the smallest eigenvalue of $\Phi(x)$ for $x \in V$. Moreover, the speed at which $c_V$ goes to zero is proportional to $a^m$ where $a = \inf_{y\in \partial V} \text{dist}(y,\partial \Omega)$.
We take $L^p(\Omega^d)$ to be the space of all measurable functions $\mathbf{u} = (u_1, \ldots, u_d)$ such that ${u}_i \in L^p(\Omega)$ for $i = 1,\ldots,d$. The norm of $L^p(\Omega^d)$ is taken to be
\begin{align*}
\|\mathbf{u} \|_{L^p(\Omega^d)} \coloneqq \sum_{i=1}^d \| u_i\|_{L^p(\Omega)}.
\end{align*}
In other words,
\begin{align*}
L^p(\Omega^d) = \underbrace{L^p(\Omega) \times \cdots \times L^p(\Omega).}_{d \text{ copies}}
\end{align*}
The space $W^{k,p}(\Omega^d)$ is defined as the subset of $\mathbf{u} = (u_1,\ldots,u_d) \in L^p(\Omega^d)$ where $u_i \in W^{k,p}(\Omega)$ for each $i = 1,\ldots, d$. We assume $\Phi : L^p(\Omega^d) \to W^{k,p}(\Omega^d)$ for some $k \in \mathbb{Z}_+$.
\begin{thm}\label{thm:range closed vanishing_matrices}
Let $\Omega \subset \mathbb{R}^d$ be open and bounded with $\mathscr{C}^{0,1}$ boundary and assume $\Phi \in \mathscr{C}^k(\bar{\Omega};\mathbb{C}^{d\times d})$ is vanishing of order $m$. Then the range of the operator $\mathbf{u} \mapsto \Phi \mathbf{u}$ is closed in $W^{k,p}(\Omega^d)$ whenever $k \geq m$ and is not closed when $k < m$.
\end{thm}
\begin{proof}
We know there exists $\mathbf{U} \in \mathscr{C}^k(\bar{\Omega}; \mathbb{C}^{d\times d})$ and $\mathbf{D} \in \mathscr{C}^k(\bar{\Omega}; \mathbb{R}^{d\times d})$ such that $\Phi = \mathbf{U}\mathbf{D}\mathbf{U}^*$, where $\mathbf{D} = \text{diag}(\varphi_1, \ldots, \varphi_d)$ and $\varphi_i \in \mathscr{C}^{k}(\bar{\Omega})$ for each $i = 1,\ldots,d$. Since $\mathbf{U}$ is one-to-one and onto $L^p(\Omega^d)$, it suffices to prove the claim for the operator $\mathbf{D}$. Now, by our definition of $W^{k,p}(\Omega^d)$, it must be the case that $ \textnormal{R} (\mathbf{D})$ is closed in $W^{k,p}(\Omega^d)$ if and only if the multiplication operators $u \mapsto \varphi_i u$ have closed range in $W^{k,p}(\Omega)$ for each $i = 1,\ldots, d$. With this in mind, we simply apply Theorem~\ref{thm:range closed vanishing} for each diagonal function $\varphi_i$, yielding the desired conclusion.
\end{proof}
\begin{thm}\label{thm:kinda hardy_matrices}
Let $\Omega \subset \mathbb{R}^d$ be a bounded open set and let $\Phi \in \mathscr{C}^k(\bar{\Omega};\mathbb{C}^{d\times d})$ be vanishing of order $m$. Assume $k \in \mathbb{N}$ is such that
\begin{align*}
k > \tfrac{d}{p} + (m-1)\big\lfloor \tfrac{d}{p} \big\rfloor,
\end{align*}
and that the boundary $\partial \Omega$ is $\mathscr{C}^k$. If $\mathbf{u} \in L^{p}(\Omega^d)$ and $\Phi \mathbf{u} \in W^{k+m,p}(\Omega^d)$ then
\begin{align}
\label{defines kappa}
\mathbf{u} \in W^{\kappa,p}(\Omega^d), \quad \text{where } \kappa \coloneqq k- m\big\lfloor \tfrac{d}{p} \big\rfloor,
\end{align}
and there exists a $c > 0$, independent of $\mathbf{u}$, such that
\begin{align}\label{est:domain estimate matrices}
\|\mathbf{u}\|_{W^{\kappa,p}(\Omega^d)} \leq c \| \Phi \mathbf{u} \|_{W^{k+m,p}(\Omega^d)}.
\end{align}
\end{thm}
\begin{proof}
We know there exists $\mathbf{U} \in \mathscr{C}^k(\bar{\Omega}; \mathbb{C}^{d\times d})$ and $\mathbf{D} \in \mathscr{C}^k(\bar{\Omega}; \mathbb{R}^{d\times d})$ such that $\Phi = \mathbf{U}\mathbf{D}\mathbf{U}^*$, where $\mathbf{D} = \text{diag}(\varphi_1, \ldots, \varphi_d)$ and $\varphi_i \in \mathscr{C}^{k}(\bar{\Omega})$ for each $i = 1,\ldots,d$. As in the above theorem, it suffices to prove the claim for the operator $\mathbf{D}$.
Given $\mathbf{u} = (u_1,\ldots, u_d) \in L^{p}(\Omega^d)$ and $\mathbf{D} \mathbf{u} \in W^{k+m,p}(\Omega^d)$ then we have $u_i \in L^p(\Omega)$ and $\varphi_i u_i \in W^{k+m,p}(\Omega)$ for each $i = 1,\ldots,d$. If $\varphi_i > 0$ on $\bar{\Omega}$ then $u_i \in W^{k+m,p}(\Omega)$, and if $\varphi_i$ is vanishing of order $j \leq m$ we apply Theorem~\ref{thm:kinda hardy} to get
\begin{align*}
u_i \in W^{\kappa_j,p}(\Omega), \quad \text{ where } \kappa_j \coloneqq k - j\big\lfloor \tfrac{d}{p} \big\rfloor.
\end{align*}
In either case, $u_i \in W^{\kappa,p}(\Omega)$ for each $i = 1,\ldots,d$. The proof of inequality \eqref{est:domain estimate matrices} mirrors that of Remark~\ref{rem:implicit bound} and is omitted.
\end{proof}
\section{Differential operators composed with vanishing operators}\label{section:differential_with}
In this section we examine differential operators that are composed with vanishing operators. By `differential operator' we mean any operator that is closed on the subspace $W^{k,p}(\Omega) \subset L^p(\Omega)$, $k \geq 1$, and maps to either $L^p(\Omega)$ or $L^p(\Omega^d)$. We pay particular attention to linear differential operators that are Fredholm or semi-Fredholm. Many of these results use compactness of nested Sobolev spaces.
\subsection{Compactness}\label{subsection:compactness}
One of the salient features of the Sobolev space $W^{k,p}(\Omega)$ is its compactness relationship with the ambient space $L^p(\Omega)$. In this section, we explore the implications of compactness on the composition of differential operators with vanishing functions.
We start with the following general result for Fredholm operators.
\begin{thm}
\label{thm:fredholm_on_compactly_embedded_domains}
Let $X$ and $Y$ be Banach spaces. If $A \in \mathcal{F}(X,Y)$ then $ \textnormal{D} (A) \subset\subset X$ if and only if its pseudo-inverse is compact from $Y$ to $X$.
\end{thm}
\begin{proof}
Since $A$ is closed we may equip $ \textnormal{D} (A)$ with the $A$-norm and convert it into a Banach space, which we call $W$.
\begin{claim}{1}
If $A \in \mathcal{F}(X,Y)$ with $W \subset\subset X$ then the pseudo-inverse of $A$ is compact from $Y$ to $X$.
\end{claim}
Given that $A$ is Fredholm from $X$ to $Y$, we know it is also Fredholm from $W$ to $Y$. Let $\tilde{A}_0$ denote the pseudo-inverse of $A : W \to Y$, and let $\iota : W \to X$ denote the inclusion map from $W$ to $X$. The assumption that $W \subset\subset X$ tells us that $\iota$ is compact, which implies $\iota \tilde{A}_0 : Y \to X$ is compact as well since $\tilde{A}_0 \in \mathcal{B}(Y,W)$. If we let $A_0$ denote the pseudo-inverse of $A : X \to Y$ then we see that $A_0 = \iota \tilde{A}_0$, so $A_0$ is compact.
\begin{claim}{2}
If $A \in \mathcal{F}(X,Y)$ and the pseudo-inverse of $A$ is compact from $Y$ to $X$ then $W \subset\subset X$.
\end{claim}
We are told $A$ is Fredholm, so we know $ \textnormal{N} (A)$ is finite dimensional and that there exists a closed subspace $X_0 \subset X$ such that $X = X_0 \oplus \textnormal{N} (A)$. Suppose $\{x_n\} \subset \textnormal{D} (A)$ with $\|x_n\|_{ \textnormal{D} (A)} \leq c$. Then for each $n$ we have the decomposition $x_n = a_n + b_n$ where $a_n \in X_0$ and $b_n \in \textnormal{N} (A)$. Since $\|a_n\|_{ \textnormal{D} (A)} \leq c$ for all $n$, $\{A a_n\}$ is a bounded sequence in $Y$. Given that $A_0$, the pseudo-inverse of $A$, is compact from $Y$ to $X$, there exists a subsequence of $\{a_n\} = \{A_0 A a_n\}$ that is convergent in $X$. Also, since $\{b_n\}$ is bounded and $ \textnormal{N} (A)$ is finite dimensional, every subsequence of $\{ b_n\}$ has a further subsequence that is convergent. Thus, we can find a subsequence of $\{b_n\}$ along the convergent subsequence of $\{a_n\}$ that is convergent. With this we can conclude that $\{x_n\} = \{a_n + b_n\}$ contains a convergent subsequence in $X$. This proves the claim and completes the proof of the theorem.
\end{proof}
As a consequence of Theorem~\ref{thm:fredholm_on_compactly_embedded_domains} we have the following.
\begin{thm}
\label{thm:not_closed_product}
Let $X$, $Y$, and $Z$ be Banach spaces. Suppose $A \in \mathcal{F}(X,Y)$ where $ \textnormal{D} (A)\subset\subset X$. If $B \in \mathcal{C}(Y,Z)$ but is not semi-Fredholm then $BA$ is not closed on its natural domain.
\end{thm}
\begin{proof}
The proof is by contradiction. Assume $BA$ is closed on its natural domain,
\begin{align*}
\textnormal{D} (BA) = \{ x \in \textnormal{D} (A) : Ax \in \textnormal{D} (B) \}.
\end{align*}
\begin{claim}{1}
There exists a $c > 0$ such that $\|x\|_{ \textnormal{D} (A)} \leq c\|x\|_{ \textnormal{D} (BA)}$ holds for all $x \in \textnormal{D} (BA)$.
\end{claim}
If $BA$ was closed on $ \textnormal{D} (BA)$ then $ \textnormal{D} (BA)$ would be a Banach space with the $BA$-norm. Since $A$ is Fredholm it must be closed on its domain $ \textnormal{D} (A)$, so $ \textnormal{D} (A)$ is also a Banach space with the $A$-norm. We know that $ \textnormal{D} (BA) \subset \textnormal{D} (A)$, so by Lemma~\ref{lem:equivalence_of_norms} there exists a $c > 0$ such that $\|x \|_{ \textnormal{D} (A)} \leq c\|x\|_{ \textnormal{D} (BA)}$ whenever $x \in \textnormal{D} (BA)$.
\begin{claim}{2}
There exists a sequence that converges in $ \textnormal{D} (BA)$ but does not converge in $ \textnormal{D} (A)$.
\end{claim}
Since $B$ is not semi-Fredholm, there exists a bounded sequence $\{x_n\} \subset \textnormal{D} (B)$ such that $\{B x_n \}$ converges but $\{x_n\}$ has no convergent subsequence. Given that $A$ is Fredholm we know $A$ has a pseudo-inverse, which we denote by $A_0$. We then set $y_n = A_0 x_n$ and notice that
\begin{align*}
Ay_n = A A_0 x_n = (I - K) x_n,
\end{align*}
where $K$ is a projection into some finite dimensional subspace of $Y$. Since $\{x_n\}$ has no convergent subsequence and $K$ projects to a finite dimensional subspace, $\{K x_n\}$ is eventually zero. Thus, $\{A y_n\}$ has no convergent subsequence and $\{BAy_n\}$ converges. Since $ \textnormal{D} (A) \subset\subset X$, we know that $A_0$ is compact by Theorem~\ref{thm:fredholm_on_compactly_embedded_domains} so $\{y_n\}$ has a convergent subsequence in $X$ (which, after relabeling, we take to be the entire sequence). Using claim 1, we have
\begin{align*}
\|y_m - y_n\|_{ \textnormal{D} (A)}
& = \|y_m - y_n\|_X + \|Ay_m - Ay_n\|_Y \\
& \leq c \|y_m - y_n\|_{ \textnormal{D} (BA)} = c \|y_m - y_n\|_X + c\|BA y_m - BAy_n\|_Z.
\end{align*}
We have established that $\{BAy_n\}$ and $\{y_n\}$ converge in $Z$ and $X$ respectively, so $\{y_n\}$ is convergent in $ \textnormal{D} (BA)$. But we know that $\{Ay_n\}$ does not converge in $Y$, hence $\{y_n\}$ cannot converge in $ \textnormal{D} (A)$. This is the desired contradiction.
\end{proof}
If $\varphi \in \mathscr{C}^1(\bar{\Omega})$ is simply vanishing, then by Theorem~\ref{thm:range closed vanishing} the range of the multiplication operator $u \mapsto \varphi u$ is not closed in $L^p(\Omega)$. Thus, $\varphi$ cannot be semi-Fredholm from $L^p(\Omega)$ to $L^p(\Omega)$. The above theorem then says $\varphi^m A$ is never closed on its natural domain for any $m > 0$. However, we can partially make up for this loss by showing that $\varphi^m A$ is closable. Before we begin we will need a few more tools from classical functional analysis.
The adjoint operator of $A$, denoted $A^*$, is a map from the dual $Y^*$ to $X^*$, where
\begin{align}\label{adjoint def}
A^*y^*(x) = y^*(Ax), \qquad \text{for all } x \in \textnormal{D} (A),
\end{align}
and some $y^* \in Y^*$. The set of appropriate $y^* \in Y^*$ for which \eqref{adjoint def} holds is $ \textnormal{D} (A^*)$.
The following two results are needed.
\begin{lem}[\cite{kato1995perturbation} Lemma 131, p. 137]
Let $X$ be a normed vector space. Suppose that a sequence $\{x_n\} \subset X$ is bounded, and $\lim l^*(x_n) = l^*(x)$ for each $l^* \in V$ where $V$ is a dense subset of $X^*$. Then the sequence $\{x_n\}$ converges to $x$ weakly.
\label{lem:weak_uniform_boundedness}
\end{lem}
\begin{thm}[\cite{schechter2002principles}, Theorems 7.35 and 7.36, p. 178]
\label{thm:densely_defined_adjoint}
Let $X$, $Y$, and $Z$ be Banach spaces, and assume that $A \in \mathcal{C}(X,Y)$ where $ \textnormal{R} (A)$ is closed in $Y$ with finite co-dimension. Let $B$ be a densely defined operator from $Y$ to $Z$. Then $(BA)^*$ exists, $(BA)^* = A^*B^*$, and both $(BA)^{*}$ and $BA$ are densely defined.
\end{thm}
With the above lemma and theorem, we can conclude the following.
\begin{thm}\label{thm:closable_product}
Let $X$, $Y$, and $Z$ be Banach spaces. Suppose $A \in \mathcal{F}(X,Y)$ and $B$ is a densely defined linear operator from $Y$ to $Z$. Then $BA$ is closable.
\end{thm}
\begin{proof}
Since $A$ is Fredholm, the domains of $BA$ and $(BA)^*$ are both dense, by Theorem~\ref{thm:densely_defined_adjoint}. Suppose we have a sequence $\{x_n\} \subset \textnormal{D} (BA)$ where $x_n \to 0$ and $B Ax_n \to z$ as $n \to \infty$. Then for each $w^* \in \textnormal{D} ((BA)^*)$,
\begin{align*}
w^* (z) = \lim_{n\to\infty} w^*(B Ax_n) = \lim_{n\to\infty} (BA)^* w^*(x_n) = 0.
\end{align*}
Since $ \textnormal{D} ((BA)^*)$ is dense in $X^*$ and $B Ax_n$ is bounded, this implies that $BAx_n \to 0$ weakly as $n\to\infty$, by Lemma~\ref{lem:weak_uniform_boundedness}. We know that weak limits must coincide with strong limits so $z = 0$. Thus $B A$ is closable.
\end{proof}
Theorems~\ref{thm:not_closed_product} and \ref{thm:closable_product} yield the following result.
\begin{thm}\label{thm:not closed but closable}
Let $\Omega \subset \mathbb{R}^d$ be an open and bounded set with $\mathscr{C}^{0,1}$ boundary. Let $\varphi \in \mathscr{C}^{1}(\bar{\Omega})$ be simply vanishing on $\partial \Omega$.
If $A : L^p(\Omega) \to L^p(\Omega)$ is Fredholm with $ \textnormal{D} (A) \subset W^{1,p}(\Omega)$, then $\varphi^m A$, for $m \geq 1$, is not closed on its natural domain but is closable.
\end{thm}
\begin{proof}
Theorem~\ref{thm:range closed vanishing} tells us that the range of the multiplication operator $u \mapsto \varphi^{m} u$ is not closed in $L^p(\Omega)$, so $\varphi^m$ is not semi-Fredholm from $L^p(\Omega)$ to $L^p(\Omega)$. Since $\Omega$ is bounded with $\mathscr{C}^{0,1}$ boundary, this implies $ \textnormal{D} (A) \subset W^{1,p}(\Omega) \subset\subset L^p(\Omega)$ by the Rellich-Kondrachov Theorem. Theorem~\ref{thm:not_closed_product} then tells us that $\varphi^m A$ is not closed on its natural domain, but by Theorem~\ref{thm:closable_product} $\varphi^m A$ is closable.
\end{proof}
Our next result makes use of the following theorem.
\begin{thm}[\cite{schechter2002principles} Theorem 7.22, p. 170] \label{thm:reflexive adjoint fredholm}
Let $X$ and $Y$ be Banach spaces. If $A \in \mathcal{F}(X,Y)$ and $Y$ is reflexive, then $A^* \in \mathcal{F}(Y^*, X^*)$ and $\text{ind}(A^*) = -\text{ind}(A)$.
\end{thm}
We now have the following:
\begin{thm}\label{thm:reflexive adjoint compactness}
Let $X$ and $Y$ be Banach spaces where $Y$ is also reflexive. If $A \in \mathcal{F}(X,Y)$ with $ \textnormal{D} (A) \subset\subset X$ then $ \textnormal{D} (A^*) \subset\subset Y^*$.
\end{thm}
\begin{proof}
By Theorem~\ref{thm:reflexive adjoint fredholm} we know that $A^* \in \mathcal{F}(Y^*, X^*)$. Also, if $x^* \in \textnormal{D} (A^*)$ then
\begin{align}\label{adjoint equation pseudo inverse}
A^*_0 A^*x^*(x) = A^*x^*(A_0x) = x^*(AA_0x) = x^*(x - K_2x) = (I - K_2^*)x^*(x).
\end{align}
Thus, from \eqref{adjoint equation pseudo inverse} and Theorems~\ref{thm:densely_defined_adjoint} and \ref{thm:equivalence_of_fredholm} we conclude:
\begin{align*}
A^* A_0^* = (A_0A)^* = I - K_1^*, \qquad A^*_0 A^* = (AA_0)^* = I - K_2^*,
\end{align*}
which implies $A_0^*$ is the pseudo-inverse of $A^*$. Since $A \in \mathcal{F}(X,Y)$, and $ \textnormal{D} (A) \subset\subset X$, we know that the pseudo-inverse $A_0$ is compact from $Y$ to $X$ by Theorem~\ref{thm:fredholm_on_compactly_embedded_domains}. Given $A_0$ is compact from $Y$ to $X$, we know $A_0^*$ is compact from $X^*$ to $Y^*$. If we then apply Theorem~\ref{thm:fredholm_on_compactly_embedded_domains} to $A^*$ we get $ \textnormal{D} (A^*) \subset\subset Y^*$.
\end{proof}
\subsection{The Spectrum} \label{section:spectrum}
Let $X$ be a Banach space and $A$ be a densely defined linear operator from $X$ to $X$. The resolvent set of $A$, denoted $\rho(A)$, is the set of all $\lambda \in \mathbb{C}$ such that $A - \lambda$ has a bounded inverse. The complement of $\rho(A)$ in $\mathbb{C}$ is called the spectrum of $A$, and is denoted as $\sigma(A)$. We let $ \sigma_{\text{p}} (A)$ denote the point spectrum of $A$:
\begin{align*}
\sigma_{\text{p}} (A) \coloneqq \{ \lambda \in \mathbb{C} : Ax = \lambda x \text{ for some } x \in \textnormal{D} (A) \}.
\end{align*}
We define the essential spectrum as:
\begin{align*}
\sigma_{\text{ess}} (A) = \bigcap_{K \in \mathcal{K}(X)} \sigma(A + K),
\end{align*}
where $\mathcal{K}(X)$ is the set of all compact operators on $X$. This set is sometimes referred to as \emph{Schechter's essential spectrum}. Another useful characterization of the essential spectrum is given in the theorem below.
\begin{thm}[\cite{schechter2002principles} Theorem 7.27, p. 172]
\label{thm:equivalence_of_schecter_spectrum}
Let $X$ be a Banach space and assume $A \in \mathcal{C}(X)$. Then $\lambda \not\in \sigma_{\text{ess}} (A)$ if and only if $A-\lambda \in \mathcal{F}(X)$ and $\text{ind}(A-\lambda) = 0$.
\end{thm}
We will also need the notion of relatively compact operators. If $A \in \mathcal{C}(X,Y)$, an operator $B : X \to Z$ is called \emph{compact relative to $A$} if $B$ is compact from the Banach space $ \textnormal{D} (A)$ to $Z$. The following theorem is a more robust version of the Fredholm Alternative since it is stated for any Fredholm operator $A$ (not just the identity operator) and the perturbations to $A$ can be any operator that is compact relative to $A$. A proof of this theorem can be found in \cite[p. 281]{kato1958perturbation} Theorem 1, or \cite[p. 162]{schechter2002principles} Theorem 7.10.
\begin{thm}[\cite{kato1958perturbation} Theorem 1, p. 281]
\label{thm:fredholm_alternative}
If $A \in \mathcal{F}(X,Y)$ and $B$ is compact relative to $A$ then $A + B \in \mathcal{F}(X,Y)$ and $\text{ind}(A+B) = \text{ind}(A)$.
\end{thm}
\begin{rem}\label{about the essential spectrum}
With the help of Theorems \ref{thm:equivalence_of_schecter_spectrum} and \ref{thm:fredholm_alternative}, we see that the essential spectrum is invariant under relatively compact perturbations. Given that the identity map on $X$ is compact relative to $A \in \mathcal{C}(X)$ whenever $ \textnormal{D} (A) \subset\subset X$, we know that either $ \sigma_{\text{ess}} (A) = \emptyset$ or $ \sigma_{\text{ess}} (A) = \mathbb{C}$. This fact makes calculating the essential spectrum of differential operators relatively easy whenever we can use the Rellich-Kondrachov Theorem.
\end{rem}
\begin{thm}
\label{thm:abstract_ess_reverse_product}
Let $X$ be a Banach space, $A \in \mathcal{C}(X)$, and $B \in \mathcal{B}(X)$ be a one-to-one operator where $ \textnormal{D} (A) \not \subset \textnormal{R} (B)$. If $ \textnormal{D} (AB)$ is dense in $X$ with $ \textnormal{D} (AB) \subset\subset X$ and $\rho(A)$ is nonempty then $ \sigma_{\text{ess}} (AB) =\mathbb{C}$.
\end{thm}
\begin{proof}
The proof is by contradiction. Suppose $ \sigma_{\text{ess}} (AB) \neq \mathbb{C}$. As mentioned in Remark~\ref{about the essential spectrum}, $ \sigma_{\text{ess}} (AB) \neq \mathbb{C}$ implies $ \sigma_{\text{ess}} (AB) = \emptyset$ since $ \textnormal{D} (AB) \subset\subset X$. This implies $AB \in \mathcal{F}(X)$ and $\text{ind}(AB) = 0$ by Theorem~\ref{thm:equivalence_of_schecter_spectrum}.
Since $ \textnormal{D} (AB) \subset\subset X$, any bounded operator on $X$ is compact relative to $AB$. In particular, $B$ is compact relative to $AB$. By Theorem~\ref{thm:fredholm_alternative}, Fredholm operators and their indices are invariant under relatively compact perturbations. Thus, for any $\eta \in \mathbb{C}$ we have $AB - \eta B \in \mathcal{F}(X)$ and
\begin{align*}
\text{ind}(AB) = \text{ind}(AB- \eta B) = 0.
\end{align*}
Now, if $\eta \in \rho(A)$ then $ \textnormal{N} (A-\eta) = \{0\}$ and $ \textnormal{R} (A-\eta) = X$, implying $A- \eta$ maps $ \textnormal{D} (A)$ to $X$. But since $B$ does not map to all of $ \textnormal{D} (A)$ we get $ \textnormal{R} ((A-\eta)B) \neq X$. In other words
\begin{align*}
\text{co-\!} \dim \textnormal{R} (A B - \eta B) > 0.
\end{align*}
Recall that $ \textnormal{N} (B) = \{0\}$ by assumption, and $ \textnormal{N} (A-\eta) = \{0\}$ when $\eta \in \rho(A)$, which gives $ \textnormal{N} \big( (A - \eta) B \big) = \{0\}$. Thus,
\begin{align*}
\text{ind}(AB)
= \text{ind}(AB -\eta B)
= \dim \textnormal{N} (AB - \eta B) - \text{co-\!}\dim \textnormal{R} (AB - \eta B) < 0,
\end{align*}
which contradicts the fact that $\text{ind}(AB) = 0$.
\end{proof}
The proof of the above theorem yields the following corollary.
\begin{cor}
\label{cor:index_reverse_product}
Let $X$ be a Banach space, $A \in \mathcal{C}(X)$, and $B \in \mathcal{B}(X)$ be a one-to-one operator where the range of $B$ is not the entirety of $ \textnormal{D} (A)$. If $AB$ is Fredholm with $ \textnormal{D} (AB) \subset\subset X$ and $\rho(A)$ is nonempty then $\text{ind}(AB) < 0$.
\end{cor}
In some cases, we are concerned with the adjoint operator $\varphi^m A^*$ instead $A\varphi^m$. For example, one might be interested in the spectral properties of the operator $L$ given by $Lu = -(1-x^2) u_{xx}$ on $\Omega = (-1, 1)$. This operator is the adjoint of $A\varphi u = -( (1-x^2)u )_{xx}$ where $A$ is the Laplacian on $L^2(\Omega)$ and $\varphi(x) = (1-x^2)$. In this case, we have the following theorem.
\begin{thm}
\label{thm:roots equal spectrum}
Let $\Omega \subset \mathbb{R}^d$ be open and bounded with $\mathscr{C}^{0,1}$ boundary, $m,k \in \mathbb{N}$ with $m < k$, and let $A$ be densely defined on $L^p(\Omega)$. Assume
\begin{enumerate}[label=(\alph*),ref=(\alph*)]
\item $A$ is closed on $L^p(\Omega)$ and $ \textnormal{D} (A) \subset W^{k,p}(\Omega)$.
\item There exists a $u \in \textnormal{D} (A)$ such that $u \notin W^{m,p}_0(\Omega) \cap W^{k,p}(\Omega)$. \label{item:range not everything}
\item The resolvent set $\rho(A)$ is non-empty.
\end{enumerate}
If $\varphi \in \mathscr{C}^k(\bar{\Omega})$ is simply vanishing then
\begin{align*}
\sigma_{\text{ess}} (A \varphi^m ) = \sigma_{\text{ess}} (\overline{\varphi^m A^*} ) =\mathbb{C}.
\end{align*}
Moreover, if either $A\varphi^m$ or $\overline{\varphi^m A^*}$ is Fredholm then
\begin{align*}
\sigma_{\text{p}} (\overline{\varphi^m A^*} ) = \mathbb{C}.
\end{align*}
\end{thm}
\begin{rem}
By Theorem~\ref{thm:not closed but closable}, ${\varphi^m A^*}$ is never closed on its natural domain when $A$ is Fredholm. Thus, we examine its closure since statements about the essential spectrum are uninformative for operators that are not closed.
\end{rem}
\begin{proof}
As usual, let $\varphi^m$ denote the multiplication operator $u \mapsto \varphi^m u$ from $L^p(\Omega)$ to $W^{k,p}(\Omega)$. The proof is broken into 4 claims.
\begin{claim}{1}
$A\varphi^m$ is closed on its natural domain and $ \textnormal{D} (A\varphi^m) \subset\subset L^p(\Omega)$.
\end{claim}
Given $\partial \Omega$ is $\mathscr{C}^{0,1}$, we see that $ \textnormal{D} (A) \subset W^{k,p}(\Omega) \subset\subset L^p(\Omega)$ by the Rellich-Kondrachov Theorem. Since $A$ is closed on $ \textnormal{D} (A)$, $A$ must be semi-Fredholm on $L^p(\Omega)$ by Theorem~\ref{thm:compactness equals closed range}. With the assumption that $m < k$, Lemma~\ref{lem:compactness implies compactness almost} implies $ \textnormal{D} (\varphi^m) \subset\subset L^p(\Omega)$, and applying Theorem~\ref{thm:compactness equals closed range} yields $\varphi^m$ is semi-Fredholm from $L^p(\Omega)$ to $W^{k,p}(\Omega)$. Since both $A$ and $\varphi^m$ are semi-Fredholm, $A\varphi^m$ is closed on its natural domain. The fact that
\begin{align*}
\textnormal{D} (A\varphi^m) \subset \textnormal{D} (\varphi^m) \subset\subset L^p(\Omega)
\end{align*}
completes the proof of the claim.
\begin{claim}{2}
$ \sigma_{\text{ess}} (A\varphi^m) = \mathbb{C}$
\end{claim}
Applying Lemma~\ref{lem:statement of range} to $\varphi^m$ shows that $ \textnormal{R} (\varphi^m) = W^{m,p}_0(\Omega) \cap W^{k,p}(\Omega)$. This and assumption \ref{item:range not everything} implies $\varphi^m$ does not map to all of $ \textnormal{D} (A)$. Given claim 1, $\rho(A)$ is non-empty, and the multiplication operator $\varphi^m : L^p(\Omega) \to L^p(\Omega)$ is bounded, we may apply Theorem~\ref{thm:abstract_ess_reverse_product} to prove $ \sigma_{\text{ess}} (A\varphi^m) = \mathbb{C}$.
\begin{claim}{3}
If $A\varphi^m$ is not Fredholm then $ \sigma_{\text{ess}} (\overline{\varphi^m A^*} ) =\mathbb{C}$.
\end{claim}
Fix $\lambda \in \mathbb{C}$. Claim 1 implies the identity is compact relative to $A\varphi^m$. Since $A\varphi^m$ is not Fredholm, $A\varphi^m - \lambda$ cannot be Fredholm by Theorem~\ref{thm:fredholm_alternative}. By Theorem~\ref{thm:reflexive adjoint fredholm}, this implies $\overline{\varphi^m A^*} - \bar{\lambda}$ is not Fredholm. Applying Theorem~\ref{thm:equivalence_of_schecter_spectrum} to $\overline{\varphi^m A^*} - \bar{\lambda}$ shows $\bar{\lambda} \in \sigma_{\text{ess}} (\overline{\varphi^m A^*})$. Noting that $\lambda \in \mathbb{C}$ was arbitrary completes the proof of the claim.
\begin{claim}{4}
If $A\varphi^m$ or $\overline{\varphi^m A^*}$ is Fredholm then $ \sigma_{\text{ess}} (\overline{\varphi^m A^*} ) = \sigma_{\text{p}} (\overline{\varphi^m A^*} ) = \mathbb{C}$.
\end{claim}
Given $L^p(\Omega)$ is reflexive, $A\varphi^m$ is Fredholm if and only if $\overline{\varphi^m A^*}$ is Fredholm by Theorem~\ref{thm:reflexive adjoint fredholm}. Since $\rho(A)$ is non-empty and $ \textnormal{D} (A\varphi^m) \subset\subset L^p(\Omega)$, Corollary~\ref{cor:index_reverse_product} then implies $\text{ind}(A\varphi^m) < 0$. Thus,
\begin{align*}
\text{ind}(\overline{\varphi^m A^*}) = -\text{ind}(A \varphi^m ) > 0.
\end{align*}
Moreover, by Theorem~\ref{thm:reflexive adjoint compactness} and claim 1, we have that
\begin{align*}
\textnormal{D} (\overline{\varphi^m A^*}) \subset\subset L^p(\Omega),
\end{align*}
so for any $\lambda \in \mathbb{C}$, we can conclude
\begin{align} \label{index by positive}
\text{ind}( \overline{\varphi^m A^*} - \lambda) = \text{ind}( \overline{\varphi^m A^*}) > 0.
\end{align}
From \eqref{index by positive} we have $\dim \textnormal{N} (\overline{\varphi^m A^*} - \lambda) > 0$, which implies $\lambda \in \sigma_{\text{p}} (\overline{\varphi^m A^*})$. By Theorem~\ref{thm:equivalence_of_schecter_spectrum}, \eqref{index by positive} also implies $\lambda \in \sigma_{\text{ess}} ( \overline{\varphi^m A^*} )$. Since $\lambda \in \mathbb{C}$ was arbitrary, we are done.
\end{proof}
\section{Acknowledgements}
Parts of this paper have grown out of work that I completed in my thesis. I would like to thank my thesis advisor, Jay Douglas Wright, for his teachings and innumerable
insightful discussions.
|
{
"timestamp": "2015-04-14T02:14:12",
"yymm": "1504",
"arxiv_id": "1504.03157",
"language": "en",
"url": "https://arxiv.org/abs/1504.03157"
}
|
\section{Appendix}
For convenience, we collect in this part the graphics
cited in Section \ref{eq:numerial-tests-2d}
and \ref{eq:numerial-tests-3d}.
\input{cpgamma-ctrgamma-with-estimates-small}
\input{cp-with-estimates-small}
\input{func-cpgamma}
\input{func-ctrgamma}
\input{switch-func-cp-rho-1}
\input{cpgamma-ctrgamma-with-estimates-3d}
\section{Introduction}
Let $T$ be a bounded domain in $\Rd$ ($d \geq 2$) with Lipschitz
boundary $\partial T$. It is well known that the Poincar\'e inequality
(\cite{Poincare1890,Poincare1894})
\begin{equation}
\| w \|_ {T} \leq {C^{\mathrm P}_{T}} \, \| \nabla w \|_{T}
\label{eq:classical-poincare-constant}
\end{equation}
holds for any
\begin{equation*}
w\in\tildeH{1}(T)
:= \Big\{ w \in H^1(T)\, \big | \, \mean{ w }_{T} = 0 \; \Big\},
\end{equation*}
where $\| w \|_ {T}$ denotes the norm in $\L{2}(T)$,
$\mean{w}_{T}: = \tfrac{1}{|T|} \int_{T} w \, \mathrm{d} x$ is the mean
value of $w$ over $T$, and $|T|$ is the Lebesgue measure of $T$. The constant
${C^{\mathrm P}_{T}}$ depends only on $T$ and $d$.
Poincar\'e-type inequalities also hold for
\begin{equation*}
w \in \tildeH{1}(T, \Gamma) :=
\Big\{ w \in H^1(T)\, \big | \, \mean{w}_{\Gamma} = 0 \;\Big\},
\end{equation*}
where $\Gamma$ is a measurable part of $\partial T$ such that
$\mathrm{meas}_{d - 1} \Gamma > 0$ (in particular, $\Gamma$ may coincide with the whole
boundary).
For any $w \in \widetilde{H}^1(T, \Gamma)$, we have two inequalities similar to
\eqref{eq:classical-poincare-constant}. The first one
\begin{equation}
\|w\|_{T} \, \leq \CPGamma \|\nabla w\|_{T}
\label{eq:Comega}
\end{equation}
is another form of the Poincar\'e inequality \eqref{eq:classical-poincare-constant},
which is stated for a different set of functions and contains a different constant,
i.e. ${C^{\mathrm P}_{T}} \leq \CPGamma$.
The constant $\CPGamma$ is associated with the minimal positive eigenvalue of the
problem
\begin{equation}
-\Delta u = \lambda u \;\; {\rm in} \;\; T; \quad
\partial_n u = \lambda \mean{u}_{T} \;\; {\rm on} \;\; \Gamma; \quad
\partial_n u = 0 \;\; {\rm on} \;\; \partial T \backslash \Gamma; \quad
\forall u \in \tildeH{1} (T, \Gamma).
\label{eq:eigenvalue-problem-cp}
\end{equation}
We note that inequalities of this type arose in finite element analysis many years
ago (see, e.g., \cite{BabuskaAziz1976}), where \eqref{eq:Comega} was considered for
simplexes in $\Rtwo$.
The
second inequality
\begin{equation}
\|w\|_{\Gamma} \, \leq \CtrGamma \|\nabla w\|_{T}
\label{eq:Cgamma}
\end{equation}
estimates the trace of $w \in \widetilde{H}^1(T, \Gamma)$ on $\Gamma$.
It is associated with the minimal nonzero eigenvalue of the problem
\begin{equation}
-\Delta u = 0 \;\; {\rm in} \;\; T; \quad
\partial_n u = \lambda u \;\; {\rm on} \;\; \Gamma; \quad
\partial_n u = 0 \;\; {\rm on} \;\; \partial T \backslash \Gamma; \quad
\forall u \in \tildeH{1} (T, \Gamma).
\label{eq:eigenvalue-problem-ctr}
\end{equation}
The problem \eqref{eq:eigenvalue-problem-ctr} is a special case of the Steklov problem
\cite{Stekloff1902}, where the spectral parameter appears in the boundary condition.
Sometimes \eqref{eq:eigenvalue-problem-ctr} is associated with the so-called
{\em sloshing problem}, which describes oscillations of a fluid in a container.
Eigenvalues and eigenfunctions of the sloshing problem have been studied in
\cite{FoxKuttler1983,BanuelosKulczyckiPolterovichSiudeja2010,KozlovKuznetsov2004,KozlovKuznetsovMotygin2004,KuznetsovKulczyckiKwasnickiNazarovPoborchiPolterovichSiudeja2014,
ArxivGirouardPolterovich}
and some other papers cited therein.
Exact values of $\CPGamma$, $\CtrGamma$, and ${C^{\mathrm P}_{T}}$ are important from
both analytical and computational points of view.
Poincar\'{e}-type inequalities are often used in analysis of nonconforming
approximations (e.g., discontinuous Galerkin or mortar methods), domain decomposition
methods (see, e.g., \cite{Klawonnatall2008,Dohrmann2008} and \cite{ToselliWidlund2005}),
a posteriori estimates \cite{RepinBoundaryMeanTrace2015}, and other applications related
to quantitative analysis of partial differential equations.
Analysis of interpolation constants and their estimates for
piecewise constant and linear interpolations over triangular finite elements can be
found in \cite{XuefengOishi2013} and literature cited therein. Finally, we note that
\cite{CarstensenGedicke2014} introduces a method of computing lower bounds
for the eigenvalues of the Laplace operator based on nonconforming (Crouzeix-Raviart)
approximations. This method yields guaranteed upper bounds of the constant in the
Friedrichs' inequality.
It is known (see \cite{PayneWeinberger1960}) that for convex domains
$${C^{\mathrm P}_{T}} \leq \tfrac{\diam (T)}{\pi}.$$
For triangles this estimate was improved
in \cite{LaugesenSiudeja2010} to
$${C^{\mathrm P}_{T}} \leq \tfrac{\diam (T)}{j_{1, 1}},$$
where $j_{1, 1} \approx 3.8317$ is the smallest positive
root of the Bessel function $J_1$. Moreover, for isosceles triangles from
\cite{Bandle1980, LaugesenSiudeja2010} it follows that
\begin{equation}
{C^{\mathrm P}_{T}} \leq \CPLS := \diam (T) \cdot \,
\begin{cases}
\tfrac{1}{j_{1, 1}} & \alpha \in (0, \tfrac{\pi}{3}],\\
\min \Big\{ \tfrac{1}{j_{1, 1}}, \tfrac{1}{j_{0, 1}}
\big(2 (\pi - \alpha) \tan(\tfrac{\alpha}{2})\big)^{-\rfrac{1}{2}} \Big\} &
\alpha \in (\tfrac{\pi}{3}, \tfrac{\pi}{2}], \\
\tfrac{1}{j_{0, 1}} \big(2 (\pi - \alpha) \tan(\tfrac{\alpha}{2})\big)^{-\rfrac{1}{2}} &
\alpha \in (\tfrac{\pi}{2}, \pi).\\
\end{cases}
\label{eq:improved-estimates}
\end{equation}
Here, $j_{0, 1} \approx 2.4048$ is the smallest positive
root of the Bessel function $J_0$.
A lower bound of ${C^{\mathrm P}_{T}}$ for convex domains in $\Rtwo$ was derived in
\cite{Cheng1975} and it reads
\begin{equation}
{C^{\mathrm P}_{T}} \,\geq\, \tfrac{\diam \,(T)}{2 \,j_{0, 1}}.
\label{eq:cheng}
\end{equation}
Analogously, work \cite{LaugesenSiudeja2009} provides lower bound
\begin{equation}
{C^{\mathrm P}_{T}} \,\geq\, \tfrac{P}{4 \,\pi},
\label{eq:ls}
\end{equation}
which improves \eqref{eq:cheng} for some cases. Here, $P$ is perimeter of $T$.
In \cite{NazarovRepin2014}, exact values of $\CPGamma$ and $\CtrGamma$ are found for
parallelepipeds, rectangles, and right triangles. Subsequently, we exploit the following
two results:
\begin{itemize}
\item[1.]
If $T$ is based on vertexes $A = (0, 0)$, $B = (h, 0)$, $C = (0, h)$ and
$\Gamma := \big \{ x_1 \in [0, h], \, x_2 = 0 \big \}$ (i.e., $\Gamma$ coincides with
one of the legs of the isosceles right triangle), then
\begin{equation}
\CPGamma = \tfrac{h}{\zeta_0} \quad \mathrm{and} \quad
\CtrGamma = \left(\tfrac{h}{\hat{\zeta}_0 \,
\tanh({\hat{\zeta}}_0)}\right)^{\rfrac{1}{2}},
\label{eq:exact-cp-cg-t-leg}
\end{equation}
where $\zeta_0$ and $\hat{\zeta}_0$ are unique roots of the equations
\begin{equation}
z \cot(z) + 1 = 0 \quad \mbox{and} \quad \tan(z) + \tanh(z) = 0,
\label{eq:roots}
\end{equation}
respectively, in the interval $(0, \pi)$ .
\item[2.]
If $T$ is based on vertexes $A = (0, 0)$, $B = (h, 0)$,
$C = \big(\tfrac{h}{2}, \tfrac{h}{2} \big)$, and $\Gamma$ coincides with the hypotenuse of the isosceles right triangle, then
\begin{equation*}
\CPGamma = \tfrac{h}{2 \zeta_0} \quad \mathrm{and} \quad
\CtrGamma = \big(\tfrac{h}{2}\big)^{\rfrac{1}{2}}.
\end{equation*}
\end{itemize}
It is worth emphasizing that values of $\CtrGamma$ for right isosceles triangles
follow from the exact solutions of the Steklov problem related to the square.
This specific case
was discussed in the work \cite{ArxivGirouardPolterovich}.
Exact value of constants in the classical Poincar\'{e} inequality are also known for
certain triangles:
\begin{itemize}
\item[1.] For the equilateral triangle
$\Tref_{\rfrac{\pi}{3}}$ based on vertexes
$\hat{A} =(0, 0)$, $\hat{B} = (1, 0)$,
$\hat{C} = \big(\tfrac{1}{2}, \tfrac{\sqrt{3}}{2}\big)$, where \linebreak
$\hat{\Gamma} := \big\{ x_1 \in [0, 1] ; \;\; x_2 = 0 \big\}$,
the constant
$$C^{\mathrm{P}}_{\Tref,\, \rfrac{\pi}{3}} = \tfrac{3}{4 \pi}$$
is derived in \cite{Pinsky1980}.
\item[2.] For the right isosceles triangles
$\Tref_{\rfrac{\pi}{4}}$ based on vertexes
$\hat{A} =(0, 0)$, $\hat{B} = (1, 0)$,
$\hat{C} = \big(\tfrac{1}{2}, \tfrac{1}{2}\big)$
and $\Tref_{\rfrac{\pi}{2}}$ based on
$\hat{A} = (0, 0)$, $\hat{B} = (1, 0)$,
$\hat{C} = (0, 1)$,
we have
$$\CPTrefpifour = \tfrac{1}{\sqrt{2}\pi} \quad \mbox{and} \quad \CPTrefpitwo = \tfrac{1}{\pi},$$
respectively.
Proofs can be found in \cite{HoshikawaUrakawa2010} and
\cite{NakaoYamamoto2001I}.
\end{itemize}
Explicit formulas of the same constants for certain
three-dimensional domains are presented in papers \cite{Berard1980} and
\cite{HoshikawaUrakawa2010}.
The above mentioned results form a basis for deriving sharp bounds of the constants
$\CPGamma$, $\CtrGamma$, and ${C^{\mathrm P}_{T}}$ for arbitrary non-degenerate
triangles and
tetrahedrons, which are typical objects in various discretization methods. In Section
\ref{sc:arbitrary-triangle}, we deduce guaranteed and easily computable bounds of
$\CPGamma$, $\CtrGamma$, and ${C^{\mathrm P}_{T}}$ for triangular domains.
The efficiency of these
bounds is tested in Section \ref{eq:numerial-tests-2d}, where $\CPGamma$ and $\CtrGamma$
are compared with
lower bounds computed numerically by solving a generalized eigenvalue problem generated
by Rayleigh quotients discretized
over sufficiently representative sets of trial functions.
In the same section, we make a
similar comparison of numerical lower bounds related to the constant ${C^{\mathrm P}_{T}}$ with obtained upper bounds and existing estimates known from
\cite{LaugesenSiudeja2009,LaugesenSiudeja2010} and \cite{Cheng1975}.
Lower bounds of the constants presented in Section \ref{eq:numerial-tests-2d} have been
computed by two independent codes: the first code is based on the MATLAB Symbolic Math
Toolbox \cite{Matlab}, and the second one uses The FEniCS Project
\cite{LoggMardalWells2012}.
Section \ref{eq:numerial-tests-3d} is devoted to tetrahedrons. We combine numerical and
theoretical estimates in order to derive two-sided bounds of the constants.
Finally,
in Section \ref{sec:example} we present an example that shows one possible application
of the estimates considered in previous sections. Here, the constants are used in order
to deduce a guaranteed and fully computable upper bound of the distance between the
exact solution of an elliptic boundary value problem and an arbitrary function
(approximation) in the respective energy space.
\section{Majorants of $\CPGamma$ and $\CtrGamma$ for triangular domains}
\label{sc:arbitrary-triangle}
Let $T$ be based on vertexes $A = (0, 0)$, $B = (h ,0)$, and
$C = \big(h \rho \cos\alpha, \, h \rho \, \sin\alpha \big)$
and
\begin{equation}
\Gamma := \big\{ x_1 \in [0, h] ; \;\; x_2 = 0 \big\},
\label{eq:arbitrary-gamma}
\end{equation}
where $\rho>0$, $h>0$, and $\alpha \in (0,\pi)$ are geometrical parameters
that fully define a triangle $T$ (see Fig. \ref{eq:2d-simplex}).
Easily computable bounds of $C^{\mathrm{P}}_{\Gamma}$ and $C^{\mathrm{Tr}}_{\Gamma}$
are presented in Lemma \ref{th:lemma-poincare-type-constants}
below, which uses mappings of reference triangles to
$T$ and well-known integral transformations
(see, e.g., \cite{Ciarlet1978}).
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=0.6]
\draw[->] (0.0, 0.0) -- (4.0, 0.0) node[anchor=west] {$x_1$};
\draw[->] (0.0, 0.0) -- (0.0, 4.0) node[anchor=west] {$x_2$};
\draw (0,0) node[anchor=north east] {$A (0, 0)$} --
(2.8,0) node[anchor=north west] {$B (h, 0)$} --
(1.8,2.5) node[anchor=south west]
{$C \big(h \rho \cos\alpha, h \rho \sin\alpha \big)$}-- (0,0);
\filldraw [black] (0,0) circle (0.8pt)
(2.8,0) circle (0.8pt)
(1.8,2.5) circle (0.8pt);
\draw (0.4,0.0) .. controls (0.3,0.3) and (0.2, 0.3) .. (0.2,0.3);
\draw (0.6,0.1) node[anchor=south] {$\alpha$};
\draw (1.5,0.5) node[anchor=west] {$T$};
\draw [ultra thick] (0,0) -- (2.8,0);
\draw (2,-1.0) node[anchor=south] {${\Gamma}$};
\end{tikzpicture}
\caption{Simplex in $\Rtwo$.}
\label{eq:2d-simplex}
\end{figure}
\begin{lemma}
\label{th:lemma-poincare-type-constants}
For any $w \in \tildeH{1}( T, \Gamma)$, the upper bounds of constants in the inequalities
\begin{alignat}{2}
\|w\|_{ T} \, & \leq\, \CPGamma \, h \,\|\nabla w\|_{ T}
\quad \mathrm{and} \quad
\|w\|_\Gamma \, & \leq\, \CtrGamma \, h^{\rfrac{1}{2}} \, \|\nabla w\|_{ T}
\label{eq:poincare-type-inequalities}
\end{alignat}
are defined as
\begin{equation*}
\CPGamma \leq \CP = \min \Big \{ \cpleg \, \CPTrefleg, \: \cphyp \, \CPTrefhyp \Big \}
\quad \mathrm{and} \quad
\CtrGamma \leq \CG = \min \Big \{ \cgleg \, \CGTrefleg, \: \cghyp \, \CGTrefhyp \Big \},
\end{equation*}
respectively.
Here,
\begin{equation*}
\cpleg = \muleg^{\rfrac{1}{2}},\quad
\cphyp = \muhyp^{\rfrac{1}{2}}, \quad
\cgleg = \big( \rho \, \sin\alpha \big)^{-\rfrac{1}{2}} \, \cpleg, \quad
\cghyp = \big( 2 \rho \,\sin\alpha\big)^{-\rfrac{1}{2}}\, \cphyp,
\end{equation*}
where
\begin{alignat}{2}
\muleg (\rho, \alpha)
= \,& \tfrac{1}{2}
\Big( 1 + \rho^2 +
\big( 1 + \rho^4 + 2 \, \rho^2 \,\cos2\alpha \big)^{\rfrac{1}{2}} \Big),
\label{eq:mu-leg}\\
\muhyp (\rho, \alpha)
= \,& 2 \rho^2 - 2 \rho \, \cos\alpha + 1 +
\big( (2 \rho^2 + 1)
(2 \rho^2 + 1 - 4 \rho \, \cos\alpha + 4 \rho^2 \, \cos2\alpha) \big )^{\rfrac{1}{2}},
\label{eq:mu-hyp}
\end{alignat}
and $\CPTrefleg \approx 0.49291$, $\CGTrefleg \approx 0.65602$ and
$\CPTrefhyp \approx 0.24646$, $\CGTrefhyp \approx 0.70711$, where $\hat{\Gamma}$ is
defined as
\begin{equation}
\hat{\Gamma} := \big\{ x_1 \in [0, 1] ; \;\; x_2 = 0 \big\}.
\label{eq:arbitrary-gamma-hat}
\end{equation}
\end{lemma}
\noindent{\bf Proof:} \:
Consider the linear mapping $\mathcal{F}_{\rfrac{\pi}{2}} :
\Tref_{\rfrac{\pi}{2}} \rightarrow T$ with
\begin{equation*}
x = \mathcal{F}_{\rfrac{\pi}{2}} \, (\hat{x}) = B_{\rfrac{\pi}{2}} \, \hat{x},
\quad \mbox{where} \quad
B_{\rfrac{\pi}{2}} =
\begin{pmatrix}
\: h & \rho h \cos\alpha \\[0.3em]
0 & \rho h \sin\alpha \:
\end{pmatrix}
, \quad
\mathrm{det} B_{\rfrac{\pi}{2}} = \rho h^2 \, \sin\alpha.
\label{eq:transformation-1}
\end{equation*}
For any $\hat{w} \in \tildeH{1}(\Tref_{\rfrac{\pi}{2}}, \Gref)$, we have the estimate
\begin{equation}
\|\, \hat{w} \,\|_{ \Tref_{\rfrac{\pi}{2}} }
\leq \CPTrefleg \, \|\, \nabla \hat{w} \,\|_{\Tref_{\rfrac{\pi}{2}}},
\label{eq:poicare-inequality-1-hatv}
\end{equation}
where $\CPTrefleg$ is the constant associated with the basic simplex
$\Tref_{\rfrac{\pi}{2}}$ based on $\hat{A} =(0, 0)$, $\hat{B} = (1, 0)$, and
$\hat{C} = (0, 1)$,
Note that
\begin{alignat}{2}
\| \, \hat{w} \, \|^2_{\Tref_{\rfrac{\pi}{2}}}
= \tfrac{1}{\rho h^2 \sin\alpha } \| \,w\, \|^2_{ T},
\label{eq:v-norm-transformation}
\end{alignat}
and
\begin{equation}
\| \,\nabla \hat w \, \|^2_{\Tref_{\rfrac{\pi}{2}}}
\leq \tfrac{1}{\rho h^2 \sin\alpha} \Int_{T}
A_{\rfrac{\pi}{2}}(h, \rho,\alpha) \nabla w \cdot \nabla w \mathrm{\:d}x,
\label{eq:grad-hatv-lower-estimate-1}
\end{equation}
where
\begin{equation*}
A_{\rfrac{\pi}{2}} (h, \rho,\alpha) = h^2 \,
\begin{pmatrix}
1 + \rho^2 \, \cos^2\alpha \qquad & \rho^2 \sin\alpha \, \cos\alpha \; \\[0.3em]
\rho^2 \sin\alpha \, \cos\alpha & \rho^2 \sin^2\alpha
\end{pmatrix}.
\end{equation*}
It is not difficult to see that
$\lambda_{\rm max} (A_{\rfrac{\pi}{2}}) = h^2 \muleg(\rho, \alpha)$,
where $\muleg(\rho, \alpha)$ is defined in \eqref{eq:mu-leg}.
From \eqref{eq:poicare-inequality-1-hatv}, \eqref{eq:v-norm-transformation}, and
\eqref{eq:grad-hatv-lower-estimate-1}, it follows that
\begin{equation}
\|\, w \,\|_{ T} \, \leq \,
\cpleg \, \CPTrefleg \, h \, \, \|\, \nabla w \,\|_{ T}, \quad
\cpleg (\rho, \alpha) = \muleg^{\rfrac{1}{2}}(\rho, \alpha).
\label{eq:poicare-inequality-1-v}
\end{equation}
Notice that $\hat{w} \in \tildeH{1} (\Tref, \Gref)$ yields
\begin{equation*}
\mean{w}_{\Gamma} := \Int_{\Gamma} w(x) \mathrm{\:d}s =
h \Int_{\Gref} w(x(\hat{x})) \, {\rm d \hat{s}} =
h \Int_{\Gref} \hat{w} \, \rm{d \hat{s}} = 0.
\end{equation*}
Therefore, above mapping keeps $w \in \tildeH{1} (T, \Gamma)$.
In view of inequality (\ref{eq:Cgamma}),
for any $\hat{w} \in \tildeH{1}(\Tref_{\rfrac{\pi}{2}}, \Gref)$ we have
\begin{equation*}
\|\, \hat{w} \,\|_{\Gref}
\leq \CGTrefleg \|\, \nabla \hat{w} \,\|_{\Tref_{\rfrac{\pi}{2}}},
\label{eq:poicare-inequality-2-hatv}
\end{equation*}
where $\CGTrefleg$ is the constant associated with the reference simplex
$\Tref_{\rfrac{\pi}{2}}$.
Since
\begin{equation*}
\| \,\hat{w}\, \|^2_{\Gref} = \tfrac{1}{h} \| \, w \, \|^2_{\Gamma},
\end{equation*}
we obtain
\begin{alignat}{2}
\|\, w \,\|_{\Gamma} \,
\leq \, \cgleg \, \CGTrefleg \, h^{\rfrac{1}{2}} \|\, \nabla w \,\|_{ T}, \quad
\cgleg (\rho, \alpha) = \Big(\tfrac{\muleg(\rho, \alpha)}
{\rho \sin\alpha}\Big)^{\rfrac{1}{2}}.
\label{eq:poincare-2-v-cgleg}
\end{alignat}
Now, we consider the mapping $\mathcal{F}_{\rfrac{\pi}{4}} :
\Tref_{\rfrac{\pi}{4}} \rightarrow T$, where $\Tref_{\rfrac{\pi}{4}}$ is based on
$\hat{A} =(0, 0)$, $\hat{B} = (1, 0)$, and $\hat{C} = (\tfrac{1}{2}, \tfrac{1}{2})$, i.e.,
\begin{equation*}
x = \mathcal{F}_{\rfrac{\pi}{4}} (\hat{x}) = B_{\rfrac{\pi}{4}} \, \hat{x},
\quad {\rm where}\quad
B_{\rfrac{\pi}{4}} =
\begin{pmatrix}
\: h & \; 2 \rho h \cos\alpha \, \scalebox{0.5}[1.0]{\( - \)} \, h \\[0.3em]
0 & \; 2 \rho h \sin\alpha \:
\end{pmatrix},
\quad
\mathrm{det} \, B_{\rfrac{\pi}{4}} = 2 \rho h^2 \, \sin\alpha,
\end{equation*}
which yields another pair of estimates for the functions in
$\tildeH{1}( T, \Gamma)$:
\begin{equation}
\|\, w \,\|_{T} \, \leq \, \cphyp \, \CPTrefhyp \, h \, \|\, \nabla w \,\|_{T}, \quad
\cphyp (\rho, \alpha) = \muhyp^{\rfrac{1}{2}}(\rho, \alpha),
\label{eq:poicare-inequality-2-v}
\end{equation}
and
\begin{alignat}{2}
\|\, w \,\|_{\Gamma} \, \leq \, \cghyp \, \CGTrefhyp \,
h^{\rfrac{1}{2}} \|\, \nabla w \,\|_{ T}, \quad
\cghyp (\rho, \alpha)
= \Big(\tfrac{\muhyp(\rho, \alpha)}{2 \rho \sin\alpha}\Big)^{\rfrac{1}{2}},
\label{eq:poincare-2-v-cghyp}
\end{alignat}
where $\muhyp(\rho, \alpha)$ is defined in (\ref{eq:mu-hyp}).
Now, (\ref{eq:poincare-type-inequalities}) follows from
(\ref{eq:poicare-inequality-1-v}), (\ref{eq:poincare-2-v-cgleg}),
(\ref{eq:poicare-inequality-2-v}), and (\ref{eq:poincare-2-v-cghyp}).
{\hfill $\square$}
\vskip10pt
Analogously to Lemma \ref{th:lemma-poincare-type-constants}, one
can obtain an upper bound of the constant in
\eqref{eq:classical-poincare-constant}. For that we consider three reference
triangles $\Tref_{\rfrac{\pi}{2}}$, $\Tref_{\rfrac{\pi}{4}}$ (defined earlier),
and
$\Tref_{\rfrac{\pi}{3}}$ based on vertexes
$A = (0, 0)$, $B = (1, 0)$, $C = (\tfrac{1}{2}, \tfrac{\sqrt{3}}{2})$.
\begin{lemma}
\label{th:lemma-poincare-constants}
For any $w \in \tildeH{1}(\T)$, the constant in
\begin{alignat}{2}
\|w\|_{\T} \, & \leq\, \CPoincare h \,\|\nabla w\|_{\T},
\label{eq:poincare-inequality}
\end{alignat}
is estimated as
\begin{equation}
\CPoincare \leq \CPMR
= \min \Big \{ \cpifour \, \CPTrefpifour , \:
\cpithree \, \CPTrefpithree, \:
\cpitwo \, \CPTrefpitwo \Big\}.
\label{eq:classical-poincare-constant-estimate}
\end{equation}
Here,
$\cpifour = \mupifour^{\rfrac{1}{2}}$, \quad
$\cpithree = \mupithree^{\rfrac{1}{2}}$,\quad
$\cpitwo = \mupitwo^{\rfrac{1}{2}}$, where
$\mupitwo$ and $\mupifour$ being defined in \eqref{eq:mu-leg} and
\eqref{eq:mu-hyp}, respectively, and
\begin{alignat}{2}
\mupithree (\rho, \alpha)
= \,& \tfrac{2}{3} (1 + \rho^2 - \rho \,\cos\alpha) +
2 \big( \tfrac{1}{9} (1 + \rho^2 - \rho \,\cos\alpha)^2 - \tfrac{1}{3} \rho^2 \sin^2\alpha \big )^{\rfrac{1}{2}},
\label{eq:mu-pi3}
\end{alignat}
and $\CPTrefpifour = \tfrac{1}{\sqrt{2}\pi}$,
$\CPTrefpithree = \tfrac{3}{4 \pi}$, and
$\CPTrefpitwo = \tfrac{1}{\pi}$.
\end{lemma}
\noindent{\bf Proof:} \:
The mapping $\mathcal{F}_{\rfrac{\pi}{2}}: \Tref_{\rfrac{\pi}{2}} \rightarrow \T$
coincides with \eqref{eq:transformation-1} from Lemma
\ref{th:lemma-poincare-type-constants}.
It is easy to see that $w \in \tildeH{1} (T)$ provides that $\hat{w} \in \tildeH{1} (\Tref)$.
The estimate
\begin{equation}
\|\, w \,\|_{\T} \, \leq \,
\cpitwo \, \CPTrefpitwo \, h \, \, \|\, \nabla w \,\|_{\T}, \quad
\cpitwo (\rho, \alpha) = \mupitwo^{\rfrac{1}{2}}(\rho, \alpha)
\label{eq:classical-poincare-inequality-1-v}
\end{equation}
is obtained by following steps of the previous proof.
From analysis of mappings
\begin{equation*}
x = \mathcal{F}_{\rfrac{\pi}{3}} (\hat{x}) = B_{\rfrac{\pi}{3}} \, \hat{x},
\quad \mbox{ where}\quad
B_{\rfrac{\pi}{3}} =
\begin{pmatrix}
\: h & \; \tfrac{h}{\sqrt{3}} (2 \rho \cos\alpha - 1) \, \scalebox{0.5}[1.0]{\( - \)} \, h \\[0.3em]
0 & \; \tfrac{2 h}{\sqrt{3}} \rho \sin\alpha \:
\end{pmatrix}
,\quad
\mathrm{det} \, B_{\rfrac{\pi}{3}} = \tfrac{2 h^2}{\sqrt{3}} \, \sin\alpha > 0,
\end{equation*}
and
\begin{equation*}
x = \mathcal{F}_{\rfrac{\pi}{4}} (\hat{x}) = B_{\rfrac{\pi}{4}} \, \hat{x},
\quad {\rm where}\quad
B_{\rfrac{\pi}{4}} =
\begin{pmatrix}
\: h & \; 2 \rho h \cos\alpha \, \scalebox{0.5}[1.0]{\( - \)} \, h \\[0.3em]
0 & \; 2 \rho h \sin\alpha \:
\end{pmatrix}
,\quad
\mathrm{det} \, B_{\rfrac{\pi}{4}} = 2 \rho h^2 \, \sin\alpha > 0,
\end{equation*}
we obtain alternative estimates
\begin{alignat}{2}
\|\, w \,\|_{\T} \, & \leq \,
\cpithree \, \CPTrefpithree \, h \, \|\, \nabla w \,\|_{\T}, \quad
\cpithree (\rho, \alpha) = \mupithree^{\rfrac{1}{2}}(\rho, \alpha),
\label{eq:classical-poincare-inequality-2-v} \\
\|\, w \,\|_{\T} \, & \leq \,
\cpifour \, \CPTrefpifour \, h \, \|\, \nabla w \,\|_{\T}, \quad
\cpifour (\rho, \alpha) = \mupifour^{\rfrac{1}{2}}(\rho, \alpha),
\label{eq:classical-poincare-inequality-3-v}
\end{alignat}
where $\mupithree(\rho, \alpha)$ and $\mupifour(\rho, \alpha)$ are defined in
\eqref{eq:mu-pi3} and \eqref{eq:mu-hyp}, respectively.
Therefore, (\ref{eq:classical-poincare-constant-estimate}) follows from combination of
(\ref{eq:classical-poincare-inequality-1-v}),
(\ref{eq:classical-poincare-inequality-2-v}), and
(\ref{eq:classical-poincare-inequality-3-v}).
{\hfill $\square$}
\section{Minorants of $\CPGamma$ and $\CtrGamma$ for triangular domains}
\label{eq:numerial-tests-2d}
\subsection{Two-sided bounds of $\CPGamma$ and $\CtrGamma$}
Majorants of $\CPGamma$ and $\CtrGamma$ provided by Lemma
\ref{th:lemma-poincare-type-constants} should be compared with the corresponding
minorants, which can be found by means of
the Rayleigh quotients
\begin{equation}
\mathcal{R}^{\mathrm{P}}_{\Gamma} [w]
= \tfrac{\|\nabla w\|_{\T}}{\|w - \mean{w}_{\Gamma} \|_{\T}}
\quad \mathrm{and} \quad
\mathcal{R}^{\mathrm{Tr}}_{\Gamma} [w]
= \tfrac{\|\nabla w\|_{\T}}{\|w - \mean{w}_{\Gamma} \|_{\Gamma}}.
\label{eq:quotients-2d}
\end{equation}
Lower bounds are obtained if the quotients are minimized on finite dimensional
subspaces
$V^N \subset \H{1}(T)$ formed by sufficiently
representative collections of suitable test functions. For this purpose, we use either
power or
Fourier series and introduce the spaces
\begin{equation*}
V^{N}_1 := \mathrm{span} \big\{\: x^{i} y^{j}\: \big\} \quad {\rm and } \quad
V^{N}_2 := \mathrm{span} \big\{\: \cos (\pi i x) \cos (\pi j y)\: \big\}, \;\;
\end{equation*}
where $i, j = 0, \ldots, N, \;\; (i, j) \neq (0, 0)$
and
$$\mathrm{dim} \, V^{N}_1 = \mathrm{dim} \, V^{N}_2 = M(N) :=
(N + 1)^2 - 1.$$
The corresponding constants are denoted by $\approxCPT$ and $\approxCGT$, where $M$ indicates
on number of basis functions in auxiliary subspace used.
Since $ V^{N}_1$ and $V^{N}_2$ are limit dense
in $\H{1}(\T)$, the respective minorants tend to the exact constants as $M(N)$ tends to infinity.
We note that
\begin{equation}
\inf\limits_{w \in \H{1}(T)} \mathcal{R}^{\mathrm{P}}_{\Gamma} [w] =
\inf\limits_{w \in \H{1}(T)}
\tfrac{\|\nabla w\|_{\T}}{\|w - \mean{w}_{\Gamma} \|_{\T}}
=
\inf\limits_{w \in \tildeH{1}(T, \Gamma)}
\tfrac{\|\nabla w\|_{\T}}{\|w\|_{\T}} = \tfrac{1}{\CPGamma}.
\end{equation}
Therefore, minimization of the first quotient in \eqref{eq:quotients-2d} on
$V^{N}_1$ or $V^{N}_2$ yields a lower bound of $\CPGamma$.
For the quotient
$\mathcal{R}^{\mathrm{Tr}}_{\Gamma} [w]$, we apply similar arguments.
Numerical results presented below are obtained with the help of two different codes
based on the MATLAB Symbolic Math Toolbox \cite{Matlab} and The FEniCS Project
\cite{LoggMardalWells2012}.
Table \ref{tab:const-convergence-from-basis} demonstrates the ratios between the
exact
constants and respective approximate values (for the selected $\rho$ and $\alpha$).
They are quite close to $1$ even for relatively small
$N$. Henceforth, we select $N = 6$ or $7$ in the tests discussed below.
\begin{table}[!ht]
\centering
\footnotesize
\begin{tabular}{cc|cc|cc}
\multicolumn{2}{c|}{$ $}
& \multicolumn{2}{c|}{ $\alpha = \tfrac{\pi}{2}$, $\rho = 1$ }
& \multicolumn{2}{c}{ $\alpha = \tfrac{\pi}{4}$, $\rho = \tfrac{\sqrt{2}}{2}$} \\
\midrule
$N$ & $M(N)$
&
${\approxCPT}/{\CPTrefleg}$
& ${\approxCGT}/{\CGTrefleg}$
& ${\approxCPT}/{\CPTrefhyp}$
& ${\approxCGT}/{\CGTrefhyp}$ \\
\midrule
1 & 3 & 0.8801 & 0.9561 & 0.8647 & 1.0000 \\
2 & 8 & 0.9945 & 0.9898 & 0.9925 & 1.0000 \\
3 & 15 & 0.9999 & 0.9998 & 0.9962 & 1.0000 \\
4 & 24 & 1.0000 & 0.9999 & 1.0000 & 1.0000 \\
5 & 35 & 1.0000 & 1.0000 & 1.0000 & 1.0000 \\
6 & 48 & 1.0000 & 1.0000 & 1.0000 & 1.0000 \\
\end{tabular}
\\[5pt]
\caption{Ratios between approximate and reference constants
with respect to increasing $N$.}
\label{tab:const-convergence-from-basis}
\end{table}
\begin{figure}[!ht]
\centering
\subfloat[$\rho = \tfrac{\sqrt{2}}{2}$]{
\includegraphics[scale=0.95]{exact-and-approx-C-PT-rho-0-70711-h-1-1-power-series}
\label{fig:2d-cpt-rho-sqrt2-2}}
\quad
\subfloat[$\rho = \tfrac{\sqrt{2}}{2}$]{
\begin{tikzpicture}[thick, scale=0.9]
\draw[dashed] (2.1213,0) arc (0:180:2.1213cm);
\draw[->] (0.0, 0.0) -- (3.5, 0.0) node[anchor=west] {$x_1$};
\draw[->] (0.0, 0.0) -- (0.0, 3.5) node[anchor=west] {$x_2$};
\draw (0.0,0.0) node[anchor=north east] {$(0, 0)$} --
(2.1213,0.0) node[anchor=north west] {$(1, 0)$} --
(0.0,2.1213) node[anchor=south west] {$\big(0, \tfrac{1}{\sqrt{2}}\big)$}-- (0,0);
\filldraw [black] (0.0,0.0) circle (0.8pt)
(2.1213,0.0) circle (0.8pt)
(0.0,2.1213) circle (0.8pt);
\draw (0,0.2) -- (0.2,0.2) -- (0.2, 0);
\draw (0.4, -0.6) node[anchor=south] {$\tfrac{\pi}{2}$};
\draw [ultra thick] (0,0) -- (3,0);
\draw (1.5,-0.8) node[anchor=south] {${\Gamma}$};
\draw [ultra thick](0.0,0.0) -- (3.0, 0.0) -- (1.5, 1.5) -- (0,0);
\filldraw [black] (0.0,0.0) circle (0.8pt)
(3.0,0.0) circle (0.8pt)
(1.5, 1.5) circle (0.8pt);
\draw (1.0,0.0) .. controls (1.0,0.2) and (0. 95, 0.5) .. node[right] {$\tfrac{\pi}{4}$} (0.8,0.72);
\draw (0.0,0.0) -- (3.0, 0.0) -- (1.0607, 1.8371) -- (0,0);
\filldraw [black] (0.0,0.0) circle (0.8pt)
(3.0,0.0) circle (0.8pt)
(1.0607, 1.8371) circle (0.8pt);
\draw (0.8, 0.0) .. controls (0.8, 0.4) and (0.7, 0.58) .. node[anchor=south] {$ $} (0.49,0.82);
\draw (0.3, 0.6) node[anchor=south] {$\tfrac{\pi}{3}$};
\draw (0.0,0.0) -- (3.0, 0.0) -- (-1.0607, 1.8371) -- (0,0);
\filldraw [black] (0.0,0.0) circle (0.8pt)
(3.0,0.0) circle (0.8pt)
(-1.0607, 1.8371) circle (0.8pt);
\draw (0.6, 0.0) .. controls (0.6,0.7) and (-0.1, 0.9) .. node[anchor=south] {$ $} (-0.4,0.7);
\draw (-0.2, 0.7) node[anchor=south] {$\tfrac{2\pi}{3}$};
\end{tikzpicture}
\label{eq:t-rho-sqrt2-2}
}\\
\subfloat[$\rho = 1$]{
\includegraphics[scale=0.95]{exact-and-approx-C-PT-rho-1-h-1-1-power-series}
\label{fig:2d-cpt-rho-1}}
\; \;
\subfloat[$\rho = 1$]{
\begin{tikzpicture}[thick, scale=0.9]
\draw[dashed] (3,0) arc (0:180:3cm);
\draw[->] (0.0, 0.0) -- (3.2, 0.0) node[anchor=west] {$x_1$};
\draw[->] (0.0, 0.0) -- (0.0, 3.5) node[anchor=west] {$x_2$};
\draw[ultra thick] (0.0,0.0) node[anchor=north east] {$(0, 0)$} --
(3.0,0.0) node[anchor=north west] {$(1, 0)$} --
(0.0,3.0) node[anchor=east] {$(0, 1)$}-- (0,0);
\filldraw [black] (0.0,0.0) circle (0.8pt)
(3.0,0.0) circle (0.8pt)
(0.0,3.0) circle (0.8pt);
\draw (0,0.2) -- (0.2,0.2) -- (0.2, 0);
\draw (0.4, -0.6) node[anchor=south] {$\tfrac{\pi}{2}$};
\draw [ultra thick] (0,0) -- (3,0);
\draw (1.5,-0.8) node[anchor=south] {${\Gamma}$};
\draw (0.0,0.0) -- (3.0, 0.0) -- (2.5981,1.5) -- (0,0);
\filldraw [black] (0.0,0.0) circle (0.8pt)
(3.0,0.0) circle (0.8pt)
(2.5981,1.5) circle (0.8pt);
\draw (1.0,0.0) .. controls (1.0,0.2) and (0. 95, 0.4) .. node[right] {$\tfrac{\pi}{6}$} (0.89,0.5);
\draw (0.0,0.0) -- (3.0, 0.0) -- (1.5,2.5981) -- (0,0);
\filldraw [black] (0.0,0.0) circle (0.8pt)
(3.0,0.0) circle (0.8pt)
(1.5,2.5981) circle (0.8pt);
\draw (0.8, 0.0) .. controls (0.8,0.4) and (0.7, 0.58) .. node[anchor=south] {$ $} (0.49,0.82);
\draw (0.78, 0.6) node[anchor=south] {$\tfrac{\pi}{3}$};
\draw (0.0,0.0) -- (3.0, 0.0) -- (-1.5,2.5981) -- (0,0);
\filldraw [black] (0.0,0.0) circle (0.8pt)
(3.0,0.0) circle (0.8pt)
(-1.5,2.5981) circle (0.8pt);
\draw (0.6, 0.0) .. controls (0.6,0.7) and (-0.1, 0.9) .. node[anchor=south] {$ $} (-0.4,0.7);
\draw (-0.2, 0.7) node[anchor=south] {$\tfrac{2\pi}{3}$};
\end{tikzpicture}
\label{eq:t-rho-1}}
\\[5pt]
\caption{Two-sided bounds of $\CPT$
for $\T$ with different $\rho$.}
\label{fig:2d-cpt}
\end{figure}
\begin{figure}[!ht]
\centering
\subfloat[$\rho = \tfrac{\sqrt{2}}{2}$]{
\includegraphics[scale=0.95]{exact-and-approx-C-GT-rho-0-70711-h-1-1-power-series}
\label{fig:2d-cgt-rho-sqrt2-2}}
\quad
\subfloat[$\rho = 1$]{
\includegraphics[scale=0.95]{exact-and-approx-C-GT-rho-1-h-1-1-power-series}
\label{fig:2d-cgt-rho-1}}
\\[5pt]
\caption{Two-sided bounds of $\CGT$
for $\T$ with different $\rho$.}
\label{fig:2d-cgt}
\end{figure}
In Figs. \ref{fig:2d-cpt-rho-sqrt2-2} and \ref{fig:2d-cpt-rho-1}, we depict $\approxCPT$
for $M(N) = 48$ (thin red line) for $T$ with $\rho = \tfrac{\sqrt{2}}{2}$,
$1$, and $\alpha \in (0, \pi)$. Guaranteed upper bounds
$\CPTleg = \cpleg \, \CPTrefleg$ and
$\CPThyp = \cphyp \, \CPTrefhyp$ are depicted by
dashed black lines. Bold blue line illustrates $\CP = \min \Big \{\CPTleg, \CPThyp \Big \}$.
Analogously in Figs. \ref{fig:2d-cgt-rho-sqrt2-2} and \ref{fig:2d-cgt-rho-1}, a red marker denotes
the lower bound $\approxCGT$ (for $M(N) = 48$) of the constant $\CtrGamma$. It
is presented together with the upper bound $\CG$ (blue marker), which is defined as minimum of
$\CGTleg = \cgleg \, \CGTrefleg$ and $\CGThyp = \cghyp \, \CGTrefhyp$.
Table \ref{tab:big-table} represents this information in the digital form.
\begin{table}[!ht]
\centering
\footnotesize
\begin{tabular}{c|cc|cc|cc|cc}
\multicolumn{1}{c|}{$ $}
& \multicolumn{4}{c|}{ $\rho = \tfrac{\sqrt{2}}{2}$ }
& \multicolumn{4}{c}{ $\rho = 1$ } \\
\midrule
$\alpha$ &
$\underline{C}^{48, \mathrm{P}}_{\Gamma}$ & $\CP$ &
$\underline{C}^{48, \mathrm{Tr}}_{\Gamma}$ & $\CG$ &
$\underline{C}^{48, \mathrm{P}}_{\Gamma}$ & $\CP$ &
$\underline{C}^{48, \mathrm{Tr}}_{\Gamma}$ & $\CG$ \\
\midrule
$\pi/18$ & 0.2429 & 0.2657 & 1.2786 & 1.5386 & 0.3245 & 0.3486 & 1.2572 & 1.6971\\
$\pi/9$ & 0.2414 & 0.2627 & 0.9289 & 1.0838 & 0.3248 & 0.3493 & 0.9058 & 1.2116\\
$\pi/6$ & 0.2389 & 0.2577 & 0.7919 & 0.8792 & 0.3268 & 0.3527 & 0.7632 & 1.0118\\
$2\pi/9$ & 0.2379 & 0.2507 & 0.7259 & 0.7543 & 0.3339 & 0.3636 & 0.6906 & 0.9201\\
$5\pi/18$ & 0.2632 & 0.2722 & 0.6945 & 0.7503 & 0.3514 & 0.3884 & 0.6529 & 0.9003\\
$\pi/3$ & 0.3008 & 0.3220 & 0.6829 & 0.8348 & 0.3809 & 0.4269 & 0.6362 & 0.8634\\
$7\pi/18$ & 0.3382 & 0.3694 & 0.6840 & 0.8432 & 0.4173 & 0.4721 & 0.6332 & 0.7840\\
$4\pi/9$ & 0.3740 & 0.4140 & 0.6947 & 0.7973 & 0.4556 & 0.5187 & 0.6404 & 0.7162\\
$\pi/2$ & 0.4075 & 0.4554 & 0.7136 & 0.7801 & 0.4929 & 0.4929 & 0.6560 & 0.6560\\
$5\pi/9$ & 0.4382 & 0.4933 & 0.7409 & 0.7973 & 0.5280 & 0.5340 & 0.6797 & 0.7162\\
$11\pi/18$ & 0.4660 & 0.5165 & 0.7779 & 0.8432 & 0.5600 & 0.5710 & 0.7125 & 0.7840\\
$2\pi/3$ & 0.4905 & 0.5361 & 0.8274 & 0.9118 & 0.5884 & 0.6037 & 0.7569 & 0.8634\\
$13\pi/18$ & 0.5115 & 0.5552 & 0.8948 & 1.0040 & 0.6129 & 0.6318 & 0.8175 & 0.9607\\
$7\pi/9$ & 0.5289 & 0.5720 & 0.9898 & 1.1292 & 0.6332 & 0.6550 & 0.9033 & 1.0874\\
$5\pi/6$ & 0.5426 & 0.5856 & 1.1334 & 1.3107 & 0.6492 & 0.6733 & 1.0332 & 1.2673\\
$8\pi/9$ & 0.5524 & 0.5956 & 1.3796 & 1.6118 & 0.6607 & 0.6865 & 1.2565 & 1.5623\\
$17\pi/18$ & 0.5583 & 0.6017 & 1.9436 & 2.2851 & 0.6676 & 0.6944 & 1.7692 & 2.2179\\
\end{tabular}
\caption{Two-sided bounds of $\CPT$ and $\CGT$ for $\T$ for $\alpha \in (0, \pi)$ and
different $\rho$.}
\label{tab:big-table}
\end{table}
Fig. \ref{fig:2d-cpt-rho-sqrt2-2} corresponds to the case $\rho = \tfrac{\sqrt{2}}{2}$.
Notice that for $\alpha = \tfrac{\pi}{4}$ the constant $\CPGamma$ is known and the computed
lower bound $\approxCPT$ (red marker) practically coincides with it (see, e.g., Fig. \ref{eq:t-rho-sqrt2-2}).
Since in this case, the mapping $\mathcal{F}_{\rfrac{\pi}{4}}$ is identical, the upper bound also
coincides with the exact value. An analogous coincidence can be observed for $\CtrGamma$
and $\approxCGT$ in Fig. \ref{fig:2d-cgt-rho-sqrt2-2}.
In Fig. \ref{fig:2d-cpt-rho-1}, the red curve, corresponding to $\approxCPT$, coincides with
the blue line of $\CPT$ at the point $\alpha = \tfrac{\pi}{2}$ (due to the fact that
for this angle $\mathcal{F}$ is the identical mapping and $T$ coincides with
$\Tref_{\rfrac{\pi}{2}}$ (see Fig. \ref{eq:t-rho-1})). Fig. \ref{fig:2d-cgt-rho-1} exposes
similar results for ${\approxCGT}$ and $\CtrGamma$ ($\CGTleg$).
\begin{figure}[!ht]
\centering
\subfloat[$\rho = \tfrac{\sqrt{3}}{2}$]{
\includegraphics[scale=0.9]{exact-and-approx-C-PT-rho-0-86603-h-1-1-power-series}
\label{fig:2d-cpt-rho-sqrt3-2}}
\quad
\subfloat[$\rho = \tfrac{3}{2}$]{
\includegraphics[scale=0.9]{exact-and-approx-C-PT-rho-1-5-h-1-1-power-series}
\label{fig:2d-cpt-rho-3-2}}
\\[5pt]
\caption{Two-sided bounds of $\CPT$
for $\T$ with different $\rho$.}
\label{fig:2d-cpt-no-ref}
\end{figure}
\begin{figure}[!ht]
\centering
\subfloat[$\rho = \tfrac{\sqrt{3}}{2}$]{
\includegraphics[scale=0.9]{exact-and-approx-C-GT-rho-0-86603-h-1-1-power-series}
\label{fig:2d-cgt-rho-sqrt3-2}}
\quad
\subfloat[$\rho = \tfrac{3}{2}$]{
\includegraphics[scale=0.9]{exact-and-approx-C-GT-rho-1-5-h-1-1-power-series}
\label{fig:2d-cgt-rho-3-2}}
\\[5pt]
\caption{Two-sided bounds of $\CGT$
for $\T$ with different $\rho$.}
\label{fig:2d-cgt-no-ref}
\end{figure}
Figs. \ref{fig:2d-cpt-no-ref} and \ref{fig:2d-cgt-no-ref} demonstrate the same bounds
for $\rho = \tfrac{\sqrt{3}}{2}$ and $\tfrac{3}{2}$.
We see that estimates of $\CPT$ and $\CGT$ are very efficient. Namely,
$\Ieff^{\rm P} := \tfrac{\CP}{\underline{C}^{48, \mathrm{P}}_{\Gamma}} \in [1.0463, 1.1300]$
for $\rho = \tfrac{\sqrt{3}}{2}$ and
$\Ieff^{\rm P} \in [1.0249, 1.1634]$ for $\rho = \tfrac{3}{2}$.
Analogously,
$\Ieff^{\rm Tr} := \tfrac{\CG}{\underline{C}^{48, \mathrm{Tr}}_{\Gamma}} \in [1.0363, 1.3388]$
for $\rho = \tfrac{\sqrt{3}}{2}$ and $\Ieff^{\rm Tr} \in [1.2917, 1.7643]$ for $\rho = \tfrac{3}{2}$.
\subsection{Two-sided bounds of ${C^{\mathrm P}_{T}}$}
The spaces $V^{N}_1$ and $ V^{N}_2$ can also be used for analysis of the quotient
$\mathcal{R}_{T} [w] = \tfrac{\|\nabla w\|_{ T}}{\|w - \mean{w}_{ T} \|_{ T}}$,
which yields guaranteed lower bounds of the constant in
(\ref{eq:classical-poincare-constant}). The respective values are denoted by $\approxC$.
These bounds are compared with
$\CPupper := \tfrac{\diam (\T)}{j_{1, 1}}$ and $\CPlower :=
\max \Big \{ \tfrac{\diam (\T)}{2 \,j_{0,1}}, \tfrac{P}{4 \,\pi} \Big \}$
(see \eqref{eq:cheng}--\eqref{eq:ls}, respectively) as well as the one derived in
Lemma \ref{th:lemma-poincare-constants}.
\begin{figure}[!ht]
\centering
\subfloat[$\rho = \tfrac{\sqrt{2}}{2}$]{
\includegraphics[scale=0.9]{2d-cp-mean-value-on-T-comparison-LS-MR-rho-0-70711-h-1-1-power-series} \label{fig:cp-rho-sqrt2-2}
}
\subfloat[$\rho = \tfrac{\sqrt{3}}{2}$]{
\includegraphics[scale=0.9]{2d-cp-mean-value-on-T-comparison-LS-MR-rho-0-86603-h-1-1-power-series} \label{fig:cp-rho-sqrt3-2}
}\\[-1pt]
\subfloat[$\rho = 1$]{
\includegraphics[scale=0.9]{2d-cp-mean-value-on-T-comparison-LS-MR-rho-1-h-1-1-power-series}
\label{fig:cp-rho-1}}
\subfloat[$\rho = \tfrac{3}{2}$]{
\includegraphics[scale=0.9]{2d-cp-mean-value-on-T-comparison-LS-MR-rho-1-5-h-1-1-power-series}
\label{fig:cp-rho-3-2}}
\\[5pt]
\caption{$\underline{C}^{48}_{\T}$, $\CPLS$, $\CPMR$, and $\CPlower$ for $\T$ with
$\alpha \in (0, \pi)$ and different $\rho$.}
\label{fig:2d-cp-rho}
\end{figure}
In Figs. \ref{fig:cp-rho-sqrt2-2}, \ref{fig:cp-rho-sqrt3-2}, and \ref{fig:cp-rho-3-2},
we present $\approxC$ (in this case $M(N) = 48$) together with $\CPMR$ (blue think line),
$\CPupper$ and $\CPlower$ for $\alpha \in (0, \pi)$, and
$\rho = \tfrac{\sqrt{2}}{2}$, $\tfrac{\sqrt{3}}{2}$, and $\tfrac{3}{2}$.
We see that $\underline{C}^{48, \rm P}_{\T}$ (red thin line) indeed lies within the admissible
two-sided bounds. From these figures,
it is obvious that new upper bounds $\CPMR$ are sharper than $\CPupper$ for $\T$ with $\rho \neq 1$.
True values of the constant lie between the bold blue and thin red lines, but closer to the
red one, which practically shows the constant (this follows from the fact
that increasing $M(N)$ does not provide a noticeable change for the line, e.g.,
for $M(N) = 63$
maximal difference with respect to figure does not exceed $1e\minus8$). Also, we note that,
the lower bound $\CPlower$ (black dashed line) is quite efficient, and, moreover, asymptotically exact for
$\alpha \rightarrow \pi$.
Due to \cite{LaugesenSiudeja2010} and \cite{Bandle1980},
we know the improved upper bound $\CPLS$
(cf. \eqref{eq:improved-estimates}) for isosceles triangles. In Fig. \ref{fig:cp-rho-1},
we compare $\approxC$ ($M(N) = 48$) with both upper bounds $\CPMR$ (from the Lemma
\ref{th:lemma-poincare-constants}) and $\CPLS$ (black doted line).
It is easy to see that $\CPLS$ (black dashed line) is
rather accurate and for
$\alpha \rightarrow 0$ and $\alpha \rightarrow \pi$ provide
almost exact estimates. $\CPMR$ (blues thick line) improves $\CPLS$ only for some $\alpha$.
The lower bound $\underline{C}^{48}_{\T}$ (red thin line) indeed converges to
$\CPLS$ as $\T$ degenerates when $\alpha$ tends to $0$.
\subsection{Shape of the minimizer}
Exact constants in (\ref{eq:Comega}) and (\ref{eq:Cgamma}) are generated by the minimal
positive eigenvalues of \eqref{eq:eigenvalue-problem-cp}
and \eqref{eq:eigenvalue-problem-ctr}. This section presents results related to
the respective eigenfunctions. In order to depict all of them in a unified form,
we use the barycentric coordinates $\lambda_i \in (0, 1)$, $i = 1, 2, 3$,
$\Sum_{i = 1}^{3}\lambda_i = 1$.
\input{func-cpgamma}
\input{func-ctrgamma}
Figs. \ref{fig:2d-eigenfunctions-cp} and \ref{fig:2d-eigenfunctions-ctr}
show the eigenfunctions computed for isosceles triangles with different angles $\alpha$
between two legs (zero mean condition is imposed on one of the legs).
The eigenfunctions have been computed in the process of finding $\approxCPT$ and
$\approxCGT$. The eigenfunctions are normalized so that the maximal
value is equal to $1$.
For $\alpha = \tfrac{\pi}{2}$, the exact eigenfunction associated
with the smallest positive eigenvalue
$\lambda^{\mathrm{P}}_{\Gamma} = \big( \tfrac{z_0}{h}\big)^{2}$
is known (see \cite{NazarovRepin2014}):
\begin{equation*}
u^{\rm P}_{\Gamma}
= \cos(\tfrac{\zeta_0 x_1}{h}) + \cos \big(\tfrac{\zeta_0 (x_2 - h)}{h}\big).
\label{eq:u-pt-exact}
\end{equation*}
Here, $\zeta_0$ is the root of the first equation in \eqref{eq:roots}
(see Fig
\ref{fig:2d-exact-up-eigenfunctions-pi-2}).
We can compare $u^{\rm P}_{\Gamma}$ with the approximate eigenfunction $u^{M, \mathrm{P}}_{\Gamma}$
computed by minimization of
$\mathcal{R}^{\mathrm{P}}_{\Gamma} [w]$ (this function is depicted in Fig. \ref{fig:2d-up-eigenfunctions-pi-2}).
Eigenfunctions related to the constant
$\approxCGT$ are presented in Fig. \ref{fig:2d-eigenfunctions-ctr}.
Again, for $\alpha = \tfrac{\pi}{2}$ we know the exact eigenfunction
\begin{equation*}
u^{\rm Tr}_{\Gamma} =
\cos (\hat{\zeta}_0 x_1) \, \cosh \big(\hat{\zeta}_0 (x_2 - h)\big) +
\cosh (\hat{\zeta}_0 \, x_1) \ \cos \big(\hat{\zeta}_0 (x_2 - h)\big),
\label{eq:u-gt-exact}
\end{equation*}
where $\hat{\zeta}_0$ is the root of second equation in \eqref{eq:roots}
(see Fig. \ref{fig:2d-exact-ut-eigenfunctions-pi-2}). This function
minimizes the quotient $\mathcal{R}^{\mathrm{Tr}}_{\Gamma} [w]$ and yields
the smallest positive eigenvalue
$\lambda^{\mathrm{Tr}}_{\Gamma} = \tfrac{\hat{\zeta}_0 \tanh (\hat{\zeta}_0)}{h}$.
It is easy to see that numerical approximation
$\approxCGT$ (for $M(N) = 48$)
practically coincides with the exact function.
\input{switch-func-cp-rho-1}
\begin{table}[!ht]
\centering
\footnotesize
\begin{tabular}{c|c|cc|cc|cc}
$$ & $$
& \multicolumn{2}{c|}{ $\tfrac{\pi}{3} - \varepsilon$ }
& \multicolumn{2}{c|}{ $\tfrac{\pi}{3}$ }
& \multicolumn{2}{c}{ $\tfrac{\pi}{3} + \varepsilon$ }\\
\midrule
$$ & $u^{M}_{\T, i}$ &
$\underline{C}^{48}_{\T, i}$ & $\lambda^{48}_{\T, i}$ &
$\underline{C}^{48}_{\T, i}$ & $\lambda^{48}_{\T, i}$ &
$\underline{C}^{48}_{\T, i}$ & $\lambda^{48}_{\T, i}$ \\
\midrule
\multirow{3}{*}{$\rho = 1$}
& $u^{48}_{\T, 1}$ & 0.2419 & 17.0951 & 0.2387 & 17.5463 & 0.2537 & 15.5404 \\
& $u^{48}_{\T, 2}$ & 0.2229 & 20.1216 & 0.2387 & 17.5463 & 0.2355 & 18.0309 \\
& $u^{48}_{\T, 3}$ & 0.1353 & 54.6024 & 0.1378 & 52.6396 & 0.1422 & 49.4818 \\
\midrule
\multirow{3}{*}{$\rho = \tfrac{\sqrt{2}}{2}$}
& $u^{48}_{\T, 1}$ & 0.23137 & 18.6804 & 0.23671 & 17.8471 & 0.24336 & 16.8850 \\
& $u^{48}_{\T, 2}$ & 0.17082 & 34.2707 & 0.17435 & 32.8970 & 0.17642 & 32.1295 \\
& $u^{48}_{\T, 3}$ & 0.1229 & 66.2058 & 0.12789 & 61.1402 & 0.13298 & 56.5493 \\
\midrule
\multirow{3}{*}{$\rho = \tfrac{3}{2}$}
& $u^{48}_{\T, 1}$ & 0.34714 & 8.2983 & 0.35523 & 7.9247 & 0.3648 & 7.5143 \\
& $u^{48}_{\T, 2}$ & 0.24485 & 16.6801 & 0.24885 & 16.1482 & 0.25125 & 15.8412 \\
& $u^{48}_{\T, 3}$ & 0.18258 & 29.9981 & 0.19084 & 27.4575 & 0.19845 & 25.3921 \\
\end{tabular}
\caption{$\approxC$ and $\lambda^{M}_{\T}$ corresponding to the first three
eigenfunctions in Fig. \ref{fig:u1-u2-u3-rho-1}.}
\label{tab:approx-3-eigenvalues-and-constants}
\end{table}
Typically, the eigenfunctions associated with minimal positive
eigenvalues expose a continuous evolution with respect to $\alpha$.
However, this is not true for the quotient $\mathcal{R}_{\T}[w]$,
where the minimizer radically changes the profile.
Fig. \ref{fig:cp-rho-1} indicates a possibility of such a rapid change
at $\alpha = \tfrac{\pi}{3}$, where the curve (related to $\underline{C}^{48}_{\T}$)
obviously becomes
non-smooth. This happens because an equilateral triangle has double eigenvalue, therefore
the minimizer of $\mathcal{R}_{\T}[w]$ over
$V^N_1$ changes its profile.
Figs. \ref{fig:u1-rho-1-pi-3-minus-eps}--\ref{fig:u3-rho-1-pi-3-plus-eps} show
three eigenfunctions $u^{48}_{\T, 1}$, $u^{48}_{\T, 2}$,
and $u^{48}_{\T, 3}$ corresponding to three minimal eigenvalues $\lambda^{48}_{\T, 1}$,
$\lambda^{48}_{\T, 2}$, and $\lambda^{48}_{\T, 3}$.
All functions are computed
for isosceles triangles and are sorted in accordance with increasing
values of the respective eigenvalues.
It is easy to see that at $\alpha=\tfrac{\pi}{3}$ the first and the second eigenfunctions
swap places.
Table \ref{tab:approx-3-eigenvalues-and-constants} presents the corresponding
results in the digital form.
It is worth noting that for equilateral triangles two minimal eigenfunctions
are known (see \cite{McCartin2002}):
\begin{alignat*}{2}
u_1 & = \cos \Big(\tfrac{2\,\pi}{3} (2\, x_1 - 1)\Big)
- \cos \Big(\tfrac{2\,\pi}{\sqrt{3}} x_2 \Big)
\cos \Big(\tfrac{\pi}{3} (2\,x_1 - 1)\Big), \\
u_2 & = \sin \Big(\tfrac{2\,\pi}{3} (2\, x_1 - 1)\Big)
+ \cos \Big(\tfrac{2\,\pi}{\sqrt{3}} x_2 \Big)
\sin \Big(\tfrac{\pi}{3} (2\,x_1 - 1)\Big).
\end{alignat*}
These functions practically coincide with the functions $u^{48}_{\T, 1}$ and $u^{48}_{\T, 2}$
presented in Fig. \ref{fig:u1-rho-1-pi-3}.
Finally, we note that this phenomenon (change of the minimal eigenfunction) does not
appear for $\rho = \tfrac{\sqrt{2}}{2}$ or $\rho=\tfrac{3}{2}$.
The eigenvalues as well
as the constants corresponding to the eigenfunctions presented in Fig.
\ref{fig:u1-u2-u3-rho-1}
are shown in the Table \ref{tab:approx-3-eigenvalues-and-constants}.
\section{Two-sided bounds of $\CPGamma$ and $\CtrGamma$ for tetrahedrons}
\label{eq:numerial-tests-3d}
\begin{figure}[ht!]
\centering
\adjustbox{valign=b}{
\begin{minipage}[b]{0.5\linewidth}
\input{3d-simplex-new}
\caption{Simplex in $\Rthree$.}
\label{eq:3d-simplex}
\end{minipage}}
\adjustbox{valign=b}{
\begin{minipage}[b]{0.45\linewidth}
\input{3d-simplex-d-coordinates}
\caption{Coordinate of the vertex $D$.}
\label{eq:3d-simplex-d-coordinate}
\end{minipage}}
\end{figure}
We orient the coordinates as it is shown in Fig. \ref{eq:3d-simplex}
and define a non-degenerate simplex in $\Rthree$ with vertexes \linebreak
$A = (0, 0, 0)$, $B = (h_1, 0, 0)$, $C = (0, 0, h_3)$, and
$D = (h_2 \, \sin\theta\, \cos\alpha, h_2 \, \sin\theta\, \sin\alpha, h_2\, \cos\alpha)$,
where
$h_1$ and $h_3$ are the scaling parameters along axis
$O_{x_1}$ and $O_{x_3}$, respectively, $AD = h_2$,
$\alpha$ is a polar angle, and $\theta$ is an azimuthal angle
(see Fig. \ref{eq:3d-simplex-d-coordinate}).
Let $\Gamma$ be defined by vertexes $A$, $B$, and $C$.
To the best of our knowledge, exact values of constants in Poincar\'{e}-type
inequalities for simplexes in $\Rthree$ are unknown. Therefore, we first consider
four basic (reference) tetrahedrons with
$h_2 = 1$, $\hat{\theta} = \tfrac{\pi}{2}$, and
$\hat{\alpha}_1 = \tfrac{\pi}{4}$,
$\hat{\alpha}_2 = \tfrac{\pi}{3}$,
$\hat{\alpha}_3 = \tfrac{\pi}{2}$, and
$\hat{\alpha}_4 = \tfrac{2\pi}{3}$.
The respective constants are found numerically with high accuracy (see Table
\ref{tab:const-convergence-from-basis-G-leg-alpha},
which shows convergence of the constants
with respect to increasing $M(N)$).
Henceforth, $\Tref_{\hat{\theta}, \hat{\alpha}}$ denotes a reference tetrahedron,
where $\hat{\theta}$ and $\hat{\alpha}$ are certain fixed angles.
By $\mathcal{F}_{\hat{\theta}, \hat{\alpha}}$ we denote the respective mapping
$\mathcal{F}_{\hat{\theta}, \hat{\alpha}}:
\Tref_{\hat{\theta}, \hat{\alpha}} \rightarrow \T$.
\begin{table}[!t]
\centering
\footnotesize
\begin{tabular}{c|cc|cc|cc|cc}
\multicolumn{1}{c|}{$ $}
& \multicolumn{2}{c|}{ $\hat{\alpha} = \tfrac{\pi}{4}$}
& \multicolumn{2}{c|}{ $\hat{\alpha} = \tfrac{\pi}{3}$}
& \multicolumn{2}{c|}{ $\hat{\alpha} = \tfrac{\pi}{2}$}
& \multicolumn{2}{c}{ $\hat{\alpha} = \tfrac{2\pi}{3}$}\\
\midrule
\multicolumn{1}{c|}{$ $} &
\multicolumn{2}{c|}{
\begin{tikzpicture}[line join = round, line cap = round]
\coordinate [label=left:] (A) at (0,0,0);
\coordinate [label=above right:] (B) at (1.2,0,0);
\coordinate [label=below:] (C) at (0,0,1.2);
\coordinate [label=above right:] (D) at ( 0.848, 0.848, 0);
\draw[->] (1.2, 0, 0) -- (1.6, 0, 0) node[below] {$\hat{x}_1$};
\draw[->] (0, 0, 0) -- (0, 1.4, 0) node[above] {$\hat{x}_2$};
\draw[->] (0, 0, 1.2) -- (0, 0, 1.6) node[right] {$\hat{x}_3$};
\draw (2.0, 1.05) node[anchor=north east] {$\Tref_{\rfrac{\pi}{4}}$};
\draw (1.0, -0.2) node[anchor=north east] {$\widehat{\Gamma}$};
\draw[-, fill=black!10, opacity=.5, dashed] (A)--(B)--(C)--(A);
\draw[-, opacity=.5] (B)--(C);
\draw[-, opacity=.5] (C)--(D);
\draw[-, opacity=.5] (B)--(D);
\draw (0.55,0.0) .. controls (0.5, 0.3) and (0.4, 0.3) .. node[right] {$\hat{\alpha}$} (0.3,0.3);
\draw[dashed] (A)--(D);
\end{tikzpicture}
}
& \multicolumn{2}{c|}{
\begin{tikzpicture}[line join = round, line cap = round]
\coordinate [label=left:] (A) at (0,0,0);
\coordinate [label=above right:] (B) at (1.2,0,0);
\coordinate [label=below:] (C) at (0,0,1.2);
\coordinate [label=above right:] (D) at ( 0.6, 1.0392, 0);
\draw[->] (1.2, 0, 0) -- (1.6, 0, 0) node[below] {$\hat{x}_1$};
\draw[->] (0, 0, 0) -- (0, 1.4, 0) node[above] {$\hat{x}_2$};
\draw[->] (0, 0, 1.2) -- (0, 0, 1.6) node[right] {$\hat{x}_3$};
\draw (2.0, 1.05) node[anchor=north east] {$\Tref_{\rfrac{\pi}{3}}$};
\draw (1.0, -0.2) node[anchor=north east] {$\widehat{\Gamma}$};
\draw[-, fill=black!10, opacity=.5, dashed] (A)--(B)--(C)--(A);
\draw[-, opacity=.5] (B)--(C);
\draw[-, opacity=.5] (C)--(D);
\draw[-, opacity=.5] (B)--(D);
\draw (0.55,0.0) .. controls (0.5, 0.3) and (0.4, 0.3) .. node[right] {$\hat{\alpha}$} (0.27,0.4);
\draw[dashed] (A)--(D);
\end{tikzpicture}
}
& \multicolumn{2}{c|}{
\begin{tikzpicture}[line join = round, line cap = round]
\coordinate [label=left:] (A) at (0,0,0);
\coordinate [label=above right:] (B) at (1.2,0,0);
\coordinate [label=below:] (C) at (0,0,1.2);
\coordinate [label=above right:] (D) at ( 0.0, 1.2, 0);
\draw[->] (1.2, 0, 0) -- (1.6, 0, 0) node[below] {$\hat{x}_1$};
\draw[->] (0, 1.2, 0) -- (0, 1.4, 0) node[above] {$\hat{x}_2$};
\draw[->] (0, 0, 1.2) -- (0, 0, 1.6) node[right] {$\hat{x}_3$};
\draw (2.0, 1.05) node[anchor=north east] {$\Tref_{\rfrac{\pi}{2}}$};
\draw (1.0, -0.2) node[anchor=north east] {$\widehat{\Gamma}$};
\draw[-, fill=black!10, opacity=.5, dashed] (A)--(B)--(C)--(A);
\draw[-, opacity=.5] (B)--(C);
\draw[-, opacity=.5] (C)--(D);
\draw[-, opacity=.5] (B)--(D);
\draw (0.3,0.0) .. controls (0.2, 0.3) and (0.1, 0.3) .. node[right] {$\hat{\alpha}$} (0.0,0.3);
\draw[dashed] (A)--(D);
\draw[dashed] (0, 0, 0.5)--(0, 0.5, 0.5)--(0, 0.5, 0);
\draw (-0.1, 0.3) node[anchor=north east] {$\hat{\theta}$};
\end{tikzpicture}
}
& \multicolumn{2}{c}{
\begin{tikzpicture}[line join = round, line cap = round]
\coordinate [label=left:] (A) at (0,0,0);
\coordinate [label=above right:] (B) at (1.2,0,0);
\coordinate [label=below:] (C) at (0,0,1.2);
\coordinate [label=above right:] (D) at ( -0.6, 1.0392, 0);
\draw[->] (1.2, 0, 0) -- (1.6, 0, 0) node[below] {$\hat{x}_1$};
\draw[->] (0, 0, 0) -- (0, 1.4, 0) node[above] {$\hat{x}_2$};
\draw[->] (0, 0, 1.2) -- (0, 0, 1.6) node[right] {$\hat{x}_3$};
\draw (2.0, 1.05) node[anchor=north east] {$\Tref_{\rfrac{2\pi}{3}}$};
\draw (1.0, -0.2) node[anchor=north east] {$\widehat{\Gamma}$};
\draw[-, fill=black!10, opacity=.5, dashed] (A)--(B)--(C)--(A);
\draw[-, opacity=.5] (B)--(C);
\draw[-, opacity=.5] (C)--(D);
\draw[-, opacity=.5] (B)--(D);
\draw (0.3,0.0) .. controls (0.2, 0.3) and (0.1, 0.3) .. node[right] {$\hat{\alpha}$} (-0.1,0.2);
\draw[dashed] (A)--(D);
\end{tikzpicture}
}\\
\midrule
$M(N)$ & $\CPTapproxhatalphahat$ & $\CGTapproxhatalphahat$
& $\CPTapproxhatalphahat$ & $\CGTapproxhatalphahat$
& $\CPTapproxhatalphahat$ & $\CGTapproxhatalphahat$
& $\CPTapproxhatalphahat$ & $\CGTapproxhatalphahat$\\
\midrule
7 & 0.32431 & 0.760099 & 0.325985 & 0.654654 & 0.360532 & 0.654654 & 0.4152099 & 0.686161 \\
26 & 0.338539 & 0.829445 & 0.340267 & 0.761278 & 0.373669 & 0.751615 & 0.4274757 & 0.863324 \\
63 & 0.341122 & 0.831325 & 0.342556 & 0.762901 & 0.375590 & 0.751994 & 0.4286444 & 0.864595 \\
124 & 0.341147 & 0.831335 & 0.342589 & 0.762905 & 0.375603 & 0.751999 & 0.4286652 & 0.864630 \\
215 & {\bf 0.341147} & {\bf 0.831335} & {\bf 0.342589} & {\bf 0.762905} & {\bf 0.375603} & {\bf 0.751999} & {\bf 0.4286652} & {\bf 0.864630} \\
\end{tabular}
\caption{$\CPTapproxhatalphahat$ and $\CGTapproxhatalphahat$ with
respect to $M(N)$ for $\Tref_{\hat{\theta}, \hat{\alpha}}$ with
$\rho = 1$, $\hat{\theta} = \tfrac{\pi}{2}$, and different
$\hat{\alpha}$.}
\label{tab:const-convergence-from-basis-G-leg-alpha}
\end{table}
Then, for an arbitrary tetrahedron $\T$, we have
\begin{alignat}{2}
\|v\|_{\T} \, & \leq\, \CPT \, h_2 \,\|\nabla v\|_{\T} \quad \mbox{and} \quad
\|v\|_\Gamma \, \leq\, \CGT \, h_2^{\rfrac{1}{2}} \,\|\nabla v\|_{\T}
\label{eq:general-poincare-type-inequalities-for-tetrahedron}
\end{alignat}
with approximate bounds
\begin{equation}
\CPT \lessapprox \CPtetr =
\min_{\hat{\alpha} = \{\rfrac{\pi}{4}, \rfrac{\pi}{3}, \rfrac{\pi}{2}, \rfrac{2\pi}{3}\}}
\Big\{ \cpalphahat \, \CPThatalphahat \Big\}
\end{equation}
and
\begin{equation}
\CGT \lessapprox \CGtetr =
\min_{\hat{\alpha} =
\{\rfrac{\pi}{4}, \rfrac{\pi}{3}, \rfrac{\pi}{2}, \rfrac{2\pi}{3}\}}
\Big\{ \cgalphahat \, \CGThatalphahat \Big\},
\label{eq:}
\end{equation}
where $\CPThatalphahat$ and $\CGThatalphahat $ are the constants related to four reference tetrahedrons from Table \ref{tab:const-convergence-from-basis-G-leg-alpha}, and
$\cpalphahat$ and $\cgalphahat$ (see \eqref{eq:ratio-3d}) are
generated by the mapping
$\mathcal{F}_{\rfrac{\pi}{2}, \hat{\alpha}}$:
$\Tref_{\rfrac{\pi}{2}, \hat{\alpha}} \rightarrow \T$. Here, the reference tetrahedrons are
defined based on $\hat{A} = (0, 0, 0)$, $\hat{B} = (1, 0, 0)$,
$\hat{C} = (0, 0, 1)$,
$\hat{D} = (\cos\hat{\alpha}, \sin\hat{\alpha}, 0)$ with $\hat{\alpha} =
\{ \tfrac{\pi}{4},\tfrac{\pi}{3}, \tfrac{\pi}{2}, \tfrac{2\pi}{3}\}$,
and
$\mathcal{F}_{\rfrac{\pi}{2}, \hat{\alpha}}(\hat{x})$ is presented by the relation
\begin{equation*}
x = \mathcal{F}_{\rfrac{\pi}{2}, \hat{\alpha}}(\hat{x}) =
B_{\rfrac{\pi}{2}, \hat{\alpha}} \hat{x}, \quad
B_{\rfrac{\pi}{2}, \hat{\alpha}} = \{ b_{ij} \}_{i, j = 1, 2, 3} = h_2
\begin{pmatrix}
\: \tfrac{h_1}{h_2} &
\; \tfrac{\nu (\rho, \alpha)}{\sin \hat{\alpha}} &
\; 0 \\[0.3em]
\: 0 &
\; \, \tfrac{\sin\alpha \sin\theta}{\sin \hat{\alpha}} \: &
\: 0 \\[0.3em]
\: 0 &
\; \, \tfrac{\cos\theta}{\sin \hat{\alpha}} \: &
\: \tfrac{h_3}{h_2} \\
\end{pmatrix},
\end{equation*}
where $\nu(\rho, \alpha) = \cos\alpha \sin\theta - \tfrac{h_1}{h_2} \cos \hat{\alpha}$,
$\mathrm{det} \, B_{\rfrac{\pi}{2}, \hat{\alpha}} = h_1 \, h_2\, h_3 \,
\tfrac{\sin\alpha \sin\theta}{\sin \hat{\alpha}}$.
By analogy with the two-dimensional case (see \eqref{eq:grad-hatv-lower-estimate-1}),
$\cpalphahat$ and $\cgalphahat$ depend on the maximum eigenvalue of the matrix
\begin{alignat*}{2}
A_{\rfrac{\pi}{2}, \hat{\alpha}} & := h_1^2
\begin{pmatrix}
\: b_{11}^2 + b^2_{12} \quad & b_{12} b_{22} \qquad & b_{12} b_{32} \\[0.5em]
b_{12} b_{22} & b^2_{22} & b_{22} b_{32} \\[0.5em]
b_{12} b_{32} & b_{22} b_{32} & b_{33}^2 + b^2_{32} \\
\end{pmatrix}
.
\end{alignat*}
The maximal eigenvalue of the matrix $A_{\rfrac{\pi}{2}, \hat{\alpha}}$ is defined by
the relation $\lambda_{\rm max} (A_{\rfrac{\pi}{2}, \hat{\alpha}})
= h_2^2 \, \mu_{\alpha, \theta, \hat{\alpha}}$ with
\begin{equation*}
\mu_{\alpha, \theta, \hat{\alpha}}
= \Big(\mathcal{E}_5^{\rfrac{1}{3}}
- \mathcal{E}_3 \mathcal{E}_5^{-\rfrac{1}{3}}
+ \tfrac{1}{3} \mathcal{E}_1 \Big),
\end{equation*}
where
\begin{alignat*}{2}
\mathcal{E}_1 & = b_{11}^2 + b_{12}^2 + b_{22}^2 + b_{32}^2 + b_{33}^2, \\
\mathcal{E}_2 & = b_{11}^2 \, b_{22}^2 + b_{11}^2 \, b_{32}^2 + b_{11}^2 \, b_{33}^2 + b_{12}^2 \, b_{33}^2 + b_{22}^2\, b_{33}^2, \\
\mathcal{E}_3 & = \tfrac{\mathcal{E}_2}{3} - \big(\tfrac{\mathcal{E}_1}{3})^2, \\
\mathcal{E}_4 & = \big(\tfrac{\mathcal{E}_1}{3})^3 - \tfrac{\mathcal{E}_1 \, \mathcal{E}_2}{3} + \tfrac{1}{2} \, b_{11}^2 \, b_{22}^2 \, b_{33}^2, \\
\mathcal{E}_5 & = \mathcal{E}_4 + (\mathcal{E}_3^3 + \mathcal{E}_4^2)^{\rfrac{1}{2}}.
\end{alignat*}
Therefore, $\cpalphahat$ and $\cgalphahat$ in
\eqref{eq:general-poincare-type-inequalities-for-tetrahedron} are as follows:
\begin{equation}
\cpalphahat = \mu^{\rfrac{1}{2}}_{\pi/2, \hat{\alpha}}, \quad
\cgalphahat =
\Big( \tfrac{\sin \hat{\alpha}}{\rho \sin\alpha \sin\theta}\Big)^{\rfrac{1}{2}} \,\cpalphahat.
\label{eq:ratio-3d}
\end{equation}
Lower bounds of the constants $\CPGamma$ and
$\CtrGamma$ are computed by minimization of
$\mathcal{R}^{\mathrm{P}}_{\Gamma} [w]$ and $\mathcal{R}^{\mathrm{Tr}}_{\Gamma} [w]$
over the set $V^N_3 \subset \H{1}(\T)$, where
\begin{equation*}
V^{N}_3 := \Big\{\:\varphi_{ijk} = x^{i} y^{j} z^k, \quad i, j, k = 0, \ldots, N,
\;\; (i, j, k) \neq (0, 0, 0) \: \Big\}
\end{equation*}
and $\mathrm{dim} V^{N}_3 = M(N) := (N + 1)^3 - 1$.
The respective results are presented
in Tables \ref{tab:cp-3d-approx-exact-constants} and \ref{tab:cg-3d-approx-exact-constants}
for $\T$ with $h_1 = 1$, $h_3 = 1$, and $\rho = 1$.
We note that exact values of constants are probably closer to the numbers presented in
left-hand side columns. For $\theta = \rfrac{\pi}{2}$, we also present
estimates of $\approxCPT$ and $\approxCGT$ (red lines) graphically in
Fig. \ref{fig:3d-cpt-cgt-constants-with-estimates-4-refs}.
\begin{table}[!ht]
\centering
\footnotesize
\begin{tabular}{c|cc|cc|cc|cc}
$$
& \multicolumn{2}{c|}{ $\alpha = \tfrac{\pi}{6}$}
& \multicolumn{2}{c|}{ $\alpha = \tfrac{\pi}{4}$}
& \multicolumn{2}{c|}{ $\alpha = \tfrac{\pi}{3}$}
& \multicolumn{2}{c}{ $\alpha = \tfrac{\pi}{2}$} \\
\midrule
$\theta$
& $\underline{C}^{M, \mathrm{P}}_{\Gamma}$ & $\CPtetr$ &
$\underline{C}^{M, \mathrm{P}}_{\Gamma}$ & $\CPtetr$ &
$\underline{C}^{M, \mathrm{P}}_{\Gamma}$ & $\CPtetr$ &
$\underline{C}^{M, \mathrm{P}}_{\Gamma}$ & $\CPtetr$ \\
\midrule
$\pi/6$ & 0.23883 & 0.49035& 0.24621 & 0.49841& 0.25870 & 0.51054& 0.29484 & 0.51308\\
$\pi/4$ & 0.23883 & 0.45388& 0.24621 & 0.46173& 0.25870 & 0.47683& 0.29484 & 0.49075\\
$\pi/3$ & 0.29666 & 0.41958& 0.31194 & 0.42259& 0.33489 & 0.43724& 0.38976 & 0.46002\\
$\pi/2$ & 0.34302 & 0.35667& 0.34112 & 0.34115& 0.34256 & 0.34259& 0.37559 & 0.37560\\
$2\pi/3$ & 0.40428 & 0.41958& 0.40562 & 0.42259& 0.40927 & 0.43724& 0.42867 & 0.46002\\
$3\pi/4$ & 0.42890 & 0.45388& 0.43110 & 0.46173& 0.43505 & 0.47683& 0.45017 & 0.49075\\
$5\pi/6$ & 0.44964 & 0.49035& 0.45193 & 0.49841& 0.45539 & 0.51054& 0.46607 & 0.51308\\
\midrule
$$
& \multicolumn{2}{c|}{ $\alpha = \tfrac{\pi}{2}$}
& \multicolumn{2}{c|}{ $\alpha = \tfrac{2\pi}{3}$}
& \multicolumn{2}{c|}{ $\alpha = \tfrac{3\pi}{4}$}
& \multicolumn{2}{c}{ $\alpha = \tfrac{5\pi}{6}$} \\
\midrule
$\theta$
& $\underline{C}^{M, \mathrm{P}}_{\Gamma}$ & $\CPtetr$ &
$\underline{C}^{M, \mathrm{P}}_{\Gamma}$ & $\CPtetr$ &
$\underline{C}^{M, \mathrm{P}}_{\Gamma}$ & $\CPtetr$ &
$\underline{C}^{M, \mathrm{P}}_{\Gamma}$ & $\CPtetr$ \\
\midrule
$\pi/6$ & 0.29484 & 0.51308& 0.33069 & 0.51792& 0.34468 & 0.52253& 0.35499 & 0.52694\\
$\pi/4$ & 0.29484 & 0.49075& 0.33069 & 0.50261& 0.34468 & 0.51308& 0.35499 & 0.52253\\
$\pi/3$ & 0.38976 & 0.46002& 0.43880 & 0.48413& 0.45742 & 0.50261& 0.47106 & 0.51792\\
$\pi/2$ & 0.37559 & 0.37560& 0.42865 & 0.42867& 0.45017 & 0.45731& 0.46607 & 0.47811\\
$2\pi/3$ & 0.42867 & 0.46002& 0.45997 & 0.48413& 0.47457 & 0.50261& 0.48598 & 0.51792\\
$3\pi/4$ & 0.45017 & 0.49075& 0.47204 & 0.50261& 0.48239 & 0.51308& 0.49064 & 0.52253\\
$5\pi/6$ & 0.46607 & 0.51308& 0.47972 & 0.51792& 0.48607 & 0.52253& 0.49115 & 0.52694\\
\end{tabular}
\\[5pt]
\caption{$\underline{C}^{M, \mathrm{P}}_{\Gamma}$ ($M(N)$ = 124) and $\CPtetr$.}
\label{tab:cp-3d-approx-exact-constants}
\end{table}
\input{cpgamma-ctrgamma-with-estimates-3d}
\begin{table}[!ht]
\centering
\footnotesize
\begin{tabular}{c|cc|cc|cc|cc}
$$
& \multicolumn{2}{c|}{ $\alpha = \tfrac{\pi}{6}$}
& \multicolumn{2}{c|}{ $\alpha = \tfrac{\pi}{4}$}
& \multicolumn{2}{c|}{ $\alpha = \tfrac{\pi}{3}$}
& \multicolumn{2}{c}{ $\alpha = \tfrac{\pi}{2}$}\\
\midrule
$\theta$ &
$\underline{C}^{M, \mathrm{Tr}}_{\Gamma}$ & $\CGtetr$ &
$\underline{C}^{M, \mathrm{Tr}}_{\Gamma}$ & $\CGtetr$ &
$\underline{C}^{M, \mathrm{Tr}}_{\Gamma}$ & $\CGtetr$ &
$\underline{C}^{M, \mathrm{Tr}}_{\Gamma}$ & $\CGtetr$ \\
\midrule
$\pi/6$ & 1.09760 & 3.78259& 0.96245 & 2.71866& 0.91255 & 2.27382& 0.93123 & 2.05449\\
$\pi/4$ & 1.09760 & 2.43897& 0.96245 & 1.78094& 0.91255 & 1.50166& 0.93123 & 1.38951\\
$\pi/3$ & 0.89122 & 1.74467& 0.79146 & 1.31130& 0.75950 & 1.12431& 0.78904 & 1.06349\\
$\pi/2$ & 0.98017 & 1.22920& 0.83132 & 0.83133& 0.76290 & 0.76291& 0.75199 & 0.75200\\
$2\pi/3$ & 1.17698 & 1.74467& 0.99473 & 1.31130& 0.90578 & 1.12431& 0.86463 & 1.06349\\
$3\pi/4$ & 1.35195 & 2.43897& 1.14144 & 1.78094& 1.03737 & 1.50166& 0.98220 & 1.38951\\
$5\pi/6$ & 1.65317 & 3.78259& 1.39424 & 2.71866& 1.26490 & 2.27382& 1.19017 & 2.05449\\
\midrule
$$
& \multicolumn{2}{c|}{ $\alpha = \tfrac{\pi}{2}$}
& \multicolumn{2}{c|}{ $\alpha = \tfrac{2\pi}{3}$}
& \multicolumn{2}{c|}{ $\alpha = \tfrac{3\pi}{4}$}
& \multicolumn{2}{c}{ $\alpha = \tfrac{5\pi}{6}$} \\
\midrule
$\theta$ &
$\underline{C}^{M, \mathrm{Tr}}_{\Gamma}$ & $\CGtetr$ &
$\underline{C}^{M, \mathrm{Tr}}_{\Gamma}$ & $\CGtetr$ &
$\underline{C}^{M, \mathrm{Tr}}_{\Gamma}$ & $\CGtetr$ &
$\underline{C}^{M, \mathrm{Tr}}_{\Gamma}$ & $\CGtetr$ \\
\midrule
$\pi/6$ & 0.93123 & 2.05449& 1.07244 & 2.39471& 1.21573 & 2.95902& 1.47044 & 4.21999\\
$\pi/4$ & 0.93123 & 1.38951& 1.07244 & 1.64324& 1.21573 & 2.01841& 1.47044 & 2.80588\\
$\pi/3$ & 0.78904 & 1.06349& 0.91773 & 1.27423& 1.04309 & 1.50833& 1.26357 & 2.11790\\
$\pi/2$ & 0.75199 & 0.75200& 0.86459 & 0.86463& 0.98220 & 1.12971& 1.19017 & 1.67033\\
$2\pi/3$ & 0.86463 & 1.06349& 0.96174 & 1.27423& 1.08134 & 1.50833& 1.30191 & 2.11790\\
$3\pi/4$ & 0.98220 & 1.38951& 1.07921 & 1.64324& 1.20686 & 2.01841& 1.44721 & 2.80588\\
$5\pi/6$ & 1.19017 & 2.05449& 1.29582 & 2.39471& 1.44268 & 2.95902& 1.72383 & 4.21999\\
\end{tabular}
\\[5pt]
\caption{$\underline{C}^{M, \mathrm{Tr}}_{\Gamma}$ ($M(N)$ = 124)
and $\CGtetr$ for different $\theta, \alpha \in (0, \pi)$.}
\label{tab:cg-3d-approx-exact-constants}
\end{table}
\section{Example}
\label{sec:example}
Constants in the Friedrichs', Poincar\'{e}, and other functional inequalities arise in various problems of numerical analysis, where we need to
know values of the respective constants associated with particular domains.
Constants in projection type estimates arise in a priori analysis
(see, e.g., \cite{Braess2001, Ciarlet1978, Mikhlin1986}). Constants in Clement's
interpolation inequalities are important for residual type a posteriori estimates
(see, e.g., \cite{AinsworthOden2000, Verfurth1996}, and \cite{CarstensenFunken2000},
where these constants have been evaluated).
Concerning constants in the trace inequalities associated
with polygonal domain, we mention the paper \cite{CarstensenSauter2004}.
Constants in functional (embedding) inequalities arise in a posteriori error estimates
of the functional type (error majorants). The details concerning last application can be
found \cite{RepinDeGruyter2008, LangerRepinWolfmayr2014,
MatculevichNeitaanmakiRepin2015,ReVarInq2010,RepinBoundaryMeanTrace2015, Repin2000,
RepinXanthis1996} and other references cited therein.
Below, we deduce an advanced version of an error majorant,
which uses constants in Poincar\'{e}-type inequalities
for functions with zero mean traces on inter-element boundaries.
This is done in order to maximally extend the space of admissible fluxes.
However, first, we shall discuss the reasons that invoke
Poincar\'{e}-type constants in a posteriori estimates.
Let $u$ denote the exact solution of an elliptic boundary value problem generated by the pair
of conjugate operators $\rm grad$ and $-\dvrg$
(e.g., the problem (\ref{eq:reactdiff1})--(\ref{eq:reactdiff4}) considered below)
and $v$ be a function in the energy space satisfying the prescribed (Dirichlet)
boundary conditions. Typically, the error $e := u - v$ is measured in terms of the energy
norm $\|\nabla\,e\|$ (or some other equivalent norm), whose square is bounded
from above by the quantities
\begin{equation*}
\IntO R (v, \dvrg q) \,e\, \mathrm{\:d}x,\quad
\IntO D (\nabla v, q) \cdot \nabla\,e\, \mathrm{\:d}x,\;
{\rm and}\;
\Int_{\Gamma_N} R_{\Gamma_N}(v, q\cdot n)\,e\,\mathrm{\:d}s,
\end{equation*}
where $\Omega$ Lipschitz bounded domain, $\Gamma_N$ is the Neumann part of the boundary $\partial \Omega$ with the
outward unit normal vector $n$, and $q$ is an approximation of the dual variable (flux).
The terms
$R$, $D$, and $R_{\Gamma_N}$ represent residuals of the differential (balance)
equation, constitutive (duality) relation, and Neumann boundary condition, respectively.
Since $v$ and $q$ are known from a numerical solution, fully computable estimates
can be obtained if these integrals are estimated by the H\"{o}lder, Friedrichs, and trace
inequalities (which involve the corresponding constants). However, for $\Omega$ with
piecewise smooth (e.g., polynomial) boundaries these
constants may be unknown. A way to avoid these difficulties is suggested by
modifications of the estimates using ideas of domain decomposition. Assume that $\Omega$
is a polygonal (polyhedral) domain decomposed into a collection of non-overlapping convex
polygonal sub-domains $\Omega_i$, i.e.,
\begin{equation*}
\overline{\Omega} :=
\bigcup\limits_{ \Omega_i \in \, \mathcal{O}_\Omega}
{\overline{\Omega}}_i,
\quad
\mathcal{O}_\Omega :=
\Big\{ \; \Omega_i \in \Omega \; \big| \;
{\Omega}_{i'} \, \cap \, {\Omega}_{i''} = \emptyset, \;
i' \neq i'', \;
i = 1, \ldots, N \; \Big \}.
\end{equation*}
We denote the set of all edges (faces) by ${\mathcal G}$ and the set of all interior
faces by ${\mathcal G}_{\rm int}$ (i.e., $\Gamma_{ij} \in {\mathcal G}_{\rm int}$,
if $\Gamma_{ij} = \overline{\Omega}_i \, \cap \, \overline{\Omega}_j$). Analogously,
${\mathcal G}_{N}$ denotes the set of edges on $\Gamma_N$. The latter set is decomposed
into $\Gamma_{N_k}:=\Gamma_N\cap\partial\Omega_k$ (the number of faces that belongs to
${\mathcal G}_{N}$ is $K_N$).
Now, the integrals associated with $R $ and $R_{\Gamma_N}$ can be replaced by sums of
local quantities
\begin{equation*}
\Sum_{\Omega_i \in {\mathcal G}} \Int_{\Omega_i}R_{\Omega} (v,\dvrg q)\,e \, \mathrm{\:d}x,\quad{\rm and}\quad
\Sum_{\Gamma_{N_k} \in {\mathcal G}_{N}} \Int_{\Gamma_{N_k}} R_{\Gamma_{N}} (v,q\cdot n)\,e\,\mathrm{\:d}s.
\end{equation*}
If the residuals satisfy the conditions
\begin{equation*}
\Int_{\Omega_i} R_{\Omega_i}(v,\, \dvrg q) \mathrm{\:d}x = 0
\quad \forall i = 1, \ldots, N, \quad
\end{equation*}
and
\begin{equation*}
\Int_{\Gamma_{N_k}} R_{\Gamma_{N}}(v,\, q\cdot n) \mathrm{\:d}s = 0,
\quad \forall k = 1, \ldots, K_N,
\end{equation*}
then
\begin{equation}
\Int_{\Omega_i} R_{\Omega}(v, \dvrg q)\,e\, \mathrm{\:d}x
\leq C^{{\mathrm P}}_{\Omega_i} \|R_{\Omega_i}(v,\dvrg q)\|_{\Omega_i}\,
\|\nabla\,e\|_{\Omega_i} \quad
\label{eq:estOmega}
\end{equation}
and
\begin{equation}
\Int_{\Gamma_{N_k}} R_{\Gamma_{N}}(v, q\cdot n)\,e\,\mathrm{\:d}s
\leq C^{{\mathrm Tr}}_{\Gamma_{N_k}} \|R_{\Gamma_{N}}(v,q\cdot n)\|_{\Gamma_{N_k}}
\|\nabla\,e\|_{\Omega_k}.
\label{eq:estGammaN}
\end{equation}
Hence, we can deduce a computable upper bound of the error that contains local constants
$C^{{\mathrm P}}_{\Omega_i}$ and $C^{{\mathrm Tr}}_{\Gamma_{N_k}}$ for simple subdomains
(e.g., triangles or tetrahedrons) instead of the global constants associated with
$\Omega$.
The constant $\CPoincare$ may arise if, e.g., nonconforming approximations are used.
For example, if $v$ does not exactly satisfy the Dirichlet boundary condition on
$\Gamma_{D_k}$, then in the process of estimation it may be necessary to evaluate terms of
the type
\begin{equation*}
\Int_{\Gamma_{D_k}} G_{D} (v)\,e\,\mathrm{\:d}s,\quad k = 1, \ldots, K_D,
\end{equation*}
where $\Gamma_{D_k}$ is a part of $\Gamma_D$ associated with a certain $\Omega_k$, and
$G_D(v)$ is a residual generated by inexact satisfaction of the boundary condition. If
we impose the requirement that the Dirichlet boundary condition is satisfied in a
weak sense, i.e.,
$
\mean{G_{D}(v)}_{\Gamma_{D_k}} = 0,
$
then each boundary integral can be estimated as follows:
\begin{equation}
\Int_{\Gamma_{D_k}} G_{D}(v)\,e\,\mathrm{\:d}s
\leq C^{{\mathrm P}}_{\Gamma_{D_k}} \|G_D(v)\|_{\Gamma_{D_k}}\|\nabla\,e\|_{\Omega_k}.
\label{eq:estGammaD}
\end{equation}
After summing up (\ref{eq:estOmega}), (\ref{eq:estGammaN}), and (\ref{eq:estGammaD}), we
obtain a product of weighted norms of localized residuals (which are known) and
$\|\nabla e\|_\Omega$. Since the sum is bounded from below by the squared energy norm,
we arrive at computable error majorant.
Now, we discuss elaborately these questions within the paradigm of the
following
boundary value problem: find $u$ such that
\begin{eqnarray}
& - \dvrg {p} + \varrho^2 u = f, \quad & {\rm in} \; \Omega, \quad
\label{eq:reactdiff1}\\
& p = A \nabla u, \quad & {\rm in} \; \Omega, \quad
\label{eq:reactdiff2}\\
& u = u_D, \quad & {\rm on} \; \Gamma_D,\
\\
& A \nabla u \cdot { n} = F \quad & {\rm on} \; \Gamma_N.
\label{eq:reactdiff4}
\end{eqnarray}
Here $f \in \L{2}(\Omega)$, $F \in \L{2}(\Gamma_N)$, $u_D \in \H{1}(\Omega)$, and $A$
is a symmetric positive definite matrix with bounded coefficients satisfying the
condition $\lambda_1 |\xi|^2 \, \leq \, A \xi \cdot \xi$, where $\lambda_1$ is a
positive constant independent of $\xi$. The generalized solution of
\eqref{eq:reactdiff1}--\eqref{eq:reactdiff4} exists and is unique in the set $V_0 + u_D$,
where
$V_0 := \Big \{ w \in \H{1}(\Omega) \; \mid \; w = 0 \; {\rm on} \; \Gamma_D \Big\}$.
Assume that $v \in V_0 + u_D$ is a conforming approximation of $u$.
We wish to find a computable majorant of the error norm
\begin{equation}
\mid\!\mid\!\mid\! e \!\mid\!\mid\!\mid^2 \,:= \| \nabla e \|^2_{A} + \| \varrho\, e \|^2,
\end{equation}
where
$
\| \nabla e \|^2_{A} := \Int_\Omega A \nabla e \cdot \nabla e \mathrm{\:d}x.
$
First, we note that the integral identity that defines $u$ can be rewritten in the form
\begin{equation}
\Int_\Omega A \nabla e \cdot \nabla w \, \mathrm{\:d}x \,
+ \Int_\Omega \varrho^2 e \, w \, \mathrm{\:d}x
= \Int_\Omega (f w - \varrho^2 v \, w - A \nabla v \cdot \nabla w) \mathrm{\:d}x \,
+ \Int_{\Gamma_N} F w \mathrm{\:d}s, \quad \forall w \in V_0.
\label{eq:weak-statement}
\end{equation}
It is well known (see Section 4.2 in \cite{RepinDeGruyter2008}) that this relation
yields a computable majorant of $\mid\!\mid\!\mid\! e \!\mid\!\mid\!\mid^2 $,
if we introduce a vector-valued function
${q} \in H(\Omega, \dvrg)$, such that ${q} \cdot n \in L^2(\Omega)$, and transform
\eqref{eq:weak-statement} by means of integration by parts relations. The majorant has
the form
\begin{equation}
\mid\!\mid\!\mid\! e \!\mid\!\mid\!\mid\,
\leq\,\|D_\Omega (\nabla v,q)\|_{A^{-1}} +
C_1 \| R (v,\dvrg {q}) \|_{\Omega} + C_2 \| R_{\Gamma_N} (v,q\cdot n) \|_{\Gamma_N},
\label{eq:global-estimate}
\end{equation}
where $C_1$ and $C_2$ are positive constants explicitly defined by $\lambda_1$,
the Friedrichs' constant $C^{\mathrm F}_{\Omega}$ in inequality \linebreak
$\|v\|_\Omega \leq \, C^{\mathrm F}_{\Omega} \|\nabla v\|_\Omega$ for functions vanishing
on $\Gamma_D$, and constant $C^{\mathrm Tr}_{\Gamma_N}$ in the
trace inequality associated with $\Gamma_N$. The integrands are defined by the
relations
\begin{equation*}
D (\nabla v,q) := A\nabla v - q, \quad
R (v,\dvrg q) := \dvrg q + f - \varrho^2 v, \quad {\rm and} \quad
R_{\Gamma_N} (v, q \cdot n) := q\cdot n - F.
\end{equation*}
In general, finding $C^{\mathrm F}_{\Omega}$ and $C^{\mathrm Tr}_{\Gamma_N}$ may not be
an easy
task. We can exclude $C_2$ if $ q$ additionally satisfies the condition ${ q}\cdot n = F$.
Then, the last term in \eqref{eq:global-estimate} vanishes. However, this condition is
difficult to satisfy, if $F$ is
a complicated nonlinear function. In order to exclude $C_1$ together with $C_2$,
we can apply domain
decomposition technique and use (\ref{eq:estOmega}) instead of the global estimate. Then, the
estimate will operate with the constants $C^{{\mathrm P}}_{\Omega_i}$ (whose upper bounds
are known for convex domains). Moreover, it is shown below that by using the inequalities
(\ref{eq:Comega}) and (\ref{eq:Cgamma}), we can essentially weaken the assumptions
required for the variable $q$.
Define the space of vector-valued functions
\begin{alignat*}{2}
\hat{H} (\Omega, {\mathcal O}_{\Omega}, \dvrg) := \, \Big\{{q} \in \L{2} (\Omega, \Rd) \; \mid & \quad {q} = {q}_i \in H(\Omega_i, \dvrg),
\;\; \mean{\dvrg {q}_i + f - \varrho^2 \, v}_{\Omega_i} = 0, \quad \forall\,\Omega_i \in {\mathcal O}_{\Omega}, \\
& \;\; \mean{({q}_i - {q}_j) \cdot { n}_{ij}}_{\Gamma_{ij}} = 0 , \quad \forall\, \Gamma_{ij} \in {\mathcal G}_{\rm int},\\
& \;\; \mean{{ q}_i \cdot { n}_k - F}_{ \Gamma_{N_k}} = 0, \quad \forall \, k = 1, \ldots, K_N \Big \}.
\end{alignat*}
We note that the space $\hat{H} (\Omega, {\mathcal O}_{\Omega},\dvrg)$ is wider than
$H(\Omega, \dvrg)$ (so that we have more flexibility in determination of
optimal reconstruction
of numerical fluxes). Indeed, the vector-valued functions in $H(\Omega,\dvrg)$ must
have continuous normal components on all $\Gamma_{ij} \in {\mathcal G}_{\rm int}$
and satisfy the Neumann boundary
condition in the pointwise sense. The functions in
$\hat{H} (\Omega, {\mathcal O}_{\Omega},\dvrg)$ satisfy much weaker conditions: namely,
the normal components are continuous only in terms of mean values (integrals) and
the Neumann condition must hold in the integral sense only.
We reform (\ref{eq:weak-statement}) by means of the integral identity
\begin{equation*}
\Sum_{\Omega_i \in {\mathcal O}_{\Omega}} \int_{\Omega_i}
\left( {q} \cdot \nabla w + \dvrg {q} \, w \right) \mathrm{\:d}x
= \Sum_{\Gamma_{ij} \, \in \, {\mathcal G}_{\rm int}} \;
\Int_{\Gamma_{ij}}\; ({ q}_i - { q}_j) \cdot { n}_{ij} \, w \mathrm{\:d}s
+ \, \Sum_{\Gamma_{N_k} \, \in \, \Gamma_N} \;
\Int_{\Gamma_{N_k}} { q}_i \cdot { n}_{i} \, w \mathrm{\:d}s,
\end{equation*}
which holds for any $ w \in V_0$ and
${q} \in \hat{H} (\Omega, {\mathcal O}_{\Omega},\dvrg)$. By setting $w = e$ in
\eqref{eq:weak-statement} and applying the H\"{o}lder inequality, we find that
\begin{multline*}
\mid\!\mid\!\mid\! e \!\mid\!\mid\!\mid^2\! \;\;
\leq \|D (\nabla v,q) \|_{A^{-1}} \|\nabla e\|_{A}
+ \Sum_{\Omega_i \in {\mathcal O}_{\Omega}} \|R(v,\dvrg q)\|_{\Omega_i}
\big\|e - \mean{e}_{\Omega_i}\big\|_{\Omega_i} \\
+ \Sum_{\Gamma_{ij} \in \, {\mathcal G}_{\rm int}}
r_{ij}(q) \big\| e - \mean{e}_{\Gamma_{ij}}\big\|_{\Gamma_{ij} }
+ \Sum_{\Gamma_{N_k} \in \Gamma_N} \rho_k(q)
\Big\| e - \mean{e}_{\Gamma_{N_k}}\Big\|_{\Gamma_{N_k}},
\end{multline*}
where
\begin{equation*}
r_{ij}(q) := \| ({ q}_i - { q}_j) \cdot { n}_{ij} \|_{\Gamma_{ij}}
\quad \mbox{and} \quad
\rho_k(q) := \| { q}_k \cdot { n}_k - F \|_{\Gamma_{N_k}}.
\end{equation*}
In view of (\ref{eq:classical-poincare-constant}) and (\ref{eq:Cgamma}), we obtain
\begin{multline}
\mid\!\mid\!\mid\! e \!\mid\!\mid\!\mid^2\! \;\;
\leq \|D (\nabla v,q) \|_{A^{-1}} \|\nabla e\|_{A}
+ \Sum_{\Omega_i \in {\mathcal O}_{\Omega}}
\|R(v,\dvrg q)\|_{\Omega_i} C^{\mathrm P}_{\Omega_i}
\|\nabla e \|_{\Omega_i} \\
+ \Sum_{\Gamma_{ij} \in \, {\mathcal G}_{\rm int}}
r_{ij}(q) C^{\mathrm{Tr}}_{\Gamma_{ij}}
\|\nabla e \|_{\Omega_i}
+ \Sum_{\Gamma_{N_k} \in \Gamma_N}
\rho_k(q) C^{\mathrm{Tr}}_{\Gamma_{N_k}}
\|\nabla e \|_{\Omega_i}.
\label{eq:estimate-2}
\end{multline}
The second term in the right hand side is estimated by the quantity
$\Re_1(v,q)\,\|\nabla e\|_\Omega$, where
\begin{equation*}
\Re^2_1(v,q) := \,\Sum_{\Omega_i \in {\mathcal O}_{\Omega}}
\tfrac{( \diam \, \Omega_i)^2}{\pi^2}\|R(v,\dvrg q)\|^2_{\Omega_i}.
\end{equation*}
We can represent any $\Omega_i \in {\mathcal O}_{\Omega}$ as a sum of simplexes such
that each simplex has one edge on $\partial\Omega_i$. Let $C^{\mathrm{Tr}}_{i, {\rm max}}$
denote the largest constant in the respective Poincar\'{e}-type inequalities
\eqref{eq:Cgamma} associated with all edges of $\partial\Omega_i$. Then, the last two
terms of (\ref{eq:estimate-2}) can be estimated by the quantity
$\Re_2 (v,q)\,\|\nabla e\|_\Omega$, where
\begin{equation*}
\Re^2_2(q):=\,\Sum_{\Omega_i \in {\mathcal O}_{\Omega}}
(C^{\mathrm{Tr}}_{i, {\rm max}})^2 \, \eta^2_i, \quad { \rm with} \quad
\eta^2_{i} =
\Sum_{\myatop{\Gamma_{ij} \in {\mathcal G}_{\rm int}}{\Gamma_{ij} \cap \partial\Omega_i \neq \varnothing}}\tfrac{1}{4} r^2_{ij}(q) +
\Sum_{\myatop{\Gamma_{k} \in {\mathcal G}_{N}}{\Gamma_{k} \,\cap \,\partial\Omega_i \neq \varnothing}}
\rho^2_k(q).
\end{equation*}
Then, (\ref{eq:estimate-2}) yields the estimate
\begin{equation*}
\mid\!\mid\!\mid\! e \!\mid\!\mid\!\mid^2\! \;\;
\leq \|D(\nabla v,q)\|_{A^{-1}} \|\nabla e\|_{A}
+ (\Re_1(v,q) + \Re_2(q) )\,\|\nabla e\|_\Omega,
\end{equation*}
which shows that
\begin{equation}
\mid\!\mid\!\mid\! e \!\mid\!\mid\!\mid\! \;\;
\leq \|D(\nabla v,q) \|_{A^{-1}}
+ \tfrac{1}{\lambda_1} \Big( \Re_1(v,q) + \Re_2(q) \Big).
\label{eq:estimate-final}
\end{equation}
Here, the term $\Re_2(q)$ controls violations of conformity of $q$ (on interior edges)
and inexact satisfaction of boundary conditions (on edges related to $\Gamma_N$). It is
easy to see that $\Re_2(q) = 0$, if and only if the quantity $q \cdot n$ is continuous on
${\mathcal G}_{\rm int}$ and exactly satisfies the boundary condition. Hence, $\Re_2(q)$
can be viewed as a measure of the `flux nonconformity'. Other terms have the same
meaning as in well-known a posteriori estimates of the functional type, namely, the
first term measures the
violations of the relation ${q} = A \nabla v$ (cf. (\ref{eq:reactdiff2})), and
$\Re_1(v,q)$ measures inaccuracy in the equilibrium (balance) equation
(\ref{eq:reactdiff1}). The right-hand side of \eqref{eq:estimate-final} contains known
functions (approximations $v$ and $q$ of the exact solution and exact flux). The
constants $C^{\mathrm{Tr}}_{i, {\rm max}}$ can be easily computed using results of Section \ref{sc:arbitrary-triangle}-\ref{eq:numerial-tests-3d}.
Finally, we note that estimates similar to \eqref{eq:estimate-final} were derived in
\cite{ReVarInq2010} for elliptic variational inequalities and in
\cite{MatculevichNeitaanmakiRepin2015} for a class of parabolic problems.
\bibliographystyle{plain}
|
{
"timestamp": "2016-02-05T02:12:50",
"yymm": "1504",
"arxiv_id": "1504.03166",
"language": "en",
"url": "https://arxiv.org/abs/1504.03166"
}
|
\section{Introduction}
\label{sec:1}
Unveiling the phase diagram of quantum chromodynamics (QCD) is among the most fundamental issues in quark-hadron physics.
So far, considerable theoretical and experimental efforts have been devoted to exploring the QCD phase diagram (for reviews see Ref.~\cite{Fukushima:2010bq});
the properties of the high temperature regime are studied
experimentally in ultra-relativistic heavy-ion collisions,
and in ab-initio lattice QCD simulations.
The latter are subject to technical difficulties at non-zero net baryon-number density, the so-called sign problem.
In the near future, data at lower beam energies, relevant for the exploration of the phase diagram at non-vanishing
baryon density, will be forthcoming. In order to exploit this opportunity in an optimal way, it is necessary
to find appropriate observables for deciphering the properties of dense and moderately hot matter in such collisions~\cite{Friman:2011zz}.
In recent theoretical studies of QCD at finite temperature and density,
various inhomogeneous chiral condensed phases have been proposed
(for a recent review see Ref.~\cite{Buballa:2014tba}).
These studies suggest that the conventional QCD phase diagram should be redrawn.
Indeed, it is possible that the phase structure at high net baryon densities and moderate temperatures is modified considerably
by the presence of inhomogeneous phases. Thus, the region of the chiral transition may be extended and the order of the phase transition may change.
These features are gleaned primarily from mean-field calculations
in the Nambu-Jona-Lasinio (NJL) and quark-meson (QM) models~\cite{Nakano:2004cd,Nickel:2009wj},
and the Dyson-Schwinger approach to dense QCD~\cite{Muller:2013tya}.
It also interesting to note that within the Gor'kov approach to chiral effective models~\cite{Nickel:2009ke,Carignano:2014jla}
it is found that the QCD critical endpoint is a Lifshitz point,
where the normal, homogeneous, and inhomogeneous chiral condensed phases meet.
In the large $N_c$ approach to dense QCD,
early studies suggested the emergence of the so-called chiral density wave~\cite{Deryagin:1992rw,Shuster:1999tn},
while in the context of quarkyonic matter~\cite{McLerran:2007qj}
another inhomogeneous phase, the so-called quarkyonic chiral spiral, was discussed~\cite{Kojo:2009ha}.
The inhomogeneous chiral condensed phases mentioned above
correspond to a one-dimensional modulation embedded in three spatial dimensions. Some
of these structures are based on extrapolations from
analytic solutions obtained for purely (1+1)-dimensional systems~\cite{Schon:2000qy}.
Possible extensions to higher-dimensional modulations have been studied,
with the result that the one-dimensional modulation tends to be favored close to the Lifshitz point~\cite{Abuki:2011pf} and/or at zero temperature~\cite{Carignano:2012sx}.
Let us start by classifying the modulations for inhomogeneous phases,
according to the convention employed in condensed matter physics.
There are basically two types of one-dimensional modulations:
one is of the Fulde-Ferrell (FF) type~\cite{Fulde:1964zz},
characterized by modulations of the phase of a complex order parameter with constant amplitude,
while the other is of the Larkin-Ovchinnikov (LO) type~\cite{larkin:1964zz},
where by contrast only the amplitude is modulated.
The FF type includes the dual chiral density wave (DCDW)~\cite{Nakano:2004cd}
and the quarkyonic chiral spiral~\cite{Kojo:2009ha}.
On the other hand,
the chiral density wave (a plane wave)~\cite{Deryagin:1992rw,Shuster:1999tn}
and periodic domain walls~\cite{Nickel:2009wj} are of the LO type.
In the present paper we focus on the DCDW, which is of FF type.
The DCDW is characterized by modulated scalar and pseudoscalar condensates
with a constant amplitude $\Delta$ and a wavenumber $q$~\cite{Nakano:2004cd},
\beq
\langle \bar{\psi}\psi \rangle =\Delta \cos{qz}, \quad
\langle \bar{\psi}i\gamma_5\tau_3\psi \rangle =\Delta \sin{qz},
\eeq
where $\psi$ is the quark field for two flavors,
and $\tau_3$ a Pauli matrix diagonal in the isospin space.
This configuration is akin to
$\sigma\pi^0$ condensation, obtained in neutron matter within the sigma model~\cite{Dautry:1979bk},
and is thus expected to smoothly connect between nuclear and quark matter.
Most studies of inhomogeneous chiral condensed phases so far are based on mean-field calculations.
Thus, effects of thermal and quantum fluctuations have yet to be studied.
In the context of pion condensation,
the stability of modulated condensates against thermal fluctuations has been studied on the basis of Landau-Peierls arguments~\cite{landau1969statical}.
It was found that in such systems there is no true long-range order with a non-vanishing order parameter~\cite{Baym:1982ca}. Instead, such systems can develop a quasi-ordered one-dimensional condensate, with correlation functions that decay algebraically in space.%
\footnote{A similar state is found, e.g., in smectic liquid crystals~\cite{AlsNielsen:1980zz,de1993physics}. See also Refs.~\cite{Shimahara1998,Radzihovsky2009} for a corresponding discussion on FFLO superconductors/superfluids.}
In this paper, we investigate the stability of the DCDW phase
against low-energy fluctuations of Nambu-Goldstone (NG) modes associated with the spontaneous symmetry breaking, along the lines of Ref.~\cite{Baym:1982ca}.
The paper is organized as follows.
In the next section,
we construct
a (3+1)-dimensional Landau-Ginzburg-Wilson effective Lagrangian
for general order parameters of the chiral condensate,
which are allowed to be spacetime dependent,
and then apply the formalism to the DCDW phase.
In Sec.~\ref{sec:3},
we discuss the symmetry breaking pattern and the corresponding NG modes in the DCDW phase.
We also present the dispersion relations for these low-energy collective excitation modes, by introducing fluctuations such as amplitudons and phonons/phasons (NG modes) on the ground state of the DCDW.
In Sec.~\ref{sec:4},
we investigate the impact of low-energy fluctuations on the order parameter by evaluating the long-range correlation functions.
Finally, section~\ref{sec:5} is devoted to summary and outlook.
\section{Landau-Ginzburg-Wilson effective Lagrangian}
\label{sec:2}
We start by introducing a $2\times 2$ matrix field $M$ as the $\left( {1}/{2},{1}/{2}\right)$ representation of the chiral $SU(2)_L \times SU(2)_R$ symmetry. In the following, we shall use the fact that $SU(2)_L \times SU(2)_R$ is isomorphic to the four-dimensional rotation group $O(4)$.
The matrix $M$ can be expressed in terms of the right and left handed quark fields $\psi_{R,L}$~\cite{lee1972chiral},
$M_{ij}= {\psi}_{L,i}\psi_{R,j}^{\dagger}$.
Under the transformations $U_{R,L}=\exp\left[ -i\left( \vec{\alpha}\pm\vec{\beta}\right) \cdot\vec{\tau}/2 \right]$, where $\vec{\tau}$ is the isospin Pauli matrix and $\vec{\alpha}$ and $\vec{\beta}$ are two three-dimensional vector parameters,
the matrix $M$ transforms (to leading order in $\vec{\alpha}$ and $\vec{\beta}$) as $M_{ij} \rightarrow M_{ij}+\frac{i}{2}\vec{\alpha}\cdot\left[\vec{\tau},M\right]_{ij} +\frac{i}{2}\vec{\beta}\cdot\left\{ \vec{\tau},M\right\}_{ij}$.
With the parameterization $M = \sigma + i\vec{\pi}\cdot \vec{\tau}$, one finds the corresponding transformation laws for $\sigma$ and $\vec{\pi}$: $\sigma \rightarrow \sigma - \vec{\beta}\cdot \vec{\pi}$ and
$\vec{\pi} \rightarrow \vec{\pi}-\vec{\alpha}\times \vec{\pi}+\vec{\beta}\,\sigma$.
Thus, the rotation with $\vec{\alpha}$ corresponds to the vector (isospin) rotation while that with $\vec{\beta}$ to an axial vector (chiral or axial isospin) rotation, respectively. We can then introduce a four-component composite field
$\phi^T = \left( \sigma, \vec{\pi} \right)$, which transforms as a four-dimensional vector under $O(4)$ rotations.
Now, we construct an effective Lagrangian density ${\mathcal L}$ with $O(4)$ symmetry in terms of $\phi$ and its derivatives
\footnote{For non-zero isospin charges,
a term of the form $\frac{1}{2} \epsilon_{ijkl} \mu_{ij} Q_{ij}$ is added to the Lagrangian. Here $\epsilon_{ijkl}$ ($\epsilon_{1234}=1$) is the antisymmetric tensor, and $\mu_{ij}=-\mu_{ji}$ the chemical potential for the isospin density
$Q_{ij}=i(\phi_i\partial_0 \phi_j-\phi_j\partial_0 \phi_i)$.}
\beq
{\mathcal L}&=&
c_2\partial_0\phi\cdot\partial_0\phi -{\mathcal V}, \label{Lag} \\
{\mathcal V}&=& a_2\phi\cdot \phi+a_{4,1}\left( \phi\cdot\phi\right)^2
+a_{4,2}\bnab \phi\cdot\bnab \phi
+a_{6,1}\bnab^2 \phi\cdot\bnab^2 \phi\nn
&&+a_{6,2}\left( \bnab \phi\cdot\bnab \phi\right)\left( \phi\cdot\phi\right)
+a_{6,3}\left(\phi\cdot\phi\right)^3
+a_{6,4}\left( \phi\cdot \bnab \phi\right)^2,
\eeq
Note that since the Lagrangian describes the excitations of a medium, we do not assume Lorentz invariance.
The potential term ${\mathcal V}$ is expanded up to sixth order in powers of the field and fourth order in its derivatives,
as required for stability of the inhomogeneous phase at the mean-field level
Hereafter we set $c_2=1$
for simplicity.
The expansion coefficients $a_{i,j}$ in ${\mathcal V}$ can be evaluated within effective chiral models, like the NJL~\cite{Nickel:2009ke} and QM~\cite{Nickel:2009wj,Carignano:2014jla} models. In the former, one finds the following relations among them $a_{4,1} = a_{4,2}$ and $\left( a_{6,1}, a_{6,2}, a_{6,4} \right) = \left( {1}/{2}, 3, 2 \right) a_{6,3}$.
\section{Low energy effective modes in DCDW phase}
\label{sec:3}
We consider an inhomogeneous time independent chiral condensate of the DCDW type,
\beq
\phi_0^T = \Delta \left( \cos{qz}, 0,0, \sin{qz}\right),
\eeq
where $\Delta$ is a constant amplitude corresponding to
$\langle \bar{\psi}e^{i\gamma_5 \tau_3 \bq\cdot\br} \psi \rangle$
and $q$ is the wavenumber of modulation in the $z$ direction.
The values of $\Delta$ and $q$ are determined by minimizing the potential term of the Lagrangian.
For the condensate $\phi_0$, the potential term reads
\beq
\mathcal{V}(\phi_0)=
a_2 \Delta^2+a_{4,1} \Delta^4
+a_{4,2} q^2\Delta^2+a_{6,1} q^4\Delta^2+a_{6,2} q^2\Delta^4
+a_{6,3}\Delta^6.
\eeq
Stability of the inhomogeneous phase is guaranteed by
\beq
a_{6,1}>0 \ ({\rm or}\ a_{6,3}>0),
\quad {\rm and} \quad a_{6,1}a_{6,3}-a_{6,2}^2/4>0.
\eeq
The stationary conditions for $q$ and $\Delta$,
$\frac{\partial {\mathcal V}}{\partial \Delta}=\frac{\partial {\mathcal V}}{\partial q}=0$, yield
\beq
&&2q\Delta^2 \left( a_{4,2}+2a_{6,1}q^2+a_{6,2}\Delta^2 \right) =0, \\
&&2\Delta \left[ a_2+a_{4,2}q^2+a_{6,1}q^4+2 \left( a_{4,1}+a_{6,2}q^2 \right) \Delta^2
+3a_{6,3}\Delta^4 \right] =0.
\eeq
They admit three types of solutions:
\begin{enumerate}
\item[(a)] Normal phase: $q=\Delta=0$,
\item[(b)] Homogeneous chiral condensed phase: $q=0$, $\Delta\neq0$,
\item[(c)] Inhomogeneous chiral condensed phase:
$q^2=-\left( a_{4,2}+a_{6,2} \Delta^2\right)/2a_{6,1}$, $\Delta\neq0$.
\end{enumerate}
For a given set of coefficients, the phase with the lowest energy, is realized on the classical level.
The coefficients implicitly depend on the medium, and are thus functions of thermodynamic
variables, like temperature and chemical potentials.
\subsection{Symmetry breaking and Nambu-Goldstone modes in DCDW phase}\label{subsec:SSBNG}
In the DCDW phase with non-vanishing $\Delta$ and $q$, the $SU(2)_L \times SU(2)_R$ chiral symmetry, as well as the translational invariance in the $z$ direction and the symmetry under rotations about the $x$ and $y$ axes are spontaneously broken.
To see the symmetry breaking pattern explicitly, we first perform infinitesimal transformations corresponding to a spatial translation in the $z$ direction with a displacement parameter $s$ and a chiral
rotation through the angles $\vec{\alpha}$ and $\vec{\beta}$:
\beq
\phi_0 \rightarrow \phi_0 +
\Delta \left(
\begin{array}{c}
-\left( sq+\beta_3 \right) \sin{qz}\\
\beta_1\cos{qz}-\alpha_2\sin{qz} \\
\beta_2\cos{qz}+\alpha_1\sin{qz} \\
\left( sq+\beta_3\right) \cos{qz}
\end{array}
\right).
\label{SSB1}
\eeq
The form of the first and the fourth components implies that $\phi_0$ is invariant under
a simultaneous spatial translation and axial isospin rotation about the $z$ axis through the angle $\beta_3$, if $qs+\beta_3=0$. Thus, in the DCDW phase a locking of axial isospin rotations with translations is realized.
In terms of symmetry generators, there are two unique orthogonal linear combinations of $s$ and $\beta_{3}$; one corresponding to a broken generator, the other to an unbroken one.
Consequently, the corresponding NG mode is generated by a transformation with $qs+\beta_3\neq 0$, i.e.,
by one whose generator is broken in the DCDW phase. In the following, we use $\beta_3=\beta_3(t, \vx)$
and $s=0$ to generate the NG mode associated with the broken generator. Similar arguments for the
spontaneous breakdown of internal and spacetime symmetries are given in Refs.~\cite{Kobayashi:2014xua,Hidaka:2014fra}.
The rotations through $\beta_1$ and $\alpha_2$ generate variations
in the second component in (\ref{SSB1}).
However, the corresponding NG modes are linearly dependent in the sense discussed in Ref.~\cite{Low:2001bw}.
In case of a vanishing wavenumber $q=0$, only $\beta_1$ is relevant. Thus, we generate the corresponding NG mode
using $\beta_1=\beta_1(t,\vx)$ and $\alpha_2=0$. Analogous arguments can be applied the third component in (\ref{SSB1}), thus eliminating $\alpha_1$ in favour of $\beta_2$.
Spatial rotations about the $x$-axis by an angle $\theta_x$ yields a transformation,
which is nonuniform in space: $z \rightarrow z \cos\theta_x + y\sin\theta_x$.
Similarly, the rotations about the $y$-axis by $\theta_y$ yields the analogous transformation.
However, the corresponding NG modes and those generated by translations are also linearly dependent~\cite{Low:2001bw}.
We conclude that, although there are eight broken generators for internal and space time symmetries in the DCDW phase, only three independent NG modes
remain. These can be chosen as the axial isospin rotations generated by $\vec{\beta}=\vec{\beta}(t,\vx)$.
\subsection{Low energy collective excitations}\label{subsec:LECE}
We now consider a general fluctuation in the DCDW phase:
\beq\label{DCDW-gen-fluct}
\phi&=&(\Delta +\delta)
\left(
\begin{array}{l}
\cos{\left( qz+\beta_3\right)}\cos{\beta_2}\cos{\beta_1} \\
\cos{\left( qz+\beta_3\right)}\cos{\beta_2}\sin{\beta_1} \\
\cos{\left( qz+\beta_3\right)}\sin{\beta_2} \\
\sin{\left( qz+\beta_3\right)}
\end{array}
\right)
=(\Delta +\delta) U(\beta_i)
\left(
\begin{array}{c}
\cos{\left( qz\right)}\\
0\\
0\\
\sin{\left( qz\right)}
\end{array}
\right).
\eeq
Here $\delta$ is the amplitude fluctuation,
the parameters $\vec{\beta}=\left\{ \beta_1,\beta_2,\beta_3\right\}$
specifies a rotation in the four-dimensional space spanned by the $\sigma$ and $\vec{\pi}$ fields. Finally, $U(\beta_{i=1,2,3}):=e^{i\beta_1 L_1}e^{i\beta_2 L_2}e^{i\beta_3 L_3}$, where $L_{1,2,3}$ are the O(4) (axial isospin) generators~\cite{lee1972chiral}.
This parametrization clearly shows that the displacement in the $z$ direction is equivalent to a chiral rotation through $\beta_{3}$. To leading order in the fluctuations, (\ref{DCDW-gen-fluct}) yields
\beq
\phi = \left( 1+\delta \right) \phi_0 + \Delta
\left(
\begin{array}{c}
-\beta_3\sin{qz}\\
\beta_1\cos{qz}\\
\beta_2\cos{qz}\\
\beta_3\cos{qz}\\
\end{array}
\right) +{\mathcal{O}}\left( \beta_i^2, \delta\beta_i, \delta^2\right),
\eeq
which exhibits the fluctuation of the amplitude in addition to the fluctuations corresponding to the NG modes.
In the following we consider local fluctuations, promoting the parameters $\delta$ and $\vec{\beta}$ to fields $\delta(x)$ and $\vec{\beta}(x)$, where we use the compact notation $x\equiv \{t, \vx \}$.
Plugging the above parametrization into the Lagrangian, we can systematically derive a low energy effective theory by expanding
in powers of the fluctuation fields $\delta$ and $\vec{\beta}$.
Up to the second order in the fields,
the Lagrangian $L=\int {\rm d}^3x{\mathcal L}$ reads
\beq
{\mathcal L}&=&
\left( \partial_0 \delta\right)^2+\Delta^2(\partial_0 \vec{\beta}_U)^2 + \Delta^2(\partial_0 \beta_3)^2 - \left( {\mathcal V}_\delta+{\mathcal V}_{\delta \beta}+{\mathcal V}_\beta\right),
\eeq
where
\beq
{\mathcal V}_\delta
&=& M^2 \delta^2
+a_{6,4}\Delta^2(\nabla\delta)^2 +4a_{6,1}q^2(\nabla_z\delta)^2 +a_{6,1}(\nabla^2\delta)^2,
\\
{\mathcal V}_{\delta \beta}
&=& 4q\Delta \left[ a_{6,2}\Delta^2\delta -2a_{6,1}\nabla^2\delta\right] \nabla_z\beta_3,
\\
{\mathcal V}_\beta
&=&
a_{6,1}\Delta^2\left( \nabla^2\vec{\beta}_U +q^2\vec{\beta}_U\right)^2 +a_{6,1}\Delta^2\left[ \left( \nabla^2\beta_3\right)^2 +4q^2\left( \nabla_z\beta_3\right)^2 \right],
\eeq
with the mass term $M^2=4(a_{4,1}+a_{6,2}q^2)\Delta^2 +12a_{6,3}\Delta^4$ and the transverse field $\vec{\beta}_U\equiv \vec{\beta}_T\cos{qz}$ where $\vec{\beta}_T = \left\{ \beta_1,\beta_2\right\}$.
In the above equations the stationary condition $a_{4,2}+a_{6,2}\Delta^2+2q^2a_{6,1}=0$ has been used.
For details of the derivation, see Appendix~\ref{app:A}.
In order to investigate the thermodynamics of the system, we now move to Euclidean space:
$t \rightarrow -i\tau$ with the period $0\le \tau \le \beta$
where $\beta=1/T$ is the inverse temperature.
For Gaussian fluctuations,
we obtain the Euclidean action in Fourier space
(for details we refer the reader to Appendix~\ref{app:B}),
\beq
-S_{E}
&=&
\int_0^\beta {\rm d}\tau \int {\rm d}^3 x {\mathcal L}_E
=
\sum\!\!\!\!\!\!\!\int{\rm d}k
\left(
\begin{array}{c}
\delta^*(k)\\
\Delta \beta_3^*(k)
\end{array}
\right)^T
\left(
\begin{array}{cc}
S_{\delta 0}^{-1}(k) & -g(k) \\
g(k) & S_{0}^{-1}(k)
\end{array}
\right)
\left(
\begin{array}{c}
\delta(k)\\
\Delta \beta_3(k)
\end{array}
\right)
\nn
&+&
\frac{1}{4}
\sum\!\!\!\!\!\!\!\int{\rm d}k
\left(
\begin{array}{c}
\Delta\vec{\beta}^*_T(k)\\
\Delta\vec{\beta}^*_T(k+2q\hat{z})
\end{array}
\right)^T
\left(
\begin{array}{cc}
S_0^{-1}(k) & G(k) \\
G(k) & S_0^{-1}(k+2q\hat{z})
\end{array}
\right)
\left(
\begin{array}{c}
\Delta\vec{\beta}_T(k)\\
\Delta\vec{\beta}_T(k+2q\hat{z})
\end{array}
\right), \nn
\eeq
where we have used the shorthand notation:
$\Sigma\!\!\!\!\!\!\!~\int{\rm d}k \equiv T\sum_n \int \frac{{\rm d}^3 k}{(2\pi)^3}$,
and $k=(\omega_n, \vec{k})$ with the Matsubara frequency $\omega_n=2\pi n T (\equiv i\omega)$.
The inverse propagators in the above matrix notation are given by
\beq
S_{\delta0}^{-1}(k)
&=&\omega^2-\left[
M^2 + {a_{6,4}\Delta^2\vec{k}^2}
+4a_{6,1}q^2(k_z)^2+a_{6,1}(\vec{k}^2)^2\right], \nn
g(k)
&=&
2iq\left[ a_{6,2}\Delta^2
+2a_{6,1}\vec{k}^2\right] k_z, \nn
S_0^{-1}(k)
&=&
\omega^2
-a_{6,1}\left[ 4q^2 k_z^2 +(\vec{k}^2)^2 \right],
\nn
\mbox{and} \quad
G(k)
&=&
\omega^2
-a_{6,1}\left( \vec{k}^2+2qk_z \right)^2. \label{disp}
\eeq
We note that for a non-vanishing wavenumber $q$, the $\delta$ and $\beta_3$ fluctuations mix. Moreover, transverse fluctuations $\vec{\beta}_T$ with different momenta $k$ and $k+2q\hat{z}$ mix, owing to the scattering of fluctuations off the background modulation.
The determinant of the first matrix, $S_{\delta0}^{-1}(k)S_{0}^{-1}(k)+g^2(k)=0$, yields the dispersion relations of the normal modes involving $\delta$ and $\beta_3$,
\beq
\omega_+^2&\simeq&M^2+a_{6,1}\left[ u^2_{z+} k_z^2+\left(\vec{k}^2\right)^2\right] +a_{6,4}\Delta^2\vec{k}^2+A \vec{k}^2k_z^2 +Bk_z^4, \\
\omega_-^2&\simeq&a_{6,1}\left[ u^2_{z-} k_z^2+\left(\vec{k}^2\right)^2\right] -A \vec{k}^2k_z^2 -Bk_z^4, \label{disp-}
\eeq
where $u_{z\pm}^2= 4q^2\left( 1\pm\frac{a_{6.2}^2\Delta^4}{a_{6.1}M^2}\right)$,
$A\equiv 4q^2\Delta^2 a_{6.2}\left( 4 M^2 a_{6.1}-\Delta^4a_{6.2}a_{6.4}\right)/M^4$, and
$B\equiv -\left( 2q\Delta^2 a_{6.2}\right)^4/M^6$.
Note that in the massless mode $\omega_-$,
the dependence on the transverse momentum is subleading, $\mathcal{O}(k^4)$.%
\footnote{The sign of $u^2_{z-}$ in $\omega_-$ is always positive if $a_{4.1}>0$, which generally holds if the transition is second order.}
Consequently, the transverse fluctuations are softer that the longitudinal ones.
Similarly, equating the determinant of the second matrix to zero,
$S_{0}^{-1}(k)S_{0}^{-1}(k)-G^2(k)=0$,
we obtain the dispersion relation for $\vec{\beta}_T$,
\beq
\omega_k^2=a_{6,1}\left[ 4q^2 k_z^2+\left( \vec{k}^2\right)^2\right]
-a_{6,1}\frac{2k_z^2\left( \vec{k}^2\right)^2}{4q^2+6qk_z+2k_z^2+\vec{k}^2}.
\label{disp1}
\eeq
The second term with the negative sign is a higher order correction of ${\mathcal O}(k^6)$, stemming from interactions with the background modulation. This term is irrelevant for the effects of low energy fluctuations and is therefore
dropped in the following discussion.
\section{Impacts of low energy fluctuations}
\label{sec:4}
At low temperatures, the low energy fluctuations about the calssical DCDW state dominate.
We evaluate the contribution of Gaussian fluctuations to the partition function
\beq
Z = \int \left[{\mathcal D}\delta\right] \left[{\mathcal D}\Delta\vec{\beta}\right] e^{-S_E}.
\eeq
Higher-order derivative corrections are dropped, with the assumption that fluctuations
at energies above some cutoff $\Lambda$ have been integrated out in the effective Lagrangian (\ref{Lag}),
which then involves only the low-energy fluctuations, $\delta$ and $\vec{\beta}$, explicitly.
We first explore the impact of low energy fluctuations on the order parameter,
\beq
\langle (\Delta+\delta) U(\beta_i)\,\phi_0 \rangle
&=&
\Delta \langle U(\beta_i)\,\phi_0 \rangle +\langle \delta\, U(\beta_i)\,\phi_0 \rangle,
\label{op1}
\eeq
where we use the compact notation
$\langle \cdots \rangle \equiv
\int \left[{\mathcal D}\delta\right] \left[{\mathcal D}\Delta\vec{\beta}\right] \cdots e^{-S_E}/Z$.
In the Gaussian approximation, the two contributions to the expectation value reduce to
\beq
\langle U(\beta_i)\,\phi_0 \rangle
&\simeq&
\left(
\begin{array}{l}
\cos(qz)e^{-\sum_i\langle \beta_i^{2} \rangle/2} \\
0 \\
0\\
\sin(qz)e^{-\langle \beta_3^{2} \rangle/2}
\end{array}
\right)
\\
{\rm and}&&\nonumber
\\
\langle \delta\, U(\beta_i)\,\phi_0 \rangle
&\simeq &
\left(
\begin{array}{l}
-\sin(qz)\langle \delta \beta_3\rangle e^{-\sum_i\langle \beta_i^{2} \rangle/2} \\
0 \\
0\\
\cos(qz)\langle \delta \beta_3\rangle e^{-\langle \beta_3^{2} \rangle/2}
\end{array}
\right) . \qquad
\eeq
Here the second order fluctuations are given by
\beq
&&\langle \delta(x) \beta_3(x)\rangle
\simeq 0,
\\
&&
\Delta^2\langle \beta_{1,2}^2(x)\rangle
\simeq \frac{1}{2}\int \frac{{\rm d}^3k}{(2\pi)^3}\frac{T}{\omega_k^2}, \quad
\label{beta1}
\\
\mbox{and}\
&&
\Delta^2\langle \beta_{3}^2(x)\rangle
\simeq \frac{1}{2}\int \frac{{\rm d}^3k}{(2\pi)^3}\frac{T}{\omega_-^2},
\label{beta12}
\eeq
where the fluctuations $\langle \beta_{1,2,3}^2(x)\rangle$ are all logarithmically divergent due to the soft modes in the transverse directions.
Details of the derivation are given in Appendix~\ref{app:C}.
Consequently, the low-energy fluctuations wash out the order parameter, i.e., they destroy the off-diagonal long-range order,
\beq
\langle (\Delta+\delta) U(\beta_i)\phi_0 \rangle =0.
\eeq
This result implies that a DCDW phase with true long-range order strictly speaking does not exist at non-zero temperature. Such a phase may, however, be realized in a modified form, with a quasi-long-range order (QLRO), analogous to that in the
Berezinsky-Kosterlitz-Thouless phase in two-dimensional systems~\cite{Berezinsky:1970fr} and in smectic liquid crystals~\cite{de1993physics}. As we discuss in the next section, the quasi-long-range order is characterized by a power-law decay of the order parameter correlation function.
At zero temperature, on the other hand,
quantum fluctuation are not strong enough to break the modulating order.
In fact, at $T=0$ the second order fluctuations are given by the infrared convergent integral
$\Delta^2\langle \beta_{1,2 \ \!\!(3)}^2(x)\rangle \simeq \frac{1}{4}\int\!\frac{{\rm d}^3k}{(2\pi)^3}\frac{1}{\omega_{k \ \!\!(-)}}$,
obtained by taking the zero temperature limit of Eqs.~(\ref{C6}) and (\ref{C12}) in Appendix~\ref{app:C}.
The results of this section imply that the transition temperature of the true DCDW phase is $T_{DCDW}=0$. Now assume that there is a critical temperature $T_{c}>0$, where the system becomes unstable with respect to the formation of a state with a modulated order parameter. Then a quasi-one-dimensionally ordered phase or a phase with true long-range order in two or three dimensions may be realized below $T_{c}$~\cite{Baym:1982ca}. To determine which phase is preferred, one must, in principle, compare their free energies. Considering the different nature of these phases, this is a challenging task.
\subsection{Long-range correlations}
We now explore the behavior of the correlation functions
in the Gaussian approximation.
Since the order parameter is vector-like,
we define correlation functions among the components:
\beq
f_{ij}(x)=\langle \phi_i(x)\phi^*_j(0)\rangle. \label{lrcf}
\eeq
These correlation functions are spatially anisotropic owing to the one-dimensional modulation of the background.
We compute the dependence of the iso-scalar correlation function on $z$.
The diagonal components which contribute to the scalar channel are of the form:
\beq
f_{11}(\hat{z}z)
&\simeq&
\frac{1}{8}\Delta^2 \cos{qz} e^{-\sum_{i=1,2,3}\langle ({\beta_i^-})^2\rangle/2},
\eeq
where $\beta_i^{-}\equiv \beta_i(z) - \beta_i(0)$.
For details we refer to Appendix~\ref{app:D}.
Similar results are obtained
for the other components:
\beq
f_{22}(\hat{z}z)
&\simeq&
\frac{1}{8}\Delta^2 \cos{qz} e^{-\sum_{i=1,2,3}\langle ({\beta_i^-})^2\rangle/2},
\\
f_{33}(\hat{z}z)
&\simeq&
\frac{1}{4}\Delta^2 \cos{qz} e^{-\sum_{i=2,3}\langle ({\beta_i^-})^2\rangle/2},
\\
\mbox{and} \ \
f_{44}(\hat{z}z)
&\simeq&
\frac{1}{2}\Delta^2 \cos{qz} e^{-\langle ({\beta_3^-})^2\rangle/2}.
\eeq
Here the exponents,
$\langle ({\beta_{i}^-})^{2}\rangle$,
exhibit the following functional form at large $z$,
\beq
\langle ({\beta_{1,2\ \!\!(3)}^-})^{2}\rangle/2
&\simeq&
\frac{T}{2\Delta^2}\int \frac{{\rm d}^3 k}{(2\pi)^3}
\frac{1 - \cos{k_zz}}{\omega_{k\ \!\!(-)}^2} \nn
&\simeq&
\frac{T}{16\pi a_{6,1}\Delta^2u_{z-}}\ln{\frac{z\Lambda^2}{2q}},
\eeq
where $\Lambda$ is an ultraviolet cutoff.
Putting it all the together, we obtain the long-range scalar correlation in the $z$ direction,
\beq
\langle \phi(z\hat{z})\cdot\phi^*(0)\rangle
&\sim & \frac{1}{2}\Delta^2 \cos{qz}\left(\frac{z}{z_0}\right)^{-T/T_0},
\eeq
where $z_0 \equiv 2q/\Lambda^2$, and $T_0 \equiv 16\pi a_{6,1}\Delta^2u_{z-}$.
In the similar manner, we compute the form of the long-range correlation function in transverse directions,
\beq
\langle \phi(x_{t}\hat{x}_{t})\cdot\phi^*(0)\rangle
&\sim & \frac{1}{2}\Delta^2 \left(\frac{x_t}{x_0}\right)^{-2T/T_0} ,
\eeq
where $x_0 \equiv \Lambda^{-1}$,
$x_t$ is the transverse distance,
and the factor $2$ in the exponent of ${x_t}/{x_0}$ reflects the number of transverse directions.
Note that, in contrast to the longitudinal direction, there is no modulation of the correlation function in the transverse directions.
In this section we have shown that quasi-long-range order of the one-dimensional DCDW phase feature algebraically decaying correlation functions at large distances. The slow decay of the spatial correlations distinguish the quasi-ordered phase from normal or disordered phases, characterized by exponential decays. Depending on the experimental resolution and finite size effects, the algebraic correlations can effectively mimic true long-range order~\cite{Berezinsky:1970fr,AlsNielsen:1980zz,Baym:1982ca}.
\section{Summary and outlook}
\label{sec:5}
In this paper we have explored the soft modes of an inhomogeneous chiral condensed phase with one-dimensional modulation, the DCDW phase.
We found that this phase exhibits a flavor-translation locking symmetry and clarified the counting of Nambu-Goldstone modes.
The dispersion relations for collective excitations, including the NG modes,
were derived. The low-energy modes
are spatially anisotropic and particularly soft in the directions transverse to the modulation,
owing to the lack of terms quadratic in the transverse momentum in the dispersion relations.
As in smectic liquid crystals, the absence of such terms is a consequence of the symmetry under rotations about any axis orthogonal to the modulation direction~\cite{landau1969statical,de1993physics}.
Moreover, we have shown that at non-zero temperatures the DCDW phase exhibits a Landau-Peierls instability, i.e., the long range order is destroyed by low-energy (long-wavelength) fluctuations of the order parameter. Nevertheless, a phase similar to the smectic phases of liquid crystals, characterized by a quasi-long-range order with algebraically decaying order parameter correlation functions, is possible. Such an ``algebraic order'' can, depending on the conditions, emulate true long-range order. In particular, this would be the case, in a finite systems, where the range of the order-parameter correlations exceeds the size of the system.
The experimental verification of ``algebraic order'' can be challenging. The slow decay of the correlations has been observed by light scattering in smectic-A liquid crystals~\cite{AlsNielsen:1980zz}, by neutron scattering in Bragg glass~\cite{Nattermann:1990} and only recently in a two-dimensional system of the Berezinsky-Kosterlitz-Thouless type~\cite{Nitsche:2014xxx}, by measuring the coherence of photons emitted in quasiparticle decays.
Whether the quasi-one-dimensionally ordered DCDW phase could be observed by an appropriate choice of probes
is still an open question.
Hence, it would be important to systematically explore how the collective modes
in the DCDW phase interact with external probes such as
hadrons (quarks) and photons (gauge fields).
There are also several theoretical issues, that deserve further study.
In particular, it is known that inhomogeneous chiral phases are favored
in systems with vector-vector type interactions,
which tend to enhance the size of the inhomogeneous area~\cite{Carignano:2010ac},
and that in the presence of an external magnetic field
the FF type phase is stabilized, also at finite temperatures, by topological aspects~\cite{Frolov:2010wn,Tatsumi:2014wka}.
Moreover, since two- and three-dimensional condensates with true long-range order are allowed at any temperature~\cite{landau1969statical}, it would be important to compare the free energy of such phases with that of a one-dimensional condensate.
It would also be interesting to understand how higher order interactions among the collective modes modify the soft modes.
These may affect the Landau-Peierls instability of inhomogeneous phases discussed here.
Finally, the topics discussed here may have an impact on the physics of compact stars. It has been speculated that various spatially inhomogeneous phases, like nuclear pasta phases~\citep{Nakazato:2009ed} and hadron-quark mixed phases~\cite{Yasutake:2009sh}, could be realized in the interior of such stars. These could have phenomenological implications, allowing, e.g., novel cooling scenarios~\cite{Tatsumi:2014cea}.
Thus, it would be interesting to study the properties of inhomogeneous chiral condensed phases under conditions relevant for neutron stars in general and compact stars with quark cores in particular, i.e., in charge neutral matter in $\beta$ equilibrium but also at nonzero isospin density~\cite{Abuki:2013vwa} and finite strangeness~\cite{Moreira:2013ura}.
These topics will be considered in future works.
TGL would like to thank R.~Yoshiike, T.~Maruyama, K.~Iida, and K.~Kamikado
for useful comments and discussions.
This work is partially supported by
Grant-in-Aid for Scientific Research on Innovative Areas thorough No.~24105008 provided by MEXT.
|
{
"timestamp": "2015-04-14T02:14:41",
"yymm": "1504",
"arxiv_id": "1504.03185",
"language": "en",
"url": "https://arxiv.org/abs/1504.03185"
}
|
\section{Introduction}
\parindent 20pt
Optimization is an essential process in scientific investigation. There are many effective and efficient methods proposed in the literature, see for example \cite{Polak1997}, \cite{MN2002}, \cite{BL2004}
and \cite{GS2007}. We propose a new algorithm that solves optimization problems from a point of view that is different than most of the major methods in the literature. The proposed method builds one dimensional paths or curves and travels on it with a constant speed to search for the global optimum.
The main idea of the proposed optimization algorithm is built upon geodesics which are a generalization of straight lines in Euclidean space that minimizes the
non-Euclidean distance between two points on a given manifold defined by the objective function.
The paths of optimization are constructed numerically by using a local quadratic approximation since there is no analytical solution to the geodesic equation in a general manifold. To avoid numerical instabilities in converting ill-conditioned matrices, the geodesics are constructed on a manifold under conformal mapping which preserves intrinsic geometrical features of the objective function. The algorithm will then follow either the geodesics or contours under conformal mapping to search for the optimum. Each contour curve could provide a bridge to a new search area and the constant speed enforced by the algorithm ensures that the search never stops or be trapped at a stationary point. Even with carefully constructed geodesics, the proposal algorithm can still be trapped within one region. In order to search another promising region, we also build a jumping mechanism by examining the values of the objective function along the geodesics to detect any potential and hidden influence to the geodesic flow from an nearby optimum.
The algorithm can be further improved by integrating with other search methods.
For example, one can use a few points along the geodesic as starting points to a Quasi-Newton algorithm to improve computational efficiency. From the Quasi-Newton outputs the algorithm is able to change its parameters adaptively for oscillating and smooth objective functions. We found that the resulting algorithm performs well in both types of functions in moderately high dimensions. Furthermore, we built a stopping criterion for the algorithm using Quasi-Newton methods by setting a threshold on the number of the maximum found within tolerance.
The remaining part of the paper is organized as follows.
In Section \ref{DiffGeo} we give an introductory review on differential geometry. Theoretical properties of the proposed algorithm are
established in Section \ref{theo}.
In Section \ref{Alg}, we give a general description of the algorithm and we also
describe the method of choosing the parameters adaptively.
Numerical results comparing the proposed algorithm
with the Quasi-Newton, genetic algorithm, wedge trust region methods and the global search function in Matlab's global optimization package are
provided in Section \ref{results}.
Finally, the conclusion is given in Section \ref{conclusion}.
\section{The Main Idea}
\label{DiffGeo}
We generalize the line search method with geodesics in order to discover multiple maxima on a manifold conformally related to $\mathbb{R}^n$.
\subsection{ Geodesics and Geodesic Equations}
We consider a topological manifold that is a second countable and locally compact Hausdorff space. It is also connected and completely regular. Detailed discussions can be found in \cite{Boothby2003} and \cite{Lee2010}.
A Riemannian metric on a smooth and differentiable manifold $M$ is a 2-tensor field ${\cal T}^2(M)$ that is symmetric and positive definite. A Riemannian metric thus determines an inner product on each tangent space $T_p(M)$, which is typically written as $g(U, V)$ for $U,V \in T_p(M)$. For an Euclidean space, the
metric matrix (or just metric henceforth) in component form is the Kronecka delta, i.e $g_{ij}=\delta_{ij}$. The inner product $g(U, V)$ in Euclidean reduces to the dot product, $\sum_{ij} \delta_{ij}U^i V^j$, where the sum is over all dimensions. In the Einstein summation convention, it is understood that repeated indices are summed over and the inner product is expressed as $g_{ij}U^i V^j$.
A geodesic is defined to be the path of minimum length for two given distinct points in a connected manifold. It is simply a straight line in Euclidean space. In a non-flat manifold, however, it is a curve and no longer a straight line.
Let $X^i(t)$ denote the local coordinate for the $i$-th dimension for a parameter $t$ which is a time step in our case.
The geodesic is then characterized by a set of partial differential equations, using the Einstein summation convention:
\begin{equation}
\frac{d^2 X^i(t)}{dt^2} + \Gamma^i_{jk} \frac{d X^j(t)}{dt} \frac{d X^k(t)}{dt} = 0,
\label{Geodesic}
\end{equation}
where $\Gamma^i_{jk}$ are Christoffel symbols defined to be
\[
\Gamma^i_{jk} = \frac{1}{2} g^{im}\bigg( \frac{\partial g_{mj}}{\partial x^k} + \frac{\partial g_{mk}}{\partial x^j} - \frac{\partial g_{jk}}{\partial x^m} \bigg),
\]
$g_{ij}$ is the metric and $g^{ij}$ is the inverse metric.
There exists a unique vector field on the tangent bundle of manifold, denoted as $TM$, whose trajectories are of the form $(\gamma(t), \gamma^{'}(t))$ where $\gamma$ is the geodesic.
Geodesics play an important role in General Relativity, see \cite{Foster2006}, where the path of a planet orbiting around a star is the projection of a geodesic of the curved 4-D spacetime geometry around the star onto a 3-D Euclidean space.
\subsection{ Conformal Mapping}
Numerical calculations of the Christoffel symbols involve the inversion of the metric and can be unstable and computationally costly. One strategy to avoid such complications is to calculate the Christoffel symbols in a manifold where the metric is easily inverted and then map the results to the manifold desired. We consider the case where the manifold containing information about the objective function is mapped from $\mathbb{R}^n$, where the metric and the inverse metric is the Kronecka delta. In such a case an analytic expression for the Christoffel symbols is available and the costly matrix inversion is avoided.
Each local neighborhood in the new manifold is holomorphic to $\mathbb{R}^n$. The resulting metric under the conformal mapping is said to be conformally related to the Euclidean metric
\[
g_{ij} = \Psi(x)^2 \delta_{ij},
\]
where the scale factor $\Psi(x) = e^{\phi(x)}$ and $\phi(x)$ is a real valued objective function. Trivially, the manifolds obtained this way are Riemannian as the metric tensors are both symmetric and positive definite. The existence of such mappings is trivial since we assume the existence of $\phi(x)$ to begin with.
\section{Theoretical Properties of the Geodesic in a Conformally Flat Metric}
\label{theo}
This section investigates the path of the geodesic by considering the direction of its tangent vector and the jumping mechanism used in the algorithm. Unless otherwise stated, the Einstein summation convention is used on all quantities in component form.
\subsection{The Geodesics under Conformal Mapping}
\label{attractor}
\paragraph{Theorem 3.1} {\bf The level curves and the gradient of the objective function $\phi$ are the attractors of the geodesics on any manifold conformally related to Euclidean space.}
\subsection{Jumping Mechanism}
Occasionally, the geodesic can be confined in the neighborhood of a local maximum. Here we discuss a method to estimate the direction of a neighboring maximum from the local maximum using a geodesic so that a jump can be implemented to restart the geodesic along that direction.
Let $l$ be the length of the geodesic, $\gamma$. We define the jumping direction to be along the vector
\[
\Delta \mathbf{x} := \frac{1}{l} \int_{\gamma} \widehat{\phi}(\mathbf{x}) \mathbf{x} \ d \mathbf{x} - \frac{1}{l} \int_{\gamma} \mathbf{x} \ d \mathbf{x},
\]
where the integral is over the geodesic and $\widehat{\phi}$ being the normalized $\phi$ over $t$, . In practice, this is approximated by the sum over all steps along the geodesic
\[
\Delta \mathbf{x} \simeq \frac{1}{T} \sum_{t=1}^{T} [ \widehat{\phi}(\mathbf{x}_t) \mathbf{x}_t - \mathbf{x}_t],
\]
where $T$ is the total number of steps and $\mathbf{x}_t \in \gamma$. This is just the difference of the weighted mean and the mean position vectors along the geodesic. Suppose that a neighboring maximum exists and the geodesic is symmetric about a local maximum (as it would usually be the case if the geodesic is trapped, for instance, as in Figure \ref{ST}). Then the weighted mean would be slightly biased towards the neighboring maximum. And so $\Delta \mathbf{x}$ would be pointed towards the neighboring maximum. We used a decreasing jump distance for each jump. This is by no means an accurate estimate of the direction to the next maximum. However it has been proven to be sufficient for our algorithm to discover the global maximum in many objective functions of multiple maxima.
\subsection{Solving the Geodesic Equation with a Quadratic Approximation}
\label{stepsizes}
In this subsection we discuss the quadratic approximation used to solve the geodesic equation iteratively. We give an estimation of the adaptive step sizes to ensure that the approximation is valid.
The geodesic equation is
\begin{equation}
\frac{d^2 x^i(t)}{dt^2} + \Gamma^i_{jk} \frac{d x^j(t)}{dt} \frac{d x^k(t)}{dt} = 0.
\label{Geq}
\end{equation}
In the neighborhood of $\mathbf{x}_t$, the (discretized) approximation to the solution of the geodesic equation is
\begin{equation}
\label{soln}
\mathbf{x}_{t+1} = \mathbf{x}_t + \mathbf{v}_t \delta t + \mathbf{c}_t (\delta t)^2,
\end{equation}
where $\mathbf{v}_t$ is the unit tangent vector of the geodesic at $\mathbf{x}_t$, $\delta t$ is the step size, and
\begin{equation}
\mathbf{c}_t = \frac{1}{2} \frac{d \mathbf{v}}{dt} = \frac{1}{2}[\nabla \phi(\mathbf{x}_t) - 2(\mathbf{v}_t \cdot \nabla \phi(\mathbf{x}_t)) \mathbf{v}_t].
\end{equation}
The tangent vector is estimated by the (normalized) difference $\mathbf{x}_{t} - \mathbf{x}_{t-1}$ and we set the initial tangent vector to be the gradient, $\mathbf{v}_{t=1} = \nabla \phi(\mathbf{x}_{t=1})$. Note that the quadratic approximation (\ref{soln}) is not valid when there exist a component $i$ such that $O(v_{ti} \delta t) \gg O(c_{ti}(\delta t)^2)$, as the approximation is only accurate when the quadratic term is small. The value of $\delta t$ when the linear term equals to the quadratic term in magnitude is $\delta t = t_C$, where
\[
t_C = \min_i \bigg|\frac{v_{ti}}{c_{ti}}\bigg|.
\]
If the quadratic term has opposite sign to the linear term, then $x^i_{t+1} = x^i_{t}$ when $\delta t = t_C$ for some component $i$. Also, when $\delta t = \frac{1}{2} t_C$, $x^i_{t+1} - x^i_t$ is maximized. Therefore, at every time step, the algorithm chooses a step size of $\delta t = \frac{1}{2} t_C$, if it is not smaller than the specified lower bound on $t$ (to be explained in the algorithm section).
Since the geodesic aligns itself with the gradient (or the level curves) as shown in the Section \ref{attractor}, in both cases, the step sizes are
\[
||\mathbf{x}_{t+1} - \mathbf{x}_t || = \frac{3}{4} \frac{1}{| \nabla \phi| }.
\]
It can be found by substituting $\delta t = \frac{1}{2} t_C$ into Equation \ref{soln} and setting $v_i = 1$ in the component parallel to the gradient (or the level curves). Furthermore, as the geodesic travels towards a maximum following the gradient, it has a linear rate of convergence similar to gradient descent.
\section{Algorithm}
\label{Alg}
The algorithm has two parts. The first is a geodesic guided optimization (GEO) algorithm. It estimates the geodesic using the quadratic approximation for a total of $T$ steps. The step size is adaptive and bounded by the validity of the quadratic approximation. Quasi-Newton (QN) optimization may be performed, using points along the geodesic as the starting points. Figure \ref{cam} shows the estimated geodesic moving through three local maxima. The algorithm returns the location of the maximum and its function value along the geodesic, or the one obtained by QN, whichever is the largest.
\begin{figure}
\caption{Geodesic traversing through multiple local maxima in the search space.}
\label{cam}
\end{figure}
\subsection{Choosing Parameters}
There are cases where the geodesic fails to reach multiple maxima, for instance, as in Figure \ref{ST}. When the objective function is highly oscillatory, the global maximum is less likely to be found by the geodesic. Furthermore, a lower bound on the step size $\delta t_{LB}$ must be specified as an input parameter to prevent the step size to become impractically small in regions of large gradient, but the choice of an appropriate lower bound for any objective function is difficult (if not impossible) to determine. Intuitively, a large $\delta t_{LB}$ would allow the geodesic to escape from local fluctuations. On the other hand, it may prevent the geodesic from visiting the global maximum.
The second part of the algorithm, Sequential GEO (SGEO), is developed to overcome these difficulties. Information from the geodesic is obtained and passed to SGEO. It includes an estimate of the direction of a neighboring local maximum, $\widehat{\Delta \mathbf{x}}$, an indicator, $k$, to denote whether the geodesic is trapped in a local maximum, and the average distance between the starting points and the end points of QN, $\bar{R}$, to determine whether the objective function is oscillatory.
\begin{figure}
\caption{Geodesic trapped in a local maximum.}
\label{ST}
\end{figure}
SGEO calls GEO sequentially with decreasing $\delta t_{LB}$ for $N$ times. In each subsequent run, $\delta t_{LB}$ is reduced by a factor of $\alpha$, determined by requiring that $\delta t_{LB}$ in the last run to be 1000 times smaller than that in the first run. The next GEO run starts from a position obtained by translating $\mathbf{x^*}$ along $\widehat{\Delta \mathbf{x}}$, with the magnitude and method of the jump determined by $k$. The two mechanisms described above assist in the escape from local maxima. In the case of oscillatory functions, QN is not performed, allowing for a larger number of GEO runs. The algorithm first assumes a non-oscillatory function, and adaptively adjusts its parameters suitable for an oscillatory function by setting a threshold on $\bar{R}$. Finally, we impose a stopping criterion to improve its computational cost. The technical details are described in the follow subsections. The only inputs needed are the number of GEO runs, $N$, total number of steps $N_T$, the stopping threshold, $N_{th}$, which only depends on the dimensionality, and the initial $\delta t_{LB}$ which is only dependent on the dimensionality and the size of the search region.
\subsection{The first component, GEO}
GEO estimates the geodesic corresponding to a conformally Euclidean metric with the conformal factor given by the objective function up to $T$ steps and evaluates the objective function $\phi(\mathbf{x}_t)$ at every time step, $t$, along the geodesic.
At each $t$, the normalized tangent vector is used to evaluate $\mathbf{x}_{t+1}$ in Equation \ref{soln}. Since the step size is at most of order $O(1 / \nabla_i \phi)$ as discussed in Section \ref{stepsizes}. The geodesic tends to get trapped in regions of large gradient. To avoid this problem we introduce a lower bound on the step size, $\delta t_{LB}$, so that the step size is $\delta t = \max \{0.5 t_C, \delta t_{LB}\}$. The lower bound also ensures that the geodesic has length of at least $T \delta t$.
Now, consider the case where $t_{LB} \gg t_C$. The trajectory is dominated by the quadratic term $\mathbf{c}_t (\delta t)^2$. But at $t=1$, the tangent vector is the unit gradient and so $\mathbf{c}_{t=1} = - \nabla \phi /2$, the geodesic moves against the gradient. An additional minus sign is introduced to $\mathbf{c}_t$ whenever $t_{LB} > 0.5 t_C$ at $t=1$. A backward geodesic that initially moves against the gradient is also estimated by using the initial condition $\mathbf{v}_{t=1} = - \nabla \phi$.
For both the forward and backward geodesics, Quasi-Newton optimization can be performed at every $T_{QN}$ steps. Whenever $\mathbf{x}_{t+1}$ is outside the search region, the algorithm sets $\mathbf{x}_{t+1}$ to be a random point sampled uniformly within the search region. The following quantities are also evaluated to pass to SGEO, the mean distance between the Quasi-Newton initial position and the solution $\bar{R}$, the normalized mean of $\phi(\mathbf{x}_t)\mathbf{x}_t - \mathbf{x}_t$ over all $t$, and an integer $k \in \{0 , 1 ,2\}$ which characterizes the degree of locality of the geodesics,
\[
k = \begin{cases}
0 & \mbox{if $\phi^*_t$ is not unique in the forward geodesic within tolerance $\forall t$. } \\
1 & \mbox{if $\phi^*_t$ is unique in only the forward geodesic within tolerance $\forall t$.} \\
2 & \mbox{if $\phi^*_t$ is unique in both geodesics within tolerance $\forall t$,}
\end{cases}
\]
where $\phi^*_t$ is the larger of $\phi(\mathbf{x}_t)$ and the optimized value with QN. The algorithm returns the maximum objective function value and its position along the geodesics as well as the information needed to pass onto SGEO.
\subsection{Algorithm 1: GEO, Geodesic Guided Optimization}
Matlab's fminunc() function is used for the Quasi-Newton optimization. Its input parameters are set as $MaxIter = 200, Tol_{f} = 0.05, Tol_X = 0.01$, where the last two quantities are the tolerances in $\phi$ and $\mathbf{x}$, respectively.
The set of input parameters for GEO is
\begin{itemize}
\item $T$, the number of time steps along the geodesic,
\item $T_{QN}$, the number of time steps between each QN call.
\item $\mathbf{x}_0$, a vector the initial point of the geodesic,
\item $\phi(\cdot)$, the objective function,
\item $\nabla \phi(\cdot)$, the gradient of the objective function,
\item $\mathbf{L}$, a vector containing the lower bounds of the search space,
\item $\mathbf{U}$, a vector containing the upper bounds of the search space,
\item $\delta t_{LB}$, the lower bound on the step size,
\item $s_{QN}$, boolean variable denoting whether QN is performed.
\end{itemize}
The algorithm is as follows
\begin{enumerate}
\item For $t = 1:T$
\begin{enumerate}
\item Calculate $\phi(\mathbf{x}_t)$ and $\nabla \phi(\mathbf{x}_t)$.
\item If $t = 1$, set $\mathbf{x}_t := \mathbf{x}_0$, $k := 1$ and $\delta \mathbf{x}_t := \nabla \phi(\mathbf{x}_t)$,
\begin{enumerate}
\item else, set $\delta \mathbf{x}_t := \mathbf{x}_t - \mathbf{x}_{t-1}$.
\end{enumerate}
\item If $mod(t, T_{QN}) = 0$ and $s_{QN} = 1$,
\begin{enumerate}
\item call $QN(\phi(\cdot), \mathbf{x}_t)$ and obtain $\{\phi^{*(1)}_t, \mathbf{x}^*\}$,
\item set $R^{(1)}_{m} := ||\mathbf{x}^* - \mathbf{x}_t||$ and set $ m := m+1$,
\item else set $\phi^{*(1)}_t := \phi(\mathbf{x}_t)$.
\end{enumerate}
\item Calculate the normalized tangent vector $\mathbf{v}_t := \delta \mathbf{x}_t / ||\delta \mathbf{x}_t||$.
\item Calculate $\mathbf{c}_t := \frac{1}{2}[\nabla \phi(\mathbf{x}_t) - 2(\mathbf{v}_t \cdot \nabla \phi(\mathbf{x}_t))\mathbf{v}_t]$.
\item Set $t_C := \min_i |v_{ti}/c_{ti}|$, where $i \in \{1,\ldots, D\}$ denotes the $i$-th component.
\item Set the step size $\delta t := \max \{\frac{1}{2} t_C, \delta t_{LB}\}$.
\item If $\delta t = \delta t_{LB}$ and $t=1$, change the sign of $\mathbf{c}_t := -\mathbf{c}_t$.
\item Calculate $x_{t+1} := \mathbf{x}_t + \mathbf{v}_t \delta t + \mathbf{c}_t (\delta t)^2$.
\item If $x_{(t+1)i} < L_i$ or $x_{(t+1)i} > U_i$ for any component $i$, sample $\mathbf{x}_{(t+1)}$ from a uniform distribution in $[\mathbf{L}, \mathbf{U}]$.
\item Calculate $\Delta \mathbf{x}^{(1)} := \frac{1}{T}[\sum_{t = 1}^T (\mathbf{x}_t \widehat{\phi}^{*(1)}_t - \mathbf{x}_t)] $, where $\widehat{\phi}^{*(1)}_t = \phi^{*(1)}_t / \sum \phi^{*(1)}_t$
\item Set $\phi^{*(1)} = \max \phi_t^{*(1)}$.
\item If $| \phi^{*(1)} - \phi_t^{*(1)}| < Tol_{f} \phi^{*(1)}$ for all $t$, then set $k =1$. Else set $k = 0 $.
\end{enumerate}
\item For the backward geodesic, step (1) is repeated with the following adjustments,
\begin{enumerate}
\item $\delta \mathbf{x}_t$ is defined as $\delta \mathbf{x}_t := -\nabla \phi(\mathbf{x}_t)$, in step (1b),
\item $\phi^{*(2)}_t := QN(\phi(\cdot), \mathbf{x}_t)$ in step (1c(i)),
\item $R^{(2)}_{m} := ||\mathbf{x}^* - \mathbf{x}_t||$ in step (1c(iii)), and
\item omitting step (1h).
\item $\Delta \mathbf{x}^{(2)} := \frac{1}{T}[\sum_{t = 1}^T (\mathbf{x}_t \widehat{\phi}^{*(2)}_t - \mathbf{x}_t)] $ in step (1k).
\item Set $\phi^{*(2)} = \max \phi_t^{*(2)}$ in step (1l).
\item If $| \phi^{*(2)} - \phi^{*(1)}| < Tol_{f} \phi^{*(2)}$ for all $t$ and $k =1$, set $k = 2 $.
\end{enumerate}
\item Set $\Delta \mathbf{x} := \frac{1}{2}[\Delta \mathbf{x}^{(1)} + \Delta \mathbf{x}^{(2)}]$.
\item Return $\phi^* := \max \{\phi^{*(1)}, \phi^{*(2)}\}, \mathbf{x}^* := \{ \mathbf{x} | \phi(\mathbf{x}) = \phi^*\}, \bar{R} := \textrm{mean} \{ \mathbf{R}^{(1)},\mathbf{R}^{(2)}\}$, $\widehat{\Delta \mathbf{x}} := \Delta \mathbf{x} / || \Delta \mathbf{x} ||$ and $k$.
\end{enumerate}
\subsection{The second component, SGEO}
SGEO runs GEO sequentially with different parameters. Let $\mathbf{U}$ and $\mathbf{L}$ be vectors denoting the upper and lower bounds of the search space and let $\Lambda = \min (\mathbf{U} - \mathbf{L})$. This algorithm checks whether the objective function is highly oscillating. We use the following criterion that an oscillatory function must satisfy: $\bar{R} < 0.1 \Lambda \sqrt{D}$ in any the first two GEO calls. The reason for limiting to just the first two GEO runs is that $\delta t_{LB}$ gets smaller after each consecutive runs and it is more likely for $\bar{R}$ to be small even for non-oscillatory functions. In the case of a high dimensional ($D > 10$) oscillatory function, no Quasi-Newton optimization is performed to allow for a higher number of GEO runs. Both of which are crucial in locating the global optimum of highly oscillating functions.
The algorithm uses a procedure similar to annealing to reduce $\delta t_{LB}$ for each GEO run. Initially, $\delta t_{LB}^{(n=0)}$ is set to be $\Lambda \sqrt{D} / 100$, where $n$ denotes the $n$-th GEO run. Then the lower bound on $\delta t$ is lowered such that $\delta t_{LB}^{n} = \alpha^n \delta t_{LB}^{(n=0)}, \alpha \in (0,1)$. The factor $\alpha$ is chosen such that $\delta t_{LB}^{(n=N)}/ \delta t_{LB}^{(n=0)} = 10^{-3}$.
After each GEO call, the initial value, $\mathbf{x}_0^{(n)}$, for the next GEO call is estimated depending on the value of $k$ passed from GEO. Intuitively, $\Delta \mathbf{x}$ would be a vector pointing roughly towards a neighboring maximum. For $k = 0$, the local geodesic is not trapped,
\[
\mathbf{x}_0^{(n+1)} = \mathbf{x}^{*(n)} + (\alpha^{n} \Lambda) \widehat{\Delta \mathbf{x}}^{(n)}.
\]
For $k=1$, the forward geodesic is trapped and we set $\mathbf{x}_0^{(n+1)}$ to be further away from $\mathbf{x}_0^{(n)}$,
\[
\mathbf{x}_0^{(n+1)} = \mathbf{x}^{*(n)} + (\alpha \Lambda) \widehat{\Delta \mathbf{x}}^{(n)}.
\]
Finally for $k=2$, when both the backward and forward geodesics are trapped, the method using $\Delta \mathbf{x}$ becomes ineffective as the objective function has similar values along the geodesics - $\Delta \mathbf{x}^{(n)}$ points in the same direction as $\mathbf{x}^{(n)}$. Therefore we simply set $\mathbf{x}_0^{(n+1)}$ to be a point reflected across the midpoint of the search space from $\mathbf{x}_0^{(n)}$,
\[
\mathbf{x}^{(n+1)}_0 := \frac{\mathbf{L} + \mathbf{U}}{2} + \alpha^n \Lambda(\frac{\mathbf{L} + \mathbf{U}}{2}- \mathbf{x}^{(n)}_0).
\]
A stopping criterion is imposed to reduce the computational cost. Let $\Phi^{(n)} = \{\phi^{*(n=1)}, \ldots, \phi^{*(n)} \}$ be a series of maxima found up to the $n$-th GEO run, $\phi^* = \max \Phi^{(n)}$ and $N^*$ be the number of elements in $\Phi^{(n)}$ that are within tolerance of $\phi^*$. The algorithm is stopped if $N^* > N_{th}(D)$, where
\[
N_{th}(D) = \begin{cases}
5 & D < 10 \\
10 & 10 \leq D < 20 \\
20 & 20 \leq D \leq 50.
\end{cases}
\]
The algorithm returns $\phi^*$ and $x^*= \{ \mathbf{x} |\phi( \mathbf{x}) = \phi^*\}$.
\subsection{Algorithm 2: SGEO, Sequential Geodesic Optimization} Let $\Lambda= \min (\mathbf{U} - \mathbf{L})$, the set of input parameters is
\begin{itemize}
\item $N$, the number of GEO runs,
\item $N_T$, total step number,
\item $\delta t_{LB}^{(n=0)}$, starting lower bound on the step size,
\item $N_{th}(D)$.
\end{itemize}
The algorithm is
\begin{enumerate}
\item Set $\Lambda := \min (\mathbf{U}-\mathbf{L})$, then set $\{ N, \alpha, N_T, \delta t_{LB}^{(n=0)}, T_{QN},s_{QN}\} = \{ 20, 0.7, 500, \Lambda \sqrt{D} / 100, 10,1\}$.
\item Calculate $T := \lfloor \frac{N_T}{N} \rfloor$ and sample $\mathbf{x}_0^{(n=1)}$ uniformly in $[\mathbf{L}, \mathbf{U}]$.
\item For $n = 1:N$
\begin{enumerate}
\item Calculate $\delta t^{(n)}_{LB} := \alpha \delta t_{LB}^{(n-1)}$.
\item Obtain $\{\phi^{*(n)}, \mathbf{x}^{*(n)}, \bar{R}, \widehat{\Delta \mathbf{x}}^{(n)},k \} $ by calling GEO($\mathbf{x}^{(n)}_0,\delta t^{(n)}_{LB},s_{QN}, T, T_{QN}$).
\item If $D > 10$, $\bar{R} < 0.1 \Lambda \sqrt{D}$, $n \leq 2$ and $s_{QN} = 1$,
\begin{enumerate}
\item set $\{ N, \alpha, N_T,s_{QN}\} := \{ 400, 0.98, 4000,0\}$ and
\item break and restart current loop with the parameters in the above step in place of those in step 1.
\end{enumerate}
\item If $k = 0$, set $\mathbf{x}^{(n+1)}_0 := \mathbf{x}^{*(n)} + (\alpha^n \Lambda) \widehat{\Delta \mathbf{x}}^{(n)}$. Else if $k = 1$, set $\mathbf{x}^{(n+1)}_0 := \mathbf{x}^{*(n)} + (\alpha \Lambda) \widehat{\Delta \mathbf{x}}^{(n)}$. Else set $\mathbf{x}^{(n+1)}_0 := \frac{\mathbf{L} + \mathbf{U}}{2} + \alpha^n \Lambda(\frac{\mathbf{L} + \mathbf{U}}{2}- \mathbf{x}^{(n)}_0)$.
\item If $n = 1$, set $\phi^* = \phi^{*(n=1)}$. Else set $\phi^* := \max \{ \phi^*, \phi^{*(n)}\}$.
\item Find $N^*$, the number of instances such that $|\phi^* - \phi^{*(n')}| < Tol_{f} \phi^*$, $n' \in \{1, \ldots, n\}$.
\item If $N^* \geq N_{th}(D)$, exit loop.
\end{enumerate}
\item Return $\phi^*$ and $\mathbf{x}^* := \{ \mathbf{x} |\phi( \mathbf{x}) = \phi^*\}$.
\end{enumerate}
\section{Numerical Experiments}
\label{results}
In this section we compare SGEO with other algorithms on test functions commonly used in the literature. The objective functions\footnote{These are obtained from http://www.sfu.ca/$\sim$ssurjano/optimization.html. We have used the log of the Hartman and Goldstein-Price functions as the values of these functions vary across five orders of magnitude within the search space.} used have dimensions ranging from 2 to 50. Sixteen of which are reasonably smooth, the other twelve are oscillatory. All calculations are performed in Matlab.
Table \ref{tab:1} shows the number of failures in locating the global maximum for existing methods. A success is defined when the estimated function value is within 5\% of the maximum value. If the maximum value is zero, a success corresponds to finding a function value less than 0.05. As it can be seen in Table \ref{tab:1}, the Global Search (GS) method in Matlab's global optimization toolbox outperforms Quasi-Newton (QN), Trust Region (TR), and Genetic Algorithm (GA).
In Table \ref{tab:2} we justify the use of QN and the jumping mechanism in SGEO. The algorithm without QN fails in high dimensions whereas without jumping the algorithm fails in oscillatory cases.
For finding a global maximum from the chosen test functions, we found that the global search method is the best among all commonly used optimization method. We therefore did an extensive comparison with the global search method in Table \ref{tab:3}. We found that SGEO can discover the true global maximum with a higher chance than GS in the objective functions tested. Our method is more accurate than the global search in many test functions. There is only one function that our method is not as good as the global search. At the same time, the computational cost represented by the number of function calls and the computational time remains similar in most cases.
\begin{table}[ht]
\caption{Number of failures over 100 runs for Quasi-Newton (QN), Trust Region (TR), Genetic Algorithm (GA), and Global Search (GS).}
\label{tab:1}
\end{table}
\begin{table}[ht]
\caption{Number of failures over 50 runs for SGEO, without Quasi-Newton (with $T=200$), and without jumping.}
\label{tab:2}
\end{table}
\begin{table}[ht]
\caption{The number of failures ($N_{failure}$), computational time, and the number of function calls ($N_{call}$) over 50 runs of SGEO and GS.}
\label{tab:3}
\end{table}
\section{Conclusion}
\label{conclusion}
A new algorithm is proposed in order to find multiple optima of a continuous objective function. The path constructed by the algorithm follows either a geodesic or a contour line. Conformal mapping and the Newton Raphston algorithm are employed to enhance computational efficiency. A built-in jumping mechanism also directs the proposed algorithm to a more promising search area. A stopping criterion is implemented if the same maximum is found too many times.
We are extending this algorithm to handle optimization in high dimensions with contestation.
\section{Acknoledgements}
This research is supported by the Discovery Grants and Discovery Accelerator Supplement from Natural Sciences and Engineering Research Council of Canada (NSERC).
\vskip 0.2in
|
{
"timestamp": "2015-04-15T02:01:07",
"yymm": "1504",
"arxiv_id": "1504.03355",
"language": "en",
"url": "https://arxiv.org/abs/1504.03355"
}
|
\section{Introduction}
\label{sec:1}
This final chapter summarizes areas of major progress in understanding galaxy bulges
and tries to distill the important unresolved issues that need further work.
I do not revisit the subjects covered by all chapters --
Madore (2015: historical review),
M\'endez-Abreu (2015: intrinsic shapes),
Falc\'on-Barroso (2015: kinematic observations),
S\'anchez-Bl\'azquez (2015: stellar populations),
Laurikainen \& Salo (2015: observations of boxy bulges),
Athanassoula 2015: modeling of boxy bulges),
Gonzalez \& Gadotti (2015: observations of the Milky Way boxy bulge),
Shen \& Li (2015: modeling of the Milky Way boxy bulge),
Cole \& Debattista (2015: nuclear star clusters), and
Combes (2015: bulge formation within MOND).
I comment briefly on Zaritsky's (2015) chapter on scaling relations.
I concentrate in this summary chapter on three main areas of progress and on two main
areas where there are unresolved difficulties:
Two additions to our picture of bulge formation are (1) formation by massive clump instabilities
in high-$z$ disks; Bournaud (2015) develops this story, but it deserves emphasis here, too, and
(2) our picture of secular evolution of galaxy disks that produces two distinct kinds of dense central
components in galaxies, disky pseudobulges (reviewed here by Fisher \& Drory 2015) and boxy pseudobulges
(discussed in four chapters listed above). Both deserve emphasis here, too.
The main areas with unresolved issues come in two varieties:
Probably the most important chapter in this book is Brooks\ts\&{\ts}Christensen (2015)
on the modeling of galaxy{\ts}--{\ts}and thus also bulge{\ts}--{\ts}formation.~These~models
that add baryonic physics to giant $n$-body simulations of the hierarchical clustering of
cold dark matter (CDM) in a $\Lambda$CDM universe define
the state of the art in the most general version of galaxy formation theory. Much has
been accomplished, and progress is rapid. Brooks \& Christensen (2015) is an excellent
review of the state of the art as seen by its practitioners. In this chapter, I
would like to add the viewpoint of an observer of galaxy archaeology. I suggest
a slightly different emphasis on the successes and
shortcomings of present models. My main purpose is to promote a dialog between theorists
and observers that may help to refine the observational constraints that are most telling and
the modeling exercises that may be most profitable. Baryonic galaxy formation is an
extraordinarily rich and difficult problem. Many groups struggle honorably
and carefully with different aspects of it. In this subject, besides a strong push on
remaining limitations such as resolution, the main need seems to me to be a broader
use of observational constraints and a consequent refinement of the physics that may succeed
in explaining them.
A second issue involves Graham's (2015) chapter on supermassive black holes.
It is inconsistent with all other work that I am aware of on this subject, including
McConnell \& Ma (2013) and Kormendy \& Ho (2013). Section 6 summarizes this subject
using results from Kormendy \& Ho (2013, hereafter KH13).
Section 7 reviews the quenching of star formation in galaxies. Many different lines of research
are converging on a consistent picture of how quenching happens.
Finally, I conclude with a personal view of the most important, big-picture issues that are
still unsolved by our developing picture of galaxy evolution.
\section{Secular Evolution and the Formation of Pseudobulges}
\label{sec:2}
Progress on bulge formation is dominated by two conceptual advances. This section revisits
secular evolution in disk galaxies. This is a major addition~that~complements our picture of galaxy
evolution by hierarchical clustering. I begin here because {\it all further discussion depends on the
resulting realization that the dense central components in galaxies come in two varieties with different
formation processes, classical and pseudo bulges.} Section 3 discusses the second conceptual advance,
the discovery of a new channel for the formation of classical bulges. This is the formation at high $z$
of unstable clumps in gas-rich disks; they sink to the center along with lots of disk gas and starburst
and relax violently. In this way, bulge formation proceeds largely as it does during major mergers.
This leads to a discussion of the merger formation of both bulges and ellipticals in Section\ts4.
Our pictures of the merger formation of classical bulges and ellipticals~and~the secular growth of
pseudobulges out of disks both got their start in the late 1970s. The importance of major mergers
(Toomre \& Toomre 1972; Toomre 1977) in a hierarchically clustering universe (White \& Rees 1978) got a
major boost from the realization that CDM halos make galaxy collision cross sections much bigger than they look.
This subject ``took off'' and rapidly came to control our formation paradigm. Secular evolution is a
more difficult subject -- slow processes are hard to study -- and it did not get a similar boost from the
CDM revolution. However, the earliest papers on the subject come from the same time period: e.{\ts}g.,
Kormendy (1979a) emphasized the importance of slow interactions between nonaxisymmetric galaxy components;
Kormendy (1979b) first pointed out the existence of surprisingly disky bulges;
Combes \& Sanders (1981) showed that boxy pseudobulges are edge-on bars.
Kormendy (1981, 1982) reviewed and extended the results on disky bulges.
This subject did not penetrate the galaxy formation folklore; rather, it remained a series of active
but unconnected ``cottage industries'' for the next two decades. Nevertheless, by the 1990s, the
concept{\ts}--{\ts}if not yet the name\ts-- of disky pseudobulges was well established (see Kormendy 1993 for a review),
and the idea that boxy bulges are edge-on bars was well accepted (see Athanassoula 2005 for a
more recent and thorough discussion). I hope it is fair to say that the comprehensive review by Kormendy
\& Kennicutt (2004) has helped to convert this subject into a recognized paradigm -- it certainly is so in this
book -- although it is still not as widely understood or taken into account as is hierarchical clustering.
Kormendy \& Kennicutt (2004) remains up-to-date and comprehensive on the basic results and on observations
of prototypical pseudobulges. However, new reviews extend and complement it. Kormendy \& Fisher (2005, 2008)
and Kormendy (2008, 2012) provide the most important physical argument that was missing in Kormendy \& Kennicutt (2004):
Essentially all self-gravitating systems evolve toward more negative total energies (more strongly bound configurations)
by processes that transport kinetic energy or angular momentum outward. In this sense, the secular
growth of pseudobulges in galaxy disks is analogous to the growth of stars in protostellar disks, the growth of
black holes in black hole accretion disks, the sinking of Jupiters via the production of colder
Neptunes in protoplanetary disks, core collapse in globular clusters, and the evolution of stars into red
(super)giants with central proto white dwarfs, \hbox{neutron stars, or stellar-mass} black holes. All of these evolution
processes are related. So secular disk evolution and the growth of pseudobulges is very
fundamental, provided that some process redistributes angular momentum~in~the~disk.
My Canary Islands Winter School lectures (Kormendy 2012) are an \hbox{up-to-date} observational review that
includes environmental secular evolution. Sellwood (2014) provides an excellent theoretical review.
Boxy pseudobulges are discussed in four chapters of this book; I concentrate on disky pseudobulges.
Fisher \& Drory (2015) review the distinction between classical and pseudo bulges from a
purely phenomenological point of view.~That~is,~they intercompare observational diagnostics to distinguish
between the two bulge types with no reference to physical interpretation. This is useful, because
it gives relatively unbiased failure probabilities for each diagnostic. They are not wholly
independent, of course, because they are intercompared. But they are independent enough in execution so that we get a
sufficient estimate of the failure probability when they are combined by multiplying the individual failure probabilities.
Kormendy \& Kennicutt (2004), Kormendy (2012), and KH13 strongly advocate the use of as
many bulge classification criteria as possible. The reason is that any one criterion has a non-zero
probability of failure. Confusion in the literature (e.{\ts}g., Graham 2011) results from the fact that
some authors use a single classification criterion (e.{\ts}g., S\'ersic index) and so get results that conflict
with those derived using multiple criteria. But we have long known that most classical bulges have $n \geq 2$,
that most pseudobulges have $n < 2$, and that there are exceptions to both criteria. No-one should be
surprised that S\'ersic index sometimes fails to correctly classify a bulge. This is the point that
Fisher \& Drory (2015) make quantitative.
Fisher \& Drory (2015) show that the failure probability of each classification criterion
that they test is typically 10\ts--\ts20\ts\%. A few criteria are completely robust (if $B/T$ $_>\atop{^\sim}$} \def\lapprox{$_<\atop{^\sim}$ 0.5,
then the bulge is classical) and a few are less reliable (star formation rate cannot be used for S0s). But,
by and large, it is reasonable to conclude that the use of $M$ criteria, each with failure probability~$\epsilon_m$,
results in a classification with a failure probability of order the product of the individual
failure probabilities, $\Pi_1^M \epsilon_m$. This becomes very small very quickly as $M$ grows even
to 2 and especially to $M > 2$. For example, essentially all bulge-pseudobulge classifications in
KH13 were made using at least two and sometimes as many as five criteria.
Fisher \& Drory (2015) also contribute new criteria that become practical as new technology
such as intergral-field spectroscopy gets applied to large samples of galaxies. These are incorporated
into an enlarged list of classification criteria below.
A shortcoming of Fisher \& Drory's approach is that it is applied without regard to galaxy Hubble types.
But we know that both many S0s and many Sbcs contain pseudobulges, but the latter all tend to be star-forming
whereas the former generally are not. This is one reason for their conclusion (e.{\ts}g.)~that high star
formation rate near the galaxy center robustly implies a pseudobulge, but no star formation near the center
fails to prove that the bulge is classical. Classification criteria that involve gas content and star formation
rate cannot be applied to S0 galaxies. Application to Sas is also fragile. Fortunately, most criteria do work
for early-type galaxies.
\subsection{Enlarged List of Bulge-Pseudobulge Classification Criteria}
\label{subsec:2}
Kormendy \& Kennicutt (2004), Kormendy (2012), and Fisher \& Drory (2015) together provide the following
improved list of (pseudo)bulge classification criteria. I note again: The failure rate for individual criteria
ranges from 0\ts\% to roughly 25\ts\%. Therefore the use of more criteria quickly gives much more reliable results.
\begin{enumerate}
\item[(1){\kern -3pt}]{If the galaxy center is dominated by young stars and gas
but there is no sign~of a merger in progress, then the bulge is mostly pseudo.
Ubiquitous star formation must be secular. Fisher \& Drory (2015) make this
quantitative: if the specific star formation rate sSFR $\geq$ $10^{-11}$ yr$^{-1}$,
then the bulge is likely to be pseudo; whereas if sSFR $<$ $10^{-11}$ yr$^{-1}$, then
the bulge is likely to be classical. Also, if the bulge is very blue,
$B - V < 0.5$, then it is pseudo. Criteria (1) cannot be used for S0s. }
\item[(2){\kern -3pt}]{Disky pseudobulges (a) generally have apparent flattening similar to that of the outer disk
or (b) contain spiral structure all the way to the galaxy center. Classical bulges
are much rounder than their disks unless they are seen almost face-on, and they cannot
have spiral structure. Criterion 2(a) can be used for S0s; 2(b) can not.}
\item[(3){\kern -3pt}]{Pseudobulges are more rotation-dominated than are classical bulges
in the \hbox{$V_{\rm max}/\sigma$\ts--\ts$\epsilon$} diagram; $V_{\rm max}$ is
maximum rotation velocity, $\sigma$ is near-central velocity dispersion, and
$\epsilon$ is ellipticity.
Integral-field spectroscopy often shows that the central surface brightness excess over the inward
extrapolation of the disk profile is a flat central component that rotates rapidly and has
small $\sigma$.}
\item[(4){\kern -3pt}]{Many pseudobulges are low-$\sigma$ outliers in the Faber-Jackson (1976) correlation between (pseudo)bulge
luminosity and velocity dispersion. Integral-field~spectra often show that $\sigma$
decreases from the disk into a pseudobulge. Fisher and Drory make this
quantitative: Pseudobulges have rather flat logarithmic derivatives of the dispersion
profile $d{\log{\sigma}}/d{\log{r}} \geq -0.1$ and $V^2/\sigma^2 \geq 0.35$.
In contrast, if $d{\log{\sigma}}/d{\log{r}} < -0.1$ or if central
$\sigma_0 > 130$ km s$^{-1}$, then the bulge is classical.
\item[(5){\kern -3pt}]{Small bulge-to-total luminosity ratios do not guarantee that a bulge
is pseudo,~but almost all pseudobulges have $PB/T$ \lapprox \ts0.35.~If $B/T$ $_>\atop{^\sim}$} \def\lapprox{$_<\atop{^\sim}$ \ts0.5,
the bulge is classical.}
\item[(6){\kern -3pt}]{Most pseudobulges have S\'ersic index $n < 2$; most classical bulges have $n \geq 2$.}
\item[(7){\kern -3pt}]{Classical bulges fit the fundamental plane correlations for elliptical
galaxies. Some pseudobulges do, too, and then the correlations are not useful for
classification. More extreme pseudobulges are fluffier than classical bulges;
they have larger effective radii $r_e$ and fainter effective surface brightnesses $\mu_e$.
These pseudobulges can be identified using fundamental plane correlations.}
\item[(8){\kern -3pt}]{In face-on galaxies, the presence of a nuclear bar shows that a pseudobulge dominates the central light.
Bars are disk phenomena. Triaxiality in giant Es involves different physics --
slow (not rapid) rotation and box (not $x_1$ tube) orbits. }
\item[(9){\kern -3pt}]{In edge-on galaxies, boxy bulges are edge-on bars; seeing one identifies a pseudobulge.
The boxy-core-nonrotating side of the ``E{\ts}--{\ts}E dichotomy'' between two kinds of elliptical
galaxies (see Section 4.1.1) cannot be confused with boxy, edge-on bars
because boxy ellipticals -- even if they occur in disk galaxies (we do not know of an example)
-- are so luminous that we would measure $B/T > 0.5$. Then point (5) would tell us that this
bulge is classical.}
\item[(10){\kern -3pt}]{Fisher \& Drory (2015) conclude that pseudobulges have weak Fe and Mg b lines:
equivalent width of [Fe $\lambda$5150\ts\AA]\ts$<$\ts3.95\ts\AA; equivalent width of [Mg{\ts}b]\ts$<$\ts2.35\ts\AA.
In their sample, no classical bulge has such weak lines. Some pseudobulges have
stronger lines, so this criterion, like most others, is not 100\ts\% reliable.}
\item[(11){\kern -3pt}]{If a bulge deviates from the [Mg b]\ts--\ts$\sigma$ or [Mg b]\ts--\ts[Fe] correlations
for elliptical galaxies by $\Delta$[Mg b] $< 0.7$ -- that is, if the [Mg] line strength is
lower than the scatter for Es -- then the bulge is likely to be pseudo (Fisher \& Drory 2015).}}
\end{enumerate}
It is important to emphasize that classical and pseudo bulges can occur together. Fisher \& Drory (2015) review examples
of dominant pseudobulges that have small central classical bulges. And some giant classical bulges contain nuclear disks (e.{\ts}g.,
NGC 3115: Kormendy {et~al.\ } 1996b; NGC 4594: Kormendy {et~al.\ } 1996a).
Criterion (9) for boxy pseudobulges works only for edge-on and near-edge-on galaxies.
In face-on galaxies, it is easy to identify the elongated parts of bars, but they also have rounder, denser
central parts, and these are not easily distinguished from classical bulges (Athanassoula 2015; Laurikainen
\& Salo 2015). So the above criteria almost certainly fail to find some pseudobulges in face-on barred galaxies.
\vskip -30pt
\centerline{\phantom{00000000000000000}}
\subsection{Secular Evolution in Disk Galaxies: Applications}
\vskip -15pt
\centerline{\phantom{00000000000000000}}
Progress in many subjects depends on a full integration of the picture of disk secular evolution into
our paradigm of galaxy evolution. Examples include the following:
\begin{enumerate}
\item[(1){\kern -3pt}]{If the smallest bulges are pseudo and not classical, then the luminosity and mass functions of
classical bulges and ellipticals are very bounded:
\hbox{$M_K$ \lapprox $-19$;} \hbox{$M_V$ \lapprox $-16$;}
$L_V$ $_>\atop{^\sim}$} \def\lapprox{$_<\atop{^\sim}$ $10^{8.5}$ $L_\odot$; stellar mass $M_{\rm bulge}$ $_>\atop{^\sim}$} \def\lapprox{$_<\atop{^\sim}$ $10^9$ $M_\odot$.
In simulations (Brooks \& Christensen 2015; Section 4 here),
the physics that makes classical bulges and ellipticals does not need to explain objects that
are smaller than the above. More accurately: If the same generic physics (e.{\ts}g.,
major mergers) is relevant for smaller objects, it does not have to produce remnants that
are consistent with low-mass extrapolations of parameter correlations for classical bulges
and ellipticals. One possible reason may be that the progenitors of that physics are very gas-rich.
\lineskip=-15pt \lineskiplimit=-15pt}
\item[(2){\kern -3pt}]{Our understanding that, below the above limits, lower-mass bulges are essentially all pseudo
makes it harder to understand how galaxy formation by hierarchical clustering of CDM
makes so many giant, classical-bulge-less (i.{\ts}e., pure-disk) galaxies. This was
the theme of the observational papers Kormendy {et~al.\ } (2010) and Fisher \& Drory (2011). It is
addressed in Brooks \& Christensen (2015). We return to this issue in Section 4.}
\item[(3){\kern -3pt}]{Understanding how supermassive black holes (BHs) affect galaxy evolution
requires an understanding that classical and pseudo bulges are different.
Classical bulges participate in the correlations between BH mass and
bulge luminosity, stellar mass, and velocity dispersion. Pseudobulges essentially do not.
This is some of the evidence that BHs coevolve with classical bulges and
ellipticals in ways to be determined, whereas BHs exist in but do not influence
the evolution of disks or of disk-grown pseudobulges. We return to this subject in
Section 6.}
\end{enumerate}
\vfill\eject
\section{Giant Clumps in High-z Gas-Rich Disks Make Classical Bulges}
The second major advance in our picture of bulge formation involves the observation that many high-$z$ disks
are very gas-rich and dominated by \hbox{$10^8$\ts--\ts$10^9$ $M_\odot$, kpc-size} star-forming clumps
(Elmegreen{\ts}et{\ts}al.\ts2005,\ts2007,\ts2009a,{\ts}b;
Bournaud{\ts}et{\ts}al.\ts2007;
Genzel{\ts}et{\ts}al.\ts2006,\ts2008,\ts2011;
F\"orster{\ts}Schreiber{\ts}et{\ts}al.\ts2009,\ts2011a,{\ts}b;
Tacconi{\ts}et{\ts}al. 2010).
These galaxies evidently accrete cold gas so rapidly that they become violently unstable. Bulgeless disks
tend to have small epicyclic frequencies $\kappa$. If the surface density $\Sigma$ rapidly grows
large and is dominated by gas with low velocity dispersion~$\sigma$, then the Toomre (1964)
instability parameter $Q = 0.30 \sigma \kappa/ G \Sigma$ \lapprox \ts1 ($G$ = graviational constant).
The observed clumps are interpreted to be the result. Theory and simulations suggest that the
clumps sink rapidly toward the center by dynamical friction.
They also dump large amounts of additional cold gas toward the center via tidal torques. The result
is violent relaxation plus a starburst that produces a classical bulge. Many papers discuss this evolution (e.{\ts}g.,
Dekel, Sari, \& Ceverino 2009;
Ceverino, Dekel, \& Bournaud 2010;
Cacciato, Dekel, \& Genel 2012;
Forbes {et~al.\ } 2014;
Ceverino {et~al.\ } 2015).
Bournaud (2015) reviews this subject in the present book. I include it here for two reasons,
it is a major advance, so it deserves emphasis in this concluding chapter, and I want to add two science points:
Figure 1 illustrates my first point:~{\it Evolution by clump sinking, inward gas transport, violent relaxation,
and starbursts proceeds much as it does in our picture of wet major mergers. That is, in practice (if not in its beginnings),
classical bulge formation from clump instabilities is a variant of our standard picture of bulge formation in wet major mergers.}
The process starts differently than galaxy mergers~-- \phantom{000000000000000}
\centerline{\null} \vfill
\begin{figure}[hb]
\special{psfile=./vanAlbada-Bournaudlo.eps hoffset=-3 voffset=-15 hscale=55 vscale=55}
\caption{Mergers of clumpy initial conditions make S\'ersic (1968)
function remnants with indices $n \sim 2${\ts}--{\ts}4. A remarkably early illustration is
the $n$-body simulation of van Albada (1982), whose initial conditions (grayscale densities)
resemble the clumpy high-$z$ galaxy UDF 1666 studied by Bournaud {et~al.\ } (2007).
Van Albada's initial conditions were parameterized by the ratio
of twice the total kinetic energy~to the negative of the potential energy.~In
equilibrium, $2T/W$\ts$=$\ts1. For smaller values,
gentle collapses ($2T/W = 0.5$) make S\'ersic profiles with~$n < 4$.
Violent collapses ($2T/W$\ts\lapprox\ts0.2) make $n$\ts$_>\atop{^\sim}$} \def\lapprox{$_<\atop{^\sim}$ 4. Clump sinking
in high-$z$ disks is inherently gentle. The hint is that the clumps merge to make classical bulges
with $n < 4$. This figure is from Kormendy (2012).
}
\end{figure}
\eject
\noindent what merges here are not finished galaxies but rather are clumps that formed quickly and temporarily in unstable disks.
Nevertheless, what follows -- although two- and not three-dimensional -- is otherwise closely similar to
a wet merger with gas inflow and a starburst. That is, it is a slower, gentler version of Arp 220.
Early models by Elmegreen {et~al.\ } (2008) confirm that gas-rich galaxy disks violently form
clumps like those observed. The clumps quickly sink, merge, and make a high-S\'ersic-index, vertically thick bulge.
It rotates slowly, and rotation velocities decrease with increasing distance above and below the disk plane.
These are properties of classical bulges, and Elmegreen and collaborators conclude that this process indeed
makes classical (not pseudo) bulges. Many of the later papers summarized above and reviewed by Bournaud (2015)
reach similar conclusions.
However, Bournaud (2015) goes on to review more recent simulations that\ts-- among other improvements --
include strong feeback from young stars. The results complicate the above picture. For example,
Genel {et~al.\ } (2012) find that ``galactic winds are critical for [clump] evolution. The giant clumps we obtain are
short-lived and are disrupted by wind-driven mass loss. They do not virialize or migrate to the galaxy centers
as suggested in recent work neglecting strong winds.'' Other simulations produce pseudobulge-like, small S\'ersic
indices. Some results are inherently robust, such as the conclusion that gas-rich, violently unstable disks
at high $z$ gradually evolve into gas-poor, secularly evolving disks at lower redshifts
(Cacciato, Dekel, \& Genel 2012;
cf.~Ceverino, Dekel, \& Bournaud 2010).
However, the conclusions from the models are substantially more uncertain than the inferences from the observations.
This is part of a problem that I emphasize in the next section:
Simulations of baryonic galaxy evolution inside CDM halos formed via $n$-body simulations of cosmological
hierarchical clustering are making rapid progress as the baryonic physics gets implemented in better detail.
But these simulations still show clearcut signs of missing important physics. In contrast, practitioners of
this art who carefully put great effort into improving the physics tend to be overconfident about its results.
We are -- I will suggest -- still in a situation where robust observational conclusions that are theoretically
squishy are more trustworthy than conclusions based on state-of-the-art simulations, at least when baryonic
physics is involved.
Another caveat is the observation that the clumps in high-$z$ disks are much less
obvious in the inferred mass distributions than they are in rest-frame optical or blue light (Wuyts {et~al.\ } 2012).
Frontier observations have opened up a popular new window on the formation of classical bulges,
but its importance is not entirely clear.
In the present subject of bulge formation, it seems provisionally plausible that
formation via high-$z$ disk instabilites and consequent clump sinking represents a significant new channel
in the formation of classical bulges. Meanwhile, a large body of work from the 1980s and 1990s continues
to tell us that major galaxy mergers make classical bulges, too. Can we distinguish the results of the
two processes? We do not yet know, but my second point is that Figure 1 provides a hint: Although results
are still vulnerable to unknown details in (for example) feedback, it seems likely that the classical
bulges produced by sinking clumps have S\'ersic indices that are systematically smaller than those made
by major galaxy mergers. This is one aspect of many that deserves further work.~See also point (8) in Section\ts8.
\vfill\eject
\section{Making Classical Bulges and Ellipticals by Major Mergers}
Brooks \& Christensen (2015) is perhaps the most important chapter in~this~book. The mainstream
of theoretical work on galaxy formation has come to be the simulation in a cosmological context first of
purely collisionless CDM but now with gloriously messy baryonic physics included. Progress is impressively rapid,
but we are far from finished. This subject is well reviewed from the perspective of its practitioners by
Brooks and Christensen. This includes a discussion of uncertainties and shortcomings in the models,
again as seen by theorists. As an observer, I have a complementary perspective on which
measurements of galaxies provide the most useful constraints on and ``targets'' for formation models.
It gives me the feeling that modelers are at least partly ``barking up the wrong tree.''
This section complements Brooks and Christensen (2015) by reviewing these observations.
Pseudobulge formation was covered in Section 2. Here, I focus on the formation of classical bulges
and ellipticals. My discussion uses the observations that classical bulges are essentially indistinguishable
from coreless-disky-rotating ellipticals (see, e.{\ts}g., Figure 4). The inference is that they
formed in closely related ways.
\vskip -25pt
\centerline{\phantom{00000000000000000}}
\subsection {Observer's Perspective on Bulge Formation Via Major Mergers}
I begin with giant ellipticals and classical bulges:~their structure and formation
are understood in the most detail. Classical bulges are identified by the criteria listed in
Kormendy \& Kennicutt (2004),
Kormendy (2012),
KH13,
Fisher \& Drory (2015), and
Section 2 here.
I know no observational reason to seriously doubt our understanding of bulges with $B/T$ $_>\atop{^\sim}$} \def\lapprox{$_<\atop{^\sim}$ 0.8.
Then, as $B/T$ drops to \lapprox \ts1/2, the situation gets less clear. Our formation
picture may still essentially be correct, but it gets less directly based on observations as $B/T$ or
bulge luminosity decreases. Meanwhile, the theoretical problem is that simulations make too many bulges,
especially big ones. In this section, I review things that we know and outline things that we do not know.
It is critically important to start with a discussion of ellipticals, because
our understanding of classical bulges must be within this context.
\vskip -25pt
\centerline{\phantom{00000000000000000}}
\subsubsection {Observed Properties of Ellipticals: Clues to Their Formation}
The observed properties of elliptical galaxies are reproduced by simulations of wet and dry mergers
in remarkable detail. These are not embedded in large-scale cosmological simulations, but this is
not a fundamental fault if the initial conditions are realistic -- galaxies with typical $z \sim 0$
gas fractions and encounter velocities that are roughly parabolic.
Kormendy {et~al.\ } (2009, hereafter KFCB) provide an ARA\&A-style review and develop some of the evidence.
Hopkins {et~al.\ } (2009a and 2009b) provide the most detailed models for wet and dry mergers, respectively.
These papers are comprehensive; a concise summary of the ``E -- E dichotomy'' in Kormendy (2009) is updated below.
The critical observation is that ellipticals come in two varieties and that bulges are similar to one (but not both)
of these varieties.
The E\ts--{\ts}E dichotomy of ellipticals into two kinds is based on these observations: \vskip 5pt
\underbar{Giant ellipticals} ($M_V$ \lapprox \ts$-21.5 \pm 1$ for $H_0 = 70$ km s$^{-1}$ Mpc$^{-1}$)
generally \hfill\break
(1) have S\'ersic function outer profiles with $n > 4$; \hfill\break
(2) have cores; i.{\ts}e., central missing light with respect to the outer S\'ersic profile;\hfill\break
(3) rotate slowly, so rotation is of little importance dynamically; hence \hfill\break
(4) are anisotropic and modestly triaxial; \hfill\break
(5) are less flattened (ellipticity $\epsilon$ $\sim${\thinspace}0.2) than smaller ellipticals; \hfill\break
(6) have boxy-distorted isophotes; \hfill\break
(7) mostly are made of very old stars that are enhanced in $\alpha$ elements (Figure 2); \hfill\break
(8) often contain strong radio sources (Figure 3), and \hfill\break
(9) contain X-ray-emitting gas, more of it in more luminous galaxies (Figure 3). \vskip 5pt
\underbar{Normal ellipticals and dwarf ellipticals like M{\ts}32} ($M_V$ $_>\atop{^\sim}$} \def\lapprox{$_<\atop{^\sim}$ \ts$-21.5$) generally \hfill\break
(1) have S\'ersic function outer profiles with $n \simeq 2$ to 3; \hfill\break
(2) are coreless -- have central extra light with respect to the outer S\'ersic profile;\hfill\break
(3) rotate rapidly, so rotation is dynamically important to their structure; \hfill\break
(4) are nearly isotropic and oblate spheroidal, albeit with small axial dispersions; \hfill\break
(5) are flatter than giant ellipticals (ellipticity $\epsilon$ $\sim${\thinspace}0.35); \hfill\break
(6) have disky-distorted isophotes; \hfill\break
(7) are made of younger stars with little $\alpha$-element enhancement (Figure 2); \hfill\break
(8) rarely contain strong radio sources (Figure 3), and \hfill\break
(9) generally do not contain X-ray-emitting gas (Figure 3). \vskip 5pt
These results are etablished in many papers (e.{\thinspace}g.,
Davies et al.~1983;
Bender 1988;
Bender et al.~1989;
Nieto et al.~1991;
Kormendy et al.~1994;
Lauer et al.~1995, 2005, 2007a, b;
Kormendy \& Bender 1996;
Tremblay \& Merritt 1996;
Gebhardt et al.~1996;
Faber et~al.~1997;
Rest et al.~2001;
Ravindranath et al.~2001;
Thomas et al.~2002a, b, 2005;
Emsellem et al.~2007, 2011;
Cappellari et al.~2007, 2011, 2013b;
KFCB;
Kuntschner {et~al.\ } 2010).
A few ellipticals are exceptions to one or more of (1) -- (9). The above summary is quoted from Kormendy (2009).
Why is this relevant here? The answer is that
classical bulges are closely similar to coreless-disky-rotating ellipticals. No bulge is similar to a
core-boxy-nonrotating elliptical as far as I know. This is a clue to formation processes.
First, though, we need to understand the difference between the two kinds of ellipticals:
How did the E{\thinspace}--{\thinspace}E dichotomy arise? The ``smoking gun'' for an explanation
is a new aspect of the dichotomy originally found in Kormendy (1999) and observed in all low-luminosity ellipticals
in the Virgo cluster by KFCB. Coreless galaxies do not have featureless power-law profiles. Rather,
all coreless galaxies in the KFCB sample show a new structural component, i.{\thinspace}e., central extra light
above the inward extrapolation of the outer S\'ersic profile. Kormendy (1999) suggested that the
extra light is produced by starbursts fed by gas dumped inward during dissipative mergers. Starbursts were predicted by merger
simulations as soon as these included gas, dissipational gas inflow, and star formation (Mihos \& Hernquist 1994).
Mihos and Hernquist were concerned that extra components had not been observed. The reason turns
out to be that we had not measured ellipticals with enough surface brightness range and spatial resolution.~Like
Faber{\ts}et{\ts}al.\ts(1997,\ts2007), KFCB suggest that the origin of the E\ts--{\ts}E dichotomy is
that core ellipticals formed in dry mergers whereas coreless ellipticals formed in wet mergers. Simulations of
dry and wet mergers reproduce the structural properties of core and extra light ellipticals in beautiful
detail (Hopkins et al.~2009a, b). And, although the formations scenarios differ, Khochfar {et~al.\ } (2011) similarly
conclude that the difference between fast and slow rotators is related to cold gas dissipation and star-formations shutdown, respectively.
Cores are thought to be scoured by supermassive black hole binaries that
were formed in major mergers. The orbit shrinks as the binary flings stars away. This decreases the surface
brightness and excavates a core
(Begelman {et~al.\ } 1980;
Ebisuzaki {et~al.\ } 1991;
Makino \& Ebisuzaki 1996;
Quinlan \& Hernquist 1997;
Faber {et~al.\ } 1997;
Milosavljevi\'c \& Merritt 2001;
Milosavljevi\'c et al.~2002;
Merritt 2006). The same process should happen during wet mergers; in fact, gas accelerates the orbital decay
(Ivanov, Papaloizou, \& Polnarev 1999;
Gould \& Rix 2000;
Armitage \& Natarajan 2002, 2005;
Escala {et~al.\ } 2004, 2005;
Dotti {et~al.\ } 2007;
Hayasaki 2009;
Cuadra {et~al.\ } 2009;
Escala \& Del Valle 2011; see
Mayer 2013 for a recent review).
However, we observe that the fraction
of the luminosity that is in extra light in low-luminosity ellipticals is larger than the fraction of the light
that is ``missing'' in the cores of high-luminosity ellipticals. KFCB suggest that core scouring is
swamped by the starburst that makes the extra light in coreless-disky-rotating ellipticals.
When did the E{\thinspace}--{\thinspace}E dichotomy arise? Figure 2 shows observation (7) that core ellipticals
mostly are made of old stars that are enhanced in $\alpha$ elements. In contrast, coreless ellipticals
are made of younger stars with more nearly solar compositions. This means (Thomas et al.~2002a, b, 2005) that the stars
in core Es formed in the first few billion years of the universe and over a period of \lapprox\ts1 Gyr,
so quickly that Type I supernovae did not have time to dilute with Fe the
\hbox{$\alpha$-enriched} gas recycled \phantom{000000000000}
\vfill
\special{psfile=./thomas-MgFeAge.ps hoffset=48 voffset=-87 hscale=36 vscale=36}
\begin{figure}
\caption{
\pretolerance=15000 \tolerance=15000
\lineskip=0pt \lineskiplimit=0pt
Alpha element overabundance ($\log$ solar units) versus relative age of the stellar
population. Red and blue points denote core and ``power law'' (i.{\ts}e., coreless) ellipticals.
The [Mg/Fe] and age data are from Thomas {et~al.\ } (2005); this figure is from KFCB.
}\end{figure}
\eject
\noindent by Type II supernovae. This does not mean that core ellipticals were made at the same time as their stars.
Mass assembly via dry mergers as required to explain their structure could have happened at
any time after star formation stopped. Our problem is to explain how star formation was
quenched so quickly and not allowed to recur. In contrast, coreless ellipticals have younger,
less-$\alpha$-enhanced stellar populations. They are consistent with a simple picture in
which a series of wet mergers with accompanying starbursts formed their stellar populations
and assembled the galaxies more-or-less simultaneously over the past 9 billion years.
Faber et al.~(2007) discuss these issues in detail. A big problem with the present state of the art
is that we know so little about mergers and merger progenitors at high $z$.
Why did the E{\thinspace}--{\thinspace}E dichotomy arise?~The key observations are:~(8)
core-boxy ellipticals often are radio-loud whereas coreless-disky ellipticals are not, and
(9) core-boxy ellipticals contain \hbox{X-ray} gas whereas coreless-disky ellipticals do not
(Bender et al.~1989). Figure 3 (from KH13) illustrates these results. KFCB suggest
that the hot gas keeps dry mergers dry and protects giant ellipticals from late star formation. This
is the operational solution to the above ``maintenance problem''. I~return to the problem of
star-formation quenching in Section 7.
\vfill
\special{psfile=./AR4-Fig36s.ps hoffset=-12 voffset=-38 hscale=58 vscale=58}
\begin{figure}
\caption{
\pretolerance=15000 \tolerance=15000
\lineskip=0pt \lineskiplimit=0pt
(Left) Correlation with isophote shape parameter $a_4$ of (top) X-ray emission from hot gas and
(bottom) radio emission (from Bender {et~al.\ } 1989). Boxy ellipticals ($a_4 < 0$) contain hot gas and
strong radio sources; disky ellipticals ($a_4 > 0$) generally do not.
(Right) KFCB update of the X-ray correlation. Detections are color-coded according to the E\ts--{\ts}E
dichotomy. The emission from X-ray binary stars is estimated by the black line (O'Sullivan, Forbes,
\& Ponman 2001); this was subtracted from the total emission in constructing the left panels.
The red line is a bisector fit to the core-boxy-nonrotating ellipticals. They statistically
reach $L_X = 0$ from hot gas at $\log{L_B} \simeq 9.4$. This corresponds to $M_V \simeq -20.4$,
a factor of 2 fainter than the luminosity that divides the two kinds of ellipticals. Thus,
if a typical core E was made in a merger of two equal-mass galaxies, then both were marginally
big enough to contain X-ray gas and the remnant immediately was massive enough so that hot gas
could quench star formation. KFCB suggest that this is why these mergers were dry.
For similar results, see Pellegrini (1999, 2005) and Ellis \& O'Sullivan (2006).
}\end{figure}
\eject
In the above story, the challenge is to keep the hot gas hot, given that X-ray gas cooling times are short (Fabian 1994).
KFCB review evidence that the main heating mechanism may be energy feedback from accreting BHs (the active galactic
nuclei [AGNs] of observation 8); these may also have helped to quench star formation. Many details of this picture require work
(Cattaneo et al.~2009). Cosmological gas infall is an additional heating mechanism
(Dekel \& Birnboim 2006). Still, Figure 3 is a crucial connection
between \hbox{X-ray} gas, AGN physics, and the E\ts--{\ts}E dichotomy.
``\underbar{Bottom line}:'' In essence, only giant, core ellipticals and their progenitors
are massive enough to contain hot gas that helps to engineer the E{\thinspace}--{\thinspace}E dichotomy.
\vskip 12pt
\begin{svgraybox}
\phantom{0000000000}
\end{svgraybox}
\vskip -68pt
\phantom{0000000000}
\subsubsection{Classical Bulges Resemble Coreless-Disky-Rotating Ellipticals}
\vskip -30pt
\phantom{0000000000}
\begin{svgraybox}
\phantom{0000000000}
\end{svgraybox}
\vskip -65pt
\phantom{0000000000}
\begin{svgraybox}
Are both kinds of ellipticals also found as bulges? So far, observations indicate that the answer is ``no''.
Classical bulges closely resemble only the coreless-disky-rotating ellipticals. There are apparent exceptions in the
literature, but all the exceptions that I know about are classification errors brought about (e.{\ts}g.)~by the very
large S\'ersic indices of some core galaxies (see KFCB Table 1 for examples and KFCB Section 5.2 for discussion).
This comment also does not include ellipticals with nuclear disks. All signs are that these involve different physics,
so these really are ellipticals, not S0 bulges.
There is physics in this conclusion. The X-ray gas prevents cooling and dissipation during any subsequent
mergers or any $z$\ts\lapprox 1 cold accretion. Plausibly, it should also prevent there from being any cold gas left over to make
a new disk after a merger is complete. Further checks, both of the observational conclusion and of the
theoretical inference, should be made.
\end{svgraybox}
\vskip -20pt
\begin{svgraybox}
\phantom{0000000000}
\end{svgraybox}
\vskip -45pt
\phantom{0000000000}
\subsubsection{The Critically Important Target for Galaxy Formation}
\vskip -30pt
\phantom{0000000000}
\begin{svgraybox}
The most fundamental distinction between galaxy types is the one between bulges $+$ ellipticals and disks.
Bulges and disks overlap over a factor of about $\sim$\ts1500 in luminosity and mass (Figure 4), but over that entire
overlap range, they are dramatically different from each other. This includes differences in specific angular momentum
(Romanowsky \& Fall 2012; Fall \& Romanowsky 2013), in orbit structure, in flattening, and in radial density profiles (disks are roughly
exponential; coreless-disky-rotating ellipticals have $n$ $_>\atop{^\sim}$} \def\lapprox{$_<\atop{^\sim}$ \ts2). At absolute magnitude $M_V \simeq -16.7$ and outer
circular-orbit rotation velocity $V_{\rm circ} \sim 85$ km s$^{-1}$, M{\ts}32 is a normal small elliptical
galaxy (KFCB). At $M_V \simeq -21.6$ and $V_{\rm circ} = 210 \pm 15$ km s$^{-1}$, M{\ts}101 is almost 100 times
more luminous but is thoroughly different from M{\ts}32 (Kormendy {et~al.\ } 2010).
\end{svgraybox}
\begin{svgraybox}
I believe that the goal of galaxy formation modeling should be to produce realistic disks and realistic ellipticals
that overlap over the observed factor of $\sim$\ts1500 in luminosity but that differ as we observe them to differ over the
whole of that range. And over the whole of that range, disks and bulges can be combined with $B/T$ and $D/T \simeq 1 - B/T$
ratios that have the observed distribution (i.{\ts}e., $B/T \sim 1$ near the upper end of the range, but it can be $\ll$\ts1
at the bottom of the range). The properties of individual disks and bulges are essentially independent of $B/T$ with
structural parameters shown in Figure\ts4.
\end{svgraybox}
\vfill
\special{psfile=./fig_final_bulges.ps hoffset=-78 voffset=-122 hscale=51 vscale=51}
\special{psfile=./fig_spiral_disks.ps hoffset=95 voffset=-122 hscale=51 vscale=51}
\begin{figure}
\caption{
\pretolerance=15000 \tolerance=15000
\lineskip=0pt \lineskiplimit=0pt
Correlations between effective radius $r_e$, effective brightness $\mu_V$,
and absolute magnitude $M_V$ for classical bulges and ellipticals (brown and pink points), for
spheroidal (Sph) galaxies and S0 disks (green points), and for spiral galaxy disks (blue points).
When bulge-disk decomposition is necessary, the two components are plotted separately. Bulges and
disks overlap from $M_V \simeq -15$ to $M_V \simeq -23$, i.{\ts}e., over a factor of about 1500. The left
panel shows (1) that Sph galaxies are distinct from bulges $+$ ellipticals and (2) that classical bulges
and ellipticals satisfy the same structural parameter correlations. The right panel adds S0 and S
galaxy disks. It shows that all disks satisfy the same structural parameter correlations
over the whole range of luminosities. Note that disks and ellipticals have similar $r_e$ and $\mu_e$
at the highest luminosities, but they have very different S\'ersic indices ($\sim 1$ and 2 to $>$\ts10,
respectively). As a result, the central surface brightnesses in bulges and ellipticals are
more than an order of magnitude higher than the central surface brightnesses of disks (Kormendy
1985, 1987). Bulges $+$ ellipticals and disks also have non-overlapping distributions of intrinsic flattening
(e.{\ts}g., Sandage, Freeman, \& Stokes 1970). From Kormendy \& Bender (2012),
}
\end{figure}
\eject
Of course, bulges and disks are not different in {\it every} parameter; e.{\ts}g., the $r_e$\ts--\ts$\mu_e$
correlations overlap at high luminosities (Figure 4). This makes sense: At the highest masses, it does not
require much dissipation to turn a disk into an elliptical, at least in terms of virial parameters. All that
is required is to scramble disk orbits into an ellipsoidal remnant. A larger amount of dissipation is
required to make the high-density centers, and in {\it central} parameters and parameter correlations, disks and
bulges $+$ ellipticals are very different (Kormendy 1985, 1987).
\vskip -10pt
\begin{svgraybox}
\phantom{0000000000}
\end{svgraybox}
\vskip -69pt
\phantom{0000000000}
\subsubsection{Critical Obserational Clue: The Problem of Giant, Pure-Disk Galaxies Depends on Environment,
Not on Galaxy Mass}
\vskip -34pt
\phantom{0000000000}
\begin{svgraybox}
The most difficult challenge in our picture of galaxy formation\ts--{\ts}I suggest\ts--{\ts}is to understand
how hierarchical clustering produces so many giant, pure-disk galaxies that have no sign of
a classical bulge. CDM halos grow by merging; fragments arrive from all directions, and not all
fragments are small. There are two parts to this problem: (1) It is difficult to understand how
cold, flat disks survive the violence inherent in the mergers that grow DM halos. And
(2) it is difficult to prevent the stars that arrive with the latest accretion victim from adding
to a classical bulge that formed in (1) from the scrambled-up disk.
Brooks \& Christensen (2015) review how the modeling community tries to solve this problem.
In spite of several decades of evidence that mergers make bulges, they do not use mergers to turn
disks into bulges. Instead, they use feedback from young stars and active galactic nuclei to ``whittle
away'' the low-angular-momentum part of the distribution of gas angular momenta and argue that this
prevents bulge formation. And they use feedback to delay disk formation until the halo is assembled.
Feedback is likely to be important in the formation of dwarf galaxies (Governato {et~al.\ } 2010), and indeed,
they essentially never have bulges (e.{\ts}g., Kormendy \& Freeman 2015, Figure 10).
However, it is difficult for me to believe that feedback, either from star formation or from AGNs, is
responsible for the difference between bulges and disks. Feedback is fundamentally an internal process
that is controlled by the galaxy's potential well depth. It is not clear how {\it only} tweaking the
feedback can make a small elliptical like M{\ts}32 (different from small disks) and a giant disk like M{\ts}101
(different from similarly giant ellipticals) with no intermediate cases. Bulge-to-total ratios vary widely,
but classical bulges are always like ellipticals no matter what the B/T ratio, and disks are always different
from ellipticals no matter what the D/T ratio. Observations do not suggest that it is primarily feedback
that results in this difference. Rather:
There is a fundamental observational clue that modelers are not using:
Whether evolution makes disks or whether it makes bulges does not depend mainly on galaxy mass.
Rather, it is a strong function of environment. Kormendy {et~al.\ } (2010) show that, in the extreme field
(i.{\ts}e., in environments like the Local Group), most giant galaxies ($V_{\rm circ} \geq 150$ km s$^{-1}$)
are pure disks. Only 2 of 19 giant galaxies closer to us than 8 Mpc have $B/T$ as big as 1/3. Only 2 more
are ellipticals. A few have smaller classical bulges, but 11 of the 19 galaxies have essentially no classical
bulge. In contrast, $>$ 2/3 of all stars in the Virgo cluster live in bulges or elliptical galaxies.
{\it There is no problem of understanding giant pure-disk galaxies in the Virgo cluster.} It is a
mature, dense environment that contains large amounts of X-ray-emitting, hot gas. Rich clusters are places
where most of the baryons live suspended in hot gas (e.{\ts}g., Kravtsov \& Borgani 2012). I argue in Section 4.1.1
that various heating processes maintain this situation for very long times. In contrast, poor groups are
environments in which accretion of cold gas from the cosmic web can dominate, as long as the galaxies involved --
i.{\ts}e., the aforementioned pure disks -- are low enough in mass so that they cannot hold onto X-ray gas.
As long as this environmental dependence is not a primary, essential part of the explanation, I believe
that attempts to solve the problem of overproduction of bulges in $\Lambda$CDM cosmology are ``barking up the wrong tree''.
\end{svgraybox}
\quad Why can't we use feedback to delay star formation until the halo is assembled? As reviewed by
Brooks \& Christensen (2015), this is commonly suggested. The counterexample is our Galaxy: The oldest stars in
the thin disk are $\sim$\ts$10^{10}$ yr old, so much of the growth of our Galaxy happened when the thin disk was
already in place (Kormendy {et~al.\ } 2010, p.~73).
\subsubsection{It is not a problem that major mergers are rare}
The prevailing theoretical paradigm is more and more converging on the view that major mergers are rare -- are,
in fact, almost irrelevant -- and that, instead, minor mergers make both bulges and ellipticals (see Naab 2013 for a review),
even some core-boxy-nonrotating ellipticals (Naab {et~al.\ } 2014). It will be clear from this writeup that, based on observational
evidence, I agree that major mergers are rare. But I disagree that they are unimportant in the formation of bulges and ellipticals.
The above papers make important points that are robust. They argue convincingly that major mergers are rare -- that
only a small fraction of galaxies undergo several of them in their recent history (say, since $z \sim 2$). And many
authors argue that most star formation does not occur during mergers; rather, it occurs in a ``main sequence''
of disks of various masses, with higher star formation rates at higher masses (e.{\ts}g.,
Schiminovich {et~al.\ } 2007;
Noeske {et~al.\ } 2007;
Elbaz {et~al.\ } 2007;
Daddi {et~al.\ } 2007;
Finlator \& Dav\'e 2008;
Karim {et~al.\ } 2011;
Peng {et~al.\ } 2010;~Rodighiero~et~al.~2011;
Wuyts {et~al.\ } 2011;
Salmi {et~al.\ } 2012;
Whitaker {et~al.\ } 2012;
Tacconi {et~al.\ } 2013;
Speagle {et~al.\ } 2014).
These authors conclude that the duty cycle of star formation is large. Therefore most star formation does not occur
in rare events. I made the same argument in Section 2: If almost all galaxies of a particular type are energetically
forming stars, then star formation must be secular; it cannot be episodic with short duty cycles. Caveat: the star
formation that is associated with mergers is not instantaneous. Puech {et~al.\ } (2014) argue that merger-induced star
formation is significant. Are these results consistent with a picture in which essentially all formation of classical bulges
and ellipticals happens via major mergers?
I believe that the answer is yes, although the details need further work. Elliptical galaxies are observed to be rare;
the morphology-density relation (Dressler 1980; Cappellari {et~al.\ } 2011) shows that they are a small fraction of all galaxies
except in rich clusters. Classical bulges are rarer than we thought, too; this is a clear conclusion of the work on disk
secular evolution. {\it Therefore the events that make bulges must be rare.} It is also not a problem if most
star formation happens~in~disks. For example, only a small fraction of the galaxy mass is contained in the extra light
components that are identified by KFCB and by Hopkins (2009a) as the parts of coreless/disky/rotating ellipticals that
formed in the most recent ULIRG-like starburst (e.{\ts}g., Genzel {et~al.\ } 2001). Most of the mass was already in stars before these late,
wet mergers. And in dry mergers, essentially all the mass was already in stars (or in X-ray gas that stays X-ray gas) and essentially
no new stars are formed.
How many mergers do we need to explain elliptical galaxies? Toomre (1977) already pointed out that a reasonable
increase in merger rate with increasing $z$ would suffice. He based this on ten mergers-in-progress that he
discussed in his paper. He assumed that such objects are identifiable for $\sim$ half a billion years. Then, if the number
of mergers in progress increased as (lookback time)$^{5/3}$ consistent~with a flat distribution of binding energies
for galaxy pairs, the result is that the number of remnants is consistent with the number of elliptical and early-type
disk galaxies. This estimate was made for the level of completeness of the Second Reference Catalogue of Bright Galaxies
(de Vaucouleurs, de Vaucouleurs, \& Corwin 1976).
Conselice (2014) reviews observational estimates of how merger rates depend on $z$. As Toomre predicted,
the major merger rate is inferred -- e.{\ts}g., from counting close pairs of galaxies -- to increase rapidly with $z$.
Observations of high-$z$ galaxies show that close binary fractions increase roughly as $(1 + z)^m$ with $m \sim 2$ to 3 (e.{\ts}g.,
Bluck {et~al.\ } 2009, 2012;
Conselice {et~al.\ } 2009;
L\'opez-Sanjuan {et~al.\ } 2013;
Tasca {et~al.\ } 2014).
ULIRGs increase in comoving energy density even faster toward higher redshift, at least out to $z = 1$ (Le Floc'h {et~al.\ } 2005).
The necessary connections between these results to establish or disprove whether bulges $+$ ellipticals are made via major
mergers have not been established. Important uncertainties include (1) the low-mass end of the mass functions for ellipticals
and especially for classical bulges, and (2) the degree to which mass-clump sinking in disks contributes. However, the above
results on merger frequencies appear at least qualitatively consistent with the conclusion that bulges and ellipticals are
made in major mergers, as the pre-2000 history of observational work established (see Schweizer 1998 for a review).
A shortcoming of many current investigations is that they concentrate on a few parameter distributions for large galaxy samples
and not specifically on the histories of bulges and disks. E.{\ts}g., they look at the statistics of what fraction of galaxies experience mergers.
Outcomes are difficult to estimate, because with samples of $10^4$ to $10^5$ $z \sim 0$ galaxies or $10^2$ high-$z$ galaxies, the typical galaxy
is only a few pixels in radius. Then it is difficult to identify and classify galaxy components.
\subsubsection{Uncertainties With Our Picture of Bulge Formation in Major Mergers}
\centerline{\phantom{00000000000000000}}
\vskip -18pt
Two major uncertainites are a concern (see also Brooks \& Christensen 2015).
Virtually all observational evidence on mergers-in-progress (e.{\ts}g.,
Toomre 1977;
Joseph \& Wright 1985;
Sanders {et~al.\ } 1988a, b;
Hibbard {et~al.\ } 1994, 1995, 1996, 2001a, b; see
Schweizer 1987, 1990, 1998 for reviews)
involves giant galaxies. And the detailed evidence is for $z \sim 0$ galaxies with gas
fractions of a few to $\sim$\ts10\ts\%. (1) We do not have comparable evidence for dwarfs. That is, we have
not studied a sample of dwarfs that fill out a merger sequence from close pairs to mergers engaged
in violent relaxation to train wrecks that are still settling down to mature objects. And (2) we do not have comparably
detailed studies of galaxies at high z that have gas fractions $_>\atop{^\sim}$} \def\lapprox{$_<\atop{^\sim}$ \ts50\%. It is possible that mergers behave
differently for such objects.
\vskip -26pt
\centerline{\phantom{00000000000000000}}
\subsubsection{The Problem of Giant, Pure-Disk Galaxies: Conclusion}
\centerline{\phantom{00000000000000000}}
\vskip -18pt
My most important suggestion in this section is that the modeling community relies too strongly on feedback
as the only way to prune excessive bulge formation. On the contrary, I suggest that environmental differences in
the amount of dynamical violence in galaxy formation histories are the central factor. I suggest that the solution
is not a to whittle away the low-angular-momentum tail of the distribution of angular momenta in forming galaxies.
Nearby galaxies dramatically show us the importance of violent relaxation. To me, the issue is: How much does
violent relaxation dominate? How much is the evolution controlled by gentle accretion? And how do the answers
depend on environment?
\vskip -34pt
\centerline{\phantom{00000000000000000}}
\section{Universal Scaling Relations For All Galaxies?}
\centerline{\phantom{00000000000000000}}
\vskip -20pt
How we best construct parameter correlations depends on what~we~want~to~learn. Projections~of the
fundamental plane correlations {\it separate} galaxy classes; e.{\ts}g., bulges$+$ellipticals from disks$+$Sphs (Figure 4).
So they teach us about differences in formation processes. In contrast, it is possible to construct parameter
correlations that make most or all galaxy types look continuous. {\it These encode less information about galaxy formation}.
E.{\ts}g., in a projection of the structural parameter correlations that encodes mass-to-light ratio, the difference
between ellipticals, spheroidals, and even irregulars largely disappears (Bender, Burstein, \& Faber 1992). Zaritsky (2015)
regards this as progress -- as replacing correlations that are flawed with ones that capture some inherent simplicity.
That simplicity is real. But it is insensitive to the power that other correlations clearly have to tell us things
about galaxy formation.
I therefore disagree, not with Zaritsky's operational results but with his motives. If you look at the
fundamental plane face-on, it contains lots of information. If you look at it edge-on, then it looks simple.
This may feel like a discovery. But it just means that you are looking at a projection that hides the information content
in the parameter plane. Other combinations of parameters make still more types of objects looks continuous and indistinguishable.
But this means that we learn still less, not more, about their nature and origin. The simple correlations are not uninteresting,
but the ones that teach us the most are the ones that correctly identify differences that turn out to have causes within
formation physics.
\section{Coevolution of Supermassive Black Holes and Host Galaxies}
The observed demographics of supermassive black holes (BHs) and their implications for the coevolution (or not)
of BHs and host galaxies are discussed in Kormendy \& Ho (2013). This is a 143-page ARA\&A review that revisits methods
used to measure BH masses $M_\bullet$ using spatially resolved stellar and gas dynamics. It also provides a detailed
analysis of host galaxy morphologies and properties. Careful treatment of the $M_\bullet$ and galaxy measurements
allows Kormendy and Ho to reach a number of new science conclusions. They are summarized in this section.
Graham (2015) reviews the same subject in the present book. Some of his review is historical,
especially up to the beginning of his Section 4.1 but also sporadically thereafter. I do not comment here on
the historical review. However, on the science, I cannot ``duck'' my responsibility as author of this concluding chapter{\kern 0.3pt}:
I disagree with most of the scientific conclusions in Graham (2015). Starting in his Section 4.1,
his discussion uses data and repeats conclusions from Graham \& Scott (2013, 2015). Problems with the 2013 data
are listed in KH13 (p.\ts555); a point made there that is not repeated further here is that
many of Graham's galaxy classifications are incorrect. Here, rather than write a point-by-point rebuttal to
Graham (2015), I first concentrate on a summary of the unique strengths of the KH13 analysis and data.
However, a few comments are added to further explain the origin of the disagreements with Graham (2015).
I then summarize the KH13 results and conclusions about $M_\bullet$\ts--{\ts}host-galaxy correlations
(Sections 6.1 and 6.2).
\begin{svgraybox}
\centerline{\null} \vskip -20pt
Before I begin, a comment is in order about how readers react to disagreements in the literature.
The most common reaction is that the subject needs more work. Specialists may know enough to decide
who is correct. But the clientele community of non-specialists who mainly want to use the results
often do not delve into the details deeply enough to decide who is correct. Rather, their reaction is
that this subject needs further work until everybody agrees that the disagreement is resolved. Sometimes,
this is an appropriate reaction, when the issues are more complicated than our understanding of the physics,
or when measurements are still too difficult, or when results under debate have low significance compared to
statistical errors or systematic effects. My reading of the community is that reactions to disagreements
on BH demographics take this form.
However, I suggest that we already know enough to decide who is correct in the disagreement
between KH13 and Graham (2015). Our ARA\&A review and the Graham \& Scott papers both
provide enough detail to judge the data and the analysis. It is particularly important to note how these
separate discussions do or do not connect up with a wide body of results in other published work, including
other chapters in this book. A strength of the Kormendy \& Ho analysis is that it connects up with -- i.{\ts}e.,
it uses and it has implications for -- a wide variety of aspects of galaxy formation.
\end{svgraybox}
Strengths of the data and suppoprting science that are used by KH13 include the following.
Some of these points are discussed more fully in the Supplemental Material of KH13.
\begin{enumerate}
\item[(1){\kern -3pt}]{BH masses based on absorption-line spectroscopy are now derived by including
halo dark matter in the stellar dynamical models. This generally leads to an upward
revision in $M_\bullet$ by a factor that can be $_>\atop{^\sim}$} \def\lapprox{$_<\atop{^\sim}$ \ts2 for core galaxies.
Kormendy and Ho use these masses. For some galaxies (e.{\ts}g., M{\ts}87), Graham
uses them; for other galaxies (e.{\ts}g., NGC 821, NGC 3377, NGC 3608, NGC 4291,
NGC 5845), he does not, even though such masses are published (Schulze \& Gebhardt 2011).
\lineskip=-15pt \lineskiplimit=-15pt}
\item[(2){\kern -3pt}]{Kormendy and Ho include new $M_\bullet$ determinations for mostly high-mass galaxies from
Rusli {et~al.\ } (2013). Graham \& Scott (2013) did not include these galaxies. It is not
clear whether they are included in Graham (2015), but observation that the highest $M_\bullet$
values plotted in his Figure 4 are $\sim 6 \times 10^9$ $M_\odot$ and not $>$\ts$10^{10}$\ts$M_\odot$
suggests that they are not included, at least in this figure.
\lineskip=-15pt \lineskiplimit=-15pt}
\item[(3){\kern -3pt}]{BH masses derived from emission-line gas rotation curves are used without correction
when the emission lines are narrow. However, when the emission lines are
wide -- often as wide in km s$^{-1}$ as the rotation curve amplitude -- some authors
have ignored the line widths in the $M_\bullet$ determinations. KH13 argue
that these BH masses are underestimated and do not use them. Graham (2015) uses them.
\lineskip=-15pt \lineskiplimit=-15pt}
\item[(4){\kern -3pt}]{All disk-galaxy hosts have $B/T$ values based on at least one and sometimes as
many as six bulge-disk decompositions. Graham \& Scott (2013) use a mean statistical correction
to derive some bulge magnitudes from total magnitudes.
\lineskip=-15pt \lineskiplimit=-15pt}
\item[(5a){\kern -3pt}]{All disk-galaxy hosts have (pseudo)bulge classifications that are based on at least two and
as many as five criteria such as those listed here in Section 2.1. Graham (2015) rejects this
approach and instead compares BH--host correlations for barred and unbarred galaxies. However,
Kormendy and Ho emphasize that some barred galaxies contain classical bulges, whereas many
unbarred galaxies contain pseudobulges. If classical and pseudo bulges correlate differently
with their BHs (Figure 7), then a division into barred and unbarred galaxies does not cleanly see this.
It should be noted that other derivations of BH--host correlations (e.{\ts}g., the
otherwise very good paper by McConnell \& Ma 2013), also do not differentiate between classical and
pseudo bulges. They compare early and late galaxy types. But many S0s contain pseudobulges,
and a few Sbcs contain classical bulges (e.{\ts}g., NGC 4258: Kormendy {et~al.\ } 2010).}
\item[(5b){\kern -3pt}]{\pretolerance=100000\tolerance=100000 The picture of disk secular evolution and the conclusion that
\hbox{pseudobulges}
are distinguishable from classical bulges is fully
integrated into the analysis. Graham (2011, 2015) does not use this picture and
argues that classical and pseudo bulges cannot reliably be
distinguished. Kormendy \& Kennicutt (2004), Kormendy (2012), Kormendy \& Ho (2013),
Fisher \& Drory (2015) in this book, and Section 2 in this summary chapter
disagree. \hbox{The subject is growing} rapidly, and whole meetings are devoted to it
(e.{\ts}g., 2012 IAU General \hbox{Assembly} Special Session 3, ``Galaxy Evolution
Through Secular Processes,'' {\tt http://bama.ua.edu/$\sim$rbuta/iau-2012-sps3/proceedings.html} and
Kormendy 2015;
XXIII Canary Islands Winter School, ``Secular Evolution of Galaxies'', Falc\'on-Barroso \&
Knapen 2012). Kormendy \& Ho make a point of distinguishing classical and pseudo bulges by purely morphological
criteria such as those given in Section\ts2.1. The fact that we then discover that
BHs correlate differently with classical and pseudo bulges is a substantial
success of the secular evolution picture.
\lineskip=-15pt \lineskiplimit=-15pt}
\item[(6){\kern -3pt}]{KH13 find that the $\log M_\bullet$ -- $M_{K,\rm bulge}$,
$\log M_\bullet$ -- $\log \sigma$ and $\log M_\bullet$\ts--\ts$\log M_{\rm bulge}$ correlations
for classical bulges and ellipticals have intrinsic scatter of 0.30, 0.29, and 0.28 dex,
respectively. This small scatter is a consequence of the care taken in (1)\ts--{\ts}(5), above,
in implementing a uniform, accurate distance scale based as much as possible on standard candles,
in correcting galaxy classifications when detailed photometry reveals errors,
and in correcting $K$-band magnitudes for systematic errors.
Given this small scatter, it was possible to discover a new result; i.{\ts}e., that five sample
galaxies that are major mergers in progress deviate from the above correlations in having undermassive
BHs for their host size (see Figure 14 in KH13). Having noted this result, the five
mergers are omitted from our correlation fits shown below. However, mergers in progress are included in
Graham (2015) and in McConnell \& Ma (2013).
\lineskip=-15pt \lineskiplimit=-15pt}
\end{enumerate}
These procedural differences plus others summarized in KH13 or omitted here for the sake
of brevity account for most of the differences in the correlation plots shown in Graham (2015) and those in KH13.
Generically, they have the following effects (ones in italics also apply to McConnell \& Ma 2013).
(1) {\it At the high-$M_\bullet$ end, Graham's BH masses are biased low, because he uses underestimated values from
emission-line rotation curves,} because he uses $M_\bullet$ values that are not corrected for effects of halo dark matter,
and because he does not consistently use the Rusli {et~al.\ } (2013) high-$M_\bullet$ galaxies. (2) At the \hbox{low-$M_\bullet$}
end, Graham's BH masses are biased low, because he includes pseudobulges. Differentiating barred and unbarred galaxies
is not sufficient to solve this problem. {\it McConnell and Ma also include pseudobulges, differentiating early- and
late-type galaxies helps, although many S0s contain pseudobulges.} (3) Graham regards M{\ts}32 as pathological and
omits~it. KFCB show that it is a normal, tiny elliptical. Including it in KH13 helps to anchor the
BH correlations at low BH masses. (4) {\it The result is that the BH--host correlations have much larger scatter in Graham (2015)
and in McConnell \& Ma (2013) than they do in KH13 (see Figures 5 and 7 below)}. Also, Graham sees a
kink in the \hbox{$\log M_\bullet$ -- $M_{K,\rm bulge}$} correlation whereas we do not, and he sees no kink in the
$\log M_\bullet$ -- $\log \sigma$ whereas we see signs of a kink at high $\sigma$ where $M_\bullet$ becomes largely
independent of $\sigma$. McConnell \& Ma (2013) and KH13 agree on the kinks (and lack of kinks) in the
$M_\bullet$\ts--{\ts}host-galaxy correlations.
\subsection{Correlations Between BH Mass and Host Galaxy Properties from Kormendy \& Ho (2013)}
This section summarizes the BH{\ts}--{\ts}host-galaxy correlations from KH13.
\vfill\eject
The procedures summarized above lead in KH13 to Table 2 for 44 elliptical galaxies and
Table 3 for 20 classical bulges and 21 pseudobulges. Figure 5 shows the resulting $\log M_\bullet$ -- $M_{K,\rm bulge}$ and
$\log M_\bullet$ -- $\log \sigma$ correlations for classical bulges and ellipticals. Mergers in progress are omitted as
explained~above, and three ``monster'' BHs that deviate above the correlations are illustrated in faint symbols but
are omitted from the fits. Also shown are symmetric, least-squares fits (Tremaine {et~al.\ } 2002) symmetrized
around $L_{K,\rm bulge}$\ts=\ts$10^{11}$\ts$L_{K\odot}$ and $\sigma_e$\ts=\ts200{\ts}km{\ts}s$^{-1}$:
\vskip -28pt
\null
$$
\log\biggl(\kern -2pt{{M_\bullet} \over {10^9{\ts}M_\odot}}\kern -2pt\biggr)
= -(0.265 \pm 0.050) - (0.488 \pm 0.033) (M_{K,\rm bulge} + 24.21); \eqno{(1)}
$$
\centerline{\null}
\vskip -42pt
\centerline{\null}
$$
\log\biggl(\kern -2pt{{M_\bullet} \over {10^9{\ts}M_\odot}}\kern -2pt\biggr)
= -(0.509 \pm 0.049) + (4.384 \pm 0.287) \log \biggl({{\sigma} \over {200~{\rm km~s}^{-1}}}\biggr). \eqno{(2)}
$$
\null
\vskip -14pt
\noindent Here, we adopt equal errors of $\Delta M_{K,\rm bulge} = 0.2$ and $\Delta \log{M_\bullet} = 0.117$, i.{\ts}e., the mean
for all fitted galaxies. Then the intrinsic scatters in Equations (1) and (2) are 0.30 dex and 0.29 dex, respectively. In
physically more transparent terms,
\vskip -5pt
$$
{{M_\bullet} \over {10^9~M_\odot}} = \biggl(0.544^{+0.067}_{-0.059}\biggr)\
\biggl({{L_{K,\rm bulge}} \over
{10^{11}{\ts}L_{K\odot}}}\biggr)^{1.22 \pm 0.08} \eqno{(3)}
$$
\null
\vskip -26pt
\null
$$
{{M_\bullet} \over {10^9~M_\odot}} = \biggl(0.310^{+0.037}_{-0.033}\biggr)\
\biggl({{\sigma} \over
{200~{\rm km~s}^{-1}}}\biggr)^{4.38 \pm 0.29} \eqno{(4)}
$$
\noindent Both relations have shifted to higher BH masses because of corrections to $M_\bullet$, because mergers in
progress are omitted, and because pseudobulges are postponed.
\vskip 10pt
\vfill
\special{psfile=./ar4-fig19s.ps hoffset=-13 voffset=-39 hscale=59.5 vscale=59.5}
\begin{figure}
\caption{
\pretolerance=15000 \tolerance=15000
\lineskip=0pt \lineskiplimit=0pt
Correlations of BH mass $M_\bullet$ with the $K$-band absolute magnitude and luminosity of the host bulge ({\it left panel})
and with its velocity dispersion at radii where $\sigma_e$ is unaffected by the BH ({\it right panel}). Black points are
for ellipticals; a white center indicates that this galaxy has a core. Red points are for classical bulges. The lines
are Equations~(1)~and~(2). Note:~the $M_\bullet$--$M_{K,\rm bulge}$ correlation remains log-linear with no kink
at high luminosities. In contrast, the biggest BH masses look essentially independent of $\sigma_e$ in ellipticals that have cores.
From KH13.
}
\end{figure}
\eject
\centerline{\null}
The $\log M_\bullet$ -- $L_{K,\rm bulge}$ correlation in Figure 5 is converted to a correlation with bulge stellar
mass $M_{\rm bulge}$ by applying mass-to-light ratios that were engineered by KH13 to be independent
of the papers that determine $M_\bullet$, to have zeropoints based on the Williams {et~al.\ } (2009) dynamical models,
but also to take variations in stellar population age into account. The resulting mass correlation is:
\vskip -18pt
\null
$$
\null\quad\quad100\biggl({{M_\bullet} \over {M_{\rm bulge}}}\biggr) = \biggl(0.49^{+0.06}_{-0.05}\biggr)\
\biggl({{M_{\rm bulge}} \over
{10^{11}{\ts}M_\odot}}\biggr)^{0.15 \pm 0.07},~~ \eqno{(5)}
$$
\null
\vskip -6pt
\noindent with an intrinsic scatter of 0.28 dex. The BH mass fraction,
$M_\bullet/M_{\rm bulge}$\ts=\ts$0.49^{+0.06}_{-0.05}$\ts\% at $M_{\rm bulge} = 10^{11}$\ts$M_\odot$,
is approximately a factor of 4 larger than we thought before the $M_\bullet$ values were corrected
(Merritt \& Ferrarese 2001;
Kormendy \& Gebhardt 2001;
McClure \& Dunlop 2002;
Marconi \& Hunt 2003;
Sani {et~al.\ } 2011).
Note again that $M_\bullet$--$L_{\rm bulge}$ is a single power law with no kink, whereas $M_\bullet$--$\sigma$
is a power law that ``saturates'' at high $M_\bullet$ (see also McConnell \& Ma 2013). That is, $M_\bullet$
becomes nearly independent of $\sigma$ in the highest-$\sigma$ galaxies that also have cores (Figure 5).
We understand why: The Faber-Jackson $L$--$\sigma$ correlation saturates at high $L$, because $\sigma$
does not grow very much once galaxies are massive enough so that all mergers are dry (Figure 6).
This is seen in simulations of dry, major mergers by (e.{\ts}g.)~Boylan-Kolchin, Ma, \& Quataert (2006)
and by Hilz {et~al.\ } (2012). Section 4.1.1 reviewed arguments why core ellipticals are remnants of dry mergers.
\vfill
\special{psfile=./ar4-fig20s.ps hoffset=38 voffset=-34 hscale=41 vscale=41}
\begin{figure}
\caption{
Faber-Jackson (1976) correlations for core ellipticals ({\it black\/}) and coreless ellipticals ({\it red\ts}). Total $V$-band
absolute magnitudes $M_{V,\rm total}$, velocity dispersions $\sigma$, and profile types are mostly from Lauer {et~al.\ } (2007b) or
otherwise from KFCB. The lines are symmetric least-squares fits to core Es ({\it black line}) and coreless Es ({\it red line})
with 1-$\sigma$ uncertainties shaded. The coreless galaxies show the familiar relation, $\sigma \propto L_V^{0.27 \pm 0.02}$.
But velocity dispersions in core ellipticals increase only very slowly with luminosity, $\sigma \propto L_V^{0.12 \pm 0.02}$.
As a result, $M_\bullet$ becomes almost independent of $\sigma$ for the highest-$\sigma$ galaxies in Figure 5.
This figure from KH13 is based on Kormendy \& Bender (2013). Lauer {et~al.\ } (2007a) and Cappellari {et~al.\ }
(2013a, b) show closely similar diagrams.
\pretolerance=15000 \tolerance=15000
\lineskip=0pt \lineskiplimit=0pt
}
\end{figure}
\eject
The pseudobulges that were postponed from Figure 5 are added to the BH--host correlations in Figure 7. Hu (2008) was
the first person to show that pseudobulges deviate from the $M_\bullet$\ts--\ts$\sigma_e$ correlation in having small BH masses.
This was confirmed with larger samples and extended to the $M_\bullet$\ts--\ts$M_{K,\rm bulge}$ and
$M_\bullet$\ts--\ts$M_{\rm bulge}$ correlations by Greene {et~al.\ } (2010) and by Kormendy, Bender, \& Cornell (2011).
Figure 7 now shows this result for the largest available sample, that of KH13.
\vfill
\special{psfile=./ar4-fig21s.ps hoffset=-13 voffset=-36 hscale=58.5 vscale=58.5}
\begin{figure}
\caption{
\pretolerance=15000 \tolerance=15000
\lineskip=0pt \lineskiplimit=0pt
Correlations of BH mass with the $K$-band absolute magnitude and luminosity of the host bulge ({\it top-left panel}), with its
stellar mass ({\it bottom panel}), and with the mean velocity dispersion of the host bulge at radii that are large enough so that
$\sigma_e$ is unaffected by the BH ({\it right panel}). Gray points are for ellipticals, red points are for classical bulges,
and blue points are for pseudobulges. The lines with shaded 1-$\sigma$ uncertainties are symmetric least-squares fits to the
classical bulges and ellipticals. In all panels, pseudobulge BHs are offset toward smaller $M_\bullet$ from the correlations
for classical bulges and ellipticals. Absent any guidance from the red and gray points, we conclude the pseudobulge BHs do not
correlate with their hosts in any way that is strong enough to imply BH-host coevolution. From KH13, who
tabulate the data and give sources.
}
\end{figure}
\eject
Hints of this result are seen in McConnell \& Ma (2013); they compare early- and late-type galaxies and note
that many late-type galaxies have undermassive BHs. This captures some of the result in Figure 7 but not all of it,
because many S0 galaxies contain pseudobulges. Similarly, Graham (2015) compares barred and unbarred galaxies and
concludes that many barred galaxies have undermassive~BHs. Again, this result is related to Figure 7 -- many (but
not all) barred galaxies contain pseudobulges, and many (but not all) unbarred galaxies contain classical bulges.
In Figure 7, the highest-$M_\bullet$ pseudobulge BHs largely agree with the correlations for classical
bulges and ellipticals; the lowest-$M_\bullet$ BHs deviate, but not by much more than an order of magnitude.
Note that the BHs that we find in pseudobulges may be only the high-$M_\bullet$ envelope of a distribution that
extends to much lower BH masses. Still, why are pseudobulge BHs even close to the correlations? KH13
argue that this natural: even one major merger converts a pseudobulge to a classical bulge, and then
merger averaging manufactures an essentially linear correlation with a zeropoint near the upper end of the
mass distribution of progenitors (see Figure\ts37 in KH13 and
Peng 2007;
Gaskell 2010, 2011;
Hirschmann {et~al.\ } 2010;
Jahnke \& Macci\`o 2011, who developed this idea).
Turning next to disks: Figure 8 confirms the conclusion reached in Kormendy \& Gebhardt (2001) and
in Kormendy, Bender, \& Cornell (2011) that BH masses are completely uncorrelated with properties of their host disks.
\vfill
\special{psfile=./ar4-fig22s.ps hoffset=42 voffset=-34 hscale=38 vscale=38}
\begin{figure}
\caption{
\pretolerance=15000 \tolerance=15000
\lineskip=0pt \lineskiplimit=0pt
Black hole mass $M_\bullet$ vs $K$-band absolute magnitude of the disk of the host galaxy. Filled circles are for galaxies
with BH detections based on spatially resolved stellar or gas dynamics; open circles are for galaxies with upper limits on
$M_\bullet$. The strongest upper limit is $M_\bullet$ \lapprox 1500 $M_\odot$ in M{\ts}33 (Gebhardt {et~al.\ } 2001). Red and
blue circles are for galaxies with classical and pseudo bulges, respectively. Green points are for galaxies with no
classical bulge and (almost) no pseudobulge but only a nuclear star cluster. From KH13, who tabulate the
data and give sources.
}
\end{figure}
\eject
M{\ts}33, with its strong upper limit on $M_\bullet$, briefly gave us the feeling that pure disks might not
contain BHs. But it was clear all along that they can have AGNs. Figure\ts8 includes bulgeless galaxies in which
we find $10^{6 \pm 1}$-$M_\odot$ BHs. The prototypical example is NGC 4395, a dwarf Sd galaxy with $M_V = -18.2$, with
no classical or pseudo bulge, but with only a nuclear star cluster that has an absolute magnitude of
$M_B \simeq -11.0$ and a velocity dispersion of \hbox{$\sigma$ \lapprox \ts30 $\pm$ 5 km s$^{-1}$} (Filippenko \& Ho 2003;
Ho {et~al.\ } 2009). And yet, NGC 4395 is the nearest Seyfert 1 galaxy known (Filippenko \& Ho 2003). It shows the
signatures of BH accretion -- broad optical and UV emission lines (Filippenko, Ho \& Sargent 1993), variable X-ray
emission (Shih, Iwasawa, \& Fabian 2003), and a compact, flat-spectrum radio core (Wrobel \& Ho 2006). Peterson {et~al.\ } (2005)
get $M_\bullet = (3.6 \pm 1.1) \times 10^5 \, M_\odot$ by reverberation mapping. This is the smallest BH mass
measured by reverberation mapping. But the BH in NGC 4395 is much more massive than $M_\bullet$ \lapprox \ts1500
$M_\odot$ in the brighter pure-disk galaxy M{\ts}33 ($M_V = -19.0$).
This is the best example of many that are revealed in the observing programs of Ho, Barth, Greene, and collaborators
and reviewed by Ho (2008) and by KH13. Other important galaxies include Pox 52
(Barth {et~al.\ } 2004;
Thornton {et~al.\ } 2008)
and Henize 2-10 (Reines {et~al.\ } 2011). Broader AGN surveys to find low-mass BHs, many of them in late-type, pure-disk galaxies, include
Greene \& Ho (2004, 2007),
Barth, Greene, \& Ho (2008), and
Dong {et~al.\ } (2012).
The general conclusion is that classical and even pseudo bulges are not necessary equipment for the
formation and nurture of supermassive BHs.
We need one more result before we discuss implications for galaxy evolution:
Very popular for more than a decade has been the suggestion that the fundamental correlation between BHs and
their host galaxies is not one with bulge properties but rather is a correlation with halo DM. This was
suggested by Ferrarese (2002) and supported by papers such as Baes {et~al.\ } (2003). The idea is attractive for galaxy
formation theory, because then halo mass is the natural parameter to control AGN feedback (e.{\ts}g., Booth \& Schaye 2010).
The most robust part of our effort to model galaxy formation is the calculation of DM hiararchical clustering. Conveniently,
DM mass is then provided by halo-finder algorithms.
However, we can now be confident that {\it halo DM does not correlate directly with $M_\bullet$ independent
of whether or not the galaxy contains a bulge} (Kormendy \& Bender 2011). This result is reviewed in detail and
with the largest galaxy sample in KH13. They list eight arguments against Ferrarese's conclusion.
Some are based on examining the proxy parameters that she used to make her arguments ($\sigma$ for $M_\bullet$
and $V_{\rm circ}$ for the DM; e.{\ts}g. we now know that $\sigma$ is not a proxy for BH mass for pseudobulge galaxies:
Figure 7 here). Some arguments are based on the direct correlation of measured $M_\bullet$ with $V_{\rm circ}$:
there is essentially no correlation unless the galaxy has a classical bulge. Perhaps the most telling argument is based
on the well determined relationship between the stellar mass $M_*$ and the DM mass $M_{\rm DM}$ of galaxies.
Behroozi, Wechsler, \& Conroy~(2013) show that $M_*/M_{\rm DM}$ reaches a maximum at $M_{\rm DM} \simeq 10^{12}$ $M_\odot$
and is smaller at both higher and lower $M_{\rm DM}$ (see also Fig.~9 here). Together with the correlation (Equation 5) between $M_\bullet$
and $M_{\rm bulge} \simeq M_*$ (exact for ellipticals and approximate for bulge-dominated galaxies), Behroozi's
result implies that the relationship between $M_\bullet$ and $M_{\rm DM}$ is complicated,
\vskip -7pt
$$ M_\bullet \propto M_{\rm DM}^{2.7}~~~{\rm at}~~M_{\rm DM} \ll 10^{12}{\ts}M_\odot~~, \eqno{(6)}$$
\vskip -9pt
\noindent but
\vskip -9pt
$$ M_\bullet \propto M_{\rm DM}^{0.34}~~~{\rm at}~~M_{\rm DM} \gg 10^{12}{\ts}M_\odot~, \eqno{(7)}$$
\vskip -1pt
\noindent with a kink in the correlation at $M_{\rm DM} \simeq 10^{12}$\ts$M_\odot$.
Meanwhile, the $M_\bullet$\ts--\ts$M_{\rm bulge}$ correlation is log linear with small scatter from the lowest to the
highest bulge masses in Figure 5. This correlation shows no kink at $M_{\rm DM} \sim 10^{12}$ $M_\odot$ corresponding
to $M_{\rm bulge} \sim 3 \times 10^{10}$ $M_\odot$ (see Figure 7). The simplicity of $M_\bullet$\ts--\ts$M_{\rm bulge}$
versus the complexity of $M_\bullet$\ts--\ts$M_{\rm DM}$ is another argument in favor of the
conclusion that BHs coevolve with bulges and ellipticals but not directly with DM halos.
\vskip -26pt
\centerline{\phantom{00000000000000000}}
\subsection{AGN Feedback and the Coevolution (Or Not) of Supermassive Black Holes and Host Galaxies}
Implications for the coevolution (or not) of BHs and host galaxies are reviewed by Kormendy \& Ho (2013).
They distinguish four modes of AGN feedback:
\begin{enumerate}
\item[(1){\kern -3pt}]{Galaxies that are not dominated by classical bulges -- even ones like NGC 4736 that contain big
pseudobulges -- can contain BHs, but these grow by low-level AGN activity that involves too little energy
to affect the host galaxy. Whether or not AGNs are turned on when we observe them, these galaxies
actively form stars and engage in secular evolution by the redistribution of gas.
Most AGNs at $z \sim 0$ and probably out to $z \sim 2$ are of this kind. They include giant galaxies
such as our Milky Way, with outer circular-orbit rotation velocites $V_{\rm circ} > 220$ km s$^{-1}$.
These galaxies are not correctly described by simple prescriptions in which gravitational potential
well depth controls AGN feedback.}
\item[(2){\kern -3pt}]{Most consistent with the prevailing emphasis on AGN feedback are classical bulges and
coreless-disky-rotating ellipticals. They satisfy the tight correlations between $M_\bullet$
and bulge properties in Figure 5. It is likely (although the engineering is not fully understood) that AGN
feedback helps to establish these $M_\bullet$--host relations during dissipative (``wet'') major mergers.
This must happen mostly at high $z$, because gas fractions in major mergers at $z \sim 0$ are small,
and indeed, mergers in progress at $z \simeq 0$ do not satisfy the $M_\bullet$ correlations.
It is important to note that even small Es with $V_{\rm circ}$ \lapprox \ts100 km s$^{-1}$ (e.{\ts}g.,
M{\ts}32) satisfy the $M_\bullet$--host correlations, whereas even giant pure disks (e.{\ts}g., M{\ts}101)
do not. Coevolution is not about potential well depth. Coevolution (or not) is determined by whether
(or not) the galaxy contains a classical bulge of elliptical -- i.{\ts}e., the remnant of at least one
major merger.
}
\item[(3){\kern -3pt}]{The highest-mass ellipticals are coreless-boxy-nonrotating galaxies whose most recent
mergers were dissipationless (``dry''). These giant ellipticals inherit any feedback magic --
including the $M_\bullet$--host relations -- from (2). In them, AGN feedback plays a different,
essentially negative role. It keeps galaxy formation from ``going to completion'' by keeping
baryons suspended in hot gas. With masses $M > M_{\rm crit}$ in Section 7, these galaxies hold
onto hot, X-ray-emitting gas that is believed to prevent cold-gas dissipation and to quench star
formation. However, X-ray gas cooling times are short,
and so -- given that we observe only weak temperature gradients -- something
must keep the hot gas hot. One such process is gas infall from the cosmological web (Dekel \&
Birnboim 2006). Another is ``maintenance-mode AGN feedback'' (see Fabian 2012 for a review).
All proposed heating processes may be important. See Section\ts7.
\pretolerance=100000\tolerance=100000}
\item[(4){\kern -3pt}]{The averaging that is inherent in galaxy mergers may significantly decrease the scatter in the
$M_\bullet$--host correlations. That is, during a merger, the progenitors' stellar masses add and so
do their BH masses. In the absence of new star formation, the effect is to decrease the correlation
scatter. Recall a conclusion in Section 4 that only a modest amount of star formation happens during
mergers. So the central limit theorem ensures that the scatter in BH correlations
with their hosts decreases as $M_\bullet$ increases via either wet or dry mergers.}
\end{enumerate}
In summary, KH13 provides the largest available database on BH detections via spatially
resolved dynamics, putting the many heterogeneous discovery papers on a homogeneous system of (for example)
distances and magnitudes, and incorporating many $M_\bullet$ corrections from the recent literature. Homogeneous
data are also provided for all BH host galaxies, including all disk-galaxy hosts, many of which had not
previously been studied. Bulge-pseudobulge classifications are provided based on multiple classification
criteria (cf.{\ts}Section\ts2.1 here), and (pseudo)bulge-disk photometric decompositions are derived for all galaxies
that did not previously have photometry. The results (their Tables~2~and~3) are an accurate enough database
to allow Kormendy \& Ho (2013) to derive a number of new conclusions about BH-host correlations and their
implications. Some of these are reviewed above. Others, such as correlations (or not) with nuclear star clusters
and globular cluster systems, are omitted here, in part to keep the length of this paper manageable, and in part
because the connection with galaxy bulges is less direct than it is for subjects that we cover.
Many of our conclusions disagree with Graham (2015). Within the subjects that I have reviewed in this paper,
I have tried to explain why. Readers are encouraged to compare the accuracy of our data sets
(particularly $M_\bullet$ measurements), our results, and the physical picture in which they are embedded.
We believe that the observational conclusions reached in KH13 are robust, and the essential implications
for galaxy evolution -- the big picture of what happens, if not the engineering details -- are well established.
Section 7 is an important example.
\vskip -25pt
\centerline{\phantom{00000000000000000}}
\section{Quenching of Star Formation}
Many papers on star formation histories begin by setting up a ``straw-man target'' that the quenching of star formation is mysterious.
In contrast, it strikes me that the literature shows encouraging convergence on a picture at least at $z < 1$ in which~well defined
processes convert ``blue cloud'' star-forming galaxies to ``red sequence'' red and dead galaxies. This section rephrases Section 6.2
to describe this picture.
The essential observation that has driven progress on this subject is summarized in Figure 9. The left panel shows the
Allen {et~al.\ } (2011) version of the Behroozi~{et~al.\ } (2013) result that led to Equations (6) and (7) in Section 6.1. I use
it because the abscissa is in the same units as in the right panel. It shows that the ratio of stellar mass to total
mass reaches a maximum at $V_{\rm circ} \sim 300${\ts}km{\ts}s$^{-1}$ or, in Behroozi {et~al.\ } (2013), at $M_{\rm DM} \sim 10^{12}$ $M_\odot$.
This maximum is $\sim$\ts1/5 of the cosmological baryon fraction, so most baryons in the universe have not yet made stars.
\hbox{Lower-mass} halos have smaller stellar fractions ({\it left panel\/}) and smaller baryon fractions ({\it right panel\/}) because -- we believe --
the baryons have increasingly been ejected from DM halos by star-formation and supernova feedback or never accreted after cosmological
reionization. But the focus here is on higher DM masses. They, too, have smaller stellar mass fractions than at the ``sweet spot''
halo mass of $10^{12}$ $M_\odot$. But Figure\ts9 ({\it right\/}) shows that these baryons are not ``missing'' at $M_{\rm DM} \gg 10^{12}$\ts$M_\odot$.
On the contrary, the total baryon fraction converges to essentially the cosmological value in the highest-mass halos, which are
halos of rich clusters of galaxies. This is the by-now well known result that, as $M_{\rm DM}$ grows above $10^{12}$ $M_\odot$ and $V_{\rm circ}$
grows above 300 km s$^{-1}$, an increasingly large fraction of the baryons are indeed present but have not made stars. Rather, they are suspended
in hot, X-ray-emitting gas, until in rich clusters of galaxies, that hot gas outmasses the stellar galaxies in the cluster by $1.0 \pm 0.3$ dex
(Kravtsov \& Borgani 2012). This has led to the essential idea of \hbox{``$M_{\rm crit}$ quenching''} of star formation by X-ray-emitting gas,
which can happen provided that the DM mass is larger than the critical mass, $M_{\rm DM}$ $_>\atop{^\sim}$} \def\lapprox{$_<\atop{^\sim}$ \ts$M_{\rm crit} \simeq 10^{12}$\ts$M_\odot$,
that is required to support the formation and retention of hot gas halos (e.{\ts}g.,
Birnboim \& Dekel 2003;
Kere\v s {et~al.\ } 2005;
Cattaneo {et~al.\ } 2006, 2008, 2009;
Dekel \& Birnboim 2006, 2008;
Faber {et~al.\ } 2007;
KFCB;
Peng {et~al.\ } 2010, 2012;
KH13,
Knobel {et~al.\ } 2015, and
Gabor \& Dav\'e 2015).
\vfill
\special{psfile=./Dai-Allens.ps hoffset=-14 voffset=-39 hscale=67 vscale=67}
\begin{figure}
\caption{
\pretolerance=15000 \tolerance=15000
\lineskip=0pt \lineskiplimit=0pt
Stellar mass fraction $M_*/(M_{\rm baryon} + M_{\rm DM})$ ({\it left\/}) and total baryon mass fraction
$M_{\rm baryon\/}/(M_{\rm baryon} + M_{\rm DM})$ ({\it right\/}) versus a circular-orbit rotation velocity
$V_{\rm circ} \sim \sqrt{G M_{\rm DM} / r}$ (Dai {et~al.\ } 2010) that approximately characterizes the total
mass distribution. Here $M_*$ is the stellar mass, $M_{\rm DM}$ is the DM halo mass, $r$ is the radius of
the halo, and $G$ is the gravitational constant. The cosmological baryon fraction has been adjusted very
slightly to $0.16 \pm 0.01$, i.{\ts}e., the mean of the WMAP and Planck measurements (Hinshaw {et~al.\ } 2013 and
Planck Collaboration 2014, respectively). Both figures originally come from Dai {et~al.\ } (2010).
}
\end{figure}
\eject
The transition mass between galaxies that should contain X-ray gas and those that should not is consistently derived
by a variety of theoretical arguments and is consistent checked via a variety of observational tests. It should occur at the
DM mass at which the hot gas cooling time is comparable to the infall time (Rees \& Ostriker 1977). Birnboim \& Dekel (2003) and Dekel \&
Birnboim (2006, 2008) argue from theory and Kere\v s {et~al.\ } (2005) find from SPH simulations that gas that is accreted during hierarchical
clustering falls gently into shallow potential wells and makes star-forming disks, whereas gas crashes violently onto giant galaxies
and is shock-heated to the virial~temperature. It is this hot gas that quenches star formation. Calculated hot-gas cooling times are
short; this led to the well known ``cooling flow problem'' (Fabian 1994).~\hbox{But X-ray} measurements of temperature profiles
now show that they are much shallower than cooling-time calculations predict in the absence of heating
(McNamara \& Nulsen 2007;
Kravtsov \& Borgani 2012;
Fabian 2012).
Debate continues about how the gas is kept hot;
Dekel \& Birnboim (2006, 2008)
suggest that the required heating is caused by continued accretion; AGN feedback is another candidate (e.{\ts}g.,
Best et al. 2006;
Best 2006, 2007a, b;
Fabian 2012;
Heckman \& Best 2014),
and dying stars return gas to the intergalactic medium at just the right kinetic temperature (Ostriker 2006).
The engineering details need to be sorted out. It is likely that all processes are important. But from
the point of view of this paper, the engineering is secondary. The important point is that the galaxies and clusters
tell us that they know how to keep the gas hot.
Many observed properties of galaxies can be understood in the context of $M_{\rm crit}$ quenching. E.{\ts}g.,
it allows semianalytic models of galaxy formation to reproduce the color bimodality of galaxies (``red sequence'' versus ``blue cloud'';
Blanton \& Moustakas 2009) as a function of redshift (Cattaneo et al. 2006, 2008, 2009).
Faber {et~al.\ } (2007) and KFCB emphasize the connection of the above results to this paper: {\it $M_{\rm crit}$ star-formation
quenching is believed to explain the difference between the two kinds of ellipticals discussed in Section 4.1.1.} I noted there that
classical bulges and coreless-disky-rotating ellipticals generally do not contain X-ray-emitting gas, whereas core-boxy-nonrotating
ellipticals contain more X-ray gas as their luminosities increase more above $L_{\rm crit} = 10^{10.2}$ $L_{B\odot}$~(Figure\ts3). Now,
$L_{\rm crit}$ corresponds to $M_V \simeq -20.9$; i.{\ts}e., 0.6 mag fainter~than~the~divide between coreless-disky-rotating
and core-boxy-nonrotating ellipticals. This is a factor of almost 2. If the most recent event that made an elliptical
was an \hbox{equal-mass} merger, then {\it the divide betweeen coreless-disky-rotating and core-boxy-nonrotating~ellipticals happens at a
luminosity below which neither of the merger progenitor galaxies should have contained \hbox{X-ray} gas and above which one or both progenitor
galaxies should have contained \hbox{X-ray} gas. Thus KFCB point out~that~the \hbox{E{\ts}--{\ts}E} dichotomy occurs at the correct luminosity
so that coreless-disky-rotating ellipticals formed in wet mergers whereas core-boxy-nonrotating ellipticals formed in dry mergers.}
Specifically, $M_V \simeq -20.7$ for merger progenitors corresponds (using $M/L_V \sim 6$) to a stellar mass of $M_* \simeq 1 \times 10^{11}$
$M_\odot$ or, using a baryon-to-total mass ratio of 1/6 (Komatsu {et~al.\ } 2009), to $M_{\rm DM} \simeq 6 \times 10^{11}$ $M_\odot$.
And the divide between coreless-disky-rotating Es and core-boxy-nonrotating Es happens at $M_{\rm DM} \simeq 10^{12}$ $M_\odot$.
So the agreement with the above picture of $M_{\rm crit}$ star-formation quenching is good.
Thus our picture of the formation of classical bulges and elliptical galaxies by wet and (at $M_{\rm DM} > 10^{12}$\ts$M_\odot$) dry
major mergers (Section 4 of this paper) is a tidy addition to our developing paradigm of star-formation quenching. Many details of the
structure of classical bulges and ellipticals (e.{\ts}g., the list in Section 4.1.1) fit into and support this paradigm.
But the paradigm is more general than just an explanation of the E{\ts}--{\ts}E dichotomy. I turn to these more general aspects next:
In a seminal paper, Peng {et~al.\ } (2010) use a few robust observations~to~derive very general conclusions about how quenching must work.
They do this completely operationally, without any need to identify the physical mechanism(s) of quenching. At redshift $z$\ts$\sim$\ts0
(Sloan Digital Sky Survey) and out to $z$\ts$\sim$\ts1 (zCOSMOS survey: Lilly {et~al.\ } 2007) the most essential observations used are (1) that the specific star
formation rate is almost independent of galaxy mass (there is a ``main sequence'' of star formation) but with rapidly decaying specific star
formation rate as $z \rightarrow 0$, and (2) that star-forming galaxies satisfy a Schechter (1976) mass function whose characteristic mass is
almost independent of $z$. From a discussion of how star formation operates to reproduce the above and other observations,
they deduce that quenching is driven by galaxy mass and by galaxy environment and that these two modes (not identified physically) are separable
and independent. Plus there must be an additional quenching mode that is associated with bulge formation via mergers.
Figure 10 connects their picture with the quenching paradigm that we review here.
\vfill
\special{psfile=./PengLillys.ps hoffset=-12 voffset=-34 hscale=57.8 vscale=57.8}
\begin{figure}
\caption{
\pretolerance=15000 \tolerance=15000
\lineskip=0pt \lineskiplimit=0pt
Powerpoint slide connecting the star-formation quenching picture of Peng {et~al.\ } (2010: central figure and its caption)
with the picture that is summarized in this paper (surrounding text).
}
\end{figure}
\eject
Peng {et~al.\ } (2010) emphasize that their analysis is operational:~it~identifies the conditions in which quenching
must operate, but it does not identify quenching mechanisms. However, with this section's background
on $M_{\rm crit}$ quenching and with results from KH13 on BH{\ts}--{\ts}host-galaxy coevolution (or not),
we can identify aspects of our developing physical picture of star-formation quenching with the conclusions of Peng {et~al.\ } (2010).
This is illustrated in Figure 10.
The masses used in Peng {et~al.\ } (2010) are estimated by integrating star formation rates and by fitting spectral energy distributions;
in essence, they are stellar masses. Figure 10 suggests that mass quenching tends to happen at masses $\sim 10^{10.5}$\ts$M_\odot$.
In Figure 7 of Peng {et~al.\ } (2010), the fraction of quenched galaxies (independent of environment) reaches 50\ts\% at $\sim 10^{10.6}$\ts$M_\odot$
and 80\ts\% at $\sim 10^{11.25}$\ts$M_\odot$. These correspond to $M_{\rm DM} \sim 10^{11.4}$ to $10^{12}$ $M_\odot$. The good agreement with
$M_{\rm crit}$ suggests that Peng's ``mass quenching'' is precisely our ``$M_{\rm crit}$ quenching'' by hot gas.
Peng {et~al.\ } (2010) conclude further that some low-mass galaxies are quenched by their environments. That is, these galaxies are
quenched because they are satellites of higher-mass objects -- ones (either individual galaxies or clusters of galaxies) that
{\it can have\/} masses $M_{\rm DM}$ $_>\atop{^\sim}$} \def\lapprox{$_<\atop{^\sim}$ $M_{\rm crit}$. {\it I suggest that Peng's ``environmental quenching''
is the same physical process as mass quenching, but in Peng's mass quenching, the X-ray gas that does the work belongs to the galaxy that
is being quenched, whereas in environmental quenching, the X-ray gas that does the work belongs to somebody else; i.{\ts}e., to
the quenched galaxy's parent giant galaxy or galaxy cluster.} This idea is verified by Peng {et~al.\ } (2012), Knobel {et~al.\ } (2015), and
Gabor \& Dav\'e (2015).
The suggested connection with KH13 then is this: Both mass and environment quenching are aspects of
point 3 in Section 6.2 -- they are effects of hot gas that is kept hot by a combination of maintenance-mode AGN feedback and other processes
such as continued infall of gas from the cosmological hierarchy and the injection of the kinetic energy of gas that is shed by dying stars.
But the above quenching processes are not sufficient. It is easy~to~explain~why\ts-- to give an example that mass quenching and
environment quenching cannot explain. {\it What quenches field S0 galaxies with masses $M$ $\ll$\ts$M_{\rm crit}$?} Kormendy \& Ho (2013)
suggest that they are quenched in the context of wet galaxy mergers that include starbursts, with energy feedback from the starburst beginning
the job of quenching and AGN feedback (Section 6.2, point 2) finishing the job. It seems natural to suggest that this is the Peng's
``merger quenching''. Observations of gas outflows~in high-$z$, star-forming galaxies such as submillimeter galaxies{\ts}--{\ts}at least some
of which are mergers{\ts}--{\ts}are reviewed in KH13. Of course, bulge-formation and $M_{\rm crit}$ quenching can be mutually
supportive (e.{\ts}g., Woo {et~al.\ } 2015).
Once star formation is quenched at $M > M_{\rm crit}$, then dry mergers preserve both the quenched state and the $M_\bullet${\ts}--{\ts}host
correlations (Section 6.2, point 4 and modes ``mass quenched then merged'', ``environment quenched then merged'', and ``merger quenched
then merged'' in Figure 10).
The biggest remaining question in our $z < 1$ picture is this: {\it In merger-quenched galaxies that have $M \ll M_{\rm crit}$, i.{\ts}e.,
in objects in which X-ray gas is not available even after the merger is finished, what preserves the quenched, red and dead state?
We do not know, but episodic, low-level AGN feedback may be~the~answer.}
\vfill
The biggest overall uncertainty is that quenching may operate \hbox{differently at $z$ $_>\atop{^\sim}$} \def\lapprox{$_<\atop{^\sim}$ {\kern 0.6pt}2}. Dekel and Birnboim argue
(1) that $M_{\rm crit}$ is higher at high $z$, when gas fractions in galaxies and gas accretion rates onto galaxies~are~both~higher~and
(2) that cold streams can penetrate hot gas at high $z$ and contribute to the growth of disks at masses that are unattainable at $z \sim 0$
(Dekel {et~al.\ } 2009).
Another difference involves the observation that most star-forming galaxies define a main sequence of star formation with few outliers,
implying that duty cycles are long and hence that star formation is not driven primarily by short-duration events such as mergers (Section 4.1.5).
When strong gas outflows are seen in star-forming galaxies at $z \sim 2$, the inference is that some combination of star formation and AGN
feedback is responsible but that these are not primarily driven by major mergers (e.{\ts}g.,
F\"orster Schreiber {et~al.\ } 2014;
Genzel {et~al.\ } 2014).
Because these processes are also associated with bulge growth in disk galaxies (Lang {et~al.\ } 2014), the most consistent interpretation
that also includes the $M_\bullet$ correlation results is that the bulge growth in these objects is by clump cluster sinking (Section 3 here).
Genzel (private communication) suggests that Peng's mass quenching may be this outflow process associated with more-or-less steady-state star formation,
AGN feedback, and classical bulge growth. On the ``plus side'', there is clearly a danger that our tidy $z < 1$ picture is basically
correct but not a description of what happens at $z \gg 1$. On the other hand, we already know that many details of galaxy structure are well
explained by the $z \sim 0$ picture. Particularly important is the natural explanation of cores in dry-merger remnants and central extra light
in wet-merger remnants (see KFCB). Alternative suggestions for quenching mechanisms at high $z$ have not addressed and solved the
problem of also explaining these aspects of $z \sim 0$ galaxy structure. This is not a proof that the suggested high-$z$ processes
are wrong.
It seems reasonable to conclude that our $z < 1$ picture of star formation quenching is robust. Mostly, it needs clarification of
engineering details. In marked contrast, star formation quenching at $z$ $_>\atop{^\sim}$} \def\lapprox{$_<\atop{^\sim}$ 2 is less well understood, although progress is rapid.
\section{A Partial Summary of Outstanding Problems}
I conclude with a summary of the most important outstanding problems. I restrict myself
to big-picture issues and do not address the myriad engineering details that are unsolved
by our present state of the art. They are, of course, vitally important. But a comprehensive list
would require a paper of its own. I therefore refer readers to earlier chapters of this book,
which discuss many of these problems in detail.
\begin{enumerate}
\item[(1){\kern -3pt}]{I emphasized in Section 4.1.3 that, to me, the most important goal is to produce
realistic classical bulges\ts$+${\ts}ellipticals and realistic disks that overlap over
a factor of $>$ 1000 in mass but that differ from each other in
ways that we observe over the whole of this range. They can combine
with any $B/T$ from 0 to 1, but the differences between bulges
and disks depend very little on $B/T$.
}
\item[(2){\kern -3pt}]{Four decades of work on $z \simeq 0$ galaxies showed convincingly that major mergers
convert disks into classical bulges and ellipticals with the observed~properties,
including S\'ersic index, fundamental plane parameter correlations,
intrinsic shape and velocity distributions, both as functions of mass,
the presence of cores or central extra light, and isophote shape. This work also
suggested that merger rates were higher in the past, and modern observations confirm
this prediction. By the mid-1990s, we had converged on a picture in which classical
bulges and ellipticals were made in major mergers. Enthusiasm for mergers was probably
overdone, but now, the community is overreacting in the opposite direction. The
successes of the 1970s--1990s are being forgotten, and -- I believe -- we
have come to believe too strongly that minor mergers control galaxy evolution. Reality
probably lies between these extremes. For today's audience, the
important comment is this: The observations that led to our picture of E formation via
major mergers have not been invalidated. I suggest that the profitable way forward is
to use what we learn from $z \simeq 0$ mergers-in-progress to explore how mergers
make bulges and ellipticals at higher $z$, including (of course) differences
caused (for example) by large gas fractions and including new ideas, such as violent
disk instabilities that make clumps that make bulges. For this still-elusive
true picture, it is OK that mergers are rare, because ellipticals are rare, too, and classical
bulges are rarer than we thought. And it is OK that most star formation does not
happen in mergers, because ellipticals are rare anyway, and because their main bodies
are made up of the scrambled-up remnants of already-stellar progenitor disks.
}
\item[(3){\kern -3pt}]{The most important unsolved problem is this: How did hierarchical clustering produce so many giant
galaxies (say, those with $V_{\rm circ}$ $_>\atop{^\sim}$} \def\lapprox{$_<\atop{^\sim}$ 150 km s$^{-1}$) with no sign of a
classical bulge? This problem is a very strong function of environment -- in field
environments such as the Local Group, most giant galaxies are bulgeless, whereas in
the Virgo cluster, most stars live in classical bulges and elliptical
galaxies. The clue therefore is that the solution involves differences in accretion
(gentle versus violent) and not largely internal physics such as star-formation or AGN feedback.
}
\item[(4){\kern -3pt}]{Calculating galaxy evolution {\it ab inito\/}, starting with $\Lambda$CDM density fluctuations,
constructing giant $n$-body simulations of halo hierarchical clustering, and then adding baryonic
physics is the industry standard today and the way of the future. It is immensely difficult
and immensely rewarding. It is not my specialty, and I have only one point to add to the
excellent review by Brooks \& Christensen: {\it Observations hint very strongly that we put
too much reliance on feedback to solve our engineering problems in producing realistic galaxies.
Observations of supermassive BH demographics tell us that AGN feedback
does not much affect galaxy structure or star formation until mergers start to make classical
bulges. And point (3) emphasizes that environment and not gravitational potential well depth
is the key to solving the problem of giant, pure-disk galaxies.}
}
\item[(5){\kern -3pt}]{We need to fully integrate our picture of disk secular evolution into our paradigm of galaxy evolution.
As observed at $z \simeq 0$, this picture is now quite detailed and successful. Essentially all of
the commonly occurring morphological features of galaxies -- bars, (nuclear, inner, and outer) rings,
nuclear bars, and pseudobulges -- are at least qualitatively explained within this picture. Some of
these details are beyond the ``targets'' of present galaxy-formation simulations. But pseudobulges
are immediately relevant, because our recognition of them has transformed our opinions about
classical bulges. They are much rarer than we thought. In particular, small classsical bulges
are {\it very\/} rare. And although some galaxies have structure that is completely determined
by the physics of hierarchical clustering, others -- and they dominate in the field -- appear to have
been structured almost exclusively by secular processes. Incorporating these processes is a challenge, because
slow processes are much more difficult to calculate than rapid processes. But secular evolution is an
ideas whose time has come (Sellwood 2014), and we need to include it in our paradigm.
}
\item[(6){\kern -3pt}]{At the same time, our {\it quantitative\/} understanding of secular evolution needs more work. For
example, we need a study similar to Dressler's (1980) work on the morphology-density relation:
We need to measure the luminosity and mass functions of disks, pseudobulges, and classical bulges$+$ellipticals,
all as functions of environmental density. At present, we have essentially
only two ``data points'' -- the extreme field (Kormendy {et~al.\ } 2010; Fisher \& Drory 2011) and
the Virgo cluster (see Kormendy {et~al.\ } 2010). This is already enough to lead to point (3) in
this list. We need corresponding studies in more environments
that span the density range from the field to the richest clusters. This will not be easy,
first because we need high spatial resolution whereas observing more environments drives us to
larger distances, and second because of point (7).
}
\item[(7){\kern -3pt}]{Our picture of disk secular evolution predicts that many galaxies should contain
both a classical and a pseudo bulge. Work on the subject has concentrated on
extremes -- on galaxies that are dominated by one kind of bulge or the
other. Samples of large numbers of galaxies will inevitably have to face
the challenge of separating at least three components (bulge, pseudobulge, and disk) and
in many cases more (bar, lens, \dots). We also need to be able to find pseudobulges
in face-on barred galaxies (see Section 2.1). But it is easy to overinterpret details
in the photometry. The best way to approach this problem is probably to begin with infrared
observations of nearly-edge-on galaxies (e.{\ts}g., Salo {et~al.\ } 2015).
}
\item[(8){\kern -3pt}]{Are classical bulges really indistinguishable from ellipticals? The structural
parameter scaling relations shown in Figure 4 (based on many authors' work) show that
they are closely similar. I use this result thoughout the present~paper. It is central
to Renzini's (1999) paraphrase of the classical morphological definition: ``A bulge is nothing
more nor less than an elliptical galaxy that happens to live in the middle of a disk.''
But not everybody agrees. Based on multi-component decompositions,
different fundamental plane correlations for classical bulges and ellipticals have been found by
Gadotti (2008,~2009,~2012) and by
Laurikainen {et~al.\ } (2010).
We need to resolve these differences.~At~stake is an understanding of whether classical bulges
and ellipticals form\ts--{\ts}as~I~suggest\ts-- by essentially the same major merger process
or whether important variations in that process produce recognizably different results. In particular,
it is not impossible that we can learn to distinguish ellipticals and perhaps {\it some\/} bulges
that form via mergers of distinct galaxies from other bulges that form via the mergers of
mass clumps that form in unstable disks. Both processes drive additional gas toward the center,
but it is possible that bulge formation via disk instabilities is intrinsically
more drawn out in time with the result (for example) that ``extra light components'' such as
those studied in Kormendy (1999), KFCB, and Hopkins {et~al.\ } (2009a) are smoothed away and
unrecognizable in the resulting classical bulges but not in disky-coreless-rotating ellipticals.
}
\item[(9){\kern -3pt}]{Returning to elliptical galaxies: KFCB present a detailed observational picture and ARA\&A-style
review of the two kinds of ellipticals in large part as seen in the Virgo cluster. Hopkins {et~al.\ }
(2009a, b) present modeling analyses of wet and dry mergers, respectively. {\it We need to know how
this very clean picture as seen in the nearest rich cluster translates into other environments.}
Much of the work published by Lauer {et~al.\ } (1995, 2005, 2007a, b), by Faber {et~al.\ } (1997), by
Kormendy \& Bender (1996, 2013), and by Bender {et~al.\ } (1989) applies to broader ranges of
environments. It suggests that the picture summarized here in Section 4.1.1 is basically
valid but that the distinction between coreless-disky-rotating and core-boxy-nonrotating galaxies
is somewhat ``blurred'' in a broader range of environments. For example, $M_V = -21.6$
cleanly separates the two kinds in Virgo, with only one partial exception (NGC 4621 at
$M_V = -21.54$ has $n = 5.36^{+0.30}_{-0.28}$ characteristic of core galaxies, but it has
a small amount of extra light near the center). However, the above papers and others show
that the two galaxy types overlap over a range of absolute magnitudes from about $M_V = -20.5$
to about $M_V = -23$. In the overlap range and occasionally outside it, some classification
criteria in Section 4.1.1 conflict with the majority. We should not be surprised that heterogeneous
formation histories can have variable outcomes; on the contrary, it is encouraging to see
as much uniformity as we see. Still, a study of how the systematics depend on environment
should be profitable.
}
\item[(10){\kern -3pt}]{Still on ellipticals and classical bulges: The SAURON and ATLAS$^{\rm 3D}$ teams have
carried out an enormous amount of truly excellent work on nearly all aspects of bulge$+$E
structure and evolution. A review is in preparation by Cappellari~(2015).
{\it It is natural to ask how the picture of bulges and ellipticals developed by the SAURON
and ATLAS$^{\rm 3D}$ papers compares with the one outlined in Section 4 here. The answer is that
they agree exceedingly well.} There are differences in emphasis, and the
large SAURON $+$ ATLAS$^{\rm 3D}$ teams address many subjects that are beyond the scope of
studies by our team or by the Nuker team. There is also one difference in analysis that
makes me uncomfortable -- in their work, they generally do not decompose galaxies into bulge
and disk parts. {\it It is therefore all the more remarkable that careful work without using
component decomposition and our work that always is based on component decomposition
converge on pictures that are so similar.} E.{\ts}g., the separate parameter correlations
for bulges and disks that are shown here in Figure~4 are visible as \hbox{pure-bulge~and~pure-disk}
boundaries of parameter correlation {\it regions\/} shown in Cappellari {et~al.\ } (2013b).
In their diagrams, the parameter space between our bulge and disk correlations is filled in
with intermediate-Hubble-type galaxies that have $0 < B/T < 1$. Similarly, Cappellari {et~al.\ }
(2011) and Kormendy \& Bender (2012) both revive the ``parallel sequence'' galaxy
classification of van den Bergh (1976), as do Laurikainen {et~al.\ } (2011).
Kormendy and Bender (2012) also add Sph galaxies (as distinct from ellipticals) to the classification.
} \vskip 1pt
\item[\phantom{(0)}{\kern -3pt}]{What may appear as a difference between Section 4 and the SAURON $+$ ATLAS$^{\rm 3D}$
work is our emphasis on many E{\ts}--{\ts}E dichotomy classification criteria versus their
distinction based only on fast versus slow rotation. However, Lauer (2012) shows that the SAURON $+$
ATLAS$^{\rm 3D}$ division into fast and slow rotators is essentially equivalent to the
division between coreless and core galaxies. The equivalence is not exact based in the rotation
amplitude parameter $\lambda_{r_e/2}$ (within 1/2 of the effective radius $r_e$) chosen by the
SAURON and ATLAS$^{\rm 3D}$ teams. But it becomes much more nearly exact if slow and fast
rotators are divided at a slighly higher rotation rate, $\lambda_{r_e/2} = 0.25$. In unpublished
work, I found an essentially equivalent result for the original SAURON kinematic classification,
in which slow rotators have $\lambda_R < 0.1$ and fast rotators have $\lambda_R > 0.1$ as defined
in Emsellem {et~al.\ } (2007). If the division is instead made at $\lambda_R = 0.175$, then core and
coreless ellipticals are separated essentially perfectly. (The only exception in KFCB is NGC 4458,
which is slowly rotating but coreless. But it is almost exactly round, and rotating galaxies that
are seen face-on will naturally look like slow rotators.) The more nuanced ATLAS$^{\rm 3D}$ look
at elliptical galaxy dynamics leads to a
revised suggestion that fast and slow rotators should be separated at
$\lambda_{r_e/2} = (0.265 \pm 0.01) \times \sqrt{\epsilon_{\rm e/2}}$
(Emsellem {et~al.\ } 2011, Equation 4). A typical $\epsilon = 0.2$ for core-boxy galaxies and
$\epsilon = 0.35$ for coreless-disky galaxies (from Tremblay \& Merritt 1996)
then implies a division at $\lambda_{r_e/2} = 0.16$ and 0.12, respectively. The typical intrinsic
ellipticaly of 0.4 found by Sandage, Freeman, \& Stokes (1970) for all ellipticals implies
$\lambda_{r_e/2} = 0.17$. These values are closer to the rotation parameters 0.25 and 0.175 that
divide core and coreless galaxies as found by Lauer (2012) and by my work, respectively.} \vskip 1pt
\item[\phantom{(0)}{\kern -3pt}]{I suggest that the best way to divide
slow rotators from fast rotators is not to pick some arbitrary value of the rotation parameter
but rather to ask the galaxies what value of the rotation parameter produces the cleanest
distinction into two kinds of galaxies as summarized in Section 4.1.1. When this is done, the
E{\ts}--{\ts}E dichotomy as discussed in this paper and the large body of work done by the SAURON
and ATLAS$^{\rm 3D}$ teams are remarkably consistent.
}\vskip 1pt
\item[(2$+$3 redux){\kern -3pt}]{A partial exception to the above conclusion is some of the $n$-body simulation work, e.{\ts}g.,
by Naab {et~al.\ } (2014). They acknowledge the importance of major mergers in some ways that
are consistent with the story advocated~in~this~paper. But their conclusion that ``The galaxies
most consistent with the class of non-rotating round early-type galaxies grow by gas-poor {\it minor\/}
mergers alone'' (emphasis added) is at best uncomfortable within the picture presented here. The
core-boxy-nonrotating galaxies have a large range of mostly homogeneous properties with respect
to which the round ones do not stand out as different (e.{\ts}g., KFCB). In particular, our
understanding of cores -- especially the tight correlations between core properties and BH masses --
depends on our picture that cores are scoured by black hole binaries that are formed in major
mergers (see KFCB and Kormendy \& Bender 2009 for both the data and a review). At best, {\it it remains
to be demonstrated that minor mergers -- which necessarily involve many small galaxies with}
(from Figure 7) {\it undermassive BHs -- can produce the very large BH masses and cores
that are seen in giant core ellipticals. Dry minor mergers cannot do better than to preserve
the $M_\bullet/M_{\rm host}$ mass ratio. Also, if many minor mergers are necessary{\ts}--{\ts}and
these galaxies are so massive that very many minor mergers are necessary to grow them{\ts}--{\ts}then
there is a danger of producing a central cluster of low-mass BHs that is never observed
as a cluster of compact radio sources and that is inherently unstable to the ejection of
objects in small-$n$ $n$-body systems} (see KH13, p.~634).
}
\item[(11){\kern -3pt}]{I conclude with two sociological points: It is worth emphasizing that galaxy evolution work
did not start in the 2000s. Many results that were derived in the 1960s -- 1990s remain valid today.
We should not forget them. We should integrate them into our current picture of galaxy evolution.}
\item[(12){\kern -3pt}]{And finally: Galaxy evolution work has changed
profoundly in the SDSS and HST eras. Before the early
1990s, {\it our goal was to understand the evolution of galaxy structure.} Now, most emphasis on galaxy
structure has disappeared. {\it Now, our goal is to understand the history of star formation in the
universe.} The main reason for this change is the common ground found between SDSS studies
of many thousands of galaxies and HST studies of very distant galaxies. Necessarily, both kinds
of studies concentrate on galaxies whose images are a few arcsec across. We do not resolve
structural details. Mainly, we measure colors and magnitudes. So galaxy evolution has
evolved into the study of the red sequence and blue cloud in the color-magnitude relation.
Star formation and its quenching are, of course, important. But it would be enormously healthy
if we could improve the dialog between SDSS$+$HST people and those -- such as this author -- who work
on nearby galaxies whose star formation histories and structures can be studied in great detail.
Conselice (2014) is an example of a paper that tries to bridge the gap. We would benefit greatly
if we could completely connect the two approaches to galaxy evolution.
}
\end{enumerate}
\begin{acknowledgement}
Many of my ideas about galaxy evolution were forged in intense and enjoyable collaborations
with Ralf Bender, Luis Ho, and the Nuker team. It is a pleasure to thank all these people and many
more who I do not have room to list for fruitful conversations over many years. I am especially
grateful to Reinhard Genzel for stimulating and insightful discussions and to
Ralf Bender,
Dimitri Gadotti, and
Eija Laurikainen
for very helpful comments on this paper.
Any errors of interpretation that remain are of course my responsibility. I thank
Steve Allen,
Xinyu Dai,
Ying-Jie Peng, and
Simon Lilly for permission to copy figures. My work on this paper
was supported by the Curtis T.~Vaughan, Jr.~Centennial Chair in Astronomy at the University of Texas.
\end{acknowledgement}
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\vfill\eject
\end{document}
|
{
"timestamp": "2015-04-15T02:00:37",
"yymm": "1504",
"arxiv_id": "1504.03330",
"language": "en",
"url": "https://arxiv.org/abs/1504.03330"
}
|
\section{Introduction}
Magnetic field and rotation are ubiquitous in stars. MiMeS
survey has observed over 550 Galactic O- and B-type stars, and detected
the surface magnetic fields of $\gtrsim 100$~G for $\sim 10$~\% of them (see
\citet{wad14} for review). Estimating the upper limit of the
currently-undetected magnetic
field to be $\sim 100$~G, \citet{wad14} argued
that the distribution
of the magnetic fields for massive stars may be bimodal: a small
population of strong magnetic fields ($\gtrsim 1$~kG) and a large
majority of weak magnetic fields ($\lesssim 100$~G).
A magnetic field of 1~kG corresponds to the magnetic
flux of $\sim 10^{27}$~G~cm$^{2}$ for a 17~$M_\odot$ star with the
radius of $\sim 8 R_\odot$ \citep{mcn65}, which is comparable to that
of magnetars.
\citet{ram13}
measured the surface rotational velocities of 216 O-type stars, and
found that 25~\% of the sample have $v \sin i > 200$~km~s$^{-1}$ while
the rest of them are slow rotators. According to stellar evolution
calculations by \citet{woo06}, if a star is rotating fast enough
initially, the rotational mixing prevents a very efficient angular
momentum transport between the helium core and the hydrogen envelope, and
the central iron core maintains a large amount of angular momentum at
pre-collapse stage.
They inferred the rotation period of a neutron
star to be 2.3--9.7~ms for such evolutions of a 16~$M_\odot$ star with
solar metallicity. Then, the high-rotational-velocity
population found by \citet{ram13} might produce proto-neutron stars
rotating with a
period similar to those of millisecond pulsars (MSPs).
The influences of magnetic field and rotation on core-collapse
supernovae have been studied as a possible agent to drive explosion
other than neutrino heating, while the latter fails to produce
energetic explosions
\citep[e.g.,][]{suw10,mue12,bru13}.
MHD core-collapse simulations done so far have placed the main focus on
rather extreme cases, viz., $B_{\textrm{pre}}\sim 10^{12}$--$10^{13}$~G
and $\Omega_{\textrm{pre}}\sim 1$~rad~s$^{-1}$ at pre-collapse, which
correspond to the magnetar-class magnetic field and MSP-class rotation
\citep[e.g.,][]{yam04,obe06,bur07,shi06,sch08,sch10,saw13a,mos14}.
In those simulations, the
magnetic field wound by differential rotation grows to dynamically
important strengths and later drives a strong outflow along the
rotation axis, reproducing the typical supernova-explosion energy of
$E_{\textrm{exp}}\sim 10^{51}$~erg.
Since the magnetic field and rotation in massive stars are
likely to have wide range of values as mentioned above, it may be
also important to study more "ordinary'' cases. In these cases
amplification mechanisms that are more efficient than the
simple winding are imperative to produce the field strength of $\sim
10^{15}$~G outside the proto-neutron star, which may be necessary to
impact on the supernova dynamics.
For non-rotating case \citet{end10,end12}
numerically studied the standing accretion shock instability, while
\citet{obe14} investigated the convection. In both cases,
the amplification is rather modest, and the impacts on dynamics
are found to be minor.
If the iron core is initially rotating rapidly, another candidate of
an efficient field amplification
mechanism in core-collapse supernovae is the magnetorotational
instability (MRI), which basically occurs in
differentially rotating systems
\citep{bal91,aki03}. Simulations of the MRI for weak seed
magnetic fields are computationally demanding, since the wavelength of the
fastest growing mode is quite small compared with
the size of the iron core, $\sim 1000$~km:
\begin{eqnarray}
\lambda_{\textrm{FGM}}
&\sim&\frac{2\pi v_{\textrm{A}}}{\Omega}\nonumber\\
&\sim& 200 \textrm{m}
\left(\frac{\rho}{10^{12}\textrm{g cm}^{-1}}\right)^{-\frac{1}{2}}
\left(\frac{B}{10^{13}\textrm{G}}\right)
\left(\frac{\Omega}{10^{3}\textrm{rad s}^{-1}}\right)^{-1},\nonumber\\
\end{eqnarray}
where $v_{\textrm{A}}$ is Alfv\'en velocity\footnote{The wavelength
of the fastest growing mode given here is one obtained for
cylindrical rotation laws, $\Omega(\varpi)$, with neglecting
buoyancy. We deal with the
general rotation laws, $\Omega(\varpi,z)$, taking the buoyancy into
account later in Section~\ref{sec.mri}}.
In fact, most of previous core-collapse simulations assuming
sub-magnetar-class magnetic fields have insufficient spatial
resolutions
to capture the MRI \citep{moi06,bur07,tak09}\footnote{In spite that
\citet{moi06} found an exponential growth of magnetic
field and claimed that the growth is due to a Tayler-type
"magnetorotational
instability'', which is completely different from one found by
\citet{bal91}. Note however, that the property of the instability
is still unclear, and no other groups succeeded to reproduce their
results to date.}.
In order to resolve the fastest growing mode,
local simulation boxes are utilized in some 2D/3D computations
\citep{obe09,mas12,gui15b,rem15}. The problems in the local
simulations, however, are the difficulties in taking into account the
effects from and feedbacks to dynamically changing structures.
\citet{saw13b} conducted the first global core-collapse
simulations for sub-magnetar-class magnetic fields with a sufficient
spacial resolution to capture the MRI albeit in 2D axisymmetry,
and found that the magnetic field is amplified by the
MRI to dynamically important strengths. In order to study its impacts
on the global dynamical, \citet{saw14} carried out similar but
longer-term simulations up to several hundred milliseconds after
bounce,
employing the simple light bulb approximation for neutrino
transfer. They found that the MRI
indirectly enhances the neutrino heating, and thus boost the explosion.
Performing 3D simulations for a thin layer on the equator,
\citet{mas15} argued another possible effect of the MRI, i.e., the
enhancement of neutrino luminosity by MRI-driven turbulence around
the proto-neutron star surface.
This paper is a sequel to \citet{saw13b} and
\citet{saw14}. We conducted 2D-axisymmetric high-resolution
simulations of core-collapse for rapidly-rotating magnetized
progenitors, changing the initial magnetic field strength and
the spatial resolution. The initial magnetic field strength assumed
here,
$B_{\textrm{pre}}\sim 10^{11}$~G, are one or two orders of magnitude
smaller than the extreme values adopted in some previous
simulations mentioned above.
The rest of the paper is organized as follows. In
Section~\ref{sec.method}, we describe the numerical method and
models. The results are presented in Section~\ref{sec.result},
and the discussion and conclusion are given in Section~\ref{sec.discon}.
\section{Numerical Methods}\label{sec.method}
We adopt a $15 M_\odot$ star \citep{woo95} for the progenitor of core-collapse simulations,
adding magnetic fields and rotations by hand. The following ideal
MHD equations and the equation of electron number density are
numerically solved by a time-explicit
Eulerian MHD code, \textit{Yamazakura} \citep{saw13a}:
{\allowdisplaybreaks
\begin{eqnarray}
&&\frac{\partial\rho}{\partial
t}+\nabla\cdot(\rho\mbox{\boldmath$v$})=0\label{eq.mhd.mass},
\\
&&\frac{\partial}{\partial t} (\rho\mbox{\boldmath$v$})+
\nabla\cdot\left(\rho\mbox{\boldmath$v$}\mbox{\boldmath$v$}-
\frac{\mbox{\boldmath$B$}\mbox{\boldmath$B$}}{4\pi}\right)\nonumber\\
&&\hspace{1pc}=-\nabla\left(p+\frac{B^2}{8\pi}\right)-\rho\nabla\Phi
\label{eq.mhd.mom},
\\
&&\frac{\partial}{\partial t}\left(e+\frac{\rho
v^2}{2}+\frac{B^2}{8\pi}\right)
\nonumber\\
&&\hspace{1pc}+\nabla\cdot
\left[
\left(e+p+\frac{\rho
v^2}{2}+\frac{B^2}{4\pi}\right)
\mbox{\boldmath$v$}
-\frac{(\mbox{\boldmath$v$}\cdot\mbox{\boldmath$B$})
\mbox{\boldmath$B$}}{4\pi}
\right]
\nonumber\\
&&\hspace{4pc}
=-\rho(\nabla\Phi)\cdot\mbox{\boldmath$v$}
+Q_E^{\textrm{abs}}+Q_E^{\textrm{em}}\label{eq.mhd.eng},
\\
&&\frac{\partial\mbox{\boldmath $B$}}{\partial t}=
\nabla\times\left(\mbox{\boldmath$v$}\times\mbox{\boldmath$B$}\right)
\label{eq.mhd.far},
\\
&&\frac{\partial n_e}{\partial
t}+\nabla\cdot(n_e\mbox{\boldmath$v$})=Q_N^{\textrm{abs}}+Q_N^{\textrm{em}}
\label{eq.ne},
\end{eqnarray}}where $Q_E^{\textrm{abs}}$ and $Q_E^{\textrm{em}}$ are
the changes of energy density due to neutrino/anti-neutrino absorptions
and emissions, respectively, and $Q_N^{\textrm{abs}}$ and
$Q_N^{\textrm{em}}$ are the similar notations for the changes of electron
number density. The other symbols have their usual meanings. The electron
fraction, $Y_e$, is given
by the prescription suggested by \citet{lie05} until bounce. After that,
where Liebend{\"o}rfer's prescription is no longer valid,
Equation~(\ref{eq.ne}) is solved to obtain $Y_e=n_e m_u/\rho$, where
$m_u=1.66\times 10^{-24}$~g is the atomic mass unit. We
assume Newtonian mono-pole gravity. A tabulated nuclear equation of
state produced by \citet{she98a, she98b} is utilized.
Computations are done with polar coordinates in two
dimensions, assuming axisymmetry and equatorial symmetry.
We take into account interactions of electron
neutrinos $\nu_e$ and anti-neutrinos $\Bar \nu_e$ with nucleons.
Instead of dealing with detailed neutrino transport, the light bulb
approximation is used as
in \citet{mur09,nor10,han12}. Taking the ultra-relativistic
limit for electrons and positrons, assuming the Fermi-Dirac distribution
with vanishing chemical potential for $\nu_e$ and $\Bar \nu_e$,
and neglecting the phase space blocking, we evaluate the source
term related to
$\nu_e$/$\Bar \nu_e$ absorption ($\nu_e+n\to e^-+p, \Bar\nu_e+p\to
e^++n$) in the energy equation (\ref{eq.mhd.eng}) as
\begin{equation}
Q_E^{\textrm{abs}}=
\frac{3\alpha^2+1}{4}
\frac{\sigma_0\langle\epsilon_{\nu_e}^2\rangle}{(m_ec^2)^2}
\frac{\rho}{m_u}
\frac{L_{\nu_e}}{4\pi r^2\langle\mu_\nu\rangle}
\left(Y_n+Y_p\right)\label{eq.qe.abs}
\end{equation}
\citep{jan01}, where $\alpha=1.26$ is the charged-current axial-vector coupling
constant, $\sigma_0=1.76\times 10^{-44}$~cm$^2$ the characteristic
cross section of weak interaction,
$\langle\epsilon_{\nu_e}^2\rangle=20.8 \left(kT_{\nu_e}\right)^2$
the mean square neutrino energy, $m_ec^2=0.511$~MeV the rest-mass
energy of electron, $L_{\nu_e}$ the neutrino luminosity, $r$ the
distance from the center, and $\langle\mu_\nu\rangle$ the so-called
flux factor. Here, we assume the same luminosity and spectral
temperature for $\nu_e$ and $\Bar \nu_e$. In the present simulations,
$L_{\nu_e}=1.0\times 10^{52}$~erg~s$^{-1}$,
$kT_{\nu_e}=4.0$~MeV, and $\langle\mu_\nu\rangle=1.0$
are chosen. Similarly, the source term related to $\nu_e$/$\Bar \nu_e$
absorption in the equation of $n_e$ (\ref{eq.ne}) is
\begin{equation}
Q_N^{\textrm{abs}}=
\frac{3\alpha^2+1}{4}
\frac{\sigma_0\langle\epsilon_{\nu_e}\rangle}{(m_ec^2)^2}
\frac{\rho}{m_u}
\frac{L_{\nu_e}}{4\pi r^2\langle\mu_\nu\rangle}
\left(Y_n-Y_p\right),\label{eq.qn.abs}
\end{equation}
where $\langle\epsilon_{\nu_e}\rangle=4.11\left(kT_{\nu_e}\right)$
is the mean neutrino energy.
The source term related to the
$\nu_e$/$\Bar \nu_e$ emission ($e^-+p\to \nu_e+n$,
$e^++n\to\Bar\nu_e+p$) in Equation~(\ref{eq.mhd.eng}) is given by
\begin{eqnarray}
Q_E^{\textrm{em}}&=&
-\left(3\alpha^2+1\right)
\frac{\pi\sigma_0 c \left(kT\right)^6}{(hc)^3(m_ec^2)^2}
\frac{\rho}{m_u}\nonumber\\
&&\times
\left[Y_n\mathcal{F}_5(-\eta_e)+Y_p\mathcal{F}_5(\eta_e)\right]\label{eq.qe.em}
\end{eqnarray}
\citep{jan01}, where $\eta_e$ is the electron chemical potential normalized
by the temperature, and
\begin{equation}
\mathcal{F}_l(\eta)\equiv\int^\infty_0 \frac{x^l}{1+\exp(x-\eta)}dx.
\label{eq.fermi}
\end{equation}
Similarly, the emission source term in Equation~(\ref{eq.ne}) is
\begin{eqnarray}
Q_N^{\textrm{em}}&=&
\left(3\alpha^2+1\right)
\frac{\pi\sigma_0 c \left(kT\right)^5}{(hc)^3(m_ec^2)^2}
\frac{\rho}{m_u}\nonumber\\
&&\times
\left[Y_n\mathcal{F}_4(-\eta_e)-Y_p\mathcal{F}_4(\eta_e)\right].\label{eq.qn.em}
\end{eqnarray}
The Fermi integrals (Equation~(\ref{eq.fermi})) are calculated as
follows:
\begin{eqnarray}
\mathcal{F}_4(\eta_e)&=&24\mathcal{I}_4(\eta_e),\\
\mathcal{F}_5(\eta_e)&=&120\mathcal{I}_5(\eta_e),\\
\mathcal{F}_4(-\eta_e)&=&
\frac{1}{5}\eta_e^5 + \frac{2}{3}\pi^2\eta_e^3 +
\frac{7}{15}\pi^4\eta_e + 24\mathcal{I}_4(\eta_e),\\
\mathcal{F}_5(-\eta_e)&=&
\frac{1}{6}\eta_e^6 + \frac{5}{6}\pi^2\eta_e^4 +
\frac{7}{6}\pi^4\eta_e^2
\nonumber\\
&&+ \frac{31}{126}\pi^6 + 120\mathcal{I}_5(\eta_e),
\end{eqnarray}
where
\begin{equation}
\mathcal{I}_l(\eta)\equiv
\sum_{m=1}^\infty \left[\frac{(-1)^{m-1}}{m^{l+1}}e^{-m\eta}\right].
\end{equation}
The source terms given by
Equations~(\ref{eq.qe.abs}), (\ref{eq.qn.abs}), (\ref{eq.qe.em}),
(\ref{eq.qn.em}) are valid only in optically thin regions, and must
decrease toward the optically thick regions. To mimic such reduction
they are multiplied by $e^{-\tau_{\textrm{eff}}}$, following \citet{mur09}.
Here, the effective optical depth is defined as
\begin{eqnarray}
\tau_{\textrm{eff}}=\int^\infty_r \kappa_{\textrm{eff}}(r) dr,
\end{eqnarray}
where the effective opacity is given as
\begin{eqnarray}
\kappa_{\textrm{eff}}= 1.2\times10^{-7}
\left(\frac{\rho}{10^{10}\textrm{g cm}^{-3}}\right)
\left(\frac{kT_{\nu_e}}{4\textrm{MeV}}\right)
\left(Y_n+Y_p\right),\nonumber\\
\end{eqnarray}
from the Equations (10), (11), and (14) of \citet{jan01}.
Before conducting high-resolution simulations to capture the
MRI, we first
follow the collapse of the 15~$M_\odot$ progenitors until several
100~ms after bounce by low-resolution simulations, whose
numerical domain spans from the radius of 100~m to 4000~km. We refer to
these simulations as background (BG) runs.
In the BG runs, the core is covered with
$N_r\times N_\theta = 720\times 60$ numerical grids, where the spatial
resolution is 0.4--23~km.
The pre-collapse cores are assumed to be
rapidly rotating with the initial angular velocity profile of
\begin{eqnarray}
\Omega(r)=\Omega_{0}\frac{r_0^2}{r_0^2+r^2},
\end{eqnarray}
where $r$ is the distance from the center of the core. The parameters
are chosen as $r_0=1000$~km and $\Omega_{0}=2.73$~rad s$^{-1}$,
corresponding to a millisecond proto-neutron star after collapse.
The initial rotational energy divided by the gravitational binding
energy is $2.5\times 10^{-3}$.
We assume that the pre-collapse magnetic fields have dipole-like
configurations produced by electric currents of a 2D-Gaussian-like
distribution centered at $(\varpi,z)=(\varpi_0,0)$,
\begin{eqnarray}\label{eq.model.dipole}
j_\phi(\varpi,z)&=&j_0e^{-\tilde{r}^2/2\sigma(\tilde \theta)^2}
\left(\frac{\varpi_0\varpi}{\varpi_0^2+\varpi^2}\right),
\end{eqnarray}
where $(\varpi,z)$ are cylindrical coordinates, $\tilde r \equiv
\sqrt{(\varpi-\varpi_0)^2+z^2}$, $\tilde \theta \equiv
\arccos (z/\tilde r)$, and
\begin{equation}
\sigma(\tilde \theta)=
\frac{\tilde r_{\textrm{dec}}}{\sqrt{1-e^2\cos{\tilde \theta}}}
\end{equation}
\citep{saw13a}. Changing $j_0$, we perform
three BG runs with different strengths of magnetic fields, where the
maximum strengths at pre-collapse, $B_{\textrm{pre}}$, are 5.0$\times
10^{10}$, 1.0$\times
10^{11}$, and $2\times 10^{11}$~G. Hereafter we refer to these
BG runs as B5e10bg, B1e11bg, and B2e11bg, respectively.
The rest of parameters are set as $\varpi_0=\tilde
r_{\textrm{dec}}=1000$~km and $e=0.5$ in all the computations.
The initial magnetic energy divided by the gravitational binding
energy is quite small, $2.1\times 10^{-6}$, even for the strongest-field
model, B2e11bg. We also computed models without magnetic
field and rotation as well as a model having rotation alone for comparison.
In order to capture the growth of MRI we conduct high-resolution
simulations with the numerical domain spanning $50<(r/$km$)<500$
(referred to as MRI runs). The initial conditions of
MRI runs are given by mapping the data of the BG runs onto the
above domain at 5~ms after bounce. In order to satisfy the divergence
free constraint on the magnetic field, not the magnetic field itself but
the vector potential is mapped as in \citet{saw13a}.
The inner and outer radial boundary
conditions for the MRI runs are given by the data of the basic runs,
except that the inner boundary conditions of $B_r$ are determined to
satisfy the divergence-free condition. The
grid spacing is such that the radial and angular grid sizes are the same,
viz. $\Delta r=r\Delta\theta$, at the innermost and outermost
cells. For each BG run, four MRI runs with different grid resolutions
are carried out.
Our choice of the resolution
at $r=50$~km, $\Delta_{50}$, (and the numbers of grids,
$N_r\times N_\theta$), is 12.5~m ($9250\times 6400$), 25~m
($4650\times 3200$), 50~m ($2300\times 1600$), and 100~m ($1160\times
800$). We label the MRI runs by the initial
field strength of the corresponding BG run followed by the
spatial resolution. For
example, the MRI run using the data of model B5e10bg and
$\Delta_{50}=12.5$~m is referred to as model B5e10$\Delta$12.5.
For a set of models involving the same initial magnetic
field, we use a term ``model series'', e.g., models series B5e10.
In dealing with the MHD equations in the polar coordinates,
we should be cautious about numerical treatments of
the coordinate singularities at the center of the core ($r=0$) and
the pole ($\theta=0$). In the vicinity of the pole, the regularity
conditions demand that the expansions of $v_{\theta}$, $v_{\phi}$,
$B_{\theta}$, and $B_{\phi}$ with respect to $\theta$ should not contain
$\theta$-independent terms, which is not necessarily
satisfied in numerical simulations. In order to numerically meet the
regularity conditions in the vicinity of the pole albeit
approximately, we remove the region of $\theta<0.3^\circ$ from the
numerical domain and impose boundary conditions based on
the regularity conditions except for $B_{\theta}$, which is determined by
the divergence-free constraint. To diffuse undesirable
fluctuations that tend to violate the regularity, we further introduce
an artificial resistivity only at the cells closest to the pole in the
form of
\begin{equation}
\eta_{\textrm{a}}= \frac{\alpha a_{\textrm{max}} \Delta^2}{l_B},
\end{equation}
where $\alpha$ is a dimensionless factor, $a_{\textrm{max}}$ the local
maximum characteristic speed, $\Delta$ the grid width, and $l_B$ the
scale height of the magnetic field.
The factor $\alpha$ is automatically controlled between 0.1--$10^3$
during the simulations depending on how well the regularity condition is
satisfied.
In order to maintain the regularity conditions approximately around the
center in the BG runs, we remove the central part within the radius of
100~m from the numerical domain and take a similar remedy.
\section{Results}\label{sec.result}
\subsection{The Growth of MRI}\label{sec.mri}
The stability condition of the axisymmetric MRI for general rotation
laws $\Omega(\varpi,z)$ is given by
\begin{eqnarray}
\mathcal{C}&\equiv&
\left(
\mathcal{G}_z\mathcal{B}_z\tan^2\theta_k
-2\mathcal{G}_z\mathcal{B}_\varpi\tan\theta_k
+ \mathcal{G}_\varpi\mathcal{B}_\varpi
+ \mathcal{R}_\varpi
\right)
/\Omega^2\nonumber\\
&>&0,
\end{eqnarray}
where $\theta_k$ is the angle between the perturbation wavenumber
$\mbox{\boldmath$k$}$ and the $z$-axis,
\begin{eqnarray}
\mbox{\boldmath$\mathcal{G}$}&\equiv&\frac{\nabla P}{\rho},\\
\mbox{\boldmath$\mathcal{B}$}&\equiv&
-\frac{1}{\Gamma}\frac{\partial\ln P}{\partial s}\Big|_{\rho,Y_e}\nabla s
-\frac{1}{\Gamma}\frac{\partial\ln P}{\partial Y_e}\Big|_{\rho,s}\nabla Y_e,\\
\mbox{\boldmath$\mathcal{R}$}&\equiv&\varpi\nabla\Omega^2,
\end{eqnarray}
and $\Gamma\equiv\partial\ln P/\partial \ln \rho |_{s,Y_e}$ \citep{bal95,obe09}.
The MRI involves two distinct modes, namely, Alfv\'en mode and buoyant
mode, where the former appears for $\mathcal{C}<0$, and the latter
only emerges for $\mathcal{C}+4<0$ \citep{urp96}. Which mode dominates
over the other for a fixed $\theta_k$ depends on the value of
$\mathcal{C}$ \citep{obe09}. For
$-8<\mathcal{C}<0$, the fastest growing mode is the Alfv\'en mode with
the wavenumber of
\begin{equation}
\mbox{\boldmath$k$}_{\textrm{FGM}}\cdot \mbox{\boldmath$v$}_{\textrm{A}}=
\cos\theta_k \Omega\frac{\sqrt{-\mathcal{C}(\mathcal{C}+8)}}{4},
\label{eq.kalf}
\end{equation}
and the growth rate of
\begin{equation}
\omega_{\textrm{FGM}}=\cos\theta_k \Omega\frac{\sqrt{-\mathcal{C}^2}}{4}.
\end{equation}
For $\mathcal{C}\le-8$, the fastest growth occurs with
\begin{equation}
\omega_{\textrm{FGM}}=\cos\theta_k \Omega\sqrt{\mathcal{C}+4},
\end{equation}
for
$\mbox{\boldmath$k$}_{\textrm{FGM}}\cdot\mbox{\boldmath$v$}_{\textrm{A}}=0$,
i.e., it is the buoyant mode.
\begin{figure}
\epsscale{1}
\plotone{f1a.jpg}
\plotone{f1b.jpg}
\caption{Color maps of the dominant modes and growth rate for model
B5e10$\Delta$12.5 at $t_{\textrm{pb}}=7$ (upper panel) and
$t_{\textrm{pb}}=12$ (lower panel). The red and blue colors,
respectively, represent the locations where buoyant mode and Alfv\'en
mode are dominant. The growth rate is multiplied by -1 for
buoyant-mode-dominant regions. The boxes in the upper panel
correspond to the plot areas of Figure~\ref{fig.bpolz}.}
\label{fig.omri}
\end{figure}
Since $\mathcal{C}$ depends on
$\theta_k$, the dominant mode differs for different directions. In
order to find the dominant mode for a fixed spatial point,
we vary $\theta_k$ numerically in the range of
$[-\pi/2:\pi/2]$. The result is shown in Figure~\ref{fig.omri} for
model B5e10$\Delta$12.5 at the postbounce time of $t_{\textrm{pb}}=7$
and 12~ms, where the red
and blue colors represent buoyant-mode- and Alfv\'en-mode-dominant
regions, respectively, and the shades of the colors indicate the
growth rate\footnote{The resolutions of color maps in this paper are
not the same as those of simulations, where the former are
reduced to decrease the size
of figures.}. It is evident that the dominant
mode is different from
location to location, and the regions dominated by the buoyant-mode
have on average larger growth rates than those dominated by the
Alfv\'en mode.
\begin{figure}
\epsscale{1.2}
\plotone{f2.jpg}
\caption{Cumulative volume fractions having $N_{\textrm{MRI}}$ smaller
than a given value for the all models.}
\label{fig.Nmri}
\end{figure}
The growth of the Alfv\'en-mode, although slower than the buoyant mode,
still may have an important effect on
the magnetic field amplification in the locations of its
dominance. It is hence important to know how
well we numerically resolve the fastest-growing Alfv\'en mode (FGAM), whose
wave number is given by Equation~(\ref{eq.kalf}).
Figure~\ref{fig.Nmri} shows for all the models the cumulative fraction
of the volume that has $N_{\textrm{MRI}}$ smaller than a given
value. Here $N_{\textrm{MRI}}$ is defined at each point to be the
ratio of the wavelength of FGAM to the grid size and is measure of how
well FGAM is resolved numerically.
We introduce to characterize the models, a factor
$\xi \equiv\left(B_{\textrm{in,max}}/10^{11}\textrm{G}\right)(\Delta_{50}/100
\textrm{m})^{-1}$, since only the initial
strength of the magnetic field and spacial resolution are different
among the current set of the models.
In fact, similar distributions is obtained for the
models having the same $\xi$ at early epochs when the magnetic field
is almost passive (see Figure~\ref{fig.Nmri}). According to
\citet{shi06}, $N_{\textrm{MRI}}\gtrsim 10$ is required to capture the
linear growth of the Alfv\'en mode. In our weakest-field model series
B5e10, the volume fractions with $N_{\textrm{MRI}}\le 10$ are 0.6,
0.27, 0.12, and 0.018 for models $\Delta$100 ($\xi=0.5$), $\Delta$50
($\xi=1$), $\Delta$25 ($\xi=2$), and $\Delta$12.5 ($\xi=4$),
respectively.
We hence believe that our highest-resolution models should be able to
capture the linear growth of the Alfv\'en mode well.
\begin{figure*}
\epsscale{1}
\plottwo{f3a.jpg}{f3b.jpg}
\plottwo{f3c.jpg}{f3d.jpg}
\plottwo{f3e.jpg}{f3f.jpg}
\caption{Color maps for the strength of the poloidal magnetic field
for models B5e10$\Delta$12.5 (left panels) and B2e11$\Delta$12.5
(right panels). The upper four panels zoom in a part of equatorial
region (presented by the large box in the upper panel of
Figure~\ref{fig.omri}), while the lower two panels that of a
middle-latitude region (the small box in the upper panel of
Figure~\ref{fig.omri}). Panels (c) and (d) are for models with
initial perturbations.}
\label{fig.bpolz}
\end{figure*}
In order to confirm that both the buoyant and Alfv\'en modes are
growing indeed in the regions predicted in Figure~\ref{fig.omri}, we
examine the wavelengths of the growing modes in these regions.
The upper panels of Figure~\ref{fig.bpolz}
show the color maps of the poloidal magnetic field
strength at $t_{\textrm{pb}}=9.5$~ms in a region around the
equator, indicated by the large box in the upper panel of
Figure~\ref{fig.omri}, where two red belts of buoyant-mode dominance are
observed\footnote{Although the upper panel of
Figure~\ref{fig.omri} is depicted for model B5e10$\Delta$12.5, its
feature is very similar among all the models at this point
of time.}.
We found that the patterns of strong-magnetic-field filaments
seen in the upper panels of Figure~\ref{fig.bpolz} have grown in
the regions of buoyant-mode dominance: in the case of model
B5e10$\Delta$12.5, this region corresponds to the right-side red
belt observed in the large box indicated in the upper panel of
Figure~\ref{fig.bpol}, which has been advected leftward
during $t_{\textrm{pb}}=7.0$--$9.5$~ms.
We compare models B5e10$\Delta$12.5 and B2e11$\Delta$12.5, which have
different initial field strengths.
As expected for the buoyant mode, the wavelengths of the growing
modes, which are evaluated from the sizes of the patterns, are nearly
identical between the two models.
The wavelengths of
the growing modes observed here may reflect the scale of the dominant
perturbation, which may come from numerical noises.
To see if this is true, we performed test simulations for the two
models in which
a perturbation of $u'=u'_0\sin (2\pi z/\lambda_{\textrm{prt}})$ is
given at the beginning of the MRI runs, where the amplitude and
the wavelength of the perturbation is set as 1\% and 500~m,
respectively. As a result, we found that the wavelengths of
growing modes are
shorter than those of the models without perturbation
(see panels (c) and (d) of
Figure~\ref{fig.bpolz}), which indicates that the observed modes
depend on the dominant scale of perturbations.
The lower panels of Figure~\ref{fig.bpolz}
zoom in the area around $\theta=35^\circ$ in the vicinity of the
inner boundary, where a pocket of buoyant-mode-dominant regions are
surrounded
by an Alfv\'en-mode-dominant region (see the small box in the
upper panel of
Figure~\ref{fig.omri}). The wavelength of the growing mode there is
shorter for the weaker initial field. In fact, the
widths of protruding magnetic flux loops in the panel (e) of
Figure~\ref{fig.bpolz} for model B5e10$\Delta$12.5
are about three times smaller than those in the panel (f) for model
B2e11$\Delta$12.5. The ratio is close to four, the value expected for
the Alfv\'en mode. With these facts, we believe that our simulations
capture both the buoyant and Alfv\'en modes correctly.
\begin{figure}[h]
\epsscale{1}
\plotone{f4a.jpg}
\plotone{f4b.jpg}
\plotone{f4c.jpg}
\caption{Evolutions of the magnetic energies of the poloidal
and toroidal components, integrated over the whole numerical domain,
for model series B5e10 (upper panel), B1e11
(middle panel), and B2e11 (lower panel). Those for the MRI
runs are plotted until the shock surface reaches the outer
boundary in each model. Only those for model B2e11$\Delta$12.5 are
plotted even after the shock passes the outer boundary in order to
see a clear saturation of the magnetic energy of the poloidal
component. The red crosses represent the moment when the shock
surface reaches the outer boundary in model B2e11$\Delta$12.5.}
\label{fig.t-emag}
\end{figure}
\begin{figure}
\epsscale{1}
\plotone{f5a.jpg}
\plotone{f5b.jpg}
\caption{Color maps for the strength of the poloidal magnetic fields
for model B5e10$\Delta$12.5 at
$t_{\textrm{pb}}=12$~ms (upper panel) and 200~ms (lower panel). The
black line in the upper panel represents the shock surface.}
\label{fig.bpol}
\end{figure}
Figure~\ref{fig.t-emag} plots the time evolutions of the magnetic
energies, integrated over the whole numerical domain for all
models. Those for the MRI runs are plotted until the shock surface
reaches the outer boundary. Only in model B2e11$\Delta$12.5, the plots
are continued for another 12~ms after the shock surface passes the
outer boundary. Since the magnetic energies flowing out of the
boundary during this 12~ms are found to be negligible, i.e., 0.81\%
and 0.24\% of the total poloidal and toroidal magnetic energies at the
end of the plot, respectively, we do not take them into account in the
following discussion.
The exponential growth of the energy of the
poloidal component, $E_{B_{\textrm{p}}}$, is apparent during the first
$\sim 10$~ms for all the MRI runs. In
each model series, the growth timescale becomes shorter (or the growth rate
is larger) for higher resolutions until it converges to
$\approx$3--3.5~ms. These timescales well match the theoretical
prediction for the buoyant-mode of $\sim$2000~rad~s$^{-1}$, which is
shown in the upper panel of
Figure~\ref{fig.omri}. This implies that the exponential growth is
dominated by the buoyant mode. Indeed, the comparison between the
lower panel of Figure~\ref{fig.omri} and the upper panel of
Figure~\ref{fig.bpol} indicates the coincidence of the locations,
where the poloidal magnetic field is preferentially amplified, with
those of buoyant-mode dominance at
$t_{\textrm{pb}}=12$~ms, around the end of the exponential growth.
From the numerical convergence we observed, it is suggested that the
high spatial resolution is required even for the buoyant mode,
in which all wavelengths grow at an equal rate.
After the exponential growth phase ceases, $E_{B_{\textrm{p}}}$
continues to increase gradually until it reaches
saturation roughly around $t_{\textrm{pb}}=$210, 270, and 160~ms for
model series B5e10, B1e11, and
B2e11, respectively (see Figure~\ref{fig.t-emag}). During this phase
the region of strong magnetic field, say, $B>10^{14}$~G,
spreads over a considerable volume inside the radius of $\sim
100$~km (see the lower panel of Figure~\ref{fig.bpol} for B5e10$\Delta$12.5).
\begin{figure}
\epsscale{1}
\plotone{f6.jpg}
\caption{The time-averaged saturation values of the magnetic energy of
the poloidal component (crosses) and the fitted curves (solid
lines) with respect
to the resolution for model series B1e11 and B2e11. The dotted lines
represent the saturation values, $a_1$.}
\label{fig.bsat}
\end{figure}
As can be seen from each panel of Figure~\ref{fig.t-emag}, the
saturated values of $E_{B_{\textrm{p}}}$ do
not converge, which may be because the turbulence is not yet fully
captured due to numerical diffusivity
\citep{saw13b}. Nevertheless, since model
series B1e11 and B2e11 show a trend of convergence, we may be
able to estimate the converged values by fitting the time-averaged
$E_{B_{\textrm{p}}}$ for the different resolution models with suitable
functions. Taking the time
averages over $t_{\textrm{pb}}=270$--330~ms
and 165--185~ms for model series B1e11 and B2e11,
respectively, we fitted the results with functions in the form of
\begin{equation}
\langle E_{B_{\textrm{p}},{\textrm{sat}}}\rangle
=a_1-a_2\exp\left({-a_3/\Delta_{50}}\right),
\end{equation}
where $a_1$, $a_2$, and $a_3$ are the parameters to be determined.
The values obtained for $a_1$ are
$1.4\times 10^{49}$ and $3.1\times 10^{49}$~erg for model series B1e11
and B2e11, respectively (see Figure~\ref{fig.bsat}). It is also found
that the saturated values of $E_{B_{\textrm{p}}}$ for
the highest-resolution models B1e11$\Delta$12.5 and B2e11$\Delta$12.5
are 91\% and 96\% of these values, respectively, i.e., they
are close to
convergence. Indeed, it is expected that if we were
able to afford twice higher resolution, we could achieve convergence.
Our results also suggest that larger initial
magnetic fields may result in larger saturated values. This is
consistent with the results obtained by \citet{haw95}, who performed
local box simulations of MRI in the context of accretion
disks. \citet{mas15} also claimed that the saturation depends on the
initial fields, however, no resolution study was done. As shown here, the
resolution dependence should properly taken into account in discussing
the saturation.
The magnetic energy of the toroidal component, $E_{B_{\textrm{t}}}$,
also shows the exponential growth in each model (see
Figure~\ref{fig.t-emag}). Since the non-axisymmetric MRI for the
toroidal components cannot be treated with the current
simulations, this is not due to the MRI but due to the winding
of the MRI-amplified poloidal component by differential rotation.
This is understood from the fact that $E_{B_{\textrm{t}}}$
continuously increases even after the poloidal component is saturated
and become one order of magnitude greater
than $E_{B_{\textrm{p}}}$. At the end of the simulations,
$E_{B_{\textrm{t}}}$ still continues to increase gradually in most of
the models. Only for models B1e11$\Delta$25 and B1e11$\Delta$12.5,
$E_{B_{\textrm{t}}}$ has nearly reached the saturated values.
Incidentally, the numerical convergence is achieved in
$E_{B_{\textrm{t}}}$ for model series B1e11 and B2e11.
\subsection{Impacts on Global Dynamics}\label{sec.dyn}
\subsubsection{Background Runs}
With our choice of $L_\nu=1.0$~erg~s$^{-1}$, the BG run with
no magnetic field and rotation fails to explode,
the shock wave being stalled at $r\lesssim 150$~km (see black line in
Figure~\ref{fig.t-rsh.bg}).
Although the shock surface is deformed by SASI-like
oscillations during the early postbounce phase, which is imprinted in
the zigzag evolution of the shock radius in
Figure~\ref{fig.t-rsh.bg} at $t_{\textrm{pb}}\lesssim 300$~ms,
it becomes almost spherically symmetric later on.
Initial rotation of $|T/W|=0.25$~\%
substantially changes the behavior of the shock
evolution. In fact, fluids at middle to low latitudes tend to
expand toward a larger
radius thanks to centrifugal forces. The
maximum shock radius gradually
increases and exceeds 200~km by $t_{\rm{pb}}=700$~ms (see the cyan line in
Figure~\ref{fig.t-rsh.bg}), at which time some parts of fluid elements
are still going outward albeit slowly. These features are
consistent with the former findings \citep{suw10,nak14,iwa14},
that the rotation helps the explosion (See, however, \citet{mar09}).
\begin{figure}
\epsscale{1}
\plotone{f7.jpg}
\caption{Time evolutions of the maximum shock radii for BG runs.}
\label{fig.t-rsh.bg}
\end{figure}
In the BG runs with both magnetic field and rotation, the shock
surface propagates
outward more easily compared with the rotation-only model, with
faster propagation speeds for stronger initial magnetic fields
(Figure~\ref{fig.t-rsh.bg}). The panels (a), (b), and (c) of
Figure~\ref{fig.betamulti} show that the regions of low plasma beta
($\beta$, the ratio of matter pressure to magnetic pressure)
appear around the mid-latitude, indicating that
the magnetic pressure plays an important role to push the shock outward.
\subsubsection{Dynamical Behavior of MRI Runs}
\begin{figure*}
\begin{minipage}{0.33\hsize}
\includegraphics[scale=0.55]{f8a.jpg}
\end{minipage}
\begin{minipage}{0.33\hsize}
\includegraphics[scale=0.55]{f8b.jpg}
\end{minipage}
\begin{minipage}{0.33\hsize}
\includegraphics[scale=0.55]{f8c.jpg}
\end{minipage}
\begin{minipage}{0.33\hsize}
\includegraphics[scale=0.55]{f8d.jpg}
\end{minipage}
\begin{minipage}{0.33\hsize}
\includegraphics[scale=0.55]{f8e.jpg}
\end{minipage}
\begin{minipage}{0.33\hsize}
\includegraphics[scale=0.55]{f8f.jpg}
\end{minipage}
\begin{minipage}{0.33\hsize}
\includegraphics[scale=0.55]{f8g.jpg}
\end{minipage}
\begin{minipage}{0.33\hsize}
\includegraphics[scale=0.55]{f8h.jpg}
\end{minipage}
\begin{minipage}{0.33\hsize}
\includegraphics[scale=0.55]{f8i.jpg}
\end{minipage}
\begin{minipage}{0.33\hsize}
\includegraphics[scale=0.55]{f8j.jpg}
\end{minipage}
\begin{minipage}{0.33\hsize}
\includegraphics[scale=0.55]{f8k.jpg}
\end{minipage}
\begin{minipage}{0.33\hsize}
\includegraphics[scale=0.55]{f8l.jpg}
\end{minipage}
\begin{minipage}{0.33\hsize}
\includegraphics[scale=0.55]{f8m.jpg}
\end{minipage}
\begin{minipage}{0.33\hsize}
\includegraphics[scale=0.55]{f8n.jpg}
\end{minipage}
\begin{minipage}{0.33\hsize}
\includegraphics[scale=0.55]{f8o.jpg}
\end{minipage}
\caption{Color maps of the plasma beta with velocity vectors
presented by arrows at $t_{\textrm{pb}}=$578, 402,
and 172~ms for model series B5e10, B1e11, B2e11, respectively.}
\label{fig.betamulti}
\end{figure*}
\begin{figure*}[t]
\plottwo{f9a.jpg}{f9b.jpg}
\caption{Color maps of the plasma beta for model
B2e11$\Delta$25 at $t_{\textrm{pb}}=$195~ms (a) and model
B1e11$\Delta$12.5 at $t_{\textrm{pb}}=$440~ms (b), which are
supplemental plots for panels (l) and (n) of
Figure~\ref{fig.betamulti}, respectively.}
\label{fig.beta2}
\end{figure*}
The dynamics change even more drastically when the spatial
resolution is increased (MRI runs).
Figure~\ref{fig.betamulti} displays the distributions of the plasma
beta for all the 15 models at $t_{\textrm{pb}}= 578$, 402,
and 172~ms for model
series B5e10 (left column), B1e11 (middle column), and B2e11
(right column), respectively. In model series B2e11, all the MRI runs
result in the formation of a collimated low-$\beta$ jet emerging
from inside the roughly-spherical shock, whereas the BG run yields an
almost spherical expansion of the shock wave. Note that the
low-$\beta$ region seen in panel (l) for model B2e11$\Delta$25 at
172~ms evolves into a collimated jet later on as shown in panel
(a) of Figure~\ref{fig.beta2}. Meanwhile, in
a weaker-field model series B1e11, the situation is
not as simple as in model series B2e11. As the resolution gets higher
from model B1e11bg to
B1e11$\Delta$100, the shape of shock surface changes from spherical to
prolate, but it returns to spherical shape when the resolution is
doubled again
(see panels (b), (e), (h) of Figure~\ref{fig.betamulti}). Another
doubling of the resolution, in
turn, brings about the formation of a collimated low-$\beta$
jet emerging from inside the spherical shock (model
B1e11$\Delta$25, panel (k)). In the highest resolution
model B1e11$\Delta$12.5, a low-$\beta$ region is observed
around the radius of 200~km in the vicinity of the pole (panel (n) of
Figure~\ref{fig.betamulti}), which may hint at a later jet
formation. Although the head of low-$\beta$ region is still lingering
around the radius of 300~km at the end of the simulation
($t_{\textrm{pb}}=$440~ms, see panel (b) of Figure~\ref{fig.beta2}),
we expect that it would propagate further and eventually forms a
collimated jet as found in model B1e11$\Delta$25 (see below).
Finally in the weakest-field model series B5e10, the
shock surfaces are roughly spherical for all the resolutions except
model B5e10$\Delta$100, which has a prolate shock, and no model shows
a jet formation until the end of the simulation.
\begin{figure*}[t]
\epsscale{1}
\plottwo{f10a.jpg}{f10b.jpg}
\plottwo{f10c.jpg}{f10d.jpg}
\caption{Color maps of the ram pressure (upper panels) and the plasma
beta (lower panels) for model B1e11$\Delta$100 (left panels) and
B1e11$\Delta$25 (right panels). The ram pressure is multiplied by
$-1$ where the radial velocity is negative.}
\label{fig.rambeta}
\end{figure*}
We first discuss the factors responsible for the different shock
morphorogies for different resolutions by comparing
models B1e11$\Delta$100 and B1e11$\Delta$25 at
$t_{\textrm{pb}}=213$~ms, several milliseconds
prior to the launch of the collimated-jet in model B1e11$\Delta$25.
The upper panels of Figure~\ref{fig.rambeta} depict the distribution
of the ram pressure for the two models. It is observed in both the
models that a vicinity of the pole is dominated by intense downflow
(blue region), outside of which modest outflow driven by
relatively-low plasma beta is seen (red region; see also lower panels).
The width of the downflow channel is found to be narrower for the lower
resolution model B1e11$\Delta$100, which would be due to less
effective MRI: the better the MRI resolved, the more efficiently
angular momentum is transferred outwards from the rotation axis, with
which the rotational support decreases further around the pole and
a broader downflow channel forms. Accordingly, the lower-$\theta$
edge of the outflow gets closer to the pole in this
model, viz. the matter is ejected more preferentially along the pole.
It is likely that this causes the prolate shock surface found in a model
B1e11$\Delta$100. Note that although relatively-low plasma beta,
$\beta\sim 1$, is seen around the bottom of the downflow channel in
both the models, it seems not enough to drive the matter outward
against the downflow (see the lower panels of
Figure~\ref{fig.rambeta}). The magnetically-driven mass ejection is
only possible for the region outside the channel with such
relatively-low plasma beta. It is found from Figure~\ref{fig.betamulti}
that the trend of broader downflow channel and thus a larger
deflection of the outflow direction from the
pole for higher resolution models is valid for a wide range of
resolution in model-series B5e10 and B1e11 as long as a jet is
absent. This suggest that our interpretation for the shock morphology
is reasonable. Note that the trend discussed here becomes no longer
valid once a jet appears, since it changes the flow structure.
\begin{figure*}[t]
\epsscale{1}
\plottwo{f11a.jpg}{f11b.jpg}
\plottwo{f11c.jpg}{f11d.jpg}
\caption{Evolution of low-$\beta$ head (upper panels), and the
downflow energy, $E_{\textrm{df}}$ and magnetic energy
responsible for jet driving, $E_{B_\textrm{jet}}$ (lower
panels). The left and right columns are for models series B1e11
and B2e11, respectively. Note that the time average over the
interval of 10~ms is taken for each plot.}
\label{fig.t-jet}
\end{figure*}
According to the above discussion, the collimated-jets seen in some
models must have been launched against the downflow.
In the case of model B1e11$\Delta$25, the low-$\beta$ clump around
$r=70$~km in the vicinity of the pole observed in panel (d) of
Figure~\ref{fig.rambeta} is a prototype of the low-$\beta$ jet seen
at a later phase (panel (k) of Figure~\ref{fig.betamulti}). We found
indeed that this low-$\beta$ clump suffers from successive depression by the
downflow until it finally forms the collimated jet.
To see the process of jet formation more in detail, we define the
``low-$\beta$ head'' by the maximum radius of the region where the
ratio of the matter
pressure to magnetic pressure, each of which is angularly averaged
over $\theta\le5^\circ$\footnote{The average of variable A is
taken as $\int A dV/\int dV$.}, is less than 0.5, and plot the
evolutions of
them for model series B1e11 and B2e11 (see the upper panels of
Figure~\ref{fig.t-jet}). For model B1e11$\Delta$25 the sequence of
the depression described above is clearly seen as an oscillatory
evolution of the low-$\beta$ head until $t_{\textrm{pb}}\approx
370$~ms, from which it grows monotonically. Model
B1e11$\Delta$12.5 shows a slower growth of
the low-$\beta$ head and more distinct feature of oscillation,
indicating that the depression by downflow is more significant. A
similar oscillation is also found for model B1e11$\Delta$100, but
the low-$\beta$ head almost stagnates at a small radius in this
case. On the contrary to the weaker field case, model series B2e11
shows no oscillation of the low-$\beta$ head, which grows almost
monotonically in all of the three models plotted in panel (b) of
Figure~\ref{fig.t-jet}\footnote{As shown, the low-$\beta$ heads
evolve more or less similarly among all the MRI runs
of model series B2e11. Although, in the right column of
Figure~\ref{fig.betamulti}, all the MRI runs except for
B2e11$\Delta$25 show a trend that the jet-head
radius is larger for a higher resolution at 172~ms, with which one may
think that model B2e11$\Delta$25 is an outlier,
Figure~\ref{fig.t-jet} represents that there is
in fact no such a trend on the whole time.}.
From the above discussion, the downflow seems the
key to the propagation of low-$\beta$ head and the eventual formation of a
collimated jet. Then the condition for the jet formation may be
obtained by comparing the kinetic energy of the downflow and the magnetic
energy responsible for the jet driving. As argued below, we indeed
found that this comparison reasonably explains the jet formation.
In the lower panels of
Figure~\ref{fig.t-jet} we plot the time evolutions of the above two
energies for model series B1e11 and B2e11, defining the former as
$E_{\textrm{df}}\equiv\int_{v_r<0}(\rho v_r^2/2)dV$ and
the latter as $E_{B_\textrm{jet}}\equiv\int_{\beta<0.5}(B^2/8\pi) dV$,
where the integrants are
nonzero only for $v_r<0$ and $\beta<0.5$, respectively, and the
integration ranges are confined to 50~km$\le r \le r_{\textrm{sh}}$ and
$\theta<20^\circ$. In model series B2e11, the evolutions of
$E_{B_\textrm{jet}}$ appears rather similar among the different
resolutions. In each model, $E_{B_\textrm{jet}}$ exceeds
$E_{\textrm{df}}$ around $t_{\textrm{pb}}\approx 100$~ms, which is found
to approximately coincide with the start of the low-$\beta$ head
propagation (see right column of Figure~\ref{fig.t-jet}). This
suggests that the $E_{\textrm{df}}$-$E_{B_\textrm{jet}}$ comparison is
indeed a rough indicator for the jet formation.
Unlike in model series B2e11, the values of
$E_{B_\textrm{jet}}$ in model series B1e11 rather diverge among
different resolutions during a late phase after
their growth nearly saturates around $t_{\textrm{pb}}\approx
250$~ms. Interestingly, while reducing the grid size, $\Delta_{50}$,
from 50~m to 25~m result in averagely-larger $E_{B_\textrm{jet}}$
during the late phase, which may be simply due to smaller numerical
diffusivity, another doubling of resolution decreases that value
(see panel (c) of Figure~\ref{fig.t-jet}). We infer that the
decrease of $E_{B_\textrm{jet}}$ observed here is caused by the
interaction of low-$\beta$ matter with downflow. Since the downflow
matter involves high-$\beta$ (see Figure~\ref{fig.rambeta}), the
plasma beta of low-$\beta$ matter increases as it hit by and mixed
with the downflow matter, which results in downturn of
$E_{B_\textrm{jet}}$. Indeed, the panel (c) of
Figure~\ref{fig.t-jet} shows that the downflow energy,
$E_{\textrm{df}}$, is averagely larger in model B1e11$\Delta$25 than
in B1e11$\Delta$12.5, which is due to more effective MRI as
discussed before, and the effect of downflow is expected to be more
standout for the latter model. Although the effect of the downflow
basically becomes more potent by increasing the resolution, whether
it is essential for the change of $E_{B_\textrm{jet}}$ would depend
on the competition with other factors. For the increase of
$E_{B_\textrm{jet}}$ from model B1e11$\Delta$50 to B1e11$\Delta$25,
it is likely that the reduction of the numerical diffusivity is more
important than the increment of the downflow effect. Meanwhile, the
fact that the $E_{B_\textrm{jet}}$ is roughly unchanged by
increasing the resolution in model series B2e11 indicates that the
downflow effect is insignificant in these models. One reason
for this would be that
the plasma beta of low-$\beta$ matter is low enough to maintain
$\beta < 0.5$, the criterion for adding up $E_{B_\textrm{jet}}$, even
after the mixing with downflow matter. We compare in
Figure~\ref{fig.beta-emag} the
$\beta$-distribution of magnetic energy contained within $\theta<20^\circ$
for models B1e11$\Delta$12.5 and B2e11$\Delta$12.5 at the moment
when $E_{\textrm{df}}$ first reaches $2.0\times 10^{48}$~erg in
each model, and
found that the latter model indeed involves more low-$\beta$
matter. Another reason that we consider important is that models
B2e11 take shorter
time to reach the saturation of magnetic energy than models B1e11
do (see Figure~\ref{fig.t-emag} and lower panels of
Figure~\ref{fig.t-jet}). Since $E_{\textrm{df}}$ gradually
increases until attenuated by the jet formation, an early
growth of magnetic energy is advantageous to alleviate the downflow
effect. In fact, the value of $E_{\textrm{df}}$ at the
moment when it is caught up with by $E_{B_\textrm{jet}}$, is
generally smaller in model
series B2e11 ($E_{\textrm{df}}=0.6-1.2\times 10^{48}$~erg) than in B1e11
($E_{\textrm{df}}=1.5-2.3\times 10^{48}$~erg).
\begin{figure}
\epsscale{1}
\plotone{f12.jpg}
\caption{$\beta$-distribution of magnetic energy contained within
$\theta<20^\circ$ for models B1e11$\Delta$12.5 and B2e11$\Delta$12.5
at the moment when $E_{\textrm{df}}$ first reaches $2.0\times
10^{48}$~erg in each model, $t_{\textrm{pb}}=198$~ms for the former and
$t_{\textrm{pb}}=140$~ms for the latter. The vertical-dotted line
represents $\beta=0.5$.}
\label{fig.beta-emag}
\end{figure}
Bearing in mind the variations of $E_{B_\textrm{jet}}$ and
$E_{\textrm{df}}$ among the different resolutions mentioned
for model series B1e11 in the above, the non-monotonic dependence of jet
formation on the resolution found in these models (the middle column
of Figure~\ref{fig.betamulti}) is also explained reasonably in terms
of the $E_{\textrm{df}}$-$E_{B_\textrm{jet}}$ comparison. For model
B1e11$\Delta$50, the fact that $E_{B_\textrm{jet}}$ is almost always
smaller than $E_{\textrm{df}}$ is consistent with the stagnation of
the low-$\beta$ head at small radii. Similar to
model series B2e11, model B1e11$\Delta$25 shows the outward
propagation of the low-$\beta$ head after $E_{B_\textrm{jet}}$
becomes comparable to $E_{\textrm{df}}$ around
$t_{\textrm{pb}}\approx 220$~ms. Contrary to the former cases,
however, $E_{B_\textrm{jet}}$ does not exceed $E_{\textrm{df}}$ so
much and sometimes even falls behind that, as expected from the
oscillatory evolution of the low-$\beta$ head. In the higher
resolution model B1e11$\Delta$12.5, the low-$\beta$ head starts to
propagate after $E_{B_\textrm{jet}}$ grows comparable to
$E_{\textrm{df}}$ around $t_{\textrm{pb}}\approx 280$~ms, but shows
remarkable oscillations as $E_{B_\textrm{jet}}$ occasionally becomes
smaller than $E_{\textrm{df}}$ by up to factor $\approx 10$, due to
the downflow effect. About 140~ms later,
however, low-$\beta$ filaments outside the downflow choke the
channel region (see panel (b) of Figure~\ref{fig.beta2}), decreasing
$E_{\textrm{df}}$ drastically and resulting in the acceleration of
the low-$\beta$ head. As mentioned before the position of the
low-$\beta$ head is still at $r<300$~km and no clear jet formation
is observed by the end of the simulation. Nevertheless, since
$E_{B_\textrm{jet}}$ exceeds $E_{\textrm{df}}$ by almost factor 10
at that time, and the latter does not increase significantly
afterward, we expect that a collimated jet will form later also in
model B1e11$\Delta$12.5.
It should be noted that since doubling the resolution from model
B1e11$\Delta$25 to B1e11$\Delta$12.5 renders the downflow effect
more significant,
which is disadvantageous for a jet formation, higher resolution runs
are necessary to understand how the dynamics converge in
terms of resolution for model series B1e11.
Since the jet formations discussed above take place close to
the pole, where the coordinates become singular, one may be worried
that the observed features are merely numerical artifacts. Although
some level of numerical noises originating from the coordinate
singularity may be inevitable in spite of the special treatment
described in Section~\ref{sec.method}, we believe that they are of physical
origin. This is because the jet is born at some distance from the
pole, $\approx 10$~km (panel (d) of Figure~\ref{fig.rambeta}), which
is much larger than the width of
the region of the special treatment, and because the evolution of
the low-$\beta$ region is reasonably understood by the above arguments.
\subsubsection{Boost of Explosion via MRI}
\begin{figure}[t]
\epsscale{1}
\plotone{f13a.jpg}
\plotone{f13b.jpg}
\plotone{f13c.jpg}
\caption{Time evolutions of the shock radii at equator for all the models.}
\label{fig.t-req}
\end{figure}
\begin{figure}[t]
\epsscale{1}
\plotone{f14.jpg}
\caption{Time evolutions of the diagnostic explosion energies for
model series B1e11. After the maximum shock position exceeds the
outer radial boundary, they are plotted by dotted lines.}
\label{fig.t-eexp}
\end{figure}
Although the variation of dynamical behavior with the
resolution seems rather complicated as described above, there is actually
one clear
trend, i.e., the faster shock expansion at the equator for the
higher-resolutions (see
Figure~\ref{fig.betamulti} and \ref{fig.t-req}). Since the equatorial region
contains a larger amount of mass compared to the polar region, the larger
explosion energy is expected for the faster shock expansion. As shown
shortly, this is indeed the case.
Figure~\ref{fig.t-eexp} shows the time evolution of the diagnostic
explosion energy, which is defined as the sum of the kinetic,
magnetic, internal, and gravitational energies over the fluid
elements that move outward with positive energies, for model series
B1e11. This clearly shows
that the diagnostic explosion energy becomes larger as
the resolution is increased.
Figure~\ref{fig.betamulti} indicates that the magnetic
effects are not necessarily lager for higher resolutions
(e.g., compare panel (e) and (h)),
which suggests that the magnetic pressure is not a key factor to boost
the explosion. Note that although the collimated jets are driven by
magnetic pressure, they give a minor contribution to the explosion energy
due to their small volumes.
As pointed out by \cite{saw14}, the increase in the explosion
energy is attributes to the more efficient neutrino heating in
higher resolution models.
\begin{figure}[t]
\epsscale{1}
\plotone{f15a.jpg}
\plotone{f15b.jpg}
\caption{Time evolutions of $\tau_{\textrm{a}}/\tau_{\textrm{h}}$
(upper panel) and the net heating rate per unit mass averaged over
the heating region (lower panel) for model series B1e11.
See text for the definition of $\tau_{\textrm{a}}$ and $\tau_{\textrm{h}}$.}
\label{fig.t-heat}
\end{figure}
\begin{figure*}[t]
\epsscale{1}
\plottwo{f16a.jpg}{f16b.jpg}
\plottwo{f16c.jpg}{f16d.jpg}
\caption{Top panels: color maps for the net heating rate per unit mass
for models B1e11$\Delta$100 (panel (a)) and B1e11$\Delta$12.5 (panel
(b)) at $t_{\textrm{pb}}=180$~ms. Panel (c): Color map for the proton
fraction, $Y_p$, for model B1e11$\Delta$12.5. Panel (d): Color map for the
strength of the poloidal magnetic fields for model B1e11$\Delta$12.5.}
\label{fig.heat}
\end{figure*}
How close to revival the stalled shock is roughly measured by
the ratio of the advection timescale, $\tau_{\textrm{a}}$,
during which matter traverses the gain region, to the heating
timescale, $\tau_{\textrm{h}}$, within which matter gains enough
energy to overcome gravity \citep{tho00}. Following \citet{dol13}, we
define the advection timescale as
\begin{equation}
\tau_{\textrm{a}}=\int^{R_{\textrm{gain}}}_{R_{\textrm{sh}}}
\frac{dr}{\langle\langle v_r\rangle\rangle},
\end{equation}
where the double angle bracket implies that the solid-angle average
over $4\pi$ as well as the time average over the interval of 10~ms are taken.
$R_{\textrm{sh}}$ is the mean shock radius, whereas
$R_{\textrm{gain}}$ is defined as the innermost radius at
which the solid-angle-averaged net heating is positive. The heating
timescale is defined as
\begin{equation}
\tau_{\textrm{h}}=
\frac{4\pi\int^{R_{\textrm{gain}}}_{R_{\textrm{sh}}}
\langle e+\frac{\rho v^2}{2}+\frac{B^2}{8\pi}+\rho\Phi\rangle r^2 dr}
{4\pi\int^{R_{\textrm{gain}}}_{R_{\textrm{sh}}}
\langle Q_E^{\textrm{em}}+ Q_E^{\textrm{abs}}\rangle r^2dr},
\end{equation}
where the single angle brackets mean that the only solid-angle average
is taken.
The upper panel of Figure~\ref{fig.t-heat} plots the evolution of
$\tau_{\textrm{a}}/\tau_{\textrm{h}}$ for model series B1e11. The
comparison of this figure with Figure~\ref{fig.t-eexp}
indicates that shock revival, which is indicated by positive explosion
energies, roughly corresponds to
$\tau_{\textrm{a}}/\tau_{\textrm{h}}\gtrsim 1$. It is also evident that higher
resolutions result in higher heating efficiency.
In \cite{saw14}, we
argued that this is due to the increase of $\tau_{\textrm{a}}$ in the
higher resolution models as a
result of more efficient angular momentum transfer, which leads to the
expansion of the heating region. This is true of the current models.
Comparison between models B1e11$\Delta$100 and
B1e11$\Delta$12.5 at $t_{\textrm{pb}}=180$~ms shows that the heating
region is thicker (see the upper panels of Figure~\ref{fig.heat}) and
the amount of angular momentum contained in the heating region is
larger for the latter model: they are
$7.0\times 10^{47}$g~cm$^{2}$s$^{-1}$ for model B1e11$\Delta$100
and $1.9\times 10^{48}$g~cm$^{2}$s$^{-1}$ for model
B1e11$\Delta$12.5 at $t_{\textrm{pb}}=180$~ms.
Besides the increment of $\tau_{\textrm{a}}$, we found in this paper
that the reduction of $\tau_{\textrm{h}}$ owing to a larger heating
rate per unit mass is also contributing to the larger
$\tau_{\textrm{a}}/\tau_{\textrm{h}}$ in the higher resolution
models. As shown in the lower panel of Figure~\ref{fig.t-heat}, the
heating rate per unit mass during $\sim 100$--$250$~ms, the period
crucial to shock revival, becomes larger as
the resolution increases. The comparison of the upper panels of
Figure~\ref{fig.heat} for models B1e11$\Delta$100 and
B1e11$\Delta$12.5 indicates that this is originated in a patch of
region with large heating rates around the equator observed in the
latter model. We evaluated the heating
and cooling rates separately and found that the relatively inefficient
cooling in the patch compared with the surroundings is responsible
for the larger net heating rate. From
Equation~(\ref{eq.qe.em}), the cooling rate per unit mass,
$Q_E^{\textrm{em}}/\rho$, is proportional to $(kT)^6$ and
$Y_n\mathcal{F}_5(-\eta_e)+Y_p\mathcal{F}_5(\eta_e)$. We found that
there is no
substantial difference in $(kT)^6$ between the patch and surroundings
but that $Y_n\mathcal{F}_5(-\eta_e)+Y_p\mathcal{F}_5(\eta_e)$ is several
times smaller in the patch.
In the surroundings, where $Y_n\sim Y_p\sim 0.2$,
$\mathcal{F}_5(-\eta_e)\sim 10$, and $\mathcal{F}_5(\eta_e)\sim 900$,
the products are $Y_n\mathcal{F}_5(-\eta_e)\sim 2$ and
$Y_p\mathcal{F}_5(\eta_e)\sim 200$, viz., the cooling is dominated by
electron capture since electrons are much more abundant than positrons.
On the other hand, the electron capture is found to be relatively
inactive in the patch due to small number of protons ($Y_p\sim 0.05$, see
panel (c) of Figure~\ref{fig.heat}) and electrons
($\mathcal{F}_5(\eta_e)\sim 500$): the product is
$Y_p\mathcal{F}_5(\eta_e)\sim 30$. We found that the positron
capture rate is small as well in the patch:
$Y_n\mathcal{F}_5(-\eta_e) \sim 20$, where $Y_n\sim 0.8$ and
$\mathcal{F}_5(-\eta_e)\sim 30$. To summarize, the the larger heating
rate in the patch is caused by poverty of protons and electrons, and
a low electron capture rate as a consequence.
The low-$Y_p$ (equivalently low-$Y_e$) region coincides with the location
where the poloidal magnetic field is relatively strong (compare panels
(c) and (d)
of Figure~\ref{fig.heat}). This suggests that low-$Y_p$ fluids
originally located at small radii are drifted along the magnetic flux
loops by MRI.
We also found that the outflow along the rotation axis observed in model
B1e11$\Delta$100 and the collimated jets found in models
B1e11$\Delta$25 and B1e11$\Delta$12.5 (see panels (e), (k), and (n) of
Figure~\ref{fig.betamulti},
respectively) also convey low-$Y_p$ matter from deep inside the
core, which is reflected to the rise of the volume-averaged net heating
rate seen after
$t_{\textrm{pb}}\sim 350$~ms for these models (see green, magenta, and
red lines in the bottom panel of Figure~\ref{fig.t-heat}).
\begin{figure}
\epsscale{1}
\plotone{f17a.jpg}
\plotone{f17b.jpg}
\caption{Color maps for the plasma beta for models B2e11H (upper panel) and
B2e11NH (lower panel) at $t_{\textrm{pb}}=229$~ms.}
\label{fig.beta1000}
\end{figure}
In order to estimate the possible influences of these effects on the
global dynamics, we carried out
two groups of additional test simulations based on models B1e11$\Delta$50 and
B2e11$\Delta$50. The first one of them are extends the radial outer
boundaries to $r=1000$~km (models B1e11H and
B2e11H). The other one is different in that the net neutrino heating
is switched off, i.e.,
$Q_E^{\textrm{abs}}+Q_E^{\textrm{em}}=\max\left[Q_E^{\textrm{abs}}+Q_E^{\textrm{em}},0\right]$
(models B1e11NH and B2e11NH).
Figure~\ref{fig.beta1000} shows the profile of
the plasma beta for models B2e11H and B2e11NH at $t_{\textrm{pb}}=229$~ms, when
the shock surface reaches $r=1000$~km in model B2e11H. Comparing
the two panels, we can immediately see the importance of the neutrino
heating: when the shock surface reaches $r=1000$~km in model B2e11H,
that in model B2e11NH has just passed the
radius of 500~km, and it arrives
at $r=1000$~km 65~ms later than model B2e11H.
Even though the collimated jets observed in these
two models are magnetically dominated, the neutrino heating plays a
significant role. The diagnostic explosion energies at
the time when the shock fronts reach the radius of 1000~km are
$4.6\times 10^{49}$ and $1.8\times10^{49}$~erg,
respectively, for models B2e11H and B2e11NH, implying that the
contribution of the neutrino heating to the explosion energy is
about 60\%. The neutrino heating is even more crucial for
weaker initial
fields. The shock surface in model B1e11NH stays within
400~km even at 678~ms after bounce while that of
B1e11H has passed the radius of 1000~km at
$t_{\textrm{pb}}=595$~ms. The diagnostic
explosion energy in model B1e11H is
$4.7\times 10^{49}$~erg at the time when the shock front reach the
radius of 1000~km, while that in model B1e11NH is negligibly small,
$\lesssim 2\times 10^{47}$~erg, through the simulation. Note that
longer time simulations following the propagation of the shock front
through the whole progenitor would be necessary to correctly measure
the explosion energy.
The small contribution of magnetic field to the explosion
energy discussed above is a substantial difference from previous MHD
simulations involving magnetar-class initial fields, in which
a magnetic field alone boosts the explosion energy up to
$\sim10^{51}$~erg, accompanying a jet with a rather-large opening
angle \citep[e.g.,][]{yam04}. This implies that in our models, the
Maxwell stress is weaker, and thus the extraction of the rotational
energy is less efficient than in those simulations.
\subsection{Effects of Inner Boundary}\label{sec.ibc}
We have seen that the MRI efficiently amplifies weak seed
magnetic fields to dynamically important strengths, having a
positive impacts on explosion. In this section, we investigate
whether the above results depends on the location of the inner
boundary, shifting it to smaller radii in model B2e11$\Delta$50.
We ran a new simulation with the inner boundary
at $r=30$~km to take the effect of strong differential
rotation beneath the radius of 50~km into account, which we refer
to as model Rin30. The
spatial resolution of this model is similar to
that of model B2e11$\Delta$50 outside the
radius of 50~km and is 30~m at the inner boundary.
The fraction of the volume where $N_{\textrm{MRI}}$ is less than 10 is only a few
percent inside the radius of 50~km, which is similar to that outside
(see Figure~\ref{fig.Nmri}).
\begin{figure}[t]
\epsscale{1}
\plotone{f18a.jpg}
\plotone{f18b.jpg}
\caption{Upper panel: time evolutions of the magnetic
energies contained in the range of $50<(r/$km$)<500$ for models
Rin30 and B2e11$\Delta$50. Lower panel: time evolutions of the
diagnostic explosion energies for models Rin30 and B2e11$\Delta$50. After
the maximum shock position exceeds the outer radial boundary, they
are plotted by dotted lines.}
\label{fig.ibc}
\end{figure}
The growth rates of MRI inside the radius of 50~km are found to be
lager than those outside on average. The upper panel of
Figure~\ref{fig.ibc} indicates that the exponential growth rate
of $E_{B_p}$ averaged over the range of $50<(r/$km$)<500$ is larger
for model Rin30 than for model B2e11$\Delta$50. This implies that the
magnetic fields amplified inside the radius of 50~km are advected
outward in model Rin30. Note that the degree of differential
rotation is even greater inside the radius of 30~km, and thus the
above effect would be more pronounced if we carried out the
simulation with a yet smaller radius of inner boundary.
On the other hand, we found that the saturation level of
$E_{B_p}$ is unaffected by the position of the inner boundary, which
may be reasonable if the saturation is determined
by the strength of the numerical diffusion, and hence the spatial
resolution, which are more or less the same (see
Section~\ref{sec.mri}). The evolution of $E_{B_t}$ is similar to that
of $E_{B_p}$.
The global dynamics does not change significantly, either, by moving the
inner boundary position. The low-$\beta$ collimated jet that emerges from
inside the roughly-spherical shock is a feature common to both models
B2e11$\Delta$50 and Rin30. Whereas the evolutions of the jets are rather
different between the two models, the shock propagation on the equator
are very similar between the two. As shown
in the lower panel of Figure~\ref{fig.ibc}, the diagnostic explosion
energies of the two models are also nearly identical.
From these results, we believe that the conclusions of the
current study are not affected by our choice of the inner boundary
position, $r=50$~km.
\section{Discussion and Conclusion}\label{sec.discon}
We have performed MHD simulations of the core-collapse of rapidly-rotating
magnetized stars in two dimensions under axisymmetry, changing both the
strength of magnetic field and the spatial resolution.
Our goal is to study the behavior of the MRI in core-collapse
supernovae and its impacts upon the global dynamics.
As a result of computations we found the followings.
The MRI greatly amplifies the seed magnetic fields even in the
dynamical background of the core-collapse. Although the dominant mode,
buoyant mode or Alfv\'en
mode, differs from location to location, the former plays a primary
role in the exponential growth phase. It is true that the linear
growth rate of the buoyant mode is independent of the wavelength, a
certain degree of high spatial resolution seems necessary to correctly
capture the exponential
growth. The magnetic energies of the poloidal component gets nearly
saturated within the simulation times in all models,
where the saturation level is higher for larger initial magnetic
fields. The magnetic energies of the toroidal component grow
continuously in most of the models, on the other hand, and the core
becomes toroidal-field dominant.
The MRI also has a grate impacts on the global dynamics.
Models in which the MRI are well resolved show faster expansions of
shock surface and obtain more powerful explosions. The formation of
collimated jet is also found in models where the initial magnetic
field is relatively strong and the MRI is well resolved.
The following two effects are found to be the key to the boost of
explosion: the first one is the expansion of the heating region due to
the outward angular momentum transfer. This makes the advection
timescale, or the time for matter to traverse the heating region, longer and
thus enhance the heating; the second effect is the drift of
low-$Y_p$ (equivalently low-$Y_e$) matter along the MRI-distorted
magnetic flux loops as well
as their ejection by the jets, from deep inside the core to the heating
region. The cooling due to electron capture is reduced in the low-$Y_p$
region, and the net heating rises as a result.
The diagnostic explosion energies obtained in the current simulations
are much smaller than the typical value of $\sim 10^{51}$~erg in
reality. Note, however, that our choice of the neutrino
luminosity, $L_{\nu_e}=L_{\bar{\nu_e}}=1.0\times10^{52}$~erg~s$^{-1}$,
which is assumed to be constant,
is quite modest. According to the core-collapse simulations by
\citet{bru13}, who employed the flux-limited diffusion with the
ray-by-ray-plus approximation for neutrino transport, both
$L_{\nu_e}$ and $L_{\bar{\nu_e}}$ are
$\sim 5.0\times 10^{52}$~erg~s$^{-1}$ at $t_{\textrm{pb}}\sim 100$~ms
and decay to $\sim 1.0\times 10^{52}$~erg~s$^{-1}$ over several hundred
milliseconds. If such an evolution of luminosity is adopted in our
simulations more energetic explosion would be obtained.
Although our choice of the initial magnetic field strength, $\sim
10^{11}$~G, is much smaller than those assumed in the former global
MRI simulations
\citep[e.g.,][]{obe06,shi06,saw13a}, most of progenitors of
core-collapse
supernovae may posses even weaker magnetic fields \citep{wad14}.
Since lower saturation magnetic fields are expected for
weaker initial magnetic fields according to our results, the impact of
MRI in "normal''
supernovae will be smaller than that found in this work. Although it
is important to study much weaker
magnetic fields, simulations will be computationally expensive and
thus currently unfeasible: a reduction of the initial magnetic field by half,
with the spatial resolution kept at the current level, demands
eight times higher computational cost for 2D time-explicit simulations.
The dependence on the initial rotation also needs to be investigated,
since the rotation speed of stars is also likely to
distribute over a wide range \citep{ram13}. We are currently undertaking
such studies, and the results will be presented elsewhere in the future.
Although our simulations are 2D under axisymmetry, supernovae
occur in three dimensions in reality, and non-axisymmetric effects such
as dynamo, three-dimensional turbulence, non-axisymmetric modes
of various instabilities may be important. One should
keep in mind that these effects possibly alter results obtained
by current 2D simulations. For example,
3D-MHD simulations performed
by \citet{mos14} demonstrated that magnetically driven-jets can be
destroyed by the $m=1$ mode of a kink-type instability, whereas
such destruction of the jet was not observed in 3D-MHD simulations by
\citet{mik08}. In order to know how essential non-axisymmetric
effects are, 3D global simulations are
mandatory. During the reviewing process of this paper,
\citet{mos15} published the results of the first global-3D
simulations of the MRI in proto-neutron stars. Under
quadrant symmetry, they had simulated the evolution of the MRI
for 10~ms, and found the formation of large-scale, strong toroidal
fields, which hints at later magnetically-driven mass ejections. Such
simulations have only just begun, and the possible 3D effects mentioned above
should be studied in detail in the future.
This requires long-term, large-domain, full 3D
simulations, which may be
marginally feasible with exa-flops computers of the next generation.
\citet{mas07} and \citet{gui15} argued that the neutrino viscosity may
hamper the
growth of MRI deep inside the core, i.e.,
$r\lesssim 30$~km for a fast rotation like ours. Applying the
magnetic field of $\sim 10^{13}$--$10^{14}$~G
obtained in our BG runs for $r\lesssim 30$~km to the
fast rotation model of \citet{gui15}, we found that the neutrino
viscosity may be
marginally important there. Since the inner boundary
condition of our MRI runs is
given by the data of the BG runs of low resolution, the artificial
suppression of MRI by numerical diffusions may effectively mimic the
damping by the neutrino viscosity. We hence believe that
full-sphere simulations including the neutrino viscosity
will not change our conclusions in this paper so much, if the viscous
process is important at all.
\acknowledgments
H.S. is grateful to Kenta Kiuchi, Nobuya Nishimura, Yuichiro
Sekiguchi, and Tomoya Takiwaki for fruitful discussion.
H.S. also thank Daisuke Yamaki and Hideki Yamamoto at RIST, Kobe
Center for useful advice about MPI and openMP parallelization.
Numerical computations in this work were carried out on Cray XC30 at
Center for Computational Astrophysics, National Astronomical
Observatory of Japan, and on HITACHI SR16000 at the Yukawa Institute
Computer Facility.
This work is supported by a Grant-in-Aid for Scientific Research from
the Ministry of Education, Culture, Sports, Science and Technology,
Japan (24103006, 24244036, 26800149).
|
{
"timestamp": "2016-01-06T02:05:33",
"yymm": "1504",
"arxiv_id": "1504.03035",
"language": "en",
"url": "https://arxiv.org/abs/1504.03035"
}
|
\section{Introduction}\label{intro}
In our recent study of
$C^*$-covariant systems $(A,G,\alpha)$
and
crossed product algebras between the full crossed product $A\rtimes_\alpha G$ and the regular crossed product $A\rtimes_{\alpha,r} G$, it turns out that various generalizations of the concept of proper actions of $G$
play an important role. We
therefore
start by taking
a closer look at this concept, and it turns out that even for a classical action of $G$ on a space $X$ we made what we believe to be new discoveries.
Classically (going back to Bourbaki \cite{topologie}), a $G$-space $X$ is called \emph{proper} if the map from $G\times X$ to $X\times X$ given by
\[
(s,x)\mapsto (x,sx)
\]
is proper, i.e., inverse images of compact sets are compact.
We call the action \emph{pointwise proper} if the map from $G$ to $X$ given by
\[
s\mapsto sx
\]
is proper for each $x\in X$.
There is also an intermediate property: $X$ is \emph{locally proper} if each point of $X$ has a $G$-invariant neighbourhood
on which $G$ acts properly.
Apparently the above terminology is not completely standard.
For a discrete group, \cite{varadarajan} uses the terms
\emph{discontinuous}, \emph{properly discontinuous}, and \emph{strongly properly discontinuous}
instead of
pointwise proper, locally proper, and proper, respectively.
Palais uses \emph{Cartan} instead of locally proper.
And \cite{koszul} uses the terms $P_2$, $P_1$, and $P$, respectively.
A characteristic property of properness (see \lemref{compact wandering} below) is sometimes referred to as ``compact sets are wandering''.
It is folklore that for proper $G$-spaces $X$ the full crossed product $C_0(X) \rtimes_{\alpha} G$
is isomorphic to the reduced crossed product $C_0(X) \rtimes_{\alpha,r} G$ (see \cite{phillipsproper} for the second countable case).
In \propref{locally regular} (perhaps also folklore) we show that this carries over to locally proper actions.
We will show in \thmref{pointwise regular} (believed to be new) that this is true also if $X$ is first countable, but the action is only assumed to be pointwise proper.
We propose the following as natural generalizations of properness to a general $C^*$-covariant system $(A,G,\alpha)$:
\begin{defn*}
\quad
\begin{itemize}
\item $(A,G,\alpha)$ is {\it s-proper} if for all $a,b\in A$ the map
\[
g\mapsto \alpha_g(a)b \text{ is in } C_0(G,A).
\]
\item
$(A,G,\alpha)$ is {\it w-proper} if for all $a\in A$, $\phi\in A^*$ the map
\[
g\mapsto \phi(\alpha_g(a)) \text{ is in } C_0(G).
\]
\end{itemize}
\end{defn*}
This is consistent with the classical case, for $A=C_0(X)$ we have
\begin{align*}
(X,G) \text{ is proper} &\iff (C_0(X),G) \text{ is s-proper } \\
(X,G) \text{ is pointwise proper} &\iff (C_0(X),G) \text{ is w-proper. }
\end{align*}
One indication that w-properness is an interesting property is the following
\begin{prop*} Suppose
$(A,G,\alpha)$ is w-proper, $\pi$ a representation of $A$, and $s\mapsto U_s$ a continuous map into the unitaries
\(but not necessarily a homomorphism\) such that $\pi(\alpha_s(a))=U_s\pi(a)U_s^*$. Then for all $\xi,\eta$ in the Hilbert space the
coefficient function
$s\mapsto \<U_s\xi,\eta\>$ is in $C_0(G)$.
\end{prop*}
We treat the classical situation of a $G$-space $X$ in Sections~\ref{spaces} and \ref{c star ramifications}, and discuss general $C^*$-covariant systems in \secref{c star general}.
For a $C^*$-covariant system $(A,G,\alpha)$,
there are various definitions of properness (by Rieffel and others) involving some integrability properties. We show in \secref{rieffel proper} that they imply s- or w-properness. The main purpose of these integrability properties is to define a suitable fixed point algebra in $M(A)$, so our properness definitions are too general for this purpose.
The natural dual concept of a $C^*$-covariant system
is that of a \emph{coaction}. As we briefly describe in \secref{coactions}, it turns out that
s- and w-properness can be defined in a similar way for coactions, and we describe some of the relevant results.
In \secref{E} we describe a general construction of crossed product algebras between
$A\rtimes_{\alpha} G$ and $A\rtimes_{\alpha,r} G$. We claim that the interesting ones are obtained by first taking as our group $C^*$-algebra $C^*(G)/I$ where $I$ is a \emph{small} ideal of $C^*(G)$ (\emph{i.e.}\ $I$ is $\delta_G$-invariant and contained in the kernel of the regular representation $\lambda$ of $C^*(G)$).
We showed in [KLQ13] that $I$ is a small ideal of $C^*(G)$ if and only if
the annihilator $E=I^\perp$ in $B(G)$ is a \emph{large} ideal,
in the sense that it is a nonzero, weak* closed, and $G$-invariant ideal of the Fourier-Stieltjes algebra
$B(G)$. There are various interesting examples, see [BG] and [KLQ13].
Now to a $C^*$-covariant system $(B,G,\alpha)$ and $E$ as above one can define an $E$-crossed product $B\rtimes_{\alpha,E} G$ between the full and the reduced crossed product. In [KLQ13] we show that if the coaction is w-proper then there is a Galois theory describing these crossed products.
Finally we mention the work by Kirchberg, Baum, Guentner, and Willet \cite{bgwexact} on the Baum-Connes conjecture. They have shown that
there is a unique minimal exact and Morita compatible functor
that assigns to a $C^*$-covariant system $(A,G,\alpha)$ a $C^*$-algebra between $A\rtimes_\alpha G$ and $A\rtimes_{\alpha,r} G$.
At least one of the authors doubts that this minimal functor is an $E$-crossed product for some large ideal $E$, although this remains an open problem.
In Sections~\ref{spaces}--\ref{full=reduced} we give a fairly detailed exposition, in particular proofs of results we believe to be new. Sections~\ref{coactions}--\ref{E} will be more
descriptive, referring to the literature for details and proofs.
\section{Actions on spaces}\label{spaces}
Throughout, $G$ will be a locally compact group,
$A$ will be a $C^*$-algebra,
and $X$ will be a locally compact Hausdorff space.
We will be concerned with actions $\alpha$
of
$G$
on
$A$,
and we just say $(A,\alpha)$ is an action
since the group $G$ will typically be fixed.
If $G$ acts on $X$ then
we sometimes call $X$ a \emph{$G$-space},
and
the \emph{associated action} $(C_0(X),\alpha)$ is defined by
\[
\alpha_s(f)(x)=f(s^{-1} x)\righttext{for}s\in G,f\in C_0(X),x\in X.
\]
Recall that, since the map $(s,x)\mapsto sx$ from $G\times X$ to $X$ is continuous,
the associated action $\alpha$ is \emph{strongly continuous} in the sense that
for all $f\in C_0(X)$ the map $s\mapsto \alpha_s(f)$ from $G$ to $C_0(X)$ is continuous for the uniform norm.
The following notation is borrowed from Palais \cite{palais}:
\begin{notn}
If $G$ acts on $X$,
then for two subsets $U,V\subset X$
we define
\[
((U,V))=\{s\in G:sU\cap V\ne\varnothing\}.
\]
\end{notn}
Note that if $U$ and $V$ are compact then $((U,V))$ is closed in $G$.
Much of the following discussion of actions on spaces is well-known;
we present it in a formal way for convenience.
We make no attempt at completeness,
but at the same time we include many proofs to make this exposition self-contained.
When a result can be explicitly found in \cite{palais}, we give a precise reference,
but lack of such a reference should not be taken as any claim of originality.
In much of the literature on proper actions the spaces are only required to be Hausdorff, or completely regular;
in the proofs we will take full advantage of our assumption that our spaces are locally compact Hausdorff.
\begin{defn}\label{proper defn}
A $G$-space $X$
is \emph{proper} if the map
$\phi:X\times G\to X\times X$ defined by
$\phi(x,s)=(x,sx)$
is proper,
i.e., inverse images of compact sets are compact.
\end{defn}
The following is routine,
and explains why properness is sometimes referred to as ``compact sets are wandering''
(e.g., \cite[Situation~2]{rieffelapplications}):
\begin{lem}\label{compact wandering}
A $G$-space $X$
is proper if and only if for every compact $K\subset X$ the set $((K,K))$ is compact,
equivalently for every compact $K,L\subset X$ the set $((K,L))$ is compact.
\end{lem}
\begin{ex}
If $H$ is a closed subgroup of $G$, then it is an easy exercise that
the action of $G$ on the homogeneous space $G/H$ by translation is proper if and only if $H$ is compact.
\end{ex}
The following result is contained in \cite[Theorem~1.2.9]{palais}.
\begin{prop}
A $G$-space $X$ is proper if and only if for all $x,y\in X$ there are neighborhoods $U$ of $x$ and $V$ of $y$ such that $((U,V))$ is relatively compact.
\end{prop}
\begin{proof}
One direction is obvious, since if the action is proper we only need to choose the neighborhoods $U$ and $V$ to be compact.
Conversely, assume the condition involving pairs of points $x,y$,
and let $K\subset X$ be compact.
To show that $((K,K))$ is compact, we will prove that any net $\{s_i\}$ in $((K,K))$ has a convergent subnet.
For every $i$ we can choose $x_i\in K$ such that $s_ix_i\in K$.
Passing to subnets and relabeling, we can assume that $x_i\to x$ and $s_ix_i\to y$ for some $x,y\in K$.
By assumption we can choose compact neighborhoods $U$ of $x$ and $V$ of $y$ such that $((U,V))$ is compact.
Without loss of generality, for all $i$ we have $x_i\in U$ and $s_ix_i\in V$, and hence
$s_i\in ((U,V))$.
Thus $\{s_i\}$ has a convergent subnet by compactness.
\end{proof}
\begin{defn}
A $G$-space $X$ is \emph{locally proper} if
it is a union of open $G$-invariant sets on which $G$ acts properly.
\end{defn}
Palais uses the term \emph{Cartan} instead of locally proper.
The forward direction of the following result is \cite[Proposition~1.2.4]{palais}.
\begin{lem}\label{UU}
A $G$-space $X$ is locally proper if and only if
every
$x\in X$
has a
neighborhood $U$
such that $((U,U))$ is compact.
\end{lem}
\begin{proof}
First assume
that the action is locally proper,
and let $x\in X$.
Choose an open $G$-invariant set $V$ containing $x$ on which $G$ acts properly.
Then choose a compact neighborhood $U$ of $x$ contained in $V$.
Then $((U,U))$ is compact by properness.
Conversely, assume the condition involving compact sets $((U,U))$.
Choose an open neighborhood $V$ of $x$ such that $((V,V))$ is relatively compact,
and let $U=GV$.
We will show that the action of $G$ on $U$ is proper.
Let $y,z\in U$.
Choose $s,t\in G$ such that $y\in sV$ and $z\in tV$.
Then we have neighborhoods $sV$ of $y$ and $tV$ of $z$, and
\[
((sV,tV))=t((V,V))s^{-1}
\]
is relatively compact.
\end{proof}
The following result displays a kind of semicontinuity of the sets $((V,V))$, and in also of the stability subgroups.
The forward direction is \cite[Proposition~1.1.6]{palais}.
\begin{prop}
A $G$-space $X$ is locally proper if and only if for all $x\in X$, the isotropy subgroup $G_x$ is compact and for every neighborhood $U$ of $G_x$ there is a neighborhood $V$ of $x$ such that $((V,V))\subset U$.
\end{prop}
\begin{proof}
First assume that the action is locally proper.
We argue by contradiction.
Suppose we have $x\in X$ and a neighborhood $U$ of $G_x$ such that for every neighborhood $V$ of $x$ there exists $s\in ((V,V))$ such that $s\notin U$.
Fix a neighborhood $R$ of $x$ such that $((R,R))$ is compact.
Restricting to neighborhoods $V$ of $x$ with $V\subset R$,
we see that
we can find nets $\{s_i\}$ in the complement $U^c$ and $\{y_i\}$ in $R$ such that
\begin{itemize}
\item $s_iy_i\in R$ for all $i$,
\item $y_i\to x$, and
\item $s_iy_i\to x$.
\end{itemize}
Then $s_i\in ((R,R))$ for all $i$,
so passing to subnets and relabeling we can assume that $s_i\to s$ for some $s\in G$.
Then $s_iy_i\to sx$, so $sx=x$.
Thus $s\in G_x$.
But then eventually $s_i\in U$, which is a contradiction.
Conversely, assume the condition regarding isotropy groups and neighborhoods thereof,
and let $x\in X$.
Since $G_x$ is compact, we can choose a compact neighborhood $U$ of $G_x$,
and then we can choose a neighborhood $V$ of $x$ such that $((V,V))\subset U$.
Then $((V,V))$ is relatively compact,
and we have shown that the action is locally proper.
\end{proof}
The following result is contained in \cite[Theorem~1.2.9]{palais}.
\begin{prop}
A $G$-space $X$ is proper if and only if
it is locally proper and
$G\backslash X$ is Hausdorff.
\end{prop}
\begin{proof}
First assume that the action is proper.
Then it is locally proper, and
to show that $G\backslash X$ is Hausdorff, we will prove that if a net $\{Gx_i\}$ in $G\backslash X$ converges to both $Gx$ and $Gy$ then $Gx=Gy$.
Since the quotient map $X\to G\backslash X$ is open,
we can pass to a subnet and relabel so that without loss of generality $x_i\to x$.
Then again passing to a subnet and relabeling we can find $s_i\in G$ such that $s_ix_i\to y$.
Choose
compact
neighborhoods $U$ of $x$ and $V$ of $y$,
so
that $((U,V))$ is compact by properness.
Without loss of generality $x_i\in U$ and $s_ix_i\in V$ for all $i$.
Then $s_i\in ((U,V))$ for all $i$,
so by compactness we can pass to subnets and relabel so that $\{s_i\}$ converges to some $s\in G$.
Then $s_ix_i\to sx$, so $sx=y$, and hence $Gx=Gy$.
Conversely, assume that the action is locally proper and $G\backslash X$ is Hausdorff.
Let $x,y\in X$.
By assumption we can choose a compact neighborhood $U$ of $x$ such that $((U,U))$ is compact.
Now choose any compact neighborhood $V$ of $y$.
To show that the action is proper, we will prove that $((U,V))$ is compact.
Let $\{s_i\}$ be any net in $((U,V))$.
For each $i$ choose $x_i\in U$ such that $s_ix_i\in V$.
By compactness we can pass to subnets and relabel so that $x_i\to z$ and $s_ix_i\to w$ for some $z\in U$ and $w\in V$.
Then by Hausdorffness we can write
\[
Gz=\lim Gx_i=\lim Gs_ix_i=Gw,
\]
so we can choose $s\in G$ such that $w=sz$.
Then $s_ix_i\to sz$, so
\[
s^{-1} s_ix_i\to z.
\]
Without loss of generality, for all $i$ we can assume that $s^{-1} s_ix_i\in U$,
so that $s^{-1} s_i\in ((U,U))$.
By compactness we can pass to subnets and relabel so that $s^{-1} s_i\to t$ for some $t\in G$.
Thus $s_i\to st$, and we have found a convergent subnet of $\{s_i\}$.
Thus $((U,V))$ is compact.
\end{proof}
\begin{ex}\label{locally proper not proper}
It is a well-known fact in topological dynamics that
there are actions
that are
locally proper but not proper,
e.g., the action of $\mathbb Z$ on
\[
[0,\infty)\times [0,\infty)\setminus \{(0,0)\}
\]
generated by the homeomorphism $(x,y)\mapsto (2x,y/2)$, where any compact neighborhood of $\{(1,0),(0,1)\}$ meets itself infinitely often.
This action is locally proper because its restriction to each of the open sets
$[0,\infty)\times (0,\infty)$ and
$(0,\infty)\times [0,\infty)$,
which cover the space,
are proper.
A closely related example is given by letting $\mathbb R$ act on the same space by
$s(x,y)=(e^sx,e^{-s}y)$.
\end{ex}
\begin{defn}
A $G$-space $X$ is \emph{pointwise proper} if for all $x\in X$ and compact $K\subset X$,
the set $((x,K))$ is compact.
\end{defn}
The above properness condition does not seem to be very often studied in the dynamics literature, and the
term we use is not standard, as far as we have been able to determine.
It is obvious that the above definition can be reformulated as follows:
\begin{lem}\label{orbit proper}
A $G$-space $X$ is pointwise proper if and only if
for every $x\in X$ the map $s\mapsto sx$ from $G$ to $X$
is proper.
\end{lem}
\begin{prop}\label{T1}
If a $G$-space $X$ is pointwise proper then orbits are closed, and hence $G\backslash X$ is $T_1$.
\end{prop}
\begin{proof}
Let $x\in X$, and suppose we have a net $\{s_ix\}$ in the orbit $Gx$ converging to $y\in X$.
Choose a compact neighborhood $U$ of $y$.
Without loss of generality, for all $i$ we have $s_ix\in U$,
and hence $s_i\in ((x,U))$.
This set is compact by pointwise properness, so passing to a subnet and relabeling we can assume that $s_i\to s$ for some $s\in G$.
Then $s_ix\to sx$, so $y=sx\in Gx$.
\end{proof}
\begin{notn}
For $x\in X$ let $G_x$ denote the isotropy subgroup.
\end{notn}
\begin{prop}
A $G$-space $X$ is pointwise proper if and only if for all $x\in X$
the isotropy subgroup $G_x$ is compact and
the map
$s\mapsto sx$ from $G$ to $Gx$
is relatively open,
equivalently,
the action of $G$ on the orbit $Gx$ is conjugate to the action on the homogeneous space $G/G_x$.
\end{prop}
\begin{proof}
First assume that the action is pointwise proper,
and let $x\in X$.
Then $G_x$ is trivially compact.
By homogeneity it suffices to show that the map
$s\mapsto sx$ from $G$ to $Gx$
is relatively open at $e$.
Let $W$ be a neighborhood of $e$.
Suppose that $Wx$ is not a relative neighborhood of $x$ in the orbit $Gx$.
Then we can choose a net $\{s_i\}$ in $G$ such that $s_ix\notin Wx$ and $s_ix\to x$.
Choose a neighborhood $U$ of $x$ such that $((U,U))$ is compact.
Without loss of generality, for all $i$ we have $s_ix\in U$, and so $s_i\in ((x,U))$.
By compactness we can pass to a subnet and relabel so that $s_i\to s$ for some $s\in G$.
Then $s_ix\to sx$. Thus $sx=x$, and so $s\in G_x$.
But then eventually $s_i\in WG_x$, which is a contradiction because
$WG_xx=Wx$.
The converse is obvious, since if $G_x$ is compact the action of $G$ on $G/G_x$ is proper.
\end{proof}
We will show that pointwise properness is weaker than local properness, but for this we need a version of \propref{T1} for local properness.
The following result is contained in \cite[Proposition~1.1.4]{palais}.
\begin{lem}
If a $G$-space $X$ is locally proper then orbits are closed.
\end{lem}
\begin{proof}
Let $x\in X$, and suppose we have a net $\{s_i\}$ in $G$ such that $s_ix\to y$.
Choose an open $G$-invariant subset $U$ containing $y$ on which $G$ acts properly.
Then the action of $G$ on $U$ is pointwise proper, so the orbit $Gx$ is closed in $U$,
and hence $y\in Gx$.
\end{proof}
\begin{cor}
If a $G$-space $X$ is locally proper then it is pointwise proper.
\end{cor}
\begin{proof}
Let $x\in X$.
Choose an open $G$-invariant set $U\subset X$ such that the action of $G$ on $U$ is proper.
Let $K\subset X$ be compact,
and put $L=K\cap Gx$.
Then $L$ is compact because $Gx$ is closed, and $L\subset U$.
Thus
$((x,K))=((x,L))$
is compact because $\{x\}$ and $L$ are compact subsets of $U$ and $G$ acts properly on $U$.
\end{proof}
\begin{ex}\label{pointwise not local}
This example is
taken from
\cite[Example~5 in Section~2]{varadarajan}.
Recall that in \exref{locally proper not proper}
we had an action of $\mathbb Z$ on the space
\[
X=\bigl([0,\infty)\times [0,\infty)\bigr)\setminus\{(0,0\}
\]
generated by the homeomorphism $(x,y)\mapsto (2x,y/2)$.
We form the quotient of $X$ by identifying
$\{0\}\times (0,\infty)$ with $(0,\infty)\times \{0\}$
via
\[
(0,y)\sim (1/y,0).
\]
Then the action descends to the identification space,
and the quotient action is pointwise proper but not locally proper.
\end{ex}
With suitable countability assumptions, there is a surprise:
\begin{cor}[Glimm]
Let $G$ act on $X$, and assume that $G$ and $X$ are second countable,
and that every isotropy subgroup is compact.
Then the following are equivalent:
\begin{enumerate}
\item the action is pointwise proper;
\item for all $x\in X$ the map $sG_x\mapsto sx$ from $G/G_x$ to $Gx$ is a homeomorphism;
\item $G\backslash X$ is $T_0$;
\item $G\backslash X$ is $T_1$;
\item every orbit is locally compact in the relative topology from $X$;
\item every orbit is closed in $X$.
\end{enumerate}
\end{cor}
\begin{proof}
Because we assume that the isotropy groups are compact, we know (1) $\iff$ (2).
Glimm \cite[Theorem~1]{glimm} proves that, in the second countable case,
(2) $\iff$ (3) $\iff$ (5).
We also know (1) $\ensuremath{\Rightarrow}$ (6) $\ensuremath{\Rightarrow}$ (4).
Finally, (4) $\ensuremath{\Rightarrow}$ (3) trivially.
\end{proof}
\section{$C^*$-ramifications}\label{c star ramifications}
Let $X$ be a $G$-space, and let $\alpha$ be the associated action of $G$ on $C_0(X)$.
In this section we examine the ramifications for the action $\alpha$ of the various properness conditions covered in \secref{spaces}.
For the state of the art in the case of proper actions, see \cite{echemeproper}.
\begin{notn}
If $\psi:X\to Y$ is a continuous map between locally compact Hausdorff spaces,
define $\psi^*:C_0(Y)\to C_b(X)$ by $\psi^*(f)=f\circ\psi$.
\end{notn}
It is an easy exercise to show:
\begin{lem}\label{Cc}
For a continuous map $\psi:X\to Y$ between locally compact Hausdorff spaces,
the following are equivalent:
\begin{enumerate}
\item $\psi$ is proper
\item $\psi^*$ maps $C_0(Y)$ into $C_0(X)$
\item $\psi^*$ maps $C_c(Y)$ into $C_c(X)$.
\end{enumerate}
\end{lem}
\begin{prop}\label{proper C0(X)}
The $G$-space $X$ is proper if and only if for all $f,g\in C_0(X)$
the map $s\mapsto \alpha_s(f)g$ from $G$ to $C_0(X)$ vanishes at infinity.
\end{prop}
\begin{proof}
First assume that the action is proper.
Since $C_c(X)$ is dense in $C_0(X)$,
by continuity it suffices to show that for all $f,g\in C_c(X)$
the map continuous $s\mapsto \alpha_s(f)g$ from $G\to C_0(X)$ has compact support.
Define $f\times g\in C_c(X\times X)$ by
\[
f\times g(x,y)=f(x)g(y).
\]
Since the map $\phi:G\times X\to X\times X$ given by $g(s,x)=(sx,x)$ is proper,
we have $\phi^*(f\times g)\in C_c(G\times X)$,
so there exist compact sets $K\subset G$ and $L\subset X$ such that
for all $(s,x)\notin K\times L$ we have
\[
0=\phi^*(f\times g)(s,x)=f\times g(sx,x)=f(sx)g(x)=\bigl(\alpha_{s^{-1}}(f)g\bigr)(x).
\]
Since $s\notin K$ implies $(s,x)\notin K\times L$,
we see that
the map $s\mapsto \alpha_s(f)g$ is supported in the compact set $K^{-1}$.
Conversely, assume the condition regarding $\alpha_s(f)g$.
To show that the action is proper, we will show that the map $\phi$ is proper,
and by \lemref{Cc} it suffices to show that
if $h\in C_c(X\times X)$ then $\phi^*(h)\in C_c(G\times X)$.
The support of $h$ is contained in a product $M\times N$ for some compact sets $M,N\subset X$,
and we can choose $f,g\in C_c(X)$ with $f=1$ on $M$ and $g=1$ on $N$.
Then $h(f\times g)=h$,
so it suffices to show that $\phi^*(f\times g)$ has compact support.
By assumption the support $K$ of $s\mapsto \alpha_s(f)g$ is compact,
and letting $L$ be the support of $g$ we see that
for all $(s,x)$ not in the compact set $K^{-1}\times L$ we have
\[
\phi^*(f\times g)(s,x)=\bigl(\alpha_{s^{-1}}(f)g\bigr)(x)=0.
\qedhere
\]
\end{proof}
\begin{prop}\label{pointwise proper C0(X)}
The $G$-space $X$ is pointwise proper if and only if for all $f\in C_0(X)$ and $\mu\in M(X)=C_0(X)^*$ the map
\[
g(s)=\int_Xf(sx)\,d\mu(x)
\]
is in $C_0(G)$.
\end{prop}
\begin{proof}
First assume that the action is pointwise proper.
Let $f\in C_0(X)$ and $\mu\in M(X)$, and define $g$ as above.
Note that $g$ is continuous since the
associated action $(C_0(X),\alpha)$ is strongly continuous.
Suppose that $g$ does not vanish at $\infty$,
and pick $\epsilon>0$ such that the closed set
\[
S:=\{s\in G:|g(s)|\ge \epsilon\}
\]
is not compact.
It is a routine exercise to verify that we can find
a sequence $\{s_n\}$ in $S$
and
a compact neighborhood $V$ of $e$ such that
the sets $\{s_nV\}$ are pairwise disjoint.
Then for each $x\in X$ we have $\lim_{n\to\infty}f(s_nx)=0$,
because for fixed $x$ and any $\delta>0$
it is an easy exercise to see that
the compact set $\{s\in G:|f(sx)|\ge \delta\}$ can only intersect finitely many of the sets $\{s_nV\}$.
Thus by the Dominated Convergence theorem $\lim_{n\to\infty}g(x_n)=0$,
contradicting $s_n\in S$ for all $n$.
The converse follows immediately by taking $\mu$ to be a Dirac measure and applying \lemref{orbit proper}.
\end{proof}
\propref{module} below is the first time we need vector-valued integration.
There are numerous references dealing with this topic.
We are interested in integrating functions $f:\Omega\to B$, where $\Omega$ is a locally compact Hausdorff space equipped with a Radon measure $\mu$ (sometimes complex, but other times positive, and then frequently infinite), and $B$ is a Banach space.
Rieffel \cite[Section~1]{integrable} handles continuous bounded functions to a $C^*$-algebra using $C^*$-valued weights.
Exel \cite[Section~2]{exelunconditional} develops a theory of \emph{unconditionally integrable} functions with values in a Banach space,
involving convergence of the integrals over relatively compact subsets of $G$.
Williams \cite[Appendix~B.1]{danacrossed} gives an exposition of the general theory of $L^1(\Omega,B)$, that
in some sense
unifies the treatments in
\cite[Chapter~3]{dunford},
\cite{bourbaki_integration},
\cite[Chapter~II]{fd1},
and
\cite[part~I, Section~III.1]{hille}.
However, Williams uses a positive measure throughout, and we occasionally need complex measures;
this poses no problem, since the theory of \cite{danacrossed} can be applied to the positive and negative variations of the real and imaginary parts of a complex measure.
We prefer to use \cite{danacrossed} as our reference for vector-valued integration,
mainly because it entails \emph{absolute integrability} rather than \emph{unconditional integrability}
(see the first item in the following list).
Here are the main properties of $L^1(\Omega,B)$ that we need:
\begin{itemize}
\item The map $f\mapsto \int_\Omega f\,d\mu$ from $L^1(\Omega,B)$ to $B$ is bounded and linear,
where $\|f\|_1=\int_\Omega \|f(x)\|\,d|\mu|(x)$.
\item If $f\in L^1(\Omega,B)$ and $\omega$ is a bounded linear functional on $B$,
then $\omega\circ f\in L^1(\Omega)$ and
\[
\omega\left(\int_\Omega f(x)\,d\mu(x)\right)=\int_\Omega \omega(f(x))\,d\mu(x).
\]
\item If $f\in L^1(\Omega)$ and $b\in B$ then
\[
\int_\Omega (f\otimes b)\,d\mu=\left(\int_\Omega f\,d\mu\right)b,
\]
where $(f\otimes b)(x)=f(x)b$.
\item Every continuous bounded function from $\Omega$ to $B$ is measurable,
and is also essentially-separably valued on compact sets,
and so is integrable with respect to any complex measure.
\end{itemize}
Of course, we refer to the elements of $L^1(\Omega,B)$ as the \emph{integrable} functions from $\Omega$ to $B$.
If $X$ is a $G$-space, then
$C_0(X)$ gets a Banach-module structure over $M(G)=C_0(G)^*$ by
\[
\mu*f(x)=\int_G f(sx)\,d\mu(s)\righttext{for}\mu\in M(G),f\in C_0(X),x\in X.
\]
Here we are integrating the continuous bounded function
$s\mapsto \alpha_s(f)$
with respect to the complex measure $\mu$.
The following is a special case of \propref{module noncom} below.
\begin{prop}\label{module}
The action on $X$ is pointwise proper
if and only if
for each $f$ the map $\mu\mapsto \mu*f$ is weak*-to-weakly continuous.
\end{prop}
\section{Properness conditions for actions on $C^*$-algebras}\label{c star general}
Propositions~\ref{proper C0(X)} and \ref{pointwise proper C0(X)}
motivate the following:
\begin{defn}\label{s-proper}
An action $(A,\alpha)$ is \emph{s-proper} if for all $a,b\in A$ the map $s\mapsto \alpha_s(a)b$ from $G$ to $A$ vanishes at infinity.
\end{defn}
Taking adjoints, we see that the above map could equally well be replaced by $s\mapsto a\alpha_s(b)$.
\begin{defn}\label{w-proper}
An action $(A,\alpha)$ is \emph{w-proper} if for all $a\in A$ and all $\omega\in A^*$ the map
\[
g(s)=\omega\bigl(\alpha_s(a)\bigr)
\]
is in $C_0(G)$.
\end{defn}
We use the admittedly nondescriptive terminology s-proper and w-proper to avoid confusion with the myriad other uses of the word ``proper'' for actions on $C^*$-algebras.
\begin{rem}\label{locally proper C0(X)}
It is
almost obvious
that
a
$G$-space $X$
is locally proper if and only if
there is a family of $\alpha$-invariant closed ideals of $C_0(X)$
that densely span $C_0(X)$ and
on each of which $\alpha$ has the property in \propref{proper C0(X)}.
In fact, we will use this in the proof of \propref{locally regular}.
This could be generalized in various ways to actions on arbitrary $C^*$-algebras,
but since
we have no applications of this
we will not
pursue it here.
\end{rem}
Propositions~\ref{proper C0(X)} and \ref{pointwise proper C0(X)}
can be rephrased as follows:
\begin{cor}\label{proper action}
A $G$-space $X$ is proper if and only if the associated action $(C_0(X),\alpha)$ is s-proper,
and is pointwise proper if and only if $\alpha$ is w-proper.
\end{cor}
\begin{rem}\label{s vs w}
If an action $(A,\alpha)$ is s-proper then it is w-proper, since by the Cohen-Hewitt factorization theorem every functional in $A^*$ can be expressed in the form $\omega\cdot a$, where
\[
\omega\cdot a(b)=\omega(ab)\righttext{for}\omega\in A^*,a,b\in A.
\]
On the other hand, \exref{locally proper not proper} implies that $\alpha$ can be w-proper but not s-proper.
\end{rem}
If $(A,\alpha)$ is an action then $A$ gets a Banach module structure over $M(G)$ by
\[
\mu*a=\int_G \alpha_s(a)\,d\mu(s)\righttext{for}\mu\in M(G),a\in A.
\]
\propref{module} is the commutative version of the following:
\begin{prop}\label{module noncom}
An action $(A,\alpha)$ is w-proper if and only if for each $a\in A$
the map $\mu\mapsto \mu*a$ is weak*-to-weakly continuous.
\end{prop}
\begin{proof}
First assume that $\alpha$ is w-proper,
and let $a\in A$.
Let $\mu_i\to 0$ weak* in $M(G)$,
and let $\omega\in A^*$.
Then
\[
\omega(\mu_i*a)=\omega\left(\int_G \alpha_s(a)\,d\mu_i(s)\right)=\int \omega(\alpha_s(a))\,d\mu_i(s)
\to 0,
\]
because the map $s\mapsto \omega(\alpha_s(a))$ is in $C_0(G)$.
Conversely, assume the weak*-weak continuity,
and let $a\in A$ and $\omega\in A^*$.
If $\mu_i\to 0$ weak* in $M(G)$, then
\[
\int_G \omega(\alpha_s(a))\,d\mu_i(s)
=\omega(\mu_i*a)
\to 0
\]
by continuity.
By the well-known \lemref{in C0} below, the element $s\mapsto \omega(\alpha_s(a))$ of $C_b(G)$ lies in $C_0(G)$.
\end{proof}
In the above proof we appealed to the following well-known fact:
\begin{lem}\label{in C0}
Let $f\in C_b(G)$.
Then $f\in C_0(G)$ if and only if for every net $\{\mu_i\}$ in $M(G)$ converging weak* to 0 we have
\[
\int f\,d\mu_i\to 0.
\]
\end{lem}
s-properness and w-properness are both preserved by morphisms:
\begin{prop}\label{morphism}
Let $\phi:A\to M(B)$ be a nondegenerate homomorphism that is equivariant for actions $\alpha$ and $\beta$, respectively.
If $\alpha$ is
s-proper
or w-proper,
then $\beta$ has the same property.
\end{prop}
\begin{proof}
First assume that $\alpha$ is s-proper.
Let $c,d\in B$.
By the Cohen-Hewitt Factorization theorem, $c=c'\phi(a)$ and $d=\phi(b)d'$ for some $a,b\in A$ and $c',d'\in B$.
Then
\begin{align*}
\beta_s(c)d
&=\beta_s(c'\phi(a))\phi(b)d'
\\&=\beta_s(c')\phi\bigl(\alpha_s(a)b\bigr)d',
\end{align*}
which vanishes at infinity because
$s\mapsto \alpha_s(a)b$ does and $s\mapsto \beta_s(c')$ is bounded.
Now assume that $\alpha$ is w-proper.
Let $b\in B$ and $\omega\in B^*$.
We must show that the function $s\mapsto \omega\circ \beta_s(b)$ vanishes at $\infty$, and it suffices to do this for $\omega$ positive.
By the Cohen-Hewitt Factorization theorem we can assume that $b=\phi(a^*)c$ with $a\in A$ and $c\in B$.
By the Cauchy-Schwarz inequality for positive functionals on $C^*$-algebras, we have
\begin{align*}
\bigl|\omega\circ \beta_s(b)\bigr|^2
&=\Bigl|\omega\bigl(\phi(\alpha_s(a^*))\beta_s(c)\bigr)\Bigr|^2
\\&\le \omega\circ\phi(\alpha_s(a^*a))\omega(\beta_s(c^*c)),
\end{align*}
which vanishes at $\infty$ since $s\mapsto \omega\circ\phi(\alpha_s(a^*a))$ does and $s\mapsto \omega(\beta_s(c^*c))$ is bounded.
\end{proof}
In \secref{coactions} we will discuss properness for coactions, the dualization of actions.
Here we record an easy corollary of \propref{morphism} that involves coactions,
because it gives a rich supply of s-proper actions.
For now we just need to recall that if $(A,\delta)$ is a coaction of $G$,
with crossed product $C^*$-algebra $A\rtimes_\delta G$,
then there is a pair of nondegenerate homomorphisms
\[
\xymatrix{
A \ar[r]^-{j_A}
&M(A\rtimes_\delta G)
&C_0(G) \ar[l]_-{j_G}
}
\]
such that $(j_A,j_G)$ is a universal covariant homomorphism.
The \emph{dual action} $\widehat\delta$ of $G$ on $A\rtimes_\delta G$ is characterized by
\begin{align*}
\widehat\delta_s\circ j_A&=j_A
\\
\widehat\delta_s\circ j_G&=j_G\circ\textup{rt}_s,
\end{align*}
where $\textup{rt}$ is the action of $G$ on $C_0(G)$ by right translation.
\begin{cor}\label{dual action}
Every dual action is s-proper.
\end{cor}
\begin{proof}
If $\delta$ is a coaction of $G$ on $A$, then the canonical nondegenerate homomorphism $j_G:C_0(G)\to M(A\rtimes_\delta G)$ is $\textup{rt}-\widehat\delta$ equivariant.
Thus $\widehat\delta$ is s-proper since $\textup{rt}$ is.
\end{proof}
\cite[Corollary~5.9]{brogue} says that
if an action of a discrete group $G$ on a compact Hausdorff space $X$
is a-T-menable in the sense of \cite[Definition~5.5]{brogue},
then every covariant representation of the associated action $(C(X),\alpha)$ is weakly contained in a representation $(\pi,U)$, on a Hilbert space $H$, such that for all $\xi,\eta$ in a dense subspace of $H$ the function $s\mapsto (U_s\xi,\eta)$ is in $c_0(G)$.
The following proposition shows that w-proper actions on arbitrary $C^*$-algebras have a quite similar
property:
\begin{prop}\label{c0 coef}
Let $(A,\alpha)$ be a w-proper action,
let $\pi$ be a representation of $A$ on a Hilbert space $H$,
and for each $s\in G$ suppose we have a unitary operator $U_s$ on $H$ such that $\ad U_s\circ\pi=\pi\circ\alpha_s$.
Then
for all $\xi,\eta\in H$ the function
\[
s\mapsto \<U_s\xi,\eta\>
\]
vanishes at infinity.
\end{prop}
\begin{proof}
We can assume that $\pi$ is nondegenerate.
Then we can factor $\xi=\pi(a)\xi'$ for some $a\in A,\xi'\in H$, and
we have
\begin{align*}
|\<U_s\pi(a)\xi',\eta\>|
&=|\<U_s\xi',\pi(\alpha_s(a^*))\eta\>|
\\&\le \|\xi'\|\<\pi(\alpha_s(aa^*)\eta,\eta\>^{1/2},
\end{align*}
so we can appeal to w-properness with $\omega\in A^*$ defined by
\[
\omega(b)=\<\pi(b)\eta,\eta\>.
\qedhere
\]
\end{proof}
\begin{rem}
Note that in the above proposition we do not require $U$ to be a homomorphism; it could be a projective representation.
\end{rem}
\begin{rem}
Thus it would be interesting to study the relation between a-T-menable actions in the sense of \cite{brogue} and pointwise proper actions.
As it stands, the connection would be subtle, because an infinite discrete group cannot act pointwise properly on a compact space.
\end{rem}
\subsection*{Action on the compacts}
The following gives a strengthening of a special case of
\propref{c0 coef}:
\begin{prop}\label{c0 coef 2}
Let $H$ be a Hilbert space, and let $\alpha$ be an action of $G$ on $\mathcal K(H)$.
For each $s\in G$ choose a unitary operator $U_s$ such that $\alpha_s=\ad U_s$.
The following are equivalent:
\begin{enumerate}
\item $\alpha$ is s-proper;\label{one}
\item $\alpha$ is w-proper;\label{two}
\item $s\mapsto \<U_s\xi,\xi\>$
vanishes at infinity
for all $\xi\in H$.\label{three}
\item $s\mapsto \<U_s\xi,\eta\>$
vanishes at infinity
for all $\xi,\eta\in H$.\label{four}
\end{enumerate}
\end{prop}
\begin{proof}
We know \eqref{one} $\ensuremath{\Rightarrow}$ \eqref{two} $\ensuremath{\Rightarrow}$ \eqref{three} by
\remref{s vs w} and \propref{c0 coef},
and \eqref{three} $\ensuremath{\Rightarrow}$ \eqref{four} by polarization.
Assume \eqref{four}.
Let $E(\xi,\eta)$ be the rank-1 operator given by $\zeta\mapsto \<\zeta,\eta\>\xi$.
For $\xi,\eta,
\gamma,\kappa
\in H$,
A routine computation shows
\begin{align*}
E(\xi,\eta)\alpha_s(E(\gamma,\kappa))
&=\<U_s\gamma,\eta\>E(\xi,\kappa)U_s^*,
\end{align*}
so
\[
\bigl\|E(\xi,\eta)\alpha_s(E(\gamma,\kappa))\bigr\|
\le \bigl|\<U_s\gamma,\eta\>\bigr|\|E(\xi,\kappa)\|,
\]
which vanishes at infinity.
Thus $s\mapsto a\alpha_s(b)$ is in $C_0(G,\mathcal K(H))$ whenever $a$ and $b$ are rank-1,
and by linearity and density it follows that $\alpha$ ia s-proper.
\end{proof}
In \propref{c0 coef 2}, when $U$ can be chosen to be a representation of $G$,
we have the following:
\begin{cor}\label{cyclic}
Let $U$ be a representation of $G$ on a Hilbert space $H$, and
let $\alpha=\ad U$ be the associated action of $G$ on $\mathcal K(H)$.
Suppose that $\xi$ is a cyclic vector for the representation $U$.
If $s\mapsto \<U_s\xi,\xi\>$ vanishes at infinity, then $\alpha$ is s-proper.
\end{cor}
\begin{proof}
As in \cite[Remark~2.7]{brogue}, it is easy to see that
for all $\eta,\kappa$ in the dense subspace
of $H$
spanned by $\{U_s\xi:s\in G\}$
the function $s\mapsto \<U_s\eta,\kappa\>$ vanishes at infinity.
Then for all $\eta,\kappa\in H$ we can find sequences $\{\eta_n\},\{\kappa_n\}$ such that
$\|\eta_n-\eta\|\to 0$,
$\|\kappa_n-\kappa\|\to 0$, and
for all $n$ the function $s\mapsto \<U_s\eta_n,\kappa_n\>$ vanishes at infinity.
Then a routine estimation shows that
the functions $s\mapsto \<U_s\eta_n,\kappa_n\>$
converge uniformly to the function $s\mapsto \<U_s\eta,\kappa\>$,
and hence this latter function vanishes at infinity.
The result now follows from \propref{c0 coef 2}.
\end{proof}
\section{Rieffel properness}\label{rieffel proper}
We will show that if an action $(A,\alpha)$ is proper in Rieffel's sense
\cite[Definition~1.2]{proper} (see also \cite[Definition~4.5]{integrable}
then it is s-proper.
Rieffel's definitions of proper action in both of the above papers involve integration of $A$-valued functions on $G$,
and we have recorded our conventions regarding vector-valued integration
in the discussion preceding \propref{module}.
In \cite{proper}, Rieffel defined an action $(A,\alpha)$ to be \emph{proper}
(and we follow \cite{BusEch4} in using the term \emph{Rieffel proper})
if
$s\mapsto \alpha_s(a)b$ is integrable
for all $a,b$ in some dense subalgebra,
plus other conditions that we will not need.
\begin{cor}\label{integrable then proper}
Let $(A,\alpha)$ be an action.
\begin{enumerate}
\item Suppose that there is a dense $\alpha$-invariant subset $A_0$ of $A$ such that for all $a,b\in A_0$ the function
\begin{equation}\label{function}
s\mapsto \alpha_s(a)b
\end{equation}
is
integrable.
Then $\alpha$ is s-proper in the sense of \defnref{s-proper}.
\item
Suppose that there is a dense $\alpha$-invariant subset $A_0$ of $A$ such that for all $a\in A_0$
and all $\omega\in A^*$
the function
\[
s\mapsto \omega(\alpha_s(a))
\]
is integrable.
Then $\alpha$ is w-proper in the sense of \defnref{w-proper}.
\end{enumerate}
\end{cor}
\begin{proof}
(1)
Since the functions \eqref{function} are uniformly continuous in norm, it follows immediately from
the elementary lemma
\lemref{barbalat}
below
that
$s\mapsto \alpha_s(a)b$ is in $C_0(G,A)$ for all $a,b\in A_0$,
and then
(1)
follows by density.
(2)
This can be proved similarly to (1), except now the functions are scalar-valued.
\end{proof}
In the above proof we referred to the following:
\begin{lem}\label{barbalat}
Let
$B$ be a Banach space, and let
$f:G\to B$ be uniformly continuous and integrable.
Then $f$ vanishes at infinity.
\end{lem}
\begin{proof}
Since the composition of $f$ with the norm on $B$ is uniformly continuous,
and $\|f\|_1=\int_G \|f(s)\|\,ds<\infty$ by hypothesis,
so this follows immediately from the scalar-valued case
(for which, see \cite[Theorem~1]{carcano}),
and which itself
is a routine adaptation of a classical result about scalar-valued functions on $\mathbb R$,
sometimes referred to as Barbalat's Lemma.
\end{proof}
In the commutative case, \corref{integrable then proper} (1) has a converse.
First, following \cite{BusEch4}, we will call an action $(A,\alpha)$
\emph{Rieffel proper} if
it satisfies the conditions of \cite[Definition~1.2]{proper}.
\begin{prop}\label{abel}
If $A=C_0(X)$ is commutative, then an action $(A,\alpha)$ is s-proper if and only if it is Rieffel proper.
\end{prop}
\begin{proof}
First assume that $\alpha$ is s-proper.
Then by \thmref{proper action} the $G$-space $X$ is proper,
and then it follows from \cite[Theorem~4.7 and and its proof]{integrable} that $\alpha$ is Rieffel proper.
Conversely, if $\alpha$ is Rieffel proper, then in particular it satisfies the hypothesis of \corref{integrable then proper} (1), so $\alpha$ is s-proper.
\end{proof}
\begin{rem}
Thus,
if the $G$-space $X$ is proper, then by \cite[Theorem~1.5]{proper}
(for the case of free action,
see also \cite[Situation~2]{rieffelapplications}, which refers to \cite{gre:smooth})
there is an ideal of $C_0(X)\rtimes_r G$
(which is known to equal $C_0(X)\rtimes G$ in this case --- see \propref{locally regular} below)
that is Morita equivalent to $C_0(G\backslash X)$.
This uses the following:
for $f\in C_c(X)$ the integral
\[
\widehat f(Gx):=\int_G f(sx)\,ds
\]
defines $\widehat f\in C_c(G\backslash X)$.
If the action on $X$ is just pointwise proper,
the integral $\int_G f(sx)\,ds$ still makes sense for $f\in C_c(X)$.
It would be interesting to know what properties persist in this case.
\end{rem}
\begin{ex}
\propref{abel} is not true for arbitrary actions $(A,\alpha)$.
For example, let $G$ be the free group $\mathbb F_n$ with $n>1$, and let $l$ be the length function.
Haagerup proves in \cite{haagerup} that for any $a>0$ the function $s\mapsto e^{-al(s)}$
is positive definite.
For $k\in\mathbb N$ define $h_k(s)=e^{-l(s)/k}$,
and let $U_k$ be the associated cyclic representation on a Hilbert space $H_k$,
so that we have a cyclic vector $\xi_k$ for $U_k$ with
\[
\<U_k(s)\xi_k,\xi_k\>=h_k(s).
\]
For each $k$, since $h_k$ vanishes at infinity the associated inner action $\alpha_k=\ad U_k$ of $G$ on $\mathcal K(H_k)$ is s-proper, by \corref{cyclic}.
We claim that not all these actions $\alpha_k$ can be Rieffel proper.
Rieffel shows in \cite[Theorem~7.9]{integrable}
that the action $\alpha$ is proper in the sense of \cite[Definition~4.5]{integrable}
if and only if the representation $U$ is square-integrable in the sense of \cite[Definition~7.8]{integrable}.
This latter definition is somewhat nonstandard,
in that it uses concepts from the theory of left Hilbert algebras.
Also, Rieffel's definition of proper action in \cite{integrable} is somewhat complicated in that it involves $C^*$-valued weights.
In this paper we prefer to deal with the more accessible definition of Rieffel-proper action in \cite[Definition~1.2]{proper},
which Rieffel shows implies the properness condition \cite[Definition~4.5]{integrable}.
Actually, we need not concern ourselves here with Rieffel's definition of square-integrable representations, rather all we need is his reassurance (see \cite[Corollary~7.12 and Theorem~7.14]{integrable} that a cyclic representation of $G$ is square-integrable in his sense if and only if it is contained in the regular representation of $G$ --- so his notion of square integrability is equivalent to the more usual one (as he assures us in his comment following \cite[Definition~7.8]{integrable}).
Suppose that for every $k\in\mathbb N$ the action $\alpha_k$ of $G$ on $\mathcal K(H_k)$ is Rieffel proper.
Then, as noted above, $\alpha_k$ is also proper in the sense of \cite[Definition~4.5]{integrable}, and so the representation $U_k$ is contained in the regular presentation $\lambda$.
Now we argue exactly as in \cite[proof of Proposition~4.4]{brogue}:
since the functions $h_k$ converge to 1 pointwise on the discrete group $G$,
for all $s\in G$ we have
\[
\<U_k(s)\xi_k,\xi_k\>\to 1,
\]
and hence
\[
\|U_k(s)\xi_k-\xi_k\|\to 0.
\]
Thus the direct sum representation $\bigoplus_k U_k$ weakly contains the trivial representation.
But since each $U_k$ is contained in $\lambda$, the direct sum is weakly contained in $\lambda$.
This gives a contradiction, since $G=\mathbb F_n$ is nonamenable.
\end{ex}
\section{Full equals reduced}\label{full=reduced}
\begin{defn}
Let $(A,\alpha)$ be an action.
We say the \emph{full and reduced crossed products of $(A,\alpha)$ are equal}
if the regular representation
\[
\Lambda:A\rtimes_\alpha G\to A\rtimes_{\alpha,r} G
\]
is an isomorphism.
\end{defn}
It is an old theorem \cite{phillipsproper}
that if $X$ is a second countable proper $G$-space then the associated action $(C_0(X),\alpha)$ has full and reduced crossed products equal.
It is folklore that the second-countability hypothesis can be removed --- see the proof of \propref{locally regular} and \remref{proper case}.
We extend this to pointwise proper actions and weaken the countability hypothesis:
\begin{thm}\label{pointwise regular}
If $X$ is a first countable pointwise proper $G$-space,
then the
full and reduced crossed products of the
associated action $(C_0(X),\alpha)$ are equal.
\end{thm}
We need some properties of the ``full = reduced" phenomenon for actions.
First, it is
frequently
inherited by invariant subalgebras:
\begin{lem}\label{faithful}
Let $(A,\alpha)$ and $(B,\beta)$ be actions,
and let $\phi:A\to M(B)$ be an injective $\alpha-\beta$ equivariant homomorphism.
Suppose that the crossed-product homomorphism
\[
\phi\rtimes G:A\rtimes_\alpha G\to M(B\rtimes_\beta G)
\]
is faithful.
If the full and reduced crossed products of $\beta$ are equal,
then the full and reduced crossed products of $\alpha$ are equal.
\end{lem}
\begin{proof}
We have a commutative diagram
\[
\xymatrix@C+30pt{
A\rtimes_\alpha G \ar[r]^-{\phi\rtimes G} \ar[d]_{\Lambda_\alpha}
&M(B\rtimes_\beta G) \ar[d]^{\Lambda_\beta}
\\
A\rtimes_{\alpha,r} G \ar[r]_-{\phi\rtimes_r G}
&M(B\rtimes_{\beta,r} G),
}
\]
and the composition $\Lambda_\beta\circ(\phi\rtimes G)$ is faithful,
and therefore $\Lambda_\alpha$ is faithful.
\end{proof}
Next,
``full = reduced'' is preserved by extensions:
\begin{lem}\label{extension}
Let $(A,\alpha)$ be an action, and let $J$ be a closed invariant ideal of $A$.
If the actions of $G$ on $J$ and on $A/J$ both
have full and reduced crossed products equal,
then the full and reduced crossed products of $\alpha$ are equal.
\end{lem}
\begin{proof}
Let $\phi:J\hookrightarrow A$ be the inclusion map, and let $\psi:A\to A/J$ be the quotient map.
We have a commutative diagram
\[
\xymatrix{
J\rtimes G \ar[r]^-{\phi\rtimes G} \ar[d]_{\Lambda_J}
&A\rtimes G \ar[r]^-{\psi\rtimes G} \ar[d]_{\Lambda_A}
&A/J\rtimes G \ar[d]^{\Lambda_{A/J}}
\\
J\rtimes_r G \ar[r]_-{\phi\rtimes_r G}
&A\rtimes_r G \ar[r]_-{\psi\rtimes_r G}
&A/J\rtimes_r G.
}
\]
The argument is a routine diagram-chase.
The vertical maps are the regular representations, which are surjective,
and moreover $\Lambda_J$ and $\Lambda_{A/J}$ are injective by hypothesis.
Since $J$ is an ideal, the map $\phi\rtimes G$ is an isomorphism onto
the kernel of $\psi\rtimes G$
\cite[Proposition~12]{gre:local}.
Further, since $J$ is an invariant subalgebra, $\phi\rtimes_r G$ is injective.
Let $x$ be in the kernel of $\Lambda_A$.
Then
\[
0=(\psi\rtimes_r G)\circ {\Lambda_A}(x)=\Lambda_{A/J}\circ (\psi\rtimes G)(x),
\]
so $x$ is in the kernel of $\psi\rtimes G$.
Thus $x\in J\rtimes G$, and
\[
0=\Lambda_A\circ (\phi\rtimes G)(x)=(\phi\rtimes_r G)\circ \Lambda_J(x),
\]
so $x=0$.
\end{proof}
Next we show that ``full = reduced'' is preserved by direct sums:
\begin{lem}\label{direct}
Let $\{(A_i,\alpha_i)\}_{i\in I}$ be a family of actions,
and assume that the full and reduced crossed products are equal for every $\alpha_i$.
Then the direct sum action
\[
\left(\bigoplus_{i\in I} A_i,\bigoplus_{i\in I} \alpha_i\right)
\]
also has full and reduced crossed products equal.
\end{lem}
\begin{proof}
By \lemref{extension}, the conclusion holds if $I$ has cardinality 2, and by induction it holds if $I$ is finite.
By \cite[Proposition~12]{gre:local},
we can regard $(\bigoplus_{i\in I}A_i)\rtimes G$
as the inductive limit of the ideals $(\bigoplus_{i\in F}A_i)\rtimes G$ for finite $F\subset I$.
Similarly (but not requiring the reference to \cite{gre:local}),
we can regard $(\bigoplus_{i\in I}A_i)\rtimes_r G$
as the inductive limit of the ideals $(\bigoplus_{i\in F}A_i)\rtimes_r G$.
For every finite $F\subset I$ we have a commutative diagram
\[
\xymatrix{
\left(\bigoplus_{i\in F}A_i\right)\rtimes G \ar@{^(->}[r] \ar[d]_{\Lambda_F}^\simeq
&\left(\bigoplus_{i\in I}A_i\right)\rtimes G \ar[d]^{\Lambda_I}
\\
\left(\bigoplus_{i\in F}A_i\right)\rtimes_r G \ar@{^(->}[r]
&\left(\bigoplus_{i\in I}A_i\right)\rtimes_r G,
}
\]
where the vertical arrows are the regular representations.
Thus $\Lambda_I$ must be an isomorphism,
by properties of inductive limits.
\end{proof}
\begin{cor}\label{separate}
Let $(A,\alpha)$ be an action,
let
$\{(A_i,\alpha_i)\}_{i\in I}$ be a family of actions for which the full and reduced crossed products are equal,
and for each $i$ let $\phi_i:A\to M(A_i)$ be an $\alpha-\alpha_i$ equivariant homomorphism.
Let
\[
\phi:A\to M\left(\bigoplus_{i\in I}A_i\right)
\]
be the associated equivariant homomorphism.
Suppose that $\bigcap_{i\in I} \ker \phi_i=\{0\}$,
and that the crossed-product homomorphism
\[
A\rtimes_\alpha G\to M\left(\biggl(\bigoplus_{i\in I}A_i\biggr)\rtimes_{\alpha_i} G\right)
\]
is faithful.
Then $\alpha$ also has full and reduced crossed products equal.
\end{cor}
\begin{proof}
This follows immediately from Lemmas~\ref{faithful} and \ref{direct}.
\end{proof}
We are almost ready for the proof of \thmref{pointwise regular},
but first we need to recall the notion of quasi-regularity,
and we only need this in the special case of closed orbits:
\begin{defn}[{special case of \cite[Page~221]{gre:local}}]
Let $G$ act on $X$, and assume that all orbits are closed.
Then the associated action of $G$ on $C_0(X)$ is \emph{quasi-regular}
if for every irreducible covariant representation $(\pi,U)$ of $(C_0(X),G)$ there is an orbit $G\cdot x$ such that
\[
\ker \pi=\{f\in C_0(X):f|_{G\cdot x}=0\}.
\]
\end{defn}
In this case, $\pi$ factors through a faithful representation $\rho$ of $C_0(G\cdot x)$ such that
the covariant pair $(\rho,U)$ is
an irreducible representation of the restricted action $(C_0(G\cdot x),\alpha)$.
By \cite[Corollary~19]{gre:local},
the action is quasi-regular
if the orbit space $G\backslash X$ is second countable or
\emph{almost Hausdorff} in the sense that every closed subset contains a dense relative open Hausdorff subset.
Here we will prove a variant of this result:
\begin{prop}\label{quasi}
If a $G$-space $X$ is pointwise proper and
first countable,
then the associated action of $G$ on $C_0(X)$ is quasi-regular.
\end{prop}
We first need a topological property of pointwise proper actions on first countable spaces:
\begin{lem}\label{nbd}
If a $G$-space $X$ is pointwise proper and
first countable,
then each orbit is a countable decreasing intersection of open $G$-invariant sets.
\end{lem}
\begin{proof}
Since orbits are closed, the quotient space $G\backslash X$ is $T_1$.
Since the quotient map is continuous and open, $G\backslash X$ is first countable.
In particular, every point is a countable decreasing intersection of open sets,
and the result follows.
\end{proof}
\begin{rem}
In \lemref{nbd} the first countability assumption could be weakened to: every point in $X$ is a $G_\delta$.
\end{rem}
It seems to us that the proof of \propref{quasi} is clearer if we separate out a special case:
\begin{lem}\label{special}
If a $G$-space $X$ is pointwise proper and
first countable,
and if there is an irreducible covariant representation $(\pi,U)$ of $(C_0(X),G)$
such that $\pi$ is faithful,
then $X$ consists of a single orbit.
\end{lem}
\begin{proof}
We can extend $\pi$ to a representation of the algebra of bounded Borel functions on $X$,
and we let $P$ be the associated spectral measure
(see, e.g., \cite[Theorem~2.5.5]{GM} for a version of the relevant theorem in the nonsecond-countable case; Murphy states the theorem for compact Hausdorff spaces, but it applies equally well to locally compact spaces by passing to the one-point compactification).
Since $(\pi,U)$ is irreducible, for every $G$-invariant Borel set $E$ we have $P(E)$ = 0 or 1.
In particular each orbit has spectral measure 0 or 1, and there can be at most one orbit with measure 1.
Claim: every nonempty $G$-invariant open subset $O$ of $X$ has spectral measure 1.
It suffices to show that $P(O)\ne 0$.
Since $O\ne\varnothing$, we can choose a nonzero $f\in C_0(X)$ supported in $O$.
Then
\[
0\ne \pi(f)=\pi(f\raisebox{2pt}{\ensuremath{\chi}}_O)=\pi(f)P(O),
\]
so $P(O)\ne 0$.
Let $x\in X$. We will show that $X=G\cdot x$.
By \lemref{nbd} we can choose a decreasing sequence $\{O_n\}$ of open $G$-invariant sets with $\bigcap_1^\infty O_n=G\cdot x$.
By the
properties of spectral measures, we have
\[
P(G\cdot x)=\lim_nP(O_n)=1.
\]
Thus every orbit has spectral measure 1, so there can be only one orbit.
\end{proof}
\begin{proof}[Proof of \propref{quasi}]
Let $(\pi,U)$ be an irreducible covariant representation of $(C_0(X),G)$ on a Hilbert space $H$.
Then $\ker\pi$ is a $G$-invariant ideal of $C_0(X)$, so there is a closed $G$-invariant subset $Y$ of $X$ such that
\[
\ker\pi=\{f\in C_0(X):f|_Y=0\}.
\]
We will show that $Y$ consists of a single orbit.
The restriction map $f\mapsto f|_Y$ is a $G$-equivariant homomorphism of $C_0(X)$ to $C_0(Y)$,
and $\ker\pi=C_0(X\setminus Y)$,
so $\pi$ factors through a faithful representation $\rho$ of $C_0(Y)$
such that $(\rho,U)$ is an irreducible covariant representation of $(C_0(Y),G)$.
Then $Y$ is a single orbit, by \lemref{special}.
\end{proof}
\begin{proof}[Proof of \thmref{pointwise regular}]
For each $x\in X$,
the orbit $G\cdot x$ is closed,
the isotropy subgroup $G_x$ is compact,
and the canonical bijection $G/G_x\to G\cdot x$ is an equivariant homeomorphism.
Thus $G_x$ is in particular amenable, so
it follows from the above and
\cite[Corollary~4.3]{qs:regularity}
(see also \cite[Theorem~3.15]{K})
the associated action of $G$ on $C_0(G\cdot x)$ has full and reduced crossed products equal.
The restriction map $\phi_x:C_0(X)\to C_0(G\cdot x)$
is equivariant,
and
we get an equivariant injective homomorphism
\[
\phi:C_0(X)\to M\left(\bigoplus_{x\in X}C_0(G\cdot x)\right).
\]
By \propref{quasi} the action of $G$ on $C_0(X)$ is quasi-regular, so every irreducible covariant representation of $(C_0(X),G)$ factors through a representation of $(C_0(G\cdot x),G)$ for some orbit $G\cdot x$.
It follows that
the crossed-product homomorphism
\[
\phi\rtimes G:C_0(X)\rtimes G\to M\left(\biggl(\bigoplus_{x\in X}C_0(G\cdot x)\biggr)\rtimes G\right)
\]
is faithful.
Therefore the theorem follows from \corref{separate}.
\end{proof}
The above strategy
can also be used to prove the following folklore result,
which is a mild extension of Phillips' full-equals-reduced theorem.
Actually, we could not find the following result explicitly recorded in the literature, but it seems to us that it must have been noticed before.
\begin{prop}\label{locally regular}
If a $G$-space $X$ is locally proper then
the associated action $(C_0(X),\alpha)$ has full and reduced crossed products equal.
\end{prop}
Note that there is no countability hypothesis on $X$.
We need the following, which
will play a role similar to that of \corref{separate} in the pointwise proper case:
\begin{cor}\label{ideals}
Let $(A,\alpha)$ be an action, and let $\{J_i\}_{i\in I}$ be a family of $G$-invariant ideals that densely span $A$.
If for every $i$ the restriction of the action to $J_i$ has full and reduced crossed products equal, then the action on $A$ has the same property.
\end{cor}
\begin{proof}
For each $i$ let $\alpha_i=\alpha|_{J_i}$,
let
$\phi_i:A\to M(J_i)$ be the $\alpha-\alpha_i$ equivariant homomorphism induced by the $A$-bimodule structure on $J_i$,
and let $\phi:A\to M(\bigoplus_{i\in I}J_i)$ be the associated equivariant homomorphism,
Since $A=\clspn_{i\in I}J_i$, we have $\bigcap_{i\in I}\ker\phi_i=\{0\}$.
Thus, by \corref{separate} we only need to show that
\[
\phi\rtimes G:A\rtimes_\alpha G\to M\left(\biggl(\bigoplus_{i\in I}J_i\biggr)\rtimes_{\alpha_i} G\right)
\]
is faithful.
Suppose that $\ker (\phi\rtimes G)\ne \{0\}$.
The ideals $J_i\rtimes_{\alpha_i} G$ densely span $A\rtimes_\alpha G$, since
the $J_i$'s densely span $A$.
Thus we can find $i\in J$ such that
\[
\{0\}\ne \ker(\phi\rtimes G)\cap (J_i\rtimes_{\alpha_i} G)=\ker(\phi|_{J_i}\rtimes G).
\]
But $\phi|_{J_i}\rtimes G$ is faithful since $\phi|_{J_i}$ is faithful and $J_i$ is a $G$-invariant ideal, so we have a contradiction.
\end{proof}
\begin{proof}[Proof of \propref{locally regular}]
First, if the $G$-space $X$ is actually proper,
then $G\backslash X$ is Hausdorff, so by \cite[Corollary~19]{gre:local} the action of $G$ on $C_0(X)$ is quasi-regular, so the conclusion follows as in the proof of \propref{pointwise regular}.
In the general case, $X$ is a union of open $G$-invariant subsets $U_i$, on each of which $G$ acts properly.
Then $C_0(X)$ is densely spanned by the ideals $C_0(U_i)$,
so by properness the associated actions $\alpha_i$ have full and reduced crossed products equal,
and hence the conclusion follows from \lemref{ideals}.
\end{proof}
\begin{rem}\label{proper case}
In the above proof we appealed to \cite[Corollary~19]{gre:local},
whose proof involved dense points in irreducible closed sets.
In the spirit of the techniques of the current paper, we offer an alternative argument: assume that $X$ is a proper $G$-space. To see that the action is quasi-regular,
as in the proof of \propref{quasi} we can assume without loss of generality that there is an irreducible covariant representation $(\pi,U)$ of $(C_0(X),G)$ such that $\pi$ is faithful.
We must show that $X$ consists of a single $G$-orbit.
Suppose $G\cdot x$ and $G\cdot y$ are distinct orbits in $X$.
By properness, the quotient space $G\backslash X$ is Hausdorff, so we can find disjoint open
neighborhoods of $G\cdot x$ and $G\cdot y$ in $G\backslash X$,
and hence nonempty disjoint open $G$-invariant sets $U$ and $V$ in $X$.
But, as in the proof of \lemref{special}, letting $P$ denote the spectral measure associated to the representation $\pi$ of $C_0(X)$, every nonempty $G$-invariant open subset $O$ of $X$ has $P(O)=1$.
Since
we cannot have two disjoint open sets with spectral measure 1,
we have a contradiction.
\end{rem}
The above methods quickly lead to another property of the crossed product.
Recall that a $C^*$-algebra is called \emph{CCR}, or \emph{liminal} \cite[Definition~4.2.1]{dix}, if every irreducible representation is by compacts.
In the second countable case, the following result is contained in \cite[Proposition~7.31]{danacrossed}.
\begin{prop}\label{ccr}
Let $X$ be a $G$-space.
In either of the following two situations, the crossed product $C_0(X)\rtimes G$ is CCR:
\begin{enumerate}
\item the action of $G$ is locally proper;
\item the action is pointwise proper and $X$ is first countable.
\end{enumerate}
\end{prop}
\begin{proof}
(1)
If the $G$-space $X$ is actually proper, then this is well-known.
To illustrate how the above methods apply, we give the following argument.
We have seen above that the action is quasi-regular,
and hence for every irreducible covariant representation $(\pi,U)$ of $(C_0(X),G)$
factors through an irreducible representation of the restriction of the action
to $(C_0(G\cdot x),G)$ for some $x\in X$.
The $G$-spaces $G\cdot x$ and $G/G_x$ are isomorphic,
and $C_0(G/G_x)\rtimes G$ is Morita equivalent to $C^*(G_x)$
by
Rieffel's version of Mackey's Imprimitivity Theorem \cite[Section~7]{rie:induced}.
Since the isotropy subgroup $G_x$ is compact, $C^*(G_x)$ is CCR, and hence the image of the integrated form $\rho\times U$, which equals the image of $\pi\times U$, is the algebra of compact operators.
In the general case, $X$ is a union of open $G$-invariant proper $G$-spaces $U_i$,
so $C_0(X)\rtimes G$ is the closed span of the CCR ideals $C_0(U_i)\rtimes G$.
Since every $C^*$-algebra has a largest CCR ideal \cite[Proposition~4.2.6]{dix},
$C_0(X)\rtimes G$ must be CCR.
(2)
By \propref{quasi} the action is quasi-regular,
and it follows as in part (1) that $C_0(X)\rtimes G$ is CCR.
\end{proof}
\begin{rem}
As remarked in
\cite[Example~2.7 (3)]{delaroche},
it follows from
\cite[Corollary~2.1.17]{ADR}
that if an action of $G$ on $X$ is proper then the action is amenable
(a condition involving approximation by positive-definite functions).
By \cite[Theorem~5.3]{delaroche}, if a $G$-space $X$ is amenable then the associated action $\alpha$ on $C_0(X)$ has full and reduced crossed products equal.
This raises a question:
is every pointwise proper action amenable?
It seems that amenability of the $G$-space is closely related to equality of full and reduced crossed products: by \cite[Theorem~3.3]{matsumura}, for an action of a discrete exact group $G$ on a compact space $X$,
if $\alpha$ has full and reduced crossed products equal then the action is amenable.
Unfortunately, this is of no help for our question, because a noncompact group cannot act pointwise properly on a compact space.
\end{rem}
\section{Properness conditions for coactions}\label{coactions}
We will now dualize the properness properties of Definitions~\ref{s-proper} and \ref{w-proper}.
To motivate how this will go, we pause to recall some basic facts regarding $C^*$-tensor products, commutative $C^*$-algebras, and actions.
For locally compact Hausdorff spaces $X,Y$ we have
the standard identifications
\[
C_0(X\times Y)=C_0(X)\otimes C_0(Y)
\]
and
\[
C_b(X)=M(C_0(X)).
\]
For a $C^*$-algebra $A$ we have
\[
A\otimes C_0(G)=C_0(G,A)
\]
and
\[
M(A\otimes C_0(G))=C_b(G,M^\beta(A)),
\]
where
$M^\beta(A)$ denotes the multiplier algebra
$M(A)$
with
the strict topology.
For an action $(A,\alpha)$ we have a homomorphism
\[
\widetilde\alpha:A\to M(A\otimes C_0(G))
\]
given by
\[
\widetilde\alpha(f)(s,x)=\widetilde\alpha(f)(s)(x)=f(sx)=\alpha_{s^{-1}}(f)(x).
\]
In fact, the image of $\widetilde\alpha$ lies in the $C^*$-subalgebra
$\widetilde M(A\otimes C_0(G))$,
where for any $C^*$-algebras $A$ and $D$
\[
\widetilde M(A\otimes D):=\{m\in M(A\otimes D):m(1\otimes D)\cup (1\otimes D)m\subset A\otimes D\}.
\]
Using the above facts,
\corref{proper action} can be restated as follows:
\begin{lem}
An action $(A,\alpha)$ is
s-proper if and only if
\[
\widetilde\alpha(A)(A\otimes 1_{M(C_0(G))})\subset A\otimes C_0(G),
\]
and is
w-proper if and only if
for all $\omega\in A^*$,
\[
(\omega\otimes\text{\textup{id}})\circ\widetilde\alpha(A)\subset C_0(G).
\]
\end{lem}
Now consider a coaction $(A,\delta)$ of $G$.
The main difference from actions is that the commutative $C^*$-algebra $C_0(G)$ is replaced by $C^*(G)$.
Here we will use the standard conventions for tensor products and coactions (see, e.g., \cite[Appendix~A]{enchilada},
in particular, the coaction is a homomorphism
\[
\delta:A\to \widetilde M(A\otimes C^*(G)).
\]
\begin{defn}\label{proper coaction}
A coaction $(A,\delta)$ is \emph{s-proper} if
\[
\delta(A)\bigl(A\otimes 1_{M(C^*(G))}\bigr)\subset A\otimes C^*(G),
\]
and is \emph{w-proper} if
for all $\omega\in A^*$ we have
\[
(\omega\otimes\text{\textup{id}})\circ\delta(A)\subset C^*(G).
\]
\end{defn}
\begin{rem}
In \cite[Definition~5.1]{exotic} we introduced
the above properness conditions,
but in that paper we used the term
\emph{proper coaction} for the above s-proper coaction,
and
\emph{slice proper coaction} for the above w-proper coaction (because it involves the slice map $\omega\otimes\text{\textup{id}}$).
After we submitted \cite{exotic}, we learned that
Ellwood had defined properness more generally for coactions of Hopf $C^*$-algebras
\cite[Definition~2.4]{ellwood}.
Indeed,
\propref{proper C0(X)}
is essentially \cite[Theorem~2.9(b)]{ellwood}.
\defnref{proper coaction} should also be compared with
Condition~(A1) in \cite[Section~4.1]{Goswami-Kuku}, which concerns discrete quantum groups and involves the algebraic tensor product.
\end{rem}
\begin{rem}
An action on $C_0(X)$ can be w-proper without being s-proper,
and a fortiori a coaction can be w-proper without being s-proper, even for $G$ abelian.
\end{rem}
\begin{rem}
(1)
Just as every action of a compact group is s-proper, every coaction of a discrete group is s-proper, because then we in fact have $\delta(A)\subset A\otimes C^*(G)$.
(2)
For any locally compact group $G$ the canonical coaction $\delta_G$ on $C^*(G)$
given by the comultiplication
is s-proper, because it is symmetric in the sense that
\[
\delta_G=\Sigma\circ\delta_G,
\]
where $\Sigma$ is the flip automorphism on $C^*(G)\otimes C^*(G)$.
\end{rem}
If $(A,\delta)$ is a coaction, then
$A$ gets a Banach module structure over
the Fourier-Stieltjes algebra $B(G)=C^*(G)^*$ by
\[
f\cdot a=(\text{\textup{id}}\otimes f)\circ\delta(a)\righttext{for}f\in B(G),a\in A.
\]
In \cite[Lemma~5.2]{exotic} we proved the following dual analogue of \lemref{module noncom}
\begin{lem}\label{module coaction}
A coaction
$(A,\delta)$ is w-proper if and only if for all $a\in A$ the map $f\mapsto f\cdot a$ is
weak*-to-weakly
continuous.
\end{lem}
\begin{proof}
See \cite[Lemma~5.2]{exotic}.
\end{proof}
s-properness and w-properness are both preserved by morphisms.
For w-properness this is proved in \cite[Proposition~5.3]{exotic}, and here it is for s-properness:
\begin{prop}\label{morphismco}
Let $\phi:A\to M(B)$ be a nondegenerate homomorphism that is equivariant for coactions $\delta$ and $\epsilon$, respectively.
If $\delta$ is
s-proper
then $\epsilon$ has the same property.
\end{prop}
\begin{proof}
We have
\begin{align*}
(B\otimes 1)\epsilon(B)
&=(B\phi(A)\otimes 1)(\phi\otimes\text{\textup{id}})(\delta(A))\epsilon(B)
\\&=(B\otimes 1)(\phi(A)\otimes 1)(\phi\otimes\text{\textup{id}})(\delta(A))\epsilon(B)
\\&=(B\otimes 1)(\phi\otimes\text{\textup{id}})\bigl((A\otimes 1)\delta(A)\bigr)\epsilon(B)
\\&\subset (B\otimes 1)(\phi\otimes\text{\textup{id}})(A\otimes C^*(G))\epsilon(B)
\\&=(B\otimes C^*(G))\epsilon(B)
\\&\subset B\otimes C^*(G)).
\qedhere
\end{align*}
\end{proof}
\begin{cor}\label{dual coaction}
Every dual coaction is s-proper.
\end{cor}
\begin{proof}
If $(A,\alpha)$ is an action, then the canonical nondegenerate homomorphism $i_G:C^*(G)\to M(A\rtimes_\alpha G)$ is $\delta_G-\widehat\alpha$ equivariant,
where $\delta_G$ is the canonical coaction on $C^*(G)$ given by the comultiplication.
Thus $\widehat\alpha$ is s-proper since $\delta_G$ is.
\end{proof}
Recall that if $(A,\delta)$ is a coaction then the
\emph{spectral subspaces} $\{A_s\}_{s\in G}$ are given by
\[
A_s=\{a\in M(A):\delta(a)=a\otimes s\},
\]
and the
\emph{fixed-point algebra} is $A^\delta=A_e$.
\begin{prop}
Suppose $A\cap A^\delta\ne \{0\}$.
Then the following are equivalent:
\begin{enumerate}
\item
$\delta$ is s-proper;
\item
$\delta$ is w-proper;
\item
$G$ is discrete.
\end{enumerate}
\end{prop}
\begin{proof}
We know (1) implies (2) and (3) implies (1).
Assume (2), and let $a_e\in A\cap A^\delta$ be nonzero.
Then
\begin{align*}
f\mapsto
&f\cdot a_e
=(\text{\textup{id}}\otimes f)\circ\delta(a_e)
=(\text{\textup{id}}\otimes f)(a_e\otimes 1)
=f(e)a_e
\end{align*}
is weak*-weak continuous from $B(G)$ to $A$,
so $f\mapsto f(e)$ is a weak* continuous linear functional on $B(G)$,
which implies
$e\in C^*(G)$,
and hence
$G$ is discrete.
\end{proof}
\begin{rem}
Of course, the above proposition applies if $A$ is unital.
Also note that when $G$ is nondiscrete a coaction $(A,\delta)$ can be s-proper and still have nonzero spectral subspaces $A_s$ (and hence nontrivial fixed-point algebra $A^\delta$, but these will be subspaces in $M(A)$ that intersect $A$ trivially.
\end{rem}
For the next lemma, recall that if $(A,\delta)$ is a coaction, then
a projection $p\in M(A)$ is called \emph{$\delta$-invariant} if
$p\in A^\delta$,
and in this case $\delta$ restricts to a coaction $\delta_p$ on the corner $pAp$:
\[
\delta_p(pap)=(p\otimes 1)\delta(a)(p\otimes 1)\in M(pAp\otimes C^*(G))\righttext{for}a\in A.
\]
\begin{lem}\label{corner}
Let $(A,\delta)$ be a coaction, and let $p$ be a $\delta$-invariant projection in $M(A)$.
If $(A,\delta)$ is s-proper, then so is the corner coaction $(pAp,\delta_p)$ defined above.
\end{lem}
\begin{proof}
This is a routine computation:
\begin{align*}
\delta_p(pAp)(pAp\otimes 1)
&\subset (p\otimes 1)\delta(A)(A\otimes 1)(p\otimes 1)
\\&\subset (p\otimes 1)(A\otimes C^*(G))(p\otimes 1)
\\&=pAp\otimes C^*(G).
\qedhere
\end{align*}
\end{proof}
For the
definitions of normalization and maximalization,
we refer to
\cite[Appendix~A.7]{enchilada}
and
\cite{ekq}.
Normalizations
and maximalizations
always exist, and are unique up to equivariant isomorphism.
\begin{prop}
For any coaction $(A,\delta)$, the following are equivalent:
\begin{enumerate}
\item $(A,\delta)$ is s-proper;
\item The normalization $(A^n,\delta^n)$ is s-proper;
\item The maximalization $(A^m,\delta^m)$ is s-proper.
\end{enumerate}
\end{prop}
\begin{proof}
It follows from \propref{morphism}
that (1) implies (2) and (3) implies (1),
and a careful examination of
the construction of the maximalization in \cite{ekq}
(particularly
Lemma~3.6
and
the proof of Theorem~3.3
in that paper)
shows that
(2) implies (3).
\end{proof}
\begin{rem}
In case the above proof seems overly fussy, note that it would not be enough to observe that
the double-dual coaction $\widehat{\widehat\delta}$ is automatically s-proper
and the maximalization $\delta^m$ is Morita equivalent to $\widehat{\widehat\delta}$,
because s-properness is \emph{not} preserved by Morita equivalence ---
otherwise every coaction of an amenable group would be s-proper!
\end{rem}
Recall from \cite[Proposition~3.1]{kmqw1} that if $\AA\to G$ is a Fell bundle then there is a coaction $\delta_\AA$ of $G$ on the (full) bundle algebra $C^*(\AA)$.
\begin{prop}\label{fell proper}
Let $\AA\to G$ be a Fell bundle. Then the coaction $(C^*(\AA),\delta_\AA)$ is s-proper.
\end{prop}
\begin{proof}
We must show that for all $a,b\in C^*(\AA)$ we have $\delta(a)(b\otimes 1)\in C^*(\AA)\otimes C^*(G)$,
and by density and nondegeneracy it suffices to take $a\in \Gamma_c(\AA)$ and $b$ of the form $f\cdot b$ for $f\in A(G)\cap C_c(G)$:
\begin{align*}
\delta(a)(f\cdot b\otimes 1)
&=\int_G \bigl(a(t)f\cdot b\otimes t\bigr)\,dt
\\&=\int_G \bigl(a(t)b\otimes tf\bigr)\,dt\righttext{(justified below)}
\\&\in C^*(\AA)\otimes C^*(G),
\end{align*}
because the integrand
\[
t\mapsto a(t)b\otimes tf
\]
is in $C_c(G,C^*(\AA)\otimes C^*(G))$.
In the above computation we used the equality
\[
a(t)f\cdot b\otimes t=a(t)b\otimes tf\righttext{for all}t\in G,
\]
which we justify as follows:
computing inside $M(C^*(\AA)\otimes C^*(G))$, we have
\begin{align*}
a(t)f\cdot b\otimes t
&=\bigl(a(t)\otimes t\bigr)\bigl(f\cdot b\otimes 1\bigr)
\\&=\bigl(a(t)\otimes t\bigr)\bigl(b\otimes f\bigr)\righttext{(justified below)}
\\&=a(t)b\otimes tf,
\end{align*}
where we must now justify the equality $f\cdot b\otimes 1=b\otimes f$:
both sides can be regarded as compactly supported strictly continuous functions from $G$ to $M(C^*(\AA)\otimes C^*(G))$, and for all $s\in G$ we have
\begin{align*}
(f\cdot b\otimes 1)(s)
&=(f\cdot b)(s)\otimes 1
\\&=f(s)b(s)\otimes 1
\\&=b(s)\otimes f(s)\righttext{(since $f(s)\in\mathbb C$)}
\\&=(b\otimes f)(s).
\qedhere
\end{align*}
\end{proof}
\begin{rem}
Let $\AA$ be a Fell bundle over $G$, and let
\[
\delta_\AA^r=(\text{\textup{id}}\otimes\lambda)\circ\delta_\AA:C^*(\AA)\to M(C^*(\AA)\otimes C^*_r(G))
\]
be the reduction of the coaction $\delta_\AA$.
\cite[Theorem~3.10]{BussCoaction} shows that $\delta_\AA^r$ is \emph{integrable} in the sense that the set of positive elements $a$ in $A$ for which $\delta_\AA^r(a)$ is in the domain of the operator-valued weight $\text{\textup{id}}\otimes\varphi$ is dense in $A^+$, where $\varphi$ is the Plancherel weight on $C^*_r(G)$.
\end{rem}
\corref{dual integrable} below is a dual analogue of
\corref{integrable then proper} (1).
To explain the terminology, we recall a few things from Buss' thesis \cite{bussthesis}.
Buss worked with reduced coactions, but as he points out in \cite[Remark~2.6.1 (4)]{bussthesis},
the theory carries over to full coactions by considering the reductions of the coactions.
Throughout, $(A,\delta)$ is a coaction of $G$.
Let $\varphi$ be the Plancherel weight on $C^*(G)$,
let $\mathcal M_\varphi^+=\{c\in C^*(G)^+:\varphi(c)<\infty\}$,
$\mathcal N_\varphi=\{c\in C^*(G):c^*c\in \mathcal M_\varphi\}$,
and $\mathcal M_\varphi=\spn \mathcal M_\varphi^+$,
so that $\mathcal M_\varphi^+$ is a hereditary cone in $C^*(G)$,
and coincides with both $\mathcal M_\varphi\cap C^*(G)^+$
and $\spn \mathcal N_\varphi^*\mathcal N_\varphi$,
and $\varphi$ extends uniquely to a linear functional on $\mathcal M_\varphi$.
Let $\text{\textup{id}}\otimes\varphi$ denote the associated $M(A)$-valued weight on $A\otimes C^*(G)$,
with associated objects
$\mathcal M_{\text{\textup{id}}\otimes\varphi}^+$,
$\mathcal N_{\text{\textup{id}}\otimes\varphi}$,
and $\mathcal M_{\text{\textup{id}}\otimes\varphi}$,
and characterized as follows:
for $x\in (A\otimes C^*(G))^+$ we have
$x\in \mathcal M_{\text{\textup{id}}\otimes \varphi}^+$ if and only if
there exists $a\in M(A)^+$ such that
\[
\theta(a)=(\text{\textup{id}}\otimes\varphi)\bigl((\theta\otimes\text{\textup{id}})(x)\bigr)
\righttext{for all}\theta\in A^{*+},
\]
in which case $(\text{\textup{id}}\otimes\varphi)(x)=a$.
We have $(\text{\textup{id}}\otimes\varphi)(a\otimes c)=\varphi(c)a$ for all $a\in A$ and $c\in \mathcal M_\varphi$.
Let $\Lambda:\mathcal N_\varphi\to L^2(G)$ be the canonical embedding associated to the GNS construction for $\varphi$, so that
$\Lambda(bc)=\lambda(b)\Lambda(c)$ for all $b\in C^*(G)$ and $c\in \mathcal N_\varphi$.
Let $\text{\textup{id}}\otimes\Lambda:\mathcal N_{\text{\textup{id}}\otimes\varphi}\to M(A\otimes L^2(G))=\mathcal L(A,A\otimes L^2(G))$
be the map associated to the KSGNS construction for $\text{\textup{id}}\otimes\varphi$,
characterized by
\[
(\text{\textup{id}}\otimes\Lambda)(x)^*\bigl(a\otimes\Lambda(c)\bigr)=(\text{\textup{id}}\otimes\varphi)\bigl(x^*(a\otimes c)\bigr)
\]
for all $x\in \mathcal N_{\text{\textup{id}}\otimes\varphi}$, $a\in A$, and $c\in \mathcal N_\varphi$.
We have $(\text{\textup{id}}\otimes\Lambda)(a\otimes c)=a\otimes \Lambda(c)$
for all $a\in A$ and $c\in \mathcal N_\varphi$,
and
\[
(\text{\textup{id}}\otimes\Lambda)(xy)=(\text{\textup{id}}\otimes \lambda)(x)(\text{\textup{id}}\otimes\Lambda)(y)
\]
for all $x\in M(A\otimes C^*(G))$ and $y\in \mathcal N_{\text{\textup{id}}\otimes\Lambda}$.
The weight $\varphi$ extends canonically to $M(C^*(G))$,
and the associated objects are denoted by
$\bar\mathcal M_\varphi^+$, $\bar\mathcal N_\varphi$, and $\bar\mathcal M_\varphi$.
Similarly for the canonical extension of $\text{\textup{id}}\otimes\varphi$ to $M(A\otimes C^*(G))$,
$\bar\mathcal M_{\text{\textup{id}}\otimes\varphi}^+$, etc.
Let
\[
A_\text{si}=\{a\in A:\delta(aa^*)\in \bar\mathcal M_{\text{\textup{id}}\otimes\varphi}^+\}.
\]
Then the coaction $\delta$ is
\emph{square-integrable} if $A_\text{si}$ is dense in $A$.
For $a\in A_\text{si}$ define
\[
\<\<a|\in M(A\otimes L^2(G))=\mathcal L(A,A\otimes L^2(G))
\]
by
\[
\<\<a|(b)=(\text{\textup{id}}\otimes\Lambda)\bigl(\delta(a)^*(b\otimes 1)\bigr),
\]
then define $|a\>\>=\<\<a|^*\in \mathcal L(A\otimes L^2(G),A)$,
and for $a,b\in A_\text{si}$ define
\[
\<\<a|b\>\>=\<\<a|\circ |b\>\>\in \mathcal L(A\otimes L^2(G)).
\]
Then $(A,\delta)$ is \emph{continuously square-integrable} if there is a dense subspace $\mathcal R\subset A_\text{si}$ such that
\[
\<\<a|b\>\>\in A\rtimes_\delta G\subset \mathcal L(A\otimes L^2(G))
\righttext{for all}a,b\in A_\text{si}.
\]
\begin{cor}\label{dual integrable}
Every continuously square-integrable coaction is s-proper.
\end{cor}
\begin{proof}
Let $(A,\delta)$ be a continuously square-integrable coaction.
\cite[Section~6.8 and Proposition~6.9.4]{bussthesis}
gives a
Fell bundle $\AA$ over $G$
and
a $\delta_\AA-\delta$ equivariant surjective homomorphism $\kappa:C^*(\AA)\to A$.
By \propref{morphism}, every quotient of an s-proper coaction is s-proper, so the corollary follows from \propref{fell proper}.
\end{proof}
\begin{rem}
Buss states on page 10 of \cite{bussthesis} that it is an open problem whether $\delta_\AA$ is maximal,
but in the second-countable case this is now known to be true \cite[Theorem~8.1]{kmqw1}.
\end{rem}
\section{$E$-crossed products}\label{E}
For an action $(B,\alpha)$, there are numerous crossed-product $C^*$-algebras.
The largest is the (full) crossed product $B\rtimes_\alpha G$ and the smallest is the reduced crossed product $B\rtimes_{\alpha,r} G$.
But there are frequently many ``exotic'' crossed products in between,
i.e., quotients $(B\rtimes_\alpha G)/J$ where $J$ is a nonzero ideal properly contained in the kernel of the regular representation $\Lambda$.
In \cite{graded}, inspired by work of Brown and Guentner \cite{brogue}, we introduced a tool that produces many (but not all) of these exotica.
Our strategy is to base everything on ``interesting'' $C^*$-algebras $C^*(G)/I$ between $C^*(G)$ and $C^*_r(G)$.
We call a closed ideal $I$ of $C^*(G)$ \emph{small} if
it is contained in the kernel of the regular representation $\lambda$
and is \emph{$\delta_G$-invariant},
i.e.,
the coaction $\delta_G$ descends to a coaction on
$C^*(G)/I$.
In \cite[Corollary~3.13]{graded} we proved that
$I$ is small
if and only if the annihilator $E=I^\perp$ in $B(G)$ is an ideal,
which will then be \emph{large} in the sense that it is nonzero, weak* closed, and $G$-invariant,
where $B(G)$ is given the $G$-bimodule structure
\[
(s\cdot f\cdot t)(u)=f(tus)\righttext{for}f\in B(G),s,t,u\in G.
\]
Large ideals automatically contain the reduced Fourier-Stieltjes algebra $B_r(G)=C^*_r(G)^*$ \cite[Lemma~3.14]{graded},
and the map $E\mapsto {}\ann E$ gives a bijection between the large ideals of $B(G)$ and the small ideals $I$ of $C^*(G)$.
For a large ideal $E$ the quotient map
\[
q_E:C^*(G)\to C^*_E(G):=C^*(G)/{}\ann E
\]
is equivariant for $\delta_G$ and a coaction $\delta^E_G$.
\begin{ex}
$E=\bar{B(G)\cap C_0(G)}$ is a large ideal, and if $G$ is discrete then $G$ has the Haagerup property if and only if $E=B(G)$ \cite[Corollary~3.5]{brogue}.
\end{ex}
\begin{ex}
For $1\le p\le \infty$, $E^p:=\bar{B(G)\cap L^p(G)}$ is a large ideal.
Of course $E^\infty=B(G)$.
For $p\le 2$ we have $E^p=B_r(G)$ \cite[Proposition~4.2]{graded} (and \cite[Proposition~2.11]{brogue} for discrete $G$).
If $G=\mathbb F_n$ for $n>1$,
it has been attributed to Okayasu \cite{okayasu} and (independently) to Higson and Ozawa (see \cite[Remark~4.5]{brogue}) that
for $2\le p<\infty$ the ideals $E^p$ are all distinct.
\end{ex}
Given an action $(B,\alpha)$,
we use large ideals to produce exotic crossed products by involving the dual coaction $\widehat\alpha$ on $B\rtimes_\alpha G$.
As in \cite{exotic}, the process is most cleanly expressed in terms of an abstract coaction $(A,\delta)$.
An ideal $J$ of $A$ is called \emph{$\delta$-invariant} if $\delta$ descends to a coaction on the quotient $A/J$.
We call an ideal $J$ \emph{small} if it is invariant and contained in the kernel of $j_A$, where $(j_A,j_G)$ is the canonical covariant homomorphism of $(A,C_0(G))$ into the multiplier algebra of the crossed product $A\rtimes_\delta G$.
For the coaction $(C^*(G),\delta_G)$, this is consistent with the above notion of small ideals of $C^*(G)$.
Recall that $A$ gets a $B(G)$-module structure by
\[
f\cdot a=(\text{\textup{id}}\otimes f)\circ\delta(a)\righttext{for}f\in B(G),a\in A.
\]
For any large ideal $E$ of $B(G)$,
\[
\mathcal J(E)=\{a\in A:f\cdot a=0\text{ for all }f\in E\}
\]
is a small ideal of $A$ \cite[Observation~3.10]{exotic}.
For a dual coaction $(B\rtimes_\alpha G,\widehat\alpha)$,
we call the quotient
\[
B\rtimes_{\alpha,E} G:=(B\rtimes_\alpha G)/\mathcal J(E)
\]
an \emph{$E$-crossed product}.
In the other direction, for any small ideal $J$ of $A$,
\[
\mathcal E(J)=\{f\in B(G):(s\cdot f\cdot t)\cdot a=0\text{ for all }a\in J,s,t\in G\}
\]
is an ideal of $B(G)$,
which is $G$-invariant by construction,
and which will be weak*-closed if the coaction is w-proper.
The following is \cite[Lemma~6.4]{exotic}:
\begin{lem}\label{galois ideal}
For any w-proper coaction $(A,\delta)$,
the above maps
$\mathcal J$ and $\mathcal E$ form a Galois correspondence
between the large ideals of $B(G)$ and
the
small ideals of $A$.
\end{lem}
By \emph{Galois correspondence} we mean that
$\mathcal J$ and $\mathcal E$ reverse inclusions,
$E\subset \mathcal E(\mathcal J(E))$ for every large ideal $E$ of $B(G)$,
and
$J\subset \mathcal J(\mathcal E(J))$ for every small ideal $J$ of $A$.
Since every dual coaction is s-proper, and hence w-proper, \lemref{galois ideal}
is applicable to $(B\rtimes_\alpha G,\widehat\alpha)$ for any action $(B,\alpha)$.
In \cite[Theorem~6.10]{exotic} we used this Galois correspondence to exhibit examples of small ideals $J$ that are not of the form $\mathcal J(E)$ for any large ideal $E$.
Buss and Echterhoff \cite[Example~5.3]{BusEch} have given examples that are better in the sense that the coaction $(A,\delta)$ is of the form $(B\rtimes_\alpha G,\widehat\alpha)$.
Consequently, there are exotic crossed products that are not $E$-crossed products for any large ideal $E$.
However, the real goal is not to look at exotic crossed products one at a time, but rather all at once:
In \cite{bgwexact}, Baum, Guentner, and Willett
define a \emph{crossed-product} as a functor
$(B,\alpha)\mapsto B\rtimes_{\alpha,\tau} G$,
from the category of actions to the category of $C^*$-algebras,
equipped with natural transformations
\[
\xymatrix{
B\rtimes_\alpha G \ar[r] \ar[d]
&B\rtimes_{\alpha,\tau} G \ar[dl]
\\
B\rtimes_{\alpha,r} G,
}
\]
where the vertical arrow is the regular representation,
such that the horizontal arrow is surjective.
For a large ideal $E$ of $B(G)$, the $E$-crossed product
\[
(B,\alpha)\mapsto B\rtimes_{\alpha,E} G
\]
gives a crossed-product functor in the sense of \cite{bgwexact}.
\cite{bgwexact} defines a crossed-product functor $\tau$ to be
\emph{exact}
if for every short exact sequence
\[
0\to
(B_1,\alpha_1)\to
(B_2,\alpha_2)\to
(B_3,\alpha_3)\to
0
\]
of actions the corresponding sequence of $C^*$-algebras
\[
0\to
B_1\rtimes_{\alpha_1,\tau} G\to
B_2\rtimes_{\alpha_2,\tau} G\to
B_3\rtimes_{\alpha_3,\tau} G\to
0
\]
is exact,
and
\emph{Morita compatible}
if for every action $(B,\alpha)$ the canonical \emph{untwisting isomorphism}
\[
(B\otimes\mathcal K_G)\rtimes G\simeq (A\rtimes G)\otimes \mathcal K_G,
\]
where $\mathcal K_G$ denotes the compact operators on $\bigoplus_{n=1}^\infty L^2(G)$,
descends to an isomorphism
\[
(B\otimes\mathcal K_G)\rtimes_\tau G\simeq (A\rtimes_\tau G)\otimes \mathcal K_G
\]
of $\tau$-crossed products.
\cite[Theorem~3.8]{bgwexact}
(with an assist from Kirchberg)
shows that there is a unique minimal exact and Morita compatible crossed product,
and \cite{bgwexact} uses this to give a promising reformulation of the Baum-Connes conjecture.
If $E$ is any large ideal of $B(G)$,
the $E$-crossed product
\[
(B,\alpha)\mapsto B\rtimes_{\alpha,E} G
\]
is a crossed-product functor in the sense of \cite{bgwexact},
and it is automatically Morita compatible \cite[Lemma~A.5]{bgwexact}.
It is an open problem whether the minimal functor of \cite{bgwexact} is an $E$-crossed product for some large ideal $E$.
The counterexamples of \cite{BusEch} do not necessarily give a negative answer, because it is unknown whether they fit into a crossed-product functor.
The state of the art regarding $E$-crossed products is depressingly meager at this early stage ---
we do not even know any examples
other than $B(G)$ itself
of large ideals $E$ for which the $E$-crossed-product functor is exact for all $G$!
Of course, by definition the $B_r(G)$-crossed product is exact for an \emph{exact} group $G$
(where $B_r(G)=C^*_r(G)^*$ denotes the reduced Fourier-Stieltjes algebra).
But nonexact groups are quite mysterious.
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\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{%
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
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{
"timestamp": "2015-04-15T02:02:58",
"yymm": "1504",
"arxiv_id": "1504.03394",
"language": "en",
"url": "https://arxiv.org/abs/1504.03394"
}
|
\section{Introduction}
Over more than last three decades, there has been tremendous progress in the technology of cooling and trapping of single-electron
alkali atomic gases and
some two-electron atomic gases. In parallel, the developments in the technology of trapping and cooling of some ions,
specially alkaline-earth cations over last several decades has led to new vistas of research activities with trapped and cold ions.
With recent experimental developments
\cite{prl:2009:vuletic,aymar:prl:2011,molphys,hudson:prl:2011,hudson:prl:2012,Zipkes1,Zipkes2,Zipkes3,Mukaiyama:pra:2013,schmid:2012:rsi,Denschlag:prl:2012,Narducci:pra:2012,schmid,rangwala:apl:2012,rangwala:natcom:2012,rangwala:pra:2013}
using hybrid ion-atom traps
\cite{hytrap}, both atoms
and ions can be confined simultaneously in a common space or an atom-trap can be
merged with an ion-trap. These hybrid systems open up prospects
for emulating solid-state physics with laser-generated periodic structure of ions \cite{prl111:2013:schmidt-kaler} and
for exploring physics of ion-controlled Josephson junction \cite{prl109:2012:schmidt-kaler,pra89:2014:negretti},
Tonk-Girardeau gas with
an ionic density ``bubble'' \cite{pra81:2010:busch}.
At a fundamental level, these new hybrid devices
facilitate the laboratory investigations on ion-atom collisions
in hitherto unexplored low energy regimes -- from milliKelvin down to microKelvin or sub-microKelvin temperature regime.
Ion-atom scattering at
low energy is important for a number of physical systems, such as
cold plasma, planetary atmospheres, interstellar clouds.
Gaining insight into the ion-atom interactions and scattering at
ultra-low energy down to Wigner threshold law regime is important to
understand charge transport\cite{cote2000} at low
temperature, radiative association \cite{aymar:prl:2011,molphys,njp17:2015:aymar}, ion-atom bound states\cite{bound},
ion-atom photoassociation\cite{Rakshit2011,jcp:2011:dulieu}, and many other related phenomena.
Several theoretical investigations
\cite{arXiv:1409.1192,pra91goodman,njp,Zhang2009,Zygelman1989,pra79,bgao,bgao2012} of atom-ion cold collisions have
been carried out in recent times. It is proposed that controlled
ion-atom cold collision can be utilized for quantum information
processes\cite{pra81:2010:calarco,contphys}. Several recent theoretical
studies have
focused on charge-transfer \cite{pccp13:2011:belyaev,pra87:2013:belyaev,jpb47:2014:McCann} and
chemical reaction processes \cite{jpb47:2014:zygelman,pra86:2012:McCann} at ultralow energies between an alkaline-earth
ion or Yb$^+$ and an alkali atom.
Ions are usually trapped by radio-frequency fields. The major
hindrance to cool trapped ions below milliKelvin temperature stems
from the trap-induced micro-motion of the ions, which seems to be
indispensable for such ion traps. Therefore, it is difficult to
achieve sub-microKelvin temperature for an ion-atom system in a
hybrid trap. But, to explore fully quantum or Wigner
threshold law regime for ion-atom collisions, it is essential to
reduce the temperature of ion-atom hybrid systems below one
micro-Kelvin. One way to overcome this difficulty is to device new
methods to trap and cool both ions and atoms in an optical trap.
With recent experimental demonstration of an optical trap for ions\cite{naturecomm}, the prospect for experimental explorations of
ultracold ion-atom collisions in Wigner threshold regime appears to
be promising.
In the contexts of ion-atom cold collisions of current interest,
there are mainly four types of ion-atom systems:(1) an alkaline-earth ion interacting with an alkali metal atom, (2) a rare earth
ion (such as Yb) interacting with an alkali metal atom, (3) an
alkali metal ion interacting with an alkali atom of the same nucleus
and (4) an alkali metal ion interacting with an alkali atom of
different nucleus. In the first two types, both the ion and the atom
have one valence electron in the outermost shell. In the third type,
the ion has closed shell structure while the atom has one valence
electron in the outermost shell. Since both the ion and atom are of
the same nucleus, there is center of symmetry for the electronic
wave function of the ion-atom pair. Because of this symmetry,
resonant charge transfer collision is possible in the third type. In
the fourth type, the ion is of closed shell structure while the atom
has one valence electron in the outermost shell. Since the nuclei of
the atom and the ion are different, there is no resonant charge
transfer collision in fourth type, though non-resonant charge
transfer is possible in all the types. For scattering in the
ground-state potentials, non-resonant charge transfer
collision is suppressed in the ultralow energy regime ($<\mu$K). \\
\begin{figure}
\includegraphics[width=10cm]{three_pot.eps}
\caption{Adiabatic potential curves for (LiRb)$^{+}$, (LiCs)$^{+}$ and (NaCs)$^{+}$}
\end{figure}
Here we study cold collision in the fourth type of ion-atom systems, though
these systems are not straightforward to obtain
experimentally. Normally, neutral alkali-metal atoms and alkaline-earth ions can be
laser-cooled and trapped. However, there are some recent experimental studies showing
different techniques \cite{schmid:2012:rsi,Denschlag:prl:2012,rangwala:apl:2012,rangwala:natcom:2012}
for trapping and cooling of closed-shell akali-metal ions.
Our primary aim is to understand ground-state elastic scattering
processes between such an ion and an alkali atom of different nuclei at ultralow energy where
inelastic charge transfer process can be ignored. A proper understanding of ground-state
elastic scattering processes\cite{bgao,bgao2012} at low energy in
an ion-atom system is important for exploring coherent control of
ion-atom systems and formation of ultracold molecular ions by
radiative processes. Due to the absence
of resonance charge transfer reaction, the fourth type system is
preferable for our purpose. However, there remains finite possibility for inelastic collisions if the
hyperfine structure of the neutral atom is taken in account. In our calculations we
do not consider hyperfine interactions.
Let a cold alkali metal ion {\bf A}$^{+}$ interact with a cold
alkali metal atom {\bf B}. Suppose, the atomic mass of {\bf B} is
smaller than that of atom {\bf A}. Then, for {\bf AB}$^{+}$
molecular system, ground-state continuum asymptotically corresponds
to the atom {\bf B} (ns) in $^{2}$S electronic state and the ion
{\bf A}$^{+}$ (complete shell) in $^{1}$S electronic state. In the
separated atom limit, the first excited molecular potential
asymptotically goes as the atom {\bf A} (ns) in $^{2}$S electronic
state and alkali ion {\bf B}$^{+}$ (complete shell) in $^{1}$S
electronic state while the second electronic excited potential
corresponds to {\bf B} (np) in $^{2}$P electronic state and alkali
ion {\bf A}$^{+}$ (complete shell) in $^{1}$S electronic state.
\begin{table}[ph]
\caption{A comparison of the spectroscopic constants for the ground (X$^{2}\Sigma^{+}$) and the first and second
excited (2$^{2}\Sigma^{+}$ and 3$^{2}\Sigma^{+}$) electronic states of (LiRb)$^{+}$ molecular ion with the available works.}
{\begin{tabular}{@{}llllllll@{}} \hline
\\[-1.8ex]
State & $R_{e}$(a.u.) &$D_{e}$(cm$^{-1}$) & T$_{e}$(cm$^{-1}$)&$\omega_{e}$(cm$^{-1}$) & $\omega_{e}\chi_{e}$(cm$^{-1}$)&$B_{e}$(cm$^{-1}$)&{references}\\[0.8ex]
\hline \\[-1.8ex]
X$^{2}\Sigma^{+}$ & 7.70 & 3912 & & 137.56 & 1.67 & 0.158098 & This work \\
& & 3705 & & & & & \cite{Bellomonte1974}\\
& 7.50 & & & & & & \cite{Carlson1980}\\
& 7.60 & 3432 & & & & & \cite{Patil2000} \\
& 7.54 & 4193 & & 139.65 & & & \cite{Azizi1}\\
2$^{2}\Sigma^{+}$ & 12.67 & 1270 & 12438 & 63.73 & 0.51 & 0.058456 & this work\\
& 12.60 & 1306 & & 63.00 & & & \cite{Azizi1} \\
3$^{2}\Sigma^{+}$ & 16.66 & 617 & 18200& 35.73 & 1.55 & 0.033769 &This work \\[0.8ex]
\hline \\[-1.8ex]
\multicolumn{8}{@{}l}{ }\\
\end{tabular}}
\label{tab1}
\end{table}
We investigate cold collisions over a wide range of energies in
three different alkali atom-alkali ion systems in the ground
molecular potentials. The colliding atom-ion pairs we consider are:
(i) Cs$^{+}$ + Li, (ii) Cs$^{+}$ + Na, and (iii) Rb$^{+}$ + Li.
Since at ultralow energy, charge transfer reaction is highly
suppressed in these systems, we study elastic scattering only.
To
calculate scattering wave functions, we need the data for
Born-Oppenheimer adiabatic potentials of the systems. In the long
range where the separation r $>$ 20 $a_{0}$ ($a_{0}$ is the Bohr
radius), the potential is given by the sum of the dispersion terms,
which in the leading order goes as $1/r^{4}$. We obtain short-range
potentials by pseudopotential method. The short-range part is
smoothly combined with the long-range part to obtain the potential
for the entire range. We then solve time-independent Schroedinger
equation for these potentials with scattering boundary conditions by
Numerov-Cooley \cite{cooley1961} algorithm. We present detailed results of scattering
cross sections for these three systems for
energies ranging
from 0.01 micro-Kelvin ($\mu$K) to 1 Kelvin (K). However, the low energy regime of our interest ranges from
sub-$\mu$K to 1 milli-Kelvin (mK). Here we choose to use Kelvin as the unit of energy to readily
convey the information how cold the system should be to explore such collision physics. Normally
the results of atomic collisions are expressed in atomic unit (a.u.).
Note that 1 a.u. of energy corresponds to 3.1577465 $\times$ 10$^5$ K or
27.21138386 electron-Volt (ev). \\
\vspace{0.5cm}
\begin{figure}
\includegraphics[width=3.65in]{pot1_b_csli_mod.eps}
\caption{(Color online) In the upper panel, the centrifugal
energies in unit of milliKelvin (mK) for $s$- (black, solid), $p$- (red, dashed) and $d$- (green,
dashed-dotted) partial waves are plotted against atom-ion separation
for $1^{2}\Sigma^{+}$ state of (LiCs)$^{+}$. The lower panel
shows the same for (NaCs)$^{+}$.}
\end{figure}
\section{Interactions and molecular properties}
Under Born-Oppenheimer approximation, we compute the adiabatic
potential energy curves of the $1^{2}\Sigma^{+}$, $2^{2}\Sigma^{+}$
and $3^{2}\Sigma^{+}$ electronic states of the three ionic molecules
(LiRb)$^{+}$, (LiCs)$^{+}$ and (NaCs)$^{+}$ using the
pseudopotential method proposed by Barthelat and Durand\cite{Barthelat1975}
in its semi-local form and used in many previous works\cite{Ghanmi1,Ghanmi2,Ghanmi2012,Berriche2003,Berriche2005}. The
interaction potentials between different alkali metal ion and an
alkali metal atom were calculated by Valance\cite{Valance1978} in
1978 by the method of pseudopotential.
Here we briefly describe spectroscopic constants and interaction potentials of the three considered atom-ion systems,
and the pseudopotential method used to obtain these molecular properties. Our final goal is to calculate
cold collisional properties of these systems using these potentials as described in the next section. \\
\vspace{0.5cm}
\begin{table}[ph]
\caption{A comparison of the spectroscopic constants for the ground (X$^{2}\Sigma^{+}$) and the first and second
excited (2$^{2}\Sigma^{+}$ and 3$^{2}\Sigma^{+}$) electronic states of (LiCs)$^{+}$ molecular ion with the work of
Khelifi et al.$^{57}$}
{\begin{tabular}{@{}llllllll@{}} \hline
\\[-1.8ex]
State & $R_{e}$(a.u.) &$D_{e}$(cm$^{-1}$) & T$_{e}$(cm$^{-1}$)&$\omega_{e}$(cm$^{-1}$) & $\omega_{e}\chi_{e}$(cm$^{-1}$)&$B_{e}$(cm$^{-1}$)&{references}\\[0.8ex]
\hline \\[-1.8ex]
1$^{2}\Sigma^{+}$ & 8.12 & 3176 & 0 & 124.46 & 1.08 & 0.138337 & This work \\
& 7.19 & 3543 & & & & & \cite{Khelifi2011} \\
2$^{2}\Sigma^{+}$ & 12.42 & 1911 & 13343 & 68.69 & 0.63 & 0.059126 & This work \\
& 12.37 & 2022 & 19553 & & & & \cite{Khelifi2011} \\
3$^{2}\Sigma^{+}$ & 18.53 & 422 & 17660 & 28.07 & 1.29 & 0.026570 & This work \\
& 18.38 & 409 & 20849 & & & & \cite{Khelifi2011} \\
\hline \\[-1.8ex]
\multicolumn{8}{@{}l}{ }\\
\end{tabular}}
\label{tab2}
\end{table}
\subsection{Adiabatic Potentials}
For the three ionic systems (LiRb)$^{+}$, (LiCs)$^{+}$ and
(NaCs)$^{+}$, the potential energy curves of the 1-3$^{2}\Sigma^{+}$
electronic states, are built from an ab initio calculation for the
internuclear distances ranging from 3 to 200 $a_0$ ($a_0$ is Bohr radius) as discussed in the sub-section 2.2.
These states dissociate
respectively, into Li(2s and 2p) + Rb$^{+}$ and Rb(5s) + Li$^{+}$,
Li(2s and 2p) + Cs$^{+}$ and Cs(6s) + Li$^{+}$, and Na(3s and 3p) +
Cs$^{+}$ and Cs(6s) + Na$^{+}$. They are displayed in figure 1(a)
for LiRb$^{+}$, 1(b) LiCs$^{+}$ and 1(c) for NaCs$^{+}$.
In the long range part
beyond $ 200 a_0$,
the potential is given by the sum of the dispersion terms. The short
range part ($\le 20 a_0$) of ab initio potential is smoothly combined with the long-range part by interpolation
to obtain the potential for entire range extending to several thousand $a_0$.
For the
three ionic systems, we remark, that the ground state has the
deepest well compared to the 2$^{2}\Sigma^{+}$ and 3$^{2}\Sigma^{+}$
excited states. Their dissociation energies are of the order of
several 1000 cm$^{-1}$. Their equilibrium positions lie at
separations that are relatively larger than those of typical neutral
alkali-alkali diatomic molecules. \\ \\
\begin{figure}
\includegraphics[width=3.65in]{cr_par_csli.eps}
\caption{(Color online) $s$-
$p$- and $d$-wave scattering cross sections in atomic unit (a.u.) or in unit of $a_0^2$ ($a_0$ is Bohr radius) are plotted against
collision energy $E$ in Kelvin (K) for $1^2\Sigma^+$ state of (LiCs)$^+$ and (NaCs)$^+$ in upper and lower panel, respectively.}
\end{figure}
The long range potential is given by the expression
\begin{equation}
V(r) = -\frac{1}{2}\left( \frac{C_{4}}{r^{4}} + \frac{C_{6}}{r^{6}}
+ \cdots \right)
\end{equation}
where $C_{4}$, $C_{6}$ correspond to dipole, quadrupole
polarisabilities of concerned atom. Hence, the long range
interaction is predominately governed by polarisation interaction.
Dipole polarisabilities for Na (3s) and Li (2s) are 162 a.u. and
164.14 a.u., respectively. One can define a characteristic length
scale of the long-range potentials by $\beta = \sqrt{2\mu
C_{4}/\hbar^{2}}$. The values of $\beta$ for the collision of Li -
Cs$^{+}$ pair and Na - Cs$^{+}$ pair are 1411.5 a.u. and 2405.8
a.u., respectively
For the ground state 1$^{2}\Sigma^{+}$ the well depth and the
equilibrium position are 2979 cm$^{-1}$ and 8.51 a.u., respectively,
for (NaCs)$^{+}$ while those for (LiCs)$^{+}$ are 3176 cm$^{-1}$ and
8.12 a.u., respectively. Reduced mass for (NaCs)$^{+}$ and
(LiCs)$^{+}$ (for $^7$Li isotope) are taken as 19.5995 a.u. and 6.6642 a.u.,
respectively. For both $^{6}$Li-Rb$^{+}$ and $^{6}$Li-Rb$^{+}$
isotopes, we used the reduced masses 5.6171 and 6.4805 a.u.
respectively.
\subsection{Pseudopotential method: Results and discussions}
The use of the pseudopotential method, for each (XY)$^{+}$ ionic
molecules (LiRb$^{+}$, LiCs$^{+}$ and NaCs$^{+}$), reduce the number
of active electrons to only one electron. We have used a core
polarization potential $V_{CPP}$ for the simulation of the
interaction between the polarizable X$^{+}$ and Y$^{+}$ cores with
the valence electron. This core polarization potential is used
according to the formulation of M\"{u}ller et al.\cite{Muller1984},
and is given by
\begin{equation}
V_{CPP} = -\frac{1}{2} \Sigma_{\lambda}
\alpha_{\lambda}\vec{f}_{\lambda} \cdot \vec{f}_{\lambda}
\end{equation}
where $\alpha_{\lambda}$ and $\vec{f}_{\lambda}$ are respectively
the dipole polarizability of the core $\lambda$, and the electric
field produced by valence electrons and all other cores on the core
$\lambda$. The electric field $\vec{f}_{\lambda}$ is defined as:
\begin{equation}
\vec{f}_{\lambda} = \Sigma_{i} \frac{\vec{r}_{i\lambda}}{r^{3}_{i}} F(\vec{r}_{i\lambda}, \rho_{\lambda})-\Sigma_{\lambda'
\neq \lambda} \frac{\vec{R}_{\lambda', \lambda}}{R^{3}_{\lambda', \lambda}}Z_{\lambda}
\end{equation}
where $\vec{r}_{i\lambda}$ and $\vec{R}_{\lambda', \lambda}$ are
respectively the core-electron vector and the core-core vector. \\
\vspace{0.5cm}
\begin{table}[ph]
\caption{A comparison of the spectroscopic constants for the ground (X$^{2}\Sigma^{+}$) and the first and second
excited (2$^{2}\Sigma^{+}$ and 3$^{2}\Sigma^{+}$) electronic states of (NaCs)$^{+}$ molecular ion with the work
of Valance$^{49}$.}
{\begin{tabular}{@{}llllllll@{}} \hline
\\[-1.8ex]
State & $R_{e}$(a.u.) &$D_{e}$(cm$^{-1}$) & T$_{e}$(cm$^{-1}$)&$\omega_{e}$(cm$^{-1}$) & $\omega_{e}\chi_{e}$(cm$^{-1}$)&$B_{e}$(cm$^{-1}$)&{references}\\[0.8ex]
\hline \\[-1.8ex]
1$^{2}\Sigma^{+}$ & 8.51 & 2979 & 0 & 68.17 & 0.32 & 0.042418 & This work \\
& 7.60 & 3388 & & & & & \cite{Valance1978} \\
2$^{2}\Sigma^{+}$ & 13.45 & 1262 & 11758 & 33.98 & 0.30 & 0.016977 & This work \\
& 14.20 & 774 & & & & & \cite{Valance1978} \\
3$^{2}\Sigma^{+}$ & 16.77 & 1393 & 18553 & 26.20 & 1.69 & 0.010920 & This work \\
& 15.00 & 732 & & & & & \cite{Valance1978} \\
\hline \\[-1.8ex]
\multicolumn{8}{@{}l}{ }\\
\end{tabular}}
\label{tab3}
\end{table}
\vspace{-1cm}
Based on the formulation of Foucrault et al.\cite{Fouc1992} the
cut-off function F($\vec{r}_{i\lambda}, \rho_{\lambda}$) is a
function of the quantum number l. In this formulation, the
interactions of valence electrons of different spatial symmetry with
core electrons are considered in a different way. The used cut-off
radii of the lowest valence s, p, d and f one-electron for the Li,
Na, Rb and Cs atoms are taken from ref.\cite{Ghanmi1,Ghanmi2012,Berriche2003,Berriche2005}. The extended
Gaussian-type basis sets used for Li, Na, Rb and Cs atoms are
respectively (9s8p5d1f/8s6p3d1f)\cite{Berriche2003}, (7s6p5d3f/6s5p4d2f)\cite{Berriche2003},
(7s4p5d1f/6s4p4d1f)\cite{Pavolini1989} and (7s4p5d1f/6s4p4d1f)\cite{Ghanmi2}. The used core
polarizabilities of Li$^{+}$, Na$^{+}$, Rb$^{+}$ and Cs$^{+}$ cores,
which are equal respectively 0.1915 $a_{0}^{3}$, 0.993 $a_{0}^{3}$,
9.245 $a_{0}^{3}$ and 15.117 $a_{0}^{3}$ are taken from\cite{Muller1984}.
Using the pseudopotential technique each molecule
is reduced to only one valence electron interacting with two cores.
Within the Born-Oppenheimer approximation an SCF calculation,
provide us with accurate potential energy curves and dipole
functions. \\
\vspace{0.5cm}
\begin{figure}
\includegraphics[width=3.65in]{totcr_lics.eps}
\caption{(Color online) Logarithm of total scattering cross section in a.u. as a function of logarithm of
energy $E$ in K for electronic
ground-state 1$^{2}\Sigma^{+}$ of (LiCs)$^{+}$ and (NaCs)$^{+}$ are plotted in upper and lower pannel, respectively.}
\end{figure}
\subsection{Spectroscopic constants}
The spectroscopic constants ($R_{e}$, $D_{e}$, T$_{e}$,
$\omega_{e}$, $\omega_{e}\chi_{e}$, $B_{e})$ of the
1-3$^{2}\Sigma^{+}$ electronic states are presented in tables 1-3
for (LiRb)$^{+}$, (LiCs)$^{+}$ and (NaCs)$^{+}$ respectively. To the
best of our knowledge, no experimental data has been found for these
systems. We compare our spectroscopic constants only with the
available theoretical results. Table 1 presents the spectroscopic
constants of (LiRb)$^{+}$ compared with the other theoretical works\cite{Bellomonte1974,Carlson1980,Patil2000,Azizi1}. As it seems
from table 1, there is a good agreement between the available
theoretical works\cite{Bellomonte1974,Carlson1980,Patil2000,Azizi1} and our ab initio study. Azizi et al.\cite{Azizi1}
reported the spectroscopic constants for many ionic alkali dimers
but only for the ground and the first excited states. They have used
in their study a similar formalism as used in our work. We remark
that our ground state equilibrium distance ($R_{e}$) presents a
satisfying agreement as well as the well depth ($D_{e}$) with the
work of Azizi et al.\cite{Azizi1}. We found for ($R_{e}$) and
($D_{e}$), respectively, 7.70 a. u. and 3912 cm$^{-1}$ and they
found 7.54 a.u. and 4193 cm$^{-1}$. The difference between the results of Azizi {\it et al.} [57]
and our values are about 2.12\% and 6.70\% for $R_{e}$ and $D_{e}$, respectively.
The same agreement is observed
between our harmonacity frequency ($\omega_{e}$) and that of Azizi
et al.\cite{Azizi1}. The difference between the two values is
2.09 cm$^{-1}$, which represents a difference of 1.49\%. There is also a very good agreement between our
equilibrium distance and that of Patil et al.\cite{Patil2000}
($R_{e}$= 7.60 a.u.), in contrast to their well depth\cite{Patil2000}, which is underestimated ($D_{e}$=3432 cm$^{-1}$).
For the first excited state, Azizi et al.\cite{Azizi1} reported
the spectroscopic constants $R_{e}$=12.60 a.u., $D_{e}$ = 1306
cm$^{-1}$ and $\omega_{e}$ = 63.00 cm$^{-1}$ to be compared with our
values of, respectively, 12.67 a.u., 1270 and 63.73 cm$^{-1}$. The differences in percentages between the values obtained by Azizi {\it et al.} [57]
and our results are 0.55\%, 2.75\% and 1.14\% for $R_e$, $D_e$ and $\omega_{e}$, respectively.\\ \\
\begin{figure}
\includegraphics[width=3.65in]{scross.eps}
\caption{(Color online) Same as in figure 4 but for Li + $^{87}$Rb$^{+}$ (1$^{2}\Sigma^{+}$)
collision. The black solid line and red dotted lines are for $^{6}$Li and $^{7}$Li,
respectively. Black dashed lines represent linear fit to both the curves for energies greater
than 10$^{-6}$ K.
Both cases follow one-third law and the intercepts are found to be 3.587 and 3.606 in a.u.
for $^{6}$Li and $^{7}$Li, respectively; where as the
theoretically calculated values are 3.3269 and 3.3476 in a.u., respectively.}
\end{figure}
The spectroscopic constants of (LiCs)$^+$ and (NaCs)$^+$, are presented respectively in tables 2 and 3 and compared
with the available theoretical results\cite{Khelifi2011,Valance1978}. To the best of our knowledge there is no experimental data
for these ionic molecules. For the (LiCs)$^+$ ionic molecule, we compare our results only with the
theoretical work of Khelifi et al.\cite{Khelifi2011} where they have used a similar method as used in our work.
They reported for the ground state 1$^2\Sigma^{+}$ the spectroscopic constants $R_e = 7.91$ a.u. and $D_e = 3543$ cm$^{-1}$ to be compared with our values of, respectively, $R_e = 8.12$ a.u. and $D_e = 3176$ cm$^{-1}$.
A rather good agreement is observed for the equilibrium distance, however their potential is much deeper.
Reasonably good agreement between our spectroscopic constant and those of Khelifi {\it et al.} \cite{Khelifi2011}
is observed for the 2$^2\Sigma^+$ and 3$^2\Sigma^+$ states. The difference between Khelifi {\it et al.} spectroscopic constants and
our results are found to be 2.58\% and 10.35\% for $R_e$ and $D_e$, respectively.
For the ground state of (NaCs)$^+$ ionic molecule, there is a good agreement between our well-depth,
as well as the equilibrium distance with the theoretical results of Valance\cite{Valance1978}.
We found a well-depth of 2979 cm$^{-1}$ located at 8.51 a u., while valance\cite{Valance1978} found 3388 cm$^{-1}$ located at 7.60 a.u..
The difference in percentages between valance’s data and our values are 12.07\% and 11.97\% for $R_e$ and $D_e$, respectively.
In contrast to the ground state,
where the agreement between our work and those of Valance\cite{Valance1978} is good,
there is a disagreement for the 2$^2\Sigma^+$ and 3$^2\Sigma^+$ states.
The two states exhibit small potential wells in Valance’s work equal, respectively, to 774 and 732 cm$^{-1}$;
while in our work we have found significantly higher well depths of 1262 and 1393 cm$^{-1}$, respectively.
Our well depth for the 2$^2\Sigma^+$ and 3$^2\Sigma^+$ states are of 63.04\% and 90.30\% larger than those of Valance [50].
The large discrepancies between the Valence’s results and our values can be explained by two reason. First, Valence used the Hellmann-type model
where the Hamiltonian is written in terms of one valence electron interacting with closed shell cores. Therefore, the core-core interaction potentials
were neglected. This is certainly true for large interatomic distances but it is not accurate enough for small separations.
Valence considered this shortcoming as not relevant to the charge exchange collision process they studied using the interaction potential
where only intermediate and long-range distances are required. Although, without considering the core-core interaction one can get correct asymptotic limits,
at intermediate distance this will affect the accuracy of the equilibrium distance and well depth. Second, the core-valence interactions were considered in our work using a formulation where,
for each atom, the core polarization effects are described by an effective potential as described previously. The latter is modified by an $l$- dependent cut-off function $F$ to consider, in a different way
the orbital symmetries. In the case of Valence, the Hellmann pseudopotential is not "$l$" dependent and the parameters are optimized to obtain the two first experimental energy levels.
Similarly, in the the absence of the core-core interaction, Valence considered that this lack of "$l$" dependence is not crucial for the charge exchange collision. In our approach,
in addition to the use of an $l$-dependent pseudoptential the parameters were optimized to reproduce with high accuracy the ionization potential and the experimental energy levels of many alkali excited states.
Moreover, in our study, the interatomic separation grid is more extended as $R$ varies from 3 to 200 a.u. with a distance step of 0.1 a.u, but Valence limited his calculation for internuclear distances ranging
from 5 a.u. to 25 a.u. It is important to note that the calculation time in our calculation is enormously reduced due to the use of the pseudopotential approach that leads these ionic molecules to one-electron systems.
Calculations are performed on a desktop Pentium IV Acer computer where the time for one single distance is about 32 seconds.
\section{Elastic collisions: Results and discussions}
Here we present results on elastic collisions of the discussed
ion-atom systems by FORTRAN code \cite{press:nrcp} using the well-known Numerov-Cooley algorithm \cite{cooley1961}.
This algorithm provides an efficient technique for numerically solving a second order differential equation. It
uses a three-point recursion relation to calculate first order derivative and makes use of exact second order derivative
provided by the equation itself. To solve time-independent Schroedinger equation as a scattering problem,
one has to propagate the Numerov-Cooley code from initially small internuclear separation $r$ to asymptotically large $r$ where
the wave function corresponds to the state of a free particle and so behaves sinusoidally. The scattering $S$-matrix is then deduced
by matching the asymptotic solution with the standard scattering boundary conditions. The correctness of the calculations is ensured
by the unitarity of the $S$-matrix. The lower the energy larger is the asymptotic boundary. To initiate the propagation of the code
from $r \simeq 0$ position, one has to set initially the values of the wave functions at two initial positions, then the code calculates
the wave function for the third position, and then taking the the values of the wave functions for the second and third position it calculates
the wave function for the fourth position, and the process continues until the asymptotic boundary is reached. The initial boundary
conditions are set by expanding the interaction potential for small $r$ and solving the Schroedinger equation analytically
for $r \rightarrow 0$ limit. A good measure for the asymptotic scattering
boundary (large $r$) can be found by setting the condition that the effective potential
$V_{eff} = - C_6/r^6 + (\hbar^2/2\mu) \ell (\ell + 1)/r^2$ for large $r$ becomes much less than
the collision energy $E$ (at least one tenth of $E$).
We use the interaction potentials calculated
above as input data. In particular, we focus on low energy collisions. Using
well-known expansion of continuum state in terms of partial waves,
the wave function $\psi_{\ell}$ ( k r) for $\ell$th partial wave is
given by
\begin{equation} \left[ \frac{d^{2}}{dr^{2}} + k^{2} -
\frac{2\mu}{\hbar^{2}} V(r) - \frac{\ell (\ell +1)}{r^{2}} \right ]
\psi_{\ell} (kr) = 0
\end{equation}
subject to the standard scattering boundary condition
\begin{equation} \psi_{\ell}(kr)\sim \sin\left[ kr - \ell\pi/2 + \eta_{l}\right]
\end{equation}
Here $r$ denotes the ion-atom separation, the wave number $k$ is
related to the collision energy $E$ by $E = \hbar^{2}$ $k^{2} /2 \mu$
and $\mu$ stands for the reduced mass of the ion-atom pair. The
total elastic scattering cross section is given by
\begin{equation}
\sigma_{el} = \frac{4\pi}{k^2}\sum_{\ell=0}^\infty
(2\ell+1)\sin^{2}(\eta_\ell).
\end{equation}
To know the relevant energy regimes where $s$- $p$- and $d$-wave
collisions are important, we have plotted in figure 2 the
centrifugal energies for first 3 partial waves ($\ell$ = 0, 1 and 2)
against ion-atom separations for ground state 1$^{2}\Sigma^{+}$ of
(NaCs)$^{+}$ and (LiCs)$^{+}$, respectively. For $d$-wave, the values
of centrifugal barrier are about 0.007 mK for (NaCs)$^{+}$ and about
0.06 mK for (LiCs)$^{+}$. These values indicate that the
potential energy barriers for low lying higher partial waves are
very low for atom-ion systems allowing tunneling of the wave
function towards the inner region of the barriers. Unlike atom-atom
systems at low energy, a number of partial waves can significantly
contribute to the ion-atom scattering cross section at low energy.
We find that in order to get convergent results at milli- and micro-Kelvin regimes,
the numerical calculation of scattering wave function needs to be extended to at least 10000 $a_0$ and $20000 a_0$,
respectively. .
In figure 3, the partial-wave cross sections for ground
1$^{2}\Sigma^{+}$ state collisions are plotted against collision
energy $E$ in K for (NaCs)$^{+}$ and (LiCs)$^{+}$. As $ k \rightarrow
0$ , $s$-wave cross section becomes independent of energy while $p$- and
$d$-wave cross sections vary in accordance with Wigner threshold laws.
According to Wigner threshold laws, as k $\rightarrow$ 0, the phase
shift for $\ell$th partial-wave $\eta_l \sim k^{2\ell+1}$ if $\ell
\leq (n-3)/2$, otherwise $\eta_{l} \sim k^{n-2}$ for a long range
potential behaving as $1/r^{n}$. Since Born-Oppenheimer ion-atom
potential goes as $1/{r^4}$ in the asymptotic limit, as $k
\rightarrow 0$, $s$-wave scattering cross section becomes independent
of energy while all other higher partial-wave cross sections
go as $\sim k^{2}$. As energy increases beyond the Wigner threshold law
regime, the $s$-wave cross section exhibits a minimum at a low energy which may be related to Ramsauer-Townsend effect.
It can be noticed from figure 3 that at Ramsauer minimum of $s$-wave scattering, $p$- and $d$-wave
elastic cross sections are finite. This means that one can explore $p$- or $d$-wave interactions of ion-atom systems
at the minimum position that occurs at a relatively low energy ($\mu$- and milli-Kelvin regime).
In the Wigner threshold regime, hetero-nuclear
ion-atom collisions are dominated by elastic scattering processes
since the charge transfer reactions are highly suppressed in this
energy regime. Furthermore, resonant charge transfer collisions do
not arise in collision of an ion with an atom of different nucleus.
It is worthwhile to compare the elastic scattering results of hetero-nuclear
alkai ion-atom system to those of homo-nuclear alkali atom-ion system Na+Na$^{+}$
\cite{cote2000}. By comparing our results in figure 3 with the figures 2 and 3 of Ref.
\cite{cote2000}, we notice that elastic scattering cross sections of both homo- and hetero-nuclear ion-atom systems
at ultralow energies are comparable and primarily governed by Winger threshold laws. However,
unlike hetero-nuclear systems, the resonant charge transfer cross section in homo-nuclear Na+Na$^{+}$ is also of the
same order as that of elastic one at ultra-low energies. Resonant charge exchange cold collisions between Yb and Yb$^{+}$
are experimentally observed and found to be a dominant process in this homo-nuclear system \cite{prl:2009:vuletic}.
In figure 4, we have plotted logarithm of total elastic scattering
cross sections ($\sigma_{tot}$) expressed in a.u. as a function of
logarithm of $E$ in K for ground state collisions between Li and
Cs$^{+}$ and between Na and Cs$^{+}$, respectively. For both
(LiCs)$^{+}$ and (NaCs)$^{+}$, to calculate $\sigma_{tot}$, we
require more than 60 partial waves to get converging results for
energies greater than 1 mK. This is because, as the collision energy increases, more and more partial waves start to contribute to
$\sigma_{tot}$. In order for a particular partial wave $\ell$ to contribute appreciably at a given energy $E$,
the centrifugal barrier height corresponding to this partial wave should not be too high
compared to the energy. An estimate of the minimum number of partial waves required for obtaining convergent results may be made by
taking the height of the centrifugal barrier to be higher than the energy
by one order of magnitude. In this context, it is worth mentioning that the centrifugal barrier height for a partial wave
of an ion-atom system is usually much lower than the corresponding height of atom-atom system, therefore larger
number of partial waves contribute to $\sigma_{tot}$ in case of an ion-atom system.
In the large energy limit, the cross
section behaves as
\begin{equation} \sigma_{tot} \sim \pi
\left( \frac{\mu C_{4}^{2}}{\hbar^{2}}\right)^\frac{1}{3} \left(1 +
\frac{\pi^{2}}{16}\right) E^{-\frac{1}{3}}\label{sig}
\end{equation}
Thus, in the large energy regime the slope of the logarithm of
$\sigma_{tot}$ as a function of $\log E$ is a straight line obeying
the equation $\sigma_{tot}(E) = - (1/3) E + c_{E}$ where the slope
of the line is $- 1/3$ and the intercept of the line on the energy
axis is solely determined by the $C_{4}$ coefficient of long-range
part of the potential, or equivalently by the characteristic length
scale $\beta$ of the potentials or the polarizability of the neutral
atom interacting with the ion. As shown in figure 4, we have
numerically verified this E$^{-1/3}$ law for both Li-Cs$^{+}$ and
Na-Cs$^{+}$ systems. The linear fit to the the plot of numerically
calculated $\sigma_{tot}$ against in the large energy regime shows
that the value of the slope is quite close to the actual value of
1/3.
Since the dipole polarizability of Li and Na is not much different, one can
expect that for both Li-Cs$^{+}$ and Na-Cs$^{+}$ systems, the
energy-dependence of $\sigma_{tot}$ at large energy should be
similar. In fact, figure 4 shows that for energies much greater 1
$\mu$K, both systems exhibit similar asymptotic energy dependence.
\begin{figure}
\includegraphics[width=3.65in]{pc_6_mod.eps}
\caption{(Color online)Partial wave cross sections for $^{6}$Li + Rb$^{+}$ (1$^{2}\Sigma^{+}$) and $^{7}$Li + Rb$^{+}$
(1$^{2}\Sigma^{+}$) collision are plotted as a function of E (in K) for $\ell$ = 0 (solid) and $\ell$ = 1 (dashed) in
upper and lower panel, respectively.}
\end{figure}
Finally, we consider the collision of $^{85}$Rb$^{+}$ with the two
isotopes of Li atom, namely, $^{6}$Li and $^{7}$Li, and the results
are shown in figures 5 and 6. The purpose here is to investigate
the isotopic effects of Li on low energy collisions with
$^{85}$Rb$^{+}$ ion . In figure 5 we have plotted logarithm (to the
base of 10) of $\sigma_{tot}$ as a function of logarithm (to the
base of 10) of E. Results clearly show that the patterns are same
for $^{6}$Li and $^{7}$Li colliding with $^{85}$Rb$^{+}$, but they
differ in magnitudes, particularly in the low energy regime. Figure
6 exhibits the variation of $s$- and $p$-wave scattering cross sections
against energy in log-log scale for both isotopes of Li colliding
with Rb$^{+}$. Again, the results are qualitatively similar for both
the isotopes, but they differ slightly quantitatively. Comparing the
upper and lower panels in figure 6, one can notice that the Wigner
threshold law regime for $^{6}$Li can be attained at slightly higher
energy than that for $^{7}$Li.
\section{Conclusions}
In conclusion, we have studied elastic scattering between alkali ion and alkali atom of different nuclei
of three ion-atom systems, namely, Li + Cs$^{+}$, Na +
Cs$^{+}$, and Li + Rb$^{+}$. We have calculated the interaction potentials and spectroscopic constants
of these systems. We have presented a detailed study of elastic collision physics over a wide range of energies, showing the
onset of Wigner threshold regime at ultra low energy and the 1/3 law of scattering at higher energy regime.
The low energy scattering results presented here
may be useful for future exploration for radiative- or photo-associative formation
of cold diatomic molecular ions from these three ion-atom systems. The
colliding ground state atom-ion pair corresponding to the continuum
of 1$^{2}\Sigma^{+}$ may be photoassociated to form a bound state in
second excited potential 3$^{2}\Sigma^{+}$ by using a laser of
appropriate frequency. This is promising because the transition is
dipole allowed and is similar to atomic transition. The formation of
such cold molecular ions by radiative processes will pave new
directions to ultracold chemistry.
\vspace{0.5cm}
\noindent
{\bf Acknowledgment} \\
This work is jointly supported by Department of Science and Technology (DST), Ministry of Science and Technology, Government of India and Ministry of
Higher Education and Scientific Research (MHESR), Government of Tunisia, under an India-Tunisia project for bilateral sientific cooperation.
\section*{References}
|
{
"timestamp": "2016-09-13T02:06:37",
"yymm": "1504",
"arxiv_id": "1504.03114",
"language": "en",
"url": "https://arxiv.org/abs/1504.03114"
}
|
\section{Introduction}
\label{sec:intro}
There is growing evidence for the existence of an intergalactic magnetic field from the observation of high energy gamma rays.
It is likely that a magnetic field in intergalactic space would have been created in the early universe, since astrophysics alone is not expected to generate fields on such large length scales.
(For a recent review on cosmic magnetic fields, see \rref{Durrer:2013pga}.)
The discovery of a primordial magnetic field (PMF) has important ramifications for cosmology as it allows one to test models of magnetogenesis, which are often tied to the physics of inflation \cite{Turner:1987bw} cosmological phase transitions \cite{Vachaspati:1991nm, Ahonen:1997wh}, and baryogenesis \cite{Cornwall:1997ms, Vachaspati:2001nb, Long:2013tha}.
The presence of a PMF after cosmological recombination can also aid in the formation of first stars \cite{1987QJRAS..28..197R} and provide the seed field for the galactic dynamo \cite{Ruzmaikin:book}.
Additionally, the existence of a PMF in our universe opens the opportunity to place indirect constraints on exotic particle physics models where the new physics couples to electromagnetism.
In this paper we will investigate the consequences of a PMF for models that contain magnetic monopoles, axions, and Dirac neutrinos with a magnetic moment.
Blazars that emit TeV gamma rays are expected to produce an electromagnetic cascade of lower energy gamma rays due to electron-positron pair production and the subsequent inverse Compton up-scattering of cosmic microwave background (CMB) photons \cite{1994ApJ...423L...5A, 1995Natur.374..430P, Neronov:2006hc, Elyiv:2009bx, Dolag:2009iv}.
In the presence of an intergalactic magnetic field, electrons and positrons directed toward the Earth can be deflected off of the line of sight, and those that are directed away from the Earth can be deflected back toward the line of sight.
As a result the point source flux is depleted in the GeV band, and the blazar acquires a halo of GeV gamma rays.
The non-observation of these GeV gamma rays was used to place a lower bound on the magnetic field
strength at the level of $B \gtrsim 10^{-16} \, {\rm G}$ \cite{Neronov:1900zz, Tavecchio:2010mk, Dolag:2010ni}.
This bound depends on modeling of the blazar flux stability and also the plasma instabilities during propagation,
and it may weaken substantially depending on these
assumptions \cite{Dermer:2010mm, Broderick:2011av, Miniati:2012ge, Schlickeiser:2012xx}.
The search for the GeV halo extended emission has been ongoing \cite{Ando:2010rb, Neronov:2010bi, Aleksic:2010gi, Alonso:2014ewa, Abramowski:2014uta}.
Most recently, Chen et al. \cite{Chen:2014rsa} have found evidence for the halo in a stacked analysis of known blazars at $\sim 1~{\rm GeV}$ energies and interpret it to be due to a field with strength $B \sim 10^{-17}-10^{-15} \, {\rm G}$.
For reference, measurements of the cosmic microwave background place an upper bound on the magnetic field strength at the level of $B \lesssim 10^{-9} \, {\rm G}$ \cite{Ade:2015cva}.
There are theoretical motivations for considering the possibility that the PMF is {\it helical}, {\it i.e.} there is an excess of power in either right- or left-circular polarization modes.
Helical magnetic fields emerge in many models of magnetogenesis \cite{Caprini:2014mja, Cornwall:1997ms, Vachaspati:2001nb, Long:2013tha}, and helicity conservation dramatically impacts the evolution of the PMF, aiding in its survival and growth \cite{Kahniashvili:2012uj}.
Recently, Tashiro et al. \cite{Tashiro:2013ita, Chen:2014qva} analyzed the diffuse gamma ray sky at 10-60~GeV energies to look for the parity-violating signature \cite{Tashiro:2013bxa, Tashiro:2014gfa} of a helical magnetic field.
They find evidence for an intergalactic magnetic field with strength $B \sim 10^{-14} \, {\rm G}$ on coherence scales $\lambda_B \sim 10 {\rm \, Mpc}$ and with left-handed helicity.
Although the results of Refs.~\cite{Tashiro:2013ita, Chen:2014qva} and \cite{Chen:2014rsa} appear inconsistent, it may be possible to reconcile them by noting that the weak bending approximation breaks down for $B \sim 10^{-14} \, {\rm G}$ for gamma rays at $\sim 1~{\rm GeV}$ energies \cite{Chowdhury:InPrep}.
Motivated by these recent results, we investigate if the existence of a helical PMF can be used to constrain other particle physics ideas in a cosmological setting.
Our analysis is sufficiently general that our results will remain relevant even if the gamma ray observation results should change or go away, {\it e.g.} with more data.
However, for the purpose of numerical estimates we will use $B \sim 10^{-14} \, {\rm G}$ and
$\lambda_B \sim 10 {\rm \, Mpc}$ as the fiducial field strength and coherence length scale, and
we will take the magnetic field to have maximal (left-handed) helicity.
In Sec.~\ref{sec:monopoles} we consider the interaction of a hypothetical abundance of cosmic
magnetic monopoles and a PMF.
The magnetic field does work on the monopoles, and its field strength is thereby depleted.
The constraints we obtain in this way are generally weaker than existing bounds but are obtained under a different set of assumptions.
These results are summarized in Fig.~\ref{fig:constraints}.
We also discuss how heavy magnetic monopoles can lead to anomalous scaling of the energy density in the PMF.
As observations of the PMF improve, they can be sensitive to the anomalous scaling and thus become a tool for further constraining magnetic monopoles.
In Sec.~\ref{sec:axions} we consider the interaction of an axion ($\varphi$) with the PMF through the coupling $g_{a\gamma} \varphi F {\tilde F}$.
In this analysis we include the cosmological plasma, and thus we study the equations of magnetohydrodynamics coupled to an axion.
Although we find that the axion has a negligible effect on the spectrum and evolution of the PMF, it is interesting to note that this conclusion is not sensitive to the assumed scale of Peccei-Quinn symmetry breaking, $f_{a}$, as long as $g_{a\gamma} \propto 1 / f_{a}$.
In turn, the PMF leaves the evolution of the axion condensate largely unaffected.
In principle the PMF damps the axion oscillations and the helicity of the PMF shifts the equilibrium point, but these effects are quadratic in the already-small magnetic field strength.
In Sec.~\ref{sec:neutrinos} we consider the interaction of the neutrino magnetic dipole moment, $\mu_\nu$, with the PMF.
Enqvist et al. \cite{Enqvist:1992di,Enqvist:1994mb} have shown that this interaction induces a spin-flip transition, which cannot be in equilibrium in the early universe without running afoul of constraints on the number of relativistic neutrino species.
Using their result with our fiducial value of $B \sim 10^{-14} \, {\rm G}$, we evaluate an upper bound on $\mu_{\nu}$, which is shown in \fref{fig:mag_mom}.
We work in the {\rm CGS} system with $\hbar = c = 1$. The unit of electric charge is
$e = \sqrt{\alpha \hidden{\hbar c}} \simeq 0.085 \hidden{\sqrt{\hbar c}}$ with $\alpha \simeq 1 / 137$ the fine structure constant,
and the unit of magnetic charge is
$e_{m} = \hidden{\hbar c \times} 1 / 2e \simeq 5.9 \hidden{\sqrt{\hbar c}}$.
The magnetic field is measured in Gauss, and $1 \, {\rm G} \simeq 6.93 \times 10^{-20} {\rm \, GeV}^2 \hidden{(\hbar c)^{-3/2}}$.
The reduced Planck mass is denoted by $M_P \simeq 2.4 \times 10^{18} {\rm \, GeV} \hidden{/c^2}$.
The metric signature is $(+---)$, and the antisymmetric tensor normalization is $\epsilon^{0123} = +1$.
\section{Magnetic monopoles}
\label{sec:monopoles}
A conservative cosmological bound on the energy density of magnetic monopoles is $\Omega_m \equiv \rho_m/\rho_{\rm cr} < 0.3$ where $\rho_m$ is the energy density in monopoles and $\rho_{\rm cr} \simeq 10^{-29} \, {\rm gm} \, c^2 / {\rm \, cm}^3$ is the critical cosmological energy density.
The number density of nonrelativistic monopoles is $n_m = \rho_{m} / m \hidden{c^2}$ with $m$ the monopole mass, and the cosmological bound implies
\begin{align}\label{eq:n0_cosmo}
n_m < 0.3 \frac{\rho_{\rm cr}}{m \hidden{c^2} } \simeq (2 \times 10^{-23} {\rm \, cm}^{-3}) \left( \frac{m \hidden{c^2}}{10^{17} {\rm \, GeV}} \right)^{-1} \, .
\end{align}
The bound grows weaker for lighter monopoles since they contribute less to the energy density for the same number density.
The existence of the galactic magnetic field leads to another indirect bound.
Magnetic monopoles tend to deplete a magnetic field in the same way that free electrons short out a conductor.
The survival of the micro-Gauss galactic magnetic field implies an upper bound on the directed flux, $\mathcal{F}$,
of magnetic charge onto the Milky Way.
Requiring that the time scale for B-field depletion is longer than the dynamo time scale of B-field regeneration ($\tau_{\rm dyn} \simeq 10^{8} {\rm \, yr}$), leads to the so called Parker bound \cite{Parker:1970xv}
\begin{align}
\mathcal{F} < 0.9 \times 10^{-16} e_{m} {\rm \, cm}^{-2} {\rm \, sec}^{-1} {\rm \, sr}^{-1} \, .
\end{align}
Assuming that monopoles have unit charge and travel with velocity $v$, the Parker bound can be expressed as an upper bound on the monopole number density:
\begin{align}\label{eq:Parker_bound}
n_m \approx \frac{(4 \pi \, {\rm \, sr}) \, \mathcal{F}}{e_m \, v} < (4 \times 10^{-23} {\rm \, cm}^{-3}) \left( \frac{v}{10^{-3} c} \right)^{-1}
\, .
\end{align}
Just as the Parker bound is predicated on the existence of a magnetic field in the Milky Way, we expect that a similar bound can be inferred from the existence of a primordial magnetic field in the early universe.
We study a gas of monopoles and antimonopoles immersed in a magnetic field that permeates the cosmological medium.
The monopoles have mass $m$, magnetic charge $e_{m}$, and they are homogeneously distributed with number density $n_{m}(t)$.
The magnetic field ${\bf B}(t,{\bf x})$ induces a Lorentz force of
\begin{equation}\label{eq:FB_def}
{\bf F}_{\rm B} = e_{m} {\bf B}
\end{equation}
on a monopole at $(t,{\bf x})$, and it begins to drift along the field line with a velocity ${\bf v}$.
The field does work by pushing the monopole, and in this way the monopole extracts energy from the magnetic field at a rate
\begin{align}\label{eq:EBdot}
\dot{E}_{\rm B} = {\bf F}_{\rm B} \cdot {\bf v} \, .
\end{align}
To solve for the evolution of the magnetic field strength we must know the monopole velocity.
Prior to electron-positron annihilation, the monopole's velocity is restricted by elastic scattering with the cosmological medium, but afterward it can free stream.
We will consider each of these cases in turn.
\subsection{Friction Dominated Regime}
In the epoch prior to $e^+ e^-$ annihilation the cosmological medium was dense with electromagnetically charged particles.
In this regime, monopoles interact with the medium through elastic scattering such as $M + e^{\pm} \to M + e^{\pm}$ with $M$ the monopole.
It is safe to assume that the monopole's rest mass is much larger than the kinetic energy of particles in the plasma.
This allows us to characterize the effective interaction with a drag force that takes the form \footnote{We obtain identical results when characterizing the interaction with a magnetic analog of Ohm's law, and the conductivity is taken to be $\sigma_{M} = e_{m}^2 n_{m} / f_{\rm drag}$. }
\begin{equation}
{\bf F}_{\rm drag} = - f_{\rm drag} {\bf v} \, .
\end{equation}
At the time of interest, the scatterers are relativistic with energy comparable to the temperature $T$ of the plasma, and they are in thermal equilibrium with number density $n \sim T^3 \hidden{(\hbar c)^{-3}}$.
For such a system the drag coefficient takes the form \cite{Goldman:1980sn, VilenkinShellard:1994}
\begin{align}
f_{\rm drag} \approx \beta \, e^2 e_{m}^2 \, g_{em} T^2 \hidden{(\hbar^{-3} c^{-4})}
\end{align}
where $g_{em}(t)$ is the number of relativistic, charged degrees of freedom in thermal equilibrium at time $t$ and $\beta$ is an $O(1)$ number related to the spin character and charge of the scatterers.
We will drop $\beta$ from this point onward since it is parametrically redundant with $g_{em}$.
Also note that $g_{em}$ and $T$ depend on time, but this can be ignored on short time scales as compared to the Hubble time scale.
The monopole's equation of motion is
\begin{equation}
m \, \dot{\bf v} = e_{m} {\bf B} - f_{\rm drag} {\bf v} \, .
\label{eq:dotv}
\end{equation}
We assume that the distance traveled by the monopole is small compared to the correlation length
$\lambda_B$ of the magnetic field, and we treat ${\bf B}$ as uniform. \eref{eq:dotv} immediately
gives the terminal velocity of the monopoles,
\begin{equation}\label{eq:vterm_def}
{\bf v}_{\rm term} = \frac{e_{m} {\bf B}}{e^2 e_{m}^2 g_{em} T^2} \hidden{\hbar^{3} c^{4}} \, ,
\end{equation}
which is achieved on a time scale
\begin{equation}\label{eq:tauterm_def}
\tau_{\rm term} = \frac{m}{e^2 e_{m}^2 g_{em} T^2} \hidden{\hbar^{3} c^{4}} \, .
\end{equation}
Comparing with the Hubble time $t_H \sim M_P / T^2$ (radiation era) we have $\tau_{\rm term} \ll t_H$ provided that $m < g_{em} M_P$, and thus the cosmological expansion is negligible.
At the present cosmic epoch, the photon temperature is $\sim 10^{-4}~{\rm eV}$ and $B \sim 10^{-14} \, {\rm G}$.
Assuming $B \propto T^2$, we get $B \simeq 10^{6} \, {\rm G}$ at $T \simeq \, {\rm MeV}$ when $g_{em} \simeq 10$.
These estimates give $v_{\rm term} \simeq 10^{-8} \hidden{c}$, which validates our uses of the
non-relativistic equation of motion. As a consequence, the distance traveled by a monopole during
$\tau_{\rm term}$ is quite small, $d_{\rm term} <10^{-8} \tau_{\rm term}$. We shall assume that the
correlation length of the magnetic field is larger, $\lambda_B > d_{\rm term}$, thus justifying our
treatment of the magnetic field as being uniform.
The magnetic field's response to this current is given by the magnetic analog of Ampere's law,
\begin{align}
\label{eq:dotB}
\dot{\bf B} = - 4 \pi \, {\bf j}_{M} = - 4\pi e_{m} n_m {\bf v} \, .
\end{align}
where $n_m$ is the number density of monopoles (assumed equal to the number density of antimonopoles).
Here we have used ${\bf E}=0$ since electric fields are screened due to the high electrical conductivity of the cosmological medium.
For typical parameters, the inter-monopole spacing is small compared to the correlation length of the magnetic field, $n_m^{-1/3} \ll \lambda_B$, and we can interpret $n_m$ and ${\bf B}$ as coarse grained quantities on this length scale.
Then we insert ${\bm v}={\bm v}_{\rm term}$ from
\eref{eq:vterm_def} into \eref{eq:dotB} to get the solution,
\begin{align}
{\bf B}(t) = {\bf B}(t_i) \, e^{- (t-t_i) / \tau_{\rm decay}}
\end{align}
where the decay time scale of the magnetic field is given by
\begin{equation}\label{eq:taudecay_def}
\tau_{\rm decay} = \frac{e^2 e_{m}^2 g_{em} T^2}{4 \pi e_{m}^2 n_m} \hidden{\hbar^{-3} c^{-4}} .
\end{equation}
In obtaining this solution, we have assumed ${\bm v}={\bm v}_{\rm term}$ which is justified if
the monopoles reach terminal velocity much more quickly than the decay time scale,
{\it i.e.} $\tau_{\rm decay} \gg \tau_{\rm term}$. As we will see below, this condition is satisfied for the
range of parameters of interest to us.
To ensure survival of the magnetic field, we require that $\tau_{\rm decay}$ is much larger than the Hubble time
at temperature $T$,
\begin{equation}
t_{H} = \frac{1}{2H} \simeq 1.5 \frac{M_P}{g_{\ast} T^2} \hidden{\hbar c^2}
\label{eq:tautH}
\end{equation}
with $H$ the Hubble parameter and $g_{\ast}$ the effective number of relativistic degrees of freedom.
Substitution of \eref{eq:taudecay_def} into $\tau_{\rm decay} > t_{H}$ now leads to a constraint on the number density of monopoles,
\begin{equation}
n_m < \frac{e^2 e_{m}^2}{6 \pi e_{m}^2} \frac{g_{em} g_{\ast} T^4}{M_P} \hidden{(\hbar^{-4} c^{-6})} \, ,
\end{equation}
when the universe had temperature $T$.
\begin{figure}[t]
\hspace{0pt}
\vspace{-0in}
\begin{center}
\includegraphics[width=0.47\textwidth]{fig_constraints--v5.pdf}
\caption{
\label{fig:constraints}
A summary of upper bounds on the magnetic monopole abundance from this work and the literature.
{\it Black:} the requirement of survival of the primordial magnetic field (``primordial Parker bound''), derived here in \eref{eq:n0_bound}.
{\it Red:} the cosmological abundance bound in \eref{eq:n0_cosmo}.
{\it Blue:} direct search constraints \cite{Agashe:2014kda} (see the magnetic monopole review).
{\it Green:} the requirement of survival of the Galactic magnetic field (Parker bound), given by \rref{Turner:1982ag}.
{\it Orange:} the requirement of survival of the Galactic seed field (``extended Parker bound''), given by \rref{Adams:1993fj}.
We take $v \simeq 10^{-3} \hidden{c}$ and assume that monopoles are unclustered, $f_{c} \simeq 1$.
If the monopoles are clustered then the Parker bound, extended Parker bound, and direct search limits move down by a factor of $f_{c} \sim 10^{5}$.
}
\end{center}
\end{figure}
The strongest bound is obtained when $T$ is smallest.
Since our calculation assumes that monopoles scatter on relativistic, charged particles with a thermal abundance, the last time at which this is possible is the epoch of $e^+ e^-$ annihilation.
At this time $T_{\rm ann} \simeq 1 \, {\rm MeV}$, $g_{em} \approx g_{\ast} \approx g_{\ast S} \simeq 10.75$.
To translate this into a bound on the monopole number density today, denoted by $n_0$, we multiply by
$(a_{\rm ann}/a_0)^3 = (g_{\ast S, 0} T_0^3 / g_{\ast S} T_{\rm ann}^3)$ with $g_{\ast S,0} \simeq 3.91$ and
$T_0 \simeq 2.3 \times 10^{-4} {\rm \, eV}$ the temperature of the microwave background photons today.
We find an upper bound on the monopole number density today,
\begin{align}\label{eq:n0_bound}
n_0 & < \frac{e^2 e_{m}^2}{6\pi e_{m}^2} \frac{g_{\ast S, 0} g_{\ast} g_{em}}{g_{\ast S}} \frac{T_{\rm ann} T_0^3}{M_p} \hidden{(\hbar^{-4} c^{-6})}
\simeq 1 \times 10^{-20} {\rm \, cm}^{-3} \, .
\end{align}
If this bound is not satisfied then any primordial magnetic field would have been exponentially depleted by the time of electron-positron annihilation.
Due to the close connection with the Parker bound for survival of galactic magnetic fields, we will refer to \eref{eq:n0_bound} as the ``primordial Parker bound.''
In \fref{fig:constraints} we compare the primordial Parker bound in \eref{eq:n0_bound} with other constraints derived previously in the literature.
Since these constraints are typically expressed as a bound on the monopole flux, we translate into a bound on the number density using $(4 \pi \, {\rm sr}) \mathcal{F} \approx f_{c} n v$.
Here $v$ is the average monopole velocity and $f_{c} = n_{\rm galaxy} / n_{\rm cosmo.}$ is the enhancement factor that accounts for clustering of monopoles in the galaxy.
For clustered monopoles $f_{c} \sim 10^{5}$, but otherwise $f_{c} \sim 1$.
For the extended Parker bound calculation of \rref{Adams:1993fj}, we take $B_{\rm seed} = 10^{-11} \, {\rm G}$.
For the direct search constraint we show a relatively conservative and robust limit of $\mathcal{F} < 10^{-15} {\rm \, cm}^{-2} {\rm \, sec}^{-1} {\rm \, sr}^{-1}$, but stronger constraints are available for specific monopole parameters \cite{Agashe:2014kda}.
From the figure, one can see that the primordial Parker bound becomes stronger than the cosmological bound for light monopoles, $m \lesssim 5 \times 10^{13} {\rm \, GeV} \hidden{/c^2}$, but it always remains weaker than the direct search constraints.
\subsection{Free Streaming Regime}
After cosmological electron-positron annihilation the number density of these scatterers decreases by a factor of $\sim 10^{-10}$.
The monopoles experience very little drag force, and they can be accelerated freely by the magnetic field.
For a uniform and static magnetic field, the solution of \eref{eq:dotv} with $f_{\rm drag} = 0$ is simply $v = e_{m} B t / m$, or for an inhomogenous field with domains of size $\lambda_{B}$ we find
\begin{equation}
v(t) \sim \frac{e_m B \lambda_B}{m} \sqrt{\frac{t}{\lambda_B}}
\label{eq:vmon}
\end{equation}
if the motion is diffusive.
The monopole becomes relativistic when $v(t_{\rm rel}) \sim 1 \hidden{\times c}$ and comparing this time with the present age of the universe gives
\begin{equation}
\frac{t_{\rm rel}}{t_0} \sim \left ( \frac{m}{e_m Bt_0} \right )^2 \frac{t_0}{\lambda_B}
\sim 10^{18} \left ( \frac{m}{M_P} \right )^2 \frac{t_0}{\lambda_B}.
\end{equation}
With $\lambda_B \sim {\rm Mpc}$ and $t_0 \sim 10 \, {\rm Gpc}$, we find that monopoles are relativistic today if $m \lesssim 10^8 {\rm \, GeV}$, and they are non-relativistic otherwise.
The above estimate ignores backreaction of the monopoles on the PMF.
To check for consistency, we compare the kinetic energy in monopoles $\rho_{\rm kin}$ to the
energy density available in the PMF $\rho_{B} = B^2 / 8 \pi$.
For relativistic monopoles we should have $\rho_B > \rho_{\rm kin} \gg m \, n_m$, and this provides an upper bound on the number density of monopoles for which the velocity estimate in \eref{eq:vmon} can be expected to hold.
Taking $B \sim 10^{-14} \, {\rm G}$ we find
\begin{equation}
n_m \ll 10^{-35} \left ( \frac{10^8 ~{\rm GeV}}{m}\right ) ~ {\rm cm^{-3}} \, .
\end{equation}
Since are interested in much larger number densities, as indicated by the bound in \eref{eq:n0_bound}, we cannot use the velocity relation in \eref{eq:vmon}, but instead the monopole and magnetic field equations will need to be evolved simultaneously.
Without friction to provide a means of energy dissipation, the monopoles cannot deplete the magnetic field strength.
Instead there is a conservative exchange of energy between the magnetic field and the kinetic energy of the monopoles.
This co-evolution can lead to an anomalous departure from the usual power law scaling behavior of the magnetic field energy density if the monopoles are non-relativistic.
This can be seen from the following argument.
In the absence of the monopole gas, the energy density in the magnetic field redshifts like radiation $\rho_{B} \sim ( 1 + z)^{4}$ where $z$ is the cosmological redshift.
Meanwhile the kinetic energy stored in a gas of non-relativistic particles redshifts more quickly.
We can write the kinetic energy density as $\rho_{\rm kin} = n_m p^2/ (2m)$ where $p$ is the typical momentum and $n_m$ is the monopole number density.
Since $p \sim (1+z)$ and $n_m \sim (1+z)^3$, the kinetic energy density redshifts like $\rho_{\rm kin} \sim (1+z)^{5}$.
If energy is transferred quickly between the monopoles and magnetic field then we might expect $\rho_{B} \sim \rho_{\rm kin.} \sim (1+z)^{9/2}$ where $9/2$ is the average of $4$ and $5$.
This result is confirmed by the full calculation, which we will now present.
Once again we consider a gas of monopoles coupled to a magnetic field, but we now include the effects of cosmological expansion.
Let $X^{\lambda}(\tau)$ be the world line of a monopole, and let $U^{\lambda}(\tau) = dX^{\lambda}/d\tau$ be its 4-velocity.
The monopole equation of motion, given previously by \eref{eq:dotv}, is now replaced by
\begin{align}
& m \left[ \frac{d U_{m}^{\lambda}}{d\tau} + \Gamma^{\lambda}_{\mu \nu} U_{m}^{\mu} U_{m}^{\nu} \right]
= e_{m} \tensor{\widetilde{F}}{^{\lambda}_{\mu}} U_{m}^{\mu} \hidden{c^{-1}}
\end{align}
where $\Gamma^{\lambda}_{\mu \nu}$ is the Christoffel symbol.
The magnetic analog of Ampere's law in \eref{eq:dotB} is now replaced by
\begin{align}
& \nabla_{\alpha} \widetilde{F}^{\alpha \beta} = 4\pi \hidden{c^{-1}} j_{M}^{\beta}
\end{align}
where $\nabla_{\alpha}$ is the covariant derivative.
The magnetic current density $j_{M}^{\mu}$ arises from monopoles with velocity $U_{m}^{\mu}$ and antimonopoles with velocity $U_{\bar{m}}^{\mu}$.
It can be written as
\begin{align}
j_{M}^{\mu} & = e_{m} \left( n_{m} \, U_{m}^{\mu} - n_{\bar{m}} \, U_{\bar{m}}^{\mu} \right) \, ,
\end{align}
and it satisfies the conservation law
\begin{align}
& \nabla_{\mu} j_{M}^{\mu} = 0 \, .
\end{align}
The spatial component of the 4-velocity is the {\it comoving} peculiar velocity $({\bf U}_{m})^{i} = U_{m}^{i}$, and with an additional factor of $a$ we form the ${\it physical}$ peculiar velocity $({\bf v}_{m})^{i} = a U_{m}^{i}$.
Neglecting the electric field and spatial gradients, the system of equations can be put into the form
\begin{subequations}\label{eq:coevolution}
\begin{align}
& \partial_{\eta} (a {\bf v}_{m}) = \frac{e_{m}}{m} (a^2 {\bf B}) \\
& a \partial_{\eta} (a^2 {\bf B}) = - 4\pi e_{m} (a^3 n_{m}) \, (a {\bf v}_{m}) \\
& \partial_{\eta} \left( a^3 n_{m} \right) = 0
\end{align}
\end{subequations}
where $d\eta = dt / a = da / Ha^2$ is the conformal time coordinate.
The third equation implies that the number of monopoles per comoving volume is conserved; $n_0 = a^3 n_{m}$ is the number density of monopoles today.
Then if not for the additional factor of $a$ in the second equation, the solutions would simply be oscillatory with angular frequency
\begin{align}
\omega_{\rm pl} = \sqrt{ \frac{4\pi e_{m}^2 n_0}{m}} \, .
\end{align}
This is just the usual formula for plasma frequency, but instead of electron charge, mass, and density, here we find the corresponding parameters for the monopole gas.
To solve these equations we must relate $a$ to $\eta$.
During the radiation era we have $\eta = \eta_{i} + (a-a_{i}) / H_{i} a_{i}^2 \approx a / H_{i} a_{i}^2$, and the solution is
\begin{subequations}\label{eq:Rera_B_and_v}
\begin{align}
{\bf B} & =
\left( \frac{a}{a_{i}} \right)^{-9/4} \left[ \frac{ J_{0}(\phi) \ {\bf B}_{1}
+ Y_{0}(\phi) \ {\bf B}_{2} }{ (a/a_{i})^{-1/4} } \right]
\\
{\bf v}_{m} & =
\left( \frac{a}{a_{i}} \right)^{-3/4} \left[ \frac{ \phi \ J_{1}(\phi) \ {\bf v}_{1}
+ \phi \ Y_{1}(\phi) \ {\bf v}_{2} }{(a/a_{i})^{1/4}} \right]
\end{align}
\end{subequations}
where
\begin{align}
\phi \equiv \frac{ 2 \tilde{\omega} \, \eta_{i}}{a_{i}} \sqrt{a} \, .
\end{align}
At late times $\phi \gg 1$, and all of the Bessel functions go to zero with an envelop $\sim \phi^{-1/2} \sim a^{-1/4}$.
Then the terms in square brackets do not scale with $a$ and we find $B \sim a^{-9/4}$ and $v_{m} \sim a^{-3/4}$.
One can check that the energy densities scale in the same way
\begin{subequations}\label{eq:anom_scaling}
\begin{align}
& \rho_{B} = \frac{|{\bf B}|^2}{8\pi} \propto a^{-9/2} \\
& \rho_{\rm kin.} = \frac{m}{2} |{\bf v}_{m}|^2 \, n_{m} \propto a^{-9/2} \, ,
\end{align}
\end{subequations}
which confirms our earlier argument.
During the matter era we have $\eta = \eta_{i} + 2 (\sqrt{a} - \sqrt{a_i}) / H_i a_i^{3/2} \approx 2 \sqrt{a} / H_i a_{i}^{3/2}$, and the solution is
\begin{subequations}\label{eq:Mera_B_and_v}
\begin{align}
{\bf B} & =
\left( \frac{a}{a_i} \right)^{-9/4}
\sum_{s=\pm1} \left( \frac{a}{a_i} \right)^{\frac{i}{4} s \sqrt{ 4 \tilde{\omega}^2 \eta_i^2 / a_i - 1} } \! {\bf B}_{s} \\
{\bf v}_{m} & =
\left( \frac{a}{a_i} \right)^{-3/4}
\sum_{s=\pm1} \left( \frac{a}{a_i} \right)^{\frac{i}{4} s \sqrt{ 4 \tilde{\omega}^2 \eta_i^2 / a_i - 1} } \! {\bf v}_{s} \, .
\end{align}
\end{subequations}
For typical parameters we have $4 \tilde{\omega}^2 \eta_{i}^2 / a_{i} \gg 1$, and the solution is oscillatory with a power law envelope.
We find the same anomalous scaling as in the radiation era, {\it cf.} \eref{eq:anom_scaling}.
If we remove the monopoles from the problem by sending $n_{m} , \tilde{\omega} \to 0$ then the would-be oscillatory factors become a power law decay, and we regain the usual scaling $B \sim a^{-2}$ and $v_{m} \sim a^{-1}$.
This anomalous scaling does not provide constraints on the monopole number density.
However, it does affect the way that we translate constraints on the magnetic field in the early universe into the value of the magnetic field today.
Measurements of the cosmic microwave background restrict the magnetic field energy density to be less than $\sim 10^{-5}$ of the photon energy density at the time of recombination \cite{Ade:2015cva}
\begin{align}
\rho_{B}(z_{\rm rec}) \lesssim 10^{-5} \rho_{\gamma}(z_{\rm rec}) \, .
\end{align}
Using the scaling relations, $\rho_{B} \sim (1+z)^{9/2}$ and $\rho_{\gamma} \sim (1+z)^{4}$ this inequality implies that the magnetic field energy density today is bounded by
\begin{align}\label{eq:tighten_CMB}
\rho_{B,0} & \lesssim 10^{-5} \rho_{\gamma,0} (1+z_{\rm rec})^{-1/2} \nonumber \\
& \simeq (3 \times 10^{-10} \, {\rm G})^2 \left( \frac{1+z_{\rm rec}}{1300} \right)^{-1/2}
\end{align}
where $\rho_{\gamma,0} \approx 2\pi^2 T_0^4/30$ is the CMB energy density today.
If it were not for the anomalous redshifting, the constraint on the B-field strength would be weaker by a factor of $(1+z_{\rm rec})^{1/4} \simeq 6$.
The anomalous scaling of the field strength in \eref{eq:anom_scaling} can become a tool in the future as measurements of the PMF improve.
By measuring the magnetic field strength at different redshifts, say by using TeV blazars at different distances, we can directly probe the anomalous scaling, and hence obtain a new handle on the relic density of non-relativistic magnetic monopoles.
\section{Axions}
\label{sec:axions}
Consider an axion $\varphi(x)$ coupled to the electromagnetic field $A_{\mu}(x)$.
The Lagrangian takes the form \cite{Kim:2008hd}
\begin{align}\label{axLag}
\mathcal{L} & = \frac{1}{2} (\partial_{\mu} \varphi)^2
- \frac{m_a^2}{2} \varphi^2 \hidden{\frac{c^2}{\hbar^2}}
- \frac{1}{16 \pi}F_{\mu\nu} F^{\mu\nu}
\nonumber \\ & \quad
- \frac{g_{a\gamma}}{16\pi} \varphi F_{\mu\nu} {\widetilde F}^{\mu\nu}
- \hidden{\frac{1}{c}} A_{\mu} j^{\mu}
\end{align}
where $\widetilde{F}^{\mu \nu} = \frac{1}{2} \epsilon^{\mu \nu \alpha \beta} F_{\alpha \beta}$ is the dual field strength tensor, and $j^{\mu} = (\hidden{c}\rho,{\bf j})$ is the electromagnetic current arising from the charged Standard Model fields.
Our analysis is sufficiently general to apply to any axion or axion-like particle described by \eref{axLag}, but as a fiducial reference point we will consider a QCD axion with Peccei-Quinn scale of $f_a \simeq 10^{10} {\rm \, GeV} \hidden{/\sqrt{\hbar c}}$, an axion mass of $m_a \hidden{c^2} \approx \Lambda_{\text{\sc qcd}}^2 / f_a \simeq 1 {\rm \, meV}$, and a photon-axion coupling constant $g_{a\gamma} \approx \alpha / (2 \pi f_a) \simeq 10^{-13} {\rm \, GeV}^{-1} \hidden{\sqrt{\hbar c}}$.
The classical axion condensate obeys the field equation
\begin{equation}
\Box \varphi + m_a^2 \varphi \hidden{\frac{c^2}{\hbar^2}} = \frac{g_{a\gamma}}{4\pi} {\bf E} \cdot {\bf B}
\label{axeq}
\end{equation}
where we have used $F_{\mu \nu} \widetilde{F}^{\mu \nu} = - 4 {\bf E} \cdot {\bf B}$.
The electromagnetic field evolves according to the constraint equation
\begin{align}
\partial_{\mu} \widetilde{F}^{\mu \nu} = 0
\end{align}
and the modified field equation
\begin{align}
\partial_{\mu} F^{\mu \nu} + g_{a\gamma} \, \partial_{\mu} \varphi \widetilde{F}^{\mu \nu} = \frac{4\pi}{c} j^{\nu} \, .
\end{align}
In terms of the electric and magnetic vector fields we have
\begin{subequations}\label{eq:EM_eqns}
\begin{align}
& {\bm \nabla} \cdot {\bf B} = 0 \label{divBeq} \\
& {\bm \nabla}\times {\bf E} + \frac{1}{c} \frac{\partial {\bf B}}{\partial t} = 0 \label{dotBeq} \\
& {\bm \nabla} \cdot {\bf E} + g_{a\gamma} {\bm \nabla} \varphi \cdot {\bf B} = 4\pi \rho \label{divEeq} \\
& {\bm \nabla} \times {\bf B} - \frac{1}{c} \frac{\partial {\bf E}}{\partial t} - g_{a\gamma} \dot{\varphi} \hidden{\frac{1}{c}} {\bf B} - g_{a\gamma} {\bm \nabla} \varphi \times {\bf E} = \frac{4\pi}{c} {\bf j} \label{dotEeq} \, .
\end{align}
\end{subequations}
Note the presence of the additional terms arising from the spatio-temporal variation of the axion field.
We seek to study the coevolution of the coupled axion and electromagnetic fields.
\erefs{axeq}{eq:EM_eqns} describe a non-dissipative system.
Dissipation is introduced as the electromagnetic field couples to charged particles in the cosmological medium, which opens an avenue for energy to be lost in the form of heat.
This coupling is parametrized by the conductivity $\sigma$ which appears in Ohm's law
\begin{equation}
{\bm j} = \sigma \left ( {\bf E} + \hidden{\frac{1}{c}} {\bf v} \times {\bf B} \right )
\label{jeq}
\end{equation}
where ${\bf v}(t,{\bf x})$ is the local velocity of the plasma.
Prior to the epoch of $e^+ e^-$ annihilation, free charge carriers were abundant and the cosmological medium had a high conductivity \cite{Baym:1997gq}
\begin{align}\label{eq:sigma_def}
\sigma \approx \hidden{\hbar^{-1}} T / \alpha
\end{align}
where $\alpha \simeq 1/137$ is the fine structure constant.
Ohm's law allows us to eliminate ${\bf j}$ and thereby reduce the system of equations in four unknowns $\{ {\bf E}, {\bf B}, {\bf j}, \varphi \}$ to a set of equations describing only three unknowns:
\begin{subequations}
\begin{align}
& \ddot{\varphi} - \hidden{c^2} \nabla^2 \varphi + m_a^2 \varphi \hidden{\frac{c^4}{\hbar^2}} = \hidden{c^2} \frac{g_{a\gamma}}{4 \pi} {\bf E} \cdot {\bf B} \label{eq:eom1a} \\
& \dot{{\bf B}} = - \hidden{c} \, {\bm \nabla} \times {\bf E} \label{eq:eom2a} \\
& \dot{{\bf E}} = \hidden{c} \, {\bm \nabla} \times {\bf B} - g_{a\gamma} \dot{\varphi} {\bf B} - \hidden{c} g_{a\gamma} {\bm \nabla} \varphi \times {\bf E} \nonumber \\
& \qquad - 4 \pi \sigma {\bf E} - 4 \pi \sigma {\bf v} \times {\bf B} \hidden{c^{-1}} \label{eq:eom3a} \, .
\end{align}
\end{subequations}
In the MHD approximation (nonrelativistic flow) we can neglect the displacement current since it is negligible compared to the curl of the magnetic field, $|\dot{{\bf E}}| / \hidden{c} |{\bm \nabla} \times {\bf B}| \sim (v/c)^2 \ll 1$ \cite{Choudhuri:1998}.
Then \eref{eq:eom3a} becomes algebraic in ${\bf E}$, and we can solve it to eliminate ${\bf E}$ from
the remaining equations.
Focusing now on a homogenous axion field, the system of equations reduces to
\begin{subequations}
\begin{align}
& \ddot{\varphi} + g_{a\gamma}^2 \frac{\eta_{\rm d} |{\bf B}|^2}{4 \pi} \dot{\varphi} + m_a^2 \varphi \hidden{\frac{c^4}{\hbar^2}} =
\hidden{c} \frac{g_{a\gamma} \eta_{\rm d}}{4 \pi} {\bf B} \cdot {\bm \nabla} \times {\bf B}
\label{axmhdeq} \\
& \dot{{\bf B}} =
{\bm \nabla} \times \left( {\bf v} \times {\bf B} \right)
+ \eta_{\rm d} \, \nabla^2 {\bf B}
+ \hidden{\frac{1}{c}} g_{a\gamma} \eta_{\rm d} \dot{\varphi} \, {\bm \nabla} \times {\bf B}
\label{mhdeq}
\end{align}
\end{subequations}
where
\begin{align}\label{eq:eta_def}
\eta_{\rm d} \equiv \hidden{c^2 \times} \frac{1}{4 \pi \sigma} \approx \frac{\alpha \hidden{\hbar c^2}}{4\pi T}
\end{align}
is the magnetic diffusivity, assumed to be homogenous.
\erefs{axmhdeq}{mhdeq} together with the Navier-Stokes equation for the plasma velocity ${\bf v}$ are the final equations to be solved. We first solve \eref{mhdeq} to determine the effect of the axion on the magnetic field, and afterward we will consider the evolution of the axion according to \eref{axmhdeq}.
\subsection{Effect of Axion on Magnetic Field}
In order to solve \eref{mhdeq} for the B-field we must know the fluid velocity, which appears in the advection term, ${\bm \nabla} \times ({\bf v} \times {\bf B})$.
Since our primary interest is in the coevolution of the magnetic field and the axion, not in the magnetohydrodynamics, we will neglect this term \footnote{Strictly speaking the advective term is negligible compared to the diffusive term, $\eta_{\rm d} \nabla^2 {\bf B}$, only when $v < \eta_{\rm d} / \lambda_B \sim \hidden{\hbar c^2} (10^3 T \lambda_B)^{-1}$ where $\lambda_{B}$ is the typical length scale of the magnetic field. Since $\lambda_{B} \gg T^{-1}$ is a macroscopic scale, this imposes a very strong upper bound on $v$, which is not easily satisfied. We expect that inclusion of the advection term, perhaps in the context of a turbulent medium, will lead to a more rich and interesting solution, but that analysis is beyond the scope of this work. }.
Since \eref{mhdeq} is linear in ${\bf B}$, and we assume that $\varphi$ is homogenous, we can solve the equation by first performing a Fourier transform.
For a given mode ${\bf k}$ let $({\bf e}_1({\bf k}),{\bf e}_2({\bf k}),{\bf e}_3({\bf k}))$ form a right-handed, orthonormal triad of unit vectors with ${\bf e}_3({\bf k}) = {\bf k}/|{\bf k}|$.
It is convenient to introduce the right- and left-circular polarization vectors by
\begin{equation}
{\bf e}^{\pm} ({\bf k}) = \frac{{\bf e}_1({\bf k}) \pm i {\bf e}_2({\bf k})}{\sqrt{2}} \, .
\end{equation}
Note that $i {\bf k} \times {\bf e}^{\pm}({\bf k}) = \pm |{\bf k}| {\bf e}^{\pm}({\bf k})$.
The mode decomposition is given by
\begin{align}
{\bf B}(t,{\bf x}) = \int \! \! \frac{d^3 k}{(2 \pi)^3} e^{i{\bf k} \cdot {\bf x}} \sum_{{\rm s}=\pm} b_{\rm s}(t,|{\bf k}|) {\bf e}^{\rm s}({\bf k}) \, .
\end{align}
With this replacement, \eref{mhdeq} becomes
\begin{align}
\dot{b}_{\pm}(t,k) = - \eta_{\rm d} k^2 {b}_{\pm}(t,k) \pm \hidden{\frac{1}{c}} g_{a\gamma} \eta_{\rm d} \, k \, \dot{\varphi} \, b_{\pm}(t,k)
\end{align}
where we have written $k = |{\bf k}|$.
The last term of this equation, essentially the chiral-magnetic effect \cite{Vilenkin:1980fu}, has
been studied previously in the context of axions \cite{Field:1998hi} and cosmology
\cite{Joyce:1997uy,Tashiro:2012mf}.
The solution is
\begin{align}\label{bpmsoln}
& b_{\pm} (t, k) = b_{\pm}(t_i, k) \ e^{-k^2 (t-t_i) \eta_{\rm d}} \ e^{\pm k / k_{\rm ax}(t)}
\end{align}
where we have defined the wavenumber
\begin{equation}\label{eq:kax_def}
k_{\rm ax}(t) \equiv \frac{2 \hidden{c}}{g_{a\gamma} \Delta \varphi(t) \eta_{\rm d}}
\end{equation}
and $\Delta \varphi(t) = \varphi(t) - \varphi(t_i)$ is the change in the axion field.
The prefactor in \eref{bpmsoln} is the initial spectrum of the magnetic field which will depend on the PMF generation mechanism.
The first exponential is the usual diffusive decay term, which exponentially suppresses modes on a length scale shorter than $k_{\rm diff}^{-1} = \sqrt{(t-t_i) \eta_{\rm d}} \simeq 10^{-1} \sqrt{\hidden{\hbar c^2} t/T}$.
The second exponential only kicks in at small length scales where $k > k_{\rm ax}$.
Then it leads to a suppression of one polarization mode and enhancement of the other, depending on the sign of $\Delta \varphi(t)$.
The value of $\Delta \varphi(t)$ depends on the solution for the axion field as well as the initial time $t_i$.
We expect that the misalignment mechanism sets the initial condition $\varphi(t_i) \sim \pm f_a$.
The subsequent evolution is determined by solving \eref{axmhdeq}, which we will turn to in the next section.
For the moment we will assume that the axion evolution is not significantly affected by the presence of the magnetic field, and the solution is the standard one: the axion remains ``frozen'' at $\varphi(t_i)$ until the time of the QCD phase transition when it begins to oscillate around $\varphi = 0$ with angular frequency $\omega = m_a \hidden{c^2/\hbar}$ \cite{Kolb:1990}.
Then we can approximate
\begin{equation}\label{eq:Dphiapprox}
\Delta \varphi(t) \approx \begin{cases}
0 & t < t_{\text{\sc qcd}} \\
{\rm s} f_{a} & t > t_{\text{\sc qcd}}
\end{cases}
\end{equation}
where ${\rm s} = {\rm sign}[\varphi(t_i)]$.
Using this approximation we can estimate $k_{\rm ax}$.
Prior to the QCD phase transition, $\Delta \varphi$ is small and $k_{\rm ax}$ is large, meaning that none of the modes receive the enhancement or suppression from the axion coupling.
This is reasonable since the axion is derivatively coupled, and as long as it is stationary there will be no effect on the magnetic field.
After the QCD transition, we can estimate
\begin{align}\label{eq:kax_estimate}
k_{\rm ax} \approx \frac{4 \pi \sigma}{g_{a\gamma} f_{a} \hidden{c}} \approx \frac{4 \pi T}{\alpha g_{a\gamma} f_{a} \hidden{\hbar c}}
\end{align}
using \erefs{eq:sigma_def}{eq:eta_def}.
Note that this result is insensitive to the Peccei-Quinn scale, and as long as $g_{a\gamma} = \alpha / (2\pi f_a)$ we have $k_{\rm ax}^{-1} \approx \alpha^2 / (8 \pi^2 T) \simeq 10^{-6}T^{-1} \hidden{\hbar c}$.
The solution in \eref{bpmsoln} can also be written as
\begin{equation}
b_{\pm} (t, k) = b_{\pm}(t_i, k) \ e^{K^2 (t-t_i) \eta_{\rm d}} \ e^{- (k \mp K)^2 (t-t_i) \eta_{\rm d}}
\end{equation}
where
\begin{equation}\label{Kformula}
K(t)
\equiv \frac{1}{k_{\rm ax} (t-t_{i}) \eta_{\rm d}}
= \frac{g_{a\gamma} \Delta \varphi(t)}{2 \hidden{c} (t-t_i)} \, .
\end{equation}
This representation of the solution is convenient, because all the spectral information is contained in the second factor.
One of the helicity modes has a Gaussian spectrum peaked at $|K(t)|>0$ with width $\sqrt{1/(t-t_i) \eta_{\rm d}}$, and the other helicity mode peaks at $k = 0$.
Estimating $k_{\rm ax}$ as above, we find that the associated length scale of the spectral peak corresponds to $K^{-1} \approx k_{\rm ax} t \eta_{\rm d} \simeq 600 \hidden{c} t$, which is larger than the scale of the cosmological horizon $d_{H} \sim \hidden{c} t$.
It appears that the presence of an axion condensate coupled to electromagnetism has a negligible impact on the evolution of a primordial magnetic field, unless there are situations in which $\Delta \varphi$ can be much larger than $f_{a}$.
We note that our analysis ignores the possibility of turbulence in the primordial plasma. It
would be of interest to include both turbulence and the axion coupling in future studies.
\subsection{Effect of Magnetic Field on Axion}
Next we will investigate the effect of a background magnetic field on the axion condensate.
We have seen that the magnetic field is approximately unmodified on length scales larger than the diffusion length, $k_{\rm diff}^{-1} \sim \sqrt{\eta_{\rm d} t}$ ({\it cf.}, \eref{bpmsoln}).
In this regime \eref{axmhdeq} can be rewritten as
\begin{align}\label{eq:a_reduced}
& \ddot{\varphi} + 2 \frac{\dot{\varphi}}{\tau_{\rm decay}} + \frac{\varphi}{\tau_a^2} = \mathcal{H}
\end{align}
where
\begin{align}
& \tau_{\rm decay} \equiv \frac{8 \pi}{g_{a\gamma}^2 \eta_{\rm d} \langle |{\bf B}|^2\rangle } \, , \\
& \tau_{a} \equiv \hidden{\frac{\hbar}{c^2}} \frac{1}{m_a} \, , \\
& \mathcal{H} \equiv \hidden{c} \frac{g_{a\gamma} \eta_{\rm d}}{4 \pi} \langle {\bf B} \cdot {\bm \nabla} \times {\bf B} \rangle ,
\end{align}
and the angled brackets $\langle \cdot \rangle$ denote spatial averaging.
The axion condensate evolves like a damped and driven harmonic oscillator, where the damping and driving forces are induced by the magnetic field background.
As we discuss below, it is interesting that the driving force is associated with the {\it helicity} of the magnetic field.
The magnetic-induced damping of axion oscillations is parametrized by the time scale $\tau_{\rm decay}$.
To determine when this damping will be relevant for the evolution of the axion, we compare it with the cosmological time scale, given by \eref{eq:tautH}.
To express $\langle |{\bf B}|^2\rangle = B^2$ in terms of the magnetic field strength today, $B_0$, we use $B = B_0 (a_0 / a)^2 \simeq 10 B_0 (T / T_0)^2$
where the factor of $10$ is related to the number of relativistic degrees of freedom in the early universe and today.
Then the ratio is found to be
\begin{align}
\frac{\tau_{\rm decay}}{t_H}
& \approx \frac{16 \pi^2 g_{\ast}}{75} \frac{T_0^4}{\alpha g_{a\gamma}^2 M_P B_0^2 T} \hidden{ \frac{1}{\hbar^2 c^4} } \\
& \simeq 10^{16} \frac{(10^{-13} {\rm \, GeV}^{-1} \hidden{\sqrt{\hbar c}})^2}{g_{a\gamma}^2} \frac{(10^{10} {\rm \, GeV})}{T} \frac{(10^{-14} \, {\rm G})^2}{B_{0}^2} \, . \nonumber
\end{align}
This estimate suggests that the magnetic-induced decay of the axion field is negligible for a typical Peccei-Quinn scale and B-field strength.
If the B-field strength today were as large as $B_{0} \sim 10^{-9} \, {\rm G}$ and the Peccei-Quinn scale was as low as $f_{a} \sim {\rm \, TeV} \hidden{\sqrt{\hbar c}}$, then $\tau_{\rm decay}$ would be comparable to the Hubble time at $T \approx f_{a} \hidden{\sqrt{\hbar c}}$.
As the temperature decreases, the magnetic-induced decay becomes less relevant.
It is interesting that the magnetic field also induces a driving force, parametrized by $\mathcal{H}$.
The pseudoscalar $\mathcal{H}$ is related to the helicity of the magnetic field.
This is perhaps more evident from the initial form of the axion field equation, \eref{axeq}, where ${\bf E} \cdot {\bf B}$ is equal to the rate of change of the helicity density $-(1/2\hidden{c}) d( {\bf A} \cdot {\bf B})/dt$ plus a divergence, which vanishes upon spatial averaging.
If the power in the magnetic field is localized on a particular length scale $\lambda_B$ we can estimate
$\langle {\bf B} \cdot {\bm \nabla} \times {\bf B} \rangle \sim B^2 / \lambda_B \sim 300 (B_0^2 / \lambda_{B_0})(T / T_0)^5$ where we used $\lambda_{B} \sim 3 \lambda_{B,0} (T_0/T)$.
Prior to the QCD phase transition we can neglect the mass and drag terms in \eref{eq:a_reduced}, and the solution is simply $\varphi = \mathcal{H} t^2 / 2$.
Since the axion is massless, there is no restorative potential, and the helical magnetic field leads to an unbounded growth of the axion condensate.
Although this analysis neglects the Hubble drag, we can estimate the maximum field excursion in one one Hubble time to be $\mathcal{H} t_H^2 / 2$.
Comparing with the Peccei-Quinn breaking scale, the corresponding angular excursion is
\begin{align}
\Delta \theta
& \approx \frac{\mathcal{H} t_H^2}{f_a}
\approx \frac{75 \hidden{\hbar^3 c^7} }{16 \pi^2} \frac{\alpha g_{a\gamma} M_P^2 B_0^2}{f_a g_{\ast}^2 T_0^5 \lambda_{B,0}} \nonumber \\
& \simeq 10^{-35}
\left( \frac{B_0}{10^{-14} \, {\rm G}} \right)^{2}
\left( \frac{\lambda_{B,0}}{10 {\rm \, Mpc}} \right)^{-1}
\nonumber \\ & \qquad \times
\left( \frac{g_{a\gamma}}{10^{-13} {\rm \, GeV}^{-1} \hidden{\sqrt{\hbar c}}} \right)
\left( \frac{f_{a}}{10^{10} {\rm \, GeV} \hidden{/\sqrt{\hbar c}}} \right)^{-1}
\, .
\end{align}
We are led to conclude that for realistic parameters, the helical PMF does not significantly impact the evolution of the axion condensate prior to the QCD phase transition.
After the QCD phase transition, the axion mass reaches its asymptotic value, and the source term displaces the minimum of the axion potential from $\varphi = 0$ to $\varphi_{\rm min} = \mathcal{H} \tau_{a}^2 = \hidden{\hbar^{2}} \mathcal{H} / m_{a}^{2} \hidden{c^4}$.
In terms of the angular coordinate:
\begin{align}\label{eq:theta_min}
\theta_{\rm min}
& \approx \frac{\mathcal{H} \tau_{a}^2}{f_a}
\approx \frac{25 \hidden{\hbar^3}}{12\pi^2 \hidden{c}} \frac{\alpha g_{a\gamma} B_0^2 T^4}{f_a m_a^2 \lambda_{B,0} T_{0}^5} \\
& \simeq 10^{-47}
\left( \frac{B_0}{10^{-14} \, {\rm G}} \right)^2
\left( \frac{\lambda_{B,0}}{10 {\rm \, Mpc}} \right)^{-1}
\left( \frac{T}{200 \, {\rm MeV}} \right)^4 \nonumber \\
& \quad \times
\left( \frac{f_{a}}{10^{10} {\rm \, GeV} \hidden{/\sqrt{\hbar c}}} \right)^{-1}
\left( \frac{g_{a\gamma}}{10^{-13} {\rm \, GeV}^{-1} \hidden{\sqrt{\hbar c}}} \right)
\left( \frac{m_{a}}{1 {\rm \, meV}} \right)^{-2} \nonumber \, .
\end{align}
The temperature dependence enters through $B \sim T^2$, $\lambda_{B} \sim 1 / T$, and $\eta_{\rm d} \sim 1 / T$, and the fractional shift is largest at high temperature where $B$ is large and $\lambda_{B}$ is small.
Immediately after the QCD phase transition, $T_{\text{\sc qcd}} \sim 200 \, {\rm MeV}$, the fractional shift is already extremely small.
Moreover if the PMF is not helical then $\mathcal{H} = 0$ and there is no shift in the axion potential.
It is interesting that the estimate of \eref{eq:theta_min} is insensitive to the Peccei-Quinn scale; as long as $g_{a\gamma} = \alpha / (2\pi f_a)$ and $m_a = \Lambda_{\text{\sc qcd}}^2 / f_a$ we have $g_{a\gamma} / f_a m_a^2 = \alpha / (2 \pi \Lambda_{\text{\sc qcd}}^4)$.
Then the primarily challenge toward obtaining a large effect is the smallness of the magnetic field strength.
Although unrelated to primordial magnetic fields, which is the motivation for this work, it would be interesting to study the axion condensate in an astrophysical system where the magnetic field is both helical and strong.
For instance, the field strength in a magnetar can grow as large as $B \sim 10^{15} \, {\rm G}$ and
the magnetic field in some astrophysical jets is known to be helical \cite{2012SSRv..169...27P}.
\subsection{Axion-Photon Interconversion}
Until this point we have focused our attention on the interplay between the axion condensate and the primordial magnetic field, and we now turn our attention to the quanta of these fields.
In the presence of a background magnetic field, the interaction in \eref{axLag} yields a mixing between axion particles and photons.
Typically the conversion is inefficient, but in the presence of a plasma the photon acquires an effective mass, and the conversion probability experiences a resonance when $m_{\gamma} = m_{a}$ \cite{Yanagida:1987nf}.
In the cosmological context, the conversion of photons into axions may lead to a dimming of the cosmic microwave background across frequencies.
Then measurements of the spectrum of the CMB can be used to place constraints on the axion-photon coupling and the magnetic field strength.
Bounds were obtained from the COBE / FIRAS measurement of the CMB spectrum in \rref{Mirizzi:2005ng}, and recently a second group \cite{Tashiro:2013yea} has extended the calculation to include forecasts for next-generation CMB telescopes, namely PIXIE and PRISM.
The latter references finds an upper bound on the product of the axion-photon coupling and the {\rm r.m.s.} magnetic field strength today:
\begin{align}\label{eq:gB_constraint}
g_{a\gamma} B_0 < 10^{-14} {\rm \, GeV}^{-1} \, {\rm nG} \quad & \text{(COBE-FIRAS data)} \nonumber \\
g_{a\gamma} B_0 < 10^{-16} {\rm \, GeV}^{-1} \, {\rm nG} \quad & \text{(PIXIE/PRISM forecast)}
\end{align}
for a light axion $m_{a} < 10^{-14} {\rm \, eV}$.
For larger axions masses the bound weakens.
Using our fiducial value for the magnetic field $B_0 \simeq 10^{-14} \, {\rm G}$, we can write
\begin{align}\label{eq:gB_constraint_2}
g_{a\gamma} < 10^{-9} {\rm \, GeV}^{-1} \, \left( \frac{B_0}{10^{-14} \, {\rm G}} \right)^{-1} \qquad & \text{(COBE-FIRAS)} \nonumber \\
g_{a\gamma} < 10^{-11} {\rm \, GeV}^{-1} \, \left( \frac{B_0}{10^{-14} \, {\rm G}} \right)^{-1} \qquad & \text{(PIXIE/PRISM)} \, .
\end{align}
These bounds are comparable to the direct search limits from the CAST helioscope \cite{Andriamonje:2007ew},
\begin{align}\label{eq:CAST}
g_{a\gamma} < 8.8 \times 10^{-11} {\rm \, GeV}^{-1}
\end{align}
for $m_{a} \lesssim 0.02 {\rm \, eV}$.
\section{Dirac Neutrinos}
\label{sec:neutrinos}
While the neutrinos are known to be massive particles, the nature of their mass remains a mystery.
If neutrinos are Dirac particles then the theory contains four light states per generation: an active neutrino $\nu_{L}$, an active antineutrino, $\bar{\nu}_{R}$, a sterile neutrino $\nu_{R}$, and a sterile antineutrino $\bar{\nu}_{L}$.
The active states interact through the weak force, and this allows them to come into thermal equilibrium in the early universe.
The sterile states, on the other hand, interact only via the Yukawa interaction with the Higgs boson, and because of the smallness of the Yukawa coupling $y_{\nu} \sim m_{\nu} / v \sim 10^{-12}$, these states are not expected to be populated.
This story is modified if a strong magnetic field permeated the early universe.
The nonzero neutrino mass implies that the neutrino will also have a nonzero magnetic moment $\mu_{\nu}$.
From Standard Model physics alone one expects \cite{Lee:1977tib, Fujikawa:1980yx}
\begin{align}\label{eq:mnu_SM}
\mu_{\nu}^{\text{\sc sm}}
\simeq (3\times 10^{-20} \mu_{B}) \frac{m_{\nu}}{0.1 {\rm \, eV} \hidden{/c^2}} \, ,
\end{align}
where $\mu_B \equiv e \hidden{\hbar}/ 2m_e \hidden{c} \simeq 83.6 {\rm \, GeV}^{-1} \hidden{(\hbar c)^{3/2}} $ is the Bohr magneton, but new physics can increase this value appreciably.
The magnetic field couples to $\mu_{\nu}$ and induces the spin-flip transitions $\nu_{L} \to \nu_{R}$ and $\bar{\nu}_{R} \to \bar{\nu}_{L}$, which can be viewed as the absorption or emission of a photon.
If the spin-flip occurs rapidly in the early universe, the sterile states would be populated, and the effective number of relativistic neutrino species would double from $N_{\nu} = 3$ to $6$.
However, this is not consistent with measured abundances of the light elements, which imply $N_{\nu} \approx 3$ at the time of nucleosynthesis \cite{Izotov:2010ca}.
We must therefore require that the spin-flip transition goes out of equilibrium prior to the QCD epoch, $T_{\text{\sc qcd}} \simeq 200 \, {\rm MeV}$, so that the subsequent entropy injection at the QCD phase transition can suppress the relative abundance of sterile states to acceptable levels \cite{Kolb:1990}.
This translates into an upper bound on the neutrino magnetic moment and magnetic field strength, which was originally discussed by Enqvist et al. \cite{Enqvist:1992di,Enqvist:1994mb}.
In the rest of this section we apply the results of \rref{Enqvist:1994mb}.
The spin-flip transition occurs with a rate
\begin{align}
\Gamma_{L \to R} = \langle P_{\nu_{L} \to \nu_{R}} \rangle \Gamma_{W}^{\rm tot}
\end{align}
where $\langle P_{\nu_{L} \to \nu_{R}} \rangle$ is the average conversion probability and $\Gamma_{W}^{\rm tot}$ is the total weak scattering rate.
The active neutrinos scatter via the weak interaction which leads to $\Gamma_{W}^{\rm tot} \simeq 30 G_{F}^2 T_{\text{\sc qcd}}^5 \hidden{\hbar^{-7} c^{-6}}$ at the QCD epoch.
The conversion probability depends on the magnetic moment and field strength as $\langle P_{\nu_{L} \to \nu_{R}} \rangle \propto \mu_{\nu}^2 B^2$, since the interaction Hamiltonian is $H_{\rm int} = - {\bm \mu}_{\nu} \cdot {\bf B}$.
The coefficient takes different values depending on the relative scale of the magnetic field domains $\lambda_{B}$ and the weak collision length $L_{W} \approx \hidden{c} (\Gamma_{W}^{\rm tot})^{-1}$.
At the QCD epoch $L_{W} \simeq 1.6 \times 10^{-2} {\rm \, cm}$, which corresponds to a length scale of $L_{W,0} \simeq 3 \times 10^{10} {\rm \, cm}$ today.
It is safe to assume that the magnetic field of interest is much larger than this length scale, and therefore $\lambda_{B} \gg L_{W}$.
To ensure that the spin-flip transition is out of equilibrium one must impose $\Gamma_{L\to R} < H$ with $H$ the Hubble parameter.
This inequality resolves to the bound (see Eq.~(37) of \rref{Enqvist:1994mb})
\begin{align}
\mu_{\nu} B(t_{\text{\sc qcd}}) < (3.5 \times 10^{2} \mu_{B} \, {\rm G}) \sqrt{\frac{L_{W}}{\lambda_{B}}} \, .
\end{align}
To express this inequality in terms of the B-field strength and correlation length today, we use $B \simeq 6 B_{0} ( T_{\text{\sc qcd}} / T_{0} )^2$ and $L_{W} / \lambda_{B} = L_{W,0} / \lambda_{B,0}$.
This leads to an upper bound on the neutrino magnetic moment:
\begin{align}\label{eq:mnu_Bbound}
\mu_{\nu} < (3 \times 10^{-16} \mu_{B}) \left( \frac{B_{0}}{10^{-14} \, {\rm G}} \right)^{-1} \left( \frac{\lambda_{B,0}}{10 {\rm \, Mpc}} \right)^{-1/2} \, .
\end{align}
If this bound is not satisfied, the sterile neutrino states will still be thermalized with the active neutrino states at the time of BBN leading to $N_{\nu} \approx 6$, which is inconsistent with the data.
If the neutrinos are Majorana particles, then the sterile states are much heavier, and this bound does not apply.
\begin{figure}[t]
\hspace{0pt}
\vspace{-0in}
\begin{center}
\includegraphics[width=0.47\textwidth]{fig_magmom_bound--v3.pdf}
\caption{
\label{fig:mag_mom}
The requirement that spin-flip transitions are out of equilibrium at the QCD epoch leads to an upper bound the neutrino magnetic moment given by \eref{eq:mnu_Bbound}.
}
\end{center}
\end{figure}
The bound in \eref{eq:mnu_Bbound} is represented graphically in \fref{fig:mag_mom}.
For comparison we show the SM prediction from \eref{eq:mnu_SM} and the direct search limits.
The strongest laboratory constraints arise from elastic $\nu-e$ scattering.
The limits are flavor-dependent, but they are typically at the level of \cite{Agashe:2014kda}
\begin{align}\label{eq:mnu_direct}
\mu_{\nu} \lesssim 10^{-10} \mu_{B}
\qquad \text{(direct)} \, .
\end{align}
From the figure we see that the indirect early universe constraint is significantly stronger than the direct constraint for $B \gtrsim 10^{-18} \, {\rm G}$.
This provides the exciting opportunity to constrain extensions of the SM that predict an enhancement to the magnetic moment of Dirac neutrinos.
\section{Summary}
\label{conclusion}
Growing evidence for the existence of an intergalactic magnetic field has motivated us to consider the effects of a primordial magnetic field on models of exotic particle physics in the early universe.
We have focused our study on magnetic monopoles, axions, and Dirac neutrinos with a magnetic moment.
We summarize our results here.
In the context of a universe containing relic magnetic monopoles, we have derived a ``primordial Parker bound'' by requiring the survival of a primordial magnetic field until the time of electron-positron annihilation.
The bound, which appears in \eref{eq:n0_bound}, gives an upper limit on the cosmological monopole number density today: $n_{0} < 1 \times 10^{-20} {\rm \, cm}^{-3}$.
This translates into an upper bound on the monopole flux in the Milky Way; if the monopoles are unclustered then $\mathcal{F} < 3 \times 10^{-14} {\rm \, cm}^{-2} {\rm \, sec}^{-1} {\rm \, sr}^{-1} (v / 10^{-3} \hidden{c})$, and if they are clustered the bound weakens by a factor of $\sim 10^{5}$.
In \fref{fig:constraints} we compare the primordial Parker bound with other constraints on relic monopoles.
If the primordial magnetic field is not generated prior to $T \simeq \, {\rm MeV}$, then this bound does not apply.
After $e^+ e^-$ annihilation the monopoles are able to free stream, and they evolve along with the magnetic field as described by the system of equations in \eref{eq:coevolution}.
The solution is an analog of the familiar plasma oscillations (``Langmuir oscillations'') seen in an electron-ion plasma.
In the regime where the plasma oscillations are fast compared to the cosmological expansion, the coupling of the monopoles to the magnetic field leads to an anomalous scaling with redshift such that $B \sim a^{-9/4}$, $v_{m} \sim a^{-3/4}$, and $\rho_{B} \sim \rho_{\rm kin.} \sim a^{-9/2}$.
The behavior of the coupled system is effectively the average of the usual scalings for radiation $\rho_{B} \sim a^{-4}$ and the kinetic energy of a non-relativistic gas $\rho_{\rm kin.} \sim a^{-5}$.
If the strength of the intergalactic magnetic field could be measured over a range of redshifts, this would allow for a direct test of the anomalous scaling, and thereby probe relic magnetic monopoles.
We have also studied the effect of a primordial magnetic field on the evolution of an axion condensate in the early universe.
We obtain an exact solution to the MHD equations for the magnetic field in the limit where the advection term is negligible and the axion is homogenous.
After Peccei-Quinn breaking but prior to the QCD phase transition, the axion field is frozen, because its mass is smaller than the Hubble scale, and since the axion is derivatively coupled, this leads to no effect on the magnetic field.
Below the QCD scale the axion field begins to oscillate, and the spectrum of the magnetic field is distorted as in \eref{bpmsoln}.
One helicity mode of the magnetic field is enhanced while the other is suppressed; this CP-violation is a consequence of the axion's pseudoscalar nature.
However, the spectral shape of the magnetic field is only affected on extremely large
length scales, as given by \eref{Kformula}, except in situations where there can be significant axion evolution prior
to the QCD epoch.
We next study the evolution of the homogenous axion condensate in the presence of a background magnetic field.
The axion behaves as a damped and driven harmonic oscillator, as seen from its equation of motion \eref{eq:a_reduced}.
The damping time scale depends on the strength of the magnetic field and the photon-axion coupling.
For typical parameters it is generally larger than the cosmological time scale, and therefore irrelevant for the evolution of the axion.
It is interesting that the driving force (source term $\mathcal{H}$) is only operative when the magnetic field has a helicity.
This can be seen directly from the interaction $\mathcal{L} \ni \varphi F \widetilde{F}$ where $F \widetilde{F} \sim {\bf E} \cdot {\bf B} \sim \dot{h}$ is related to the rate of change of magnetic helicity $h = {\bf A} \cdot {\bf B}$.
Prior to the QCD phase transition when the axion was effectively massless, the axion field equation reduces to $\ddot{\varphi} = \mathcal{H}$.
In principle a very strong magnetic field could cause the axion to grow without bound as $\varphi(t) = \mathcal{H} t^2/2$, by drawing energy from the magnetic field.
For typical parameters, however, this growth occurs on a time scale that is much longer than the cosmological time.
It may still be the case that helical magnetic fields occurring in astrophysical environments are strong enough to lead to observable signatures.
We have also considered the resonant conversion of CMB photons into axions, which leads to a distortion of the CMB blackbody spectrum \cite{Yanagida:1987nf, Mirizzi:2005ng, Tashiro:2013yea}.
Using constraints on spectral distortions from current and anticipated future CMB telescopes, \rref{Tashiro:2013yea} obtained an upper bound on the axion-photon coupling. For our fiducial magnetic field strength this translates
into $g_{a\gamma} \lesssim 10^{-9} {\rm \, GeV}^{-1}$ with current data, and a forecast of $g_{a\gamma} \lesssim 10^{-11} {\rm \, GeV}^{-1}$
for experiments presently under discussion (see \eref{eq:gB_constraint_2}).
Finally we turn to the effect of the primordial magnetic field
on Dirac neutrinos, which carry a magnetic moment.
In the presence of a magnetic field, left-handed neutrinos can be converted into right-handed neutrinos.
If this spin-flip process is in equilibrium in the early universe, the right-handed states would be populated, and the effective number of relativistic neutrino species would double from $3$ to $6$, which is inconsistent with observations.
Requiring that this process is out of equilibrium at the time of the QCD phase transition leads to an upper bound on the neutrino magnetic moment and magnetic field strength.
Drawing on the work of \rref{Enqvist:1994mb}, we find the limit in \eref{eq:mnu_Bbound}, which implies $\mu_{\nu} < 3 \times 10^{-16} \mu_{B}$ for our fiducial magnetic field parameters $B_{0} = 10^{-14} \, {\rm G}$ and $\lambda_{B} = 10 {\rm \, Mpc}$. As seen in \fref{fig:mag_mom}, this bound is significantly stronger than the direct search limits over most of the parameter space.
\acknowledgments
We are grateful to David Marsh and Hiroyuki Tashiro for comments.
TV gratefully acknowledges the Clark Way Harrison Professorship at Washington University
during the course of this work. This work was supported by the DOE at ASU.
A.J.L. was also supported in part by the National Science Foundation under grant number PHY-1205745.
\bibstyle{aps}
|
{
"timestamp": "2015-04-28T02:03:39",
"yymm": "1504",
"arxiv_id": "1504.03319",
"language": "en",
"url": "https://arxiv.org/abs/1504.03319"
}
|
\section{Introduction}\label{sec:intro}
The past few decades have seen many searches for the most chemically
primitive, metal-poor stars in the Galaxy.
Stars with $\rm [Fe/H]\footnote{In the standard
notation [A/B] = log$_{10}$(N$_{\rm A}$/N$_{\rm B}$) $-$
log$_{10}$(N$_{\rm A}$/N$_{\rm B}$)$_{\odot}$, where $N_{\rm
A}/N_{\rm B}$ is the
ratio of elements A and B by number, relative to that in the Sun
($\odot$).}\lesssim-3.0$
are very rare and much coveted because of the information they provide
about conditions in the early universe. These stars are likely some
of the first low-mass stars to form in the universe after the first
chemical enrichment episodes occurred with the supernova deaths of
metal-free Population III (Pop III) stars.
Early theoretical work on the characterstics of Pop III stars
indicated that they were short-lived, very massive ($\gtrsim$100
M$_{\odot}$) objects \citep{abel_sci,brommARAA}. More recent work
has shown that the mass range of Pop III stars may have spanned
$\sim$3 orders of magnitude, leading to the possibility that some
low-mass ($\sim$1 M$_{\odot}$) stars may have survived to the present day
\citep{hirano2014,stacy14,susa2014}. Independent of whether a relic Pop III
star is ever found,
the chemical compositions of the
most metal-poor stars in the local universe
provide a record of this first stellar
generation.
The metallicity distribution function (MDF) of the most
metal-poor stars in our Galaxy presents a history of the formation
process of the Milky Way. It is a key constraint of any chemical
evolution model that attempts to describe this process (e.g.,
\citealt{hartwick1976}). Early surveys for the most metal-poor
stars in the halo (see below) indicated that the number of stars
smoothly declined with metallicity (a factor of 10 in decline for
every 1 dex in $\rm[Fe/H]$) down to at least $\rm[Fe/H]$$\sim -$3.5.
Lower than this, some samples indicated a sharp cut-off at $\rm[Fe/H]$\ =
$-$3.6, with very few stars more metal-poor than this value
\citep{schoerck, li_mdf}. However, this cut-off is not seen in
other samples \citep{yong13_III}.
Searches for metal-poor stars
in part have been driven to populate the most extreme metal-poor end
of the MDF. We refer the reader to Frebel \& Norris (2015, ARA\&A, in
press) and references therein for an overview of the complexities
involved in its interpretation.
Historically, surveys searched for extremely metal-poor (EMP) stars
with $\rm [Fe/H] \leq -3$ in the Galaxy halo. These surveys
exploited the stars' tendency to have large proper motions (e.g.,
\citealt{Ryanetal:1991, carneyetal:1996}) or the wide-field
capabilities of Schmidt telescopes. Objective prism observations of
millions of stars, carried out by such landmark surveys as the HK
Survey \citep{BPSII} and the Hamburg-ESO Survey \citep{hes4} on
Schmidt telescopes led to the medium-resolution spectroscopic
follow-up of thousands of EMP star candidates \citep{norrisUBVpho,frebel_bmps,
schoerck, li_mdf, placco_gband}. Of these, of order
several hundred have been followed up with high-resolution spectroscopy
and detailed element abundance analyses
(e.g., \citealt{McWilliametal, norris96data, Ryan96, aoki_mg,
Francois03, cohen04, cayrel2004, lai2008, hollek11, norris13_I,
placco2013_magII, cohen2013, roederer_313stars}).
More recently, medium-resolution spectroscopy of $\sim$10$^{5}$
stars obtained by the
Sloan Extension for Galactic Understanding and Exploration
(SEGUE-I; \citealt{segueI})
and SEGUE-II
extensions of Sloan Digital
Sky Survey \citep{york_sdss} have led to the identification of
hundreds more EMP stars.
Dozens of these have been observed with high resolution
spectroscopy (e.g.,
\citealt{aoki2008,Bonifacio2012,aoki2013,caffau2013,toposI}).
A search for extremely metal-poor stars is also underway with
the Large sky Area Multi-object fiber Spectroscopic Telescope
(LAMOST; \citealt{zhao_lamost,cui_lamost}),
and high-resolution spectroscopic follow-up of
the first candidates has recently been reported \citep{LAMOST_emp}.
EMP star candidate selection in objective prism surveys is based on
the strength of the Ca II K line at 3933 \AA\ in stellar spectra.
This calcium line serves as a useful proxy for overall stellar
metallicity. It is also possible to identify EMP candidates in pure
photometric searches, but the determination of metallicity
from broadband colors is difficult due their decreased sensitivity to
metallicity at low [Fe/H] (however, see \citealt{sc14}).
The SkyMapper Southern Sky Survey \citep{keller}, being carried out with the
SkyMapper 1.3m telescope at Siding Spring Observatory in Australia, is
a new survey that takes a rather hybrid approach. It
combines the efficiency of an all-sky photometric survey with the
power of metallicity measurements through narrow-band photometry of
the Ca II K line, similar to what has been done in objective prism surveys.
SkyMapper's filter system
is comprised of a
{\it ugriz} set with the addition of a narrow Str\"{o}mgren-like filter centered on the Ca II
K line \citep{skymapper_filter}. The combination of colors including this narrow filter
provides constraints
on stellar effective temperature, surface gravity and metallicity. A
``metallicity color index'' therefore allows for the identification of
metal-poor candidates from the photometry of the $\sim$5
billion stars potentially observable by the survey. For more details about the
survey techniques and candidate selection, see Keller et al.\ (2015,
in preparation).
The most promising SkyMapper metal-poor candidates are selected for
follow-up spectroscopic observation at both medium- and
high-resolution.
3198 candidates were selected from 5,452,735 stars in 195 SkyMapper
fields. 1127 were followed up with medium-resolution
spectroscopy, and 259 with high-resolution spectroscopy. Indeed, one such candidate
was verified as being the most Fe-poor star to-date via medium and high-resolution
spectroscopic follow-up. The discovery and abundance pattern of SMSS~
J031300.36$-$670839.3, with $\rm [Fe/H] < -7.1$, has already been reported \citep{keller_thestar}.
The addition of this star raises the number of stars known to have
$\rm [Fe/H] \leq -4.5$ to six \citep{HE0107_Nature, HE1327_Nature,
he0557, caffau2011, keller_thestar, hansen2014}, and the SkyMapper EMP
star candidate selection technique shows the promise of finding more
of these stars. A corresponding survey for the most metal-poor and
oldest stars in the Galactic bulge using SkyMapper photometry and
AAOmega/AAT multi-object spectroscopy is also underway (P.I.\
M.\ Apslund; see \citealt{howes2014} for first results).
In this work, we present results of the high resolution
spectroscopic follow-up of other metal-poor stars candidates identified by
SkyMapper from 2011 to 2013 November. As described in S.\ Keller
et al.\ (2015, in preparation), the SkyMapper photometry used to select
these candidates was obtained during the commissioning phase of the survey.
Section~\ref{sec:obs} describes the target selection, observations and data,
and Section~\ref{sec:analysis} describes our method of analysis.
Results are presented in Section~\ref{sec:results}, while a discussion and
summary are given in Sections~\ref{sec:disc} and \ref{sec:summary}.
\section{Target Selection, Observations and Data Reduction}\label{sec:obs}
Most of the candidate metal-poor stars were first observed with the
Wide Field Spectrograph on the ANU 2.3m telescope, providing
medium-resolution (R$\sim$3000) optical spectra.
Stellar parameters and metallicities were estimated based on a
comparison of the spectra to a library of synthetic spectra
(S.\ Keller et al., in preparation).
However,
for some early Magellan observing runs (2012 February, 2012 May),
candidates were selected based on metallicity estimates from their
SkyMapper photometry alone. At that time, the color-metallicity calibration of the
SkyMapper filter set was still being developed and improved, so
a number of the candidates turned out to be metal-rich.
A preliminary analysis of 160 stars from these early campaigns found 46 stars to have $\rm[Fe/H]
\ge -1$, 85 stars with $-2 \le \rm[Fe/H] < -1$, and 29 stars to have
$\rm[Fe/H]< -2$.
We have therefore made a metallicity cut, and here are presenting
the results only for 24 stars with $\rm [Fe/H]\lesssim-2.2$, as measured
from high-resolution spectra. For the later observing runs
(2012 September forward), all candidates were selected from
medium-resolution spectroscopy, and we analyzed all stars observed in
these runs including the metal-rich ones. (Ten stars have $\rm[Fe/H]$\ $> -2.2$.)
Nine candidates were selected based on preliminary photometry that did
not pass subsequent quality cuts. As such, these objects do not have SMSS
photometry or coordinates and instead we have adopted 2MASS identifiers and
coordinates \citep{2MASS}.
\begin{figure*}[!ht]
\begin{center}
\includegraphics[clip=true,width=18cm]{f1.pdf}
\figcaption{ \label{spec}
Portions of the MIKE spectra for four
stars in our sample around the Ca II H and K lines. LTE stellar
parameters and metallicities determined in our analysis are
also indicated (``T$_{\rm eff}$/$\log$\,{\it g}/v$_{\rm t}$, $\rm[Fe/H]$''). Note the variations in line strength with
decreasing [Fe/H] (top to bottom).}
\end{center}
\end{figure*}
The spectra analyzed in this work were obtained with the {\it Magellan}
Inamori Kyocera Echelle (MIKE) spectrograph on the 6.5m {\it Magellan} Clay
Telescope at Las Campanas Observatory \citep{mike}. Observations
spanned multiple
campaigns from 2011 to 2013 November. Depending on
sky conditions, spectra were obtained with either the $0\farcs7$ or
$1\farcs0$ slits, resulting in spectral resolutions ($\rm R \equiv
\lambda/\Delta\lambda$) of R$\sim$35,000 in the blue and R$\sim$28,000
in the red, and R$\sim$28,000 in the blue and R$\sim$22,000 in the
red, for the smaller and larger slit sizes, respectively.
Exposure
times generally ranged from 300 to 1800 s, depending on the
brightness of the target, to obtain ``snapshot'' spectra with which
to confirm EMP star candidates.
For some of the fainter, more promising
targets, multiple (2$-$4) exposures were obtained to increase the
signal-to-noise ratios (S/N). All spectra
span nearly the full optical range, 3350-9000\AA, but the
S/N was
generally too low blueward of $\sim$3800\AA\ for analysis.
Details of the observations are given
in Table~\ref{Tab:obs}, including star ID number, J2000
coordinates, {\it g} magnitude, and {\it g$-$i} color from the
SkyMapper observations (see S. Keller et al.\ 2015, in preparation for details of the
SkyMapper photometric system). Also included are
the UT date, the total
exposure time in seconds, the slit size used, the measured radial
velocity (see next Section), and the S/N ratios (per pixel) measured at
4500 and 6000\,\AA. The average~($\pm$1$\sigma$) S/N ratios of the sample
are 35($\pm$15) and 55($\pm$21) at 4500 and 6000\,\AA, respectively.
All spectra were reduced using the CarPy data reduction
pipeline\footnote{See http://code.obs.carnegiescience.edu/mike.}
described in \citet{kelson03}.
Individual exposures were combined to increase S/N,
individual orders were merged, and the blue and red spectra combined and
then continuum normalized to create one continuous spectrum
per star.
Example MIKE spectra for four stars are presented in
Figure~\ref{spec}, all obtained with the $0\farcs7$ slit. Their stellar
parameters and metallicities, as determined in our analysis, are also
indicated. All told, the total sample of analyzed stars is 122.
For candidates selected based on metallicity
estimates from medium-resolution
spectroscopy, agreement between [Fe/H]$_{\rm MRS}$ and
[Fe/H]$_{\rm HRS}$ measured
in this work generally agree at the
0.3 dex level (Keller et al.\ 2015, in preparation).
\section{Analysis}\label{sec:analysis}
Our abundance analysis software \citep{smh}
incorporates the Castelli \& Kurucz 1D LTE hydrostatic model atmosphere grid
\citep{castelli_kurucz} and the version of the LTE abundance analysis program MOOG
that includes treatment of Rayleigh scattering
(\citealt{moog, sobeck11}). Our linelist was that compiled by
\citet{roederer10}, though we only considered lines within the
wavelength range 3500$-$6500\AA.
Radial velocities for our stars were determined
via cross-correlation of the Ca triplet ($\lambda$8450-8700) and/or Mg
Ib ($\lambda$5150-5200) region of the spectra against that of
a high S/N, rest frame MIKE spectrum of the metal-poor
standard star HD~140283.
Velocity errors due to the cross-correlation technique are generally
small (0.1$-$0.3 km s$^{-1}$). Based on repeat observations of standard stars
such as HD~122563, G64$-$12 and HD~13979 during each observing run, we
estimate a zero-point offset to our radial velocity scale of 1$-$2 km s$^{-1}$,
with our values being smaller than ones in the literature for these
stars.
Four stars were observed in two separate observing runs, and two stars
were observed in three different campaigns. The radial
velocities determined from the different spectra of these stars also
show differences
of 1$-$2 km s$^{-1}$, with the exception of the star SMSS~J022410.38$-$534659.9,
which
shows a variation of $\sim$14 km s$^{-1}$. This star may be a single-lined
spectroscopic binary, however, its stellar parameters (see next Section)
place it on the edge of the instability strip in the
Hertzsprung-Russell diagram, so it may instead be a variable.
For those stars observed with the same slit size in multiple
observing runs, all the spectra were combined to increase S/N before
the abundance analysis. For those stars observed with different slit
sizes, the spectrum with the highest S/N was analyzed; in cases where
S/N levels were comparable, the $0\farcs7$ spectrum was
analyzed\footnote{Generally, these candidates were photometrically
selected more than once and given
two different identifiers, and were only identified as duplicates
during the high-resolution spectroscopic analysis.}.
Heliocentric
radial velocities for each of the stars are given in
Table~\ref{Tab:obs}. As can be seen, many have large heliocentric
velocities as
expected for halo stars. All spectra were shifted
to rest wavelength for the abundance analysis described in the next
section.
\subsection{Determination of Stellar Parameters}\label{sec:stelpars}
The stellar parameters for each star were determined solely from its
MIKE spectrum
using the standard spectroscopic techniques: effective temperature (T$_{\rm eff}$)
by removal of any slope of Fe\,I abundance with excitation
potential (E.P.), $\log$\,{\it g}\
by matching Fe\,I and Fe\,II abundances, microturbulence
(v$_{\rm t}$) by removal of
any slope of Fe\,I abundance with reduced equivalent width (REW).
In this process, individual lines with abundances $\sim$2$\sigma$
away from
the mean were visually inspected, reassessed for measurement quality,
and if necessary, rejected (due to blending, uncertainty in continuum
placement, etc.). Our general tolerances were as follows: slope of
log $\epsilon$(Fe\,I) versus E.P. $<$ 0.005 dex/eV; [Fe\,II/H] $-$
[Fe\,I/H] $<$ 0.05 dex, and slope of log $\epsilon$(Fe\,I) versus log(RW) $<$ 0.005.
The [M/H] of the model atmosphere was set to [Fe\,I/H]$+$0.25, as
described in \citet{teff_calib}.
Spectroscopic effective
temperatures are generally cooler than photometric temperatures
due to departures from local thermodynamic
equilibrium (LTE;
\citealt{johnson2002_23stars,cayrel2004,lai2008,hollek11,lind2012,cohen2013}).
The use of 1D models as opposed to time-dependent 3D or temporally
and spatially averaged 3D ($<$3D$>$) models can also lead to this effect
\citep{asplund_araa,bergemann12}. Too-cool
temperatures translate into smaller $\log$\,{\it g}\ and larger v$_{\rm t}$\ values
than would be found using photometric temperatures.
We have adopted the
effective temperature correction presented in \citet{teff_calib}. It places
spectroscopically-determined temperatures on a scale similar to that found by
photometric temperature methods. This correction is appropriate for the
program stars, and the majority of them span the metallicity range for which the correction has
been
tested ($-3.3<\rm [Fe/H]<-2.5$).\footnote{Recall the caveat in that paper
that the calibration may not be valid for stars with
$\rm [Fe/H]<-4.0$. The lowest metallicity of any star in this
sample is $-$4. We note that although this correction was
not tested for stars with $\rm [Fe/H]>-2.5$, such stars have been
known to have similar discrepancies between their photometric and
spectroscopic T$_{\rm eff}$\ values (see, e.g., \citealt{johnson2002_23stars}).}
The final adopted
(corrected) spectroscopic temperatures were checked by visual inspection of
H$\alpha$ and H$\beta$ line profiles, in comparison to stars
of previously determined
effective temperature. Surface gravity and v$_{\rm t}$\ were then adjusted to
maintain ionization balance and remove any trend of Fe\,I abundance
with line strength, as necessary.
\begin{figure}[!ht]
\begin{center}
\includegraphics[clip=true,width=8cm]{f2.eps}
\figcaption{ \label{f_isochrone}
The Hertzsprung-Russell diagram for the current sample, plotted
with isochrones from \citet{Y2_iso} and a horizontal branch
isochrone from \citet{basti_HB}. LTE parameters are shown as
filled circles, and NLTE parameters are open red circles.}
\end{center}
\end{figure}
Stellar parameters for our program stars are presented in
Table~\ref{Tab:stellpar}. Note that the metallicities are relative to
the solar abundance from \citet{asplund09}. We have also calculated
1D non-LTE (NLTE) $\log$\,{\it g}\ and $\rm[Fe/H]$\ values for the stars following the method
described in \citet{ruchti2013}, and using the NLTE grid of
\citet{lind2012}. These values are also given in Table~\ref{Tab:stellpar}.
Figure~\ref{f_isochrone} shows the positions of the stars in this
study in the Hertzsprung-Russell diagram. 12
Gyr $\alpha$-enhanced isochrones with [Fe/H] = $-$3.0, $-$2.5, $-$2.0,
and $-$1.5 from \citet{Y2_iso}, and a 12-Gyr, [Fe/H] = $-$2.2
BaSTI horizontal branch isochrone \citep{basti_HB} are also shown.
Filled symbols indicate the LTE $\log$\,{\it g}\ values, and open circles
represent the NLTE $\log$\,{\it g}\ values. As expected, $\log$\,{\it g}\ values
calculated in NLTE are $\sim$0.4-0.5 dex larger than the LTE gravities.
\begin{figure*}[ht!]
\begin{center}
\includegraphics[clip=true,width=15cm]{f3.pdf}
\figcaption{ \label{f_csynth}
Syntheses of the CH bands at 4305$-$4317\AA\ (top) and
4320$-$4330\AA\ (bottom) in the star SMSS~J023139.43$-$523957.4.
Its stellar parameters are also listed as ``T$_{\rm eff}$/$\log$\,{\it g}/v$_{\rm t}$''.
The
synthetic spectra have C abundances varying in steps of 0.2 dex.
The best fit, log $\epsilon$(C) = 5.89, or [C/Fe] = $+$0.40, is
the solid black line. Spectra with C abundances $\pm$ 0.2 dex
around this value are shown as red dashed lines.}
\end{center}
\end{figure*}
All the stars in this sample are giant stars, many of which lie
on the upper part of the giant branch. While this means that the
abundances of elements modified by stellar evolution (i.e., Li, C, N,
O, s-process) will not reflect
the primordial values of some of the stars, we do not have to worry about
systematic differences in abundances between dwarfs and giants as found
in some studies (see, e.g., \citealt{yong13_II}).
The likely reasons for the lack of dwarfs in this sample are discussed
in Keller et al.\ (2015, in prep.).
\subsection{Element Abundance Determination}\label{sec:abund_analysis}
Equivalent widths (EW) of lines in our line list were
measured via
fits of Gaussian profiles to absorption features in the spectra of the
program stars. All measures were visually inspected. Any that were
identified as outliers in the analysis process were further
scrutinized and, where appropriate, adjusted or rejected.
The Fe\,I and Fe\,II EWs were used to determine the
stellar parameters (see Section~\ref{sec:stelpars}); in high quality
spectra of generally more metal-rich stars,
$\sim$200 Fe\,I and $\sim$20 Fe\,II lines were
used, while in the lower S/N
star spectra, as few as 22 Fe\,I and $\sim$2 Fe\,II lines
could be measured. \citet{teff_calib} compared EW measures for lines
in our line list using the same technique in this current work to
literature measures for the standard star HD~122563. The agreement
was excellent (differences less than 0.25 m\AA\ in the mean).
The stars subject to repeat observations also allow for a
quantitative estimate of the robustness of our EW measures. For each
star, the EW's measured from each spectrum were directly compared.
The mean difference, in the sense
$\rm \Delta EW = (EW1 - EW2)\pm(\sigma/\sqrt N))$,
ranged from $0.5\pm1.3$ m\AA\ to $1.7\pm0.5$ m\AA, with standard
deviations ranging from 8 m\AA\ to 29 m\AA.
The number of lines available for EW measurement varied widely for the
other elements, with as many as 29 Ti\,I and 46 Ti\,II lines available,
but typically only one line of, e.g., Al\,I and Si\,I.
Chemical abundances for
elements were determined using the measured EW values and the stellar
parameters found from the iron lines.
Table~\ref{tab_ew_stub} gives the EW measures for all program
stars, along with the measured log$\epsilon$ abundances.
The following absorption features were
analyzed via spectrum synthesis: 4313\AA, 4323\AA\ (CH, G-band);
4246\AA\ (Sc\,II); 4030\AA, 4033\AA, and 4034\AA\ (Mn\,I);
4077\AA, 4215\AA\ (Sr\,II); 4554\AA, 4934\AA\ (Ba\,II); and 4129\AA\ (Eu\,II).
Additional Mn\,I and Sc\,II lines were synthesized for 54 stars in the
sample, along with the Al\,I 3944\AA\ feature.
In each synthesized region, the
abundances of elements other than the one of interest were fixed to
the value determined via EW.
Synthetic spectra were generated with
MOOG and then convolved with a Gaussian to match the resolution of the
MIKE spectra. Where necessary, the continuum-placement of the data
was adjusted and the spectrum radial-velocity shifted to
correct for subtle wavelength differences.
Abundances were then determined by minimizing the difference between
the observed and synthetic spectra by eye.
The uncertainty of the spectral matching was then
determined by decreasing the stepsize (in abundance space) between
three synthetic spectra until the best match could no longer be
uniquely
identified. Example syntheses are given in Figure~\ref{f_csynth}.
Abundance results from spectrum synthesis (excluding CH) are also
given in Table~\ref{tab_ew_stub}.
Element abundances of the stars in this sample are presented in
Table~\ref{tab_abund}, relative to the solar abundances of
\citet{asplund09}.
Abundance upper limits obtained via spectrum synthesis are
identifed as such in the Table.
\subsection{Error Analysis}\label{sec:errors}
The abundance uncertainties in our analysis are a combination of both
random uncertainties (e.g., in the EW measures,
etc.) and systematic uncertainties (due to continuum-placement, the adopted temperature
scale, model atmosphere grid, etc.). Based on the spectroscopic
techniques used here to determine the stellar parameters, we estimate
their uncertainties to be $\sim$100 K, 0.3 dex and 0.2 km s$^{-1}$\ for
T$_{\rm eff}$, $\log$\,{\it g}\ and v$_{\rm t}$, respectively. The contribution of each of
these to the abundance uncertainty of each element was determined by
varying each parameter by its uncertainty and recalculating the
abundance.
Table~\ref{tab_unc} lists the abundance uncertainties of
individual elements for both a warmer and cooler example star from our sample.
The random uncertainty ($\sigma$) listed in the third column is
the
standard error in the mean of individual line abundances for each
element.
In the cases where only one
line was measureable, this value is placed conservatively at 0.2
dex, as appropriate for the low S/N ratio of many of our ``snapshot''
spectra. The last column shows the quadratic sum of the individual
uncertainties.
\section{Results}\label{sec:results}
In this section, we review the abundance results for this SkyMapper
sample for individual elements, divided roughly by group in the
periodic table. Unless stated otherwise, the
abundances presented here are LTE abundances.
First though, Figure~\ref{f_mdf} shows the distribution of
metallicities of the SkyMapper metal-poor candidates in different
observing campaigns.
The top left panel shows only 24 most metal-poor
stars of the first 160 followed up with high-resolution spectroscopy,
as previously discussed. The remaining panels save the last include all of the
stars observed in each campaign. Stars observed in multiple campaigns (see
Table~\ref{Tab:obs}) are distributed according to their first
observation date.
The bottom right panel shows the distribution of
the entire sample. In each panel, the mean and median $\rm[Fe/H]$\ values
are indicated by cyan solid and dashed lines, respectively.
As can be seen, the mean and median $\rm[Fe/H]$\ values do not vary much from
$\rm[Fe/H]$\ $\sim -$2.8 in each panel, though the metal-poor tail is
especially evident in the total sample (bottom right panel).
Despite the fact that the photometric
candidate selection technique was improved during the accumulation of
this sample, no obvious improvement is seen in the distributions of the individual panels of
Figure~\ref{f_mdf}. This is largely due to the relative rarity of
stars with $\rm[Fe/H]$\ $< -3.5$ in the Milky Way halo and the necessity of
observing
more metal-rich targets due to the lack of more interesting metal-poor
candidates in some runs (e.g., 2013 May).
That said, 92 of the 122 stars (75\%) have
$\rm[Fe/H]$\ $\le -2.5$; 51 have $-3 < $\rm[Fe/H]$ \le -2.5$ (42\%); 32 have $-3.5 < $\rm[Fe/H]$
\le -3$ (26\%); and nine have $\rm[Fe/H]$ $\le -3.5$ (7\%). Keep in mind
these numbers have not been corrected for any biases. Indeed,
since these candidates were selected from commissioning data, the
distributions in Figure~\ref{f_mdf}
should not be interpreted as {\it the} metallicity distribution function of
stars identified in the SkyMapper Survey. This will be the subject of
future work.
\begin{figure}[!ht]
\begin{center}
\includegraphics[clip=true,width=8cm]{f4.pdf}
\figcaption{ \label{f_mdf}
The (LTE) metallicity distribution function of the Skymapper sample,
separated by observation date, with the total sample in the
bottom right hand panel. The bin size is 0.25 dex everywhere.
The mean and median $\rm[Fe/H]$ values are indicated by solid and
dashed cyan lines, respectively. Note that the ``early 2012''
distribution is incomplete at $\rm[Fe/H] \gtrsim -2.5$.}
\end{center}
\end{figure}
\subsection{Lithium}
The Li\,I 6707\AA\ feature was detected in the spectra of 24 stars.
We determined LTE Li abundances via spectrum synthesis using the line
list of \citet{hobbs99} and assuming a pure $^{7}$Li component.
These are given in Table~\ref{tab_li}, along with
measured EWs (in m\AA) and NLTE Li abundances
calculated using the grid of \citet{lind_li}.
Figure~\ref{f_li} shows log $\epsilon$(Li) = A(Li) (LTE: crosses, NLTE: open circles) as a
function of T$_{\rm eff}$\ and $\rm[Fe/H]$, along with those of giant stars from the sample of
\citet{spite2005}. (We only show their stars with Li measures, no upper
limits.) As can be seen, the distributions of Li abundances for the
two samples are similar. The second most metal-poor star in our
sample, \2\ ($\rm[Fe/H]$ $\sim -$4), has the largest Li abundance, A(Li) =
2.0 (LTE), but at a depletion level appropriate for its T$_{\rm eff}$.
Note that NLTE corrections to the Li abundances are small
for these stars: no more than 0.12 dex.
\begin{figure}[!ht]
\begin{center}
\includegraphics[clip=true,width=8cm]{f5.pdf}
\figcaption{ \label{f_li}
The Li abundances of stars showing $\lambda$6707 absorption as a
function of T$_{\rm eff}$ (top panel) and LTE $\rm [Fe/H]$ (bottom
panel). LTE and NLTE abundances are shown as crosses and open
circles, respectively. Giant stars with Li measures from the
sample of \citet{spite2005} are shown as filled red circles.}
\end{center}
\end{figure}
\subsection{Carbon}\label{sec:carbon}
During the ascent up the red giant branch, the surface C abundance of
a star
decreases due to dredge-up of CN-processed material.
\citet{placco14} provide C abundance
corrections as a function of surface gravity and metallicity
to take this effect into account.
We have corrected the carbon abundances of our sample stars
accordingly since we are interested in the stars' natal carbon
abundances and whether their birth gas clouds were particularly
enhanced in carbon. In Table~\ref{tab:cfe_corr} we provide uncorrected and
corrected [C/Fe] values. The corrections from \citet{placco14}
were calculated adopting [N/Fe] = 0.0.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=8cm]{f6.pdf}
\figcaption{ \label{f_carbon} Top panel:
Corrected [C/Fe] abundances versus (LTE) [Fe/H] for our sample (crosses)
compared to the sample of
\citet{yong13_II} (red circles; also corrected). Upper limits are denoted by
arrows.
Bottom panel: Corrected [C/Fe] abundances versus surface gravity.
The CEMP definition of
\cite{aoki_cemp_2007} is indicated by a dotted line.}
\end{center}
\end{figure}
Corrected carbon abundances are shown as crosses in
Figure~\ref{f_carbon}, along with upper limits as arrows.
For comparison, in this and the following figures we
plot our results against those of the {\it giant} stars
in \citet{yong13_II} (red circles, also corrected).\footnote{We consider here the giant stars from
both their literature compilation and their own sample.
We note that the stellar parameter determination by \citet{yong13_II}
and the line list they used were different from those used here and so
systematic differences between results may exist.}
The bottom panel of Figure~\ref{f_carbon} shows [C/Fe] for this
sample plotted against surface gravity.
The carbon abundances exhibited by the SkyMapper sample
are overall typical for stars found in the halo.
\begin{figure*
\begin{center}
\includegraphics[clip=true,width=18cm]{f7.pdf}
\figcaption{ \label{f_light}
LTE [Na/Fe] and [Al/Fe] ratios versus (LTE) $\rm[Fe/H]$\ for the SkyMapper sample (crosses)
compared to the sample of \citet{yong13_II} (red circles) and
the literature compilation of \citet{frebel10} (orange circles). A
least-squares fit to SkyMapper stars with (LTE) $\rm [Fe/H] \leq -2.5$,
excluding 2$\sigma$ outliers, is indicated by the cyan line.
Parameters of the least-squares fit are also given.}
\end{center}
\end{figure*}
Also indicated in the Figure is the definition for carbon enhanced metal-poor
(CEMP) stars of $\rm [C/Fe]\ge0.7$ (dotted line), following \citet{aoki_cemp_2007}.
Considering the corrected carbon abundances, we determine the
frequency of CEMP stars to be 20\% (24/120) for the total sample.
The frequency is 21\% (24/113) for stars with $\rm [Fe/H] \le -2.0$, 26\% (24/91)
for $\rm [Fe/H] \le -2.5$, 39\% (16/41) for $\rm [Fe/H] \le -3.0$, and
56\% (5/9) for stars with $\rm [Fe/H] \le -3.5$.
For comparison, using 505 metal-poor stars from the literature
with $\rm [Fe/H] \le -2.0$ and corrected carbon abundances,
\citet{placco14} determined these frequencies to be 20\%, 24\%, 43\%
and 60\%,
respectively. Our values agree very well with theirs.
We note that CEMP-s and CEMP-rs stars have been excluded from the
Placco et al. sample; since our sample does not contain any of these
stars either, the comparison between these samples is
appropriate.
Interestingly, our sample contains only seven stars with $\rm [C/Fe]>1.0$,
of which five have $\rm [C/Fe]\sim1.0$. Star SMSS~J173823.36$-$145701.0 has a corrected carbon
abundance of $\rm [C/Fe] = 1.33$, which is the highest in the sample.
(Its uncorrected $\rm [C/Fe]$ is 0.60; the
gravity is low, $\log$\,{\it g}\ = 0.75, which leads to the large correction.)
It has $\rm [Fe/H] =-3.58$. \2,
at $\rm[Fe/H]$\ = $-$3.94, has the second highest [C/Fe] ratio of $+$1.21
(no carbon correction because of $\log$\,{\it g}\ = 2.95). These two, together
with the other five CEMP stars, are thus prominent examples of CEMP-no
group: they lack enhanced neutron-capture element
abundances, and their other [X/Fe] ratios are comparable to those of
typical halo stars at similar metallicity
\citep{fnARAA}. Since five of the seven
stars have $\rm[Fe/H]$\ $< -$3.4, we confirm that CEMP-no stars
preferentially appear at the lowest metallicities, i.e., below $\rm[Fe/H]$\ $< -$3.0.
It is worthwhile asking why no stars with [C/Fe] $\gtrsim$2 appear in
our sample. Such stars typically exhibit large enhancements of
s-process elements like Ba (CEMP-s stars, mentioned above). Very
strong G-band absorption may change the colors of a star, moving it
out of the range used for candidate selection. This is currently under
investigation (S.\ Keller et al.\ 2015, in preparation).
\subsection{Na and Al}
Figure~\ref{f_light} presents the LTE [X/Fe] ratios for Na and Al versus
LTE [Fe/H] for our sample. Also shown are the giant star sample from
\citet{yong13_II} and the Milky Way halo star literature sample from
\citet{frebel10} (orange circles).\footnote{In this and following figures that include
both the \citet{yong13_II} and \citet{frebel10} samples, all stars
in the former sample have been excluded from the latter. The
\citet{frebel10} sample is a mixture of dwarf and giant stars.}
In this and following figures, we have performed a linear regression
analysis on the [X/Fe] versus [Fe/H] distributions of our sample in
order to compare our results to other studies in the literature.
Following \citet{yong13_II}, we restricted the sample to
$\rm [Fe/H] \leq -2.5$ and calculated the rms scatter of
points about that fit. Stars with [X/Fe] ratios more than 2$\sigma$
away from the fit were excluded and the linear regression redone. The
resulting line of best fit, excluding the 2$\sigma$ outliers, is shown
in each panel (cyan line), along with its slope, the slope error, and the
rms scatter about the slope. Also shown are the mean [X/Fe] ratios
and standard deviations, the
total number of stars, and the number of stars used in the fit.
As can be seen, the SkyMapper stars exhibit
the $>$1 dex spread in [Na/Fe] found in other studies of metal-poor stars.
The [Na/Fe]$\sim$0 for stars with $\rm [Fe/H] > -2$ is also
consistent with other stars in this metallicity range. There is no
significant change in [Na/Fe] as a function of [Fe/H], as indicated by
the flat slope for stars with $\rm [Fe/H] \leq -2.5$. A star's
spectrum is often contaminated with Na D absorption from the
interstellar medium. The 5889/5895\AA\ Na lines of stars with very small
($\lesssim10$ km s$^{-1}$) radial velocities were inspected for possible
contamination with ISM features and were discarded when necessary.
\begin{figure}[!h]
\begin{center}
\includegraphics[clip=true,width=8cm]{f8.pdf}
\figcaption{ \label{f_na_lte_nlte}
LTE [Na/Fe] (crosses) and NLTE [Na/Fe] (red open circles) plotted
as a function of LTE [Fe/H] for the SkyMapper sample. Note not all
stars have NLTE abundances. Differences
between LTE and NLTE abundances are of order 0.7$-$1 dex. See text
for more information.}
\end{center}
\end{figure}
We have calculated NLTE Na abundances as described in \citet{lind_na}.
Five
stars fell outside the grid of \citet{lind_na}, and therefore do not
have NLTE abundances. We have confirmed with detailed calculations
for a few stars that these NLTE abundances
are appropriate at the level of $\sim$0.05 dex, in spite of differences in
model atmospheres use in this work \citep{castelli_kurucz},
and in \citealt{lind_na} (MARCS; \citealt{marcs}). LTE abundances of this work were
found to differ by up to 0.3 dex from those calculated using the
method of \citet{lind_na} for
strong lines. Figure~\ref{f_na_lte_nlte} plots LTE and NLTE [Na/Fe]
values (crosses and open circles, respectively), versus LTE [Fe/H].
(Note that the NLTE [Na/Fe] values were calculated using both NLTE Na
and Fe abundances.) The large ($\sim$0.7-1 dex) differences reflect
the negative (NLTE-LTE) corrections for Na and the positive (NLTE-LTE)
corrections for Fe.
The Al abundances for our stars are based only on measurement of the
3961 \AA\ Al\,I line for roughly half of the sample, while the
other half included measurement of the 3944 \AA\ feature. No
systematic abundance offsets were found between stars with one and two measured features.
Based on Figure~\ref{f_light}, the LTE
[Al/Fe] ratios of our sample are also comparable to those of
\citet{yong13_II}, though the
standard deviation in [Al/Fe] is roughly 1.5 times as large as in
their work. This is not surprising given the low S/N of
some of our spectra, especially below 4000 \AA.
\citet{al_nlte} found NLTE corrections as large as $+$0.65 dex are
necessary for Al abundances of cool, metal-poor stars. A correction
of this magnitude would bring
[Al/Fe] values in Figure~\ref{f_light} within $\sim$0.1 dex of solar.
Such ratios are more consistent with predictions of chemical evolution
models (e.g., \citealt{kobayashi2006}) than the LTE stellar
abundances, as has been noted before.
\subsection{$\alpha$-Elements}\label{sec:alpha}
\begin{figure*
\begin{center}
\includegraphics[clip=true,width=18cm]{f9.pdf}
\figcaption{ \label{f_alpha}
Same as Figure~\ref{f_light}, but for the $\alpha$-elements.}
\end{center}
\end{figure*}
\begin{figure}[!h]
\begin{center}
\includegraphics[clip=true,width=8cm]{f10.pdf}
\figcaption{ \label{f_tidiff_ntiI}
[TiII/Fe] $-$ [TiI/Fe] differences versus the number of Ti\,I
lines measured in each star. There is a marked increase in
scatter when N(Ti\,I) $\leq$ 5 (dotted line). The most
metal-poor ($\rm[Fe/H]$ $\leq -3.4$) and metal-rich ($\rm[Fe/H]$ $>-2.5$)
stars are indicated by circles and squares, respectively. See
text for more information.}
\end{center}
\end{figure}
The LTE [X/Fe] ratios for the $\alpha$-elements (Mg, Ca, Si, Ti) versus
LTE [Fe/H] are presented in Figure~\ref{f_alpha}.
Ti\,I and Ti\,II abundances\footnote{Strictly, these are
[Ti/Fe]$_{\rm Ti\,I}$ and [Ti/Fe]$_{\rm Ti\,II}$, but we denote them as
[TiI/Fe] and [TiII/Fe] for convenience.}
are plotted separately, with an additional panel that
shows the difference between them as a function of metallicity.
As
can be seen, the agreement between them is good, with the mean
difference comparable to the dispersion about the means of both species.
This is similar to the agreement found by \citet{yong13_II} for their
giant star sample.
Looking more closely, Ti\,I and Ti\,II abundances have different slopes in
Figure~\ref{f_alpha}, while the difference between them at low $\rm[Fe/H]$\
is different from that at high $\rm[Fe/H]$\ (bottom right). The cause of these features is
illustrated in Figure~\ref{f_tidiff_ntiI}, where the difference
between Ti\,II and Ti\,I abundances is plotted against the number of
Ti\,I lines measured per star. As can be seen, the scatter in
$\Delta$[X/Fe] increases by a factor of two when N(Ti\,I) $\leq$5,
and that the most metal-poor stars preferentially have fewer
measurable Ti\,I lines. A star in our sample has an average of 30
Ti\,II lines measured in its spectrum, compared to only 13 Ti\,I
lines. Consequently, the [Ti\,II/Fe] ratios in Figure~\ref{f_alpha}
are more reliable. For stars that have N(Ti\,I) $>$ 5,
([Ti\,II/Fe] $-$ [Ti\,I/Fe]) = 0.11$\pm$0.10.
We do not apply
NLTE corrections to any $\alpha$-element abundances for our sample,
but summarize the magnitudes of corrections appropriate for our stars.
NLTE corrections for Ti\,II abundances are
expected to be $\sim$0.05 dex or less, while
corrections for Ti\,I abundances are larger for metal-poor stars
($+$0.1$-$0.2 dex; \citealt{bergemann_ti}). The difference
between our LTE Ti\,II and Ti\,I abundances are consistent the
magnitude of these corrections.
Uncertainties in atomic data for
individual lines likely also impact the scatter in abundances for both
species, though we note that improved atomic data are now available
\citep{lawler_tiI, wood_tiII}.
NLTE corrections for Mg are at the level of $\sim$0.1 dex
\citep{gehren2004_nlte, mashonkina_mg}, while for Ca\,I lines they can be as large as
$+$0.3 dex for stars like those in our sample
(\citealt{mashonkina_ca}, but see also \citealt{starkenburg10}).
\begin{figure*
\begin{center}
\includegraphics[clip=true,width=18cm]{f11.pdf}
\figcaption{ \label{f_fepeak}
Same as Figure~\ref{f_light}, but for the Fe-peak elements.
For Zn (bottom right panel), the red symbols are
those of \citet{cayrel2004} and
\citet{heresII}, while for the remaining elements they are from
\citet{yong13_II}. As in other figures, the orange symbols are
from \citet{frebel10}. Literature sample upper limits are shown as triangles.}
\end{center}
\end{figure*}
Overall,
the SkyMapper targets exhibit typical halo star abundance patterns,
with relatively small ($\sim$0.1 dex) dispersion in, e.g., Mg, Ca and Ti abundances, and
larger scatter in Si. These dispersions are comparable to or smaller
than the
standard deviations of individual line abundances for most stars in
our sample.
The intrinsically small
dispersion in
$\alpha$-element abundances over a wide range of metallicity is well-documented in the
literature (e.g., \citealt{cayrel2004}) and implies that their
nucleosynthetic yields have remained remarkably
constant throughout the earliest phases of chemical evolution in the universe.
The larger
scatter in Si abundances is at least partly due to the difficulty in
obtaining a robust measure for this element; for many stars in our
sample it is based solely on one weak Si line (at 4102.9\AA, on the
wing of H$\delta$) that was not
measurable in all stars. The blended 3905\AA\ Si\,I line was analyzed via spectrum
synthesis in a portion of our sample; no systematic offset between
abundances of stars based on one or both lines was observed.
In addition to the very small ($<$0.1 dex) dispersion in [X/Fe] versus
[Fe/H] in metal-poor stars, previous studies have found that
the $\alpha$-elements show flat trends
with slopes consistent with zero. For all the $\alpha$-elements
in Figure~\ref{f_alpha} save Mg the magnitudes of the
slopes are equivalent to the rms scatter.
\subsection{Fe-peak Elements}\label{sec:fepeak}
Figure~\ref{f_fepeak} shows the trends with [Fe/H] for the Fe-peak
elements Sc, Cr, Mn, Co, Ni and Zn.
\begin{figure*
\begin{center}
\includegraphics[clip=true,width=15cm]{f12.pdf}
\figcaption{ \label{f_ncap}
Same as Figure~\ref{f_light}, but for Sr and Ba. Upper limits
are denoted by arrows and triangles for our sample and the
literature sample, respectively.}
\end{center}
\end{figure*}
As mentioned previously, Sc abundances are based
on spectrum synthesis of only the Sc\,II 4246\AA\ line for $\sim$40
stars in our sample, while for the remaining $\sim$50 as many as four
other lines were also analyzed. A comparison of the Sc\,II abundance
determined from the 4246\AA\ line to the mean abundance of the other
lines found a 0.08 dex ($\sigma$=0.19) offset, in the sense that the
other line abundances were larger. We have therefore added 0.08 dex
to the Sc\,II abundance for stars in which only the 4246\AA\ line was measured.
There is an unexplained systematic offset in
the zero-point of our Sc abundances compared to that of
\citet{yong13_II}: our mean [Sc/Fe] = $-$0.11 is $\sim$0.3 dex lower
than the value for their giant sample, although the values are comparable within the
standard deviations of the two samples (0.17 dex for ours, 0.14 dex
for theirs, see their Figure 22). This offset is also visible
relative to the larger literature compilation and in our analysis of
the standard star HD~122563 compared to literature studies (see Section~\ref{sec:srbastar}). \citet{yong13_II}
include hyperfine splitting in their Sc analysis, as we do here.
There is an 0.08 dex difference between their adoped log gf for Sc\,II
4246 \citep{kurucz&bell1995} and ours \citep{lawler_sc}, which is
accounted for by our 0.08 dex correction to that line's
abundance. Differences in log gf values from the above sources for the other lines considered
here range from $+$0.03 to $-$0.20, and if anything, should make our
abundances slightly larger than those of \citet{yong13_II}.
\citet{cayrel2004} and \citet{lai2008}, among others, found that the Mn\,I
$\lambda$4030 resonance lines had lower abundances than other Mn\,I lines by
as much as 0.4 dex. For the $\sim$40 stars in which we measured both
resonance and non-resonance Mn\,I lines, we found a difference of
$\Delta$(non-res.$-$res.) = $+$0.44 (s.e.m. 0.03) dex. We have
therefore applied a $+$0.44 dex correction to the abundances measured
from the Mn\,I 4030, 4033 and 4034\AA\ lines in all stars.
\citet{bergemann_mn} have demonstrated that the systematic offset
between resonance and non-resonance Mn\,I lines can be explained by
NLTE effects. They found NLTE corrections for resonance lines as
large as $\sim$ $+$0.7 dex for warm, metal-poor stars, while corrections
for other Mn\,I lines as large as $+$0.4 dex are possible. NLTE
[Mn/Fe] ratios for this SkyMapper sample would therefore be much
closer to the solar ratio.
As for the general abundance distributions shown in
Figure~\ref{f_fepeak}, the SkyMapper stars have
the same trends of [X/Fe] versus [Fe/H] and the same scatter as
the literature samples. The scatter with [Fe/H] is smallest for Cr and Ni,
while Mn and Co show (opposite to each other) trends of [X/Fe] with
[Fe/H]. \citet{cayrel2004} remarked upon the similar behavior of
[Cr/Fe] and [Mn/Fe] increasing with increasing [Fe/H] for their sample (both with quite small
scatter), and the same can be seen in the \citet{yong13_II} giant
sample.
In our sample, [Cr/Fe] and [Mn/Fe] values show similar trends with
comparable scatter ($\sim$0.15 dex). We also note that our mean
[X/Fe] values for Cr, Co and Ni agree very well with those of
\citet{yong13_II}, while our mean $\rm [Mn/Fe] = -0.42$ is $\sim$0.15
dex larger than theirs.
As \citet{yong13_II} did not include Zn, the literature sample we plot
in the bottom right panel of Figure~\ref{f_fepeak} is that of
\citet{cayrel2004} and \citet{heresII}. The SkyMapper stars show a
similar trend and scatter in [Zn/Fe] as in those samples, however, we
have found more stars exhibiting subsolar [Zn/Fe] ratios.
One star in Figure~\ref{f_fepeak} exhibits an [X/Fe] ratio very different from
the rest of the SkyMapper and literature samples. SMSS~J093829.27$-$070520.9
appears to have $\rm [Mn/Fe] \sim +0.7$. However, its spectrum has
S/N $\sim$10 at $\lambda$4000, and this abundance is based on
measurement of only
two Mn\,I resonance lines and has a standard deviation of 0.49 dex.
Consequently, its enhanced [Mn/Fe] ratio should be treated with skepticism.
\subsection{Neutron-Capture Elements}\label{sec:ncap}
The neutron-capture species considered in this analysis are Sr, Ba and
Eu. The first two are predominantly formed via the s-process in
low-mass AGB stars, while Eu is almost entirely formed via the
r-process \citep{sneden_araa, jacobson13}. The large variation of [X/Fe]
versus [Fe/H] ($>$1 dex) for neutron-capture elements, in strong constrast to the relative constancy of the $\alpha$-elements,
has also been well-established in the literature \citep{aoki05, heresII,
lai2008, roederer_ubiqrproc, yong13_II, cohen2013, spite_ncap2013,
roederer_ncap, roederer14_9stars}. Our sample shows similar
behavior (Figure~\ref{f_ncap}). Over the $\sim$2.5 dex range of
[Fe/H] spanned by our sample, there is evidence of the
dispersion in [X/Fe] increasing with decreasing [Fe/H] as found in the
literature (2$-$3 dex below $\rm [Fe/H] = -3$ compared to 1$-$2
dex at higher [Fe/H] for Sr and Ba in Figure~\ref{f_ncap}).
We have found no s-process
stars in our sample, even though the mean [Fe/H] of our sample is that
of typical s-process metal-poor stars (e.g.,
\citealt{placco2013_magI}).
This is consistent with the lack of stars with [C/Fe] $\gtrsim +$2, which
along with enhanced [s/Fe], is a signature of pollution from an AGB
companion (Section~\ref{sec:carbon}).
\begin{figure
\begin{center}
\includegraphics[clip=true,width=8cm]{f13.pdf}
\figcaption{ \label{f_srba}
[Sr/Ba] versus [Ba/Fe] (top) and [Fe/H] (bottom) for our sample and literature stars.
A conservative errorbar of 0.2 dex in [Ba/Fe] and 0.28 dex in
[Sr/Ba] is indicated in the upper right.
The location of \5, which exhibits the largest [Sr/Ba]
ratio of our sample (Section~\ref{sec:srbastar}) is labelled in both
panels.
The range of [Sr/Ba]
increases with decreasing [Ba/Fe], but some Ba-poor stars with
the solar r-process $\rm [Sr/Ba] = -0.5$ (dashed line) ratio are also
present. Though the number of stars with $\rm[Fe/H]$\ $< -3.5$ is
small, their presence is at odds with recent claims that there
is a cut-off in [Sr/Ba] in this metallicity range
\citet{aoki_srba}.}
\end{center}
\end{figure}
The top panel of Figure~\ref{f_srba} shows the [Sr/Ba] ratios for our sample as a
function of their [Ba/Fe], which compares the relative abundances
of light and heavy neutron-capture elements.
Except for the star \5\ which has [Sr/Ba]
$>$ 2 (see Section~\ref{sec:srbastar}),
our sample follows the same behavior as
those of, e.g., \citet{spite_ncap2013} and \citet{cohen2013}. The Ba-poor objects
show the largest range of [Sr/Ba] ratios, while the most Ba-rich
objects show less scatter. There are three Ba-poor stars
(with $\rm [Ba/Fe] < -1.0$) that exhibit the solar system r-process
[Sr/Ba] = $-$0.5. The Eu 4129\AA\ line was not
measureable in any of their spectra; therefore, if they do follow
the solar system r-process pattern, their level of r-process element
enrichment would be extremely low. The upper limits to their [Eu/Fe]
ratios are less than 0.4, at which level they would just be considered r-I
stars (see below).
Based on a large literature sample, \citet{aoki_srba} recently claimed
that there is a dearth of stars with measurable [Sr/Ba] ratios below
$\rm[Fe/H]$\ $< -3.5$. \citet{placco2013_magII} suggested this was due to
the small/incomplete sample of stars in this metallicity regime; the
bottom panel of Figure~\ref{f_srba} lends support to this argument
(see also \citealt{LAMOST_emp}).
Roughly half of the stars below $\rm[Fe/H]$ $< -3.5$ exhibit large
($\gtrsim$1) [Sr/Ba] ratios.
\begin{figure
\begin{center}
\includegraphics[clip=true,width=8cm]{f14.pdf}
\figcaption{ \label{f_eu}
[Eu/Fe] versus [Fe/H] for our sample (crosses) compared to
the \citet{frebel10} literature compilation (orange symbols). Upper limits
are indicated as arrows or triangles. The [Eu/Fe] ranges for r-process
enhanced (r-II and r-I) stars are indicated by dotted lines.}
\end{center}
\end{figure}
We detected the Eu 4129\AA\ feature in a number of our MIKE spectra,
and obtained upper limits on the Eu abundances for all other stars for
which it was not detected. These abundances are shown in
Figure~\ref{f_eu}. As Eu abundances were not included in the study of
\citet{yong13_II}, we only include the \citet{frebel10}
literature sample in Figure~\ref{f_eu}. Again, we see a similar
distribution of [Eu/Fe] with [Fe/H] in our study to that in the
literature. Note that most of our Eu abundances are upper limits
(denoted as arrows).
R-process enhanced stars are identified based on their Eu abundance:
strongly r-processed enhanced
so-called r-II stars have $\rm [Eu/Fe] > 1.0$, while mildly r-process
enhanced r-I stars have $0.3 \leq \rm [Eu/Fe] \leq 1.0$ (and both classes
have $\rm [Ba/Eu] < 0$; \citealt{heresII}).
These values are indicated with dotted lines
in Figure~\ref{f_eu}. Of the stars in our sample for which we have
bona fide Eu measures, four have $\rm[Eu/Fe] \ge
1$, while another 22 qualify as r-I stars. The metallicity range of
the r-II stars is $-2.77 \le \rm[Fe/H] \le -2.17$. The star with the
largest enhancement ([Eu/Fe] = $+$1.75), SMSS~J175046.30$-$425506.9,
also happens to be the most metal-rich of the r-II stars. Further
analysis of these r-process enhanced stars is ongoing.
We end the discussion with some remarks about NLTE effects on the
neutron-capture element abundances of metal-poor stars.
NLTE Sr\,II abundances are expected to differ from LTE values by no
more than 0.1 dex in the relevant stellar parameter regime
\citep{andrievsky_sr,bergemann_sr}. NLTE corrections to Ba\,II
abundances (from, e.g., the $\lambda$4554 line) can range from roughly
$-$0.10 to $+$0.25 dex, and are dependent upon the Ba abundance
\citep{andrievsky_ba}. However, as noted by, e.g., \citet{cohen2013} and
\citet{andrievsky_ba}, the magnitude of the scatter in metal-poor star Sr and Ba
abundances is far greater than can be attributed to NLTE effects, and
so they have little bearing on any interpretation of the data.
NLTE Eu abundances can be larger than the LTE
values by as much as $\sim$0.1 dex (\citealt{mashonkina2003}), though
to our knowledge Eu NLTE calculations have been done for dwarf stars only.
\subsection{Known Stars Recovered by SkyMapper}\label{sec:recover}
The coordinates of all the stars in Table~\ref{Tab:obs} were uploaded to the
Simbad\footnote{http://simbad.u-strasbg.fr/simbad/} database to check
for any that have been previously studied.
We used a search radius of 30$\arcsec$ around the stellar coordinates.
Eight stars were found to have an entry in the database: four stars were
found in the RAdial Velocity
Experiment survey (RAVE; data release 4) \citep{rave_dr4}, three were found in
various Hamburg-ESO survey studies, and the last is identified (as a star)
in the Millennium Galaxy Catalogue \citep{MGC03}. Table~\ref{tab:reobs} lists these
stars along with their alternate identifications and reference
studies.
The two most metal-poor stars in our sample, \5\ and
\2\ (with $\rm[Fe/H]$\ =
$-$3.97 and $-$3.94, respectively), are in fact rediscoveries.
\2\ was included in the sample of \citet{norris13_I} and \cite{yong13_II}, and
our stellar parameters and element abundances for this star are in excellent agreement
with their values. \5\ was identified in the RAVE survey, but the
stellar parameters found by \citet{rave_dr4} are very different from
ours: T$_{\rm eff}$/$\log$\,{\it g}/$\rm[Fe/H]$\ = 3600/4.5/$-$0.63 as opposed to
4846/1.60/$-$3.97.
Stellar parameters are determined from RAVE R$\sim$7500 spectra
($\lambda$8410-8795) using sophisticated algorithms that match the
data to a grid of synthetic spectra \citep{rave_dr4}.
\citet{rave_dr4} give a set of stellar parameters and data
characteristics (S/N, radial velocity measurement error, etc.) that
serves as quality checks to ensure the results of the RAVE pipeline
are robust and reliable. \5\ fails to meet both the S/N
($>$20 pixel$^{-1}$) and the T$_{\rm eff}$ ($>$3800 K) requirements. Of the
three other RAVE stars in our sample (Table~\ref{tab:reobs}), two meet
all quality criteria while the RAVE pipeline did not converge for \4.
For the two stars that pass muster, our T$_{\rm eff}$\ values agree within 180 K of the
RAVE values and our $\rm[Fe/H]$\ values agree within 0.15 dex.
Differences between $\log$\,{\it g}\ values are
quite large, however: 0.6 and 2.4 dex for \1\ and \8, respectively.
No systematic offset in any parameter is present.
\7\ and \3\ were studied by \citet{cayrel2004} and \citet{heresII},
respectively. For the former, our stellar parameters agree very well with those
of \citet{cayrel2004}, within 120 K, 0.15 dex, 0.25 km s$^{-1}$\ and 0.1 dex
in T$_{\rm eff}$, $\log$\,{\it g}, v$_{\rm t}$\ and $\rm[Fe/H]$, respectively. The agreement with
\citet{heresII} is not as good in the case of \3: our T$_{\rm eff}$\ is 250 K
cooler, and our $\log$\,{\it g}\ and $\rm[Fe/H]$\ values are 0.7 and 0.4 dex lower,
respectively.
As for radial velocity measures, our value for \2\ agrees with that
found by \citet{norris13_I} within 1.4 km s$^{-1}$. For \3\, our measure is
3.7 km s$^{-1}$\ larger than in \citet{heresII}, while \citet{cayrel2004} does not
provide a radial velocity measurement for \7. For those stars in
common with RAVE, our measures are $-$22
(for \5)
to $+$23 (for \1)
km s$^{-1}$\ different, in
the sense (This Study $-$ RAVE). Our measure for
\4\
is 7.5
km s$^{-1}$\ smaller than RAVE's, while there is only 0.4 km s$^{-1}$\ difference
for\\ \8. According to \citet{rave_dr4}, radial velocities measured
from RAVE spectra in the S/N range of these stars ($\sim$10-40) agree
within 5$-$8 km s$^{-1}$\ to literature values, though differences as large
as $\sim$20 km s$^{-1}$\ are possible (their Figure 34).
Given the long base-line between our measures and theirs (the RAVE
observations were taken in 2004 and 2006), it is possible that at least
\1\ and \5\ are binary systems.
\subsection{Comparison to Literature Samples}
A quantitative comparison of our analysis to those of other studies
can be made by inspection of the linear regression analyses carried
out by different groups on different samples. The results of the
regression analysis on this SkyMapper sample have been included in
Figures~\ref{f_light}--\ref{f_ncap}; for convenience, they are presented in
Table~\ref{tab_lsq} along with those of
\citet{yong13_II, cayrel2004, cohen2013}. Figure~\ref{f_slopes}
presents the values from Table~\ref{tab_lsq}
graphically.\footnote{All groups considered here confined their
regression analysis to stars with $\rm [Fe/H] < -2.5$. Note that
the slopes given for \citet{cohen2013} in Table~\ref{tab_lsq}
were calculated using [X/Fe] ratios at $\rm [Fe/H] = -3$ and $\rm
[Fe/H] = -3.5$ for their CEMP-no stars (columns 3 and 4 in
their Table 13).}
The errorbars on the points represent the uncertainty
of the slope, as given in this work and those of \citet{yong13_II} and
\citet{cayrel2004} (we note that the slope uncertainties in the latter
are smaller than the symbol in the figure).
\begin{figure
\begin{center}
\includegraphics[clip=true,width=8cm]{f15.pdf}
\figcaption{ \label{f_slopes}
Top panel:
The slopes of lines of best fit for each element [X/Fe]. Here, our
linear regression analysis (black squares) is compared to that of
\citet{yong13_II} (red circles), \citet{cayrel2004} (blue triangles)
and \citet{cohen2013} (stars). For all but the last sample, errorbars
on the points represent the slope uncertainties. Note that the
uncertainties on the \citet{cayrel2004} slopes are smaller than the
symbols in the figure. Bottom panel: the difference between
individual study mean [X/Fe] ratio for their stellar sample and the
mean [X/Fe] ratio of all four studies, for elements $\rm Z=11-30$.}
\end{center}
\end{figure}
Generally, the numerical values of the slopes in our analysis agree
with those of the literature studies within 2$\sigma$ for most of the
elements presented here. Some elements show a large range of slopes:
namely Na, Mg, Al, Co and Zn. Another way of comparing results from
different studies is to compare the mean [X/Fe] ratios found for stars
with $\rm [Fe/H] < -2.5$, and this is shown Table~\ref{tab_means}.
The bottom panel of Figure~\ref{f_slopes} shows the difference between
the [X/Fe] ratio found for a particular stellar sample and the mean
$<\rm [X/Fe]>$ ratio of the four studies in Table~\ref{tab_means}. The
errorbars on the points are the standard errors of the mean. Here one
can see evidence of the systematic offsets between our study and
others for some elements noted
earlier, namely for Sc. For most of the elements, however, the
mean [X/Fe] ratios found by different studies agree within a factor of
two of their standard errors, though our Na and Mg values are higher
than those of the other samples considered here.
\section{SkyMapper Metal-Poor Stars of Interest}\label{sec:disc}
\subsection{A New ``Fe-enhanced'' Metal-poor Star}\label{sec:ferich}
One star, SMSS~J034249.53$-$284216.0 ($\rm [Fe/H] = -2.28$),
has subsolar [Mg/Fe], [Ca/Fe], [Sc/Fe], [Ti\,/Fe]
and [Ti\,II/Fe] ratios, the lowest of the entire sample. In fact, it
is has $\rm [X/Fe]<0$ for all elements save Si and Eu. Its
S/N ratio (mean $\sim$30) is less than the median value for the
sample, but by no means the lowest, and the abundances for
most elements are based on the measure of several lines, so these
results are robust.
There is a growing number of metal-poor stars in the literature that
show similar low [X/Fe] ratios
\citep{nissen_schuster, spite2000, ivans_alphapoor,
cayrel2004, honda04, cohen_huang2010,
bonifacio2011, venn2012, caffau2013, yong13_II}.
They have been called ``$\alpha$-poor''
or ``Fe-rich'' metal-poor stars. The latter designation is likely
more appropriate for those stars that show deficiencies in numerous
other species in addition to the $\alpha$-elements. Indeed the
element abundance patterns of such stars look similar to those of more
typical metal-poor stars, but shifted as a result of an additional Fe component.
\begin{figure}[!ht]
\begin{center}
\includegraphics[clip=true,width=9cm]{f16.pdf}
\figcaption{ \label{f_ferich}
The abundance difference, in the sense ([X/Fe] $-$ [X/Fe]$_{\rm Ref}$) for
``Fe-enhanced'' stars relative to that of the mean [X/Fe] ratios
found in our study (Table~\ref{tab_means}). The star in this study,
SMSS~J034249.53$-$284216.0, is indicated by cyan squares and cyan
bold line. For simplicity, lines connecting individual element
abundances are only drawn for stars where most of the species have
been measured; some stars in this figure only have [$\alpha$/Fe]
reported in the literature. The patterns for all the stars are
generally similar. References for the
literature sample include:
\citet{ivans_alphapoor,yong13_II,cayrel2004,caffau2013,spite2000,bonifacio2011}.}
\end{center}
\end{figure}
Figure~\ref{f_ferich} plots the element
abundance pattern of SMSS~J034249.53$-$284216.0 (cyan squares, cyan bold line),
along with other stars exhibiting low [X/Fe] ratios in the
literature, relative to the mean abundances from the SkyMapper
sample\footnote{These are taken to represent [X/Fe] ratios for typical
halo stars with $\rm[Fe/H]$\ $< -2.5$.}
(Table~\ref{tab_means}). We restrict the literature stars in
Figure~\ref{f_ferich} to have $\rm[Fe/H]$\ $< -2$, though we note that many
other ``Fe-enhanced'' stars exist in the literature at higher
metallicities (e.g., \citealt{nissen_schuster,
ivans_alphapoor,cohen_huang2010, venn2012,ivans_alphapoor, bonifacio2011}).
As has been noted
in the literature (e.g., \citealt{yong13_II}), there is some scatter in
the abundances of these stars. The average [X/Fe] offset from the
mean SkyMapper sample abundances in Figure~\ref{f_ferich} is $-$0.40 dex, with a
1$\sigma$ scatter of 0.16 dex (for SMSS~J034249.53$-$284216.0,
the offset is $-$0.52 dex).
While there is scatter in the abundance patterns, the stars in general
show sub-solar [X/Fe] ratios for all elements except for the Fe-peak
elements Cr and Mn.
A natural explanation for the Fe-enhancements exhibited by these
stars is that they formed from gas preferentially enriched with SNe Ia
products rather than just SNe II (e.g., \citealt{cayrel2004, caffau2013,
yong13_II}). Such environments exist in dwarf galaxies
(indeed some known Fe-enhanced stars are in dwarf galaxies
\citep{venn2012, cohen_huang2010}), leading to the possibility that
the most metal-poor of the Fe-enhanced stars in the halo originated in dwarf galaxies. That said, recent work by \citet{kobayashi2014_alphapoor} has shown
that the scatter and low element abundance ratios of stars in
\citet{caffau2013} and \citet{cohen2013} with $\rm [Fe/H] \leq -3$ are
well-matched by single core-collapse SN or hypernova yields, making
a dwarf galaxy origin unnecessary.
This single enrichment scenario
likely does not hold for the more metal-rich stars,
including SMSS~J034249.53$-$284216.0 with $\rm [Fe/H] = -2.3$.
For these, the Fe-enhancements may be due to variations in
the progenitor masses and associated timescales of Type Ia supernovae.
For now, these few stars ($\sim$1\%$-$2\%\ of hundreds of halo stars so far
subject to high-resolution spectroscopic study)
indicate inhomogeneities in chemical evolution at the time
of their formation, in contrast to the apparent wide-spread
homogeneity in the bulk of metal-poor star formation (recall the small
scatter and lack of correlation in [$\alpha$/Fe] for the metal-poor
star sample of \citealt{cayrel2004}).
As more such stars are
found, it will be possible to investigate and better quantify the
degree of inhomogeneity in star formation and chemical evolution in
the early universe.
\subsection{A new $\rm [Fe/H] \sim -4$ star with high [Sr/Ba]}\label{sec:srbastar}
Although \5\ first appeared in the RAVE catalog (\citealt{rave_dr4}; see
Section~\ref{sec:recover}), our work demonstrates for
the first time that it is an extremely metal-poor star, with
$\rm [Fe/H] = -3.97\pm0.14$. With $\rm [C/Fe] = +0.07$ ($+$0.25
after applying the \citet{placco14} correction), it is not one of the CEMP
stars identified in Section~\ref{sec:carbon}, and its $\alpha$- and
Fe-peak element [X/Fe] ratios are normal (see Figures~\ref{f_alpha} and
\ref{f_fepeak}). However, there are no barium lines detectable in its
fairly high S/N spectrum and an upper limit of $\rm [Ba/Fe] < -0.91$
was obtained.
In contrast, Sr lines are quite strong, giving a robust measure of
$\rm [Sr/Fe] = +1.08$. This [Sr/Fe] ratio is compatible with the most
Sr-rich stars of comparable metallicity as seen in
\citet{roederer_ncap} (his Figure 2).
The measured upper limit of [Eu/Fe] is $+$1.15 and
unfortunately not helpful in further constraining the origin of the
neutron-capture elements in this star. Using Equation 6 of
\citet{hansen2014_rproc} to predict the Eu abundance from the Ba upper
limit, [Eu/Fe] $< +$0.07.
\begin{figure}[!ht]
\begin{center}
\includegraphics[clip=true,width=9cm]{f17.pdf}
\figcaption{ \label{f_3178}
The LTE element abundance pattern for star
\5\ relative to that
of HD~122563 (abundances given in Table~\ref{tab:122563}). Note that
the abundance patterns are quite similar for most elements, save for
the neutron capture species.
The large [X/Fe] ratios for Sr, Y, and Zr in \5\ are most striking,
while its Ba abundance
is just an upper limit. Two upper limits are indicated for [Eu/Fe],
connected by a dashed line: $+$1.15 and $+$0.07, as measured in the
spectrum and as predicted using the relation of
\citet{hansen2014_rproc}, respectively.}
\end{center}
\end{figure}
Figure~\ref{f_srba} shows that \5\ exhibits one of largest [Sr/Ba]
ratios currently known for a metal-poor star in the Milky Way
halo.\footnote{We note that stars in the ultra-faint dwarf galaxy Segue-1 exhibit
extremely low upper limits to their Sr and Ba abundances that point
to intriguing neutron capture element enrichment episodes that are
different from the Milky Way halo stars considered here
\citep{frebel_segue1}.}
To our knowledge, only one
other star is known to have [Sr/Ba] $\gtrsim$2: SDSS J1422$+$0031,
with $\rm[Fe/H]$\ = $-$3.03 and [Sr/Ba] = $+$2.2 \citep{aoki2013}.
Together, these two stars are the most extreme examples of the
growing number of extremely metal-poor stars that show large
($\gtrsim$0.8 dex)
enhancements of the light neutron-capture element Sr relative to
the heavier neutron-capture element Ba, as have been
found in several studies (e.g.,
\citealt{honda04,aoki05,lai2008,hollek11,aoki2013,cohen2013,placco2013_magII}).
Such stars are
generally taken as evidence for an extra neutron-capture element
production mechanism in addition to the main r-process as the source
of the heaviest elements in the early universe (e.g.,
\citealt{travaglio, honda06, sneden_araa, jacobson13}).
Mechanisms such as the Light Element Primary Process
(LEPP; \citealt{travaglio}), the
weak r-process \citep{ishimaru2005},
the weak s-process \citep{heil_weaks} and the truncated r-process \citep{boyd_tr}
have
been invoked to explain the existence of stars with large enhancements
of Sr, Y, and Zr relative to Ba and Eu.
We inspected the spectrum of \5
for the presence of other
neutron-capture species absorption lines, and were able to detect
several Y and Zr lines, but no lines of species belonging to the second
peak (e.g., Ba, La, Ce, Nd) or higher. Spectrum synthesis of four Y\,II and three
Zr\,II lines resulted in [Y/Fe] = $+$0.80$\pm$0.26 and [Zr/Fe] =
$+$1.06$\pm$0.16 (s.d.). \5\ is therefore strongly enhanced in first peak
neutron-capture species, with no detectable presence of heavier species.
Figure~\ref{f_3178}
shows the abundance pattern of
SMSS~J022423.27$-$573705.1 relative to that of
HD~122563, the poster star exhibiting such light neutron-capture element
enhancements with [Sr/Ba] = $+$0.76 \citep{honda06}. To minimize any systematics in this
comparison, we have carried out our own abundance analysis of
HD~122563, the results of which are presented in
Table~\ref{tab:122563}. (We refer the reader to \citet{teff_calib} for details
regarding the data, but note that the analysis presented here is
separate from the results in that work.)
These two stars show similar abundance
patterns for most elements save for Sr, Y and Zr.
It is clear from this figure that whatever
the source(s) that produced this pattern of heavy elements (i.e., the
LEPP \citep{travaglio}),
it operated even more strongly in the enrichment that led to the
formation of \5\ than for HD~122563.
Rapidly rotating, low metallicity massive stars (``spinstars'') have been considered as
a possible source of light neutron-capture elements in the early universe, and models of such have been able to reproduce
the s-process element enhancements of low-metallicity field stars and
globular cluster stars (e.g., \citealt{pignatari},
\citealt{chiappini11}, \citealt{frischknecht}).\footnote{See, however,
the results of \citet{ness2014} which do not support the spinstar
origin scenario in the case of globular cluster NGC 6522 \citep{chiappini11}.}
The
abundance pattern produced by the 25 M$_{\odot}$,
$\rm [Fe/H] = -3.8$ model of \citet{frischknecht} agrees relatively
well with that of SMSS~J022423.27$-$573705.1
(Figure~\ref{f_3178}) for the elements in common (Co, Ni, Sr; see
their Figure 1). They
do not give production factors for the elements Cr and Mn, which are
both low in our star. Their model also predicts a
yield of Zn relatively larger than Co and Ni, but we could not
detect Zn lines in the spectrum of SMSS~J022423.27$-$573705.1.
Of the three stars in our sample with $\rm [Fe/H] < -3.5$, only one
star has a detectable Zn line in its spectrum. An upper limit EW
measure for
the Zn~I $\lambda$4810 in the spectrum of
SMSS~J022423.27$-$573705.1 corresponds to $\rm [Zn/Fe] < +0.8$,
which, together with the other element abundances, is consistent with
the pattern from \citet{frischknecht}.
It is not straightforward to
compare the abundances of elements below the Fe-peak (Mn and lower)
to the models of \citet{frischknecht}, because these models do not
include element production in the supernova explosion itself
(R.\ Hirschi, 2014, private communication).
As more of
these stars are found, and the abundances of larger numbers of
neutron-capture elements are measured in them, it will be easier to
disentangle the presence of different production mechanisms and to
identify their production sites.
\section{Summary and Conclusions}\label{sec:summary}
We have presented a detailed chemical element abundance analysis of
the first SkyMapper metal-poor star candidates that were observed at high
spectroscopic resolution. Based on a 1D LTE element
abundance analysis, the stellar parameters and element
abundances for these stars show them to be bona fide metal-poor halo
stars, as indicated by how well they match the abundance patterns of
halo stars in the literature.
The main finding of this study is the verification of EMP star
candidates selected based on photometry from the SkyMapper Southern
Sky Survey and medium-resolution spectroscopy. Excluding previously
known extremely metal-poor stars in our sample, we have confirmed 38
new stars to have $\rm [Fe/H] < -3.0$, eight of which have $\rm [Fe/H]
< -3.5$. More importantly, the EMP candidate selection technique
based on the SkyMapper photometry has been improved over the
course of this program, and indeed the most iron-poor star
known to date (with $\rm [Fe/H] < -7$; \citealt{keller_thestar}), was
confirmed by its high-resolution Magellan-MIKE spectrum
during the accumulation of the sample presented here.
Concerning the abundances of particular elements or of particular
stars in the study presented here, we have found the following:
\begin{itemize}
\item
Eight stars previously known in the literature have been recovered by
the SkyMapper survey; six of which were previously known to be
extremely metal-poor. We find reasonable to excellent agreement with
the results of other studies for four of these objects:
T$_{\rm eff}$\ within 250 K; $\rm[Fe/H]$\ within 0.4 dex. One star,
which was not previously identified as metal-poor, turns out to be the
most metal-poor star in our sample, with $\rm[Fe/H]$\ = $-$3.97.
\item
After correcting stellar C abundances for evolutionary effects,
24 stars are classified as CEMP stars based on the criterion of
\citet{aoki_cemp_2007}. Considering only stars with
$\rm [Fe/H] \leq -3$, this results in a CEMP fraction of 39\%, in good
agreement with other studies. Seven stars have $\rm [C/Fe] > 1$ and
are classified as CEMP-no stars. Of these, five have $\rm[Fe/H]$\ $< -3$.
\item Our most metal-poor star with $\rm[Fe/H]$\ = $-$3.97,
has $\rm [Sr/Ba] \gtrsim
2$, showing an extreme ratio of light to heavy neutron-capture
element abundances. This indicates that
the weak r-process (or other mechanism) can yield more extreme
light neutron-capture element enhancements than previously thought.
\item One star with $\rm [X/Fe] \leq 0$ for all elements save Si
and Eu, is likely a member of the growing population of
``Fe-enhanced'' metal-poor stars in the literature.
\item Four stars have r-process enhancements $\rm [Eu/Fe] > 1$ and
are classified as r-II stars, while another 22
appear to be at least mildly r-process enhanced
based on their [Eu/Fe] ratios. The relative fractions of r-I (22/122
= 18\%) and r-II stars (4/122 = 3\%) are comparable to those found
by \citet{heresII} ($>$14\% and 3\%, respectively). We caution
however that the metallicity ranges of the two samples are different
(\citealt{heresII} had no stars with $\rm[Fe/H]$\ $< -3.5$), so the similarity
of the r-I/II fractions may be coincidental.
\end{itemize}
These results successfully demonstrate the
capability of the SkyMapper survey to find more stars at the
very metal-poor end of the Milky Way halo MDF, as well as stars exhibiting
interesting abundance signatures. The increased sample size of these
metal-poor stars will improve our understanding of chemical
enrichment in the early epochs of the universe, as well as reveal
insight into the nature of the Population III stars that were the
first seeds of chemical enrichment.
\acknowledgements{This research has made
use of the SIMBAD database, operated at CDS, Strasbourg, France and
of NASA's Astrophysics Data System Bibliographic
Services.
This publication also makes use of data products from the Two Micron
All Sky Survey, which is a joint project of the University of
Massachusetts and the Infrared Processing and Analysis
Center/California Institute of Technology, funded by the
National Aeronautics and Space Administration and the
National Science Foundation.
We thank the referee for helpful suggestions that
improved the presentation of this work.
R.\ Hirschi is thanked for informative discussions regarding
the models of \citet{frischknecht}.
A.F.\ acknowledges support from NSF grant AST-1255160.
A.C.\ was partially supported by the European Union FP7
programme through ERC grant 320360.
M.S.B, G.D.C. and S.K. acknowledge support from the Australian
Research Council through Discovery Projects grant
DP12010137. M.A. has been supported by an Australian Research
Council Laureate fellowship (grant FL110100012).
K.L.\ acknowledges the European Union FP7-PEOPLE-2012-IEF grant
No.\ 328098. B.P.S.\ has been supported by an Australian Research
Council Laureate fellowship (grant FL0992131). The work of
J.M.P. and Q.Y. was supported by the MIT UROP program, and
J.M.W. was supported by the Research Science Institute at MIT.
Australian access to the Magellan Telescopes was supported through
the Collaborative Research Infrastructure Strategy of the Australian
Federal Government.}
|
{
"timestamp": "2015-07-14T02:16:33",
"yymm": "1504",
"arxiv_id": "1504.03344",
"language": "en",
"url": "https://arxiv.org/abs/1504.03344"
}
|
\section{Introduction}
A \emph{histogram} computed on a dataset $D$ is a vector of counts, such that each record in $D$ affects at most a single histogram element, called \emph{bin}. The histogram constitutes one of the most basic statistical tools for describing the dataset distribution. A \emph{range-sum} query over a histogram returns the sum of values of a set of \emph{contiguous} bins. Our goal is to publish a histogram on $D$ that satisfies \emph{$\epsilon$-differential privacy} \cite{dwork11}. This paradigm entails perturbing the bins prior to their publication, so that each individual record in $D$ is protected. We aim at minimizing the total error incurred by the perturbation when answering \emph{arbitrary} (i.e., not known a priori) range-sum queries. For example, consider a database $D$ with medical records, and a histogram on $D$ where each bin contains the number of patients of a certain age. Assuming bins sorted on age, a range-sum query over this histogram returns the number of patients in the range. In this scenario, it is important that the published histogram does not violate the privacy of any individual patient; at the same time the range-sum results should be accurate.
Differentially private histogram publication has been studied extensively. The existing schemes can be divided into two categories. ``Data-aware'' methods exploit the underlying dataset distribution \cite{acs12,xu12,zhang14,li14}. They \emph{smooth} the histogram prior to its perturbation by grouping \textit{similar} bins and replacing their values with the group \emph{average}. This yields reduced perturbation per bin. However, these mechanisms exhibit superquadratic time complexities, which may be prohibitive in time-critical applications. ``Data-oblivious'' methods build a perturbed \emph{aggregate tree} on top of the histogram, and answer range-sum queries by summing a small number of tree nodes, instead of numerous individual bins falling in the range \cite{cormode12,hay10,qardaji13,xiao11}. Such approaches are very efficient, running in time linear to the number of histogram bins, but may yield low utility for some practical datasets.
To the best of our knowledge, this is the first work that aims at efficiency, without compromising utility. Towards this goal, we address the problem in a principled, \emph{modular} approach. Specifically, we first identify three building blocks, which we call \emph{modules}, namely \emph{Smoothing}, \emph{Hierarchy Level}, and \emph{Fixed Queries}; the first is inspired by the data-aware techniques, the second works on a single level in the aggregate tree, and the third is based on techniques such as the matrix mechanism \cite{li10}, which has been applied to increase utility for fixed query workloads \cite{li14}. We then formulate a \emph{scheme} as a combination of these modules, integrated with certain components, called \emph{connectors}. The latter do not affect privacy, but serve to properly format the inputs of the modules. Subsequently, we express the existing state-of-the-art methods in our modular framework, and discover opportunities for optimization. Finally, we devise novel efficient and effective schemes by composing the modules non-trivially. Concretely, our contributions are:
\begin{itemize}
\vspace{-0.1cm}
\item We introduce a modular framework on differentially private histograms for range-sum queries. Our approach offers multiple benefits: (i)~using a small set of simple modules and connectors, we can analyze all existing methods, and devise novel schemes with variable efficiency and utility, (ii)~given the privacy level of each module, we can easily derive the privacy of arbitrarily complex schemes, and (iii)~each module can be optimized separately; furthermore, potential future optimizations can be incorporated to an existing scheme with minimal effort.
\vspace{-0.06cm}
\item We analyze an important submodule of the Smoothing module, namely the grouping method, which essentially solves an optimization problem. We point out the two objective functions most heavily used in the literature, propose optimizations and evaluate their effect on utility.
\vspace{-0.06cm}
\item We design novel schemes based on the defined modules, including the first mechanism that seamlessly combines the data-aware and -oblivious methodologies. In addition, we efficiently adapt schemes for fixed query workloads (e.g., the matrix mechanism) to arbitrary range-sums, via a simple but powerful technique based on \emph{prefix sums} \cite{ho97}.
\vspace{-0.06cm}
\item We provide a thorough experimental evaluation that compares the best existing methods with our new solutions, testing over three real datasets with different characteristics. We exhibit that there is a trade-off between algorithmic efficiency and utility across the various approaches.
\vspace{-0.1cm}
\end{itemize}
The remainder of the paper is organized as follows. Section \ref{sec:background} includes preliminary information and surveys the related work. Section \ref{sec:modular} introduces our modular framework. Section \ref{sec:grouping} investigates grouping in depth. Section \ref{sec:schemes} presents our proposed mechanisms. Section \ref{sec:experiments} experimentally evaluates all schemes, whereas Section \ref{sec:conclusion} concludes our work.
\section{Background}\label{sec:background}
Section \ref{subsec:setting} formulates our setting, and includes the necessary primitives on differential privacy. Section \ref{subsec:related} surveys differentially private histogram publication.
\subsection{Setting and Primitives}\label{subsec:setting}
Let $\mathcal{D}$ be a collection of datasets. We define a family of functions $\mathcal{F} = \{F_j: \mathcal{D} \rightarrow \mathcal{H}\}$, such that for all $j$ and all $D \in \mathcal{D}$, $F_j(D) = \mathbf{h} \in \mathcal{H}$ is an (ordered) vector called \emph{histogram}. An element of $\mathbf{h}$ is termed \emph{bin} and consists of a value and a label, where $\mathbf{h}[i]$ represents the $i^\textrm{th}$ bin value of $\mathbf{h}$. All histograms have the property that any record in $D$ increments \emph{at most a single} $\mathbf{h}[i]$ by $1$. Finally, we call $F_j$ a \emph{histogram algorithm}. For instance, let $D\in \mathcal{D}$ be a dataset of medical records. Then, $F_1 \in \mathcal{F}$ may produce histogram $\mathbf{h}_1$ such that $\mathbf{h}_1[i]$ is the number of patients in $D$ having age $i$, and $F_2 \in \mathcal{F}$ may produce histogram $\mathbf{h}_2$ such that $\mathbf{h}_2[i]$ is the number of patients in hospital with id $i$. Observe that, the presence of a patient in $D$ increments at most one bin by $1$ in both histograms.
Our goal is to publish a $n$-element histogram $\mathbf{h}$ produced by some \emph{fixed} algorithm $F$ on a $D \in \mathcal{D}$, while satisfying $\epsilon$-\emph{differential privacy} and allowing \emph{arbitrary range-sum queries} on its bins with high utility. Specifically, we define a range-sum query as a range of bins $[i_l, i_u]$, $1 \leq i_l \leq i_u \leq n$, which returns the sum $\sum_{i=i_l}^{i_u} \mathbf{h}[i]$. In our example above, a range-sum query on $\mathbf{h}_1$ could be $[10,20]$, asking for the number of patients between $10$ and $20$ years old. We assume that the range queries are \textit{not known} prior to the publication of the histogram.
To achieve $\epsilon$-differential privacy, we apply a mechanism $M$ on the histogram, which perturbs it in a way that satisfies the following definition, adapted from \cite{dwork06orig}.
\begin{definition}\label{def:epsilon}
A mechanism $M:\mathcal{H}\rightarrow \mathcal{\hat{H}}$ satisfies $\boldsymbol{\epsilon}$\textbf{\emph{-differential privacy}} for a histogram algorithm $F \in \mathcal{F}$, if for all sets $\hat{H} \subseteq \mathcal{\hat{H}}$, and every pair $D,D' \in \mathcal{\mathcal{D}}$ where $D'$ is obtained from $D$ by removing a record ($D,D'$ are called neighboring), it holds that
$$
\mathrm{Pr}[M(F(D))\in \hat{H}] \leq e^\epsilon \cdot \mathrm{Pr}[M(F(D')) \in \hat{H}]
$$
\end{definition}
Intuitively, $\epsilon$-differential privacy guarantees that the perturbed histogram $\hat{H}$ will be the same with high probability (tunable by $\epsilon$), regardless of whether a patient agrees to participate in the publication or not. Equivalently, the sensitive information of any patient cannot be inferred from the published data.
\begin{definition}\label{def:sens}
The \emph{\textbf{sensitivity}} of any histogram algorithm $F \in \mathcal{F}$ is $\Delta(F) =\max_{D, D' \in \mathcal{D}}\left\|F(D)-F(D')\right\|=1$
for all neighboring $D,D' \in \mathcal{D}$.
\end{definition}
In other words, the sensitivity of $F$ represents how much the histogram $F(D)$ changes when a record is deleted from $D$. Since any record contributes $1$ to at most a single bin, the sensitivity is $1$ for any histogram algorithm $F \in \mathcal{F}$.
The most basic technique to achieve $\epsilon$-differential privacy is to add Laplace noise to the histogram bins using the Laplace Perturbation Algorithm (${\sf LPA}$ \cite{dwork06,dwork11}). Let $\mathit{Lap}(\lambda)$ be a random variable drawn from a Laplace distribution with mean zero and scale parameter $\lambda$. ${\sf LPA}$ achieves $\epsilon$-differential privacy through the mechanism outlined in the following theorem, adapted from \cite{dwork06}.
\begin{theorem}\label{theo:laplace_mech} Let $F \in \mathcal{F}$ and define $\mathbf{h} \stackrel{\textrm{\emph{def}}}{=} F(D)$. A mechanism $\mathcal{M}$ that adds independently generated noise from a zero-mean Laplace distribution with scale parameter $\lambda=\Delta(F)/\epsilon=1/\epsilon$ to each of the values of $\mathbf{h}$, i.e., which produces transcript $\mathbf{\hat{h}} = \mathbf{h} + \langle \mathit{Lap}(1/\epsilon) \rangle^n$,
enjoys $\epsilon$-differential privacy.
\end{theorem}
With ${\sf LPA}$, a range-sum query $[i_l, i_u]$ is processed on the noisy $\mathbf{\hat{h}}$ and returns $\sum_{i=i_l}^{i_u} \mathbf{\hat{h}}[i]$. The Laplace noise injected in each bin introduces error, which is aggregated when the noisy bin values are added. For large ranges, this error may completely destroy the utility of the answer. Numerous works (overviewed in Section~\ref{subsec:related}) introduce alternative mechanisms for improving the utility of the output histograms in the case of range-sum queries.
Finally, we include a \emph{composition} theorem (adapted from \cite{pinq}) that is useful for our proofs. It concerns executions of multiple differentially private mechanisms on non-disjoint and disjoint inputs.
\begin{theorem}\label{theo:comp}
Let $M_1, \ldots, M_r$ be mechanisms, such that each $M_i$ provides $\epsilon_i$-differential privacy. Let $\mathbf{h}_1$, $\ldots$, $\mathbf{h_r} \in \mathcal{H}$ be histograms created on pairwise non-disjoint (resp. disjoint) datasets $D_1, \ldots$, $D_r$, respectively. Let $M$ be another mechanism that executes $M_1(\mathbf{h}_1), \ldots,$ $M_r(\mathbf{h}_r)$ using independent randomness for each $M_i$, and returns their outputs. Then, $M$ satisfies
$\left(\sum_{i=1}^r{\epsilon_i}\right)$-differential privacy (resp. $\left(\max_{i=1}^r{\epsilon_i}\right)$-differential privacy).
\end{theorem}
The above theorem allows us to view $\epsilon$ as a \emph{privacy budget} that is distributed among the $r$ mechanisms. Moreover, note that the theorem holds even when $M_i$ receives as input the private outputs of $M_1, \ldots, M_{i-1}$ \cite{pinq}.
\subsection{Differentially Private Histograms}\label{subsec:related}
Existing literature on differentially private histograms for range-sum queries aims at improving upon {\sf LPA} in terms of utility. We divide the approaches into two categories; \textit{data-aware} that utilize \textit{smoothing}, and \textit{data-oblivious} that rely on \textit{hierarchical} tree structures.
\medskip
\noindent\textbf{Data-aware methods.}
These approaches first smooth the histogram, typically either by grouping similar bin values and substituting them with their average, or by performing a smoothing filter such as the Discrete Fourier Transform (DFT). Subsequently, they apply Laplace noise similar to ${\sf LPA}$ to the averages or the DFT coefficients. Range-sum queries are processed by summing the histogram bin values in the query range. Smoothing reduces the sensitivity and, hence, the injected Laplace noise, but adds approximation error. Consequently, smoothing methods are effective if the Laplace noise error reduction exceeds the smoothing approximation error. The bin grouping algorithm assigns scores to a set of potential grouping strategies, and selects the one with the minimum score, in a manner that does not compromise differential privacy. Existing approaches differ in the set of examined strategies, the scoring function, and the selection process.
The SF algorithm \cite{xu12} follows the grouping and averaging paradigm. Specifically, given as input a \emph{fixed} parameter $k$ and privacy budgets $\epsilon$, $\epsilon'$, SF initially finds a set of $k$ groups of \emph{contiguous} bins through an $\epsilon'$-differentially private process. Subsequently, it smooths the bin values based on the grouping, and adds Laplace noise generating $(\epsilon-\epsilon')$-differentially private histogram. Due to linear composition (Theorem \ref{theo:comp}), the SF mechanism achieves $\epsilon$-differential privacy. The grouping sub mechanism of SF operates on the original histogram and determines the $k$ groups such that the estimated \textit{squared} error is minimized. This error is expressed as the sum of (i) the squared approximation error due to smoothing, and (ii) the squared error from injecting Laplace noise with scale $1/(\epsilon-\epsilon')$ prior to publication. It then applies the exponential mechanism \cite{mcsherry07} in order to alter the group borders and achieve $\epsilon'$-differential privacy. Note that, due to this step, the total error of SF eventually deviates from the actual minimum. The grouping submodule of SF runs in $O(n^2)$.
Acs et al. \cite{acs12} present two mechanisms, EFPA and P-HP. EFPA is an improvement of \cite{rastogi10}, which smooths the histogram using a subset of its DFT coefficients perturbed with Laplace noise, while guaranteeing that the output histogram satisfies $\epsilon$-differential privacy. P-HP is a grouping and averaging method that improves SF \cite{xu12}. In particular, instead of receiving the number of groups $k$ as input, it discovers the optimal value of $k$ on-the-fly. Contrary to SF, it utilizes an \textit{absolute} error metric. The grouping algorithm of P-HP runs also in $O(n^2)$, but similarly to SF does not examine all possible groups. P-HP is shown to outperform both EFPA and SF in terms of utility \cite{acs12}.
Motivated by \cite{kellaris13} (for a different setting), AHP \cite{zhang14} first applies ${\sf LPA}$ to the histogram with scale $1/\epsilon'$, and \emph{sorts} the resulting bins in descending or ascending order. Subsequently, it executes a grouping and averaging technique that is different from SF and P-HP. Specifically, it operates on already $\epsilon'$-differentially private data and, hence, does not need to apply the exponential mechanism. Moreover, it finds the grouping that minimizes the \textit{squared} error metric expressed as a function of the noisy data, rather than the original histogram (and, thus, similar to \cite{xu12,acs12}, it does not guarantee the actual minimum error). Note that the ordering attempts to minimize the approximation error, since it results in groups with more uniform bin values. The authors present two algorithms; one that evaluates all possible groups and runs in $O(n^3)$ time, and a greedy one that considers only a subset of the possible options and runs in $O(n^2)$. They conducted experiments using the latter, and demonstrated that AHP offers better utility than P-HP.
DAWA \cite{li14} comprises of two stages. The first stage executes a
smoothing technique, while the second an optimized version of the matrix mechanism \cite{li10}. Its grouping and averaging submodule invests $\epsilon'$ budget to reduce the \textit{absolute} error metric similar to \cite{acs12}. However, instead of executing the exponential mechanism, it adds noise to the costs of the groups used in the selection process on-the-fly. The authors present two instantiations; the first evaluates all possible groupings and runs in $O(n^2\log n)$ time, whereas the second considers only a subset and runs in $O(n\log^2 n)$. The output of the smoothing procedure is fed to the matrix mechanism. The latter belongs to a category of schemes \cite{li10,yuan12,hardt12} that take as input a set of \textit{pre-defined} range-sum queries, and assign more privacy budget to the bins affecting numerous queries. DAWA can be adapted to our setting of \textit{arbitrary} queries in two ways; either by completely ignoring the second stage, resulting in time complexity $O(n^2\log n)$ (or $O(n\log^2 n)$ in the approximate version), or
by feeding all the $O(n^2)$ possible queries to the input of the matrix mechanism, yielding time complexity $O(n^3\log n)$.
\medskip
\noindent\textbf{Data-oblivious methods.} These schemes build an aggregate tree on the original histogram; each bin value is a leaf, and each internal node represents the sum of the leaves in its subtree. In order to achieve $\epsilon$-differential privacy, they add Laplace noise to each node, which is proportional to the tree height (since each bin value is incorporated in all the sums along its path to the root). A range-sum query is processed by identifying the maximal subtrees that exactly cover the range, and summing the values stored in their roots. Compared to ${\sf LPA}$, the hierarchical methods essentially increase the sensitivity from $1$ to $\log n$, but sum fewer noisy values when processing the range-sum, reducing the aggregate error. For a range-sum covering $m$ bins, these methods induce $O(\log m \log n)$ error, as opposed to ${\sf LPA}$ that inflicts $O(m)$ error. Therefore, the hierarchical methods exhibit benefits for large ranges. Moreover, their time complexity is $O(n)$.
Hay et al. \cite{hay10} build a binary aggregate tree and inject Laplace noise uniformly across all nodes. In addition to constructing the final range-sum from the roots of the maximal subtrees that cover the range, they also explore other node combinations. Independently from \cite{hay10}, Privelet \cite{xiao11} builds a Haar wavelet tree and adds Laplace noise, achieving practically the same effect as \cite{hay10}. Based on the observation that the privacy budget should not be divided equally among all levels, Cormode et al. \cite{cormode12} enhance \cite{hay10} with a geometric budget allocation technique. Qardaji et al. \cite{qardaji13} survey the above approaches, concluding that the theoretical optimal fan-out of the tree is $16$. They experimentally showed that \cite{hay10}, when combined with the budget allocation of \cite{cormode12} and their optimal fan-out, outperforms Privelet and SF.
\medskip
\noindent\textbf{Discussion.} Data-oblivious methods are fast, but may have low utility for practical datasets. Data-aware schemes avoid this by exploiting the underlying data, but they may be prohibitively slow. For example, assuming the squared error metric, the lowest time complexity achieved by any method is $O(n^3)$. Attempts to boost performance via approximation, by ignoring possible groupings, compromise utility in an unpredictable way. Moreover, the naive adaptation of DAWA to arbitrary queries, by feeding all the $O(n^2)$ possible range-sums, is impractical.
Finally, all the discussed methods involve common components. For instance, data-aware schemes only differ in their grouping technique (e.g., different error metrics in the scoring function), whereas data-oblivious methods only differ in the tree fanout and the budget allocation across the levels. These design decisions are orthogonal; e.g., we could use the tree fan-out of one method with the budget allocation policy of another. Going one step further, novel methods could combine the merits of both data-aware and -oblivious schemes.
Motivated by the above, in this work we formulate a principled approach, which defines the core privacy techniques as primitive modules. Our framework allows (i)~the careful study and optimization of each individual module, (ii)~the construction of efficient and effective schemes via the seamless combination of these modules, and (iii)~the effortless adaptation of additional modules, such as the matrix mechanism, in our problem setting.
\section{Modular Framework}\label{sec:modular}
Section \ref{subsec:defs} formulates the concept of \emph{module} along with related notions. Section \ref{subsec:modules} describes the module instantiations utilized to construct range-sum schemes. Section \ref{subsec:modular_related} demonstrates how the existing state-of-the-art range-sum schemes (used as competitors in our experiments) can be expressed in our modular framework.
\subsection{Definitions}\label{subsec:defs}
There are two types of building blocks in our approach: the \emph{module} and the \emph{connector}, formulated in the next two definitions.
\begin{definition}\label{def:module}
A \textbf{\emph{module}} is a mechanism that takes as input a sensitive histogram $\mathbf{h} \in \mathcal{H}$ and a vector of public parameters $\mathbf{p}$, and outputs a differentially private histogram $\mathbf{\hat{h}} \in \mathcal{\hat{H}}$. The privacy level (i.e., $\epsilon$) of the module depends on $\mathbf{h}, \mathbf{p}$, and its internal mechanics.
\end{definition}
\begin{definition}\label{def:connector}
A \textbf{\emph{connector}} is an algorithm that takes as input a vector of public parameters $\mathbf{p}$, and either $H \subseteq \mathcal{H}$ (i.e., sensitive histograms) or $\hat{H} \subseteq \mathcal{\hat{H}}$ (i.e., differentially private histograms), and outputs another vector of parameters $\mathbf{p}'$, along with sets $H' \subseteq \mathcal{H}$ and $\hat{H}' \subseteq \mathcal{\hat{H}}$. It must obey two constraints: (i) it must spend \emph{no} privacy budget, and (ii) if it takes as input some $\mathbf{h} \in \mathcal{H}$, all its outputs must be consumed by modules.
\end{definition}
Simply stated, modules are responsible for perturbing sensitive data with noise, whereas connectors \emph{connect} modules (and optionally also other connectors). The connectors essentially format the data prior to feeding them to the modules. The public parameters facilitate determining the amount of noise added by a module. The second condition of the connectors is due to technical purposes in our proofs, which will become clear later in this section. Hereafter, we denote a module by $M$ and a connector by $C$. Finally, note that a module may be further comprised of other modules and connectors, in which case we refer to it as \emph{composite}. The motivation behind distinguishing connectors from modules is to compartmentalize the components related to privacy within the scope of a module, so that we facilitate the understanding of its privacy level and possible optimization.
\begin{definition}
A \textbf{\emph{range-sum scheme}} consists of a directed acyclic graph (DAG) of modules and connectors, and a query processor. It takes as input a histogram $\mathbf{h} \in \mathcal{H}$, public parameters $\mathbf{p}$, and privacy budget $\epsilon$. The DAG of modules and connectors outputs a \emph{structure} $\mathbf{\hat{S}}$ (e.g., a histogram or tree) that satisfies $\epsilon$-differential privacy, which is fed to the query processor. The latter uses the structure to answer arbitrary range-sum queries.
\end{definition}
Note that the above definition can capture even \textit{iterative} schemes, such as MWEM \cite{hardt12}, as follows. We decompose a loop into modules, and then \textit{serialize} the loop by repeating its modules as many times as the number of loop iterations. We do not delve into more details, as we do not deal with iterative schemes in this work.
The next theorem formulates $\epsilon$-differential privacy for a range-sum scheme. Intuitively, it states that the connectors do not affect privacy at all. The privacy level of the entire scheme depends \emph{solely} on the modules and, thus, it suffices to analyze each module individually.
\begin{theorem}\label{theo:scheme}
Let a range-sum scheme comprised of modules $M_1, M_2, \ldots, M_r$ and connectors $C_1, C_2, \ldots, C_l$. Suppose that $M_1, M_2, \ldots, M_r$ work on sensitive inputs derived from pairwise non-disjoint (resp. disjoint) datasets, and each $M_i$ satisfies $\epsilon_i$-differential privacy. Then, the scheme satisfies $\left(\sum_{i=1}^r{\epsilon_i}\right)$-differential privacy (resp. $\left(\max_{i=1}^r{\epsilon_i}\right)$-differential privacy).
\end{theorem}
\begin{proof}
We distinguish two cases, assuming for now that the connectors take single inputs and produce single outputs: (i)~A connector $C$ takes as input a differentially private histogram $\mathbf{\hat{h}} \in \mathcal{\hat{H}}$ from a module $M_i$. Since $C$ spends zero privacy budget by definition, its output will retain the privacy level of the input, independently of the computations it performs. Therefore, we can devise a module $M_i'$ that encompasses $M_i$ and $C$, and retains the $\epsilon_i$-differential privacy of $M_i$. (ii)~A connector $C$ takes as input a sensitive histogram $\mathbf{h} \in \mathcal{H}$ from the scheme input. By definition, $C$ can only produce a sensitive histogram $\mathbf{h}' \in \mathcal{H}$ as output and direct it to a module $M_i$. Hence, we can trivially merge $C$ with $M_i$ to create a module $M_i'$ that retains the $\epsilon_i$-differential privacy of $M_i$.
Replicating connectors to simulate multiple inputs and outputs, and executing the processes described in the above two cases repeatedly, from a DAG of $M_1, \ldots, M_r$ modules and $C_1, \ldots, C_l$ connectors we can derive an equivalent DAG of mechanisms $M_1', \ldots, M_r'$, where $M_i'$ satisfies $\epsilon_i$-differential privacy. Due to Theorem \ref{theo:comp}, the scheme satisfies $\left(\sum_{i=1}^r{\epsilon_i}\right)$-differential privacy (resp. $\left(\max_{i=1}^r{\epsilon_i}\right)$-differential privacy).
\\
\end{proof}
\vspace{-0.1pt}
As a final remark on privacy, recall the second constraint we imposed on the connector. If $\mathbf{h}$ were the input of a connector $C$ whose output was not directed to a module, $C$ could have been allowed to send $\mathbf{h}$ to the output of the range-sum scheme, violating differential privacy. The constraint prevents this case.
The benefits of modularity are threefold: (i)~novel schemes with variable efficiency and utility can be developed based on a small set of simple modules and connectors, (ii)~given the privacy level of each module and using Theorem \ref{theo:scheme}, we can easily prove the privacy of complex schemes, and (iii)~the modules can be optimized independently, and incorporate potential future improvements.
In the following, we deconstruct existing techniques into modules and connectors in order to investigate their performance bottlenecks, and identify opportunities for improvement. The internals of composite modules and schemes are illustrated using figures, depicting a module with a rectangle, a connector with a diamond, and a query processor with a parallelogram.
\subsection{Atomic Modules}\label{subsec:modules}
The three basic modules in our framework are \textit{Smoothing}, \textit{Hierarchy Level}, and \textit{Fixed Queries}. These modules are composite, i.e., they consist of other modules and connectors. However, they are used as \emph{atomic}\footnote{\scriptsize The submodules and connectors of an atomic module are never used outside of this particular module.} blocks when analyzing existing and novel schemes in later sections. We next explain each module in turn.
\medskip
\noindent\textbf{Smoothing module.} This module constitutes a building block for the data-aware techniques. It imposes an order on the bins of the input histogram, groups and averages bins, applies noise, and outputs the perturbed histogram. Figure~\ref{fig:smoothing} depicts the internal mechanics of the Smoothing module in more detail. Its input consists of the initial histogram $\mathbf{h}$, and public parameters $\mathbf{p}$ that include a vector $\mathbf{L}$, an error metric $\mu$ (absolute or squared), and three privacy budgets $\epsilon_1, \epsilon_2, \epsilon_3$. Each element of $\mathbf{L}$ has the form $\langle g_i, v_i \rangle$, where $g_i$ is some encoding for a group of histogram bins, and $v_i$ quantifies the error in $g_i$ due to the subsequent addition of noise (the value of $v_i$ will be elaborated shortly).
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{smoothing-eps-converted-to.pdf}
\caption{Smoothing module} \label{fig:smoothing}
\end{figure}
The module consists of three submodules, called Ordering, Grouping, and Noise Addition. Ordering receives the histogram $\mathbf{h}$ and budget $\epsilon_1$, and works as in \cite{zhang14}; it adds (Laplace) noise with scale $\lambda=1/\epsilon_1$ to each bin value, and sorts them in descending order. It then forwards the noisy sorted histogram $\mathbf{\hat{h}}_o$ to the Grouping submodule.
The Grouping submodule spends budget $\epsilon_2$ to discover the groups for its input histogram, considering $\mathbf{L}$ and $\mu$. $\mathbf{L}$ (i)~describes the \emph{permissible groups}, and (ii)~includes error values $v_i$ that parameterize $\mu$. A permissible group $g_i$ can only contain \emph{contiguous} bins, and is encoded simply by a range of elements in $\mathbf{\hat{h}}_o$, but is independent of the corresponding bin labels or values. After determining the groups, the submodule incorporates a group id into each bin label. Finally, it outputs the result, which is denoted by $\mathbf{\hat{h}}_{o,avg}$. The tasks performed by Grouping are elaborated further in Section~\ref{sec:grouping}.
Noise Addition receives the original histogram $\mathbf{h}$, budget $\epsilon_3$, and the output $\mathbf{\hat{h}}_{o,avg}$ of the Grouping submodule. It groups the bins of $\mathbf{h}$ according to the (augmented with group ids) bin labels in $\mathbf{\hat{h}}_{o,avg}$, and averages their values. Then, it adds noise to the respective average with scale $1/(\epsilon_3\cdot |g_i|)$. Finally, it sets the noisy average of every group $g_i$ as the value of the bins in $g_i$, and outputs the noisy smoothed histogram $\mathbf{\hat{h}}$, which satisfies $\epsilon_3$-differential privacy\footnote{\scriptsize Contrary to {\sf LPA}, Smoothing distributes the noise \emph{non-uniformly} over the bins of $\mathbf{h}$. This can be thought of as splitting $\mathbf{h}$ into $|G|$ \emph{disjoint} histograms, each corresponding to a $g_i \in G$ and, due to averaging, having sensitivity $1/|g_i|$. Due to Theorem \ref{theo:laplace_mech}, injecting noise with scale $1/(\epsilon_3|g_i|)$ renders each histogram $\epsilon_3$-differentially private. Due to Theorem \ref{theo:comp}, Smoothing is also $\epsilon_3$-differentially private.}. Note that the bin labels in $\mathbf{\hat{h}}$ incorporate the group ids of $\mathbf{\hat{h}}_{o,avg}$.
Ordering is $\epsilon_1$-, Grouping is $\epsilon_2$-, and Noise Addition is $\epsilon_3$-differentially private, and they all operate on histograms derived from pairwise non-disjoint inputs. Hence, due to Theorem \ref{theo:comp}, Smoothing satisfies $(\epsilon_1+\epsilon_2+\epsilon_3)$-differential privacy. If we set $\epsilon_1 = 0$ ($\epsilon_2 = 0$), the Ordering (Grouping) submodule acts as a connector. Specifically, Ordering just outputs the input histogram, whereas Grouping outputs the best strategy without spending privacy budget. However, based on Definition \ref{def:connector}, it is not permitted to simultaneously set $\epsilon_1 =0$ and $\epsilon_2 = 0$; in that case, Ordering would forward a sensitive histogram $\mathbf{h}$ to another connector. Finally, if we set $\epsilon_3 = 0$, the Noise Addition submodule adds noise with infinite scale to each group average. Although the returned $\mathbf{\hat{h}}$ contains useless values, its bin labels incorporate the grouping information from the Grouping submodule.
\medskip
\noindent\textbf{Hierarchy Leve
.} This is a typical component of the data-oblivious schemes. Recall that these methods build an aggregate tree on the original histogram. Every level of the tree can be viewed as a separate histogram. The Hierarchy Level module operates on a histogram of a specific tree level. Figure \ref{fig:hier} illustrates its internal parts. The module receives a histogram $\mathbf{h}$, a vector $\mathbf{L}$, privacy budget $\epsilon$, tree height $t$, and a tree level $\ell$. It consists of two connectors (Scale Budget and Scalar Product), and a submodule Noise Addition.
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{hierarchical-eps-converted-to.pdf}
\caption{Hierarchy Level module} \label{fig:hier}
\end{figure}
In our implementation, the Scale Budget connector allocates the privacy budget based on the tree level, using the method of \cite{cormode12} to maximize utility. It receives as input the triplet $(\epsilon, t, \ell)$, and outputs $\alpha \epsilon$, i.e., it determines a parameter $\alpha$ that scales budget $\epsilon$. The Scalar Product connector takes as input $\alpha \epsilon$ and public vector $\mathbf{L}$, and simply outputs their scalar product $(\alpha \epsilon) \mathbf{L}$. This essentially distributes the budget assigned for the level (potentially) non-uniformly over the bins. The output $(\alpha \epsilon) \mathbf{L}$ is forwarded to the Noise Addition submodule, which adds noise with scale $1/((\alpha \epsilon)\mathbf{L}[i])$ to the $i^\textrm{th}$ bin of the histogram, and outputs the resulting noisy histogram $\mathbf{\hat{h}}$.
The $\mathbf{L}$ parameter is selected, so that the Hierarchy Level module is $(\alpha \epsilon)$-differentially private. In our schemes, we distinguish two cases: (i) $\mathbf{L} = \mathbf{1}^n$, and every bin receives the same noise with scale $1/(\alpha \epsilon)$. (ii) $\mathbf{L}[i] = 0$ for some bins, in which case the module adds noise with infinite scale. Observe that in both cases, the added noise achieves $(\alpha \epsilon)$-differential privacy.
\medskip
\noindent\textbf{Fixed Querie
.} This module is the building block of methods that target at range-sum queries known a priori. It receives as input a histogram $\mathbf{h}$, a privacy budget $\epsilon$, and a range-sum query workload $\mathbf{W}$. It executes an off-the-shelf mechanism such as MWEM \cite{hardt12} or the matrix mechanism \cite{li10}, and outputs the noisy histogram $\mathbf{\hat{h}}$. Figure~\ref{fig:matrix} shows the Fixed Queries module, instantiated with the optimized matrix mechanism submodule of \cite{li14}, used in our implementation.
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{matrix-eps-converted-to.pdf}
\caption{Fixed Queries module} \label{fig:matrix}
\end{figure}
\subsection{Modularizing Existing Schemes} \label{subsec:modular_related}
In this section we show how the existing approaches can be constructed using modules and connectors.
\medskip
\noindent\textbf{Smoothing scheme.} All data-aware mechanisms \cite{xu12,acs12,zhang14,li14} described in Section~\ref{subsec:related} are captured by the scheme of Figure~\ref{fig:S}, which is a simple combination of the Smoothing module with a Query Processor. The latter receives the noisy histogram output by the Smoothing module, and replies to range-sum queries. The queries are processed by adding the bins falling in the query range.
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{S-eps-converted-to.pdf}
\caption{Smoothing scheme} \label{fig:S}
\end{figure}
Depending on the choice of public parameters $\mathbf{p}$, we can have the following alternative scheme instantiations:
\begin{itemize}
\item \emph{With/without ordering:} We can deactivate (activate) the Ordering submodule by setting the value of $\epsilon_1$ to $0$ ($>0$). For instance, if we set $(\epsilon_1, \epsilon_2, \epsilon_3) = (\epsilon/2, 0, \epsilon/2)$ we reproduce AHP \cite{zhang14}. Note that $\epsilon_2=0$ because the Grouping submodule operates directly on noisy data and does not need to inject extra noise (i.e., it acts as a connector). On the other hand, if we set $(\epsilon_1, \epsilon_2, \epsilon_3) = (0, \epsilon/4, 3\epsilon/4)$, we reproduce the smoothing scheme of DAWA \cite{li14}. Observe that either case results in an $(\epsilon_1 + \epsilon_2 + \epsilon_3=\epsilon)$-differentially private Smoothing module. Since this is the only module in the scheme, the Smoothing scheme is also $\epsilon$-differentially private.
\item \emph{Exact/approximate grouping:} The Grouping submodule can be implemented either as an exact or an approximate algorithm. In the first case, public parameter $\mathbf{L}$ includes \emph{all} possible groups of contiguous bins. In the second case, $\mathbf{L}$ contains a \emph{proper subset}, which reduces the running time. In both cases, all $v_i$ values in $\mathbf{L}$ are set to $1/\epsilon_3$, which is the expected error incurred by the Noise Addition submodule.
\item \textit{Absolute/squared error metric:} There are also two options for the error metric $\mu$ utilized by the Grouping submodule; absolute as in \cite{acs12,li14} or squared as in \cite{xu12,zhang14}. As shown later in the paper, this choice impacts both utility and performance.
\end{itemize}
\noindent\textbf{Hierarchical scheme.} The scheme captures data-oblivious methods. As shown in Figure \ref{fig:H}, it consists of connectors $C_1$ and $C_2$, $t$ Hierarchy Level modules (where $t$ depends on the input public parameters), and a Query Processor. It receives as input a histogram $\mathbf{h}$, privacy budget $\epsilon$, and public parameters $\mathbf{L}$ and $f$, where $\mathbf{L} = \mathbf{1}$ and $f$ is the fan-out of the tree\footnote{\scriptsize In our implementation, we set $f=16$ because it is optimal in terms of utility for range-sum queries \cite{qardaji13}.}. Connector $C_1$ initially receives $\mathbf{h}$, $\epsilon$, $\mathbf{L}$ and $f$. Based on $\mathbf{h}$ and $f$, it creates an aggregate tree, and determines the tree height $t$. It next perceives each level of the tree as a histogram $\mathbf{h}_\ell$ for $\ell=1, \ldots, t$. Finally, it splits the budget $\epsilon$ into $t$ budgets $\epsilon/t$ and forwards $\mathbf{h}_\ell$ and $(\mathbf{1}, \epsilon/t, t, \ell)$ to the $\ell^\textrm{th}$ Hierarchy Level module.
The $\ell^\textrm{th}$ Hierarchy Level module sends a noisy histogram $\mathbf{\hat{h}}_\ell$ to $C_2$. The latter assembles a noisy tree $\mathbf{\hat{T}}$ from these histograms and forwards it to the Query Processor. In order to maximize utility, in our implentation the Query Processor answers range-sum queries by combining nodes from the noisy tree using the method of \cite{hay10} (see Section~\ref{subsec:related}).
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{H-eps-converted-to.pdf}
\caption{Hierarchical scheme} \label{fig:H}
\end{figure}
Each Hierarchy Level module $\ell$ offers $\alpha_\ell \epsilon/t$-differential privacy. Moreover, the modules work on non-disjoint sensitive inputs. As such, due to Theorem \ref{theo:scheme}, the Hierarchical scheme offers $(\sum_{\ell=1}^t{\frac{\alpha_\ell\epsilon}{t}})$-differential privacy. Note that the $\ell^\textrm{th}$ Hierarchy Level module sets its $\alpha_\ell$ as defined in \cite{cormode12} (through a closed formula based on $t, \ell$), in a way that \emph{guarantees} that $\sum_{\ell=1}^t{\alpha_\ell}=t$. Consequently, the Hierarchical scheme satisfies $\epsilon$-differential privacy.
\medskip
\noindent \textbf{DAWA-like scheme.} This scheme is a generalization of DAWA \cite{li14}. Recall that, in addition to a smoothing stage, DAWA employs the matrix mechanism, which receives as input all the possible range-sum queries. We abstract these two stages, so that any smoothing and fixed-queries scheme can be combined to realize DAWA's concept.
Figure \ref{fig:DAWA} depicts the mechanics of the scheme. It consists of modules Smoothing and Fixed Queries, a connector, and a Query Processor. It receives as input histogram $\mathbf{h}$, budget $\epsilon$, and parameters $(\mathbf{W}, \mathbf{L},\mu, \epsilon/4, 3\epsilon/4)$. Following \cite{li14}, budget $\epsilon/4$ is allocated to the Smoothing module, and $3\epsilon/4$ to Fixed Queries. Vector $\mathbf{W}$ holds all possible $O(n^2)$ range-sum queries; $\mu$ defines the utilized error metric by the Smoothing; $\mathbf{L}$ contains the permissible groups. For each group $g_i$ in $\mathbf{L}$, $v_i$ is set to $4/(3\epsilon)$, which is the expected error due to the subsequent noise addition by the Fixed Queries module.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{DAWA-eps-converted-to.pdf}
\caption{DAWA-like scheme} \label{fig:DAWA}
\end{figure}
The Smoothing module takes as input $\mathbf{h}$ and parameters $(\mathbf{L},\mu, 0,\epsilon/4,0)$, and outputs a noisy histogram $\mathbf{\hat{h}}_{g}$ that incorporates the group labels. Connector $C$ receives $\mathbf{\hat{h}}_{g}$, $\mathbf{W}$ and $\mathbf{h}$. It smooths $\mathbf{h}$ using $\mathbf{\hat{h}}_{g}$, and creates $\mathbf{h}_{avg}$. Then, it modifies workload $\mathbf{W}$ to $\mathbf{W}_{avg}$ to reflect the queries on $\mathbf{h}_{avg}$. The technical details of this conversion are included in \cite{li14}. Finally, it feeds $(\mathbf{h}_{avg}, \mathbf{W}_{avg})$ to Fixed Queries, which receives budget $3\epsilon/4$. This module computes and forwards a noisy histogram $\mathbf{\hat{h}}$ to the Query Processor, which answers range-sums by summing the bins included in the query range.
The Smoothing module satisfies $\epsilon/4$-differential privacy, while the Fixed Queries module $3\epsilon/4$-differential privacy. Both work on non-disjoint inputs and, therefore, the whole scheme satisfies $\epsilon$-differential privacy.
\section{Grouping and Metrics}\label{sec:grouping}
The Grouping submodule of Smoothing determines the way the bins are privately grouped. In all existing schemes, this is modeled as an optimization problem where the resulting grouping must minimize a certain error metric. In this section, we first present in detail the two error metrics used in the literature, namely \textit{absolute} \cite{acs12, li14} and \textit{squared} \cite{xu12, zhang14} error, explain their usage, and analyze the overall time complexity of Grouping in each case. Next, we introduce an \textit{optimal} way to compute the squared error, which (i) reduces the time complexity of the current best method by a factor of $n$, and (ii) improves the accuracy of Smoothing.
Recall that Grouping takes as input a histogram $\mathbf{\hat{h}}_o$, a privacy budget $\epsilon_2$, public vector $\mathbf{L}$, and an error metric $\mu$. Its goal is to find the groups that minimize $\mu$, while satisfying $\epsilon_2$-differential privacy. Let $G$ be a \emph{grouping strategy}, i.e., a set of $|G|$ groups of \textit{contiguous} bins that cover all histogram bins and are mutually disjoint. Let $b_j$ be a bin value, and $\bar{g}_i$ the average of the bins in group $g_i \in G$, i.e., $\bar{g}_i = \sum_{b_j \in g_i} b_j/|g_i|$.
The total error has two components. The first is due to the smoothing process and depends on the difference between the value $b_j$ of a bin and the average $\bar{g}_i$ of the group in which it belongs. The second component is due to the noise injected by the module that succeeds grouping. For each group $g_i$, $\mathbf{L}$ contains a value $v_i$ that corresponds to the latter. The \textit{absolute} and \textit{squared} error metrics combine the two components in different ways. Both metrics represent the collective error per bin, rather than the final error in a range-sum query. However, they are in general good indicators of accuracy and their minimization is likely to maximize utility.
\medskip
\noindent \textbf{Absolute error.} This metric is defined in \cite{acs12,li14} as:
\begin{equation}\label{eq:error1}
err_1=\sum_{i=1}^{|G|}{\left(\sum_{b_j\in g_i}{|b_j-\bar{g}_i|}+v_i\right)}
\end{equation}
The state-of-the-art algorithm that uses the absolute error is the Smoothing module of DAWA \cite{li14}, which works as follows. It first calculates the cost $c_i = \sum_{b_j\in g_i}{|b_j-\bar{g}_i|}+v_i$ of each group $g_i$ in Equation \ref{eq:error1} by utilizing a binary search tree in $O(\log n)$ time. Then, it adds noise with scale $1/(\epsilon_2|g_i|)$ to $c_i$ producing $\hat{c}_i = \sum_{b_j\in g_i}{|b_j-\bar{g}_i|}+v_i+Lap(1/(\epsilon_2|g_i|))$. Finally, it finds the groups that minimize $\hat{err_1} = \sum_{i=1}^{|G|}{\hat{c_i}}$ using dynamic programming in $O(n^2)$ time. The authors prove that an optimization algorithm that operates with such noisy costs ensures $\epsilon_2$-differential privacy.
The total time of Grouping is dominated by that of computing the costs of all the $O(n^2)$ groups, which is $O(n^2 \log n)$.
\medskip
\noindent \textbf{Squared error.} This metric is defined in \cite{xu12,zhang14} as:
\begin{equation}\label{eq:error2}
err_2=\sum_{i=1}^{|G|}{\left(\sum_{b_j\in g_i}{(b_j-\bar{g}_i)^2}+v_i^2\right)}
\end{equation}
The state-of-the-art grouping algorithm that utilizes the squared error is AHP \cite{zhang14}, which works as follows. It adds noise with scale $1/\epsilon_2$ to each bin of the initial histogram, and computes cost $\hat{c_i} = \sum_{\hat{b_j}\in g_i}{(\hat{b_j}-\bar{g}_i)^2}+v_i^2$, where $\hat{b_j}$ is a noisy bin value, and $\bar{g}_i$ the average of a group of noisy bins. Finally, it finds the groups that minimize $\hat{err_2} = \sum_{i=1}^{|G|}{\hat{c_i}}$. The algorithm satisfies $\epsilon_2$-differential privacy because it operates on values perturbed with noise scale $1/\epsilon_2$. Its time complexity is $O(n^3)$.
The following theorem provides a lower bound on the time complexity of Grouping, in the case that all possible groups of contiguous bins are considered. The lower bound applies to \textit{both} error metrics.
\begin{theorem}\label{theo:lower}
A grouping algorithm on a histogram with $n$ bins runs in $\Omega(n^2)$.
\end{theorem}
\begin{proof}
The number of all the possible groups is $\Theta(n^2)$. This is because we have $n$ groups of size $1$, $n-1$ groups of size $2$, and so on (recall that a permissible group can only consist of contiguous bins). Thus, the total number of groups is $n+(n-1)+(n-2)+\ldots+1=\frac{n(n+1)}{2}$. It suffices to prove that there is an input for which any algorithm must check all the possible groups at least once.
We build a histogram such that every group $g_i$ contributes cost $\hat{c}_i = |g_i|$ (i.e., equal to its cardinality) to the error metric. In this scenario, \emph{any} grouping strategy $G$ minimizes the error metric, since \emph{every} $G$ leads to error $\sum_{g_i \in G} \hat{c}_i = n$. Now suppose that we reduce the cost of a random group $g_j$ to $(|g_j|-\delta)$ for some $\delta>0$. Any grouping strategy that includes $g_j$ will result in error $n-\delta$, whereas any other will result in $n$. Therefore, the grouping strategy $G^*$ that minimizes the error metric \emph{must} include $g_j$. Since $g_j$ is a random group, the algorithm that finds $G^*$ must check the $\hat{c}_i$ of \emph{every} group $g_i$ in order to find $g_j$. This concludes our proof.
\end{proof}
We next present an algorithm that minimizes the squared error $\hat{err_2}$ in $O(n^2)$. Therefore, due to the lower bound in Theorem \ref{theo:lower}, our algorithm is \emph{optimal}. Given that $\hat{g}_i = \frac{\sum_{\hat{b}_j\in g_i}{\hat{b}_j}}{|g_i|}$, we observe that the cost of each group can be rewritten as follows.
\begin{equation*}\label{eq:cost2}
\hat{c_i} = \sum_{\hat{b}_j\in g_i}{\left(\hat{b_j}-\bar{g}_i\right)^2}+v_i^2=
\sum_{\hat{b}_j\in g_i}{\hat{b_j}^2}-\frac{\left(\sum_{\hat{b}_j\in g_i}{\hat{b}_j}\right)^2}{|g_i|}+v_i^2
\end{equation*}
Based on the above equation, we can efficiently compute the cost of each group using the following procedure. Initially, we add noise with scale $1/\epsilon_2$ to every histogram bin. In a \textit{pre-processing stage}, we build vector $\mathbf{v}_1$ that stores the noisy bin values $\hat{b}_j$, and vector $\mathbf{v}_2$ that stores their squares $\hat{b}^2_j$. Subsequently, we construct the \textit{prefix sums} for each vector. Specifically, the prefix sums for $\mathbf{v}_1$ ($\mathbf{v}_2$) is a vector $\mathbf{v}_1'$ ($\mathbf{v}_2'$), such that $\mathbf{v}_1'[j] = \sum_{i=1}^j \mathbf{v}_1[i]$ ($\mathbf{v}_2'[j] = \sum_{i=1}^j \mathbf{v}_2[i]$). The pre-processing takes $O(n)$ time. For each group $g_i$ over contiguous bins $l, l+1, \ldots, u$, we can compute $\sum_{\hat{b}_j \in g_i} \hat{b}_j$ as $\mathbf{v}_1'[u] - \mathbf{v}_1'[l-1]$ and $\sum_{\hat{b}_j \in g_i} \hat{b}_j^2$ as $\mathbf{v}_2'[u] - \mathbf{v}_2'[l-1]$ in $O(1)$ time. Thus, the cost of any group requires $O(1)$ time. Since there are $O(n^2)$ possible groups, we can calculate all their costs in $O(n^2)$. Finally, in order to find the grouping strategy that minimizes $\hat{err_2}$, we employ the dynamic programming procedure of \cite{li14}, which runs in $O(n^2)$ time. Therefore, our algorithm has total running time $O(n^2)$.
We conclude this section with an improvement on the accuracy yielded by the use of the squared error. Recall that our algorithm computes the group costs on the noisy histogram in order to ensure $\epsilon_2$-differential privacy. Thus, the grouping strategy that minimizes $\hat{err}_2$, may not minimize $err_2$ (defined on the original bins). In order to alleviate the effects of the extra noise in $\hat{err}_2$ we exploit the following observation. Using a similar approach as in the proof of Lemma 1 in \cite{xu12}, we can show that each group is expected to have its cost increased due to noise by $2\frac{|g_i|-1}{\epsilon_2^2}$, i.e., proportionally to the group size. The additional error leads to smaller groups for $\hat{err}_2$ minimization, compared to $err_2$. To mitigate this, we reduce the calculated cost $\hat{c_i}$ of each group $g_i$ by $2\frac{|g_i|-1}{\epsilon_2^2}$, before feeding it to the dynamic programming procedure. Compared to its direct competitor AHP \cite{zhang14}, our algorithm improves the utility by up to $70\%$ and the complexity by $n$.
\section{Novel Schemes}\label{sec:schemes}
We design two schemes based on our modular framework. The first, called \textit{Subtree Smoothing}, constitutes the first approach that seamlessly combines smoothing with aggregate trees, running in $O(n)$ time. The second, called \textit{Smoothed Prefix Sums}, reduces the time complexity of DAWA by a factor of $n$, while maintaining its utility.
\subsection{Subtree Smoothing Scheme}\label{subsec:subtree_smoothing}
Recall that the Hierarchical scheme builds an aggregate tree in order to compose the range-sum answer from a small number of noisy values, thus reducing the error resulting from noise aggregation as opposed to ${\sf LPA}$. However, due to the publication of multiple non-disjoint histograms (one per level), it must add more noise per level than ${\sf LPA}$. On the other hand, the Smoothing scheme reduces the sensitivity of a set of bins via grouping and averaging, thus lowering the required noise. Our Subtree Smoothing scheme builds an aggregate tree similar to the Hierarchical scheme (thus reducing the error from noise aggregation), but \emph{smooths entire subtrees} via grouping and averaging similar to the Smoothing scheme (thus reducing the per-level, per-bin noise).
Figure \ref{fig:subtree_example} illustrates the main idea. The scheme runs the Smoothing module \emph{only once} for the leaf level (i.e., for $\mathbf{h}$), setting as permissible groups only those that correspond to the leaves of \emph{full} subtrees. Suppose that the black nodes in the figure comprise a group in the returned grouping strategy. We refer to the root of the subtree corresponding to a group as the \emph{group root}. Next, the scheme creates the aggregate tree, \emph{pruning} the nodes under the group roots (black nodes). Subsequently, it feeds each level of this aggregate tree to a Hierarchy Level module, which outputs a noisy histogram. The final noisy histograms comprise a noisy tree. Finally, the scheme puts the pruned nodes back to the tree, deriving their values from their corresponding group root. Specifically, the value in the group root is distributed evenly across the nodes of the same level in the subtree. This is equivalent to smoothing the nodes at each level of the subtree via averaging.
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{subtree_example-eps-converted-to.pdf}
\caption{Subtree smoothing example} \label{fig:subtree_example}
\end{figure}
Figure \ref{fig:subtree_smoothing} illustrates the modules of the Subtree Smoothing scheme. There is a single Smoothing module, $t$ Hierarchy Level modules, two connectors, and a Query Processor. The input of the scheme includes the sensitive histogram $\mathbf{h}$, privacy budget $\epsilon$, and public parameters $\mathbf{p}=(\mathbf{L},\mu, \epsilon/4, 3\epsilon/4, f)$. $\mathbf{L}$ is the set of permissible groups for the Smoothing module and their associated $v_i$ values; $\mu$ is the error metric; $\epsilon/4$ is the budget allocated to the Smoothing module; $3\epsilon/4$ is the budget distributed evenly to the $t$ Hierarchy Level modules; $f$ is the fan-out of the aggregate tree, and $t$ is its derived height. Following \cite{qardaji13}, we set $f=16$ .
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{subtree_smoothing-eps-converted-to.pdf}
\caption{Subtree Smoothing scheme} \label{fig:subtree_smoothing}
\end{figure}
The Smoothing module takes as input $\mathbf{h}$ and parameters $(\mathbf{L},\mu, 0, \epsilon/4, 0)$, and outputs a noisy histogram $\mathbf{\hat{h}}_{g}$. $\mathbf{L}$ contains the groups formed by bins that can be leaves of full subtrees in the final aggregate tree. Their $v_i$ values are all set to $4t/(3\epsilon)$. Ordering must \emph{always} be deactivated because the final aggregate tree is built considering the order of the bins in $\mathbf{h}$. If Ordering were activated, Grouping could select a group $g_i$, whose bins are not the leaves of a full subtree on $\mathbf{h}$ (since Ordering may permute the bins of $\mathbf{h}$). Therefore, $g_i$ could not determine a group root to smooth a subtree, thus violating the scheme. The Grouping submodule takes budget $\epsilon/4$. The Noise Addition submodule receives $0$ budget; the returned $\mathbf{\hat{h}}_{g}$ contains only the grouping information, which is used later.
Connector $C_1$ receives $\mathbf{h}$, $\mathbf{\hat{h}}_{g}$, budget $3\epsilon/4$ and fan-out $f$. It builds the aggregate tree utilizing $\mathbf{h}$, $\mathbf{\hat{h}}_{g}$, $f$, and considers each tree level $\ell$ as a histogram $\mathbf{h}_\ell$ to be sent to the $\ell^\textrm{th}$ Hierarchy Level module. For every pruned node $j$ in the aggregate tree (i.e., black node in Figure \ref{fig:subtree_example}) at level $\ell$, it sets its scalar to $\mathbf{L}_\ell[j] = 0$, and for any other node $j'$ it sets $\mathbf{L}_\ell[j'] = 1$.
The $\ell^\textrm{th}$ Hierarchy Level module receives the histogram $\mathbf{h}_\ell$. For each bin $b_j$ in $\mathbf{h}_{\ell}$, it adds noise with scale $\frac{4t}{ \alpha_\ell \cdot 3\epsilon \cdot \mathbf{L}_\ell[j]}$. If $b_j$ is a node to be pruned, then $\mathbf{L}_\ell[j]=0$, and $b_j$ is perturbed with infinite noise, while a special annotation is added to its label. This is essentially equivalent to completely disregarding the pruned nodes. Otherwise, $\mathbf{L}_\ell[j]=1$ and the module adds noise with scale $\frac{4t}{ \alpha_\ell \cdot 3\epsilon}$. This procedure ensures that each Hierarchy Level module satisfies $\frac{\alpha_\ell \cdot 3\epsilon}{4t}$-differential privacy (by a direct application of Theorems \ref{theo:laplace_mech} and \ref{theo:comp}).
Connector $C_2$ receives the noisy histograms from the Hierarchy Level modules. First, it assembles a noisy aggregate tree from the histograms. Next, it substitutes the values of the nodes that received infinite noise in the Hierarchy Level modules, with the values derived from their group root, as we explained in the context of Figure~\ref{fig:subtree_example}. Finally, it forwards the resulting tree $\mathbf{\hat{T}}$ to the Query Processor, which answers range-sum queries using the technique of \cite{hay10}.
We next analyze the privacy of the scheme. The Smoothing module spends budget $\epsilon/4$ and satisfies $\epsilon/4$-differential privacy. Each of the $t$ Hierarchy Level modules satisfies $\frac{\alpha_\ell \cdot 3\epsilon}{4t}$-differential privacy as explained above. Moreover, all the modules work on non-disjoint inputs. Due to Theorem~\ref{theo:scheme}, and recalling that the $\alpha_\ell$ values are selected according to \cite{cormode12} such that $\sum_{i=1}^t \alpha_\ell=t$, the whole scheme satisfies $\epsilon$-differential privacy.
Finally, the running time of the scheme depends on the error metric $\mu$. Observe that the number of groups examined by Grouping is equal to the number of nodes in the aggregate tree, i.e., $O(n)$. For the case when $\mu$ is the absolute error, the running time of Grouping is $O(n \log n)$, using the smoothing algorithm of \cite{li14}. If $\mu$ is the squared error metric, the complexity is $O(n)$ using our optimal algorithm from Section \ref{sec:grouping}. Each Hierarchy Level module runs in time linear in the number of input nodes, thus, all the $t$ Hierarchy Level modules run collectively in $O(n)$. In our experiments, we demonstrate that the error metric in Subtree Smoothing does not significantly affect the utility. Hence, we fix $\mu$ to the more efficient square error, which yields total running time $O(n)$.
\medskip
\noindent\textbf{Utility Optimization.}
Instead of completely disregarding the nodes of a pruned subtree, we can actually utilize them to reduce the noise of its root. Specifically, for each level of the pruned subtree, we sum the node values and add noise, producing a noisy estimation of the root. Subsequently, we use the \textit{average} of these estimations as the root noisy value. The mechanism then proceeds as described above, i.e., the root value is distributed evenly among the subtree nodes. This reduces the \textit{squared} error of the root value by $t'$, where $t'$ is the height of the subtree. We omit the proof due to its simplicity.
\subsection{Smoothed Prefix Sums Scheme}\label{subsec:smoothed_prefix_sums}
This scheme is based on \emph{prefix sums} \cite{ho97}. A prefix sum query over $\mathbf{h}$ is simply described by an index $j$, and returns the sum of bins $b_1, \ldots, b_j$, i.e., $\sum_{i=1}^j \mathbf{h}[i]$. There are $n$ prefix sums, hereafter represented by a vector $\mathbf{s}$ such that $\mathbf{s}[j] = \sum_{i=1}^j \mathbf{h}[i]$ for $j=1, \ldots, n$. Moreover, observe that \emph{any arbitrary} range-sum query can be always computed by the subtraction of \emph{exactly two} prefix sums; for instance, range-sum $[i_l, i_u]$ is answered as $\mathbf{s}[i_u] - \mathbf{s}[i_l-1]$.
The Smoothed Prefix Sum scheme takes advantage of the fact that there are $n$ prefix sums, as opposed to $O(n^2)$ possible range queries, to improve the complexity of DAWA-like methods by a factor of $n$. It considers the prefix sums as the fixed workload $\mathbf{W}$, and produces a noisy histogram $\mathbf{\hat{h}}$. The latter enables the computation of a vector of noisy prefix sums $\mathbf{\hat{s}}$, such that $\mathbf{\hat{s}}[i] = \sum_{i=1}^j \mathbf{\hat{h}}[i]$. Vector $\mathbf{\hat{s}}$ is fed to the Query Processor, which computes in $O(1)$ time any range-sum $[i_l, i_u]$ as $\mathbf{\hat{s}}[i_u] - \mathbf{\hat{s}}[i_l-1]$. Since the Fixed Queries module leads to highly accurate $\mathbf{\hat{s}}[i]$, the range-sum result is expected to have very low error.
Figure \ref{fig:AM} depicts the Smoothed Prefix Sums scheme. The only differences with respect to the DAWA-like scheme of Figure \ref{fig:DAWA} are (i) the workload $\mathbf{W}$, which now contains the prefix sums, and (ii) an extra connector $C_2$ before the Query Processor, which converts the output histogram $\mathbf{\hat{h}}$ into a prefix sums array $\mathbf{\hat{s}}$.
\begin{figure}[!ht]
\centering
\includegraphics[width=\linewidth]{AM-eps-converted-to.pdf}
\caption{Smoothed Prefix Sums scheme} \label{fig:AM}
\end{figure}
\begin{figure*}[!b]
\vspace{-0.3cm}
\begin{center}
\centering
\subfigure[\emph{Citations}]{
\includegraphics[width=0.28\linewidth]{cite-eps-converted-to.pdf}
\label{plot:citedist}
}
\subfigure[\emph{Rome}]{
\includegraphics[width=0.28\linewidth]{rome-eps-converted-to.pdf}
\label{plot:romedist}
}
\subfigure[\emph{GoWalla}]{
\includegraphics[width=0.28\linewidth]{gw-eps-converted-to.pdf}
\label{plot:gwdist}
}
\vspace{-0.3cm}
\caption{Data distribution}
\vspace{-0.3cm}
\label{plot:dist}
\end{center}
\vspace{-0.3cm}
\end{figure*}
The Smoothing module satisfies $\epsilon/4$-differential privacy, while the Fixed Queries module $3\epsilon/4$-differential privacy. Both work on non-disjoint inputs and, therefore, the whole scheme satisfies $\epsilon$-differential privacy. Its time complexity is $O(|\mathbf{W}|n\log n) = O(n^2\log n)$, since now $|\mathbf{W}| = n$. The expected error is at most two times larger than that of DAWA because Smoothed Prefix Sums subtracts two noisy values from the prefix sums array to answer a range query, while DAWA essentially returns a value for the same range. However, in our experiments we demonstrate that the utility of Smoothed Prefix Sums is practically the same as that of the DAWA scheme.
A remark concerns allocation of budget $\epsilon$ to the various modules. In this work, we followed the empirical allocation policies of the existing schemes. Determining the optimal allocation is out of our scope, but we consider it as an interesting problem for future work. Finally, except for ${\sf LPA}$ and the Hierarchical scheme, where the expected error is expressed theoretically, the rest of the schemes are highly \emph{data-dependent}. Therefore, their utility must be experimentally evaluated under different real settings, a task we undertake in the next section.
\section{Experimental evaluation}\label{sec:experiments}
In this section we evaluate the methods of Table \ref{tab:summary} in terms of utility and efficiency. ${\sf LPA}$ corresponds to the Laplace Perturbation Algorithm. ${\sf H}$ implements the Hierarchical scheme, using all optimizations of \cite{qardaji13}. ${\sf S_1}$ incorporates the smoothing algorithm of DAWA \cite{li14}, based on the absolute error metric. ${\sf \tilde{S}}$ is the approximate version of ${\sf S_1}$ that considers only a subset of the possible groups \cite{li14}. ${\sf S_2}$ applies smoothing using the squared error metric, and it utilizes the quadratic algorithm and utility optimization described in Section \ref{sec:grouping}. ${\sf S_o}$ orders the bin values \cite{zhang14}, and then uses ${\sf S_2}$ for smoothing. ${\sf DAWA}$ \cite{li14} is implemented with input all the possible range queries. ${\sf SUB}$ is our Subtree Smoothing scheme based on the squared error metric as it offers similar utility to the absolute metric, and is faster by a $\log n$ factor. It also contains the utility optimization technique described at the end of Section \ref{subsec:subtree_smoothing}. ${\sf SPS}$ is our Smoothed Prefix Sum scheme. For both ${\sf SPS}$ and ${\sf DAWA}$, we use the absolute error metric, but the choice of metric does not affect either their utility, or performance.
Our evaluation includes all the dominant techniques in their respective settings. Specifically, the smoothing module of DAWA (${\sf S_1}$) offers better utility and time complexity than previous methods that are based on the absolute error metric and check all possible groupings \cite{li14}. Its approximate counterpart ${\sf \tilde{S}}$ also dominates its competitor P-HP, which in turn has been shown to outperform EFPA and SF \cite{acs12} (details about these methods can be found in Section \ref{subsec:related}). Among the exhaustive techniques based on the squared error, the state-of-the-art is AHP \cite{zhang14}, which is dominated by ${\sf S_2}$ in terms of running time and utility, as explained in Section \ref{sec:grouping}. Moreover, the approximate methods using the squared error metric have the same quadratic complexity as ${\sf S_2}$, while at best they can reach the same utility. Finally, the optimizations incorporated in ${\sf H}$ have been shown to yield the best hierarchical method in the survey of \cite{qardaji13}.
\begin{table}[t]
\centering
\vspace{-0.2cm}
\caption{Summary of schemes}\label{tab:summary}
\begin{scriptsize}
\begin{tabular}{c | c | c }
\textbf{Scheme} & \textbf{Abbrv} & \textbf{Time} \\
\hline
\hline
Laplace perturbation algorithm & {${\sf LPA}$} & $O(n)$ \\
Hierarchical scheme & ${\sf H}$ & $O(n)$ \\
Smoothing with absolute error metric & {${\sf S_1}$} & $O(n^2 \log n)$ \\
Approximate Smoothing & ${\sf \tilde{S}}$ & $O(n\log^2 n)$ \\
Smoothing with squared error metric & ${\sf S_2}$ & $O(n^2)$\\
Smoothing with ordering & {${\sf S_o}$} & $O(n^2)$ \\
DAWA & ${\sf DAWA}$ & $O(n^3\log n)$ \\
Subtree smoothing & ${\sf SUB}$ & $O(n)$ \\
Smoothed prefix sums & ${\sf SPS}$ & $O(n^2\log n)$ \\
\end{tabular}
\end{scriptsize}
\vspace{-0.5cm}
\end{table}
We implemented all methods of Table \ref{tab:summary} in Java, and conducted experiments on an Intel Core i5 CPU 2.53GHz with 4GB RAM, running Windows 7. Following the literature, we assess utility using the \emph{Mean Squared Error} (MSE), fixing $\epsilon = 1$. The cardinality of the range-sum queries varies between $10\%$ and $50\%$ of the input histogram size. Every reported result is the average of 100 executions, each containing 2000 random queries of the selected cardinality.
We used three real datasets, henceforth referred to as \emph{Citations} \cite{cite}, \emph{Rome} \cite{rome}, and \emph{GoWalla} \cite{gowalla}. In \emph{Citations}, we created a histogram of $2414$ bins as in \cite{qardaji13}, where each bin $b_i$ is the number of papers cited $i$ times. A range-sum query $[i_l,i_u]$ returns the papers cited between $i_l$ and $i_u$ times. The \emph{Rome} dataset consists of $14420$ bins, where each bin $b_i$ is the number of cars on a specific road at time instance $i$. A range-sum query asks for the traffic at this road segment during a time interval. Finally, \emph{GoWalla} consists of user check-ins at $2791$ locations. We sorted the locations in ascending order of their $x$-coordinates as in \cite{qardaji13}, and viewed them as histogram bins. A range-sum query returns the number of users in a vertical geographical strip.
\emph{Citations}, \emph{Rome}, and \emph{GoWalla} feature considerably different distributions, depicted in Figures \ref{plot:citedist}, \ref{plot:romedist}, and \ref{plot:gwdist}, respectively. \emph{Citations} is very sparse, and its consecutive bin values are similar, especially for bins that correspond to numerous citations (most such bins have 0 values). \emph{Rome} exhibits high fluctuations at specific contiguous bins (reflecting peak hours), and includes numerous small values (reflecting non-peak hours). Finally, \emph{GoWalla} contains almost random values, since the number of check-ins is independent of the value of the $x$-coordinate.
\begin{figure*}[!h]
\begin{center}
\centering
\subfigure[\emph{MSE}]{
\includegraphics[width=0.28\linewidth]{citemse-eps-converted-to.pdf}
\label{plot:citemse}
}
\subfigure[\emph{Running Time}]{
\includegraphics[width=0.28\linewidth]{citetime-eps-converted-to.pdf}
\label{plot:citetime}
}
\subfigure[\emph{Running Time vs. Error}]{
\includegraphics[width=0.28\linewidth]{skyl-eps-converted-to.pdf}
\label{plot:citesky}
}
\vspace{-0.3cm}
\caption{\emph{Citations}}
\vspace{-0.3cm}
\label{plot:cite}
\end{center}
\vspace{-0.3cm}
\end{figure*}
\begin{figure*}[!h]
\vspace{-0.3cm}
\begin{center}
\centering
\subfigure[\emph{MSE}]{
\includegraphics[width=0.28\linewidth]{romeMSE-eps-converted-to.pdf}
\label{plot:romemse}
}
\subfigure[\emph{Running Time}]{
\includegraphics[width=0.28\linewidth]{rometime-eps-converted-to.pdf}
\label{plot:rometime}
}
\subfigure[\emph{Running Time vs. Error}]{
\includegraphics[width=0.28\linewidth]{skyl2-eps-converted-to.pdf}
\label{plot:romesky}
}
\vspace{-0.3cm}
\caption{\emph{Rome}}
\vspace{-0.3cm}
\label{plot:rome}
\end{center}
\vspace{-0.3cm}
\end{figure*}
Figure \ref{plot:citemse} plots the MSE for \emph{Citations}, when varying the range size (expressed as a fraction of the number of bins). ${\sf SPS}$ and ${\sf DAWA}$ achieve the highest accuracy. The error of ${\sf S_1}, {\sf \tilde{S}}$, and ${\sf SUB}$ is up to two times higher, while that of ${\sf H}, {\sf LPA}$, ${\sf S_2}$ is more than an order of magnitude larger. ${\sf S_o}$ exhibits the worst performance because the noise injected by ordering yields a poor grouping strategy. The low MSE of ${\sf SPS}$ and ${\sf DAWA}$ is mainly due to their effective combination of smoothing and the matrix mechanism. Their almost identical error confirms our claim in Section \ref{subsec:smoothed_prefix_sums} that feeding prefix sums to the matrix mechanism of the Fixed Queries module ($\sf SPS$) leads to the same practical utility as providing all the possible ranges ($\sf DAWA$). In general, all smoothing techniques perform well because consecutive bins have similar values, leading to groups with low error (this dataset yields a small number of large groups). This also explains the marginal difference of ${\sf S_1}$ and ${\sf \tilde{S}}$; ${\sf \tilde{S}}$ can easily find a good grouping strategy even though it does not explore all possible groups. In contrast, ${\sf S_2}$ performs worse than ${\sf S_1}$ and ${\sf \tilde{S}}$, as the squared error metric is sensitive to some small fluctuations in the dataset, which leads to unnecessarily small groups. Methods that do not rely on aggregate trees (i.e., ${\sf LPA}$, ${\sf S_1}, {\sf \tilde{S}}$, ${\sf S_2}$ and ${\sf S_o}$) are affected by the range size, as the number of noisy values participating in the calculation of the range-sum (and, thus, the resulting error accumulation) increases linearly with the size. On the other hand, the range has small effect on the utility of hierarchical methods, which increases logarithmically with the range size.
Figure \ref{plot:citetime} evaluates the CPU-time as a function of the data size. In order to reduce the data size to a percentage $x\%$, we select the first $x\%$ values and the corresponding bins. $\sf H$ and $\sf LPA$ are the fastest methods as expected by their linear complexity. $\sf SUB$ is slightly more expensive because of the additional smoothing step at the leaf level of the aggregate tree. The next method in terms of efficiency is ${\sf \tilde{S}}$, with complexity $O(n\log^2 n)$, followed by the quadratic ${\sf S_2}$ and ${\sf S_o}$. ${\sf S_1}$ and ${\sf SPS}$ have almost the same running time due to their identical complexity $O(n^2\log n)$. ${\sf DAWA}$ ($O(n^3\log n)$) is more than an order of magnitude more expensive than any other method.
In order to demonstrate the utility-efficiency trade-off, Figure \ref{plot:citesky} plots the error (x-axis) versus time (y-axis), when fixing the range-sum size to $30\%$ of the bins and using the entire dataset. The best solutions on both aspects lie closest to the axes origin. Although $\sf SPS$ and $\sf DAWA$ feature the best utility, they are also the most expensive. However, $\sf DAWA$ is dominated by $\sf SPS$, which is much more efficient. On the other hand, fast methods such as $\sf LPA$ and $\sf H$ incur high error. In between the two extremes lies $\sf SUB$, which is almost as fast as $\sf H$ and $\sf LPA$, but exhibits $3.5$ times lower error than $\sf H$ and an order of magnitude lower than $\sf LPA$.
Figure \ref{plot:romemse} assesses the utility of the schemes on the \emph{Rome} dataset. The results for $\sf DAWA$ are omitted, since it failed to terminate within a reasonable time (in fact, we estimated that it would take approximately three months to finish for this dataset). Similar to Figure \ref{plot:citemse}, ${\sf SPS}$ is the best scheme, reducing the error of the next best solutions (${\sf H}$ and ${\sf SUB}$) by up to $70\%$. ${\sf H}$ and ${\sf SUB}$ have almost identical error. Compared to Figure \ref{plot:citemse}, smoothing-based techniques have inferior performance because, due to the high fluctuations of the dataset, it is difficult to find effective grouping strategies. As opposed to \textit{Citations}, \emph{Rome} yields a large number of small groups. ${\sf S_1}$, ${\sf S_2}$ and ${\sf \tilde{S}}$ achieve gains through smoothing only for small ranges. ${\sf S_2}$ outperforms ${\sf S_1}$ and ${\sf \tilde{S}}$ because the squared error distinguishes small fluctuations, whereas the absolute error erroneously merges small groups into larger ones. Similarly, ${\sf \tilde{S}}$ results in up to $50\%$ worse error than ${\sf S_1}$ and up to an order of magnitude worse than ${\sf S_2}$ because the arbitrary set of the examined groups of the former does not include the groups that minimize the error. Finally, ${\sf S_o}$ and ${\sf LPA}$ are the worst techniques.
Figure \ref{plot:rometime} measures the CPU-time versus the data size. The results and the relative order of the schemes are consistent with Figure \ref{plot:citetime}. $\sf DAWA$ can only run for up to $40\%$ of the dataset. Figure \ref{plot:romesky} plots the error versus efficiency for \textit{Rome}. Again $\sf SPS$ and $\sf LPA$ lie at the two extremes of utility and efficiency, respectively. $\sf SUB$ and $\sf H$ provide the best trade-off, dominating all the schemes but $\sf SPS$.
Figure \ref{plot:gwmse} depicts the MSE on \emph{GoWalla} as a function of the range size. \emph{GoWalla} features almost random bin values. Consequently, smoothing spends privacy budget, without finding groups that lead to noise reduction. The schemes that depend solely on smoothing, i.e., ${\sf S_1}$, ${\sf S_2}$, ${\sf S_o}$ and ${\sf \tilde{S}}$, are outperformed even by ${\sf LPA}$. On the other hand, ${\sf SUB}$, ${\sf DAWA}$ and ${\sf SPS}$ are more robust to the dataset characteristics, since they inherit the benefits of the aggregate tree and matrix mechanism, respectively. For this dataset, a simple aggregate tree generated by ${\sf H}$ is the best method.
\begin{figure*}[!h]
\begin{center}
\centering
\subfigure[\emph{MSE}]{
\includegraphics[width=0.28\linewidth]{gwMSE-eps-converted-to.pdf}
\label{plot:gwmse}
}
\subfigure[\emph{Running Time}]{
\includegraphics[width=0.28\linewidth]{gwtime-eps-converted-to.pdf}
\label{plot:gwtime}
}
\subfigure[\emph{Running Time vs. Error}]{
\includegraphics[width=0.28\linewidth]{skyl3-eps-converted-to.pdf}
\label{plot:gwsky}
}
\vspace{-0.3cm}
\caption{\emph{GoWalla}}
\vspace{-0.3cm}
\label{plot:gw}
\end{center}
\vspace{-0.3cm}
\end{figure*}
Figure \ref{plot:gwtime} plots the CPU-time. The relative performance is similar to the previous diagrams. ${\sf DAWA}$ terminates because \emph{GoWalla} is smaller than \emph{Rome} (2791 versus 14420 bins). Figure \ref{plot:gwsky} shows the error-efficiency trade-off for \textit{GoWalla}. $\sf H$ dominates all solutions in both aspects, whereas $\sf SUB$ lies close to $\sf H$.
\medskip
\noindent \textbf{Summary.} Our experiments demonstrate that data-aware techniques lead to considerable error reductions for datasets that have similar values in consecutive bins. However, the gains of smoothing vanish in datasets with numerous high fluctuations. In these scenarios, data-oblivious methods are preferable because they do not waste privacy budget on smoothing. In between these two extremes lie schemes that integrate smoothing with other modules (i.e., ${\sf SUB}$ and ${\sf SPS}$), and are more robust to the dataset characteristics. Specifically, ${\sf SPS}$ performed identically to ${\sf DAWA}$ in terms of utility, while reducing the complexity by a factor of $n$ to $O(n^2 \log n)$. On the other hand, for time-critical applications (e.g., real-time traffic), where even ${\sf SPS}$ may be too slow, ${\sf SUB}$ achieves comparable accuracy, while having the lowest running time ($O(n)$) among all the data-aware methods.
\section{Conclusion}\label{sec:conclusion}
This paper introduces a modular framework for differentially private histogram publication. We first express existing methods in the framework, and identify opportunities for optimization. We then design a new optimal algorithm for smoothing, which improves utility and reduces the running time of the current state-of-the-art. Next, we develop new schemes that combine heterogeneous privacy techniques, previously deemed unrelated. Finally, we experiment on three datasets with diverse characteristics. Our results confirm that our modular approach enables the design of schemes that (i) are tailored to the data characteristics of the application at hand, and (ii) offer a desirable tradeoff between efficiency and utility.
\bibliographystyle{abbrv}
|
{
"timestamp": "2015-04-15T02:05:52",
"yymm": "1504",
"arxiv_id": "1504.03440",
"language": "en",
"url": "https://arxiv.org/abs/1504.03440"
}
|
\section{Introduction}
\label{sect:intro}
Massive stars are the engines of the galaxies. This small part of the stellar population has a considerable influence on its environment both mechanically and by radiation. Indeed, their strong ionizing fluxes power the \ion{H}{ii} regions. Furthermore, their powerful winds and their deaths as supernovae enrich their surroundings with heavy chemical elements and reshape the interstellar medium, triggering new generations of stars. Because their lifetime is relatively short, massive stars are excellent tracers of star formation. Therefore, constraining their properties gives us the opportunity to better understand how they form and how they evolve.
The fates of these stars are mainly governed by their initial mass, but also to some extent by their rotation and their mass loss. In addition to giving more oblate shapes to the stars, the rotation modifies their temperature gradient at the surface and also induces an internal mixing that affects the angular momentum and chemical element transport. The rotation thus influences the surface abundances and the lifetimes of the massive stars on the main sequence \citep{mm00}. Several observational studies analysed these two aspects of rotation. On the one hand, \citet{hun08,hun09} reported that a large part of B-type stars in the Large Magellanic Cloud (LMC) presented a clear trend between their projected rotational velocity (\vsini) and their surface nitrogen content. This trend was detected in about 60\% of the B star population. The remaining 40\% consists of B stars with slow rotation and high nitrogen enrichment and unenriched B stars with fast rotation. These groups are currently not explained by the evolutionary models of single massive stars which include rotational mixing. On the other hand, numerous investigations have shown that the rotation brings modifications to the evolutionary tracks and to the isochrones of massive stars \citep[see e.g.][]{mm00,mm03,brott11}. Besides rotation, the stellar winds also play a crucial role in the evolution of massive stars. Indeed, these objects lose a significant fraction of their mass during their life. However, the exact quantity of ejected mass by a massive star remains difficult to determine because the true structure of these winds, and notably their homogeneity, is still poorly known. Although direct \citep{eversberg98} and indirect \citep{bouret05} evidence highlighted the presence of outward-moving inhomogeneities in the winds of O-type stars, the shapes, the sizes, and the optical depths of these inhomogeneities (or clumps) still remain unanswered questions. Investigations of the structure of these winds seem, however, to indicate that the clumps are spherical \citep{sun11,her12} rather than flattened as proposed by \citet{feld03}. The presence of clumps in the stellar winds thus modifies the determination of mass-loss rates of massive stars since many mass-loss diagnostics are dependent on the density squared (such as notably H$\alpha$).
To bring more constraints on these properties, we have determined and analysed the fundamental parameters of nineteen O-type stars belonging to four OB associations in the Cygnus complex. This area is an active star-forming region that includes a huge number of massive stars. These targets have already been presented in \citet[][hereafter Paper I]{mah13} who established their multiplicity and analysed the distribution of the orbital parameters of the binary systems. Because we know their multiplicity, in the present paper we can be more accurate on the determination of valuable physical parameters of these objects. Although all these targets are formed in the Cygnus region, they constitute a non-homogeneous sample of main-sequence band, giant, and supergiant massive stars. Among this sample of stars, we also detected in Paper I four binary systems: three SB2s and one SB1, all with orbital periods shorter than 10 days. Therefore, for the SB2s, we disentangled the observed spectra of these systems to obtain the individual spectrum of each component to investigate it with an atmosphere code as we do for single stars.
In the following section, we present the observations of these targets. The atmosphere code and the main UV/optical diagnostics are described in Sect.~\ref{sec:mod}. The results and a discussion are given in Sect.~\ref{sec:res} and in Sect.~\ref{sec:dis}, respectively. Finally, our conclusions are provided in Sect.~\ref{sec:conc}.
\section{Observations and data reduction}
\label{sect:obs}
We focus on the same data as presented in Paper I. In addition, we retrieve the IUE spectra (SWP with a dispersion of 0.2~\AA) to model the stellar winds of five objects. These data were obtained between 1979 and 1992 with exposure times from 650 s to 8400~s. These spectra cover the [1200--1800]~\AA\ wavelength domain, containing the \ion{C}{iv}~1548--1550 and \ion{N}{v}~1240 resonance P-Cygni profiles as well as the \ion{O}{v}~1371 line, used to estimate the wind clumping in early O stars. We also add to our sample four FUSE spectra retrieved from the archives. Because of the interstellar absorption, we only focus on the [1100--1200]~\AA\ wavelength region, and mainly on the \ion{P}{v}~1118--1128, \ion{C}{iv}~1169, and \ion{C}{iii}~1176 lines. Unfortunately, the FUV and UV spectra are not available for all the stars in our sample. Therefore, the mass-loss rates of the stars for which none of these spectra exists will only be derived on the basis of the H$\alpha$ line.
\begin{sidewaystable*}[ph!]
\caption{Journal of optical and UV observations}
\label{tab:obs}
\begin{tabular}{l|cccc|ccc|ccc}
\hline\hline
& \multicolumn{4}{c|}{Optical} & \multicolumn{3}{c}{IUE}& \multicolumn{3}{c}{FUSE}\\
\hline
Star & Instrument & Spectral range & HJD & Obs. date & data ID & HJD & Obs. date & data ID & HJD & Obs. date \\
& & [\AA] & [d] & & & [d] & \\
\hline
\multicolumn{11}{c}{Cyg\,OB1}\\
\hline
HD\,193443 & -- & [4450--4900] & \multicolumn{2}{c|}{disentangled spectra} & -- & -- & -- & -- & -- & -- \\
HD\,228989 & -- & [4450--4900] & \multicolumn{2}{c|}{disentangled spectra} & -- & -- & -- & -- & -- & -- \\
HD\,229234 & Espresso & [4000--6750] & 2\,454\,992.7849 & 10 Jun 2009 & -- & -- & -- & -- & -- & -- \\
HD\,193514 & Elodie & [4000--6800] & 2\,453\,600.4087 & 17 Aug 2005 & SWP18145 & 2\,445\,241.2984 & 28 Sep 1982 & E0820701 & 2\,453\,165.3275 & 08 Jun 2004\\
HD\,193595 & Espresso & [4000--6750] & 2\,454\,989.8582 & 07 Jun 2009 & -- & -- & -- & -- & -- & -- \\
HD\,193682 & Espresso & [4000--6750] & 2\,454\,989.9081 & 07 Jun 2009 & SWP09022 & 2\,444\,376.4267 & 16 May 1980 & E0820301 & 2\,453\,365.5448 & 09 Jun 2004\\
HD\,194094 & Espresso & [4000--6750] & 2\,455\,820.7063 & 16 Sep 2011 & -- & -- & -- & -- & -- & -- \\
HD\,194280 & Aur{\'e}lie & [4450--4900] & 2\,454\,407.2709 & 02 Nov 2007 & -- & -- & -- & -- & -- & -- \\
HD\,228841 & Espresso & [4000--6750] & 2\,454\,989.8092 & 07 Jun 2009 & -- & -- & -- & -- & -- & -- \\
\hline
\multicolumn{11}{c}{Cyg\,OB3}\\
\hline
HD\,190864 & Elodie & [4000--6800] & 2\,452\,134.4733 & 12 Aug 2001 & SWP06946 & 2\,44\,4168.7353 & 22 Oct 1979 & E0820501 & 2\,453\,148.7486 & 23 May 2004\\
HD\,227018 & Aur{\'e}lie & [4450--4900] & 2\,454\,717.3188 & 07 Sep 2008 & -- & -- & -- & -- & -- & -- \\
HD\,227245 & Espresso & [4000--6750] & 2\,455\,727.7497 & 15 Jun 2011 & -- & -- & -- & -- & -- & -- \\
HD\,227757 & Espresso & [4000--6750] & 2\,455\,726.7373 & 14 Jun 2011 & -- & -- & -- & -- & -- & -- \\
\hline
\multicolumn{11}{c}{Cyg\,OB8}\\
\hline
HD\,191423 & Elodie & [4000--6800] & 2\,453\,247.3339 & 29 Aug 2004 & SWP16212 & 2\,445\,000.2270 & 30 Jan 1982 & E0821301 & 2\,453\,165.7979 & 09 Jun 2004\\
HD\,191978 & Elodie & [4000--6800] & 2\,453\,247.4196 & 29 Aug 2004 & SWP46532 & 2\,448\,976.0517 & 19 Dec 1992 & -- & -- & -- \\
HD\,193117 & Espresso & [4000--6750] & 2\,454\,991.9784 & 09 Jun 2009 & -- & -- & -- & -- & -- & -- \\
\hline
\multicolumn{11}{c}{Cyg\,OB9}\\
\hline
HD\,194334 & Espresso & [4000--6750] & 2\,455\,053.7053 & 10 Aug 2009 & -- & -- & -- & -- & -- & -- \\
HD\,194649 & -- & [4450--4900] & \multicolumn{2}{c|}{disentangled spectra} & -- & -- & -- & -- & -- & -- \\
HD\,195213 & Aur{\'e}lie & [4450--4900] & 2\,454\,717.5787 & 07 Sep 2008 & -- & -- & -- & -- & -- & -- \\
\hline
\end{tabular}
\end{sidewaystable*}
For the targets detected as binary systems in Paper I, we used the disentangled spectra corrected for the brightness ratio which were already presented in Paper I. We note that we used a programme based on the \citet{gl06} technique to compute the individual spectra of both components of a binary system, but also to refine the radial velocities by applying a cross-correlation technique. These spectra can thus be considered as mean spectra for the components of the binary systems. To have as many diagnostic lines as possible, we favour the optical spectra with the widest wavelength coverage, as well as the highest spectral resolution and the highest signal-to-noise ratio. The journal of optical and FUV/UV observations is presented in Table~\ref{tab:obs}.
\section{Modelling}
\label{sec:mod}
We use the code CMFGEN \citep{hm98} for the quantitative analysis of the optical and FUV/UV spectra. CMFGEN provides non-LTE atmosphere models including winds and line-blanketing. CMFGEN needs as input an estimate of the hydrodynamical structure that we construct from TLUSTY models (taken from the OSTAR2002 grid of \citealt{lh03}) connected to a $\beta$ velocity law of the form $v = v_{\infty}(1 - R/r)^{\beta}$, where $v_{\infty}$ is the wind terminal velocity. For the stars having only optical spectra, we adopted $\beta = 0.8$, which represents a typical value for O dwarfs \citep[see e.g.][]{rep04}. Our final models include the following chemical elements: \ion{H}{i}, \ion{He}{i-ii}, \ion{C}{ii-iv}, \ion{N}{ii-v}, \ion{O}{ii-vi}, \ion{Ne}{ii-iii}, \ion{Mg}{ii}, \ion{Si}{ii-iv}, \ion{S}{iii-vi}, \ion{P}{iv-v}, \ion{Ar}{iii-iv}, \ion{Al}{iii}, \ion{Fe}{ii-vii}, and \ion{Ni}{iii-v} with the solar composition of \citet{gas07} unless otherwise stated. CMFGEN also uses the super-level approach to reduce the memory requirements. On average, we include about 1600 super levels for a total of 8000 levels. For the formal solution of the radiative transfer equation leading to the emergent spectrum, a microturbulent velocity varying linearly with velocity from 10~\kms\ to $0.1 \times v_{\infty}$ was used. We include X-ray emission in the wind since this can affect the ionization balance and the strength of key UV diagnostic lines. In practice, we adopt a temperature of three million degrees and we adjust the flux level so that the X-ray flux coming out of the atmosphere matches the observed $L_{\mathrm{X}}/L_{\mathrm{bol}}$ ratio. We simply adopt the canonical value $L_{\mathrm{X}}/L_{\mathrm{bol}} = 10^{-7}$ \citep{san06b,naze09}.
In practice, we proceed as follows to derive the stellar and wind parameters.
\begin{itemize}
\item Effective temperature: we use the classical ratio between the strengths of \ion{He}{i} and \ion{He}{ii} lines to determine \teff. The main indicators are the \ion{He}{i}~4471 and \ion{He}{ii}~4542 lines, but additional diagnostics can be built with the \ion{He}{i}~4026, \ion{He}{i}~4389, \ion{He}{i}~4713, \ion{He}{i}~4921, \ion{He}{i}~5876, \ion{He}{ii}~4200, and \ion{He}{ii}~5412 lines. When possible we also use the \ion{C}{iv}~1169 to \ion{C}{iii}~1176 line ratio, which has been shown to provide a useful temperature diagnostic \citep{heap06}. The typical uncertainty on the \teff\ determination is 1000~K, except for the binary components for which the uncertainty on \teff\ is 2500~K.
\item Gravity: the wings of the Balmer lines H$\beta$, H$\gamma$, and H$\delta$ are the main indicators of \logg. Generally, an accuracy of about 0.1~dex on \logg\ is achieved. However, because of side effects of the disentangling programme, \logg\ of the binary components has an uncertainty of 0.25~dex.
\item Wind terminal velocity: the blueward extension of UV P-Cygni profiles provides $v_{\infty}+v_{\mathrm{max}}$ where $v_{\mathrm{max}}$ is the maximum turbulent velocity. Using the above relation for microturbulent velocity gives us a direct determination of \vinf\ with an accuracy of 100~\kms. For stars without UV spectra, we use as input the standard values given by \citet{pri90} according to their spectral types.
\item Mass-loss rate: we use two diagnostics to constrain the mass-loss rate, the UV P-Cygni profiles and H$\alpha$. For the UV domain, we focus mainly on \ion{N}{v}~1240, \ion{Si}{iv}~1394--1403, and \ion{C}{iv}~1548--1551, and when FUSE spectra are available, we consider \ion{C}{iv}~1169 and \ion{C}{iii}~1176 as additional diagnostics to those lines for mass-loss rate determination. A single value of $\dot{M}$ should allow a good fit of both types of lines even though a study by \citet{mar12} has shown that this was not always the case. In our analysis, we reach a general agreement between UV and H$\alpha$ lines by playing on other parameters such as the clumping or the $\beta$. However, when no FUV/UV data are available, we fix both values to 1.0 (homogeneous model) and 0.8 \citep[see][]{rep04}, respectively. We also stress that some objects were only observed on [4450--4900] \AA\ wavelength domain. Since no sufficient diagnostic exists to determine the mass-loss rate on this region, we also use as input the values provided by \citet{muijres2012} as mass-loss rate values for our models.
\end{itemize}
Given the complexity of the parameter space for CMFGEN models, it is not possible for us to describe all the effects of every parameter change on the synthetic spectra, but we refer to dedicated papers such as \citet{hil03}, \citet{bouret05}, or \citet{mar11}. In our analysis, we decide to vary the different parameters until we obtain the solution which provided the best $\chi^2$ fit. To this end we generated a non-uniform grid composed of between 10 and 40 models depending on the wavelength coverage of the observed spectra (and thus of the number of parameters to determine). From these best-fit models, we reproduce the UV-Optical-Infrared spectral energy distribution (SED) for each star (Fig.~\ref{SED_1}) from the UBVJHK fluxes (see Table~\ref{tab:phot}) and we include the UV fluxes for the stars that have FUSE and/or IUE spectra to constrain the distance and the extinction. The galactic reddening law of \citet{car89} is used. We also derive $R_V$ when the UV spectrum is available. When this is not the case, we use $R_V$ values from \citet{pat01,pat03} and if no value exists, we fix it to 3.1. We then recompute the $M_V$ with the new extinction and the new distance values and we derive the luminosities on the basis of the $M_V$ and of the bolometric corrections given in \citet{mar06}. Once the new luminosity is obtained, we compute a new model with this value to improve the best-fit models. The uncertainties calculated on the luminosities are mainly due to the poorly-known distances. To quantify these uncertainties, we computed the largest difference between the obtained luminosity and the luminosity computed with the minimum or maximum mean distances of the associations listed in \citet{hum78}.
The projected rotational velocities of the stars are obtained from the Fourier transform method \citep{sim07}. The synthetic spectrum is then convolved by \vsini.
The rotationally broadened synthetic line profiles usually provided a good match to the observed profile for the fast rotators. In the case of moderately to slowly rotating objects, some amount of macroturbulence has to be introduced to correctly reproduce the line profiles of several features (e.g. \ion{He}{i}~4713, \ion{C}{iv}~5812, \ion{He}{i}~5876). The need for extra broadening is well documented \citep{how97,ryans2002,howarth07, nieva07,martins10,fraser10}, but its origin is unclear. \citet{aerts09} notably suggested that non-radial pulsations could trigger large-scale motions (hence macroturbulence), but this needs to be confirmed by a study covering a wider parameter space. A study by \citet{sim10} also showed that the amplitude of macroturbulence was correlated to the amplitude of line profile variability in a sample of OB supergiants. In practice, we introduce macroturbulence by convolving our rotationally broadened synthetic spectra with a Gaussian profile, thus mimicking isotropic turbulence. This is obviously a very simple approach, but it significantly improves the quality of the fits. In practice, we used \ion{He}{i}~4713 as the main indicator
of macroturbulence since it is present with sufficient signal-to-noise ratio in all the stars in our sample. Moreover, secondary indicators are the \ion{O}{iii}~5592 and \ion{He}{i}~5876 lines.
To determine the nitrogen content of our stars, we rely mainly on the \ion{N}{iii} lines between 4500 and 4520~\AA. They are present in absorption in the spectra of all stars, they are not affected by winds and they are strong enough for the abundance determination. The uncertainties are of the order of 50\%. They are estimated from the comparison of the selected \ion{N}{iii} lines to models with various N content. The uncertainties do not take any systematics related to atomic data into account.
When UV spectra are available (both FUSE and IUE), the degree of inhomogeneities of the stellar winds is determined. Clumping is implemented in CMFGEN by means of a volume filling factor $f$ following the law $f = f_{\infty} + (1 - f_{\infty})e^{-v/v_{\mathrm{cl}}}$, where $f_{\infty}$ is the maximum clumping factor at the top of the atmosphere and $v_{\mathrm{cl}}$ a parameter indicating the position where the wind starts to be significantly clumped. As shown by \citet{bouret05}, \ion{P}{v}~1118--1128, \ion{O}{v}~1371, and \ion{N}{iv}~1720 are UV features especially sensitive to wind inhomogeneities. We use these lines to constrain $f_{\infty}$ in the stars in our sample. Unfortunately, the last two lines are mainly good indicators for early O-type stars. Therefore, we are not able to use these two lines to constrain the clumping factor for HD\,191978 given the absence of a FUSE spectrum.
\section{Results}
\label{sec:res}
The derived stellar and wind parameters are listed in Table~\ref{cygnus_param}. The CMFGEN best-fit models are given in the Appendix. Below, we briefly comment on each star.
\begin{table*}
\caption{UBVJHK photometric parameters of the stars in our sample}
\label{tab:phot}
\centering
\begin{tabular}{lcccccccccccc}
\hline\hline
Star & Cluster & U & B& V& J& H& K& R$_V$& E(B$-$V)& Dist. & M$_V$ & Dist. \\
& & & & & & & & & & (SED) & & \citet{hum78}\\
\hline
\multicolumn{13}{c}{Cyg\,OB1}\\
\hline
HD\,193443& & 7.08$^{b}$& 7.61& 7.26& 6.40& 6.34& 6.34& 3.10& 0.61& $-$ & -5.93 & 1.82 \\
HD\,193514& & 7.32$^{a}$& 7.84& 7.43& 6.35& 6.24& 6.18& 3.00& 0.74& 1.71 & -5.96 & 1.90 \\
HD\,193595& Ber\,86 & 8.50$^{b}$& 9.08& 8.78& 7.92& 7.87& 7.85& 3.02& 0.69& 1.85 & -4.64 & 1.82 \\
HD\,193682& & 8.39$^{b}$& 8.87& 8.42& 7.30& 7.19& 7.14& 2.82& 0.83& 1.95 & -5.37 & 1.82 \\
HD\,194094& & 9.18$^{b}$& 9.57& 9.05& 7.74& 7.59& 7.58& 2.99& 0.89& 2.65 & -5.73 & 2.42 \\
HD\,194280& & 8.94$^{a}$& 9.08& 8.42& 6.74& 6.60& 6.49& 2.95& 1.06& 2.67 & -6.84 & 2.20 \\
HD\,228841& Ber\,86 & 9.08$^{a}$& 9.50& 9.01& 7.91& 7.78& 7.75& 2.78& 0.88& 2.70 & -5.59 & 2.09 \\
HD\,228989& Ber\,86 & 10.29$^{b}$& 10.49& 9.83& 8.15& 7.97& 7.90& 3.10& 0.92& $-$ & -3.81 & 1.43 \\
HD\,229234& NGC\,6913& 9.35$^{d}$& 9.58& 8.94& 7.34& 7.18& 7.10& 2.87& 1.07& 1.50 & -5.02 & 1.59 \\
\hline
\multicolumn{13}{c}{Cyg\,OB3}\\
\hline
HD\,190864& NGC\,6871& 7.20$^{a}$& 7.93& 7.78& 7.26& 7.28& 7.26& 2.45& 0.58& 2.15 & -5.30 & 2.29 \\
HD\,227018& NGC\,6871& 8.73$^{b}$& 9.35& 9.00& 8.04& 7.95& 7.90& 3.28& 0.71& 2.75 & -5.53 & 3.15 \\
HD\,227245& & 9.81$^{b}$& 10.25& 9.69& 8.21& 8.07& 7.99& 3.26& 0.91& 2.15 & -4.94 & 2.29 \\
HD\,227757& & 8.72$^{e}$& 9.41& 9.27& 8.77& 8.78& 8.76& 3.06& 0.53& 2.15 & -4.02 & 2.36 \\
\hline
\multicolumn{13}{c}{Cyg\,OB8}\\
\hline
HD\,191423& & 7.43$^{a}$& 8.19& 8.03& 7.73& 7.72& 7.79& 3.00& 0.46& 2.90 & -5.66 & 2.82 \\
HD\,191978& & 7.38$^{a}$& 8.15& 8.03& 7.76& 7.80& 7.80& 2.70& 0.48& 2.75 & -5.46 & 2.84 \\
HD\,193117& & 8.87$^{c}$& 9.29& 8.77& 7.24& 7.09& 6.97& 3.18& 0.87& 2.25 & -5.76 & 2.37 \\
\hline
\multicolumn{13}{c}{Cyg\,OB9}\\
\hline
HD\,194334& & 9.38$^{c}$& 9.54& 8.82& 6.99& 6.82& 6.74& 2.94& 1.13& 1.19 & -4.88 & 1.04 \\
HD\,194649& & 9.82$^{b}$& 9.93& 9.07& 6.86& 6.63& 6.51& 3.10& 1.12& $-$ & -4.80 & 1.20 \\
HD\,195213& & 9.29$^{b}$& 9.55& 8.82& 6.67& 6.44& 6.26& 3.27& 1.13& 1.00 & -4.88 & 1.20 \\
\hline
\end{tabular}
\tablebib{
$a$:~\citet{mai04}; $b$:~\citet{wes82}; $c$:~\citet{gua92}; $d$:~\citet{fer83};
$e$:~\citet{kre12}.}
\end{table*}
\subsection{The Cyg\,OB1 association}
\hspace{0.5cm}{\it HD\,193443: }In Paper I, we estimated the brightness ratio to $3.9\pm0.4$. The disentangled spectra corrected for this ratio are relatively well fitted. However, the wavelength coverage does not allow us to determine the wind parameters. The individual stellar parameters of the primary are more reminiscent of a giant star, whilst the secondary has the characteristics of a main-sequence object. This confirms the luminosity classes of both components mentioned in Paper I.
{\it HD\,228989: }We found in Paper I a brightness ratio of $1.2\pm0.1$ between the primary and the secondary. The spectral lines of the disentangled spectra are relatively well fitted even though the \ion{He}{ii}~4686 line of both stars stronger than with the synthetic profiles. This is probably due to a poor estimation of the wind parameters, but, as stressed for HD\,193443, the absence of other diagnostic lines prevents us from providing a good determination of the wind parameters.
{\it HD\,229234: }This object was reported as SB1 in Paper I. However, since the signature of the putative companion is not detectable in the observed spectra, we consider that the these spectra are only produced by the primary star, suggesting that the derived parameters correspond to an upper limit. The fit is relatively good and indicates stellar parameters similar to those of a giant star. This agrees with the spectral classification established in Paper I.
{\it HD\,193514: }The parameters derived for HD\,193514 are similar to those of \citet{rep04}. However, the wind parameters are slightly different. Indeed, we need a clumping factor of $0.01$ to correctly reproduce the FUV lines. This thus affects the mass-loss rate of the star even though $\dot{M}_{\mathrm{H}\alpha}/\sqrt{f}$ remains quite close in both analyses. The emissions of the \ion{N}{iii}~4634--41 lines in the spectrum of HD\,193514 are poorly reproduced, but it is probably not due to the N content, given that the triplet at 4500--4520~\AA\ and the \ion{N}{iv}~1720 line are well fitted. Moreover, we stress that an increase of the mass-loss rate does not improve the quality of the fit.
{\it HD\,193595: }The fit of this object is of good quality. The core of the H$\alpha$ line is filled in with some emission that is likely to be partially of nebular origin. The stellar parameters agree with a main-sequence luminosity class as established in Paper I.
{\it HD\,193682: }We use FUSE and IUE spectra to constrain the wind parameters (see Table~\ref{cygnus_param}). The FUSE spectrum is correctly fitted by our CMFGEN model. The IUE spectrum was observed with a low dispersion and the small aperture. Therefore, the quality of the observation is very low and we must strongly convolve the synthetic spectrum to achieve such a quality. The \ion{C}{iv}~1548-1550 doublet appears too saturated in comparison to the observation, whilst the fit of the \ion{N}{iv}~1718 line seems acceptable. Finally, the fit in the optical range is of excellent quality, even though the \ion{He}{i}~5876 line is too strong compared to the observations. Unlike \citet{rep04}, the red wings of the \ion{He}{ii}~4686 and H$\alpha$ lines are rather well reproduced. The line profiles show an important macroturbulence. We find that a combination of \vsini\ of about 150~\kms\ with a macroturbulent velocity of about 70~\kms\ leads to much better fits than do purely rotationally broadened profiles.
{\it HD\,194094: }No UV spectrum is available for this star, we therefore restrict ourselves to the optical range. The core of the Balmer lines is too deep in our synthetic spectra except for H$\alpha$. Moreover, the \ion{He}{ii}~4686 is poorly fitted and appears stronger in the synthetic model.
{\it HD\,228841: }A good fit of the optical wavelength domain is achieved, except for the \ion{He}{ii}~4686 and the \ion{He}{i}~5876 lines. The best fit is obtained with profiles only broadened by rotation, i.e. without any macroturbulence.
{\it HD\,194280: }The Aur{\'e}lie spectrum only allows us to determine the stellar parameters. The spectral classification is similar to an OC\,9.7I star according to the tables of \citet{mar05}. From the CMFGEN best-fit model, we find a C overabundance and a N depletion, estimated to about 3.3 and 0.2 times the solar abundances\footnote{$(\mathrm{C/H})_{\odot} = 2.45\,10^{-4}$ and $(\mathrm{N/H})_{\odot} = 6.02\,10^{-5}$ \citep[expressed by number,][]{gas07}}, respectively, as reported by \citet{wal00}. To obtain a fit of good quality, we also need to increase the helium abundance to $(\mathrm{He/H}) = 0.18$ (by number) i.e. almost twice the solar value ($(\mathrm{He/H})_{\odot} = 0.10$). However, additional diagnostic lines are required to confirm such abundances.
\subsection{The Cyg\,OB3 association}
\hspace{0.5cm}{\it HD\,190864: }A good quality of fit is achieved. We need a volume filling factor of $0.06$ and a $\beta$ of 1.5 to reach a good accuracy for the wind parameters in both UV and optical domains. As for HD\,193514, $\dot{M}_{\mathrm{H}\alpha}/\sqrt{f}$ is consistent with the value determined by \citet{rep04}. Moreover, as for HD\,193514, the synthetic model fails to correctly fit the \ion{N}{iii}~4634--41 lines.
{\it HD\,227018: }The [$4450-4900]~\AA$ wavelength domain is properly fitted, which gives us a very good determination of the stellar parameters.
{\it HD\,227245: }The synthetic model correctly reproduces the observations which leads to a good accuracy on the determination of the stellar parameters. Furthermore, the \ion{He}{ii}~4686 and H$\alpha$ lines are well fitted, thereby corresponding to a reliable determination of the mass-loss rate of the star on the basis of these diagnostic lines..
{\it HD\,227757: }We need a slow rotation rate to correctly reproduce the photospheric lines. We also see that the \ion{O}{ii} lines are present in the spectrum of the star. The stellar parameters, listed in Table~\ref{cygnus_param}, are in agreement with those determined by \citet{mar14}. The luminosity derived for that object and its mass-loss rate determined from the H$\alpha$ line could indicate that HD\,227757 is a weak-wind star (see \citealt{mar05b} and \citealt{mar09} for other examples of weak-wind O-type stars). However, without any UV spectrum for this star, we cannot confirm this assumption.
\subsection{The Cyg\,OB8 association}
\hspace{0.5cm}{\it HD\,191423: }We need to broaden the synthetic model by a rotational velocity of 410~\kms to reach a sufficient quality of the fit. Moreover, the model also yields, through the adjustment of the carbon lines at 4070 and 4650~\AA, a large depletion in C (about 0.1 times the solar abundance, i.e. $3.5 \times 10^{-5}$ in number) and an enrichment in N (about 9 times the solar abundance, i.e. $5.4 \times 10^{-4}$ in number). The wind parameters derived on the UV P-Cygni profiles indicate a low terminal velocity (about 600~\kms). We also see two weak emissions in the external wings of H$\alpha$. These emissions could be created by a disk of matter ejected by the high rotational velocity of the star. We stress that the \ion{P}{v}~1118--1128 and \ion{N}{iv}~1718 lines appear slightly weak in our model in comparison to the observed spectra.
{\it HD\,191978: }The fit is of an excellent quality, except for the red wing of the \ion{C}{iv}~1548--50 lines and for the \ion{He}{ii}~4686 line which is too weak compared to the observations. As we already stressed, the \ion{O}{v}~1371 and \ion{N}{iv}~1718 lines are not good diagnostic lines to estimate the clumping factor in late O-type stars. Furthermore, given the lack of FUSE spectrum for this star, we adopt $f=1.0$ (homogeneous model) for HD\,191978. We need a rotational velocity similar to the macroturbulent velocity ($\vsini = 40$~\kms and $\vmac = 32$~\kms) to correctly reproduce the line profiles.
{\it HD\,193117: }The cores of the Balmer lines are too deep in the best-fit model. This could come from a nebular emission at least for H$\alpha$. The other lines are rather well fitted. We also stress that although this star appears to be evolved, its nitrogen content is solar.
\subsection{The Cyg\,OB9 association}
\hspace{0.5cm}{\it HD\,194649: }In Paper I we determined a brightness ratio of $4.7 \pm 0.4$. The fit of the primary disentangled spectrum is good, except for the core of the H$\beta$ line. The secondary component being fainter in the composite spectra, its spectrum appears much noisier when corrected for the relative brightness and the disentangling process fails to correctly reconstruct the wings of H$\beta$. Nonetheless, the lines are relatively well reproduced by the synthetic model. Although the spectral classifications indicate an evolved primary star and a main-sequence secondary star, \logg\ determined on the basis of disentangled spectra fail, once again, to characterize this luminosity class.
{\it HD\,194334: }A good fit is achieved for this object. Only the cores of the Balmer lines and \ion{N}{iii}~4634--41 are less well reproduced. The fits of the \ion{He}{ii}~4686 and H$\alpha$ lines indicate a good determination of the mass-loss rate of the star.
{\it HD\,195213: }As for several stars in our sample, the cores of the Balmer lines are poorly reproduced. However, H$\alpha$ shows an emission component in its centre which probably originates from nebular emission. Furthermore, to reach a good fit of the helium lines, we have to increase the helium abundance to $0.15$ in number.
\begin{sidewaystable*}[htbp]
\caption{Stellar and wind parameters derived for the O-type stars in four Cygnus OB associations.}\label{cygnus_param}
\small
\begin{tabular}{llrrrrrrrrrrrrrrrr}
\hline
Source & SpT & \teff\ & \lL\ & \logg & \logg$_{\mathrm{true}}$\tablefootmark{a} & R & log($\dot{M}_{\mathrm{theo}}$)\tablefootmark{b} & log($\frac{\dot{M}_{\mathrm{H}\alpha}}{\sqrt(f)}$)\tablefootmark{c} & $f_{\infty}$ &\vinf & $\beta$ & \vsini\ & \vmac\ & M$_{\mathrm{evol}}$\tablefootmark{d} & M$_{\mathrm{spec}}$\tablefootmark{e} & M$_{\mathrm{dyn}}$\tablefootmark{f} & N/H\\
& & [kK] & & & & [\rsun] & & & &[\kms] & & [\kms] & [\kms] & [\msun] & [\msun] & [\msun] & [$\times 10^{-4}$]\\
\hline
\multicolumn{18}{c}{Cyg\,OB1}\\
\hline
HD\,193443P& O\,9III& 31.2 & $5.35\pm0.13 $ & 3.65 & 3.66 & $16.2_{-4.3}^{+6.0}$ & $\sim~-6.8$ & -- & -- & -- & -- & 105 & 38 & $30.0_{-4.8}^{+4.7}$ & $43.8_{-30.4}^{+103.0}$ & $1.0 \pm 0.1$ & $0.6\pm0.3$\\[3pt]
HD\,193443S& O\,9.5V& 29.7 & $4.76\pm0.23 $ & 3.75 & 3.76& $9.1_{-3.2}^{+5.0}$ & -- & -- & -- & -- & -- & 100 & 27 & $18.7_{-5.3}^{+4.6}$ & $17.4_{-13.2}^{+57.2}$ & $0.4 \pm 0.1$ & $0.6\pm0.3$\\[3pt]
HD\,193514& O\,7--7.5III(f)& 34.5 & $5.65\pm0.09$ & 3.50 & 3.51& $18.7_{-2.8}^{+3.3}$ & $-6.5$ & $-5.6$ & 0.03 & 2190 & 1.5 & 90 & 27 & $45.7_{-4.9}^{+6.8}$ & $41.3_{-17.5}^{+30.7}$ & -- & $2.0\pm1.0$\\[3pt]
HD\,193595& O\,7V& 35.5 & $5.11\pm0.05$ & 3.75 & 3.75& $9.5_{-1.0}^{+1.2}$& -- & $-8.7$ & 1.0 & -- & -- & 50 & 28 & $26.5_{-2.1}^{+2.1}$ & $18.7_{-6.8}^{+10.9}$ & -- & $3.0\pm1.5$\\[3pt]
HD\,193682& O\,5III(f)& 39.4 & $5.50\pm0.09$ & 3.75 & 3.77& $12.1_{-1.7}^{+2.0}$ & $-5.8$ & $-5.7$ & 0.01 & 2650 & 1.0 & 150 & 73 & $41.4_{-4.7}^{+5.4}$ & $31.4_{-13.1}^{+22.5}$ & -- & $6.0\pm3.0$\\[3pt]
HD\,194094& O\,8III& 29.9 & $5.45\pm0.36$ & 3.25 & 3.26& $19.8_{-7.6}^{+12.3}$ & $-6.7$ & $-7.0$ & 1.0 & -- & -- & 65 & 32 & $33.0_{-10.1}^{+23.8}$ & $25.9_{-18.0}^{+59.7}$ & -- & $1.0\pm0.5$\\[3pt]
HD\,194280& O\,9.7I& 26.9 & $5.61\pm0.36$ & 3.10 & 3.11& $29.4_{-11.4}^{+18.6}$ & $<~-6.5$ & -- & -- & -- & -- & 92 & 41 & $39.2_{-12.9}^{+25.4}$ & $41.1_{-28.8}^{+96.8}$ & -- & $0.1\pm0.1$\\[3pt]
HD\,228841& O\,7 & 34.5 & $5.42\pm0.37$ & 3.50 & 3.62& $14.4_{-5.5}^{+9.0}$ & $\sim~-6.4$ & $-7.1$ & 1.0 & -- & -- & 317 & 0 & $33.7_{-10.1}^{+22.7}$ & $31.4_{-21.9}^{+72.9}$ & -- & $5.4\pm2.7$\\[3pt]
HD\,228989P& O\,8.5V& 29.4 & $4.43\pm0.14$ & 3.50 & 3.56& $6.3_{-1.8}^{+2.6}$ & -- & -- & -- & -- & -- & 145 & 3 & $15.9_{-3.7}^{+3.7}$ & $ 5.3_{-3.8}^{+13.3}$ & $7.0 \pm 0.2$ & $1.0\pm0.5$\\[3pt]
HD\,228989S& O\,9.7V& 27.8 & $4.28\pm0.15$ & 3.75 & 3.78& $6.0_{-1.7}^{+2.6}$ & -- & -- & -- & -- & -- & 120 & 3 & $13.0_{-3.4}^{+3.8}$ & $ 7.7_{-5.6}^{+20.6}$ & $6.1 \pm 0.2$ & $3.0\pm1.5$\\[3pt]
HD\,229234& O\,9III + ...& 32.0 & $5.09\pm0.20$ & 3.60 & 3.61& $11.4_{-2.9}^{+3.9}$ & $\sim~-6.8$ & $-6.5$ & 1.0 & -- & -- & 85 & 38 & $24.8_{-4.1}^{+4.7}$ & $19.4_{-10.8}^{+24.6}$ & -- & $0.6\pm0.3$\\[3pt]
\hline
\multicolumn{18}{c}{Cyg\,OB3}\\
\hline
HD\,190864& O\,6.5III(f)& 38.0 & $5.35\pm0.14$ & 3.75 & 3.76& $10.9_{-2.1}^{+2.6}$ & $-6.3$ & $-6.4$ & 0.06 & 2250 & 1.0 & 80 & 43 & $34.1_{-3.8}^{+5.1}$ & $24.9_{-12.0}^{+23.2}$ & -- & $5.0\pm3.0$\\[3pt]
HD\,227018& O\,6.5 & 37.3 & $5.50\pm0.18$ & 4.00 & 4.00& $13.5_{-3.1}^{+4.0}$ & $-6.9$ & -- & 1.0 & -- & -- & 70 & 35 & $37.2_{-4.8}^{+13.4}$ & $66.7_{-35.2}^{+75.0}$ & -- & $1.0\pm0.5$\\[3pt]
HD\,227245& O\,7III& 36.0 & $5.18\pm0.14$ & 3.75 & 3.75& $10.0_{-1.9}^{+2.4}$ & $\sim~-6.4$ & $-6.7$ & 1.0 & -- & -- & 65 & 19 & $27.7_{-2.3}^{+6.3}$ & $20.8_{-10.1}^{+19.7}$ & -- & $3.0\pm0.5$\\[3pt]
HD\,227757& O\,9V& 32.0 & $4.73\pm0.13$ & 3.75 & 3.75& $7.6_{-1.4}^{+1.8}$ & -- & $-8.2$ & 1.0 & -- & -- & 45 & 3 & $18.6_{-4.8}^{+3.0}$ & $11.8_{-5.6}^{+10.9}$ & -- & $0.6\pm0.3$\\[3pt]
\hline
\multicolumn{18}{c}{Cyg\,OB8}\\
\hline
HD\,191423& ON\,9IIIn& 30.6 & $5.42\pm0.28$ & 3.50 & 3.67& $18.3_{-5.9}^{+8.7}$ & $\sim~-6.8$ & $-6.4$ & 0.01 & 600 & 0.9 & 410 & 0 & $34.7_{-10.3}^{+12.5}$ & $54.6_{-34.6}^{+95.1}$ & -- & $5.4\pm2.5$\\[3pt]
HD\,191978& O\,8III& 33.2 & $5.35\pm0.23$ & 3.75 & 3.75& $14.3_{-4.0}^{+5.5}$ & $-6.7$ & $-8.7$ & 1.0 & 1600 & -- & 40 & 32 & $32.1_{-7.4}^{+9.4}$ & $42.2_{-24.7}^{+59.8}$ & -- & $3.0\pm1.5$\\[3pt]
HD\,193117& O\,9III& 30.1 & $5.36\pm0.11$ & 3.30 & 3.31& $17.6_{-3.1}^{+3.8}$ & $\sim~-6.8$ & $-6.3$ & 1.0 & -- & -- & 80 & 28 & $30.4_{-4.1}^{+5.1}$ & $23.2_{-10.7}^{+19.9}$ & -- & $0.6\pm0.3$\\[3pt]
\hline
\multicolumn{18}{c}{Cyg\,OB9}\\
\hline
HD\,194334& O\,7--7.5III(f)& 33.9 & $5.19\pm0.07$ & 3.50 & 3.51& $11.4_{-1.5}^{+1.7}$& $\sim~-6.5$ & $-6.2$ & 1.0 & -- & -- & 62 & 32 & $26.6_{-2.4}^{+2.4}$ & $15.3_{-6.1}^{+10.2}$ & -- & $3.0\pm1.5$\\[3pt]
HD\,194649P& O\,6III(f) & 38.0 & $5.14\pm0.15$ & 3.75 & 3.75& $8.6_{-2.2}^{+3.1}$ & $-6.1$ & -- & -- & -- & -- & 60 & 67 & $28.1_{-4.2}^{+7.8}$ & $15.3_{-10.6}^{+35.1}$ & $4.9 \pm 0.4$ & $3.0\pm1.3$\\[3pt]
HD\,194649S& O\,8V & 33.0 & $4.36\pm0.27$ & 3.75 & 3.76& $4.6_{-1.7}^{+2.8}$ & -- & -- & -- & -- & -- & 60 & 27 & $9.6_{-9.0}^{+9.4}$ & $4.5_{-3.5}^{+15.9}$ & $1.9 \pm 0.1$ & $1.0\pm0.5$\\[3pt]
HD\,195213& O\,7III(f)& 35.3 & $5.19\pm0.22$ & 3.50 & 3.51& $10.5_{-2.8}^{+3.8}$ & $\sim -6.4$ & $-6.0$ & 1.0 & -- & -- & 85 & 28 & $27.6_{-6.1}^{+5.7}$ & $13.2_{-7.6}^{+17.8}$ & -- & $6.0\pm3.0$\\[3pt]
\hline
\end{tabular}
\tablefoot{\tablefoottext{a}{$g_{\mathrm{true}}$: gravity corrected from the contribution of the centrifugal force i.e. $g_{\mathrm{true}} = g + (\vsini)^2/R$}
\tablefoottext{b}{log($\dot{M}_{\mathrm{theo}}$: theoretical mass-loss rate estimated by \citet{muijres2012} for a given spectral classification}
\tablefoottext{c}{log($\dot{M}_{\mathrm{H}\alpha}$: mass-loss rate (where the correction of the clumping $\sqrt{f}$ is not applied) computed from the H$_\alpha$ line. }
\tablefoottext{d}{M$_{\mathrm{evol}}$: mass estimated from the HR diagram}
\tablefoottext{e}{M$_{\mathrm{spec}}$: mass computed from $M = g\,R^2/G$}
\tablefoottext{f}{M$_{\mathrm{dyn}}$: minimum mass of a binary component ($M^3\,\sin i$)}}
\end{sidewaystable*}
\section{Discussion}
\label{sec:dis}
\subsection{Distances and ages of the OB associations}
The stellar parameters derived in Sect.~\ref{sec:res} provide the positions of the stars in the Hertzsprung-Russell (HR) diagram. However, these locations are dependent on the real distances of the stars. As we have already mentioned, the luminosities were computed in the present paper from the distance modulus given by \citet{hum78} and refined through the fit of the SED of each object. When no initial value of the distance of the stars is available in the Humphreys catalogue, we have taken the mean value of the corresponding OB association. The observed $V$ and $(B-V)$ values are from \citet{kha09}, whilst the bolometric corrections were taken from \citet{mar06} as a function of the spectral classification of the stars. The error-bars on the luminosities of the presumably single stars are mainly determined by the uncertainties on the distances. For the binaries, an additional uncertainty is also provided by the brightness ratio between both components of the system.
For this analysis, we use the evolutionary tracks of \citet{brott11} computed for single stars. Given that rotation plays a major role in the evolution of the stars, we select the tracks computed with initial rotational velocities of about 100~\kms\ (black lines in Fig.~\ref{cygnus_track}) and of about 400~\kms\ (red lines in Fig.~\ref{cygnus_track}). We stress that an initial rotational velocity of 300~\kms \ is generally considered as a "standard" value for O-type stars. In this analysis, we do not investigate all the sample stars together because they are not from the same forming regions and so do not constitute a homogeneous O-type star population.
\begin{figure*}[htbp]
\begin{center}
\subfigure[Cyg\,OB1]{
\includegraphics[scale=0.45,bb=29 8 536 400,clip]{CygOB1.eps}
\label{cygob1_track}}
\subfigure[Cyg\,OB3]{
\includegraphics[scale=0.45,bb=29 8 536 400,clip]{CygOB3.eps}
\label{cygob3_track}}
\subfigure[Cyg\,OB8]{
\includegraphics[scale=0.45,bb=29 8 536 400,clip]{CygOB8.eps}
\label{cygob8_track}}
\subfigure[Cyg\,OB9]{
\includegraphics[scale=0.45,bb=29 8 536 400,clip]{CygOB9.eps}
\label{cygob9_track}}
\caption{Positions of the Cygnus O-type stars in the HR diagram. Evolutionary tracks and isochrones are from \citet{brott11} and were computed for $v_{\mathrm{rot}_{\mathrm{init}}} = 100$~\kms (black lines) and for $v_{\mathrm{rot}_{\mathrm{init}}} = 400$~\kms (red lines). Isochrones, represented by dashed lines, correspond to ages ranging from 2 to 8~Myrs with a step of 2~Myrs.}\label{cygnus_track}
\end{center}
\end{figure*}
The distances taken from \citet{hum78} are close to the largest estimates of \citet{uyaniker2001}. This yields the largest values to the luminosities of the stars. By comparing the luminosities computed in the present paper with the observational values (considered as standard) reported by \citet{mar05}, we see that the stars belonging to Cyg\,OB9 as well as HD\,227245 or both components of HD\,228989 are fainter than the standard expectations. On the contrary, HD\,193117 and HD\,191423 belonging to the Cyg\,OB8 association are brighter than expected. This makes the estimation of their distance questionable. It is clear that these standard values are dependent on other stellar parameters that are not the same for the stars in our sample such as, e.g., $\logg$. To match the standard values for the luminosity, we should increase the mean distance of Cyg\,OB9 to a value of 2.0~kpc and decrease that of Cyg\,OB8 to a mean value of 1.9~kpc.
These assumptions could have a great impact on the location of the stars in the HR diagram and could thus affect the real determination of their age. Indeed, from these possible distances for Cyg\,OB8 and Cyg\,OB9 (about 1.9 and 2.0~kpc, respectively), the stars of Cyg\,OB8 could be slightly older (between 4 and 6 Myrs) than what we find in Fig.\,\ref{cygob8_track}, whilst those of Cyg\,OB9 could have an estimated age between 2 and 4 Myrs, i.e. slightly younger. Furthermore, the location of the secondary star of HD,194649 would be more realistic than what we see in Fig.\,\ref{cygob9_track}. This means that the ages of the stars in Cyg\,OB8 do not seem to confirm the estimates of $\sim 3$~Myrs computed from the evolutionary tracks by \citet{uyaniker2001} for this association, but this assumption must still be verified on the basis of a larger sample of stars belonging to this association.
From the distances quoted in Table~\ref{tab:phot}, we estimate the evolutionary ages of O-type stars in Cyg\,OB1 and in Cyg\,OB3 between 2 and 7 Myrs and between 3 and 5 Myrs, respectively, according to the parameters reported in Table~\ref{cygnus_param}.
We must be careful about these HR diagrams, because the real ages of the stars can be slightly different depending on their initial rotational velocity. Therefore, some stars such as HD\,193682 or HD\,228841 (with $\vsini = 150$ and 317~\kms, respectively) could be older than inferred from their position in the HR diagram \citep[see][for further details]{brott11}.
Determining the real positions of the O stars from the HR diagram is difficult and requires a good knowledge of their real distances. Therefore, our study stresses the importance of astrometrical missions such as Gaia to better understand the evolutionary properties of massive stars.
\subsection{The N content}
Nitrogen is an important element to analyse and to understand the rotational mixing in the evolutionary processes of a massive star. Theoretical studies \citep[e.g.][]{mm00} have shown that rotationally induced mixing can affect the chemical composition of the surface layer of massive stars even though they are still on the main-sequence band. The theory thus predicts that the larger the rotational velocity of a star, the higher its nitrogen abundance. Several observational studies on B stars in the LMC (notably the {\it ESO VLT-FLAMES} survey \citealt{evans2005,evans2006}) have shown that at least 60\% of the B stars display the pattern expected from evolutionary models which include the rotational mixing. However, simulations of LMC early B-type stars made by \citet{brott11b} failed to reproduce the remaining 40\%, which are split into two different groups of stars, one containing slowly rotating, nitrogen-enriched objects and another one containing rapidly rotating un-enriched objects. The analysis of \citet{riv12} also indicated a large number of O-type stars in the LMC with large enrichment in nitrogen and with a low rotation rate, supporting the same conclusions as for the B stars in the LMC. These authors stressed, however, that the problem of the group containing the highly-enriched slow-rotating stars was more severe for the O-type star population than for the B-type star one. A possible explanation concerning the presence of the group with high-enriched slow-rotating stars was advanced by \citet{mor06} who have analysed ten slowly rotating galactic early $\beta$-Cephei B-type stars and found that out of four heavily enriched stars, three had a magnetic field.
In the present paper, we determine the N content of nineteen O stars belonging to four Cygnus OB associations. This result is combined and put in perspective with several other investigations of O-type stars in the Galaxy (\citealt{mar12}, \citealt{bouret12}, and \citealt{mar14}). Although this constitutes a set of heterogeneous studies, all the analyses use the same tool to determine the stellar parameters. We show in Fig.~\ref{fig:Ncontent}, the nitrogen content as a function of the projected rotational velocities of 90 O-type stars, the so-called Hunter diagram \citep{hun09}. We stress that we remove from our sample all the binary components as well as a few stars listed in \citet{mar14} that are binaries, e.g. HD\,93250 \citep{san11b} or HD\,193443 (Paper~I). We tentatively define possible groups similar to those introduced by \citet{hun08,hun09}. To do that, the group constituted of stars with intermediate rotation rates (between 50 and 110~\kms) and a high enrichment in nitrogen is assumed to be empty as noted by \citet{brott11b}. In this diagram, there is a clear outlier (HD\,194280) that seems to belong to none of the different groups. We stress though that defining their exact location requires a more sophisticated theoretical work similar to that of \citet{brott11b} for massive stars in the LMC.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=8.cm,bb=25 5 536 406,clip]{vsini_N.eps}
\caption{Nitrogen surface abundance (in units of 12 + $\log$(N/H)) as a function of projected rotational velocity, the so-called Hunter diagram. O stars in our sample are shown in black, the O-stars of NGC\,2244 \citep{mar12} in red, stars studied by \citet{bouret12} in blue, and the MiMes O-type stars \citep{mar14} in green. Grey lines define the possible locations of the different groups.}\label{fig:Ncontent}
\end{center}
\end{figure}
In Fig.~\ref{N_L_plot}, we show the N content of our initial targets versus their luminosity. We clearly see a trend of higher N content for more luminous stars. As already stressed by \citet{mar12}, the fast rotators do not appear as outliers. Two clear outliers are shown in this figure: HD\,194280, as we have already mentioned, with a very low nitrogen abundance and, to a lesser extent, the secondary component of HD\,228989.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=8.cm,bb=26 9 535 408,clip]{N_L.eps}
\caption{Nitrogen surface abundance (in units of 12 + $\log$(N/H)) as a function of stellar luminosity. Dots represent the single O stars in our sample, whilst triangles are the binary components. Evolutionary tracks are (from left to right) computed for stars with 10, 15, 20, and 40~$M_{\odot}$ and initial velocities of about 300~\kms.}\label{N_L_plot}
\end{center}
\end{figure}
\subsection{Stellar masses}
The differences between the evolutionary masses, i.e. the masses determined from the HR diagram and the evolutionary tracks, and the spectroscopic masses, i.e. those determined from the atmospherical parameters (\teff, \logg, and $L$), are recurrent issues in astrophysics. This so-called "mass discrepancy" refers to a systematic overestimate of the former relative to the latter. \citet{rep04} already noted that for stars with masses smaller than $50~M_{\odot}$ a parallel relation to the 1:1 relation in the $M_{\mathrm{evol}}-M_{\mathrm{spec}}$ diagram could be followed. More recently, \citet{mar12}, from a small sample of O-type stars in NGC~2244 and in Mon~OB2, reported on a clear trend of mass discrepancy for stars with $M< 25~M_{\odot}$, consistent with the results of \citet{rep04}.
In Paper I we also determined the dynamical mass of each component of the binary systems. We present them in Table~\ref{cygnus_param} as a reminder. We can thus compare them to their evolutionary and spectroscopic masses. The evolutionary masses have been estimated for each star (presumably single or individual component) by a bi-interpolation of the evolutionary tracks on \teff\ and on $\log(L/L_{\odot})$, whilst the spectroscopic masses have been computed from $M = g\,R^2/G$, where $G$ is the universal gravitational constant. We stress that these masses were computed from the effective gravities corrected for the effects of the centrifugal forces caused by rotation ($g_{\mathrm{true}}$ in Table~\ref{cygnus_param}) by following the approach of \citet{herrero1992}. We clearly see a real disagreement for the primary star of HD\,228989. First, the spectroscopic mass is smaller than the dynamical mass that is supposed to be the minimum mass. Then, the mass ratio computed from the spectroscopic values indicates that the secondary component is more massive than the primary one which is the opposite of the results provided in Paper I. This assumes that the distance and/or the gravities of both components are poorly estimated. The reason for a poor determination of the \logg\ is that the disentangling process does not perfectly reproduce the wings of the Balmer lines because of the broadness of these lines and because the two individual line profiles are not completely resolved throughout the orbit.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=8.cm,bb=24 2 537 407,clip]{mass_discrepancy.eps}
\caption{Comparison of the evolutionary masses and the spectroscopic masses of the presumably single stars. }\label{fig:mass}
\end{center}
\end{figure}
When we only focus on the single stars, we find that the objects in our sample (Fig.~\ref{fig:mass}) with a mass smaller than $30~M_{\odot}$ present a clear mass discrepancy as already detected for O-type stars belonging to NGC\,2244 and Mon\,OB2 \citep{mar12}. For the stars in our sample with a higher mass, the uncertainties do not allow us to conclude whether or not the mass estimates are significantly different. Two ingredients play an important role in the determination of these uncertainties: the distance and \logg. As we have already mentioned, the distances of the stars are still poorly known. It affects the stellar luminosities and thus the stellar radii since the \teff\ is thought to be well constrained. Concerning \logg, we made several tests to correctly fit the wings of the Balmer lines on {\'e}chelle spectra and also on the Aur{\'e}lie spectra (see Paper I, for a complete list of the data). Our estimation of the stellar gravity thus seems quite robust, therefore minimizing the contribution of the \logg\ to the uncertainty budget of $M_{\mathrm{spec}}$.
\subsection{Mass-loss rates}
We also estimate the mass-loss rates (Table~\ref{cygnus_param}) for most objects in our sample, but the lack of UV spectra does not allow us to determine all the wind parameters of these stars. Often, the terminal velocity, $f$, and the $\beta$ parameters cannot be estimated from only the H$\alpha$ line. Moreover, when no FUSE data were available, we only assumed that the stellar winds are homogeneous ($f = 1.0$).
The mass-loss rates obtained from atmosphere models and included in Table~\ref{cygnus_param} can be compared to the theoretical values quoted by \citet{muijres2012}. These values are given as a function of spectral type, but some of them are missing notably for late main-sequence stars. For most of our objects for which a homogeneous wind was assumed, the agreement between estimated and theoretical values is relatively good. Only few stars (HD\,228841, HD\,193514, HD\,227018, HD\,193117, and HD\,191978) have mass-loss rates which differ by more than about 0.4~dex. Although the quantities $\dot{M}/\sqrt{f}$ remain quite close between the observations and the theory, we can see that the observed mass-loss rates in general appear slightly larger than the theoretical values.
\section{Conclusion}
\label{sec:conc}
We analysed by means of the CMFGEN atmosphere code, the fundamental properties of fifteen presumably single stars and of the individual components of four binary systems belonging to different Cygnus OB associations: Cyg\,OB1, Cyg\,OB3, Cyg\,OB8, and Cyg\,OB9. In addition to the optical spectra, we retrieved, for several single stars, the UV spectra (FUSE and/or IUE) to better constrain the wind parameters of these objects.
The analysis of the individual parameters showed that the luminosities of several objects do not agree with the standard luminosities of stars with similar spectral classification. This is most likely because the distances of these stars are poorly constrained. Therefore, it is more likely that the positions of certain stars should be shifted upwards in the HR diagram. A direct consequence is that the stars in Cyg\,OB8 and Cyg\,OB9 could be younger than we have determined. However, if the distance used in our analysis is correct, this would mean that Cyg\,OB8 is older than 3~Myrs as quoted by \citet{uyaniker2001}.
We compared the N content of the stars in our sample to other galactic O-type stars. The resulting Hunter diagram shows the same structure (the same five groups) as for the O and B-type stars in the LMC \citep{hun08,hun09,riv12}.
By comparing the evolutionary and the spectroscopic masses, we also pointed out a discrepancy between the masses for the stars in our sample. This overestimate of the evolutionary masses relative to the spectroscopic masses is particularly obvious for stars with masses smaller than $30~M_{\odot}$, as was already noted by \citet{mar12}. Above this value, the masses agree with each other within the uncertainties. Although this sample of stars takes into account objects more evolved than those in the investigation of \citet{mar12}, we obtain similar general conclusions.
\begin{acknowledgements}
We wish to thank the anonymous referee for his clear report and his constructive remarks that have considerabily improved this paper. This research was supported by the Fonds National de la Recherche Scientifique (F.N.R.S.), the PRODEX XMM/Integral contract (Belspo), and the Communaut\'e fran\c caise de Belgique -- Action de recherche concert\'ee -- A.R.C. -- Acad\'emie Wallonie-Europe. We thank John Hillier for making CMFGEN available and Fabrice Martins for his advice.
\end{acknowledgements}
|
{
"timestamp": "2015-04-14T02:12:42",
"yymm": "1504",
"arxiv_id": "1504.03107",
"language": "en",
"url": "https://arxiv.org/abs/1504.03107"
}
|
\section{Introduction}
Hemisystems in finite geometry have a short but eventful history, dating back to B. Segre's definition of them as a special case of a regular system in 1965 \cite{MR0213949}.
A hemisystem on the Hermitian space $\mathrm{H}(3,q^2)$, $q$ odd, is a set of lines $\mathcal{L}$ on $\mathrm{H}(3, q^2)$ such that every point in $\mathrm{H}(3, q^2)$ has half of the lines incident with it in $\mathcal{L}$. Hemisystems are of great interest
because they give rise to partial quadrangles, strongly regular graphs and cometric $Q$-antipodal association schemes \cite[\textsection 7.5.1]{MR3092674}. In his 1965 treatise, Segre gave an example of a hemisystem on $\mathrm{H}(3, 3^2)$ and proved that
it was unique (up to equivalence) on this Hermitian space \cite{MR0213949}.
For forty years after their introduction, no new examples of hemisystems were found, and Thas conjectured that none existed on $\mathrm{H}(3,q^2)$ for $q >3$ \cite[\textsection 9.5]{buekenhout1995handbook}.
This conjecture was proven false when Penttila and Cossidente discovered a new infinite family of hemisystems in 2005 \cite{cossidente2005hemisystems}.
In 2011, Penttila and Williford introduced the notion of
\textit{relative hemisystems}, an analogous concept to hemisystems that exist on $\mathrm{H}(3,q^2)$, for $q$ even \cite{penttila2011new}. They were motivated by the desire to construct an example of an infinite family of primitive cometric association schemes that do not arise from distance regular graphs.
Prior to their paper, only sporadic examples of such association schemes were known \cite[\textsection 1]{penttila2011new}.
Let $Q$ be a generalised quadrangle of order $(q^2, q)$, containing a generalised quadrangle $Q'$ of order $(q,q)$, where $q$ is a power of two.
Each of the lines in $Q$ meet $Q'$ in either $q+1$ points or are disjoint from it. A subset $\mathcal{R}$ of the lines in $Q \setminus Q'$ is a \textit{relative hemisystem} of $Q$ with respect to $Q'$ if
for every point $P$ in $Q \setminus Q'$, exactly half the lines through $P$ disjoint from $Q'$ lie in $\mathcal{R}$. Notice that this definition is well defined, because the number of lines through $P$ disjoint from $Q'$ is $q$.
Penttila and Williford concluded their paper with an open question on the existence of non-isomorphic relative hemisystems
on $\mathrm{H}(3,q^2)$, $q$ even. Cossidente resolved this question two years later by constructing an infinite family of non-isomorphic relative hemisystems,
each admitting $\mathrm{PSL}(2,q)$ as an automorphism group \cite{MR3081646}, and another admitting a group of order $q^2(q+1)$, for each $q\geqslant 8$, a power of two \cite{cossidente2013new}. In 2014, Cossidente and Pavese constructed a relative hemisystem arising from a Suzuki-Tits ovoid on $\mathrm{H}(3,64)$ \cite{MR3252665}. They conjectured that this relative hemisystem is sporadic.
Cossidente \cite{MR3081646} shows that his construction generates a new infinite family by showing that he has created more relative hemisystems than the number generated by the Penttila-Williford construction. Through finding all of the relative hemisystems invariant under $\mathrm{PSL}(2,q)$ for varying values of $q$, we have shown that there are actually several more inequivalent infinite families of relative hemisystems that arise from this construction (see Remark \ref{remark:Cossidente}). For $q = 16$, we found by computer that there are five inequivalent relative hemisystems that each admit $\mathrm{PSL}(2, q)$, and we conjecture that the number of inequivalent infinite families increases with $q$.
In this paper, we state a set of new sufficient criteria for relative hemisystems, which unifies all currently known infinite families of relative hemisystems.
We begin with some background, and briefly recall the constructions of the three known infinite families. We go on to prove a series of results, culminating in
Theorem \ref{majortheorem} which states sufficient conditions to determine a relative hemisystem. Using these, we provide new explicit proofs of the Penttila-Williford and Cossidente results.
Finally, we briefly discuss a computational result classifying all of the relative hemisystems on $\mathrm{H}(3,4^2)$, and some results partially classifying the relative hemisystems on $\mathrm{H}(3,8^2)$.
\section{Background information}
A finite generalised quadrangle of order $(s,t)$ is an incidence structure of points and lines such that:
\begin{itemize}
\item Any two points are incident with at most one line.
\item Every point is incident with $t+1$ lines.
\item Every line is incident with $s+1$ points.
\item For any point $P$ and line $\ell$ that are not incident, there is a unique point $P'$ on $\ell$ that is collinear with $P$.
\end{itemize}
If we take the point-line dual of a generalised quadrangle of order $(s,t)$, we obtain another generalised quadrangle, of order $(t,s)$.
There are many examples of generalised quadrangles; for a good reference for the finite cases, see \cite{payne2009finite}.
For the purposes of this paper, we are interested in two families of generalised quadrangles.
\begin{itemize}
\item \textbf{$\mathrm{H}(3,q^2)$}, the set of all totally isotropic lines and points with respect to a Hermitian form on the projective space $\mathrm{PG}(3,q^2)$. These are generalised quadrangles of order $(q^2, q)$, for $q$ a prime power.
\item \textbf{$\mathrm{W}(3,q)$}, the set of all totally isotropic points and lines of a symplectic form on $\mathrm{PG}(3,q)$. These are generalised quadrangles of order $(q,q)$, for $q$ a prime power.
\end{itemize}
In this paper we will only work with fields with even characteristic, even though some results may hold for odd prime powers. For this reason, from now on we will assume that $q = 2^k$ for some positive integer $k$.
For a given $q$, we can find an embedding of $\mathrm{W}(3,q)$ in $\mathrm{H}(3,q^2)$. For a detailed description of the construction of this embedding, the reader is directed to \cite[Section 4.5]{kleidman1990subgroup} and our construction in Section \ref{newproofs}.
This gives us the setup needed for the construction of a relative hemisystem. When constructing a relative hemisystem,
we are only concerned with the set of points of $\mathrm{H}(3,q^2)$ which are outside of $\mathrm{W}(3,q)$ and the lines of $\mathrm{H}(3,q^2)$ disjoint from $\mathrm{W}(3,q)$.
For conciseness, we call the former the set of \textit{external points},
denoted $\mathcal{P}_E$, and the latter the set of \textit{external lines}, denoted $\mathcal{L}_E$.
Throughout the paper, we will also make reference to the collineation group of the Hermitian space $\mathrm{H}(3,q^2)$, which shall be denoted as the projective unitary group $\mathrm{P} \Gamma \mathrm{U}(4,q)$.
We use $\Omega / G$ to denote the set of orbits of a set $\Omega$ under a group $G$. Also recall that if $N$ is a normal subgroup of $G$, then $G$ acts on $\Omega / N$ in its action on sets.
We will say that a group $G$ acts \textit{semiregularly} in its action on a set $\Omega$ if the only elements in $G$ that fix an element of $\Omega$ are those in the kernel of the action. Note that definition of semiregular may be different to definitions in other areas of the literature.
Quadrics are also integral to many of the results in this paper. Quadrics in a three dimensional projective space are the totally singular points and
lines (if they exist) of a quadratic form. In $\mathrm{PG}(3,q)$, there are two sorts of non-singular quadrics -- hyperbolic and elliptic.
Hyperbolic quadrics, denoted $\mathrm{Q}^+(3,q)$ are quadrics which contain totally singular lines. Otherwise, the quadric is elliptic, denoted $\mathrm{Q}^-(3,q)$. For further background on quadrics, see \cite[\S22]{MR1363259}.
Suppose $q$ is even and $\mathcal{Q}^+$ is an irreducible hyperbolic quadric of $\mathrm{PG}(3,q^2)$ that shares a tangent plane with $\mathrm{H}(3,q^2)$ at a common point. Then the intersection of $\mathcal{Q}^+$ and $\mathrm{H}(3,q^2)$ has size $q^2+1$ and is an elliptic quadric $\mathrm{Q}^-(3,q)$ \cite{quadherm}.
\section{The known families of relative hemisystems}
Here, we give brief descriptions of the constructions of the known infinite families of relative hemisystems.
\subsection{The Penttila-Williford relative hemisystems}
\label{PWRHs}
The first example of an infinite family of relative hemisystems, admitting $\mathrm{P} \Omega ^- (4,q)$ as an automorphism group, was given by Penttila and Williford in their paper introducing the concept \cite{penttila2011new}.
Their construction \cite[Theorem 5]{penttila2011new} considers the action of the normaliser of a Singer cycle of $\mathrm{P} \Omega^-(4,q)$ on $\mathrm{H}(3,q^2)$. They use it to prove that
$\mathrm{P} \Omega^-(4,q)$,
$q$ even, $q \geqslant 2$ has two orbits on external lines. It transpires that these orbits form two relative hemisystems, $H_1$ and $H_2$. They further show that there exists an involution $t$, which fixes the points of $\mathrm{W}(3,q)$ and switches $H_1$ and $H_2$ \cite{penttila2011new}.
\subsection{The Cossidente relative hemisystems}
\label{CRHs}
Apart from the Penttila-Williford family of relative hemisystems, the only other known infinite families of relative hemisystems are the two discovered by Cossidente \cite{MR3081646, cossidente2013new} in 2013.
Both of these families are \textit{perturbations} of the Penttila-Williford relative hemisystems.
The first family on $\mathrm{H}(3,q^2)$, $q$ even and $q>4$, admits the linear group $\mathrm{PSL}(2,q)$ as an automorphism group \cite{MR3081646}. Cossidente constructed it by taking the two Penttila-Williford
relative hemisystems $H_1$ and $H_2$ and considering the stabiliser of a conic section of the elliptic quadric $\mathrm{Q}^-(3,q)$ in $\mathrm{W}(3,q)$ fixed by $\mathrm{P} \Omega^-(4,q)$.
This stabiliser is isomorphic to $\mathrm{PSL}(2,q)$ and
does not act transitively on $H_1$ and $H_2$. Cossidente then uses the involution $t$, switching $H_1$ and $H_2$ from the Penttila-Williford proof \cite[Theorem 5]{penttila2011new}
to delete some orbits of $H_1$ under $\mathrm{PSL}(2,q)$ and replace them with their images under $t$. Since the number of ways this can be done
outnumbers the number of Penttila-Williford relative hemisystems, Cossidente has constructed a new infinite family.
The second infinite family of relative hemisystems discovered by Cossidente admits a group of order $q^2(q+1)$ for each $q$ even and $q>4$.
The construction of this infinite family is very similar to the last. Choose a point $P$
of an elliptic quadric $\mathrm{Q}^-(3,q)$, which is an ovoid of $\mathrm{W}(3,q)$. Let $M$ be the subgroup of the stabiliser of $P$ in $\mathrm{P} \Omega^-(4,q)$ of order $q^2(q+1)$.
Instead of orbits under $\mathrm{PSL}(2,q)$, Cossidente considers orbits of $H_1$ and $H_2$ under $M$, deleting orbits of $H_1$ and replacing them by their image under the involution $t$.
Since the number of relative hemisystems invariant under $M$ exceeds that of the Penttila-Williford relative hemisystems, Cossidente must have found another infinite family of relative hemisystems.
\section{New sufficient conditions for relative hemisystems}
\label{magic}
Let $\mathcal{Q}^+$ be a hyperbolic quadric which intersects a Hermitian space $\mathrm{H}(3,q^2)$ in an elliptic quadric isomorphic to $\mathrm{Q}^-(3,q)$.
The stabiliser of $\mathcal{Q}^+$ in $\mathrm{P} \Gamma \mathrm{U} (4,q)$ is isomorphic to the orthogonal group $\mathrm{PSO}^-(4,q)$ \cite{penttila2011new};
and the subgroup of $\mathrm{P} \Gamma \mathrm{U}(4,q)_{\mathcal{Q}^+}$ that stabilises the two families of reguli of $\mathcal{Q}^+$ is
isomorphic to $\mathrm{P} \Omega^-(4,q)$ \cite{penttila2011new}. Penttila and Williford prove that $\mathrm{P}\Omega ^-(4,q)$ has two orbits on external lines, and $\mathrm{PSO}^-(4,q)$ is transitive on
external lines.
Taking $\ell \in \mathcal{L}_E$, the size of the orbit of $\ell$ under $\mathrm{PSO}^-(4,q)$ is twice as large as the orbit of $\ell$ under $\mathrm{P}\Omega ^-(4,q)$.
By the Orbit-Stabiliser Theorem,
$|\mathrm{PSO}^-(4,q) : \mathrm{PSO}^-(4,q)_{\ell}| = 2|\mathrm{P}\Omega ^-(4,q): \mathrm{P}\Omega ^-(4,q)_{\ell}|$.
Since $|\mathrm{PSO}^-(4,q):\mathrm{P} \Omega^-(4,q)| = 2$ \cite[Table 2.1d]{kleidman1990subgroup}, it follows that $\mathrm{PSO}^-(4,q)_{\ell} = \mathrm{P} \Gamma \mathrm{U}(4,q)_{\mathcal{Q}^+, \ell} = \mathrm{P}\Omega ^-(4,q)_{\ell}$.
Therefore, the size of the orbit of $\ell^{\mathrm{P}\Omega ^-(4,q)}$ under $\mathrm{PSO}^-(4,q)$ is
$$ |\mathrm{PSO}^-(4,q) : \mathrm{PSO}^-(4,q)_{\ell}\mathrm{P}\Omega^-(4,q)| = |\mathrm{PSO}^-(4,q) : \mathrm{P}\Omega^-(4,q)| =2$$ and hence $\mathrm{PSO}^-(4,q) = \mathrm{P} \Gamma \mathrm{U}(4,q)_{\mathcal{Q}^+}$
acts semiregularly on the orbits of $ \mathrm{P}\Omega ^-(4,q)$ on external lines.
\begin{proposition}
\label{cond2prop}
Let $G$ be a subgroup of $\mathrm{P} \Gamma \mathrm{U}(4,q)_{\mathcal{Q}^+}$, where $\mathcal{Q}^+$ is a hyperbolic quadric meeting $\mathrm{H}(3,q^2)$ in an elliptic quadric.
Let $D$ be the subgroup of $\mathrm{P} \Gamma \mathrm{U}(4,q)_{\mathcal{Q}^+}$ that stabilises the two families of reguli of $\mathcal{Q}^+$. If $G$ is not contained in $D$, then $G$ acts semiregularly on the orbits of $G\cap D$ on external lines.
\end{proposition}
\begin{proof}
Suppose the contrary, that $G$ fixes an orbit $\ell^{G\cap D}$. Then $\ell^G = \ell^{G\cap D}$ and by the Orbit-Stabiliser Theorem,
$$|G\cap D: (G\cap D)_\ell | =| \ell^{G\cap D} | = | \ell^G | = | G : G_\ell |$$
and hence
\begin{equation}
\label{prop1eqn}
| G : G\cap D | = | G_\ell : (G\cap D)_\ell |.
\end{equation}
Now from the discussion at the beginning of this section, we have $\mathrm{P} \Gamma \mathrm{U}(4,q)_{\mathcal{Q}^+,\ell} = D_\ell$.
So, $G$ is a subgroup of $\mathrm{P} \Gamma \mathrm{U}(4,q)_{\mathcal{Q}^+}$, and $G_{\ell}$ is a subgroup of $D_{\ell}$. Hence $G_\ell = (G \cap D)_\ell$, and so by Equation \ref{prop1eqn}, $|G:G \cap D| = 1$.
This implies that $G = G \cap D$ and $G$ is a subgroup of $D$. This is a contradiction by the definitions of $D$ and $G$. Therefore, $G$ must act semiregularly on the orbits of $G\cap D$ on external lines.
\end{proof}
\begin{theorem}
\label{majortheorem}
Suppose $G < \overline{G}$, where $\overline{G}$ is a subgroup of the collineation group of $\mathrm{H}(3,q^2)$ stabilising $\mathrm{W}(3,q)$. Further suppose that $\overline{G}$ and $G$ satisfy the following conditions:
\begin{enumerate}
\item $| \overline{G}{:} G | = 2$,
\item $\overline{G}$ acts semiregularly on $\mathcal{L}_E/G$,
\item $\mathcal{P}_E / \overline{G} = \mathcal{P}_E / G$.
\end{enumerate}
Write $\mathcal{L}_E / \overline{G} = \{ \ell_1^{\overline{G}}, \ell_2^{\overline{G}}, \dots \ell_n^{\overline{G}} \} $, where each $\ell_i$ is a representative from a distinct orbit of $\overline{G}$ on $\mathcal{L}_E$. Then $\bigcup_{i=1}^n \ell_i^G$ is a relative hemisystem \index{relative hemisystem}.
\end{theorem}
\begin{proof}
Let $X$ be an external point. For $\ell \in \mathcal{L}_E$, we define the \textit{line orbit incidence number} \index{line orbit incidence number} as
$$n_{X, \ell}^G = | \{ m \in \ell^G \mid X\in m \} |$$
First note that for all $g \in G$, \begin{equation}\label{loin} n_{X, \ell}^G = n_{X^g, \ell}^G \end{equation} because $X \in m$ if and only if $ X^g \in m^g$.
Now, since $ |\overline{G}{:} G|=2$, for all $\ell \in \mathcal{L}_E$ there exist $m_1, m_2 \in \mathcal{L}_E$ such that $\ell^{\overline{G}} = m_1^G \cup m_2^G$. Then, we can find $t \in \overline{G}$ such that
$(m_1^G)^t = m_2^G$. Since $\overline{G}$ acts semiregularly on $\mathcal{L}_E/G$, the orbits of $\overline{G}$ on $\mathcal{L}_E/G$ have size two and
$G$ is the kernel of the action.
Therefore, there exists an element $t \in \overline{G}$ such that for all $\ell \in \mathcal{L}_E$, we have $ \ell^{\overline{G}} = \ell^G \dot{\cup} (\ell^t)^G$.
We then have $n_{X, \ell}^{\overline{G}} = n_{X, \ell^t}^G + n_{X, \ell}^G$.
Consider $n_{X, \ell^t}^G = | \{ m \in (\ell^t)^G \mid X\in m \}|$. Since $G$ has index two and is therefore a normal subgroup of $\overline{G}$, $n_{X, \ell^t}^G = | \{ n^t\in (\ell^G)^t \mid X\in n^t \}| =
| \{ n\in (\ell^G) \mid X^{t^{-1}}\in n \}|$. Now, since $\overline{G}$ and $G$ have the same orbits on external points by Condition 3, , there exists $u \in G$ such that $X^{t^{-1}} = X^u$. So $n_{X, \ell^t}^G = | \{ n\in (\ell^G) \mid X^u\in n \}|$ = $n_{X^u, \ell}^G = n_{X,\ell}^G$, by \eqref{loin}.
Therefore,
\begin{equation}
\label{majorhalf}
n_{X, \ell}^{\overline{G}} = 2 n_{X, \ell}^G
\end{equation}
Consider the orbit representatives $\ell_1 , \ell_2, \dots \ell_n$ of $\mathcal{L}_E/ \overline{G}$.
The number $q$ of external lines incident with $X$ is then equal to the sum of the line orbit incidence numbers $n_{X,\ell_i}^{\overline{G}}$, for $i\in \{1, \dots n \}$.
Then, from Equation \ref{majorhalf}, $q/2 = \sum_{i=1}^n n_{X, \ell_i}^G$. So the number of lines of $\bigcup_{i=1}^n \ell_i^G$ incident with $X$ is $q/2$.
Therefore, since $X$ was an arbitrary external point, $\bigcup_{i=1}^n \ell_i^G$ is a relative hemisystem.
\end{proof}
We remark that Theorem \ref{majortheorem} is similar to the technique developed by Bayens \cite[\textsection 4.4]{MR3167190} to construct hemisystems of higher dimensional Hermitian spaces.
When we are dealing with the Penttila-Williford \index{relative hemisystem}relative hemisystems and perturbations of them, we can condense the criteria given in Theorem \ref{majortheorem} to two sufficient criteria to determine a relative hemisystem.
We state these conditions in the following corollary to Theorem \ref{majortheorem}.
\begin{corollary}
\label{Cosscoroll}
Suppose $\overline{G}$ is a subgroup of $\mathrm{PSO}^-(4,q)$ and $G$ is the intersection of $\overline{G}$ and $\mathrm{P} \Omega ^- (4,q)$.
Further suppose that $\overline{G}_P$ is not contained in $\mathrm{P} \Omega ^-(4,q)$ for all external points $P \in \mathcal{P}_E$.
Then $(G,\overline{G})$ satisfies the conditions given in Theorem \ref{majortheorem} and thus determines a relative hemisystem.\index{relative hemisystem}
\end{corollary}
\begin{proof
First notice that if $\overline{G}_P$ is not contained in $\mathrm{P} \Omega ^-(4,q)$ for all external points $P \in \mathcal{P}_E$, then there exists an element $g \in \overline{G}$ such that $g \notin \mathrm{P} \Omega ^-(4,q)$. So $\overline{G}$ is not contained in $\mathrm{P} \Omega ^- (4,q)$.
We have $|\overline{G}{:} G| = |\overline{G} {:} \overline{G} \cap \mathrm{P} \Omega ^- (4,q)|= |\overline{G} \cdot \mathrm{P} \Omega ^- (4,q){:}\mathrm{P} \Omega ^- (4,q)|= |\mathrm{PSO}^-(4,q){:}\mathrm{P} \Omega ^- (4,q)| = 2$.
Let $\ell \in \mathcal{L}_E$. Now, the discussion at the beginning of Section \ref{magic} implies that for any $\ell \in \mathcal{L}_E$, $\mathrm{PSO}^-(4,q)_{\ell} = \mathrm{P} \Omega ^-(4,q)_{\ell}$.
Thus $\overline{G}_\ell=\mathrm{PSO}^- (4,q)_\ell$ is contained in $\mathrm{P} \Omega ^-(4,q)$. Therefore, $G_\ell = \overline{G}_\ell \cap \mathrm{P} \Omega ^- (4,q)=\overline{G}_\ell$.
Now, since $\overline{G}_P$ is not contained in $\mathrm{P} \Omega ^-(4,q)$, $\overline{G}_P \neq \overline{G}_P \cap \mathrm{P} \Omega ^- (4,q)$. So the stabiliser of $P$ under $\overline{G}$ is not equal to the stabiliser under $G$. By the Orbit-Stabiliser Theorem,
$$\frac{|P^{\overline{G}}|}{|P^G|} = \frac{|\overline{G}{:}\overline{G}_P|}{|G{:}G_P|} = \frac{|\overline{G}{:}G|}{|\overline{G}_P{:}G_P|}.$$
Since $|\overline{G}{:}G| = 2$, we have $|\overline{G}_P{:}G_P| = 2$ and hence $P^{\overline{G}} = P^G.$
Therefore, $(G,\overline{G})$ satisfies the conditions of Theorem \ref{majortheorem}.
\end{proof}
\section{New proofs of the known infinite families}
\label{newproofs}
Let $\psi(x_1,x_2) = x_1^2 + \upsilon^{q+1} x_2^2 + x_1x_2$ be a form with $\upsilon \in \mathrm{GF}(q^2)$ satisfying $\upsilon^q + \upsilon = 1$. Then, $\psi$ is irreducible over $\mathrm{GF}(q)$.
Next, notice that the totally singular points and lines of the form
\begin{equation}
\label{gammaquad}
\mathcal{Q}^+{:} \, x_1^2 + \upsilon^{q+1} x_2^2 +x_1x_2 + x_3x_4
\end{equation}
define a hyperbolic quadric that intersects the Hermitian space defined by the form $x_1x_2^{q} + x_2x_1^q +x_3x_4^q+x_4x_3^q$ over $\mathrm{GF}(q^2)$
in an elliptic quadric. This elliptic quadric's defining equation is simply the equation for $\mathcal{Q}^+$ restricted to $\mathrm{GF}(q)$.
In this section, we will represent lines in array form, as the span of two projective points.
The reguli of the hyperbolic quadric are:
\begin{align*}
\mathscr{R}_1 & = \lbrace \left[ \begin{smallmatrix} \upsilon^q&1 &0&\lambda \\ \lambda \upsilon&\lambda &1&0 \\ \end{smallmatrix} \right]\mid \lambda \in \mathrm{GF}(q^2) \rbrace \cup\lbrace \left[ \begin{smallmatrix} 0&0&0&1 \\ \upsilon&1&0&0\\ \end{smallmatrix} \right] \rbrace,\\
&\\
\mathscr{R}_2 & = \lbrace \left[ \begin{smallmatrix} \upsilon^q&1&\lambda&0 \\ \lambda \upsilon&\lambda &0 &1 \\ \end{smallmatrix} \right] \mid \lambda \in \mathrm{GF}(q^2) \rbrace \cup\lbrace \left[ \begin{smallmatrix} 0&0&1&0 \\ \upsilon&1&0&0\\ \end{smallmatrix} \right]\rbrace.
\end{align*}
\begin{proposition}
The Penttila-Williford family of relative hemisystems, admitting $\mathrm{P}\Omega^-(4,q)$ as an automorphism group for each $q$ even, satisfies Corollary \ref{Cosscoroll}, with associated groups $G = \mathrm{P} \Omega ^-(4,q)$ and $\overline{G} = \mathrm{PSO}^-(4,q)$, which fix $\mathcal{Q}^+$.
\end{proposition}
\begin{proof
\label{pwproof}
Let $\mathrm{H}(3,q^2)$ be the Hermitian space defined by the form $x_1x_2^{q} + x_2x_1^q +x_3x_4^q+x_4x_3^q$ over $\mathrm{GF}(q^2)$. The embedded
symplectic space $\mathrm{W}(3,q)$ is the restriction of the Hermitian form to $\mathrm{GF}(q)$. Recall that $\overline{G} = \mathrm{PSO}^-(4,q)$ is isomorphic to the stabiliser of $\mathcal{Q}^+$, and $G = \mathrm{P} \Omega ^-(4,q)$ is isomorphic to
the stabiliser in $\overline{G}$ of the reguli of $\mathcal{Q}^+$.
Consider $g \in \overline{G}$ defined by
\[
g = \left( \begin{smallmatrix} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&0&1 \\ 0&0&1&0 \\ \end{smallmatrix} \right).
\]
We claim that $g$ does not fix the reguli of the hyperbolic quadric $\mathcal{Q}^+$.
Finding the image of $\mathscr{R}_1$ under $g$ gives us
\[
\left[ \begin{smallmatrix} \upsilon^q&1 &0&\lambda \\ \lambda \upsilon&\lambda&1&0 \\ \end{smallmatrix} \right]^g = \left[ \begin{smallmatrix} \upsilon^q&1&\lambda&0 \\ \lambda \upsilon&\lambda &0 &1 \\ \end{smallmatrix} \right]
\]
for $\lambda \in \mathrm{GF}(q^2)$, and
\[
\left[ \begin{smallmatrix} 0&0&0&1 \\ \upsilon&1&0&0\\ \end{smallmatrix} \right]^g = \left[ \begin{smallmatrix} 0&0&1&0 \\ \upsilon&1&0&0\\ \end{smallmatrix} \right]
\]
which are exactly the lines of $\mathscr{R}_2$. Since $g$ has order two, $g^{-1} = g$ and so $\mathscr{R}_2$ must map to $\mathscr{R}_1$ under the action of $g$.
Therefore, since $G$ stabilises the reguli of the hyperbolic quadric from the beginning of this section, $g \in \overline{G} \setminus G$.
Now, notice that $P_\omega = (\omega, 0, 1,1) \in \mathrm{H}(3,q^2)$ for all $\omega \in \mathrm{GF}(q^2)$, and if we take $\omega \in \mathrm{GF}(q^2) \setminus \mathrm{GF}(q)$, then $P_\omega$ is an external point. Then $P_\omega^g = (\omega,0,1,1)^g = (\omega,0,1,1)$ and therefore $g$ fixes $P_\omega$.
So $g \in \overline{G}_{P_\omega}$, but $g \notin G_{P_\omega}$ because $g \notin G$. Therefore, $\overline{G}_{P_\omega} \neq G_{P_\omega} = \overline{G}_{P_\omega} \cap G$ and $\overline{G}_{P_\omega}$ is not contained
in $G$. Finally, from \cite{penttila2011new}, $G=\mathrm{P} \Omega^-(4,q)$ is transitive on external points. It immediately follows that $\overline{G}=\mathrm{PSO}^-(4,q)$ is transitive on external points as well.
This implies that $\overline{G}_Q$ is not contained in $G$ for all external points $Q \in \mathcal{P}_E$.
Therefore, $(G, \overline{G})$ determine a relative hemisystem for every $q$ even by Corollary \ref{Cosscoroll}, and this relative hemisystem belongs to the Penttila-Williford family of relative hemisystems.
\end{proof}
We now prove that the Cossidente relative hemisystems satisfy the condition in Corollary \ref{Cosscoroll}. We begin by defining collineations $\tau$ and $\phi$ as follows:
\begin{align*}
\tau{:} (x_1,x_2,x_3,x_4) & \mapsto (x_1+ x_2,x_2,x_3,x_4),\\
\phi{:} (x_1,x_2,x_3,x_4) & \mapsto (x_1^q,x_2^q,x_3^q,x_4^q).
\end{align*}
\begin{theorem}
The first family of Cossidente relative hemisystems\index{relative hemisystem} admitting $\mathrm{PSL}(2,q)$ as a setwise stabiliser (described in \cite{MR3081646}) satisfies Corollary \ref{Cosscoroll}. The associated groups are $G = \mathrm{PSL}(2,q) \times \langle \tau\phi \rangle$ and $\overline{G} =
\mathrm{PSL}(2,q) \times \langle \tau, \phi \rangle$.
\end{theorem}
\begin{proof
Let $\mathrm{H}(3,q^2)$ be the Hermitian space in $\mathrm{PG}(3,q^2)$ with defining Gram matrix
\[
H = \left( \begin{smallmatrix} 0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1\\
0 & 0 & 1 & 0 \end{smallmatrix} \right).
\]
Note that this is the Gram matrix for the Hermitian space defined at the beginning of this section.
Let $\mathcal{Q}^+$ be the hyperbolic quadric described in Equation \ref{gammaquad}.
The Baer subspace that contains the symplectic space $\mathrm{W}(3,q)$ and the elliptic quadric $\mathcal{Q}^- = \mathcal{Q}^+ \cap \mathrm{H}(3,q^2)$ consists of points whose coordinates lie solely in $\mathrm{GF}(q)$.
Notice that $\mathrm{H}(3,q^2)$ and the Baer subspace are fixed under both $\tau$ and $\phi$.
Recall from Section \ref{CRHs} that the construction of this family of relative hemisystems stemmed from stabilising a conic of the elliptic quadric $\mathrm{Q}^-(3,q)$
fixed by $\mathrm{P} \Omega^-(4,q)$ in $\mathrm{W}(3,q)$ \cite{MR3081646}. Clearly, $\tau$ and $\phi$ preserve the form defining $\mathcal{Q}^+$ given in Equation \ref{gammaquad}.
Let us consider the application of $\tau$ to the regulus $\mathscr{R}_1$.
\[
\left[ \begin{smallmatrix} 0&0&0&1 \\ \upsilon&1&0&0\\ \end{smallmatrix} \right]^\tau = \left[ \begin{smallmatrix} 0&0&0&1 \\ \upsilon+1&1&0&0\\ \end{smallmatrix} \right]
= \left[ \begin{smallmatrix} 0&0&0&1 \\ \upsilon^q &1&0&0\\ \end{smallmatrix} \right] \in \mathscr{R}_2.
\]
Now, consider the application of $\phi$ to the reguli. For $\mathscr{R}_1$, we have the following:
\begin{align*}
\left[ \begin{smallmatrix} 0&0&0&1 \\ \upsilon&1&0&0\\ \end{smallmatrix} \right]^\phi &= \left[ \begin{smallmatrix} 0&0&0&1 \\ \upsilon^q&1&0&0\\ \end{smallmatrix} \right] \in \mathscr{R}_2.
\end{align*}
Therefore, both $\phi$ and $\tau$ map $\mathscr{R}_1$ to $\mathscr{R}_2$. Since $\phi$ and $\tau$ have order 2, $\mathscr{R}_2$ must map to $\mathscr{R}_1$ under each of $\tau$ and $\phi$.
It follows that their product $\tau\phi$ fixes reguli.
Let $K = \langle \tau, \phi \rangle$, which is isomorphic to $\mathbb{Z}_2 \times \mathbb{Z}_2$.
Now, take the collineation group $J$ isomorphic to $\mathrm{PSL}(2,q)$ that fixes the hyperplane $\pi:x_2=0$. We may represent this as matrices of the form
\[
\left( \begin{smallmatrix}
1&0&0&0\\
\sqrt{bf}+1&1&\sqrt{be}&\sqrt{cf}\\
\sqrt{bc}&0&b&c\\
\sqrt{ef}&0&e&f\end{smallmatrix} \right),
\]
where $b,c,e,f \in \mathrm{GF}(q)$ and $bf+ce=1$.
Define $\overline{G}= J \times K$ and $G = J \times \langle \tau\phi \rangle$.
We claim that these groups satisfy the conditions of Corollary \ref{Cosscoroll}.
Firstly, notice that $G$ is contained in the intersection of $\overline{G}$ and $\mathrm{P} \Omega ^-(4,q)$ because $\mathrm{PSL}(2,q)$ is a subgroup of $\mathrm{P} \Omega ^-(4,q)$ \cite{wilson2009finite},
and $\tau\phi$ fixes the reguli of $\mathcal{Q}^+$, just as $\mathrm{P} \Omega ^-(4,q)$ does. Furthermore, if $g \in \overline{G} \cap \mathrm{P} \Omega ^-(4,q)$, then $g$ must fix the reguli of $\mathcal{Q}^+$, since $\mathrm{P} \Omega ^-(4,q)$ does. Therefore, $g \in G$ and we have shown that $G = \overline{G} \cap \mathrm{P} \Omega ^-(4,q)$.
Furthermore, $\overline{G}$ is not contained in $\mathrm{P} \Omega ^-(4,q)$ because $\tau$ and $\phi$ do not fix the reguli of $\mathcal{Q}^+$. We must now show for all external points $P$ that $\overline{G}_P$ is not contained in $\mathrm{P} \Omega ^-(4,q)$.
Let us first consider the external points that lie on the plane $\pi$. Consider the line $\ell$ satisfying $x_2=x_4=0$. A simple calculation shows that the stabiliser of $\ell$ in $J$ consists only of collineations represented by matrices of the form
\[
M_{b,e}:=
\begin{pmatrix}
1&0&0&0\\
0&1&\sqrt{be}&0\\
0&0&b&0\\
\sqrt{e/b}&0&e&1/b
\end{pmatrix}, \]
where $b,e\in \mathrm{GF}(q)$ and $b \neq 0$. Now if $\xi,\zeta \in\mathrm{GF}(q^2)\backslash\mathrm{GF}(q)$, then $(1,0,\xi,0)$ will be mapped to $(1,0,\zeta,0)$ by $M_{b,e}$ if and only if $b\xi=\zeta$. We can therefore define an equivalence relation on the points which lie on $\ell$ by $(1,0,\xi,0) \sim (1,0,\zeta,0) \Leftrightarrow \zeta = b\xi$, for some $b \in \mathrm{GF}(q) \setminus \{0\}$. In other words, two points on $\ell$ are related if they lie in the same orbit under $J$. Each equivalence class will have size $q-1$, and since the orbits of $J$ partition the external points of $\ell$, we find that $\ell$ must meet $q$ orbits of $J$ on external points.
Note that the stabiliser of $(1,0,\xi,0)$ in $J$ is then $\{M_{1,e} \mid e \in \mathrm{GF}(q) \}$, which is a set of size $q$. By the Orbit-Stabiliser Theorem, the orbit of each point $(1,0,\xi,0)$, with $\xi\notin \mathrm{GF}(q)$, has size $q^2-1$. Since there are $(q^2-q)(q+1)$ external points in $\pi$, we see that the totally isotropic points $(1,0,\xi,0)$ satisfying $\xi\notin \mathrm{GF}(q^2)$ form a complete set of orbit representatives for $J$ acting on the external points in $\pi$.
Also notice that $\phi$ does not fix $\ell$, but $\tau$ does, and therefore $\tau$ lies in $\overline{G}_P$ for $P \in \ell$. Since $\tau$ switches reguli, it follows that $\overline{G}_P \nsubseteq \mathrm{P} \Omega ^-(4,q)$ for all $P \in \ell$, and since these points are orbit representatives for the action of $J$ on external points on $\pi$, $\overline{G}_P \nsubseteq \mathrm{P} \Omega ^-(4,q)$ for all external points $P$ on $\pi$.
Let $\mathcal{C}$ be the intersection of $\mathcal{Q}^+$ with $\pi$, which is a conic.
Now consider the external points which do not lie on $\pi$ or are collinear to a point on the conic $\mathcal{C}$.
We take the line $n$ defined by the span of the point $Q = (1,0,1,0)$, which lies on $\pi$ and the point $R=(0,1,0,1)$. Notice that every point on $n$ except for $Q$ lies outside $\pi$. These points can be written in the form $(u,1,u,1)$, where $u \in \mathrm{GF}(q^2)\setminus \mathrm{GF}(q)$.
In order to compute the stabiliser of each of these points in $J$, we consider
\begin{align*}
(u,1,u,1) \left( \begin{smallmatrix}
1&0&0&0\\
\sqrt{bf}+1&1&\sqrt{be}&\sqrt{cf}\\
\sqrt{bc}&0&b&c\\
\sqrt{ef}&0&e&f\end{smallmatrix} \right) & = \left( (\sqrt{bc}+1)u + \sqrt{bf} +1 + \sqrt{ef}, 1, \sqrt{be} + e + bu, f + \sqrt{cf} + cu \right).
\end{align*}
Now, $f + \sqrt{cf} + cu =1$ and since $u \in \mathrm{GF}(q^2)\setminus \mathrm{GF}(q)$ and $c,f \in \mathrm{GF}(q)$, we must have $c=0$. Therefore, $f=1$. Recall that we also have the relation $bf+ce=1$, which implies that $b=1$. Finally,
$ u = u(1+ \sqrt{bc}) +1 + \sqrt{(b+e)f} = u + 1 + \sqrt{1+e}$ and therefore, $e=0$. Substituting these values of $b,c,e,f$ into the matrix, we have the identity matrix, and therefore the stabiliser of each of the points is trivial.
Furthermore, by the Orbit-Stabiliser theorem, the orbit of each of these points must have size $q(q^2-1)$. Notice that since there are $q^2+1$ points on every line in $\mathrm{H}(3,q^2)$, we have
\[
\sum_{S \in n \setminus\{ Q\}} |S^{J}| = q(q^2-1)\times q^2.
\]
Recall that there are a total of $q(q^2-1)(q^2+1)$ external points in $\mathrm{H}(3,q^2)$, and so only $q(q^2-1)$ external points are not covered by these orbits. These are the external points that are collinear with the conic $\mathcal{C}$. Therefore, the set of points on $n$ (excluding $Q$) form a transversal of the orbits of $J$ on external points which are not collinear with points of the conic $\mathcal{C}$, or lie on $\pi$. Now, consider the following element of $\overline{G}$:
\[
C = \left( \begin{smallmatrix}
1 & 0 & 0 & 0 \\
1 & 1 & 1 & 0 \\
0 & 0 & 1 & 0\\
1 & 0 & 1 & 1 \end{smallmatrix} \right)\\
= \left( \begin{smallmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 1 & 0 \\
0 & 0 & 1 & 0\\
1 & 0 & 1 & 1\\
\end{smallmatrix} \right)\tau\\
=D\tau. \\
\]
Notice that $D \in J$ and $C$ fixes all of the points $(u,1,u,1)$ on $n \setminus \{ Q \}$. But $C$ is not in $G$ because it swaps the reguli of $\mathcal{Q}^+$ (it is the product of an element of $J$ which fixes reguli, and $\tau$ which swaps them). Therefore, for all external points $P$ which do not lie on $\pi$ and are not collinear to any points on $\mathcal{C}$, $|\overline{G}_P : G_P| \neq 1$ and so $\overline{G}_P \nsubseteq \mathrm{P} \Omega ^-(4,q)$.
Finally, consider the $q(q^2-1)$ points which are collinear with some point of the conic $\mathcal{C}$. Let $m$ be the line spanned by $R = (1,0,0,0)$, which is the nucleus of the conic, and the point $(0,1,0,\gamma)$, where $\gamma \in \mathrm{GF}(q^2) \setminus \mathrm{GF}(q)$ such that $\gamma^q + \gamma = 1$. Define $W$ to be the set of $q$ external points incident with $\ell$. These can be written as $W= \{(0,1,0,\gamma) \} \cup \{ (1,v,0, \gamma v) : v\in \mathrm{GF}(q) \setminus \{ 0\} \}$. We now calculate the stabiliser of a point in $W$:
\begin{align*}
(1,v,0, \gamma v) \left( \begin{smallmatrix}
1&0&0&0\\
\sqrt{bf}+1&1&\sqrt{be}&\sqrt{cf}\\
\sqrt{bc}&0&b&c\\
\sqrt{ef}&0&e&f\end{smallmatrix} \right) & = \left(1+v(\sqrt{bf}+1) + \gamma v\sqrt{ef}, v, v\sqrt{be}+\gamma e v, v+\sqrt{cf} + \gamma fv\right).
\end{align*}
A simple calculation shows that in order for this to be equal to $ (1,v,0, \gamma v)$, we must have $c=0, e=0, f=1, b=1$. Substituting these values in gives us the identity matrix, and so no orbit has fixed points under $J$. So the size of $W^{J} = q |J| = q^2(q^2-1)$.
Therefore, the points in $W$ form a transversal of the orbits of $J$ on external points that are collinear with a point of $\mathcal{C}$ . Consider the following element $A$ of $\overline{G}$:
\[
A = \left( \begin{smallmatrix}
1 & 0 & 0 & 0 \\
1 & 1 & 0 & 1 \\
1 & 0 & 1 & 1\\
0 & 0 & 0 & 1 \end{smallmatrix} \right)\phi = B \phi.
\]
Clearly, $B\in J$, and so $A$ switches the reguli of $\mathcal{Q}^+$. Let us consider the action of $A$ on points of $W$.
\[
(1,v,0, \gamma v) \left( \begin{smallmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 1 \\
1 & 0 & 1 & 1\\
0 & 0 & 0 & 1 \end{smallmatrix} \right)\phi
= (1, v, 0, v(\gamma^q +1) )\\
= (1,v,0,\gamma v).
\]
Likewise,
\[
(0,1,0, \gamma) \left( \begin{smallmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 1 \\
1 & 0 & 1 & 1\\
0 & 0 & 0 & 1 \end{smallmatrix} \right)\phi = (0, 1, 0, 1 + \gamma)^\phi
= (0,1,0,\gamma).
\]
Therefore, $A$ fixes $W$ pointwise, and so is contained in $\overline{G}_P$ for all $P \in W$. But since $A$ switches the reguli of $\mathcal{Q}^+$, it cannot lie in $\mathrm{P} \Omega ^-(4,q)$, and so neither can $\overline{G}_P$ for all external points $P$ collinear with a point of the conic $\mathcal{C}$. Therefore, $G$ and $\overline{G}$ satisfy the conditions of Corollary \ref{Cosscoroll} and hence determine a relative hemisystem.
\end{proof}
\begin{remark}\label{remark:Cossidente}
We remark that although Cossidente describes one infinite family of relative hemisystems using his construction in \cite{MR3081646}, there are actually several more inequivalent infinite families of relative hemisystems that
arise from this construction. For $q=16$, we found by computer that there are five inequivalent relative hemisystems that admit $\mathrm{PSL}(2,q)$, and we conjecture that the number of inequivalent examples admitting this group increases with $q$.
\end{remark}
Cossidente's second family of relative hemisystems, admitting a group of order $q^2(q+1)$, for each $q$ even, described in \cite{cossidente2013new}, also satisfy the conditions of Corollary \ref{Cosscoroll}. To prove this, we first provide a concrete construction
of this family of relative hemisystems. As before, we consider the Hermitian space $\mathrm{H}(3,q^2)$ defined by the form $x_1x_2^{q} + x_2x_1^q +x_3x_4^q+x_4x_3^q$ over $\mathrm{GF}(q^2)$, with the
embedded symplectic space $\mathrm{W}(3,q)$ defined as the restriction of the form to $\mathrm{GF}(q)$.
We explicitly define the following two hyperbolic quadrics.
\begin{equation}
\label{hypquad1}
Q_1^+(3,q^2): \alpha x_1^2 + \beta x_2^2 + x_1x_2 + x_3x_4,
\end{equation}
\begin{equation}
\label{hypquad2}
Q_2^+(3,q^2): \beta x_1^2 + \alpha x_2^2 + x_1x_2 + x_3x_4
\end{equation}
where $\alpha \in \mathbb{F}_{q^2}$, $\beta = \alpha+1$ and $\alpha + \alpha^q + 1 = 0$.
The reguli of $Q_1^+(3,q^2)$ are as follows:
\begin{align*}
\tilde{\mathscr{R}_1} & = \lbrace \left[ \begin{smallmatrix} \alpha&\alpha&0&\lambda\sqrt{\alpha} \\ \lambda\alpha&\lambda\beta&\sqrt{\alpha}&0 \\ \end{smallmatrix} \right] \mid \lambda \in \mathrm{GF}(q^2) \rbrace \cup \lbrace \left[ \begin{smallmatrix} 0&0&0&1\\ \alpha & \beta & 0 & 0\\ \end{smallmatrix} \right]\rbrace\\
&\\
\tilde{\mathscr{R}_2} & = \lbrace \left[ \begin{smallmatrix} \alpha&\alpha &\lambda\sqrt{\alpha}&0 \\ \lambda\alpha&\lambda \beta&0&\sqrt{\alpha} \\ \end{smallmatrix} \right] \mid \lambda \in \mathrm{GF}(q^2) \rbrace \cup \lbrace \left[ \begin{smallmatrix} 0&0&1&0\\ \alpha & \beta & 0 & 0\\ \end{smallmatrix} \right]\rbrace.\\
\end{align*}
\begin{theorem}
Suppose $\overline{M}$ is the stabiliser in $\mathrm{PGU}(4,q)$ of the two hyperbolic quadrics $Q_1^+(3,q^2)$ and $Q_2^+(3,q^2)$ described above. Now, let $M$ be the stabiliser in $\overline{M}$ of a class
of reguli in $Q_1^+(3,q^2)$.
Then $M$ is the group admitted by Cossidente's second family of relative hemisystems, and $M$ and $\overline{M}$ satisfy Corollary \ref{Cosscoroll}.
\end{theorem}
\begin{proof}
Firstly, we claim that $\overline{M} = M :Z$, where $Z$ is the group generated by the involution $z$ defined by $(x_1, x_2, x_3, x_4) \mapsto (x_2^q, x_1^q, x_4^q, x_3^q)$. Firstly, notice that $|\overline{M} : M| = 2$, so $M$ is a normal subgroup of $\overline{M}$.
The action on lines of $\tilde{\mathscr{R}}_1$ (for instance) is permutation isomorphic to the action on a projective line $\mathrm{PG}(1,q^2)$. Now, $M$ fixes two
lines, say $\ell_1$ and $\ell_2$, which are in the intersection of $Q_1^+(3,q^2)$ and $Q_2^+(3,q^2)$, with $\ell_1 \in \tilde{\mathscr{R}}_1$ and $\ell_2 \in \tilde{\mathscr{R}}_2$.
The group $M$ fixing $\ell_1$ is permutation isomorphic to $\mathrm{AGL}(1,q^2)$ fixing a point on $\mathrm{PG}(1,q^2)$. Now, $\mathrm{AGL}(1,q^2)$ is transitive on the remaining points of $\mathrm{PG}(1,q^2)$ \cite[\textsection 7.7]{MR1409812},
and so $M$ is transitive on $\tilde{\mathscr{R}_1} \setminus \{ \ell_1\}$. Therefore, to show that $z$ maps $\tilde{\mathscr{R}}_1$ to $\tilde{\mathscr{R}}_2$,
it is sufficient to prove it for $\ell_1$ and another line of $\tilde{\mathscr{R}}_1$. Let $\ell_{\infty , 1} = \left[ \begin{smallmatrix} 0&0&0&1\\ \alpha & \beta & 0 & 0\\ \end{smallmatrix} \right]$.
Now,
\[
\ell_1^z = \left[ \begin{smallmatrix} 1 & 1 &0&0 \\ 0&0&1&0\\ \end{smallmatrix} \right]^q \left( \begin{smallmatrix}
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1\\
0 & 0 & 1 & 0 \end{smallmatrix} \right)
= \left[ \begin{smallmatrix} 1 & 1 &0&0 \\ 0&0&0&1\\ \end{smallmatrix} \right] \in \tilde{\mathscr{R}}_2,
\]
\[
\ell_{\infty , 1}^z = \left[ \begin{smallmatrix} 0&0&0&1\\ \alpha & \beta & 0 & 0\\ \end{smallmatrix} \right]^q \left( \begin{smallmatrix}
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1\\
0 & 0 & 1 & 0 \end{smallmatrix} \right)
= \left[ \begin{smallmatrix} 0&0&1&0\\ \alpha & \beta & 0 & 0\\ \end{smallmatrix} \right] \in \tilde{\mathscr{R}}_2.
\]
A similar argument yields that the image of any line in $\tilde{\mathscr{R}}_2$ under $z$ is contained in $\tilde{\mathscr{R}}_1$.
Therefore, the involution $z$ switches the reguli of $Q_1^+(3,q^2)$ and so $M \cap Z = 1$. Since $z$ stabilises the two hyperbolic quadrics $Q_1^+$ and $Q_2^+$, we have $Z \leqslant M$, and therefore $\overline{M}= M :Z$.
Let $D_1$ be the subgroup of $\mathrm{P} \Gamma \mathrm{U}(4,q)_{Q_1^+}$ that stabilises two families of reguli on $Q_1^+$.
Similarly, let $D_2$ be the subgroup of $\mathrm{P} \Gamma \mathrm{U}(4,q)_{Q_2^+}$ that stabilises two families of reguli on $Q_2^+$.
We now prove that the orbits of $\overline{M}$ and $M$ on external points are identical.
Since $\overline{M} = M :Z$, it is sufficient to prove that for all $x \in \mathcal{P}_E, x^z \in x^M$.
Since $D_1$ is transitive on $\mathcal{P}_E$, this is equivalent to showing that
for all $g\in D_1$, we have $(P_0^g)^z \in (P_0^g)^M$ for all $P_0 \in \mathcal{P}_E$.
We claim that given a point $P_0 \in \mathcal{P}_E$, we have $P_0^z = P_0^m$ and $P_0 = (P_0^z)^z = (P_0^m)^m$ for some $m \in M$.
Since $D_1$ acts transitively on $\mathcal{P}_E$, it is sufficient for us to prove this claim for a specific point.
Let $P_0 = (1,0,1,0)$. Then we have the following
$$ P_0^z = (1,0,1,0)^q
\left( \begin{smallmatrix}
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1\\
0 & 0 & 1 & 0 \end{smallmatrix} \right) = (0,1,0,1).$$
Now, we must find an involution $m \in M$ such that $P_0^m = P_0^z = (0,1,0,1)$. This is equivalent to finding $m \in M$ such that $P_0^{zm^{-1}} = P_0^{zm}=P_0$. Take $m$ to be the following collineation:
\[
m = \phi \left( \begin{smallmatrix}
1 & 0 & 1 & 1 \\
0 & 1 & 1 & 1 \\
1 & 1 & 1 & 0\\
1 & 1 & 0 & 1 \end{smallmatrix} \right)
\]
where $\phi$ is the automorphism $x \mapsto x^q$. This collineation has the required property and so it only remains to show that $m \in M$.
To show that $m \in M$, we must show that it is an involution and that it fixes the two hyperbolic quadrics that define $M$ and also each of the reguli in the intersection of the two hyperbolic quadrics.
Firstly recalling that we are working with a field with characteristic 2,
\[
m^2 = \left( \begin{smallmatrix}
1 & 0 & 1 & 1 \\
0 & 1 & 1 & 1 \\
1 & 1 & 1 & 0\\
1 & 1 & 0 & 1 \end{smallmatrix} \right) \left( \begin{smallmatrix}
1 & 0 & 1 & 1 \\
0 & 1 & 1 & 1 \\
1 & 1 & 1 & 0\\
1 & 1 & 0 & 1 \end{smallmatrix} \right) = I_4.
\]
Secondly, we must show that $m$ fixes the two hyperbolic quadrics $Q^+_1(3,q^2)$ and $Q_2^+(3,q^2)$.
Recall the defining quadratic forms of the two hyperbolic quadrics $Q_1^+(3,q^2)$ and $Q_2^+(3,q^2)$ given in Equations \ref{hypquad1} and \ref{hypquad2} respectively.
The image of any point in $Q_1^+(3,q^2)$ under $m$ lies in $Q_1^+(3,q^2)$, so
$m$ fixes $Q^+_1(3,q^2)$, and by symmetry, it fixes $Q_2^+(3,q^2)$ as well.
Finally, we show that $m$ fixes the reguli of $Q_1^+(3,q^2)$. Again, we only need to test two lines from each regulus -- the line that is fixed by $M$ and another line.
For $\ell_1, \ell_{\infty , 1} \in \tilde{\mathscr{R}}_1$:
\[
\ell_1^m = \left[ \begin{smallmatrix} 1 & 1 &0&0 \\ 0&0&1&0\\ \end{smallmatrix} \right]^q
\left( \begin{smallmatrix}
1 & 0 & 1 & 1 \\
0 & 1 & 1 & 1 \\
1 & 1 & 1 & 0\\
1 & 1 & 0 & 1 \end{smallmatrix} \right)
= \left[ \begin{smallmatrix} 1 & 1 &0&0 \\ 1&1&1&0\\ \end{smallmatrix} \right] = \left[ \begin{smallmatrix} 1 & 1 &0&0 \\ 0&0&1&0\\ \end{smallmatrix} \right]\in \tilde{\mathscr{R}}_1,
\]
\[
\ell_{\infty , 1}^m = \left[ \begin{smallmatrix} 0&0&0&1\\ \alpha & \beta & 0 & 0\\ \end{smallmatrix} \right]^q \left( \begin{smallmatrix}
1 & 0 & 1 & 1 \\
0 & 1 & 1 & 1 \\
1 & 1 & 1 & 0\\
1 & 1 & 0 & 1 \end{smallmatrix} \right)
= \left[ \begin{smallmatrix} 1&1&0&1\\ \beta & \alpha & 0 & 0\\ \end{smallmatrix} \right]
= \left[ \begin{smallmatrix} \sqrt{\alpha}&\sqrt{\alpha}&0&\sqrt{\alpha}\\ \alpha & \beta & 1 & 0\\ \end{smallmatrix} \right] \in \tilde{\mathscr{R}}_1.
\]
Using a similar argument, $m$ fixes the reguli of $\tilde{\mathscr{R}}_2$ as well.
Therefore, $m$ preserves the reguli and so $m\in M$.
Now that we have proved the claim, we continue with the proof as before. We have $P_0^{gz} \in P_0^{gM}$ if and only if $P_0^{gzg^{-1}} \in P_0^{gMg^{-1}}$ for all $g \in D_1$.
This is equivalent to $P_0$ having identical orbits under $\overline{M}^g$ and $M^g$ for all $g \in D_1$. By the Orbit-Stabiliser Theorem and since $|\overline{M} {:} M| = 2$, it follows that $| \overline{M}^g_{P_0}| = 2|M^g_{P_0}|$.
Finally, this holds if and only if $|\overline{M}_{P_0}| = 2|M_{P_0}|$, which is true by the claim proven above. Since $M_{P_0} = \overline{M}_{P_0} \cap \mathrm{P}\Omega^-(4,q)$, we have shown that $\overline{M}_{P_0} \nsubseteq \mathrm{P}\Omega^-(4,q)$ and
since $D_1$ is transitive, this holds for all external points $P\in \mathcal{P}_E$.
Therefore, by Corollary \ref{Cosscoroll}, $M$ and $\overline{M}$ determine a relative hemisystem.
\end{proof}
Interestingly, we have found by computation in GAP \index{GAP}\cite{GAP4} that the relative hemisystem on $\mathrm{H}(3,q^2)$ arising from a Suzuki-Tits ovoid does not satisfy the criteria for Theorem \ref{majortheorem}.
We leave as an open question whether there are more relative hemisystems on $\mathrm{H}(3,q^2)$ fitting the criteria given in Theorem \ref{majortheorem}.
\section{A classification of the relative hemisystems on $\mathrm{H}(3,4^2)$}
Using a function written in GAP \cite{GAP4} and interfacing with Gurobi \cite{gurobi}, we were able to easily enumerate all of the relative hemisystems on the Hermitian space $\mathrm{H}(3,4^2)$. This result was previously unknown, or at least unmentioned in the literature.
\begin{proposition}
There are 240 examples of relative hemisystems on the Hermitian space $\mathrm{H}(3,4^2)$, all of which are equivalent to the Penttila-Williford example on that Hermitian space.
\end{proposition}
Unfortunately, using the same approach does not allow us to classify all of the relative hemisystems on $\mathrm{H}(3, q^2)$ for $q \geqslant 8$ because the numbers of external points and external lines are significantly larger, making the computational problem much harder to solve. However, we were able to enumerate all of the relative hemisystems on $\mathrm{H}(3,8^2)$ with certain symmetry hypotheses.
We generated a computation tree using a branching technique outlined in \cite{Martis}, which essentially uses a partially ordered set of orbit representatives on $k$-tuples (closed under taking subsets) to decrease the search space.
This reduced the number of equivalent relative hemisystems found during computation, and made the search for relative hemisystems significantly more efficient.
\begin{proposition}
A relative hemisystem of $\mathrm{H}(3,8^2)$ is equivalent to one of the four known examples, or it has a trivial stabiliser.
\end{proposition}
\section*{Acknowledgements}
The authors would like to express their thanks to Prof. Gordon Royle for his assistance in computation for this paper, and Dr. Angela Aguglia for her insight into intersections of hyperbolic
quadrics with Hermitian spaces. The first author acknowledges the support of the Australian
Research Council Future Fellowship FT120100036. The second author acknowledges the support of a Hackett Postgraduate Research Scholarship. The third author acknowledges the support of the
Australian Research Council Discovery Grant DP120101336.
\nocite{fining}
\bibliographystyle{abbrv}
|
{
"timestamp": "2015-09-29T02:20:45",
"yymm": "1504",
"arxiv_id": "1504.03435",
"language": "en",
"url": "https://arxiv.org/abs/1504.03435"
}
|
\section{Introduction}
A weighing matrix of order $n$ and weight $k$ is an
$n \times n$ $(1,-1,0)$-matrix $W$ such that
$W W^T=kI_n$, where $I_n$ is
the identity matrix of order $n$ and $W^T$ denotes the transpose of $W$.
A weighing matrix of order $n$ and weight $n$
is a Hadamard matrix.
Two weighing matrices $W_1,W_2$ of order $n$ and weight $k$ are said to be
{\em unbiased} if $(1/\sqrt{k})W_1 W_2^{T}$ is also
a weighing matrix of order $n$ and weight $k$~\cite{HKO} (see also~\cite{BKR}).
Unbiased weighing matrices of order $n$ and weight $n$
are unbiased Hadamard matrices (see~\cite{HKO}).
Weighing (Hadamard) matrices $W_1,W_2,\ldots,W_f$ are said to be
{\em mutually unbiased}
if any distinct two of them are unbiased.
Generalizing the above concept, recently the concept ``quasi-unbiased'' for weighing matrices
has been introduced by Nozaki and the second author~\cite{NSpre}.
Namely, two weighing matrices $W_1,W_2$ of order $n$ and weight $k$ are said to be
{\em quasi-unbiased for parameters $(n,k,l,a)$} if
$(1/\sqrt{a}) W_1 W_2^T$ is a weighing matrix of weight $l$.
It follows from the definition that $l=k^2/a$.
In addition, weighing matrices $W_1,W_2,\ldots,W_f$ are said to be
{\em mutually quasi-unbiased weighing matrices
for parameters $(n,k,l,a)$}
if any distinct two of them are quasi-unbiased for parameters $(n,k,l,a)$.
Mutually quasi-unbiased weighing matrices were defined
from the viewpoint of a connection with spherical codes~\cite{NSpre}.
This notion was introduced to show that Conjecture~23 in~\cite{BKR}
is true.
Only quasi-unbiased weighing matrices are previously known
for parameters $(n,n,n/2,2n)$, where $n=2^{2k+1}$ and $k$ is a positive
integer~\cite[Section~4]{BKR} and \cite[Section~4]{NSpre},
and for parameters $(n,2,4,1)$, where $n$ is an even positive
integer with $n \ge 4$~\cite[Section~3]{NSpre}.
Suppose that $n=2^m$, where $m$ is an integer with $m \ge 2$.
Let $C$ be a binary $[n,k]$ code satisfying the following two conditions:
\begin{align}
\label{eq:C1}
&\{i \in \{0,1,\ldots,n\}\mid A_i(C) \ne 0\}=\{0,n/2\pm a,n/2,n\}, \\
\label{eq:C2}
&\text{$C$ contains the first order Reed--Muller code $RM(1,m)$ as a
subcode},
\end{align}
where $A_i(C)$ denotes the number of codewords of weight $i$ in $C$,
and $a$ is a positive integer with $0<a<n/2$.
Then it follows from~\cite[Proposition~2.3 and Lemma~4.2]{NSpre}
that $C$ constructs a set of $2^{k-m-1}$ mutually quasi-unbiased
weighing matrices for parameters $(n,n,(n/2a)^2,4a^2)$.
In this note, we study binary $[2^m,k]$ codes satisfying the two
conditions~\eqref{eq:C1} and~\eqref{eq:C2}.
The weight distribution of the above code
is determined using an integer $a$ given in~\eqref{eq:C1}.
We give a classification of binary codes $C$
satisfying the two conditions~\eqref{eq:C1} and~\eqref{eq:C2}
for lengths $8,16$.
We also give a classification of
binary maximal codes $C$ (with respect to the subspace relation)
satisfying the two conditions~\eqref{eq:C1} and~\eqref{eq:C2}
for length $32$.
As an application, sets of $8$ mutually quasi-unbiased
weighing matrices for parameters $(16,16,4,64)$
and $4$
mutually quasi-unbiased weighing matrices for parameters $(32,32,4,256)$
are constructed for the first time.
All computer calculations in this note
were done by {\sc Magma}~\cite{Magma}.
\section{Mutually quasi-unbiased weighing matrices and codes} \label{sec:2}
We begin with definitions on codes used throughout this note.
A binary $[n,k]$ {\em code} $C$ is a $k$-dimensional vector subspace
of $\mathbb{F}_2^n$,
where $\mathbb{F}_2$ denotes the finite field of order $2$.
All codes in this note are binary.
A $k \times n$ matrix whose rows form a basis of
$C$ is called a {\em generator matrix} of $C$.
The parameters $n$ and $k$ are called the {\em length} and the {\em dimension} of $C$,
respectively.
For a vector $x=(x_1,\ldots,x_n)$,
the set $\{i \mid x_i \ne 0\}$ is called
the {\em support} of $x$.
The {\em weight} $\wt(x)$ of a vector $x$ is
the number of non-zero components of $x$.
The minimum non-zero weight of all codewords in $C$ is called
the {\em minimum weight} of $C$, which is denoted by $d(C)$.
Two codes $C$ and $C'$ are {\em equivalent}
if one can be
obtained from the other by permuting the coordinates.
A code $C$ is {\em doubly even} (resp.\ {\em triply even})
if all codewords of $C$ have weight divisible by four (resp.\ eight).
The \textit{dual code} $C^{\perp}$ of a code
$C$ of length $n$ is defined as
$
C^{\perp}=
\{x \in \mathbb{F}_2^n \mid x \cdot y = 0 \text{ for all } y \in C\},
$
where $x \cdot y$ is the standard inner product.
A code $C$ is called
{\em self-orthogonal} (resp.\ {\em self-dual}) if
$C \subset C^{\perp}$ (resp.\ $C = C^{\perp}$).
A {\em covering radius} $\rho(C)$ of $C$ is
$
\rho(C)=\max_{x\in\mathbb{F}_2^n}\min_{c\in C}\wt(x-c).
$
The {\em first order Reed--Muller codes $RM(1,m)$} for all positive integer $m$ are defined recursively by
\begin{align*}
RM(1,1)&=\mathbb{F}_2^2,\\
RM(1,m)&=\{(u,u),(u,u+{\bf 1}) \in \mathbb{F}_2^{2^m}\mid u\in RM(1,m-1)\} \text{ for } m>1,
\end{align*}
where $\bf{1}$ denotes the all-one vector of suitable length.
Mutually quasi-unbiased weighing matrices
for parameters $(n,n,(n/2a)^2,4a^2)$ are constructed from
$[n,k]$ codes $C$
satisfying the two conditions~\eqref{eq:C1} and~\eqref{eq:C2},
where $n=2^m$ and $m$ is a positive integer as
follows~\cite[Proposition~2.3 and Lemma~4.2]{NSpre}.
Define $\psi$ as a map from $\mathbb{F}_2^n$ to
$\{\pm1\}^n$ $(\subset \mathbb{Z}^n)$ by
$\psi((x_i)_{i=1}^n)=(\alpha_i)_{i=1}^n$, where $\alpha_i=-1$ if $x_i=1$ and $\alpha_i=1$ if $x_i=0$.
Note that $\wt(x-y)=j$ if and only if the standard inner product of
$\psi(x)$ and $\psi(y)$ is $n-2j$ for $x,y\in \mathbb{F}_2^n$.
Let $\{u_1,u_2,\ldots,u_{2^{k-m-1}}\}$ be a set of complete representatives of $C/RM(1,m)$.
Since $\{i\in \{0,1,\ldots,n\} \mid A_i(RM(1,m)) \ne 0\}=\{0,n/2,n\}$,
$\psi(u_i+RM(1,m))$ is antipodal, that is, $-\psi(u_i+RM(1,m))=\psi(u_i+RM(1,m))$.
Hence, there exists a subset $X_i$ of $\psi(u_i+RM(1,m))$
such that $X_i\cup(-X_i)=\psi(u_i+RM(1,m))$ and $X_i\cap(-X_i)=\emptyset$.
For $1\leq i\leq 2^{k-m-1}$, define $H_i$ to be an $n\times n$
$(1,-1)$-matrix whose rows consist of the vectors of $X_i$.
Any two different vectors in $X_i$ are orthogonal
for $1\leq i\leq 2^{k-m-1}$, which means that the matrix $H_i$ is a Hadamard matrix for $1\leq i\leq 2^{k-m-1}$.
Let $x_i$ be a vector in $X_i$.
The assumption of~\eqref{eq:C1}
implies that $\wt(\psi^{-1}(x_i)-\psi^{-1}(x_j))=n/2,n/2\pm a$ $(i \ne j)$,
namely,
the inner product of $x_i$ and $x_j$ $(i \ne j)$ is $0,\mp 2a$ respectively, where $a$ is the integer given in~\eqref{eq:C1}.
This shows that for any distinct $i,j\in \{1,2,\ldots, 2^{k-m-1}\}$,
$(1/2a)H_i H_j^T$ is a $(1,-1,0)$-matrix and thus it is a
weighing matrix of weight $(n/2a)^2$.
Therefore, Hadamard matrices $H_1,H_2,\ldots,H_{2^{k-m-1}}$ are mutually quasi-unbiased weighing matrices for parameters $(n,n,(n/2a)^2,4a^2)$.
\begin{remark}\label{rem}
Since $n/2a$ must be an integer, $a$ is a divisor of $2^{m-1}$.
\end{remark}
\begin{proposition} \label{prop:WD}
Suppose that $n=2^m$, where $m$ is an integer with $m \ge 2$.
Let $C$ be an $[n,k]$ code
satisfying the two conditions~\eqref{eq:C1} and~\eqref{eq:C2}.
Then the weight distribution of $C$ is given by
\begin{align*}
(&A_0(C),A_{n/2-a}(C),A_{n/2}(C),A_{n/2+a}(C),A_n(C))\\&=(1,(2^{k-m-1}-1)l,2n-2+(2^{k-m-1}-1)(2n-2l),(2^{k-m-1}-1)l,1),
\end{align*}
where $l=(n/2a)^2$.
\end{proposition}
\begin{proof}
We denote the set of complete representatives of $C/RM(1,m)$
by $\{u_1,u_2,\ldots,u_{2^{k-m-1}}\}$ described as above, where
we assume that $u_1=\boldsymbol{0}$.
In addition, we denote the mutually quasi-unbiased weighing
matrices for parameters $(n,n,(n/2a)^2,4a^2)$ by $H_1,H_2,\ldots,H_{2^{k-m-1}}$
described as above.
Since $(1/2a)H_1 H_i^T$ is a weighing matrix of weight $l$ for
$1<i\leq 2^{k-m-1}$, $0$ appears $n-l$ times
in the first row of $(1/2a)H_1 H_i^T$.
Since the first row of $H_1$ is the all-one vector, this implies that the number of codewords of weight $n/2$ in $u_i+RM(1,m)$ for $i>1$ is $2n-2l$.
Thus, $A_{n/2}(C)=2n-2+(2^{k-m-1}-1)(2n-2l)$ holds.
Since $C$ contains the all-one vector, we have the desired weight distribution.
\end{proof}
\begin{remark}
The minimum weight of $C$ determines the weight distribution
of $C$.
Indeed, the minimum weight determines $a$, and thus $l$. Since $k$ and $m$ are given, the weight distribution is determined.
\end{remark}
\section{Codes satisfying the conditions~\eqref{eq:C1} and~\eqref{eq:C2}}
In this section,
we give a classification of codes $C$ of length $2^m$
satisfying the two conditions~\eqref{eq:C1} and~\eqref{eq:C2}
for $m=3,4$.
We also give a classification of
maximal codes $C$ of length $32$
satisfying the two conditions~\eqref{eq:C1} and~\eqref{eq:C2}.
\subsection{Length 8}
The case $m=3$ is somewhat trivial, but
we only give the result for the sake of completeness.
Note that $RM(1,3)$ is equivalent to the extended Hamming $[8,4,4]$ code
$e_8$.
The complete coset weight distribution of $e_8$ is listed
in~\cite[Example~1.11.7]{Huffman-Pless}.
From~\cite[Example~1.11.7]{Huffman-Pless},
$RM(1,3)$ has
seven (nontrivial) cosets of minimum weight $2$.
In addition, every $[8,5]$ code $C$ satisfying
the conditions~\eqref{eq:C1} and~\eqref{eq:C2}
can be constructed as $\langle RM(1,3), x \rangle$, where $x$ is
a coset leader of the seven cosets.
We verified by {\sc Magma} that
there exists a unique $[8,5]$ code $C_{8,5}$
satisfying the conditions~\eqref{eq:C1} and~\eqref{eq:C2}.
This was done by the {\sc Magma} function {\tt IsIsomorphic}.
Similarly, we verified by {\sc Magma} that
$C_{8,5}$ has
three (nontrivial) cosets of minimum weight $2$, and
there exists a unique $[8,6]$ code $C_{8,6}$
satisfying the conditions~\eqref{eq:C1} and~\eqref{eq:C2}.
It is trivial that the even weight $[8,7]$ code $C_{8,7}$ is the unique
$[8,7]$ code satisfying the conditions~\eqref{eq:C1} and~\eqref{eq:C2}.
We remark that
$\{i\in \{0,1,\ldots,8\} \mid A_i(C) \ne 0\}=\{0,4\pm 2,4,8\}$
for $C=C_{8,i}$ $(i=5,6,7)$.
\subsection{Length 16}
The next case is $m=4$.
First we fix the generator matrix of the first order Reed--Muller
$[16,5,8]$ code $RM(1,4)$ as follows:
\[
\left(
\begin{array}{c}
1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 \\
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 \\
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 \\
0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 \\
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 \\
\end{array}
\right).
\]
Every $[16,6]$ code $C$ satisfying the conditions~\eqref{eq:C1} and~\eqref{eq:C2}
can be constructed as $\langle RM(1,4), x \rangle$, where $x$ is an element
of a set of complete representatives of $\mathbb{F}_2^{16}/RM(1,4)$,
satisfying that $x+RM(1,4)$ has minimum weight $4$ or
$6$ (see Remark~\ref{rem}).
In this way,
we found all $[16,6]$ code $C$ satisfying the
conditions~\eqref{eq:C1} and~\eqref{eq:C2}, which must be checked further for equivalences
to complete the classification.
We verified by {\sc Magma} that any $[16,6]$ code
satisfying the conditions~\eqref{eq:C1} and~\eqref{eq:C2}
is equivalent to one of the two inequivalent codes
$C_{16,6,1}$ and $C_{16,6,2}$.
This was done by the {\sc Magma} function {\tt IsIsomorphic}.
The minimum weights $d(C)$ and the constructions
of the two codes $C$ are listed in Table~\ref{Tab:16C}.
This table means that $C_{16,6,1}$ and $C_{16,6,2}$
can be constructed as
$\langle RM(1,4), x_{16,6,1} \rangle$ and
$\langle RM(1,4), x_{16,6,2} \rangle$, respectively,
where the supports of $x_{16,6,1}$ and $x_{16,6,2}$ are
listed in Table~\ref{Tab:16V}.
\begin{table}[thb]
\caption{$[16,k]$ codes satisfying~\eqref{eq:C1} and~\eqref{eq:C2}}
\label{Tab:16C}
\begin{center}
{\small
\begin{tabular}{c|c|c|l}
\noalign{\hrule height0.8pt}
$k$ & Codes $C$ & $d(C)$ & \multicolumn{1}{c}{Vectors} \\
\hline
6&$C_{16,6,1}$ &6& $x_{16,6,1}$ \\%&1& 0& 16& 30& 16& 0& 1\\
&$C_{16,6,2}$ &4& $x_{16,6,2}$ \\%&1& 4& 0& 54& 0& 4& 1\\
7&$C_{16,7,1}$ &6& $x_{16,6,1}$, $x_{16,7,1}$ \\%&1& 0& 48& 30& 48& 0& 1\\
&$C_{16,7,2}$ &4& $x_{16,6,2}$, $x_{16,7,2}$ \\%&1& 12& 0& 102& 0& 12& 1\\
8&$C_{16,8,1}$ &4& $x_{16,6,2}$, $x_{16,7,2}$, $x_{16,18,1}$ \\
&$C_{16,8,2}$ &4& $x_{16,6,2}$, $x_{16,7,2}$, $x_{16,18,2}$ \\
\noalign{\hrule height0.8pt}
\end{tabular}
}
\end{center}
\end{table}
\begin{table}[thb]
\caption{Vectors in Table~\ref{Tab:16C}}
\label{Tab:16V}
\begin{center}
{\small
\begin{tabular}{c|l|c|l}
\noalign{\hrule height0.8pt}
& \multicolumn{1}{c|}{Supports} &
& \multicolumn{1}{c}{Supports} \\
\hline
$x_{16,6,1}$ & $\{1,8,12,14,15,16\}$&$x_{16,7,2}$ & $\{1,8,10,15\}$\\
$x_{16,6,2}$ & $\{1,2,15,16\}$ &$x_{16,8,1}$ & $\{2,3,13,16\}$\\
$x_{16,7,1}$ & $\{1,4,5,7,9,10\}$ &$x_{16,8,2}$ & $\{4,5,12,13\}$\\
\noalign{\hrule height0.8pt}
\end{tabular}
}
\end{center}
\end{table}
Let $D$ be a doubly even $[n,k]$ code satisfying the
conditions~\eqref{eq:C1} and~\eqref{eq:C2}.
Every $[n,k+1]$ code $C$ satisfying the conditions~\eqref{eq:C1}
and~\eqref{eq:C2} with $D \subset C$
can be constructed as $\langle D, x \rangle$, where $x$ is an element
of a set of complete representatives of $D^\perp/D$,
satisfying that $0 \ne \wt(x) \in \{i\in \{0,1,\ldots,n\} \mid A_i(D) \ne 0\}$,
since $D$ is self-orthogonal and
$\{i\in \{0,1,\ldots,n\} \mid A_i(C) \ne 0\}
=\{i\in \{0,1,\ldots,n\} \mid A_i(D) \ne 0\}$.
This observation reduces the number of codes which need
be checked for equivalences.
This observation is applied to doubly even codes
$C_{16,6,2}$ and $C_{16,7,2}$.
Using an approach similar to the previous subsection along with
the above observation,
we completed the classification of codes
satisfying the conditions~\eqref{eq:C1} and~\eqref{eq:C2}
for dimensions $7$ and $8$.
In this case, the only results are listed in
Tables~\ref{Tab:16C} and~\ref{Tab:16V}.
We verified by {\sc Magma} that $C_{16,7,1}$ has covering
radius $4$.
This was done by the {\sc Magma} function {\tt CoveringRadius}.
Thus, $C_{16,7,1}$ is a
maximal code (with respect to the subspace relation).
Since $C_{16,8,1}$ and $C_{16,8,2}$ are doubly even self-dual codes,
there exists no $[16,k]$ code satisfying the conditions~\eqref{eq:C1}
and~\eqref{eq:C2} for $k \ge 9$.
Therefore, we have the following:
\begin{proposition}
If there exists a $[16,k]$ code satisfying the conditions~\eqref{eq:C1}
and~\eqref{eq:C2}, then $k \in \{6,7,8\}$.
Up to equivalence,
there exist two $[16,k]$ codes satisfying the conditions~\eqref{eq:C1}
and~\eqref{eq:C2}
for $k=6,7,8$.
\end{proposition}
By the construction of quasi-unbiased
weighing matrices described in Section~\ref{sec:2},
we have the following:
\begin{corollary}
There exists a set of at least $8$ mutually quasi-unbiased
weighing matrices for parameters $(16,16,4,64)$.
\end{corollary}
A set of $4$ mutually quasi-unbiased
weighing matrices for parameters $(16,16,16,16)$
is also constructed.
It is known that the maximum size
among sets of mutually quasi-unbiased
weighing matrices for the parameters
is $8$ \cite[Proposition~6]{CS73} and \cite[Theorem~5.2]{DGS2}.
\subsection{Length 32}
For the next case $m=5$, the classification of
maximal codes satisfying the conditions~\eqref{eq:C1}
and~\eqref{eq:C2} was done
by a method similar to that for the cases $(n,k)=(16,7), (16,8)$.
\begin{proposition}\label{prop:32}
If there exists a maximal $[32,k]$ code satisfying the
conditions~\eqref{eq:C1} and~\eqref{eq:C2}, then $k \in \{9,10,11\}$.
Up to equivalence,
there exist $92$ maximal $[32,9]$ codes satisfying the
conditions~\eqref{eq:C1} and~\eqref{eq:C2},
there exist $102$ maximal $[32,10]$ codes satisfying the
conditions~\eqref{eq:C1} and~\eqref{eq:C2}, and
there exist two maximal $[32,11]$ codes satisfying the
conditions~\eqref{eq:C1} and~\eqref{eq:C2}.
\end{proposition}
By the construction of quasi-unbiased
weighing matrices described in Section~\ref{sec:2},
we have the following:
\begin{corollary}
There exists a set of at least $4$
mutually quasi-unbiased weighing matrices for parameters $(32,32,4,256)$.
\end{corollary}
A set of $8$ mutually quasi-unbiased weighing matrices for
parameters $(32,32,16,64)$ is also constructed.
It is known that the maximum size
among sets of mutually quasi-unbiased
weighing matrices for the parameters
is $32$~\cite[Theorems~4.1, 4.4]{NSpre}.
\begin{table}[thb]
\caption{Maximal $[32,k]$ codes satisfying~\eqref{eq:C1} and~\eqref{eq:C2}}
\label{Tab:32C}
\begin{center}
{\small
\begin{tabular}{c|l|c}
\noalign{\hrule height0.8pt}
$k$ & \multicolumn{1}{c|}{Codes $C$} & $d(C)$ \\
\hline
9& $C_{32,9,1},\ldots,C_{32,9,91}$ &12 \\
& $C_{32,9,92}$ & 8 \\
10& $C_{32,10,1},\ldots,C_{32,10,101}$ &12 \\
& $C_{32,10,102}$ & 8 \\
11&$C_{32,11,1},C_{32,11,2}$ & 12\\
\noalign{\hrule height0.8pt}
\end{tabular}
}
\end{center}
\end{table}
We denote the $92$ inequivalent
maximal $[32,9]$ codes given in Proposition~\ref{prop:32}
by $C_{32,9,i}$ ($i=1,2,\ldots,92$).
We denote the $102$ inequivalent
maximal $[32,10]$ codes given in Proposition~\ref{prop:32}
by $C_{32,10,i}$ ($i=1,2,\ldots,102$).
We denote the two inequivalent
maximal $[32,11]$ codes given in Proposition~\ref{prop:32}
by $C_{32,11,i}$ ($i=1,2$).
The minimum weights of the codes given in
Proposition~\ref{prop:32} are listed in Table~\ref{Tab:32C}.
\begin{table}[thb]
\caption{Maximal $[32,9]$ codes satisfying~\eqref{eq:C1} and~\eqref{eq:C2}}
\label{Tab:32C-9}
\begin{center}
{\small
\begin{tabular}{c|l|l}
\noalign{\hrule height0.8pt}
&\multicolumn{1}{c|}{Codes} & \multicolumn{1}{c}{Vectors} \\
\hline
$x_7$
&$C_{32,9,1},\ldots,C_{32,9,90}$ &$x_{32,7,1}$ \\
&$C_{32,9,91}$ &$x_{32,7,2}$ \\
&$C_{32,9,92}$ &$x_{32,7,3}$ \\
\hline
$x_8$
&$C_{32,9,1}, \ldots, C_{32,9,15}$ &$x_{32,8,1}$ \\
&$C_{32,9,16}, \ldots, C_{32,9,22}$ &$x_{32,8,2}$ \\
&$C_{32,9,23}, \ldots, C_{32,9,51}$ &$x_{32,8,3}$ \\
&$C_{32,9,52}, \ldots, C_{32,9,76}$ &$x_{32,8,4}$ \\
&$C_{32,9,77}, C_{32,9,78}, C_{32,9,79}$ &$x_{32,8,5}$ \\
&$C_{32,9,80}, C_{32,9,81}, C_{32,9,82}$ &$x_{32,8,6}$ \\
&$C_{32,9,83}, C_{32,9,84}, C_{32,9,85}$ &$x_{32,8,7}$ \\
&$C_{32,9,86}, C_{32,9,87}$ &$x_{32,8,8}$ \\
&$C_{32,9,88}$ &$x_{32,8,9}$ \\
&$C_{32,9,89}, C_{32,9,90}$ &$x_{32,8,10}$\\
&$C_{32,9,91}$ &$x_{32,8,11}$\\
&$C_{32,9,92}$ &$x_{32,8,12}$\\
\hline
$x_9$
&$C_{32,9,i}$ $(i=1,2,\ldots,92)$ &$x_{32,9,i}$ \\
\noalign{\hrule height0.8pt}
\end{tabular}
}
\end{center}
\end{table}
\begin{table}[thbp]
\caption{Maximal $[32,10]$ codes satisfying~\eqref{eq:C1} and~\eqref{eq:C2}}
\label{Tab:32C-10}
\begin{center}
{\footnotesize
\begin{tabular}{c|l|l|l|l}
\noalign{\hrule height0.8pt}
&\multicolumn{1}{c|}{Codes} & \multicolumn{1}{c|}{Vectors }
&\multicolumn{1}{c|}{Codes} & \multicolumn{1}{c}{Vectors } \\
\hline
$x_7$
&$C_{32,10,1},\ldots,C_{32,10,101}$ &$x_{32,7,1}$ &
$C_{32,10,102}$ & $x_{32,7,3}$ \\
\hline
$x_8$
&$C_{32,10,1},\ldots, C_{32,10,30} $ &$x_{32,8,1}$ &
$C_{32,10,31}, \ldots, C_{32,10,73} $ &$x_{32,8,2}$ \\
&$C_{32,10,74}, \ldots, C_{32,10,89} $&$x_{32,8,3}$ &
$C_{32,10,90}, \ldots, C_{32,10,98} $ &$x_{32,8,4}$ \\
&$C_{32,10,99}, C_{32,10,100} $ &$x_{32,8,5}$ &
$C_{32,10,101}$ &$y_{32,8,1}$ \\
&$C_{32,10,102}$ &$x_{32,8,12}$ & \\
\hline
$x_9$
&$ C_{32,10,1}$ & $y_{32,9,1}$ &
$ C_{32,10, 2}, C_{32,10, 3} $ & $y_{32,9,2}$ \\
&$ C_{32,10, 9}, C_{32,10,10}, C_{32,10,11} $ & $y_{32,9,3}$ &
$ C_{32,10, 4} $ & $y_{32,9,4}$ \\
&$ C_{32,10, 5}, C_{32,10, 6} $ & $y_{32,9,5}$ &
$ C_{32,10, 7}, C_{32,10, 8} $ & $y_{32,9,6}$ \\
&$ C_{32,10,12}, C_{32,10,13} $ & $y_{32,9,7}$ &
$ C_{32,10,14}, C_{32,10,15}, C_{32,10,16} $ & $y_{32,9,8}$ \\
&$ C_{32,10,17}, C_{32,10,18} $ & $y_{32,9,9}$ &
$ C_{32,10,19}, C_{32,10,20} $ & $y_{32,9,10}$ \\
&$ C_{32,10,21} $ & $y_{32,9,11}$ &
$ C_{32,10,22} $ & $y_{32,9,12}$ \\
&$ C_{32,10,23}, C_{32,10,24} $ & $y_{32,9,13}$ &
$ C_{32,10,25}, C_{32,10,26} $ & $y_{32,9,14}$ \\
&$ C_{32,10,27} $ & $y_{32,9,15}$ &
$ C_{32,10,28} $ & $y_{32,9,16}$ \\
&$ C_{32,10,29} $ & $y_{32,9,17}$ &
$ C_{32,10,30} $ & $y_{32,9,18}$ \\
&$ C_{32,10,31}, C_{32,10,32} $ & $y_{32,9,19}$ &
$ C_{32,10,33} $ & $y_{32,9,20}$ \\
&$ C_{32,10,34}, \ldots, C_{32,10,37} $ & $y_{32,9,21}$ &
$ C_{32,10,38} $ & $y_{32,9,22}$ \\
&$ C_{32,10,39}, C_{32,10,40} $ & $y_{32,9,23}$ &
$ C_{32,10,41} $ & $y_{32,9,24}$ \\
&$ C_{32,10,42}, C_{32,10,43} $ & $y_{32,9,25}$ &
$ C_{32,10,44} $ & $y_{32,9,26}$ \\
&$ C_{32,10,45}, C_{32,10,46}, C_{32,10,47} $ & $y_{32,9,27}$ &
$ C_{32,10,48} $ & $y_{32,9,28}$ \\
&$ C_{32,10,49}, C_{32,10,50} $ & $y_{32,9,29}$ &
$ C_{32,10,51}, C_{32,10,52} $ & $y_{32,9,30}$ \\
&$ C_{32,10,53}, \ldots, C_{32,10,57} $ & $y_{32,9,31}$ &
$ C_{32,10,58} $ & $y_{32,9,32}$ \\
&$ C_{32,10,59}, C_{32,10,60} $ & $y_{32,9,33}$ &
$ C_{32,10,61} $ & $y_{32,9,34}$ \\
&$ C_{32,10,62}, C_{32,10,63}, C_{32,10,64} $ & $y_{32,9,35}$ &
$ C_{32,10,65} $ &$y_{32,9,36}$ \\
&$ C_{32,10,66} $ &$y_{32,9,37}$ &
$ C_{32,10,67}, C_{32,10,99} $ &$y_{32,9,38}$ \\
&$ C_{32,10,68} $ &$y_{32,9,39}$ &
$ C_{32,10,69} $ &$y_{32,9,40}$ \\
&$ C_{32,10,70} $ &$y_{32,9,41}$ &
$ C_{32,10,71}, C_{32,10,72} $ &$y_{32,9,42}$ \\
&$ C_{32,10,73} $ &$y_{32,9,43}$ &
$ C_{32,10,74} $ &$y_{32,9,44}$ \\
&$ C_{32,10,75} $ &$y_{32,9,45}$ &
$ C_{32,10,76}, C_{32,10,77} $ &$y_{32,9,46}$ \\
&$ C_{32,10,78},C_{32,10,79},C_{32,10,80}$&$y_{32,9,47}$ &
$ C_{32,10,81} $ &$y_{32,9,48}$ \\
&$ C_{32,10,82} $ &$y_{32,9,49}$ &
$ C_{32,10,83} $ &$y_{32,9,50}$ \\
&$ C_{32,10,84} $ &$y_{32,9,51}$ &
$ C_{32,10,85} $ &$y_{32,9,52}$ \\
&$ C_{32,10,86} $ &$y_{32,9,53}$ &
$ C_{32,10,87} $ &$y_{32,9,54}$ \\
&$ C_{32,10,88} $ &$y_{32,9,55}$ &
$ C_{32,10,89} $ &$y_{32,9,56}$ \\
&$ C_{32,10,90} $ &$y_{32,9,57}$ &
$ C_{32,10,91} $ &$y_{32,9,58}$ \\
&$ C_{32,10,92}, C_{32,10,93} $ &$y_{32,9,59}$ &
$ C_{32,10,94} $ &$y_{32,9,60}$ \\
&$ C_{32,10,95} $ &$y_{32,9,61}$ &
$ C_{32,10,96} $ &$y_{32,9,62}$ \\
&$ C_{32,10,97} $ &$y_{32,9,63}$ &
$ C_{32,10,98} $ &$y_{32,9,64}$ \\
&$ C_{32,10,100}$ &$y_{32,9,65}$ &
$ C_{32,10,101}$ &$y_{32,9,66}$ \\
&$ C_{32,10,102}$ &$y_{32,9,67}$ & \\
\noalign{\hrule height0.8pt}
\end{tabular}
}
\end{center}
\end{table}
To describe the codes given in Proposition~\ref{prop:32},
we fix the generator matrix of the first order Reed--Muller
$[32, 6, 16]$ code $RM(1,5)$ as follows:
\[
\left(
\begin{array}{c}
10010110011010010110100110010110\\
01010101010101010101010101010101\\
00110011001100110011001100110011\\
00001111000011110000111100001111\\
00000000111111110000000011111111\\
00000000000000001111111111111111
\end{array}
\right).
\]
The codes $C_{32,9,i}$ ($i=1,2,\ldots,92$)
are constructed as
$\langle RM(1,5),x_7,x_8,x_9 \rangle$,
where Table~\ref{Tab:32C-9} indicates
$x_7,x_8,x_9$ and the supports are listed in Table~\ref{Tab:32V}.
The codes $C_{32,10,i}$ ($i=1,2,\ldots,102$)
are constructed as
$\langle RM(1,5),x_7,x_8,x_9,x_{10} \rangle$,
where Table~\ref{Tab:32C-10} indicates
$x_7,x_8,x_9,x_{10}$ and the supports are
listed in Table~\ref{Tab:32V}.
The codes $C_{32,11,i}$ ($i=1,2$)
are constructed as follows:
\[
\begin{split}
C_{32,11,1}=&\langle RM(1,5),x_{32,7,2}, z_{32,8,1}, z_{32,9,1}, z_{32,10,1}, z_{32,11,1}\rangle,\\
C_{32,11,2}=&\langle RM(1,5),x_{32,7,2}, z_{32,8,1}, z_{32,9,2}, z_{32,10,2}, z_{32,11,2}\rangle,
\end{split}
\]
where the supports of the vectors are listed in Table~\ref{Tab:32V}.
Finally, we compare our codes with some known codes and
we discuss the maximality of our codes.
It follows from the weight distributions that
$C_{32,9,92}$ (resp.\ $C_{32,10,102}$) is equivalent to
the unique maximal triply even $[32,9]$ (resp.\ $[32,10]$) code,
which is given in~\cite[Table~2]{BM}.
It follows that $C_{32,9,92}$ and $C_{32,10,102}$ are maximal.
We verified by {\sc Magma} that
$C_{32,9,1},C_{32,9,2},\ldots,C_{32,9,90}$ have covering radius $ \le 11$,
$C_{32,10,1},C_{32,10,2},\ldots,C_{32,10,101}$ have covering radius $10$ and
$C_{32,11,1},C_{32,11,2}$ have covering radius $8$.
This shows that these codes are maximal.
We verified by {\sc Magma} that $C_{32,11,2}$
is equivalent to the extended BCH $[32,11,12]$ code.
\section*{Postscript}
After this work, we continued the study of quasi-unbiased weighing matrices
obtained from (not necessary linear) codes in \cite{AHS}.
\section*{Acknowledgments}
The authors would like to thank Hiroshi Nozaki for helpful discussions.
The authors would also like to thank the anonymous referees for
their valuable comments leading to several improvements of this note.
This work is supported by JSPS KAKENHI Grant Number 23340021.
|
{
"timestamp": "2016-08-11T02:03:51",
"yymm": "1504",
"arxiv_id": "1504.03502",
"language": "en",
"url": "https://arxiv.org/abs/1504.03502"
}
|
\section{\label{sec:introduction}Introduction}
Neutrino flavor conversion in vacuum \cite{Pontecorvo:1967fh, Gribov:1968kq},
in matter \hbox{\cite{Wolfenstein:1977ue, Wolfenstein:1979ni, Mikheev:1986gs,
Mikheev:1986if, Kuo:1989qe, Dighe:1999bi}},
or self-induced flavor conversion in a gas of interacting neutrinos
\cite{Pantaleone:1992eq, Samuel:1993uw, Pantaleone:1994ns, Sawyer:2005jk, Duan:2006an,
Hannestad:2006nj, Balantekin:2006tg, Dasgupta:2009mg, Duan:2010bg, Friedland:2010sc,
Banerjee:2011fj, Galais:2011gh, Pehlivan:2011hp, Raffelt:2011yb, Cherry:2012zw,
deGouvea:2012hg, Raffelt:2013rqa, Duan:2014gfa, Mirizzi:2015fva}
provide a rich phenomenology of very practical experimental and
astrophysical importance. The data leave no room for doubt that
neutrinos have small but nonvanishing masses. One consequence is that
neutrinos have small electromagnetic dipole and transition moments
\cite{Giunti:2014ixa}. These lead to spin and spin-flavor
oscillations in strong electromagnetic fields
\cite{Werntz:1970, Cisneros:1970nq, Lim:1987tk, Akhmedov:1987nc}.
Actually, polarized matter or
matter currents alone instigate spin and spin-flavor transitions of
massive neutrinos, having effects similar to electromagnetic fields
\hbox{\cite{Egorov:1999ah, Grigoriev:2002zr, Studenikin:2004bu, Vlasenko:2013fja, Cirigliano:2014aoa}}.
For neutrinos streaming from a supernova core, the background medium
may contain currents. Moreover, the neutrino stream itself
provides an unavoidable nonisotropic background. In addition,
self-induced flavor conversion in an interacting neutrino gas requires
unstable modes in flavor space (run-away solutions). If such solutions
exist, even small perturbations or otherwise small effects can grow
exponentially. In this sense, it is never obvious if a seemingly small
effect can get amplified by an instability to play an important role
after all. Therefore, it is interesting to study if an interacting
neutrino gas can amplify helicity conversion effects
\cite{Vlasenko:2014bva} which otherwise are very small.
Flavor oscillations lead to correlations building up between
neutrinos of different flavor. If $a^\dagger_\alpha$ is the creation
operator of a neutrino in flavor state $\alpha$ with a certain
momentum $\vec{p}$, the initially prepared system can be described by the
occupation number $\langle a^\dagger_\alpha a_\alpha\rangle$. One way
of looking at flavor oscillations is that ``flavor off-diagonal''
occupation numbers of the type $\langle a^\dagger_\alpha
a_\beta\rangle$ develop and oscillate
\cite{Dolgov:1980cq, Rudzsky:1990, Sigl:1992fn}. One unifies these
expressions in a density matrix $\rho$ with components
\smash{$\rho_{\alpha\beta}=\langle a^\dagger_\beta a_\alpha\rangle$}. It
evolves according to the commutator equation $i\dot\rho=[{\sf
H},\rho]$, where ${\sf H}$ is the Hamiltonian matrix, consisting of
oscillation frequencies. For vacuum oscillations we have ${\sf H}={\sf
M}^2/2E$, where ${\sf M}^2$ is a matrix of squared neutrino
masses. Similar descriptions pertain to spin and spin-flavor
oscillation, where the indices now indicate various states
of spin and/or flavor.
It was recently stressed that yet another form of correlations,
hitherto neglected in the context of neutrino propagation, can build
up in nonisotropic media \cite{Volpe:2013uxl, Vaananen:2013qja, Serreau:2014cfa}.
If $a^\dagger_{\vec{p}}$ is the creation operator of a
massless neutrino in mode $\vec{p}$ and \smash{$b^\dagger_{-\vec{p}}$} the one for an
antineutrino with opposite momentum, correlators of the form
\smash{$\kappa_{\vec{p}}=\langle b_{-\vec{p}}a_{\vec{p}}\rangle$} and
\smash{$\kappa^\dagger_{\vec{p}}=\langle a^\dagger_{\vec{p}}b^\dagger_{-\vec{p}}\rangle$}
will build up, the latter corresponding to the creation of a
particle-antiparticle pair with vanishing total momentum. Because
massless neutrinos and antineutrinos have opposite
helicity, this pair has total spin~1 so that its creation requires a
medium current transverse to $\vec{p}$ to satisfy angular-momentum
conservation. This requirement is analogous to the case of helicity
transitions where we also need a transverse current or magnetic field
for the same reason. Including flavor and spin degrees of freedom
expands the ``pair correlations'' $\kappa$ and $\kappa^\dagger$ to
become matrices similar to $\rho$.
To develop more intuition about the meaning of the pair correlations,
we consider a single mode $\vec{p}$ of neutrinos
and $-\vec{p}$ of antineutrinos. We define \smash{$\rho_{\vec{p}}=\langle
a^\dagger_{\vec{p}} a_{\vec{p}}\rangle$} and for antineutrinos
\smash{$\bar\rho_{\vec{p}}=\langle b^\dagger_{-\vec{p}} b_{-\vec{p}}\rangle$}
involving the opposite momentum. Following the earlier literature
\cite{Volpe:2013uxl, Vaananen:2013qja, Serreau:2014cfa}, we unify
these expressions in an extended density matrix
\begin{equation}\label{eq:simple-R-equation}
{\sf R}=\begin{pmatrix}\rho&\kappa\cr\kappa^\dagger&1-\bar\rho\cr\end{pmatrix}
=\begin{pmatrix}
\langle a^\dagger_{\vec{p}} a_{\vec{p}}\rangle&\langle b_{-\vec{p}}a_{\vec{p}}\rangle\cr
\langle a^\dagger_{\vec{p}}b^\dagger_{-\vec{p}}\rangle&\langle b_{-\vec{p}} b^\dagger_{-\vec{p}}\rangle\cr
\end{pmatrix}\,,
\end{equation}
which obeys an equation of motion of the form
\cite{Volpe:2013uxl, Vaananen:2013qja, Serreau:2014cfa}
\begin{equation}\label{eq:EOM1}
i\dot{\sf R}=[{\sf H},{\sf R}]\,.
\end{equation}
If the background is a current
moving in the transverse direction with velocity $\beta$, the
Hamiltonian matrix is found to be
\begin{equation}\label{eq:simplehamiltonian}
{\sf H}=E\,\begin{pmatrix}1&0\cr 0&-1\cr\end{pmatrix}
+V\,\begin{pmatrix}1&-\beta\cr-\beta&1\cr\end{pmatrix}\,,
\end{equation}
where $E=|\vec{p}|$. For $\nu_\mu$ or $\nu_\tau$ neutrinos the usual matter
potential is \smash{$V=G_{\rm F} n_n/\sqrt{2}$}, where $n_n$ is the neutron density.
This commutator equation has the same structure that one
encounters for the evolution of any two-level system and in
particular for two-flavor or helicity oscillations,
of course with a different matrix ${\sf H}$ for each
case. However, what specifically are the two states that are being
mixed by the matter current in the pair-correlation case?
The answer
becomes evident if one considers the evolution of states
rather than correlators. Our simple
system is described by the four basis states $|00\rangle$,
$|10\rangle$, $|01\rangle$ and $|11\rangle$, where the first entry
refers to $\nu(\vec{p})$ and the second to $\bar\nu(-\vec{p})$. A homogeneous
background medium cannot mix states of different total momentum, so the
single-particle states must evolve independently as $i\partial_t|10\rangle=
(E+V)|10\rangle$ and $i\partial_t|01\rangle=
(E-V)|01\rangle$, i.e., they simply suffer the usual energy shift by
the weak potential of the medium. This leaves us
with $|00\rangle$ and $|11\rangle$ which both have zero momentum and
therefore can be mixed by a homogeneous medium.
The former has spin~0, the latter spin~1, so for the medium to mix
them, it must provide a transverse vector in the form of a current or
a spin polarization. If $A_{00}$ and $A_{11}$ are the
amplitudes of $|00\rangle$ and $|11\rangle$, respectively, we will
show later that Eq.~(\ref{eq:EOM1}) corresponds to
\begin{equation}\label{eq:0011}
i\partial_t\begin{pmatrix}A_{00}\cr A_{11}\cr\end{pmatrix}
=\begin{pmatrix}0&\beta V\cr\beta V&2E\cr\end{pmatrix}
\begin{pmatrix}A_{00}\cr A_{11}\cr\end{pmatrix}\,.
\end{equation}
Therefore, it is the empty and the completely filled states that are
being mixed and that oscillate. The true ground state
of our system is not $|00\rangle$, but a suitable combination of $|00\rangle$
and $|11\rangle$ which follows from diagonalizing the matrix in Eq.~(\ref{eq:0011}).
As we have noted, any two-level system is equivalent to an abstract
spin-$\frac{1}{2}$ system. In two-flavor oscillations, the ``spin''
represents the two flavor states. In the pair-correlation case, ``spin
up'' means ``empty'' and ``spin down'' means ``full with a pair.''
This interpretation is analogous to Anderson's ``pseudo spin'' devised
to describe Cooper pairs in the context of superconductivity
\cite{Anderson:1958zza}. A coherent superposition of these two spin
states, represented in our case by the pair correlations, corresponds
to a coherent superposition of $|00\rangle$ and $|11\rangle$.
In analogy to the example of superconductivity,
another way to think about these phenomena is in terms of Bogolyubov
transformations of the creation and annihilation operators. If we think of a
single momentum mode $\vec{p}$ of mixed neutrinos in vacuum, the operators $a_{\nu_e}$
and $a_{\nu_\mu}$ in the flavor basis are rotated by a unitary transformation with
mixing angle $\vartheta$ to form new operators $c_\vartheta a_{\nu_e} + s_\vartheta a_{\nu_\mu}$
and $c_\vartheta a_{\nu_e}-s_\vartheta a_{\nu_\mu}$, and similarly for the creation operators,
to form a new set of canonically anticommuting operators, now describing
neutrinos in the mass basis. Describing flavor oscillations in terms of
time-dependent Bogolyubov transformations can be especially illuminating to
understand quantum statistics in mixing phenomena for both bosons and fermions
\cite{Raffelt:1991ck}. Pair correlations correspond to the
same idea where the mixing is between $a_{\vec{p}}$ and \smash{$b^\dagger_{-\vec{p}}$} with a
mixing angle corresponding to the unitary transformation that diagonalizes
the matrix in Eq.~(\ref{eq:0011}). The state $|00\rangle$ defined in
the Bogolyubov-transformed basis is the ground state of
the system and no longer oscillates into the new $|11\rangle$ state.
The goal of our paper is two-pronged. On the one hand we reconsider
the mean-field equations of motion for massive neutrinos propagating
in a background medium that can consist of matter and neutrinos, and
that is homogeneous but not isotropic. Besides the usual flavor
oscillations in matter, the resulting phenomena include spin and
spin-flavor oscillations as well as pair correlations.
As a second goal, we provide a phenomenological discussion of the
interpretation of the pair correlations in the context of neutrino
oscillation problems in dense media. Ultimately, our community needs
to develop an understanding if, from a practical perspective, we need
to worry about pair correlations and helicity oscillations in the
supernova context.
The supernova environment is characterized by
small neutrino energies of at most some 200~MeV (for degenerate
$\nu_e$), i.e., small compared to $W$ and $Z$ masses so that it
suffices to describe neutrino interactions in terms of an effective
current-current Hamiltonian. In the early Universe, where the chemical
potentials of background particles are small, one has to worry about
corrections from the electroweak gauge-boson propagators even at low
temperatures \cite{Notzold:1987ik}. The supernova environment, in
contrast, has large densities of background particles and this concern
is moot.
On the mean-field level, the current of background particles is a
classical quantity. For example, the neutral-current interaction of a
neutrino with neutrons is given by the Hamiltonian density ${\cal
H}=\sqrt{2}G_{\rm F}\,[\bar\nu\gamma_\mu P_L\nu]\,I_n^\mu$, where $G_{\rm F}$ is
the Fermi constant, $\nu$ is the neutrino Dirac field, $P_L$ is the
left-handed projector, and $I_n^\mu$ is the neutron current. If the
current is homogeneous, $H=\int d^3{\bf x}\,{\cal H}$ is effectively a
``forward'' Hamiltonian: it couples, e.g.,
\smash{$a^\dagger$} and $a$ of equal momenta. Following the previous
literature \cite{Rudzsky:1990, Sigl:1992fn, Serreau:2014cfa}, the
evolution of, e.g., the annihilator for a neutrino of
mass eigenstate $i$ in mode $\vec{p}$
is given by the Heisenberg equation of motion $i\partial_t
a_i(t,\vec{p})=[a_i(t,\vec{p}),H]$. It is then straightforward to
find the equations of motion of bilinears of the form
\smash{$a_i^\dagger(t,\vec{p})a_j(t,\vec{p})$}, of their expectation value
\smash{$\langle a_i^\dagger(t,\vec{p})a_j(t,\vec{p})\rangle$}, of the entire
matrix $\rho$, and then of the extended matrix ${\sf R}$ which also
includes pair correlations.
It is largely a
cumbersome bookkeeping exercise to obtain, for neutrinos with mass,
all the components of the Hamiltonian matrix ${\sf H}$
appearing in Eq.~(\ref{eq:EOM1}) when ${\sf R}$ involves
all components of spin and flavor. We perform this task separately
for Dirac neutrinos in Sec.~\ref{sec:dirac}, for Majorana neutrinos in Sec.~\ref{sec:majorana}, and
for Weyl neutrinos (massless two-component case) in Sec.~\ref{sec:weyl}.
These derivations closely parallel the recent paper by Serreau and Volpe
\cite{Serreau:2014cfa} and we will largely follow their notation to
avoid confusion. In the Majorana case, we find a small correction, but
otherwise our results agree.
The density matrix formalism
allows one to treat helicity oscillations induced by magnetic fields
and by matter currents on equal footing for both Dirac and Majorana
fermions. We derive the mean-field Hamiltonian induced by electromagnetic
fields in Sec.~\ref{sec:magnetic}. Concerning helicity oscillations
induced by matter currents, which we analyze in Sec.~\ref{sec:helicity}, our results
coincide with those of Volpe and Serreau, and parallel those of Vlasenko, Fuller, and
Cirigliano \cite{Vlasenko:2013fja, Cirigliano:2014aoa,
Vlasenko:2014bva} as far as the mean-field limit is concerned. These
authors have derived the neutrino kinetic equations starting from
first field-theoretic principles and have carried the results beyond
the mean-field limit to include (nonforward) collision terms,
generalizing previous derivations \cite{Dolgov:1980cq, Rudzsky:1990,
Sigl:1992fn}. We note in passing that one of their findings ---
helicity oscillations in a nonisotropic matter background --- had been
anticipated in several papers by Studenikin and collaborators who have
worked out the one-to-one correspondence to the effect of
electromagnetic fields \cite{Egorov:1999ah, Studenikin:2004bu}. Of course,
Vlasenko, Fuller, and Cirigliano also included
neutrino-neutrino interactions as an agent of helicity conversion and
carried their results beyond the mean-field limit.
Pair correlations have been studied in detail in condensed matter and
nuclear physics, as well as in the context of pair creation in quantum field theory.
On the other hand, in neutrino physics these concepts are less familiar. They
have been addressed only in a handful of papers in the context of leptogenesis,
where pair correlations have been studied from first principles
in a series of papers by Fidler, Herranen, Kainulainen and Rahkila
\cite{Herranen:2008di,Herranen:2008hi,Herranen:2009zi,Herranen:2011zg, Fidler:2011yq}.
In the context of neutrino propagation in supernovae, the only discussions so far
appear in a series of papers
by Volpe and collaborators \cite{Volpe:2013uxl,Vaananen:2013qja,Serreau:2014cfa}.
We address phenomenological aspects of pair correlations in
Sec.~\ref{sec:partantipart} and compare them to helicity correlations.
Finally, in Sec.~\ref{sec:summary} we summarize the results and present
our conclusions.
\section{\label{sec:dirac}Dirac neutrino}
Our first goal is to derive the components of the Hamiltonian matrix
${\sf H}$ which governs the evolution equation (\ref{eq:EOM1}) for the
extended density matrix ${\sf R}$ including flavor, helicity,
and pair correlations. In this rather technical section, we begin
with the conceptually simplest case of three neutrino flavors which
are assumed to have Dirac masses. Therefore, helicity correlations
involve the sterile components of the neutrino field, which otherwise
are completely decoupled.
\subsection{Two-point correlators and kinetic equations}
In the simplest approximation, one can describe the state of a
neutrino gas in terms of one-particle distribution functions. They are
extended to include flavor and helicity coherence effects by promoting
the one-particle distribution functions to density matrices
\cite{Dolgov:1980cq, Rudzsky:1990, Sigl:1992fn, Vlasenko:2013fja,
Serreau:2014cfa}. In terms of the usual creation and annihilation
operators, their components are
\begin{subequations}
\label{RhoDef}
\begin{align}
(2\pi)^3\delta(\vec{p}{-}\vec{k})\rho_{ij,sh} (t,\vec{p})&=
\langle a^\dagger_{j,h}(t,+\vec{k}) a_{i,s}(t,+\vec{p})\rangle\,,\\
(2\pi)^3\delta(\vec{p}{-}\vec{k})\bar{\rho}_{ij,sh} (t,\vec{p})&=
\langle b^\dagger_{i,s}(t,-\vec{p}) b_{j,h}(t,-\vec{k})\rangle\,,
\end{align}
\end{subequations}
where $i$ and $j$ are flavor indices in the mass basis, and $s$ and
$h\in\{+,-\}$ denote helicities. In this convention, the density matrix
for antineutrinos $\bar{\rho}_{ij,sh}(t,\vec{p})$ for momentum
${\vec{p}}$ actually corresponds to the occupation numbers of
antineutrinos with physical momentum $-\vec{p}$.
This convention is necessary to combine $\rho$ and $\bar\rho$ with the
pair correlations which are defined as \cite{Vaananen:2013qja,Serreau:2014cfa}
\begin{subequations}
\label{KappaDef}
\begin{align}
(2\pi)^3\delta(\vec{p}{-}\vec{k})\kappa_{ij,sh} (t,\vec{p})&=
\langle b_{j,h}(t,-\vec{k}) a_{i,s}(t,+\vec{p})\rangle\,,\\
(2\pi)^3\delta(\vec{p}{-}\vec{k})\kappa^\dagger_{ij,sh} (t,\vec{p})&=
\langle a^\dagger_{j,h}(t,+\vec{p}) b^\dagger_{i,s}(t,-\vec{k})\rangle\,,
\end{align}
\end{subequations}
and which involve opposite-momentum modes.
The kinetic equations for Eqs.~\eqref{RhoDef} and~\eqref{KappaDef} are
obtained with the Heisenberg equation of motion. As we will show
below, in the mean-field approximation, and assuming spatial
homogeneity, the Hamiltonian of charged- and neutral-current
neutrino interactions can be written in the compact form
\begin{align}
\label{Heff}
H_{\rm mf}=\int d^3{\bf x}\,\bar{\nu}_i(t,\vec{x}) \Gamma_{ij}\nu_j(t,\vec{x})\,,
\end{align}
where summation over repeated indices is implied.
The kernel takes account of the background medium and is
\begin{align}
\Gamma_{ij} = \gamma_\mu P_L V^\mu_{ij}\,,
\end{align}
where $P_L=(1-\gamma_5)/2$ is the usual left-chiral projector.
The current of background matter $V^\mu_{ij}$ will be defined in Eq.~\eqref{SigmaDirac}.
The momentum-mode
decomposition of a Dirac neutrino field reads
\begin{align}\label{eq:Dirac-decomposition}
\nu_i(t,\vec{x}\,)=\int_{\vec{p},s} e^{i\vec{p}\cdot\vec{x}}\nu_{i,s}(t,\vec{p})\,,
\end{align}
where \smash{$\int_{\vec{p},s}$} denotes the phase-space integration \smash{$\int
d^3\vec{p}/(2\pi)^3$} and the summation over helicities.
In the mass basis, the individual momentum modes are
\begin{align}
\label{nudecompositiondir}
\nu_{i,s}(t,\vec{p}) &= a_{i,s}(t,\vec{p})u_{i,s}(\vec{p})
+b^\dagger_{i,s}(t,-\vec{p})v_{i,s}(-\vec{p})\,.
\end{align}
The chiral spinors $u$ and $v$ are given in Appendix~\ref{sec:spinorproducts},
and the creation and annihilation operators satisfy the usual equal-time
anticommutation relation,
\begin{align}
\label{AnticommRels}
\{a_{i,s}(t,\vec{p}),a^\dagger_{j,h}(t,\vec{k})\}=(2\pi)^3 \delta(\vec{p}{-}\vec{k})\delta_{ij}\delta_{sh}\,.
\end{align}
Similar relations hold for the antiparticle operators $b$ and
$b^\dagger$.
As a next step, we contract the kernels $\Gamma_{ij}$ with the
spinors appearing in the mean-field Hamiltonian \eqref{Heff},
leading to the matrices \cite{Serreau:2014cfa}
\begin{subequations}
\label{GammasDef}
\begin{align}
\Gamma^{\nu\nu}_{ij,sh}(\vec{p})&\equiv\bar{u}_{i,s}(\vec{p})\Gamma_{ij}u_{j,h}(\vec{p})\,,\\
\Gamma^{\nu\bar\nu}_{ij,sh}(\vec{p})&\equiv\bar{u}_{i,s}(\vec{p})\Gamma_{ij}v_{j,h}(-\vec{p})\,,\\
\Gamma^{\bar\nu\nu}_{ij,sh}(\vec{p})&\equiv\bar{v}_{i,s}(-\vec{p})\Gamma_{ij}u_{j,h}(\vec{p})\,,\\
\Gamma^{\bar\nu\bar\nu}_{ij,sh}(\vec{p})&\equiv\bar{v}_{i,s}(-\vec{p})\Gamma_{ij}v_{j,h}(-\vec{p})\,,
\end{align}
\end{subequations}
in component form. We can now bring Eq.~\eqref{Heff} to the desired
form bilinear in the creation and annihilation operators
\begin{align}
\label{HeffCrAn}
H_{\rm mf}=\int_{\vec{p}}\,\, \Bigl[& a^\dagger_{i,s}(\vec{p}) \Gamma^{\nu\nu}_{ij,sh}(\vec{p})
a_{j,h}(\vec{p})\nonumber\\[-2mm]
&{}+a^\dagger_{i,s}(\vec{p}) \Gamma^{\nu\bar\nu}_{ij,sh}(\vec{p})b^\dagger_{j,h}(-\vec{p})\nonumber\\
&{}+b_{i,s}(-\vec{p}) \Gamma^{\bar\nu\nu}_{ij,sh}(\vec{p}) a_{j,h}(\vec{p})\nonumber\\
&{}+b_{i,s}(-\vec{p}) \Gamma^{\bar\nu\bar\nu}_{ij,sh}(\vec{p})b^\dagger_{j,h}(-\vec{p})\Bigr]\,,
\end{align}
where we have omitted the time arguments to shorten the
notation. Summation over repeated indices is implied.
Using the Heisenberg equation of motion with this Hamiltonian
one finds the extended equation of motion $i\dot{\sf R}=[{\sf H},{\sf R}]$,
see also Eq.\,\eqref{eq:EOM1}. The extended density matrix,
Eq.\,\eqref{eq:simple-R-equation}, and the Hamiltonian, Eq.\,\eqref{eq:simplehamiltonian},
generalize to \cite{Volpe:2013uxl}
\begin{align}
\label{RHstructure}
\sf{R}=
\begin{pmatrix}
\rho & \kappa\\
\kappa^\dagger & \mathds{1}-\bar{\rho}
\end{pmatrix}\quad {\rm and} \quad
\sf{H}=
\begin{pmatrix}
\sf{H}^{\nu\nu} & \sf{H}^{\nu\bar\nu}\\
\sf{H}^{\bar\nu\nu} & \sf{H}^{\bar\nu\bar\nu}
\end{pmatrix}\quad \,,
\end{align}
where the submatrices ${\sf H}^{\nu\nu}=\Gamma^{\nu\nu}$, ${\sf H}^{\nu\bar\nu}=\Gamma^{\nu\bar\nu}$
etc.\, and $\rho$,
$\kappa$, etc.\ are $6{\times}6$ matrices in helicity and flavor space. The product
between such matrices in the commutator is defined in the obvious way
\smash{$(A\cdot B)_{ij,sh}\equiv A_{in,sr} B_{nj,rh}$} with a summation over repeated indices.
In the following we write the matrix structure in the form of $2{\times}2$ matrices in helicity space,
\begin{equation}
\label{spinflavorstruct}
\begin{pmatrix}
\framebox{\hbox to 0pt{$\vphantom{+}$}$--$}_{~ij}&\framebox{$-+$}_{~ij}\\[6pt]
\framebox{$+-$}_{~ij}&\framebox{$++$}_{~ij}
\end{pmatrix}\,,
\end{equation}
where each entry is itself a $3{\times}3$ matrix in flavor space.
\subsection{Hamiltonian in the mean-field approximation}
After having established the overall structure of the kinetic equations
we now turn to the interactions contributing to neutrino refraction in the
supernova environment. In this subsection we only consider charged- and
neutral-current neutrino interactions, whereas the analysis of the electromagnetic
interactions is postponed to Sec.\,\ref{sec:magnetic}.
\subsubsection{Charged-current interaction}
We begin with charged-current (cc) interactions with
background charged leptons. In the low-energy limit and after a Fierz
transformation, the usual current-current Hamiltonian density is
\begin{align}
\label{HCC}
\mathcal{H}^{\rm cc}&= \sqrt{2} G_{\rm F}\sum\limits_{\alpha,\beta}
\bigl[\bar{\nu}_\alpha\gamma^\mu P_L\nu_\beta\bigr]
\bigl[\bar{\ell}_\beta\gamma_\mu (1-\gamma^5) \ell_\alpha\bigr]\,,
\end{align}
where $\alpha,\beta \in \{e,\mu,\tau\}$ are flavor indices.
To obtain the neutrino mean-field Hamiltonian we
replace the second bracket by its expectation value. In the supernova
environment, the temperature is too low to support a substantial density of
muons or tauons, and we use only the electron background. Then we find in the
mass basis
\begin{align}
\label{HCCeff}
\mathcal{H}_{\rm mf}^{\rm cc}&= \sqrt{2}G_{\rm F}\sum\limits_{i,j}
\bigl[\bar{\nu}_i\gamma^\mu P_L\nu_j\bigr]
\bigl[U^\dagger_{ie} I_{\rm cc}^\mu U_{ej}\bigr] \,,
\end{align}
where $U$ is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. We have
introduced a linear combination of vector and axial-vector charged electron
currents,
\begin{align}
\label{Jlept}
I_{\rm cc}^\mu\equiv c_V\langle \bar{e} \gamma^\mu e \rangle
-c_A \langle \bar{e}\gamma^\mu \gamma^5 e \rangle\,,
\end{align}
where $c_V=c_A=1$. Because electrons are the only background particles
contributing to charged-current interactions and to simplify the notation, an ``$e$''
index is implied in $I_{\rm cc}^\mu$. If the electrons are not polarized, the
axial current vanishes and $I_{\rm cc}^\mu=J^\mu_{e}$, the ``convective''
electron current.
\subsubsection{Neutral-current interaction with matter}
The neutral-current (nc) interactions with matter are described in the mass
basis by the Hamiltonian density
\begin{align}
\label{HNC}
\mathcal{H}^{\rm nc}&= \sqrt{2}G_{\rm F}\sum\limits_{i,f}\bigl[
\bar{\nu}_i\gamma_\mu P_L\nu_i\bigr]\bigl[\bar{\psi}_f\gamma^\mu (c_V^f{-}c_A^f\gamma^5)\psi_f]\,,
\end{align}
where $f$ denotes electrons, protons, and neutrons. The resulting
contribution to the mean-field Hamiltonian is
\begin{align}
\label{HNCeff}
\mathcal{H}^{\rm nc}_{\rm mf}&= \sqrt{2}G_{\rm F}\sum\limits_{i}\bigl[
\bar{\nu}_i\gamma_\mu P_L\nu_i\bigr]\!\bigl[I_{\rm nc}^\mu+I_p^\mu+I_n^\mu],
\end{align}
where $I_{\rm nc}^\mu$ denotes the electron neutral current (index $e$
implied), whereas the other contributions refer to protons and neutrons as
explicitly indicated.
These currents are defined in analogy to Eq.~\eqref{Jlept} with the
appropriate coupling constants. For electrons, they are given by
$c_V=-\frac12+2\sin^2\theta_W$ (Weinberg angle $\theta_W$) and
$c_A=-\frac12$. For protons, \smash{$c_V=\frac12-2\sin^2\theta_W$}, i.e., the same as
for electrons with opposite sign, and for neutrons \smash{$c_V=-\frac12$}. For the
nucleon axial vector one often uses $c_A=\pm1.26/2$ in analogy to their
charged current. However, the strange-quark contribution to the nucleon spin
as well as modifications in a dense nuclear medium leave the exact values
somewhat open \cite{Raffelt:1993ix, Horowitz:2001xf}.
In an unpolarized and electrically neutral environment, the axial currents
disappear and the electron and proton contributions to the convective neutral
current cancel such that in Eq.~\eqref{HNCeff} we have $I_{\rm nc}^\mu+
I_p^\mu+I_n^\mu=-\frac12J_n^\mu$, where $J^\mu_n$ is the neutron convective
current. Neutrino refraction in such a medium depends only on the charged
electron current and the neutral neutron current.
\subsubsection{Neutrino-neutrino interaction}
The most complicated interaction is the neutral-current neutrino-neutrino
one. It is described in the mass basis by the Hamiltonian density
\begin{align}
\label{Hself}
\mathcal{H}^{\nu\nu}&= \frac1{\sqrt{2}}G_{\rm F}\sum\limits_{ij}
\bigl[\bar{\nu}_i\gamma_\mu P_L\nu_i\bigr]\bigl[\bar{\nu}_j\gamma^\mu P_L\nu_j\bigr]\,.
\end{align}
To obtain the mean-field Hamiltonian bilinear in the neutrino fields we need
to replace products of two of the four neutrino fields in this expression by
their expectation value.
The only combinations that do not violate lepton number are of the type
$\langle\bar{\nu}_i \nu_j\rangle$ and $\langle\nu_i \bar{\nu}_j\rangle$,
where $i$ and $j$ can be equal or different. We denote the corresponding
mean field as
\begin{align}
\label{Inu}
I^\mu_{ij}\equiv \langle \bar{\nu}_j\gamma^\mu P_L\nu_i \rangle\,.
\end{align}
To simplify notation we avoid an explicit ``neutrino'' and ``nc'' index,
i.e., expressions of the type $I^\mu_{ij}$ always refer to the neutral neutrino current
for the mass states $i$ and $j$. An explicit expression in terms of the
density matrices and pair correlators will be given in
Eq.~\eqref{InuExplicit} below.
For the $i=j$ contractions it is sufficient to take the
expectation value of one of the square brackets in Eq.~\eqref{Hself}, leading
to the mean-field Hamiltonian
$\sqrt{2}G_{\rm F}\sum_{ij} \bigl[\bar{\nu}_i\gamma_\mu P_L\nu_i\bigr]\,
I^\mu_{jj}$. For the $i\neq j$ contractions we use the Fierz identity to rewrite
the Hamiltonian as $[\bar{\nu}_i\gamma_\mu P_L\nu_j][\bar{\nu}_j\gamma^\mu P_L\nu_i]$ in
Eq.\,\eqref{Hself}, leading to the contribution
$\sqrt{2}G_{\rm F}\sum_{ij} \bigl[\bar{\nu}_i\gamma_\mu P_L\nu_j\bigr] I^\mu_{ij}$.
Altogether, we find
\begin{align}
\label{HselfEff}
{\cal H}^{\nu\nu}_{\rm mf}=
\sqrt{2}G_{\rm F} \sum\limits_{ij}\bigl[\bar{\nu}_i\gamma_\mu P_L\nu_j\bigr]
\bigl[I_{ij}^\mu+\delta_{ij}\,\textstyle{\sum_k} I^\mu_{kk}\bigr]
\end{align}
for the neutrino-neutrino mean-field Hamiltonian.
\subsection{Components of the Hamiltonian matrix \boldmath{$\sf H$}}
Adding up Eqs.~\eqref{HCCeff}, \eqref{HNCeff}, and \eqref{HselfEff} we find
the overall mean-field current
\begin{align}
\label{SigmaDirac}
V^\mu_{ij} = \sqrt{2}G_{\rm F}\bigl[&
U^\dagger_{ie} I_{\rm cc}^\mu U_{ej}
+\delta_{ij}(I_{\rm nc}^\mu+I_p^\mu+I_n^\mu)\nonumber\\
&{}+I^\mu_{ij}+\delta_{ij}\,\textstyle{\sum_k} I^\mu_{kk}\bigr]\,.
\end{align}
The spinor contractions defined in Eq.~\eqref{GammasDef} lead to the
components of the Hamiltonian matrix ${\sf H}$ of the form
\begin{subequations}
\label{GammaDiracFull}
\begin{align}
\label{GammaDiracnunu}
{\sf H}^{\nu\nu}_{ij,sh}&=(\gamma_\mu P_L)^{\nu\nu}_{ij,sh}V^\mu_{ij}
+\delta_{sh}\delta_{ij}E_{i}\,,\\
{\sf H}^{\nu\bar\nu}_{ij,sh}&=(\gamma_\mu P_L)^{\nu\bar\nu}_{ij,sh}V^\mu_{ij}\,,\\
{\sf H}^{\bar\nu\nu}_{ij,sh}&=(\gamma_\mu P_L)^{\bar\nu\nu}_{ij,sh}V^\mu_{ij}\,,\\
{\sf H}^{\bar\nu\bar\nu}_{ij,sh}&=(\gamma_\mu P_L)^{\bar\nu\bar\nu}_{ij,sh}V^\mu_{ij}
-\delta_{sh}\delta_{ij}E_{i}\,,
\end{align}
\end{subequations}
where $E_i=(\vec{p}^2+m_i^2)^\frac12$ is the neutrino energy, and
we have identified \smash{${\sf H}^{\nu\nu}=\Gamma^{\nu\nu}$},
\smash{${\sf H}^{\nu\bar\nu}=\Gamma^{\nu\bar\nu}$},
etc. We have used
the compact notation
\begin{subequations}\label{eq:contractiondefinition}
\begin{align}
(\gamma_\mu P_L)^{\nu\nu}_{ij,sh} &\equiv\bar{u}_{i,s}(+\vec{p})\gamma_\mu P_L u_{j,h}(+\vec{p})\,,\\
(\gamma_\mu P_L)^{\nu\bar\nu}_{ij,sh} &\equiv\bar{u}_{i,s}(+\vec{p})\gamma_\mu P_L v_{j,h}(-\vec{p})\,,\\
(\gamma_\mu P_L)^{\bar\nu\nu}_{ij,sh} &\equiv\bar{v}_{i,s}(-\vec{p})\gamma_\mu P_L u_{j,h}(+\vec{p})\,,\\
(\gamma_\mu P_L)^{\bar\nu\bar\nu}_{ij,sh}&\equiv\bar{v}_{i,s}(-\vec{p})\gamma_\mu P_L v_{j,h}(-\vec{p})\,.
\end{align}
\end{subequations}
Later we will use similar expressions for contractions with other Dirac structures.
The neutrino mean-field current itself contains spinor contractions of this type and
can be expressed in terms of the density matrices and pair correlations as
\begin{align}
\label{InuExplicit}
I^\mu_{ij}=\int_{\vec{p},s,h} \Bigl[&(\gamma^\mu P_L)^{\nu\nu}_{ji,hs}\rho_{ij,sh}
\nonumber\\[-4pt]
&{}+(\gamma^\mu P_L)^{\nu\bar\nu}_{ji,hs}\kappa^\dagger_{ij,sh}\nonumber\\[2pt]
&{}+(\gamma^\mu P_L)^{\bar\nu\nu}_{ji,hs}\kappa_{ij,sh}
\nonumber\\
&{}+(\gamma^\mu P_L)^{\bar\nu\bar\nu}_{ji,hs}(\delta_{ij}\delta_{sh}-\bar{\rho}_{ij,sh})
\,\Bigr]\,.
\end{align}
Notice that in this case there is no implied summation over $i$ and $j$.
The fourth term contains a divergent vacuum contribution that must be renormalized.
We finally work out the spinor contractions explicitly to lowest order in neutrino
masses. To this end we introduce
\begin{align}
n^\mu=
\left(1,\hat{\vec p}\right)\,,\quad
\bar{n}^\mu=
\left(1,-\hat{\vec p}\right)\,,
\quad
\epsilon^\mu=
\left(0,\hat{\vec \epsilon}\right)\,,
\end{align}
where $\hat{\vec p}$ is a unit vector in the momentum direction
and the complex polarization vector $\hat{\vec \epsilon}$ spans
the plane orthogonal to~${\vec p}$ (see Appendix \ref{sec:spinorproducts} for more
details). We also use $\phi$ to denote the polar angle
of $\vec{p}$ in spherical coordinates. To lowest order in $m_i$, the
spinor contractions are then found to be
\begin{subequations}\label{Xinunu}
\begin{align}
(\gamma_\mu P_L)^{\nu\nu}_{ij,sh} &\approx
\begin{pmatrix}
n_\mu & -e^{+i\phi}\frac{m_j}{2p}\epsilon^*_\mu \\
-e^{-i\phi}\frac{m_i}{2p}\epsilon_\mu & 0
\end{pmatrix}\,,\label{eq:contraction-nunu}\\[3pt]
(\gamma_\mu P_L)^{\nu\bar\nu}_{ij,sh} &\approx
\begin{pmatrix}
-e^{+i\phi}\frac{m_j}{2p}n_\mu & \epsilon^*_\mu \\
0 & -e^{-i\phi}\frac{m_i}{2p}\bar{n}_\mu
\end{pmatrix}
\,,\\[3pt]
(\gamma_\mu P_L)^{\bar\nu\nu}_{ij,sh} &\approx
\begin{pmatrix}
-e^{-i\phi}\frac{m_i}{2p}n_\mu & 0 \\[2pt]
\epsilon_\mu & -e^{+i\phi}\frac{m_j}{2p}\bar{n}_\mu
\end{pmatrix}
\,,\\[3pt]
(\gamma_\mu P_L)^{\bar\nu\bar\nu}_{ij,sh}&\approx
\begin{pmatrix}
0 & -e^{-i\phi}\frac{m_i}{2p}\epsilon^*_\mu \\
-e^{+i\phi}\frac{m_j}{2p}\epsilon_\mu & \bar{n}_\mu
\end{pmatrix}\,,
\end{align}
\end{subequations}
where we use the notation introduced in Eq.\,\eqref{spinflavorstruct}.
As an example, the $\nu\nu$ term, Eq.~\eqref{eq:contraction-nunu},
reads explicitly
\begin{subequations}
\label{gnuPLexpl}
\begin{align}
\framebox{\hbox to 0pt{$\vphantom{+}$}$--$}_{~ij}^{~\nu\nu}&=
\begin{pmatrix}
1 & 1 & 1\\
1 & 1 & 1\\
1 & 1 & 1
\end{pmatrix}
\,n_\mu\,,\\
\framebox{$-+$}_{~ij}^{~\nu\nu}&=
- \frac{1}{2p}
\begin{pmatrix}
m_1 & m_2 & m_3\\
m_1 & m_2 & m_3\\
m_1 & m_2 & m_3\\
\end{pmatrix}
e^{+i\phi}\epsilon^*_\mu\,,\\
\framebox{$+-$}_{~ij}^{~\nu\nu}&=
-\frac{1}{2p}
\begin{pmatrix}
m_1 & m_1 & m_1\\
m_2 & m_2 & m_2\\
m_3 & m_3 & m_3\\
\end{pmatrix}
e^{-i\phi}\epsilon_\mu\,,\\
\framebox{$++$}_{~ij}^{~\nu\nu}&=0
\,.
\end{align}
\end{subequations}
These results agree with those
obtained in Ref.~\cite{Serreau:2014cfa}.
\section{\label{sec:majorana}Majorana neutrino}
From a theoretical perspective, it is quite natural for neutrino
masses to be of Majorana type. In this case, the two helicity states
of a given family coincide with the $\nu$ and $\bar\nu$ states, the
mass term violates lepton number, and there are no sterile degrees of
freedom. We work out the modifications of the results of the previous
section for the Majorana case, concentrating again on technical
issues.
\subsection{Two-point correlators and kinetic equations}
In the Majorana case, the momentum decomposition of the neutrino field
looks the same as for the Dirac case
Eq.~\eqref{eq:Dirac-decomposition}. However, because there are no
independent antiparticle degrees of freedom, the field mode $\vec p$
has the simpler form
\begin{align}
\label{nudecompositionmaj}
\nu_{i,s}(t,\vec{p})&= a_{i,s}(t,\vec{p})u_{i,s}(\vec{p})
+a^\dagger_{i,s}(t,-\vec{p})v_{i,s}(-\vec{p})\,.
\end{align}
The creation and annihilation operators satisfy the same anticommutation relations
Eq.~\eqref{AnticommRels} and the bispinors are the same as in the
Dirac case.
The definitions of the two-point correlation functions are different
because of the different particle content,
\begin{subequations}
\label{MajCorrelators}
\begin{align}
(2\pi)^3\delta(\vec{p}{-}\vec{k})&\rho_{ij,sh} (\vec{p})
=\langle a^\dagger_{j,h}(+\vec{k}) a_{i,s}(+\vec{p})\rangle\,,\\
(2\pi)^3\delta(\vec{p}{-}\vec{k})&\bar{\rho}_{ij,sh} (\vec{p})
=\langle a^\dagger_{i,s}(-\vec{p}) a_{j,h}(-\vec{k})\rangle\,,\\
(2\pi)^3\delta(\vec{p}{-}\vec{k})&\kappa_{ij,sh} (\vec{p})
=\langle a_{j,h}(-\vec{k}) a_{i,s}(+\vec{p})\rangle\,,\\
(2\pi)^3\delta(\vec{p}{-}\vec{k})&\kappa^\dagger_{ij,sh} (\vec{p})
=\langle a^\dagger_{j,h}(+\vec{p}) a^\dagger_{i,s}(-\vec{k})\rangle\,,
\end{align}
\end{subequations}
where all operators are taken at the same time $t$. In the Dirac case,
$\kappa^\dagger$ has no additional information relative to
$\kappa$. Here we have additional redundancies
\begin{subequations}
\label{RhoMajProperties}
\begin{align}
\bar{\rho}_{ij,sh} (t,\vec{p})&=\rho_{ji,hs} (t,-\vec{p})\,,\\
\kappa_{ij,sh} (t,\vec{p})&=-\kappa_{ji,hs} (t,-\vec{p})\,,
\end{align}
\end{subequations}
which reflect that Majorana neutrinos have half as many degrees of freedom as
Dirac ones. Note that in the Majorana case, the pair correlations violate
total lepton number.
The mean-field Hamiltonian, bilinear in the neutrino
creation and annihilation operators, has the same form Eq.~\eqref{Heff} as in
the Dirac case. However, as we will demonstrate below, the kernel has a more general structure,
\begin{align}
\label{MajKernel}
\Gamma_{ij} = \gamma_\mu P_L V^\mu_{ij} + P_L V^R_{ij}+ P_R V^L_{ij}\,.
\end{align}
The first piece, $V^\mu_{ij}$, is defined as in Eq.~\eqref{SigmaDirac}.
In addition, there are two scalar pieces
\begin{align}
\label{Vpot}
V^{L,R}_{ij}=\sqrt{2}G_{\rm F} I^{L,R}_{ij}\,,
\end{align}
depending, as we will see, on the left-chiral and right-chiral
neutrino mean-field scalar background
\begin{align}
\label{VLVRdef}
I^{L,R}_{ij} = \langle \bar{\nu}_j P_{L,R} \nu_i\rangle\,.
\end{align}
These scalar pieces are missing in the previous literature.\footnote{In a private
communication, the authors of Ref.~\cite{Serreau:2014cfa} agree that these terms
should indeed be present in the Majorana case. Of course, the presence of these
terms does not modify the overall structure of the kinetic equations.} Their
explicit form in terms of the density matrices and pair correlators will
be given in Eq.~\eqref{IL}.
The mean-field Hamiltonian can be written in a form similar to Eq.~\eqref{HeffCrAn},
\begin{align}
\label{HeffCrAnMaj}
H_{\rm mf}= \int_{\vec{p},s,h}& \Bigl[a^\dagger_{i,s}(\vec{p}) \Gamma^{\nu\nu}_{ij,sh}(\vec{p}) a_{j,h}(\vec{p})\nonumber\\[-4pt]
&{}+a^\dagger_{i,s}(\vec{p}) \Gamma^{\nu\bar\nu}_{ij,sh}(\vec{p}) a^\dagger_{j,h}(-\vec{p})\nonumber\\[2pt]
&{}+a_{i,s}(-\vec{p}) \Gamma^{\bar\nu\nu}_{ij,sh}(\vec{p}) a_{j,h}(\vec{p})\nonumber\\
&{}+a_{i,s}(-\vec{p}) \Gamma^{\bar\nu\bar\nu}_{ij,sh}(\vec{p}) a^\dagger_{j,h}(-\vec{p})\Bigr]\,,
\end{align}
where the matrices $\Gamma^{\nu\nu}$, $\Gamma^{\nu\bar\nu}$, etc.\ are the spinor
contractions defined in Eq.\,\eqref{GammasDef}. Using the Heisenberg equation of
motion with the Hamiltonian Eq.~\eqref{HeffCrAnMaj} one recovers the equation of
motion $i\dot{\sf R}=[{\sf H},{\sf R}]$, where $\sf R$ and $\sf H$ have the same
structure as in Eq.\,\eqref{RHstructure}. The components of the effective Hamiltonian
now read \cite{Serreau:2014cfa}
\begin{subequations}
\label{GammasMajDef}
\begin{align}
{\sf H}^{\nu\nu}_{ij,sh}(\vec{p})& = \Gamma^{\nu\nu}_{ij,sh}(\vec{p}) -
\Gamma^{\bar\nu\bar\nu}_{ji,hs}(-\vec{p})\,,\\
\label{GammanubarnuMaj}
{\sf H}^{\nu\bar\nu}_{ij,sh}(\vec{p})& = \Gamma^{\nu\bar\nu}_{ij,sh}(\vec{p}) -
\Gamma^{\nu\bar\nu}_{ji,hs}(-\vec{p})\,,\\
\label{GammabarnunuMaj}
{\sf H}^{\bar\nu\nu}_{ij,sh}(\vec{p})& = \Gamma^{\bar\nu\nu}_{ij,sh}(\vec{p}) -
\Gamma^{\bar\nu\nu}_{ji,hs}(-\vec{p})\,,\\
{\sf H}^{\bar\nu\bar\nu}_{ij,sh}(\vec{p})& = \Gamma^{\bar\nu\bar\nu}_{ij,sh}(\vec{p}) -
\Gamma^{\nu\nu}_{ji,hs}(-\vec{p})\,.
\end{align}
\end{subequations}
Not all of these components are independent. In particular
\begin{subequations}
\label{GammaMajProperties}
\begin{align}
{\sf H}^{\bar\nu\bar\nu}_{ij,sh}(\vec{p})=-{\sf H}^{\nu\nu}_{ji,hs}(-\vec{p})\,,\\
{\sf H}^{\nu\bar\nu}_{ij,sh}(\vec{p})=-{\sf H}^{\nu\bar\nu}_{ji,hs}(-\vec{p})\,,
\end{align}
\end{subequations}
so only two of the four submatrices of $\sf H$ are independent.
\subsection{Neutrino-neutrino mean-field Hamiltonian}
The Majorana neutrino interaction with matter is described by the same charged-
and neutral-current Hamiltonian densities Eqs.~\eqref{HCC} and~\eqref{HNC} which lead
to the same mean-field currents of electrons and nucleons---see Eqs.~\eqref{HCCeff}
and \eqref{HNCeff}.
The neutrino-neutrino interaction in the Majorana case is also described by Eq.~\eqref{Hself}.
However, Majorana neutrinos violate lepton-number conservation,
and in addition to the four lepton-number-conserving
combinations considered in Sec.~\ref{sec:dirac} one should also take into account the
lepton-number-violating combinations \smash{$\langle\nu_i\nu_j\rangle$} and
\smash{$\langle\bar\nu_i \bar{\nu}_j\rangle$} which were not included in the
previous literature.
To calculate these additional contractions, we use the definition of the charge-conjugate
field \smash{$\nu^c\equiv C\bar{\nu}^T$}, where $C$ is the charge-conjugation matrix which
has the property $C^T C=1$. Using this definition
\smash{$\bar{\nu}=(\nu^c)^TC$} and \smash{$\nu=C\gamma^0(\nu^c)^*$}, which further
implies \smash{$\bar{\nu}\gamma^\mu P_L\nu=-\overline{\nu^c}\gamma^\mu P_R \nu^c$}.
Therefore, we can rewrite the Hamiltonian as \smash{$-\bigl[\bar{\nu}_i\gamma^\mu P_L\nu_i\bigr]
\bigl[\overline{\nu^c_j}\gamma_\mu P_R \nu^c_j\bigr]$}
in Eq.~\eqref{Hself}. The Fierz identity \cite{Nishi:2004st} $(\gamma^\mu P_L)[\gamma_\mu P_R]=2(P_R][P_L)$
further allows us to rewrite it as \smash{$2\bigl[\bar{\nu}_i P_R\nu^c_j\bigr]
\bigl[\overline{\nu^c_j} P_L\nu_i\bigr]$}, where another sign change was induced by
anticommuting the neutrino fields. Taking the expectation value of one of the
square brackets we obtain for the new contribution to the mean-field Hamiltonian density
\begin{align}
\label{Heff3}
{\cal H}_{\rm mf}^{\nu\nu}= \sqrt{2}G_{\rm F}\sum\limits_{ij}
\bigl(&\bigl[\bar{\nu}_i P_R\nu^c_j\bigr]I^L_{ij}
+\bigl[\overline{\nu^c_i} P_L\nu_j\bigr]I^R_{ij} \bigr)\,.
\end{align}
These new terms supplement the expression for the effective
Majorana Hamiltonian obtained in the previous literature \cite{Serreau:2014cfa}.
In Appendix~\ref{sec:contractions} we reproduce this result using
two-component notation.
Two comments are in order here. First,
for Majorana fer\-mi\-ons $\nu^c=\nu$ and therefore the resulting contribution to the
kernel reduces to the last two terms in Eq.\,\eqref{MajKernel}, while the definition
of left- and right-chiral neutrino backgrounds reduces to Eq.\,\eqref{Vpot}.
Second, \smash{$\bar{\nu} P_R\nu^c=\bar{\nu}_L\nu_L^c$} and
\smash{$\overline{\nu^c} P_L\nu=\overline{\nu^c_L}\nu_L$},
where \smash{$\nu_L\equiv P_L\nu$}, which are nothing but
components of the Majorana mass term.
\subsection{\label{EffHamMaj}Components of the Hamiltonian matrix \boldmath{$\sf H$}}
The new contributions stemming from neutrino-neutrino interactions can be expressed
in terms of the (anti)particle densities and pair correlators,
\begin{align}
\label{IL}
I^L_{ij}=\int_{\vec{p},s,h}\Bigl[&\,(P_L)^{\nu\nu}_{ji,hs}\rho_{ij,sh}
+(P_L)^{\bar\nu\bar\nu}_{ji,hs}(\delta_{ij}\delta_{sh}-\bar{\rho}_{ij,sh})\nonumber\\[-5pt]
&{}+(P_L)^{\bar\nu\nu}_{ji,hs}\kappa_{ij,hs}
+(P_L)^{\nu\bar\nu}_{ji,hs}\kappa^\dagger_{ij,sh}\Bigr]\,,
\end{align}
where we have again suppressed the common arguments $\vec p$ and $(t,\vec{p})$. The notation
for the scalar contractions
$(P_L)^{\nu\nu}_{ij,sh}$ etc.\ is analogous to Eq.~\eqref{eq:contractiondefinition}, except
that now there is no $\gamma^\mu$ included.
To lowest order in the small neutrino masses we find, using the explicit form of the chiral spinors
of Appendix~\ref{sec:spinorproducts},
\begin{subequations}
\label{XiL}
\begin{align}
(P_L)^{\nu\nu}_{ij,sh}&\approx
\begin{pmatrix}
\frac{m_i}{2p} & 0 \\
0 &\frac{m_j}{2p}
\end{pmatrix}\,,\\
(P_L)^{\nu\bar\nu}_{ij,sh}
&\approx
\begin{pmatrix}
0 & 0 \\
0 & -e^{-i\phi}
\end{pmatrix}\,,\\
(P_L)^{\bar\nu\nu}_{ij,sh}
&\approx
\begin{pmatrix}
e^{-i\phi} & 0 \\
0 & 0
\end{pmatrix}\,,\\
(P_L)^{\bar\nu\bar\nu}_{ij,sh}&\approx
\begin{pmatrix}
-\frac{m_j}{2p} & 0 \\
0 & -\frac{m_i}{2p}
\end{pmatrix}\,.
\end{align}
\end{subequations}
The components of $(P_R)$ can be obtained from these results using the relations
$(P_R)^{\nu\nu}_{ij,sh}=[(P_L)^{\nu\nu}_{ji,hs}]^*$ and
$(P_R)^{\nu\bar\nu}_{ij,sh}=[(P_L)^{\bar\nu\nu}_{ji,hs}]^*$, as well
as similar relations for the remaining two components.
Using the definitions Eq.~\eqref{GammasMajDef} combined with
Eq.~\eqref{Xinunu} and the corresponding definition for the scalar case
we obtain for the $\nu\nu$ component of $\sf H$
\begin{align}
\label{GammaMnunu}
{\sf H}^{\nu\nu}_{ij,sh}&(\vec{p})=\delta_{sh}\delta_{ij} E_i\nonumber\\
&+(\gamma_\mu P_L)^{\nu\nu}_{ij,sh}(\vec{p})V^\mu_{ij}-
(\gamma_\mu P_L)^{\bar\nu\bar\nu}_{ji,hs}(-\vec{p})V^\mu_{ji}\nonumber\\
&+(P_L)^{\nu\nu}_{ij,sh}(\vec{p})V^R_{ij}-
(P_L)^{\bar\nu\bar\nu}_{ji,hs}(-\vec{p})V^R_{ji}\nonumber\\
&+(P_R)^{\nu\nu}_{ij,sh}(\vec{p})V^L_{ij}-
(P_R)^{\bar\nu\bar\nu}_{ji,hs}(-\vec{p})V^L_{ji}\,.
\end{align}
The second line generalizes the Dirac result of Eq.~\eqref{GammaDiracnunu}
to the Majorana case and has been obtained in Ref.~\cite{Serreau:2014cfa}. The third and
fourth lines stem from the the contractions Eq.~\eqref{Heff3} and supplement the
previous results.
The $\bar\nu\bar\nu$ term follows from the identity Eq.~\eqref{GammaMajProperties}.
For the $\nu\bar\nu$ component we find
\begin{align}
\label{GammaMnunubar}
{\sf H}^{\nu\bar\nu}_{ij,sh}(\vec{p})
&=(\gamma_\mu P_L)^{\nu\bar\nu}_{ij,sh}(\vec{p})V^\mu_{ij}-
(\gamma_\mu P_L)^{\nu\bar\nu}_{ji,hs}(-\vec{p})V^\mu_{ji}\nonumber\\
&+(P_L)^{\nu\bar\nu}_{ij,sh}(\vec{p})V^R_{ij}-
(P_L)^{\nu\bar\nu}_{ji,hs}(-\vec{p})V^R_{ji}\nonumber\\
&+(P_R)^{\nu\bar\nu}_{ij,sh}(\vec{p})V^L_{ij}-
(P_R)^{\nu\bar\nu}_{ji,hs}(-\vec{p})V^L_{ji}\,.
\end{align}
The $\bar\nu\nu$ component follows from replacing
$\nu\bar\nu$ with $\bar\nu\nu$ everywhere in this result.
An inspection of Eqs.~\eqref{IL} and \eqref{XiL} shows that the last two lines of Eq.~\eqref{GammaMnunu}
contain terms proportional to $\kappa$ and $\kappa^\dagger$ that are linear
in the neutrino masses, and additionally terms quadratic in the neutrino masses which we neglect here.
A peculiar feature of Eq.~\eqref{GammaMnunubar} is
that its last two lines contain terms proportional to $\kappa$ and $\kappa^\dagger$ that
are not suppressed by the neutrino masses and therefore do not vanish when we set
the masses to zero. This is somewhat surprising because we expect that
Dirac and Majorana neutrinos are equivalent for $m_\nu\to0$.
Therefore, the components of ${\sf H}$ must coincide in this limit.
We return to this question in Sec.~\ref{sec:weyl},
where we study the case of massless two-component neutrinos and demonstrate that
in the massless limit these additional terms, which are proportional to the
lepton-number-violating correlators, are not produced if they are zero initially.
On the other hand, one important finding of our paper
is that for a Majorana neutrino with an arbitrary small mass, lepton-number-violating
correlators are automatically produced and, in turn, induce the additional scalar background terms of
the mean-field Hamiltonian which then affect the dynamics of the density matrices.
\section{\label{sec:weyl}Weyl neutrino}
In the previous section we have found that the additional scalar contributions
to the mean-field Hamiltonian, that naturally arise for Majorana neutrinos,
do not vanish in the massless limit. This is somewhat surprising
because we expect no difference between Dirac and Majorana neutrinos in this
case. To clarify this paradox we study a single generation of
massless neutrinos. The equations presented in this section will also be
used later to study particle-antiparticle coherence.
\subsection{Standard two-point correlators and kinetic equations}
In the Weyl case, the momentum decomposition of the neutrino field
looks the same as for the Dirac case Eq.~\eqref{eq:Dirac-decomposition}.
However, because a Weyl fermion has only two degrees of freedom the
field mode $\vec p$ does not carry a spin index,
\begin{align}
\label{nudecompositionweil}
\nu(t,\vec{p})&= a(t,\vec{p})u_{_-}(\vec{p})+ b^\dagger(t,-\vec{p})v_{_+}(-\vec{p})\,.
\end{align}
It is automatically left-chiral because the right-chiral components of the chiral
spinors $u_{_-}(\vec{p})$ and $v_{_+}(-\vec{p})$ vanish in the massless limit,
see Appendix \ref{sec:spinorproducts}.
If we require lepton-number conservation then the only correlators that we can
define are
\begin{subequations}
\label{RhoKappaWeylCons}
\begin{align}
(2\pi)^3\delta(\vec{p}-\vec{k})\rho_{{\scalebox{0.9}[0.9]{-}\smi}} (\vec{p})&=
\langle a^\dagger(\vec{k}) a(\vec{p})\rangle\,,\\
(2\pi)^3\delta(\vec{p}-\vec{k})\bar{\rho}_{{\scalebox{0.75}[0.75]{+}\spi}} (\vec{p})&=
\langle b^\dagger(-\vec{k}) b(-\vec{p})\rangle\,,\\
(2\pi)^3\delta(\vec{p}-\vec{k})\kappa_{{\scalebox{0.9}[0.9]{-}\scalebox{0.75}[0.75]{+}}} (\vec{p})&=
\langle b(-\vec{k}) a(\vec{p})\rangle\,,\\
(2\pi)^3\delta(\vec{p}-\vec{k})\kappa^\dagger_{{\scalebox{0.75}[0.75]{+}\scalebox{0.9}[0.9]{-}}} (\vec{p})&=
\langle a^\dagger(\vec{p}) b^\dagger(-\vec{k})\rangle\,.
\end{align}
\end{subequations}
Note that we keep helicity indices in these definitions to distinguish
the lepton-number-conserving correlators from the lepton-number-violating ones,
which we introduce below.
We can extract the explicit form of the kinetic equations for these correlators
from Eq.\,\eqref{eq:EOM1},
\begin{subequations}
\label{RhoKappaEqWeyl}
\begin{align}
\label{RhoKappaEqWeylnunu}
i\dot{\rho}_{{\scalebox{0.9}[0.9]{-}\smi}}&={\sf H}^{\nu\nu}_{{\scalebox{0.9}[0.9]{-}\smi}}\rho_{{\scalebox{0.9}[0.9]{-}\smi}}-
\rho_{{\scalebox{0.9}[0.9]{-}\smi}}{\sf H}^{\nu\nu}_{{\scalebox{0.9}[0.9]{-}\smi}}+{\sf H}^{\nu\bar\nu}_{{\scalebox{0.9}[0.9]{-}\scalebox{0.75}[0.75]{+}}}\kappa^\dagger_{{\scalebox{0.75}[0.75]{+}\scalebox{0.9}[0.9]{-}}}-
\kappa_{{\scalebox{0.9}[0.9]{-}\scalebox{0.75}[0.75]{+}}}{\sf H}^{\bar\nu\nu}_{{\scalebox{0.75}[0.75]{+}\scalebox{0.9}[0.9]{-}}}\,,\\
\label{RhoKappaEqWeylnubarnubar}
i\dot{\bar\rho}_{{\scalebox{0.75}[0.75]{+}\spi}}&={\sf H}^{\bar\nu\bar\nu}_{{\scalebox{0.75}[0.75]{+}\spi}}\bar{\rho}_{{\scalebox{0.75}[0.75]{+}\spi}}-
\bar{\rho}_{{\scalebox{0.75}[0.75]{+}\spi}}{\sf H}^{\bar\nu\bar\nu}_{{\scalebox{0.75}[0.75]{+}\spi}}
-{\sf H}^{\bar\nu\nu}_{{\scalebox{0.75}[0.75]{+}\scalebox{0.9}[0.9]{-}}}\kappa_{{\scalebox{0.9}[0.9]{-}\scalebox{0.75}[0.75]{+}}}
+\kappa^\dagger_{{\scalebox{0.75}[0.75]{+}\scalebox{0.9}[0.9]{-}}}{\sf H}^{\nu\bar\nu}_{{\scalebox{0.9}[0.9]{-}\scalebox{0.75}[0.75]{+}}}\,,\\
\label{KappaEq}
i\dot{\kappa}_{{\scalebox{0.9}[0.9]{-}\scalebox{0.75}[0.75]{+}}}&={\sf H}^{\nu\nu}_{{\scalebox{0.9}[0.9]{-}\smi}}\kappa_{{\scalebox{0.9}[0.9]{-}\scalebox{0.75}[0.75]{+}}}-
\kappa_{{\scalebox{0.9}[0.9]{-}\scalebox{0.75}[0.75]{+}}}{\sf H}^{\bar\nu\bar\nu}_{{\scalebox{0.75}[0.75]{+}\spi}}
-{\sf H}^{\nu\bar\nu}_{{\scalebox{0.9}[0.9]{-}\scalebox{0.75}[0.75]{+}}}\bar{\rho}_{{\scalebox{0.75}[0.75]{+}\spi}}-
\rho_{{\scalebox{0.9}[0.9]{-}\smi}}{\sf H}^{\nu\bar\nu}_{{\scalebox{0.9}[0.9]{-}\scalebox{0.75}[0.75]{+}}}\nonumber\\
&+{\sf H}^{\nu\bar\nu}_{{\scalebox{0.9}[0.9]{-}\scalebox{0.75}[0.75]{+}}}\,,
\end{align}
\end{subequations}
where we omit the arguments $(t,\vec{p})$, which are common to all the functions,
to shorten the notation. Note that for a single neutrino generation the first two
terms in Eqs.~\eqref{RhoKappaEqWeylnunu} and \eqref{RhoKappaEqWeylnubarnubar} cancel
each other and we have retained them only to keep the resemblance with the general
form of the kinetic equations.
A peculiar feature of Eq.~\eqref{KappaEq} is that $\kappa$,
i.e.\ the coherence between $|00\rangle$ and $|11\rangle$ states, is automatically induced
provided that the mean-field Hamiltonian $\sf H$ has nonzero off-diagonals. The
off-diagonals can be induced even if all neutrino two-point functions are zero initially
by, for instance, a transverse neutron current.
The explicit form of the mean-field Hamiltonian can be obtained from
Eq.~\eqref{Xinunu} by setting the masses to zero,
\begin{subequations}
\label{GammaWeyl}
\begin{align}
\label{GammaWeylnunu}
{\sf H}^{\nu\nu}_{{\scalebox{0.9}[0.9]{-}\smi}}(\vec{p})&=E+V^0-\hat{\vec p}\vec{V}\,,\\
\label{GammaNuBarnu}
{\sf H}^{\nu\bar\nu}_{{\scalebox{0.9}[0.9]{-}\scalebox{0.75}[0.75]{+}}}(\vec{p})&=-\hat{\vec\epsilon}^* \vec{V}\,,\\
{\sf H}^{\bar\nu\nu}_{{\scalebox{0.75}[0.75]{+}\scalebox{0.9}[0.9]{-}}}(\vec{p})&=-\hat{\vec\epsilon}\, \vec{V}\,,\\
{\sf H}^{\bar\nu\bar\nu}_{{\scalebox{0.75}[0.75]{+}\spi}}(\vec{p})&=- E+V^0+\hat{\vec p}\vec{V}\,,
\end{align}
\end{subequations}
where $E=\abs{\vec{p}}$. Note that the $\hat{\vec p}\vec{V}$ term in Eq.~\eqref{GammaWeylnunu} accounts for
the enhancement (suppression) of the mean-field potential for the matter flowing
antiparallel (parallel) to the neutrino momentum. This has been pointed out in
Ref.~\cite{Grigoriev:2002zr}.
It remains to express the neutrino current $I^\mu$ in terms of the density
matrices and pair correlations. For its time component we obtain from
Eq.\,\eqref{InuExplicit}, \smash{$I^0=\int_{\vec p} \ell$},
where $\ell(t,\vec{p})\equiv \rho(t,\vec{p})-\bar{\rho}(t,-\vec{p})$
has the meaning of lepton number in mode $\vec p$. For the spatial
components we find
\begin{align}
\label{Inuvec}
\vec{I}&=\int_{\vec p}\bigl[
\hat{\vec p}\,\ell+\hat{\vec \epsilon}\,\kappa
+\hat{\vec \epsilon}^*\kappa^\dagger\bigr]\,,
\end{align}
which coincides with the result of Ref.~\cite{Serreau:2014cfa}.
\subsection{Lepton-number-violating correlators and kinetic equations}
If we allow for \smash{$\langle\nu\nu\rangle$} and
\smash{$\langle\bar\nu \bar{\nu}\rangle$} contractions then, similarly to the
Majorana case, the mean-field Hamiltonian receives contributions of the type
Eq.\,\eqref{Heff3}. Because Weyl fields satisfy the condition $P_L\nu=\nu$
we can rewrite Eq.\,\eqref{Heff3} as
\begin{align}
\label{HeffWeyl}
\mathcal{H}^{\nu\nu}_{\rm mf}= \sqrt{2}G_{\rm F}\!\sum\limits
\bigl(\bigl[\bar{\nu} \nu^c\bigr]I^L
\!+\!\bigl[\overline{\nu^c} \nu\bigr]I^R \bigr)\,
\end{align}
(see Sec.\,\ref{sec:majorana} and Appendix \ref{sec:contractions} for more details) where now
\begin{align}
\label{SigmaWeyl}
I^L= \langle \overline{\nu^c} \nu\rangle\,\quad {\rm and}\quad
I^R = \langle \bar{\nu} \nu^c\rangle\,.
\end{align}
As has been mentioned above $\overline{\nu^c} \nu$ and $\bar{\nu} \nu^c$
have the structure of the Majorana mass term, which is known to violate lepton number.
Therefore, we expect that also for the
Weyl neutrino the mean-field Hamiltonian Eq.~\eqref{HeffWeyl} leads to lepton number violation. However, for
Weyl neutrinos the inclusion of these additional terms is somewhat artificial because,
as we show below, these correlations are not produced if they are zero initially.
They are considered here to better understand the Majorana case, where they are
naturally produced by the lepton-number-violating interactions.
The contribution of Eq.\,\eqref{HeffWeyl} to the mean-field Hamiltonian is given by
\begin{align}
\label{HLeptViol}
H_{\rm mf}= \int_{\vec{p}} \bigl[&a^\dagger(\vec{p}) \Gamma^{\nu\bar\nu}_{{\scalebox{0.9}[0.9]{-}\smi}}(\vec{p}) a^\dagger(-\vec{p})\nonumber\\[-2mm]
+\, &b^\dagger(\vec{p}) \Gamma^{\nu\bar\nu}_{{\scalebox{0.75}[0.75]{+}\spi}}(\vec{p}) b^\dagger(-\vec{p})\nonumber\\
+\, &a(-\vec{p}) \Gamma^{\bar\nu\nu}_{{\scalebox{0.9}[0.9]{-}\smi}}(\vec{p}) a(\vec{p})\nonumber\\
+\, &b(-\vec{p}) \Gamma^{\bar\nu\nu}_{{\scalebox{0.75}[0.75]{+}\spi}}(\vec{p}) b(\vec{p})\bigr]\,,
\end{align}
and strongly resembles the mean-field Hamiltonian of Majorana neutrinos
Eq.\,\eqref{HeffCrAnMaj}.
From the structure of Eq.\,\eqref{HLeptViol} it is evident that, as expected,
it leads to the violation of lepton number. To take this into account we
are forced to introduce the following lepton-number-violating correlators,
\begin{subequations}
\label{RhoKappaWeylViol}
\begin{align}
(2\pi)^3\delta(\vec{p}-\vec{k})\kappa_{{\scalebox{0.9}[0.9]{-}\smi}} (\vec{p})&=
\langle a(-\vec{k}) a(\vec{p})\rangle\,,\\
(2\pi)^3\delta(\vec{p}-\vec{k})\kappa_{{\scalebox{0.75}[0.75]{+}\spi}} (\vec{p})&=
\langle b(-\vec{k}) b(\vec{p})\rangle\,,\\
(2\pi)^3\delta(\vec{p}-\vec{k})\kappa^\dagger_{{\scalebox{0.9}[0.9]{-}\smi}} (\vec{p})&=
\langle a^\dagger(\vec{p}) a^\dagger(-\vec{k})\rangle\,,\\
(2\pi)^3\delta(\vec{p}-\vec{k})\kappa^\dagger_{{\scalebox{0.75}[0.75]{+}\spi}} (\vec{p})&=
\langle b^\dagger(\vec{p}) b^\dagger(-\vec{k})\rangle\,,
\end{align}
\end{subequations}
which also resemble the Majorana definitions Eq.\,\eqref{MajCorrelators}.
These correlators are dictated by the structure of
the Hamiltonian Eq.\,\eqref{HLeptViol} and are the only lepton-number-violating
correlators we consider in this section. If we wanted to consider all other
possible correlators we would be back to the Majorana case with zero neutrino
masses.
The lepton-number-violating correlators contribute to the dynamics of the
lepton-number-conserving ones,
\begin{subequations}
\label{RhoKappaEqLeptViol}
\begin{align}
i\dot{\rho}_{{\scalebox{0.9}[0.9]{-}\smi}}&=\ldots\nonumber\\
&+\bigl(\Gamma^{\nu\bar\nu}_{{\scalebox{0.9}[0.9]{-}\smi}}-[\Gamma^{\nu\bar\nu}_{{\scalebox{0.9}[0.9]{-}\smi}}]^T\bigr)\kappa^\dagger_{{\scalebox{0.9}[0.9]{-}\smi}}
-\kappa_{{\scalebox{0.9}[0.9]{-}\smi}}\bigl(\Gamma^{\bar\nu\nu}_{{\scalebox{0.9}[0.9]{-}\smi}}-[\Gamma^{\bar\nu\nu}_{{\scalebox{0.9}[0.9]{-}\smi}}]^T\bigr)\,,\\
i\dot{\bar\rho}_{{\scalebox{0.75}[0.75]{+}\spi}}&=\ldots\nonumber\\
&-\bigl(\Gamma^{\bar\nu\nu}_{{\scalebox{0.75}[0.75]{+}\spi}}-[\Gamma^{\bar\nu\nu}_{{\scalebox{0.75}[0.75]{+}\spi}}]^T\bigr)\kappa_{{\scalebox{0.75}[0.75]{+}\spi}}
+\kappa^\dagger_{{\scalebox{0.75}[0.75]{+}\spi}}\bigl(\Gamma^{\nu\bar\nu}_{{\scalebox{0.75}[0.75]{+}\spi}}-[\Gamma^{\nu\bar\nu}_{{\scalebox{0.75}[0.75]{+}\spi}}]^T\bigr)\,,
\end{align}
\end{subequations}
where ellipses denote terms on the right-hand side of Eq.\,\eqref{RhoKappaEqWeyl},
and the superscript $T$ stands for transposition of the flavor and helicity indices, as well as
inversion of the momentum. Comparing Eq.\,\eqref{RhoKappaEqLeptViol} with
Eqs.\,\eqref{GammanubarnuMaj} and \eqref{GammabarnunuMaj} we see that we automatically
recover the ``Majorana'' definitions of the Hamiltonian matrix.
Note that to avoid confusion with the definitions of the elements of the mean-field
Hamiltonian, which are different for Dirac and Majorana neutrinos, we write the
right-hand side of Eq.\,\eqref{RhoKappaEqLeptViol} directly in terms of spinor contractions
defined in Eq.\,\eqref{GammasDef}.
The dynamics of $\kappa_{{\scalebox{0.9}[0.9]{-}\scalebox{0.75}[0.75]{+}}}$, see Eq.\,\eqref{KappaEq}, does not receive
any corrections. The reason is that the components of the mean-field Hamiltonian
needed to form the right spin combination with the lepton-number-violating correlators
in Eq.~\eqref{KappaEq} are zero for Weyl neutrinos. The kinetic equations for the
lepton-number-violating pair correlations read
\begin{subequations}
\begin{align}
i\dot{\kappa}_{{\scalebox{0.9}[0.9]{-}\smi}}&=
\Gamma^{\nu\nu}_{{\scalebox{0.9}[0.9]{-}\smi}}\kappa_{{\scalebox{0.9}[0.9]{-}\smi}}-
\kappa_{{\scalebox{0.9}[0.9]{-}\smi}}\bigl(-[\Gamma^{\nu\nu}_{{\scalebox{0.9}[0.9]{-}\smi}}]^T\bigr)
-\rho_{{\scalebox{0.9}[0.9]{-}\smi}}\bigl(\Gamma^{\nu\bar\nu}_{{\scalebox{0.9}[0.9]{-}\smi}}-[\Gamma^{\nu\bar\nu}_{{\scalebox{0.9}[0.9]{-}\smi}}]^T\bigr)\nonumber\\
&-\bigl(\Gamma^{\nu\bar\nu}_{{\scalebox{0.9}[0.9]{-}\smi}}-[\Gamma^{\nu\bar\nu}_{{\scalebox{0.9}[0.9]{-}\smi}}]^T\bigr)[\rho_{{\scalebox{0.9}[0.9]{-}\smi}}]^T
+\bigl(\Gamma^{\nu\bar\nu}_{{\scalebox{0.9}[0.9]{-}\smi}}-[\Gamma^{\nu\bar\nu}_{{\scalebox{0.9}[0.9]{-}\smi}}]^T\bigr)\,,\\
i\dot{\kappa}_{{\scalebox{0.75}[0.75]{+}\spi}}&= \bigl(-[\Gamma^{\bar\nu\bar\nu}_{{\scalebox{0.75}[0.75]{+}\spi}}]^T\bigr)\kappa_{{\scalebox{0.75}[0.75]{+}\spi}}
-\kappa_{{\scalebox{0.75}[0.75]{+}\spi}}\Gamma^{\bar\nu\bar\nu}_{{\scalebox{0.75}[0.75]{+}\spi}}
-\bigl(\Gamma^{\nu\bar\nu}_{{\scalebox{0.75}[0.75]{+}\spi}}-[\Gamma^{\nu\bar\nu}_{{\scalebox{0.75}[0.75]{+}\spi}}]^T\bigr)\bar{\rho}_{{\scalebox{0.75}[0.75]{+}\spi}}\nonumber\\
&-[\bar{\rho}_{{\scalebox{0.75}[0.75]{+}\spi}}]^T\bigl(\Gamma^{\nu\bar\nu}_{{\scalebox{0.75}[0.75]{+}\spi}}-[\Gamma^{\nu\bar\nu}_{{\scalebox{0.75}[0.75]{+}\spi}}]^T\bigr)
+\bigl(\Gamma^{\nu\bar\nu}_{{\scalebox{0.75}[0.75]{+}\spi}}-[\Gamma^{\nu\bar\nu}_{{\scalebox{0.75}[0.75]{+}\spi}}]^T\bigr)\,.
\end{align}
\end{subequations}
Their form can be guessed from Eq.\,\eqref{KappaEq} by replacing components
of the mean-field Hamiltonian with their ``Majorana'' counterparts, taking into account
that \smash{$\Gamma^{\bar\nu\bar\nu}_{{\scalebox{0.9}[0.9]{-}\smi}}=\Gamma^{\nu\nu}_{{\scalebox{0.75}[0.75]{+}\spi}}=0$}, and
replacing $\bar{\rho}_{{\scalebox{0.9}[0.9]{-}\smi}}$ by \smash{$[\rho_{{\scalebox{0.9}[0.9]{-}\smi}}]^T$} as well as
$\rho_{{\scalebox{0.75}[0.75]{+}\spi}}$ by \smash{$[\bar{\rho}_{{\scalebox{0.75}[0.75]{+}\spi}}]^T$}.
Using the explicit form of the chiral spinors (see Appendix \ref{sec:spinorproducts})
we obtain
\begin{subequations}
\label{GammaWeylnunuCorr}
\begin{align}
\Gamma^{\nu\bar\nu}_{{\scalebox{0.9}[0.9]{-}\smi}}(\vec{p})&=+e^{+i\phi}V^L\,,\\
\Gamma^{\nu\bar\nu}_{{\scalebox{0.75}[0.75]{+}\spi}}(\vec{p})&=-e^{-i\phi}V^R\,,\\
\Gamma^{\bar\nu\nu}_{{\scalebox{0.9}[0.9]{-}\smi}}(\vec{p})&=+e^{-i\phi}V^R\,,\\
\Gamma^{\bar\nu\nu}_{{\scalebox{0.75}[0.75]{+}\spi}}(\vec{p})&=-e^{+i\phi}V^L\,,
\end{align}
\end{subequations}
where \smash{$V^{L(R)}=\sqrt{2}G_{\rm F} I^{L(R)}$} are defined analogously to Eq.\,\eqref{Vpot}.
Let us now recall that \smash{$I^L$} and \smash{$I^R$} are produced
only by neutrino self-in\-teractions and are proportional to the lepton-number-violating pair correlations,
\begin{align}
I^L=\int_{\vec p} e^{-i\phi}\bigl[\kappa_{{\scalebox{0.9}[0.9]{-}\smi}}-\kappa^\dagger_{{\scalebox{0.75}[0.75]{+}\spi}}\bigr]\,,
\end{align}
and a similar expression for \smash{$I^{R}$}. Thus, if the lepton-number-violating
correlators are zero initially,
then the components in Eq.~\eqref{GammaWeylnunuCorr} are zero and $\kappa_{\scalebox{0.9}[0.9]{-}\smi}$ and
$\kappa_{\scalebox{0.75}[0.75]{+}\spi}$ remain zero in the course of the system's evolution. For this reason
for Weyl neutrinos the inclusion of lepton-number-violating correlators
is rather artificial because they could only exist if they were put in by hand initially.
This observation
explains why similar contributions do not vanish for Majorana neutrinos in the limit of
zero neutrino masses. While such lepton-number-violating correlators can be introduced
by hand as an initial condition, they can dynamically evolve only in the presence
of a nonvanishing Majorana mass.
\section{\label{sec:magnetic}Electromagnetic background fields}
A supernova environment is characterized not only by matter currents, but also by strong magnetic fields. Electromagnetic fields polarize both background media and the vacuum. Although neutrinos do not couple directly to the electromagnetic fields, they feel the induced polarization. The coupling to a polarized background medium has been treated in the previous sections. We now turn to the interaction with the vacuum polarization.
The effect of vacuum polarization is described by electromagnetic form factors. The most prominent examples, the magnetic and electric dipole moments, are inevitable for massive neutrinos and have to be included to obtain consistent evolution equations linear in the neutrino mass. The main effects of electromagnetic fields are spin and spin-flavor oscillations, which can be significant. We treat Dirac and Majorana neutrinos separately.
\subsection{General vertex structure}
The coupling of neutrinos to an external vector potential $A^\mu$ can be written as an effective vertex
$\mathcal{H}^{\rm em}= A_{\mu}\bar \nu \Gamma^{\mu} \nu$, where $\Gamma^{\mu}$ contains all
irreducible combinations of Lorentz vectors and pseudovectors generated by external momenta and
Dirac matrices. Neglecting a hypothetical minicharge, in coordinate space the most general Hamiltonian
density can be reduced to
\begin{align}
\label{eq:moment}
\mathcal{H}^{\rm em}&=\frac{1}{2}F_{\mu\nu}\,\bar \nu_i \bigl( f^{ij}_M \sigma^{\mu\nu} +if^{ij}_E\sigma^{\mu\nu}\gamma_5 \bigr)\nu_j\nonumber\\
&+\partial^\nu\! F_{\mu\nu} \, \bar \nu_i \bigl( f^{ij}_Q\gamma^\mu +f^{ij}_A\gamma^\mu\gamma_5\bigr) \nu_j\, ,
\end{align}
where the electromagnetic field-strength tensor is defined as usual, $F^{\mu\nu}=\partial^\mu A^\nu - \partial^\nu A^\mu$, and $\sigma^{\mu\nu}=\frac{i}{2} [\gamma^{\mu},\gamma^{\nu}]$. The form factors are $f_M$ (magnetic), $f_E$ (electric), $f_Q$ (reduced charge~\cite{Giunti:2014ixa}), and $f_A$ (anapole). The form factors carry generation indices. Diagonal elements describe the usual electromagnetic properties of a neutrino in the mass basis, and reduce to electromagnetic \emph{moments} in the static limit. The off-diagonal elements describe transitions between neutrinos of different masses. Some components of the Hamiltonian matrix have been calculated in Refs.~\cite{Giunti:2014ixa, Dvornikov:2011dv}.
Maxwell's equations tell us that $\partial_\nu F^{\mu\nu}=-J^\mu_{\text{em}}$, where $J^\mu_{\text{em}}$ is some charged matter background that sources electromagnetic fields. In supernovae, the sources are electrons and protons. In the Standard Model with massless neutrinos, the value for the anapole moment has to be $f_A=-f_Q$ to reproduce the left-chiral form of the interaction. For models with neutrino masses, the Hamiltonian matrix might obtain contributions that are not purely left-chiral, but we assume that these are always small so that we can neglect them. The charge and anapole form factors then only yield radiative corrections to the left-chiral tree-level coupling in Eq.~\eqref{HCC}. We neglect these moments because we are not interested in corrections to leading-order effects. However, for completeness, we give the spinor contractions for right-chiral currents in Appendix~\ref{sec:anapole}.
\subsection{Dipole moments of Dirac neutrinos}
To study the dipole moments, we first turn to the somewhat simpler case of Dirac neutrinos. A Dirac neutrino has diagonal magnetic and electric moments. Because we assume neutrinos to carry no charge, $\mu=f_M(0)$ is defined as the magnetic moment and $\epsilon=f_E(0)$ as the electric dipole moment~\cite{Giunti:2014ixa}. In the minimal extension of the Standard Model, the magnetic moments are found to be~\cite{Shrock:1982sc}
\begin{subequations}
\begin{align}
\label{eq:mmon1}
\mu_{ij} & = \frac{3e\sqrt{2}G_{\rm F} (m_i+m_j)}{ 2(4\pi)^2}\left(\delta_{ij}-\frac{m_\tau^2}{2m_W^2}\mathcal{F}_{ij}\right)\, ,\\
\label{eq:mmon2}\epsilon_{ij} & = i\frac{3e\sqrt{2}G_{\rm F} }{2 (4\pi)^2}(m_i-m_j)\left(\frac{m_\tau^2}{2m_W^2}\right)\mathcal{F}_{ij}\, ,
\end{align}
\begin{align}
\mathcal{F}_{ij} & =\sum_{\alpha=e,\mu,\tau} U^\dagger_{i \alpha} \left(\frac{m_\alpha}{m_\tau}\right)^2U_{\alpha j} \, \label{fij},
\end{align}
\end{subequations}
where $m_\tau$ is the tau mass. Note that the electric dipole moment does not have a diagonal component because it would violate \textit{CP}~\cite{Giunti:2014ixa}, and that the transition electric dipole moment carries a phase relative to the transition magnetic dipole moment. Numerically, the above expressions yield for the diagonal magnetic moments
\begin{equation}\label{eq:muD}
\mu_{ii}\simeq 3.2\times 10^{-19} \left(\frac{m_i}{{\rm{eV}}}\right)\mu_\text{B}\, ,
\end{equation}
where $\mu_\text{B}$ is the Bohr magneton. The transition moments are
\begin{subequations}
\begin{align}
\mu_{ij}\simeq -3.9\times 10^{-23}\mathcal{F}_{ij} \left(\frac{m_i+m_j}{{\rm{eV}}}\right)\mu_\text{B}\,,\\
\epsilon_{ij}\simeq 3.9\, i\times 10^{-23}\mathcal{F}_{ij}\left(\frac{m_i-m_j}{{\rm{eV}}}\right)\mu_\text{B}\, .
\end{align}
\end{subequations}
Note that the transition moments are much smaller than the diagonal moments due to Glashow-Iliopoulos-Maiani suppression.
\subsection{Hamiltonian matrix for Dirac neutrinos}
We treat electromagnetic effects on the same footing as background matter. To this end, we have to evaluate the components of the Hamiltonian matrix, which, for Dirac neutrinos, are equal to the spinor contractions in Eq.~\eqref{GammasDef}. For the contractions, we need to evaluate the Lorentz structure of the vertex in Eq.~\eqref{eq:moment}.
Considering only magnetic and electric form factors, the Hamiltonian reduces to
\smash{$\frac{1}{2}F_{\mu\nu}\bar \nu_i \bigl( f^{ij}_M \sigma^{\mu\nu} +if^{ij}_E\sigma^{\mu\nu}\gamma_5 \bigr)\nu_j$,}
which depends on the electric and magnetic fields, $\vec{E}$ and $\vec{B}$, through $F^{\mu\nu}$. The Lorentz structure can be decomposed into the contractions \smash{$\left(i\gamma^0\vec{\gamma}\right)_{ij,sh}$ and $\left(\gamma^0\vec{\gamma}\gamma_5\right)_{ij,sh}$}, the latter appearing through the identity \smash{$\epsilon^{abc}\gamma^0\gamma^c\gamma_5=\sigma^{ab}$} with spatial indices $a,b,c=1,2 $ or $3$, and the asymmetric tensor \smash{$\epsilon^{abc}$}. These contractions are three-vectors that are contracted with the electric and magnetic fields. We calculate the contractions in momentum space.
Explicitly, the coupling of the magnetic field through the magnetic form factor (superscript $\mu \rm B$) has the structures
\begin{subequations}\label{eq:nunumuB}
\begin{align}
\mathsf{H}^{\mu\text{B} \nu \nu}_{ij,sh}=&-\bigl(\gamma^0\vec{\gamma}\gamma_5\bigr)^{\nu\nu}_{ij,sh}f^{ij}_{M}(q^2)\vec{B}\, ,\\
\mathsf{H}^{\mu\text{B} \nu \bar\nu}_{ij,sh}=&-\bigl(\gamma^0\vec{\gamma}\gamma_5\bigr)^{\nu\bar\nu}_{ij,sh}f^{ij}_{M}(l^2)\vec{B}\, ,\\
\mathsf{H}^{\mu\text{B} \bar\nu \nu}_{ij,sh}=&-\bigl(\gamma^0\vec{\gamma}\gamma_5\bigr)^{\bar\nu\nu}_{ij,sh}f^{ij}_{M}(l^2)\vec{B}\, ,\\
\mathsf{H}^{\mu\text{B} \bar\nu \bar\nu}_{ij,sh}=&-\bigl(\gamma^0\vec{\gamma}\gamma_5\bigr)^{\bar\nu\bar\nu}_{ij,sh}f^{ij}_{M}(q^2)\vec{B}\, ,
\end{align}
\end{subequations}
where we identify ${\sf H}^{\nu\nu}=\Gamma^{\nu\nu}$, ${\sf H}^{\nu\bar\nu}=\Gamma^{\nu\bar\nu}$ , etc.,
and the minus sign in the metric $g^{\mu\nu}=\text{diag}(1,-1,-1,-1)$
has already been taken care of. In Eq.~\eqref{eq:nunumuB}, the form factors still depend on the momentum transfer. For the $\nu\nu$ and $\bar\nu\bar\nu$ components, the form factors contain $q^\mu=p^\mu_{\text{out}}-p^\mu_{\text{in}}$, where $q^\mu\to 0$ in the forward-scattering limit. These components are then proportional to the dipole moments. For the neutrino-antineutrino components of the $\mathsf{H}$ matrices, the argument of the form factor contains $l^2$ with $l^\mu =p^\mu_{\text{out}}+p^\mu_{\text{in}}$, the sum of neutrino and antineutrino momenta. In the forward-scattering limit this reduces to $l^2=(2E)^2$, and the dependence of the form factors on the four-momentum is important.
The coupling of the magnetic field to the electric form factor (superscript $\epsilon \rm B$) is
\begin{subequations}
\begin{align}
\mathsf{H}^{\epsilon\text{B} \nu \nu}_{ij,sh}=&-\bigl(i\gamma^0\vec{\gamma}\bigr)^{\nu\nu}_{ij,sh}f^{ij}_{E}(q^2)\vec{B}\,,\\
\mathsf{H}^{\epsilon\text{B} \nu \bar\nu}_{ij,sh}=&-\bigl(i\gamma^0\vec{\gamma}\bigr)^{\nu\bar\nu}_{ij,sh}f^{ij}_{E}(l^2)\vec{B}\,,\\
\mathsf{H}^{\epsilon\text{B} \bar\nu \nu}_{ij,sh}=&-\bigl(i\gamma^0\vec{\gamma}\bigr)^{\bar\nu\nu}_{ij,sh}f^{ij}_{E}(l^2)\vec{B}\,,\\
\mathsf{H}^{\epsilon\text{B} \bar\nu \bar\nu}_{ij,sh}=&-\bigl(i\gamma^0\vec{\gamma}\bigr)^{\bar\nu\bar\nu}_{ij,sh}f^{ij}_{E}(q^2)\vec{B}\,.
\end{align}
\end{subequations}
The coupling of an electric field to the magnetic form factor is
\begin{subequations}
\begin{align}
\mathsf{H}^{\mu\text{E} \nu \nu}_{ij,sh}=&\bigl(i\gamma^0\vec{\gamma}\bigr)^{\nu\nu}_{ij,sh}f^{ij}_{M}(q^2)\vec{E}\,,\\
\mathsf{H}^{\mu\text{E} \nu \bar\nu}_{ij,sh}=&\bigl(i\gamma^0\vec{\gamma}\bigr)^{\nu\bar\nu}_{ij,sh}f^{ij}_{M}(l^2)\vec{E}\, ,\\
\mathsf{H}^{\mu\text{E} \bar\nu \nu}_{ij,sh}=&\bigl(i\gamma^0\vec{\gamma}\bigr)^{\bar\nu\nu}_{ij,sh}f^{ij}_{M}(l^2)\vec{E}\, ,\\
\mathsf{H}^{\mu\text{E} \bar\nu \bar\nu}_{ij,sh}=&\bigl(i\gamma^0\vec{\gamma}\bigr)^{\bar\nu\bar\nu}_{ij,sh}f^{ij}_{M}(q^2)\vec{E}\, ,
\end{align}
\end{subequations}
which is indicated by $\mu \rm E$, and to the electric form factor, $\epsilon \rm E$,
\begin{subequations}\label{eq:nunuepE}
\begin{align}
\mathsf{H}^{\epsilon\text{E} \nu \nu}_{ij,sh}=&-\bigl(\gamma^0\vec{\gamma}\gamma_5\bigr)^{\nu\nu}_{ij,sh}f^{ij}_{E}(q^2)\vec{E},\\
\mathsf{H}^{\epsilon\text{E} \nu \bar\nu}_{ij,sh}=&-\bigl(\gamma^0\vec{\gamma}\gamma_5\bigr)^{\nu\bar\nu}_{ij,sh}f^{ij}_{E}(l^2)\vec{E},\\
\mathsf{H}^{\epsilon\text{E} \bar\nu \nu}_{ij,sh}=&-\bigl(\gamma^0\vec{\gamma}\gamma_5\bigr)^{\bar\nu\nu}_{ij,sh}f^{ij}_{E}(l^2)\vec{E},\\
\mathsf{H}^{\epsilon\text{E} \bar\nu \bar\nu}_{ij,sh}=&-\bigl(\gamma^0\vec{\gamma}\gamma_5\bigr)^{\bar\nu\bar\nu}_{ij,sh}f^{ij}_{E}(q^2)\vec{E}.
\end{align}
\end{subequations}
One can see that a magnetic field couples to both, the electric and the magnetic form factor. Also electric fields couple to both form factors. This can be understood as follows. In the neutrino rest frame, the magnetic field only couples to the magnetic dipole moment, and the electric field only couples to the electric dipole moment (if any), as suggested by the nomenclature. Lorentz covariance then demands that both electric and magnetic fields couple to the magnetic form factor in a system where the neutrino moves with nonzero velocity. A moving neutrino also exhibits spin precession in a pure electric field through its magnetic moment~\cite{Okun:1986uf}.
The Lorentz structure of Eqs.~\eqref{eq:nunumuB}--\eqref{eq:nunuepE} can now be readily calculated. In contrast to the previous sections, we neglect all contributions proportional to the mass since the magnetic and electric form factors are small and, in the models considered here, proportional to the neutrino mass already. The $(\gamma^0\vec{\gamma}\gamma_5)$ components are
\begin{subequations}
\begin{align}
\label{M1}
\bigl(\gamma^0\vec{\gamma}\gamma_5\bigr)^{\nu\nu}_{ij,sh} &\approx \left(\begin{array}{cc}0 & e^{+i\phi}\hat{\vec{\epsilon}}^{*}\\
e^{-i\phi}\hat{\vec{\epsilon}}& 0\end{array} \right)\, ,\\
\label{M2}
\bigl(\gamma^0\vec{\gamma}\gamma_5\bigr)^{\nu\bar\nu}_{ij,sh} &\approx \left(\begin{array}{cc}-e^{+i\phi}\hat{\vec{ p}} &0\\ 0& -e^{-i\phi}\hat{ \vec{p}}\end{array} \right)\,,\\
\label{M3}
\bigl(\gamma^0\vec{\gamma}\gamma_5\bigr)^{\bar\nu\nu}_{ij,sh} &\approx \left(\begin{array}{cc}-e^{-i\phi}\hat{ \vec{p}} & 0\\ 0& -e^{+i\phi}\hat{ \vec{p}}\end{array} \right)\,,\\
\label{M4}
\bigl(\gamma^0\vec{\gamma}\gamma_5\bigr)^{\bar\nu\bar\nu}_{ij,sh} &\approx \left(\begin{array}{cc}0 & -e^{-i\phi}\hat{\vec{\epsilon}}^{*}\\
-e^{+i\phi}\hat{\vec{\epsilon}}& 0\end{array} \right)\,.
\end{align}
\end{subequations}
The remaining Lorentz structures are of the form $\left(i\gamma^0\vec{\gamma}\right)$. They read
\begin{subequations}
\begin{align}
\label{E1}
\bigl(i\gamma^0\vec{\gamma}\bigr)^{\nu\nu}_{ij,sh} &\approx \left(\begin{array}{cc}0& ie^{+i\phi} \hat{\vec{\epsilon}}^{*}\\
-ie^{-i\phi} \hat{\vec{\epsilon}}&0\end{array} \right)\,,\\
\label{E2}
\bigl(i\gamma^0\vec{\gamma}\bigr)^{\nu\bar\nu}_{ij,sh}&\approx \left(\begin{array}{cc}-ie^{+i\phi}\hat{\vec{p}} & 0\\
0& ie^{-i\phi}\hat{\vec{p}}\end{array} \right)\, ,\\
\label{E3}
\bigl(i\gamma^0\vec{\gamma}\bigr)^{\bar\nu\nu}_{ij,sh}&\approx \left(\begin{array}{cc}ie^{-i\phi}\hat{\vec{p}} & 0\\
0& -ie^{+i\phi}\hat{ \vec{p}}\end{array} \right)\, ,\\
\label{E4}
\bigl(i\gamma^0\vec{\gamma}\bigr)^{\bar\nu\bar\nu}_{ij,sh} &\approx \left(\begin{array}{cc}0 & ie^{-i\phi} \hat{\vec{\epsilon}}^{*}\\
-ie^{+i\phi} \hat{\vec{\epsilon}}& 0\end{array} \right)\, .
\end{align}
\end{subequations}
To this level of approximation, the $\nu\bar\nu$ and $\bar\nu\nu$ components are diagonal in helicity space, i.e., electric and magnetic fields mainly couple spin-0 neutrino-antineutrino pairs. Because the diagonal is proportional to $\hat{\vec{p}}$, the relevant field components are those parallel to the momentum of the neutrinos. The $\nu\nu$ and $\bar\nu\bar\nu$ components are off-diagonal in helicity space. The dominant effect of magnetic and electric fields on neutrinos and antineutrinos is spin precession. Here the transverse components of the electromagnetic fields contribute. The longitudinal components enter on the diagonals in the next order of the expansion in $m/E$ and are therefore omitted.
\subsection{Dipole moments of Majorana neutrinos \label{sec:MajMom}}
For Majorana neutrinos, electromagnetic transitions always contain two contributions, e.g.,
\begin{align}\label{eq:amp}
\langle \nu_{\vec{p}_{\text{out}}}|\mathcal{H}^{\text{em}}|\nu_{\vec{p}_{\text{in}}}\rangle
=A_\mu\left( \bar u_{\vec{p}_{\text{out}}}\Gamma^\mu u_{\vec{p}_\text{in}} - \bar v_{\vec{p}_{\text{in}}}\Gamma^\mu v_{\vec{p}_\text{out}}\right)\, .
\end{align}
This difference of two amplitudes leads to the cancellation of all the diagonal moments except for the anapole moment~\cite{Giunti:2014ixa}. This can also be understood by noting that the last two terms (including the minus sign) in Eq.~\eqref{eq:amp} are charge conjugates of each other. Because the Lorentz structure of the magnetic, electric, and charge form factors are \textit{C}-odd the combination vanishes. The Lorentz structure of the anapole moment is \textit{C}-even and does not cancel.
Because the magnetic moment of the Majorana neutrino vanishes, it does not couple directly to a magnetic field. However, magnetic fields polarize the background medium, and this effect does lead to helicity oscillations, see Sec.~\ref{sec:majorana}.
Electromagnetic moments of neutrinos depend on the details of the mechanism that creates the neutrino mass. When neglecting the model-dependent amplitudes, one can compare the moments of Dirac and Majorana neutrinos. The main differences are that the Majorana amplitudes contain Majorana PMNS matrices, which may contain more phases than Dirac PMNS matrices, and that Eq.~\eqref{eq:amp} has to be taken into account for Majorana neutrinos.
After these adjustments, the off-diagonal form factors of Majorana neutrinos can be obtained from Eq.~\eqref{eq:mmon1}. One finds that they depend on the relative \textit{CP}-phases of two neutrino species~\cite{Pal:1981rm}. The relative phase can either be equal or opposite, i.e., the ratio is $\pm 1$. For neutrinos with equal \textit{CP}-phases, the magnetic transition moments vanish~\cite{Pal:1981rm}, while for opposite \textit{CP}-phase the magnetic transition moments are nonzero and can be obtained from Eq.~\eqref{eq:mmon1} by substituting $\mathcal{F}_{ij}$ with $2i\text{Im}\mathcal{F}_{ij}$.
For \emph{electric} dipole moments, the role of the \textit{CP}-phases is inverted. Opposite \textit{CP}-phases force the electric transition moments to vanish, while for equal \textit{CP}-phases the electric transition moments are nonzero and are obtained by substituting $\mathcal{F}_{ij}$ with $2\text{Re}\mathcal{F}_{ij}$~\cite{Pal:1981rm} in Eq.~\eqref{eq:mmon2}.
\subsection{Hamiltonian matrix for Majorana neutrinos}
The density matrix formalism naturally reproduces the results for the electromagnetic moments discussed in the last section. Similarly to Eq.~\eqref{eq:amp}, each component of the Hamiltonian matrix has two contributions from $\Gamma$ contractions, e.g.,
${\sf H}^{\nu\nu}_{ij,sh}(\vec{p})=\Gamma^{\nu\nu}_{ij,sh}(\vec{p})-\Gamma^{\bar\nu\bar\nu}_{ji,hs}(-\vec{p})$;
see Eq.~\eqref{GammasMajDef}.
The spinor contractions $\Gamma^{\nu\nu}$ and $\Gamma^{\bar\nu\bar\nu}$ have the same structure as for Dirac neutrinos;
see Eq.~\eqref{GammasDef}. Again neglecting the model dependence, the only difference is that the Dirac PMNS matrix have to be replaced by the Majorana PMNS matrix. For example, a magnetic field coupling to a Majorana neutrino through the magnetic form factor yields
\begin{equation}
\begin{aligned}
{\sf H}^{\nu\nu}_{ij,\scalebox{0.9}[0.9]{-}\scalebox{0.75}[0.75]{+}} =& -\bigl[f^{ij}_{M}(q^2)-c.c.)\bigr]e^{+i\phi}\hat{\vec{\epsilon}}^{*}\vec{B}\\
=&-2i\text{Im}[f^{ij}_{M}(q^2)]e^{+i\phi}\hat{\vec\epsilon}^{*}\vec{B}\, ,
\end{aligned}
\end{equation}
where we have used the Hermiticity of the form factors. In the static limit, $2i\text{Im}[f^{ij}_{M}]$ is the magnetic moment of Majorana neutrinos~\cite{Shrock:1982sc}. It is zero for equal \textit{CP}-phases since $\mathcal{F}_{ij}$ becomes real. It is nonvanishing for opposite \textit{CP}-phases because $\mathcal{F}_{ij}$ becomes imaginary. An analogous argument holds for the electric dipole moment.
\section{\label{sec:helicity}Helicity coherence}
In this section we neglect pair correlations and
discuss helicity coherence effects. To separate the latter from the
usual flavor coherence effects, we consider only one neutrino generation.
Furthermore, for definiteness we assume that neutrinos are Dirac particles.
\subsection{Order-of-magnitude estimate}
Two different mean-field backgrounds cause spin oscillations and create spin coherence: matter and neutrino currents, and electromagnetic fields. However, it is not clear which of these is dominant in a supernova. In the following we perform a crude estimate.
For Dirac neutrinos without pair correlations, the kinetic equations of neutrinos and antineutrinos decouple,
$i\dot{\rho}=[{\sf H}^{\nu\nu},\rho]$ and $i\dot{\bar\rho}=
[{\sf H}^{\bar\nu\bar\nu},\bar{\rho}]$, and, for one family, we only have to look at a $2\times2$ subsystem of the full evolution equation.
We start with a matter background with nonrelativistic velocity $\beta$, which flows orthogonal to the neutrino's momentum. The Hamiltonian matrix reads
\begin{align}
\label{GnunuMajExpl}
{\sf H}^{\nu\nu} &\approx
V\begin{pmatrix}
1 & \frac{m}{2p}\beta \\
\frac{m}{2p}\beta & 0
\end{pmatrix}\,,
\end{align}
where $V$ is the usual matter potential. For instance for $\nu_\mu$ or $\nu_\tau$
it is given by \smash{$V=G_{\rm F} n_n/\sqrt{2}$}, where
$n_n$ is the neutron density. We have omitted the neutrino kinetic
energy because it is diagonal in helicity space, and for a single generation
trivially cancels in the commutator.
The dependence
of the diagonal terms of the Hamiltonian on the parallel flux and the
dependence of the off-diagonal terms on the orthogonal flux were discussed
in Refs.~\cite{Lobanov:2001ar,Studenikin:2004bu}.
In a derivation similar to the one that leads to Eq.~\eqref{GnunuMajExpl}, we obtain the $2\times2$
subsystem of the Hamiltonian matrix for a neutrino in a transverse magnetic field
\begin{align}
\label{GnunuMajExplMagn}
{\sf H}^{\nu\nu} &\approx
-\mu B
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}\,
\end{align}
(see Sec.~\ref{sec:magnetic} for more details). Spin coherence is instigated by
the off-diagonals of Eqs.~\eqref{GnunuMajExpl} and~\eqref{GnunuMajExplMagn},
and to find the relative importance of the matter and magnetic contributions
it is sufficient to estimate their relative size. Typical magnetic fields in
a supernova are of order $10^{12} \,\rm{G}$ and much larger in magnetars.
Using the standard value for the magnetic moment given in Eq.~\eqref{eq:muD},
and assuming a neutrino mass of $0.1\, {\rm eV}$, we find for the contribution
of the magnetic field $\mu B \sim 10^{-16}\, {\rm eV}$. For a typical neutron
mass density $10^{12}~\rm{g/cm}^3$, which corresponds to a number density
$n_n\sim 10^4 \, \rm{MeV}^3$, the matter potential is of the order of the
neutrino mass, $V\sim 0.1\, {\rm eV}$. Thus, for a typical momentum $p\sim 30\, {\rm MeV}$
we obtain $V\beta m/(2p) \sim 10^{-10}\beta\, {\rm eV}$. For maximal background
velocities of $3000\, \rm{km/s}$, $\beta\sim 0.01$, the matter contribution
dominates. Surprisingly, the magnetic field is only important if the background
moves very slowly, if the matter density has decreased sufficiently, or if
the magnetic moment is enhanced.
Turning now to the density matrix, the size of the off-diagonal elements depends on the initial
conditions and history of the evolution. To obtain a rough estimate, we can assume that the system has
reached equilibrium and, hence, its previous evolution is irrelevant. In equilibrium,
the system is in an eigenstate of the Hamiltonian, i.e., $\mathsf{H}^{\nu\nu}$ and $\rho$
commute. This condition alone allows us to express the off-diagonals of the density matrix in
terms of the diagonals and components of the Hamiltonian matrix,
\begin{align}
\label{EqSolHelCoh}
\rho_{\scalebox{0.9}[0.9]{-}\scalebox{0.75}[0.75]{+}}&=\frac{\mathsf{H}^{\nu\nu}_{\scalebox{0.9}[0.9]{-}\scalebox{0.75}[0.75]{+}}}{\mathsf{H}^{\nu\nu}_{\scalebox{0.9}[0.9]{-}\smi}-\mathsf{H}^{\nu\nu}_{\scalebox{0.75}[0.75]{+}\spi}}
(\rho_{\scalebox{0.9}[0.9]{-}\smi}-\rho_{\scalebox{0.75}[0.75]{+}\spi})\,.
\end{align}
Keeping only the (dominant) matter contribution, Eq.~\eqref{GnunuMajExpl}, we find
$\rho_{\scalebox{0.9}[0.9]{-}\scalebox{0.75}[0.75]{+}}=(\rho_{\scalebox{0.9}[0.9]{-}\smi}-\rho_{\scalebox{0.75}[0.75]{+}\spi})\,m\beta/2p\sim m\beta/2p$.
For $m\sim 0.1\, {\rm eV}$ and a typical momentum $p\sim 30\, {\rm MeV}$ this results in
$\rho_{\scalebox{0.9}[0.9]{-}\scalebox{0.75}[0.75]{+}}\sim 10^{-11}$, where we have used $\beta\sim 0.01$.
The same
result can be obtained by noting that if $\rho$ and $\mathsf{H}^{\nu\nu}$ commute, they can be
simultaneously diagonalized by a rotation that mixes positive- and negative-helicity states.
The rotation angle is $\tan 2\vartheta = m\beta/p$. Considering e.g.\ the $\rho_{\scalebox{0.9}[0.9]{-}\smi}=1$
eigenstate of the diagonalized Hamiltonian and rotating back to the basis where the Hamiltonian
has the form Eq.~\eqref{GnunuMajExpl} we find to leading order
\begin{align}
\label{rhoestimate}
{\sf \rho}&\sim
\begin{pmatrix}
1 & \frac{m}{2p}\beta \\
\frac{m}{2p}\beta & 0
\end{pmatrix}
\sim
\begin{pmatrix}
1 & 10^{-11} \\
10^{-11} & 0
\end{pmatrix}\,.
\end{align}
The corrections to the diagonals are not included in Eq.~\eqref{rhoestimate}
because they are of the order of $\delta \rho_{\scalebox{0.9}[0.9]{-}\smi}\sim
\delta \bar\rho_{\scalebox{0.75}[0.75]{+}\spi}\sim \rho^2_{\scalebox{0.9}[0.9]{-}\scalebox{0.75}[0.75]{+}} \sim 10^{-22}$
and are therefore negligibly small.
\subsection{Resonant enhancement}
For a magnetic field, the diagonal elements of the Hamiltonian matrix, Eq.~\eqref{GnunuMajExplMagn},
are zero for very relativistic neutrinos. This allows for the
magnetic fields to completely flip the spin of a population of neutrinos.
On the other hand, the diagonals of Eq.~\eqref{GnunuMajExpl} are in general nonzero
and suppress a complete conversion. In general, the matter contribution is given by
\begin{align}
\label{GnunuMajExplRel}
{\sf H}^{\nu\nu} &\approx
\begin{pmatrix}
V^0-V_{\shortparallel} & \frac{m}{2p}V_\perp \\
\frac{m}{2p} V_\perp & 0
\end{pmatrix}\,,
\end{align}
see Eq.~\eqref{GammaMnunu},
where $V_{\shortparallel}\equiv \hat{\vec p}\vec{V}$ and
$V_{\perp}\equiv \hat{\vec \epsilon}\vec{V}$ are components of the matter flux
parallel and orthogonal to the neutrino momentum. Thus, if there are relativistic currents parallel to
the momentum of the neutrino such that the diagonals vanish, Eq.~\eqref{EqSolHelCoh} implies
that a resonant enhancement of the spin conversion is possible.
The possibility to generate the spin conversion
by an orthogonal flux of matter, and the cancellation of the matter
effect for relativistic matter moving along the direction of the neutrino momentum
were first discussed in Refs.~\cite{Lobanov:2001ar,Studenikin:2004bu}
on the basis of the Lorentz-covariant Bergmann-Michel-Telegdi equation.
In Refs.~\cite{Vlasenko:2013fja,Vlasenko:2014bva} these effects have also
been studied using the formalism of nonequilibrium quantum field theory.
In the context of resonant leptogenesis the formation
of flavor and helicity correlations in medium and the derivation of
flavor-covariant transport equations able to account for helicity
correlations has been discussed in Ref.~\cite{Dev:2014laa}.
For vanishing diagonals,
Eq.~\eqref{GnunuMajExplRel} can be rotated into its diagonal form with a rotation angle
$\vartheta=\pi/4$. In other words, mixing of the helicity states becomes maximal, similarly
to the Mikheyev-Smirnov-Wolfenstein resonance mixing, and hence in equilibrium
\begin{align}
{\sf \rho}&\sim
\begin{pmatrix}
\sfrac{1\!}2 & \sfrac{1\!}2 \\[1mm]
\sfrac{1\!}2 & \sfrac{1\!}2
\end{pmatrix}\,,
\end{align}
where we have again assumed that the system is in an eigenstate of the diagonalized
Hamiltonian. Outside of the core, a supernova is far from equilibrium,
but nonlinear feedback can enhance the spin-flipping processes~\cite{Vlasenko:2014bva}.
Making use of Eq.~\eqref{SigmaDirac}, we can rewrite
the resonance condition, $\mathsf{H}^{\nu\nu}_{\scalebox{0.9}[0.9]{-}\smi}-\mathsf{H}^{\nu\nu}_{\scalebox{0.75}[0.75]{+}\spi}=
V^0-V_{\shortparallel}=0$, in the form \cite{Vlasenko:2014bva}
\begin{align}
\label{SpinCoherenceResonance}
Y_e+\frac43\left(Y_\nu-\frac{V_{\shortparallel}}{2 n_b}\right)=\frac13\,,
\end{align}
where $Y_e\equiv n_e/n_B$ and $Y_\nu=(n_\nu-n_{\bar\nu})/n_B$ are the electron
and neutrino asymmetry fractions respectively and $n_B$ is the baryon number density,
The resonance condition can potentially be fulfilled in or near the proto-neutron
star in a core-collapse supernova, or near the central region of a compact object
merger; see Ref.~\cite{Vlasenko:2014bva} and references therein.
\subsection{Lorentz covariance}
Helicity coherence builds up only if the
off-diagonal elements of the Hamiltonian matrix differ from zero. On the other hand,
because the off-diagonals are proportional to the component of the matter flow orthogonal
to the neutrino momentum, one can always find a frame where the off-diagonals
vanish and no helicity coherence builds up. In other words, at first
sight physical results seem to depend on the frame. This raises the question of
Lorentz covariance of the kinetic equations.
To be specific, let us consider the following simple example. We have two identical
observers moving with velocity $\beta$ with respect to each other. In the frame
of the first observer, the neutrino has momentum $\vec{p}$ along the $z$ axis and
the matter is at rest, i.e. $V_{\shortparallel}=V_\perp=0$,
\begin{align}
\label{Hframe1}
{\sf H}^{\nu\nu}&\approx V
\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix}\,.
\end{align}
Thus no helicity coherence builds up. In the frame of
the second observer which moves with velocity $\beta$ along the $x$ axis the
Hamiltonian is no longer diagonal,
\begin{align}
\label{Hframe2}
{\sf H}^{\nu\nu}&\approx\frac{V}{\gamma}
\begin{pmatrix}
1 & \frac{m}{2p}\beta \\
\frac{m}{2p}\beta & 0
\end{pmatrix}\,,
\end{align}
and we expect helicity coherence to build up. Here $\gamma$ is the usual Lorentz
factor and $V$ and $p$ denote the potential and neutrino momentum in the frame
of the first observer.
Do the Hamiltonian matrices Eqs.\,\eqref{Hframe1} and \eqref{Hframe2} lead to different
physical results ? The answer is no, but to demonstrate this point we need to take
into account that a helicity state is also not Lorentz in\-variant. Let the neutrino
be in a state of definite helicity in the frame of the first observer, e.g.
$|p\hat{\vec z},\scalebox{2}[1]{-}\rangle$, where $p$ is the absolute value of the neutrino
momentum and $\hat{\vec z}$ is the unit vector along the $z$ axis. The
corresponding density matrix reads
\begin{align}
\label{rhoframe1}
\rho=|p\hat{\vec z},\scalebox{2}[1]{-}\rangle \langle p\hat{\vec z},\scalebox{2}[1]{-}|=
\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix}\,.
\end{align}
The Hamiltonian matrix Eq.~\eqref{Hframe1} and the density matrix commute and therefore the latter is
constant in time. The boost to the frame of the second observer transforms
$|p\hat{\vec z},\scalebox{2}[1]{-}\rangle$ into a mixed helicity state with momentum
$\vec{q}$,
\begin{align}
|\psi\rangle = c_{\theta\hspace{-0.8pt}/\hspace{-0.6pt}2} |\vec{q},\scalebox{2}[1]{-}\rangle -
s_{\theta\hspace{-0.8pt}/\hspace{-0.6pt}2} |\vec{q},+\rangle\,,
\end{align}
where $\theta$ is the angle of Wigner rotation around $\hat{\vec y}$ with
$\tan\theta=-m\beta/p$. Note that the rotation angle vanishes in the limit
of zero neutrino mass which reflects chirality conservation. The density
matrix develops off-diagonal elements,
\begin{align}
\label{rhoframe2}
\rho=|\psi\rangle\langle \psi|=\frac{1}{2}
\begin{pmatrix}
1+c_{\theta} & - s_\theta \\[1.5mm]
- s_\theta & 1-c_{\theta}
\end{pmatrix}
\,.
\end{align}
The Hamiltonian matrix and the density matrix again commute. In other words, the second
observer sees a mixed helicity state which, as expected, is also time
independent. This result reflects Lorentz covariance of the kinetic equations,
the lesson being that one has to transform the initial conditions consistently to obtain covariant results.
Let us now consider this result from a slightly different viewpoint. In each
frame, we can diagonalize the effective Hamiltonian by performing a Bogolyubov
transformation that mixes annihilation (creation) operators of the positive- and
negative-helicity states, $a_{s}\rightarrow c_\vartheta\,
a_{s}+s_\vartheta\, a_{-s}$. In particular Eq.~\eqref{Hframe2} is diagonalized
by a Bogolyubov transformation with the angle $\tan 2\vartheta=m\beta/p$. This
transformation brings the density matrix Eq.~\eqref{rhoframe2} back to the form
Eq.~\eqref{rhoframe1}. In other words, there is a connection between the Lorentz and
Bogolyubov transformations. In particular, if in every frame we diagonalize the
Hamiltonian then the transformed density matrix remains invariant under the boosts.
To summarize, as far as helicity coherence is concerned, both the Hamiltonian
and the density matrix transform under Lorentz boosts in such a way that the kinetic
equation is Lorentz covariant. We will rely on this result in
the discussion of particle-antiparticle coherence whose Lorentz transformation
properties are not as evident as for the helicity coherence.
\section{\label{sec:partantipart}Particle-antiparticle coherence}
In this section we discuss particle-antiparticle coherence.
In contrast to helicity coherence, which requires nonzero neutrino
masses, and flavor coherence, which in addition to nonzero masses
requires the existence of several neutrino generations, particle-antiparticle
coherence arises already for a single massless neutrino generation.
As has been discussed in the previous section, for a massless neutrino the only
``natural'' correlators are $\rho_{\scalebox{0.9}[0.9]{-}\smi}$, $\rho_{\scalebox{0.75}[0.75]{+}\spi}$ and
$\kappa_{\scalebox{0.9}[0.9]{-}\scalebox{0.75}[0.75]{+}}$. To shorten the notation in this section we
suppress the spin indices.
\subsection{Quantum-mechanical example}
To clarify the meaning of the particle-antiparticle coherence, let us first
study in more detail the simple quantum-mechanical example briefly discussed
in the Introduction.
We consider a system that can be in a linear combination of one of four pure states.
These are i) the empty state $|00\rangle$ without particles; ii) the paired
state $|11\rangle=a^\dagger(\vec{p}) b^\dagger(\scalebox{2}[1]{-}\vec{p})|00\rangle$, which
contains a neutrino with momentum $\vec{p}$ and an antineutrino with momentum
$\scalebox{2}[1]{-}\vec{p}$; iii) the one neutrino state $|10\rangle$; and iv) the one antineutrino
state $|01\rangle$. Note that in all these states the antineutrinos
stream in the direction opposite to that of neutrinos. A general state
can be expressed in terms of these four states,
$ |\psi\rangle= A_{00}|00\rangle+A_{11}|11\rangle+A_{10}|10\rangle+A_{01}|01\rangle$,
where the coefficients $A_{ij}$ are time dependent and normalized to unity,
$|A_{00}|^2+|A_{11}|^2+|A_{10}|^2+|A_{01}|^2=1$.
In analogy to Eq.~\eqref{HeffCrAn} we write the Hamiltonian in the form
\begin{align}
\label{twoModeH}
H&=a^\dagger(\vec{p})\mathsf{H}^{\nu\nu} a(\vec{p}) +a^\dagger(\vec{p}) \mathsf{H}^{\nu\bar\nu} b^\dagger(\scalebox{2}[1]{-}\vec{p})\nonumber\\
&+b(\scalebox{2}[1]{-}\vec{p}) \mathsf{H}^{\bar\nu\nu} a(\vec{p})- b^\dagger(\scalebox{2}[1]{-} \vec{p})\mathsf{H}^{\bar\nu\bar\nu} b(\scalebox{2}[1]{-}\vec{p})\,.
\end{align}
The Schr\"odinger equation for the coefficients $A_{ij}$ then splits into
three independent equations,
\begin{subequations}
\label{twoModeSchrod}
\begin{align}
i\partial_t
\begin{pmatrix}
A_{00}\\
A_{11}
\end{pmatrix}
&=\begin{pmatrix}
0 & \mathsf{H}^{\bar\nu\nu}\\
\mathsf{H}^{\nu\bar \nu}& \mathsf{H}^{\nu\nu}-\mathsf{H}^{\bar\nu\bar\nu}
\end{pmatrix}
\begin{pmatrix}
A_{00}\\
A_{11}
\end{pmatrix}\,,\\
i\partial_t A_{10} & = \mathsf{H}^{\nu\nu}A_{10}\,, \\
i\partial_t A_{01} & = -\mathsf{H}^{\bar\nu\bar\nu}A_{01}\,.
\end{align}
\end{subequations}
Thus the evolution of the single-particle states completely
decouples because a homogeneous background medium cannot mix states of
different total momentum. On the other hand, the $|00\rangle$ and $|11\rangle$
states have zero momenta and therefore can be mixed by a homogeneous medium
through the $\mathsf{H}^{\bar\nu\nu}$ term of the Hamiltonian. However, the
$|00\rangle$ and $|11\rangle$ states have different angular momentum. Hence,
an anisotropic background medium, e.g. a transverse matter flux, is needed to
absorb the angular momentum and to mix the two states.
To make the connection to the density matrix equations, we note that the
number of neutrinos and antineutrinos is given by $\rho = |A_{11}|^2+|A_{10}|^2$
and $\bar\rho= |A_{11}|^2+|A_{01}|^2$ respectively. Their time evolution can be
derived from Eq.~\eqref{twoModeSchrod} and takes the form expected from
Eq.~\eqref{RhoKappaEqWeyl},
\begin{subequations}
\label{eq:Weylrho}
\begin{align}
\dot \rho = -2 \text{Im}\left(\mathsf{H}^{\bar\nu\nu} \kappa\right)\, ,\\
\label{eq:Weylrho2}
\dot {\bar\rho} = -2 \text{Im}\left(\mathsf{H}^{\bar\nu\nu} \kappa\right)\,,
\end{align}
\end{subequations}
if we identify $\kappa= A_{00}^*A_{11}$. Equation~\eqref{twoModeSchrod} also
leads to an evolution equation for $\kappa$
\begin{align}\label{eq:Weylka}
i\dot \kappa & = \left(\mathsf{H}^{\nu\nu}-
\mathsf{H}^{\bar\nu\bar\nu}\right)\kappa +
\mathsf{H}^{\nu \bar \nu}\left(1-\rho -\bar \rho\right)\,,
\end{align}
which can be obtained by using the normalization of the state $|\psi\rangle$.
Equation~\eqref{eq:Weylka} is again consistent with Eq.~\eqref{RhoKappaEqWeyl}
and coincides with the result of Ref.~\cite{Serreau:2014cfa} in the one-flavor
limit.
From these kinetic equations
we can infer that while $\rho$ and $\bar\rho$ are not separately conserved in
the presence of nonzero $\kappa$, their difference is conserved \cite{Serreau:2014cfa}.
Because $\rho(t,\vec{p})$ describes neutrinos with momentum $\vec{p}$ whereas $\bar\rho(t,\vec{p})$
describes antineutrinos with momentum $-\vec{p}$, the conservation of $\rho-\bar\rho$
implies that $\kappa$ induces the production
of neutrino-antineutrino pairs with opposite momentum.
The kinetic equation for $\kappa$
describes a driven harmonic oscillator with frequency
$\mathsf{H}^{\nu\nu}-\mathsf{H}^{\bar \nu \bar \nu}\sim 2E$. Hence
$\kappa$ oscillates with twice the neutrino energy as expected.
From the definition $\kappa= A_{00}^*A_{11}$ we see that nonzero
parti\-cle-antiparticle coherence means that the system is not in an
eigenstate of the unperturbed Hamiltonian, but instead in a mixture of
the $|00\rangle$ and $|11\rangle$ states, i.e., in a squeezed state.
Such states do not have a definite particle number. This observation clarifies the physical
meaning of particle-antiparticle coherence.
\subsection{Order-of-magnitude estimate}
As a next step we perform an order-of-magnitude estimate of $\kappa$.
For a single neutrino generation the extended density matrix
reduces to a $2{\times}2$ matrix of the form, see Eq.~\eqref{eq:simple-R-equation},
\begin{align}
\label{Rmassless}
{\sf R}=
\begin{pmatrix}
\rho & \kappa\\
\kappa^\dagger & 1-\bar{\rho}
\end{pmatrix}\,,
\end{align}
where now $\rho$ and $\bar\rho$ are real numbers and $\kappa$ is a complex number.
We again start with an example of a matter
background with nonre\-la\-tivistic velocity $\beta$, which flows orthogonal
to the neutrino's momentum. Then, as follows from Eq.\,\eqref{GammaWeyl}, the
Hamiltonian matrix reads, see also Eq.~\eqref{eq:simplehamiltonian},
\begin{align}
\label{Hnonrel}
\mathsf{H}&=
E
\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}+
V
\begin{pmatrix}
1 & -\beta \\
-\beta & 1
\end{pmatrix}\,.
\end{align}
Unlike for helicity coherence, the neutrino kinetic energy $E$ no longer cancels out in the commutator.
Similarly to the case of spin coherence we can get a crude estimate of the $\kappa$
magnitude by assuming that the system has reached equilibrium and hence $\dot \kappa=0$.
Equation~\eqref{eq:Weylka} then gives
\begin{align}
\label{kappaeqsol}
\kappa= -\frac{\mathsf{H}^{\nu\bar\nu}}{\mathsf{H}^{\nu\nu}-\mathsf{H}^{\bar\nu\bar\nu}}
(1-\rho-\bar{\rho})\,.
\end{align}
If we insert this result into Eq.~\eqref{eq:Weylrho} and use the Hermiticity of the
Hamiltonian matrix, we see that indeed $\dot \rho=\dot {\bar \rho} =0$.
Let us assume for a moment that the neutrino-neutrino interactions are small
compared to the neutrino interaction with matter. For typical supernova parameters
$V\sim 0.1\,\rm{eV}$ and $E\sim 30\,\rm{MeV}$ and we then find $V/E\sim 10^{-9}$. Thus to a good
approximation $\mathsf{H}^{\nu\bar\nu}/(\mathsf{H}^{\nu\nu}-\mathsf{H}^{\bar\nu\bar\nu})
\sim V\beta/2E\sim 10^{-11}$, where we have used $\beta \sim 0.01$. Because typically
$|1-\rho-\bar{\rho}|\sim 1$ we conclude that the ``natural''
size of the particle-antiparticle coherence is $\kappa\sim 10^{-11}$.
The same result can be obtained by noting that in equilibrium $\sf R$ and $\sf H$ commute
and can be simultaneously diagonalized by a Bogolyubov transformation that mixes
neutrinos of momentum $\vec{p}$ with antineutrinos of momentum $-\vec{p}$. Under
this transformation the creation and annihilation operators transform as
\smash{$a(\vec{p})\rightarrow e^{-i\phi\hspace{-0.8pt}/\hspace{-0.6pt}2}c_\vartheta\, a(\vec{p})+
e^{i\phi\hspace{-0.8pt}/\hspace{-0.6pt}2}s_\vartheta \,
b^\dagger(-\vec{p})$} and
\smash{$b^\dagger(-\vec{p})\rightarrow e^{i\phi\hspace{-0.8pt}/\hspace{-0.6pt}2}c_\vartheta\,
b^\dagger(-\vec{p})-e^{-i\phi\hspace{-0.8pt}/\hspace{-0.6pt}2}s_\vartheta\, a(\vec{p})$} res\-pe\-ctively,
where the phase $\phi=\arg {\sf H}^{\nu\bar\nu}$ and the rotation angle is given by $\tan 2\vartheta=2|{\sf H}^{\nu\bar\nu}|
/({\sf H}^{\nu\nu}-{\sf H}^{\bar\nu\bar\nu})\sim V\beta/E$. In the basis where the
Hamiltonian is diagonal, the
system is described by (anti)neutrino densities, which we denote by $\varrho$ and $\bar\varrho$
respectively, and a pairing correlator, which we denote by $\varkappa$. From the transformation
properties of the creation/annihilation operators, we can infer the following relations
\begin{subequations}
\label{TwoBases}
\begin{align}
\rho&=c^2_\vartheta \varrho-c_\vartheta s_\vartheta \varkappa
-c_\vartheta s_\vartheta\varkappa^\dagger +s^2_\vartheta (1-\bar{\varrho})\,,\\
\bar\rho &=c^2_\vartheta \bar\varrho-c_\vartheta s_\vartheta \varkappa
-c_\vartheta s_\vartheta \varkappa^\dagger+s^2_\vartheta (1-\varrho)\,,\\
\label{kappaBogolyubov}
\kappa &=e^{i\phi}\left[c^2_\vartheta \varkappa+c_\vartheta s_\vartheta \varrho
-c_\vartheta s_\vartheta (1-\bar\varrho)-s^2_\vartheta \varkappa^\dagger\right]\,,
\end{align}
\end{subequations}
see Ref.~\cite{Vaananen:2013qja} for a detailed discussion.
Eigenstates of the diagonalized Hamiltonian are characterized by $\varkappa=0$.
Assuming, e.g., that the system is in an eigenstate of the diagonalized Hamiltonian
with some $\varrho$ and $\bar\varrho$, and rotating back to the basis where the
Hamiltonian has the form Eq.~\eqref{Hnonrel}, we find to leading order
\begin{align}
\label{Rexample}
{\sf R}&\sim
\begin{pmatrix}
\varrho & \frac{V\beta}{2E} \\[1mm]
\frac{V\beta}{2E} & 1-\bar\varrho
\end{pmatrix}\sim
\begin{pmatrix}
\varrho & 10^{-11} \\[1mm]
10^{-11} & 1-\bar\varrho
\end{pmatrix}\,,
\end{align}
which again leads to the tiny $\kappa\sim s_\vartheta\sim 10^{-11}$.
Pair correlations themselves are not measurable, and only their effect on the
number densities can be observed. A quick inspection of Eq.~\eqref{TwoBases}
shows that in equilibrium the difference between e.g. $\rho$ and $\varrho$ is of the order
of $s^2_\vartheta\sim \kappa^2$. In other words, the induced corrections to $\rho$
and $\bar \rho$ are quadratic in $\kappa$.
This can also be understood from Eq.~\eqref{eq:Weylka}.
If the system has not yet reached equilibrium, then $\kappa$ oscillates around its
stationary value Eq.~\eqref{kappaeqsol}, provided that the components of the Hamiltonian
matrix only vary slowly with time compared to $\mathsf{H}^{\nu\nu} - \mathsf{H}^{\bar\nu\bar\nu}$. This assumption allows us to approximate the evolution of $\kappa$ as a driven harmonic oscillator with an amplitude that depends on the initial conditions. Assuming that pairing correlations are not created during neutrino production, the amplitude is of the order of the equilibrium value, Eq.~\eqref{kappaeqsol}.
We then find again that the mean number density created by pairing correlations is $\sim \kappa^2$.
Therefore, the inclusion of the
particle-antiparticle coherence leads to a negligibly small $\delta \rho\sim
\delta \bar\rho \sim \kappa^2 \sim 10^{-22}$ .
\subsection{Including neutrino-neutrino interactions}
In the previous subsection we have estimated the ``natural'' size of $\kappa$ assuming that
the neutrino-neutrino
interactions are negligible. However, in a supernova the neutrino
density is very large and the neutrino background may play an important
role. This complicates the estimate of $\kappa$ because
$\mathsf{H}^{\nu\bar\nu}$ in Eq.~\eqref{kappaeqsol} itself depends on
$\kappa$ once we include neutrino-neutrino interactions,
\begin{equation}
\label{eq:selfnu}
\mathsf{H}^{\nu\bar\nu}= -V\beta-2\sqrt{2}G_{\rm F}\,
\hat{\vec\epsilon}^{*}\!\!\int_{\vec{q}}[\hat{\vec{q}} \ell+\hat{\vec\epsilon}\kappa
+\hat{\vec\epsilon}^{*}\kappa^\dagger]\,,
\end{equation}
see Eqs.~\eqref{GammaWeyl} and \eqref{Inuvec}. A further complication
arises from the fact that the stationary value for $\kappa$ of one momentum mode
$\vec p$ depends on the pair correlations of all other momentum modes $\vec q$.
Note also that the phase-space integral in Eq.~\eqref{eq:selfnu} is
unbounded. Pairing correlations with a momentum typical for the supernova
environment couple to pairing correlations of arbitrary high momentum. This
pushes us beyond the limitations of the Fermi approximation, and in principle
a fully renormalizable theory has to be studied to make sense of these
high-momentum modes. To stay within the realm of applicability of the effective
theory, we use a phenomenological cutoff $|\vec{q}|= M_W$ in the phase-space integrals.
To estimate the contribution of the $\kappa$ terms to the integral in
Eq.~\eqref{eq:selfnu}, we take into account that pair correlators of
different momentum modes oscillate incoherently such that we can
replace $\kappa$ by its approximate mean value,
$\kappa\approx-\mathsf{H}^{\nu\bar\nu}/2E$, where we use that $V\ll E$ and
assume $\rho+\bar\rho\ll 1$ in Eq.~\eqref{kappaeqsol}. To proceed we recall
that $\mathsf{H}^{\nu\bar\nu}=-\hat{\vec\epsilon}^{*}\vec{V}$, see
Eq.~\eqref{GammaWeyl}, where $\vec{V}$ is the total potential that includes
matter and neutrino contributions. Note further that $\vec V$
is momentum independent. With these substitutions, the integrals involving
$\kappa$ in Eq.~\eqref{eq:selfnu} read
\begin{equation}
\label{eq:cons1}
\text{Re}\int_{\vec{q}}\hat{\vec\epsilon}\kappa
\sim
\text{Re} \int_{\vec{q}}\hat{\vec\epsilon}\frac{\hat{\vec\epsilon}^{*}\!\cdot\!\vec{V}}{2E}
=\frac{\sqrt{2}}2\frac{G_{\rm F} M_W^2}{3\pi^2}\vec{V} \, ,
\end{equation}
where we have integrated up to the cutoff $|\vec{q}|= M_W$. Let us introduce the notation
\begin{equation}\label{eq:Hnol}
\mathsf{H}^{\nu\bar\nu}_0= -V\beta-2\sqrt{2}G_{\rm F}\,
\hat{\vec\epsilon}^{*}\!\!\int_{\vec{q}} \hat{\vec{q}} \ell\,.
\end{equation}
Then using Eq.~\eqref{eq:cons1} we can write Eq.~\eqref{eq:selfnu} as
\begin{equation}
\mathsf{H}^{\nu\bar\nu}\approx\mathsf{H}^{\nu\bar\nu}_0\biggl(1-\sqrt{2}\frac{G_{\rm F} M_W^2}{3\pi^2}\biggr)^{-1}\, .
\end{equation}
In other words the $\kappa$ terms in Eq.~\eqref{eq:selfnu} effectively lead to a renormalization of
the total potential produced by the matter and neutrino backgrounds. Numerically, the correction is small,
$\sqrt{2}G_{\rm F} M_W^2/(3\pi^2)\approx 3\times 10^{-3}$, and can be neglected.
In a supernova the neutrino density is comparable to that of matter.
Whereas each individual neutrino is relativistic, the bulk velocity of
the neutrino background is also comparable to the matter velocity. Thus,
the neutrino density contribution to Eq.~\eqref{eq:Hnol} is not expected to be larger than the matter
contribution. Furthermore, because the direction of the neutrino background
flux is more likely to be parallel to the momenta of individual neutrinos,
whereas the build up of the particle-antiparticle coherence requires a current
component orthogonal to the neutrino momentum, there is an additional
suppression as compared to the matter effect. All in all, the estimates
of $\kappa$ presented above remain essentially unaltered by the inclusion
of the neutrino-neutrino interactions.
\subsection{Resonance condition}
As follows from Eq.~\eqref{kappaeqsol}, particle-antiparticle coherence can
be resonantly enhanced if $\mathsf{H}^{\nu\nu}=\mathsf{H}^{\bar\nu\bar\nu}$.
In general for a relativistic matter flow that also includes the neutrino
flux, the Hamiltonian matrix reads, see Eq.~\eqref{GammaWeyl},
\begin{align}
\mathsf{H}&=
E
\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}+
\begin{pmatrix}
V^0-V_{\shortparallel} & -V_\perp \\
-V_\perp & V^0+V_{\shortparallel}
\end{pmatrix}\,,
\end{align}
where, as before, $V_{\shortparallel}\equiv \hat{\vec p}\vec{V}$ and
$V_{\perp}\equiv \hat{\vec \epsilon}\vec{V}$ are components of the matter flux
parallel and orthogonal to the neutrino momentum. The resonance condition then
translates into $E=V_{\shortparallel}$. Even assuming a relativistic matter flow,
for typical supernova parameters $V_{\shortparallel}/E\sim 10^{-9}$. In other
words, the resonance condition cannot be fulfilled in a supernova and there is
no reason to expect $\kappa$ to be larger than the estimate presented above.
Note also that for $V \sim E$ not only does the Fermi approximation break
down, but also the perturbative description is no longer applicable. In other
words it is in principle not possible to hit the resonance without rendering
the developed formalism meaningless.
\subsection{\label{sec:inconditions}Initial conditions}
All physical processes in which neutrinos are created have time scales much larger
than the time scale of $\kappa$ oscillation. Hence, even during the production
process neutrinos would adiabatically adapt to the propagation basis with respect to
pair correlations. On the
other hand, the time scales of flavor and helicity oscillation are much larger than
those associated with production and detection. This separation of time scales
is crucial for the idea that neutrinos are produced in an eigenstate of interaction, i.e.,
in a coherent superposition of propagation eigenstates.
For the same physical reason, as neutrinos stream away from the supernova, they
have enough time to adiabatically
adapt to the external background. Thus, $\kappa$ does not oscillate but instead
closely tracks its equilibrium value. This makes dynamical equations for $\kappa$
essentially superfluous. As the neutrinos
leave the supernova, the mean pair correlations approach zero adiabatically and
decouple from the evolution of $\rho$ and $\bar\rho$.
\subsection{Lorentz covariance}
In the early Universe, the rest frame of the plasma is the only natural reference
frame and the question of Lorentz transformation properties of the pair
correlators does not arise \cite{Fidler:2011yq}. In a supernova environment the
situation is more complicated. In particular, the comoving frame of the matter
can in some cases be more convenient than the rest frame of a distant observer.
Similarly to helicity coherence, the particle-antiparticle coherence builds up
only if the off-diagonal components of the Hamiltonian matrix are not zero. However,
because the off-diagonals are proportional to the component of the matter flow
orthogonal to the neutrino momentum, their value depends on the frame. In particular,
one can find a frame where the off-diagonals vanish and no particle-antiparticle
coherence builds up.
Let us consider the same example as in Sec.~\ref{sec:helicity}. We have
two identical observers moving with velocity $\beta$ with respect to each other.
In the frame of the first observer, the neutrino has momentum $\vec{p}$ along
the $z$ axis and the matter is at rest, i.e. $V_{\shortparallel}=V_\perp=0$,
\begin{align}
\label{Hframe1pa}
\mathsf{H}&=E
\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}+
V
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}
\,.
\end{align}
Thus no particle-antiparticle coherence builds up. In the frame of the second
observer which moves with velocity $\beta$ along the $x$ axis the Hamiltonian
is no longer diagonal,
\begin{align}
\label{Hframe2pa}
\mathsf{H}&=\gamma E
\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}+
V
\begin{pmatrix}
\sfrac{1\!}{\gamma} & -\beta \\
-\beta & \sfrac{1\!}{\gamma}
\end{pmatrix}
\,,
\end{align}
and we expect helicity coherence to build up. In other words, physical results
seem to depend on the frame.
As we have learned from the analysis of an analogous problem for helicity
coherence, the kinetic equations are covariant only if the initial conditions
also transform under the boost. Pair correlations ``couple''
neutrinos of opposite momenta. The notion of opposite momenta is not Lorentz
invariant and is violated by, e.g., a boost orthogonal to the neutrino momentum.
This alone implies that the initial conditions, which include specifying
$\kappa$ for all momentum modes, are not Lorentz invariant. At the same
time the very fact that the definition of $\kappa$ involves two momentum
modes makes it rather difficult to derive the corresponding Lorentz
transformation rules and we will not attempt the derivation here.
We have argued in the previous subsection that neutrinos are
produced and propagate in an eigenstate with respect to particle-antiparticle coherence. In
Sec.~\ref{sec:helicity} we have observed that if in every frame we
diagonalize the Hamiltonian then the (transformed) eigenstate of the
Hamiltonian remains invariant under the boosts. Here we assume that
the same holds true also for particle-antiparticle coherence.
As a particularly interesting example let us assume that in the frame
of the first observer the system is in the vacuum state of the interacting
Hamiltonian, i.e. $\rho=\bar\rho=\kappa=0$,
\begin{align}
\label{Rframe1pa}
{\sf R}&=
\begin{pmatrix}
0 & 0 \\
0 & 1
\end{pmatrix}\,.
\end{align}
The Hamiltonian matrix Eq.~\eqref{Hframe1pa} and the extended density
matrix Eq.~\eqref{Rframe1pa} commute and therefore the latter is constant
in time. According to our assumption, after diagonalizing Eq.~\eqref{Hframe2pa}
by a Bogolyubov transformation, the transformed $\sf R$ takes the form
Eq.~\eqref{Rframe1pa}. Transforming back to the initial basis we obtain,
\begin{align}
\label{Rframe2pa}
{\sf R}&=
\frac12
\begin{pmatrix}
1-c_\vartheta & s_\vartheta \\
s_\vartheta & 1+c_\vartheta
\end{pmatrix}\,,
\end{align}
where $\vartheta$ is the angle of the Bogolyubov transformation that diagonalizes
Eq.~\eqref{Hframe2pa}, $\tan 2\vartheta=(\beta V/\gamma E)/[1-\beta^2(V/E)]$. By
construction the Hamiltonian matrix Eq.~\eqref{Hframe2pa} and the extended density
matrix Eq.~\eqref{Rframe2pa} commute and the latter is time inde\-pendent as well.
A perplexing feature of Eq.~\eqref{Rframe2pa} is that it seems to describe a
state with a nonzero number of particles and antiparticles.
Whereas the first observer would see neither neutrinos nor antineutrinos, the
second observer that moves with respect to the first one with a \emph{constant}
velocity $\beta$ seems to observe a nonzero density of neutrinos and antineutrinos.
Put in other words, the empty space perceived by the first observer appears
to be filled with neutrino-antineutrino pairs in the frame of the second observer.
However, it is not entirely clear if the (anti)particle densities in Eq.~\eqref{Rframe2pa}
describe electroweak interaction eigenstates and thus would actually manifest themselves
via, e.g., particle production or momentum transfer to nuclei in scattering
processes.
\subsection{Interpretation of the Bogolyubov transformation}
To better understand the meaning of the Bogolyubov transformation,
we solve the equation of motion for a massless
neutrino field coupled to a constant classical current $V^\mu$, and
demonstrate that this solution reproduces the results obtained using
the Bogolyubov transformation.
In the Fermi limit
\smash{$\mathcal{L}=\nu^\dagger_{\dot\alpha}\bar{\sigma}^{\mu,\dot{\alpha}\alpha}(i\partial_\mu-V_\mu)\nu_\alpha$}.
Varying the Lagrangian with respect to the neutrino field, we obtain the equation of
motion, $\bar{\sigma}^{\mu,\dot{\alpha}\alpha}(i\partial_\mu-V_\mu)\nu_\alpha=0$.
Its solution can be written in a form similar to Eq.~\eqref{nudecompositionweil},
\begin{align}
\nu(t,\vec{p})&=a(t,\vec{p})\chi_{_-}(\hat{\vec{p}}_{\vec{V}})
+b^\dagger(t,-\vec{p})\chi_{_+}(\hat{\vec{p}}_{\vec{V}})\,,
\end{align}
where \smash{$a(t,\vec{p})=a(t,\vec{p})e^{-i\omega_+t}$} and
\smash{$b^\dagger(t,-\vec{p})=b^\dagger(-\vec{p})e^{i\omega_-t}$} satisfy the usual anticommutation relations,
$\hat{\vec{p}}_{\vec{V}}$ is the unit vector in the direction of
$\vec{p}-\vec{V}$, and the energy spectrum is given by $\omega_\pm\equiv
|\vec{p}-\vec{V}|\pm V^0$.
Using the orthogonal vectors $\hat{\vec p}$ and $\hat{\vec \epsilon}$
we can write $\omega_\pm$ in the form
\smash{$\omega_\pm=\sqrt{(E-\hat{\vec{p}}\vec{V})^2+|\hat{\vec{\epsilon}}\vec{V}|^2}
\pm V^0$}, which reproduces the eigenvalues of the Hamiltonian matrix
Eq.~\eqref{GammaWeyl}. The spinor contractions, see Eq.~\eqref{GammasDef},
now include $\chi_{_\mp}(\hat{\vec{p}}_{\vec{V}})$. By construction,
$\Gamma^{\nu\bar\nu}$ and $\Gamma^{\bar\nu\nu}$ vanish once we use the
solution of the equations of motion. The diagonal elements can be
expanded in terms of $\chi_{_\mp}(\hat{\vec{p}})$. For example for
$\Gamma^{\nu\nu}$ we obtain
\begin{align}
\label{chiexpansion}
\chi^\dagger_{_-}(\hat{\vec{p}}_{\vec{V}})
\bar{\sigma}^{\mu}\chi_{_-}(\hat{\vec{p}}_{\vec{V}})&=
c_1 n^\mu(\hat{\vec{p}})+\mathrm{Re}[c_2\epsilon^\mu(\hat{\vec{p}})]\,.
\end{align}
Multiplied by $V_\mu$, Eq.~\eqref{chiexpansion} reproduces the interaction
part of the $\mathsf{H}^{\nu\nu}$ element of the
diagonalized Hamiltonian matrix. The decomposition coefficients
\begin{align}
\label{c1and2}
c_1\equiv \frac{E-\hat{\vec{p}}\vec{V}}{|\vec{p}-\vec{V}|}\,\quad \mathrm{and} \quad
c_2\equiv -\frac{\hat{\vec{\epsilon}}^*\vec{V}}{|\vec{p}-\vec{V}|}\,,
\end{align}
are related to the angle of the Bogolyubov transformation by $c_1=\cos 2\vartheta$
and $|c_2|= \sin 2\vartheta$ respectively.
In other words, diagonalizing the Hamiltonian matrix by a Bogolyubov transformation
in every frame is equivalent to using the equation of motion. This equivalence suggests interpreting
physical particle densities as propagation eigenstates of the full
Hamiltonian in line with the discussion in Sec.~\ref{sec:inconditions}.
\section{\label{sec:summary}Summary and conclusions}
Neutrino flavor conversion is important in supernovae, yet a full
understanding remains elusive, largely because of neutrino-neutrino
refraction and concomitant self-induced flavor conversion, an effect caused
by run-away modes of the interacting neutrino gas. The difficulties in
developing a robust phenomenological understanding of even this relatively
simple case explains the reluctance to add further complications. Yet other
effects could be important as well, caused by inhomogeneities and
anisotropies of the medium and by magnetic fields, especially if one broadens
the view to include, for example, magnetars or neutron-star mergers. It is
often thought that helicity conversion effects will be small, at least if
neutrino dipole moments have no additional contributions beyond those
provided by their masses, yet one should remain open to such possibilities.
Finally, beyond flavor and helicity correlations, it has been stressed
recently that pair correlations could also become important.
Motivated by these concerns, we have studied extended kinetic equations that
describe flavor, helicity, and pair correlations, limiting ourselves to the
mean-field level, i.e., considering only propagation effects for freely
streaming neutrinos. Based on the ``forward Hamiltonian'' of neutrinos
interacting with a background medium, we have derived the various terms and
have given explicit results up to lowest order in the neutrino mass,
similar to previous studies in the literature. For
Dirac neutrinos, we confirmed previous results and have extended them to
include magnetic-field effects. For Majorana
neutrinos, we found a small correction to the mean-field Hamiltonian which
arises from lepton-number-violating contractions that appear only in the
Majorana case. To analyze the behavior of these additional terms in the limit
of vanishing neutrino masses, we have also studied extended kinetic equations
for Weyl neutrinos.
The density matrix formalism allows one to treat helicity oscillation induced
by matter currents and by magnetic fields on equal footing for both Dirac
and Majorana neutrinos. We have derived the mean-field Hamiltonian induced by
electromagnetic fields and compared it to that induced by matter currents.
Somewhat surprisingly, for typical supernova parameters, matter currents
dominate over magnetic fields. In principle, resonant enhancements can be
achieved, for example by relativistic flows of matter and background
neutrinos.
Flavor and helicity oscillations can be complicated in detail, but they are
conceptually straightforward. Their importance arises because charged-current
interactions produce neutrinos in flavor eigenstates, and all interactions
produce them in almost perfect helicity states. This nonequilibrium
distribution which is produced, for example, in the neutrino-sphere region of
a supernova, subsequently evolves coherently and leads to the various flavor
and helicity oscillation phenomena.
Concerning pair correlations, the mean-field equations produce similar
oscillation equations. In the simplest case of massless neutrinos, the pair
correlations are between neutrinos and antineutrinos of opposite momenta and
the oscillations are between the empty state and the one filled with a
neutrino and antineutrino. However, one probably cannot separate production
from subsequent propagation. The oscillation frequency is here twice the
neutrino energy, so in contrast to flavor and helicity oscillations, there is
no separation of scales between the energy of the state and the oscillation
frequency. Probably, as far as pair correlations are concerned, one should picture neutrinos as being produced in
eigenstates of propagation in the medium and not as eigenstates of the
interaction Hamiltonian. Flavor and helicity oscillations become important only
because one produces a coherent superposition of different
propagation eigenstates. As this crucial characteristic appears to be missing
for pair correlations, we are tempted to suspect that pair correlations
remain a small correction to neutrino dispersion.
In the simplest case, helicity and pair correlations build up only in
anisotropic media because angular-momentum conservation forbids mixing of
states with different spin. If the anisotropy is a convective matter
current, then there is a seeming paradox. In the frame with the current
we expect correlations to build up. On the other hand, we may study
these effects in the rest frame of the medium where no
correlations build up due to isotropy of the background. As far as
helicity correlations are concerned, this paradox is
resolved by noting that the handedness of massive neutrinos is not Lorentz
invariant. Transforming both the mean-field background and the neutrino
states to a different frame, e.g., the rest frame of the medium, leads to
consistent physical results. For pair correlations, physical results must
also be the same in all frames, yet it is less obvious how to show this
point explicitly because the correlated modes of opposite momentum are
different ones in every frame. Note, however, that in the
supernova context, there is not necessarily a natural coordinate system for
the study of neutrino propagation. Explicitly including production and
detection processes, i.e., the collision terms in the kinetic equation, may
shed more light on this question.
The ultimate ambition of fully understanding neutrino propagation in dense
environments and strong magnetic fields requires a more complete development
of its theoretical underpinnings. Our paper is meant as a contribution toward
this overall goal.
\section*{Acknowledgments}
We would like to thank I. Izaguirre, S. Chakraborty, A. Dobrynina, and
C. Volpe for fruitful discussions.
We acknowledge partial support by the Deutsche Forschungsgemeinschaft (DFG)
under Grant No.\ EXC-153 (Excellence Cluster ``Universe''), and by the
Research Executive Agency (REA) of the European Union under Grant No.\
PITN-GA-2011-289442 (FP7 Initial Training Network ``Invisibles'').
\begin{appendix}
\section{\label{sec:spinorproducts}Chiral spinors}
Following the conventions of Ref.~\cite{Dreiner:2008tw}, which differ
from the ones used in Ref.~\cite{Serreau:2014cfa} by the overall sign of
$\gamma^0$, the Dirac matrices
in the Weyl representation, which is used in this work, are
\begin{align}
\gamma^\mu=\left(
\begin{tabular}{cc}
$0$ & $\sigma^\mu$\\
$\bar{\sigma}^{\mu}$ & $0$
\end{tabular}
\right)\,,
\end{align}
where $\sigma^\mu=(1,\vec{\sigma})$ and
$\bar{\sigma}^\mu=(1,-\vec{\sigma})$. Here $\vec\sigma$ is a
three-vector Pauli matrix and $0$ and $1$ are $2{\times}2$
zero and unity matrices respectively.
The chiral projectors are
\begin{align}
\label{PLPR}
P_L=
\begin{pmatrix}
1 & 0\\
0 & 0
\end{pmatrix}
\,,\quad
P_R=
\begin{pmatrix}
0 & 0\\
0 & 1
\end{pmatrix}\,.
\end{align}
The charge-conjugation matrix is
\begin{align}
\label{C}
C=-i\gamma^2\gamma^0=
\begin{pmatrix}
+\varepsilon & 0\\
0 & -\varepsilon
\end{pmatrix}\,,
\end{align}
where
\begin{equation}
\varepsilon=
\begin{pmatrix}
0&-1\\
+1&0
\end{pmatrix}\,.
\end{equation}
Notice that in two-component form, one usually writes
$\varepsilon_{\alpha\beta}$ for this antisymmetric $2{\times}2$ matrix and
\smash{$\varepsilon^{\dot\alpha\dot\beta}$} for $-\varepsilon$ appearing in
the lower right position of $C$.
In the Weyl representation and with these conventions, the Dirac bispinors are
\begin{subequations}
\label{ChiralAmplitudes}
\begin{align}
u_i(\vec{p},s)&=
\begin{pmatrix}
{\cal N}^i_{p,s}\,\chi_s(\hat{\vec p})\\[1mm]
{\cal N}^i_{p,-s}\,\chi_s(\hat{\vec p})
\end{pmatrix}
\,,\\
v_i(\vec{p},s)&=s
\begin{pmatrix}
-{\cal N}^i_{p,-s}\,\chi_{-s}(\hat{\vec p})\\[1mm]
{\cal N}^i_{p,s}\,\chi_{-s}(\hat{\vec p})
\end{pmatrix}
\,,
\end{align}
\end{subequations}
where $\hat{\vec p}$ is the unit vector in the direction of $\vec{p}$,
$p\equiv |\vec p|$, $s=\pm$ is a helicity index, and
\begin{align}
\label{Ndef}
{\cal N} ^i_{p,s}=\sqrt{\frac{E_{i}-sp}{2E_{i}}}\approx
\delta_{s-}+\frac{m_i}{2p}\delta_{s+}\,,
\end{align}
where $E_i=(p^2+m_i^2)^{1/2}$ is the energy of a neutrino with mass $m_i$.
We may describe the modes of the neutrino field in spherical
coordinates where the momentum components are
$\hat{\vec p}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$. In
this case, the standard two-component helicity spinors are explicitly
\begin{subequations}
\begin{align}
\chi_{_+}(\hat{\vec p})&=
\begin{pmatrix}
\cos\frac{\theta}{2}\\
e^{i\phi}\sin\frac{\theta}{2}
\end{pmatrix}\,,\\
\chi_{_-}(\hat{\vec p})&=
\begin{pmatrix}
-e^{-i\phi}\sin\frac{\theta}{2}\\
\cos\frac{\theta}{2}
\end{pmatrix}\,.
\end{align}
\end{subequations}
They satisfy the orthogonality
condition $\chi^\dagger_{s}(\hat{\vec p})\chi_{h}(\hat{\vec p})=\delta_{sh}$.
The matrix elements of $\bar\sigma^\mu$ are then found by direct
evaluation to be
\begin{subequations}
\begin{align}
\chi^\dagger_{_-}(\hat{\vec p})\bar{\sigma}^\mu \chi_{_-}(\hat{\vec p})&=
n^\mu=(1,\hat{\vec p})\,,\\
\chi^\dagger_{_+}(\hat{\vec p})\bar{\sigma}^\mu \chi_{_+}(\hat{\vec p})&=
\bar n^\mu=(1,-\hat{\vec p})\,,\\
\chi^\dagger_{_+}(\hat{\vec p})\bar{\sigma}^\mu \chi_{_-}(\hat{\vec p})&=
-e^{-i\phi}\epsilon^\mu=-e^{-i\phi}(0,\hat{\vec \epsilon})\,,\\
\chi^\dagger_{_-}(\hat{\vec p})\bar{\sigma}^\mu \chi_{_+}(\hat{\vec p})&=
-e^{i\phi}\epsilon^{\mu*}=-e^{i\phi}(0,\hat{\vec \epsilon}^*)\,,
\end{align}
\end{subequations}
where $\epsilon^\mu$ is a polarization vector orthogonal to
$n^\mu$. The explicit components in spherical coordinates are
\begin{align}
\hat{\vec \epsilon}=
\begin{pmatrix}
e^{i\phi}\cos^2\frac{\theta}{2}-e^{-i\phi}\sin^2\frac{\theta}{2}\\[1mm]
-i\bigl(e^{i\phi}\cos^2\frac{\theta}{2}+e^{-i\phi}\sin^2\frac{\theta}{2}\bigr)\\[1mm]
-\sin\theta
\end{pmatrix}
\,.
\end{align}
Note that the vectors $n^\mu$ and $\epsilon^\mu$ depend on $\hat{\vec p}$,
but we do not show this dependence explicitly to simplify the notation.
\section{\label{sec:contractions}Neutrino-neutrino mean-field Hamiltonian}
Because Majorana and Weyl neutrinos have two degrees of freedom, in many
cases it is more convenient to use two-component notation. For Majorana
neutrinos,
\begin{align}
\label{MajTwoComp}
\nu_i=
\begin{pmatrix}
\nu_{i,\alpha}\\
\nu_i^{\dagger\dot{\alpha}}
\end{pmatrix}
\quad\hbox{and}\quad
\bar{\nu}_i=\left(
\nu_i^\alpha\,\,\nu^\dagger_{i,\dot\alpha}
\right)\,,
\end{align}
where $\nu_{i,\alpha}$ and $\nu^\dagger_{i,\dot{\alpha}}$ are two-component
fields. They are related by Hermitian conjugation and transform under the
\smash{$\left(\frac12,0\right)$} and \smash{$\left(0,\frac12\right)$} representations of the
Lorentz group respectively. To emphasize the different transformation properties,
the conjugated fields, by convention, always carry a dotted spinor index.
The spinor indices $\alpha$ and $\dot{\alpha}$
are raised (lowered) using the spinor metric matrices \smash{$\varepsilon^{\alpha\beta}$}
and \smash{$\varepsilon^{\dot\alpha\dot\beta}$} (\smash{$\varepsilon_{\alpha\beta}$}
and \smash{$\varepsilon_{\dot\alpha\dot\beta}$}). Left-handed Weyl
fields satisfy the condition $P_L\nu=\nu$. Their explicit form can be obtained from
Eq.~\eqref{MajTwoComp} by applying the chiral projectors.
Rewritten in terms of the two-component fields, the neutrino-neutrino
Hamiltonian density of Eq.~\eqref{Hself} is
\begin{align}
\mathcal{H}^{\nu\nu}&= \frac{G_{\rm F}}{\sqrt{2}}\sum\limits_{ij}
\bigl[\nu^\dagger_{i,\dot{\alpha}}\bar{\sigma}^{\mu,\dot{\alpha}\alpha}\nu_{i,\alpha}\bigr]
\bigl[\nu^\dagger_{j,\dot{\beta}}\bar{\sigma}_\mu^{\dot{\beta}\beta}\nu_{j,\beta}\bigr]\,.
\end{align}
Taking expectation values of products of two of the four neutrino fields and bearing
in mind that fermions anticommute we obtain for the mean-field Hamiltonian
\begin{align}
\label{HeffTwoComponent}
\mathcal{H}^{\nu\nu}_{\rm mf}&= \frac{G_{\rm F}}{\sqrt{2}}\sum\limits_{ij} \bar{\sigma}^{\mu,\dot{\alpha}\alpha}\bar{\sigma}_\mu^{\dot{\beta}\beta}\nonumber\\
&\times \Bigl[2\,\nu^\dagger_{i,\dot{\alpha}}\nu_{i,\alpha}\, \langle \nu^\dagger_{j,\dot{\beta}}\nu_{j,\beta} \rangle
-2\,\nu^\dagger_{i,\dot{\alpha}} \nu_{j,\beta}\, \langle \nu^\dagger_{j,\dot{\beta}} \nu_{i,\alpha}\rangle\nonumber\\
&\kern1.5em
+\nu^\dagger_{i,\dot{\alpha}} \nu^\dagger_{j,\dot{\beta}}\, \langle \nu_{j,\beta} \nu_{i,\alpha} \rangle
+ \nu_{i,\alpha} \nu_{j,\beta}\, \langle \nu^\dagger_{j,\dot{\beta}} \nu^\dagger_{i,\dot{\alpha}} \rangle\Bigr].
\end{align}
Translating back to the four-component notation we obtain
$[\bar{\nu}_i \gamma^\mu P_L \nu_i] \langle \bar{\nu}_j \gamma_\mu P_L \nu_j\rangle$
for the first term in Eq.~\eqref{HeffTwoComponent}. Using a Fierz identity \cite{Dreiner:2008tw},
$\bar{\sigma}^{\mu,\dot{\alpha}\alpha}\bar{\sigma}_\mu^{\dot{\beta}\beta}=
-\bar{\sigma}^{\mu,\dot{\alpha}\beta}\bar{\sigma}_\mu^{\dot{\beta}\alpha}$,
and translating back to the four-component notation we can represent the second term
in a similar form,
$[\bar{\nu}_i \gamma^\mu P_L \nu_j] \langle \bar{\nu}_j \gamma_\mu P_L \nu_i\rangle$.
By raising and lowering the spinor indices and reordering the fields the third
term can be rewritten as
$\sigma^{\mu}_{\dot{\alpha}\alpha}\bar{\sigma}_\mu^{\dot{\beta}\beta}
\nu^\dagger_{j,\dot{\beta}}\nu_{i}^{\dagger\,\dot{\alpha}}\,
\langle \nu_{i}^{\alpha} \nu_{j,\beta}\rangle$.
Using another Fierz identity \cite{Dreiner:2008tw},
$\sigma^{\mu}_{\dot{\alpha}\alpha}\bar{\sigma}_\mu^{\dot{\beta}\beta}
=2\delta_\alpha\,^\beta\delta^{\dot \beta}\,_{\dot \alpha}$,
and translating back to four-component notation we can rewrite
the third term in the form
$2[\bar{\nu}_j P_R C \bar{\nu}^T_i]\langle \nu^T_i C P_L \nu_j \rangle$.
Collecting all terms we obtain in four-component notation
\begin{align}
\mathcal{H}^{\nu\nu}_{\rm mf}=\sqrt{2}G_{\rm F}\sum_{ij}\Bigl(&[\bar{\nu}_i \gamma^\mu P_L \nu_i] \langle \bar{\nu}_j \gamma_\mu P_L \nu_j\rangle\nonumber\\[-3mm]
&+ [\bar{\nu}_i \gamma^\mu P_L \nu_j] \langle \bar{\nu}_j \gamma_\mu P_L \nu_i\rangle\nonumber\\[4pt]
&+ [\bar{\nu}_i P_R C \bar{\nu}^T_j]\langle \nu^T_j C P_L \nu_i \rangle \nonumber\\
&+ [\nu^T_i C P_L \nu_j ]\langle \bar{\nu}_j P_R C \bar{\nu}^T_i\rangle\Bigr)\,.
\end{align}
Using the definition of the charge-conjugate field, \smash{$\nu^c\equiv C\bar{\nu}^T$},
and the resulting $\overline{\nu^c}=\nu^T C$ we can further simplify
and write the last two terms in a form which coincides with
Eq.~\eqref{Heff3},
\begin{align}
\mathcal{H}^{\nu\nu}_{\rm mf}=\sqrt{2}G_{\rm F}\sum_{ij}\Bigl(&[\bar{\nu}_i \gamma^\mu P_L \nu_i] \langle \bar{\nu}_j \gamma_\mu P_L \nu_j\rangle\nonumber\\[-3mm]
&+ [\bar{\nu}_i \gamma^\mu P_L \nu_j] \langle \bar{\nu}_j \gamma_\mu P_L \nu_i\rangle\nonumber\\[4pt]
&+ [\bar{\nu}_i P_R \nu^c_j]\langle \overline{\nu^c_j} P_L \nu_i \rangle \nonumber\\
&+ [\overline{\nu^c_i} P_L \nu_j ]\langle \bar{\nu}_j P_R \nu^c_i\rangle\Bigr)\,.
\end{align}
Thus, the effective Hamiltonian obtained using the two-com\-ponent notation is identical
to the one obtained using the four-component notation, as expected.
\section{\label{sec:anapole}Right-chiral currents}
For completeness, we provide the contractions of the Lorentz structure of right-chiral currents $\left(\gamma^\mu P_R\right)$, which might arise in, e.g., beyond the Standard Model theories with right-handed currents. The contractions are
\begin{subequations}\label{GPR}
\begin{align}
(\gamma_\mu P_R)^{\nu\nu}_{ij,sh} &\approx
\begin{pmatrix}
0 & e^{+i\phi}\frac{m_i}{2p}\epsilon^*_\mu \\
e^{-i\phi}\frac{m_j}{2p}\epsilon_\mu & n_\mu
\end{pmatrix}\,,\\[3pt]
(\gamma_\mu P_R)^{\nu\bar\nu}_{ij,sh} &\approx
\begin{pmatrix}
e^{+i\phi}\frac{m_i}{2p}\bar n_\mu & 0 \\
\epsilon_\mu & e^{-i\phi}\frac{m_j}{2p}n_\mu
\end{pmatrix}
\,,\\[3pt]
(\gamma_\mu P_R)^{\bar\nu\nu}_{ij,sh} &\approx
\begin{pmatrix}
e^{-i\phi}\frac{m_j}{2p}\bar n_\mu & \epsilon^*_\mu \\[2pt]
0 & e^{+i\phi}\frac{m_i}{2p}n_\mu
\end{pmatrix}
\,,\\[3pt]
(\gamma_\mu P_R)^{\bar\nu\bar\nu}_{ij,sh}&\approx
\begin{pmatrix}
\bar{n}_\mu & e^{-i\phi}\frac{m_j}{2p}\epsilon^*_\mu \\
e^{+i\phi}\frac{m_i}{2p}\epsilon_\mu & 0
\end{pmatrix}\,.
\end{align}
\end{subequations}
\end{appendix}
|
{
"timestamp": "2015-06-18T02:10:54",
"yymm": "1504",
"arxiv_id": "1504.03230",
"language": "en",
"url": "https://arxiv.org/abs/1504.03230"
}
|
\section{INTRODUCTION}
\label{sec:intro}
The Australian Dark Energy Survey (OzDES)\footnote{Or: {\it Optical
redshifts for the Dark Energy Survey}.} has been designed to
provide efficient spectroscopic follow-up of targets identified from imaging by the Dark
Energy Survey \citep[DES;][]{2005IJMPA..20.3121F,2014SPIE.9149E..0VD}. OzDES extends DES by enabling
new science goals that cannot be achieved without spectroscopic
information --- such as supernova cosmology and reverberation mapping
of active galactic nuclei (AGN). We also enhance DES by providing an
important source of calibration data for photometric redshifts, which
are the cornerstone for the majority of the DES science programs.
The 3.9m Anglo-Australian Telescope (AAT) used by OzDES and the CTIO~4m
Blanco telescope used by DES are ideal partners for spectroscopy and
imaging because they have well matched $\sim$2 degree diameter fields of view (see
Fig.~\ref{fig:fov}). The AAT and CTIO~4m are similar in several
respects --- both are 4m class telescopes that were commissioned in
1974 and both have recently been rejuvenated with powerful new
instrumentation: in the case of CTIO it is the 570 mega-pixel Dark
Energy Camera \citep[DECam;][]{2012PhPro..37.1332D,2012SPIE.8446E..11F}, while on the AAT it is the new efficient
AAOmega spectrograph \citep{2004SPIE.5492..410S} coupled with the Two
Degree Field (2dF) 400-fibre multi-object fibre-positioning system
\citep{2002MNRAS.333..279L}.
The DES program consists of a wide-field survey covering 5000 square
degrees, as well as a rolling survey of ten fields that cover a total of 30
square degrees \citep{2014SPIE.9149E..0VD}. These ten fields (see Table~\ref{tab:field_centers} for the sky coordinates) are targeted repeatedly over the
course of the survey with a cadence of approximately 6 days in order
to find transient objects, such as supernovae, and monitor variable
objects, such as AGN. OzDES repeatedly targets these ten fields,
selecting objects that range in brightness from $m_r\sim17$ to
$m_r\sim25$ mag, a range of more than a thousand in flux density. Once a
redshift is obtained, we deselect the target (with some exceptions).
Objects that lack a redshift are
observed until a redshift is measured.
This tactic allows us to obtain
redshifts for targets far fainter than ever previously achieved with
the AAT. Together this means we can run an efficient survey of bright
objects while simultaneously acquiring spectra for much fainter
objects.
\begin{table}
\caption{Center coordinates of the ten DES SN fields.}
\label{tab:field_centers}
\begin{tabular*}{80mm}{@{\extracolsep{\fill}}lccc}
\hline
Field Name & R.A. (h m s) & Decl. ($\circ$ $^\prime$ ${''}$) & Comment\textsuperscript{*} \\
\hline
~E1 & 00 31 29.9 & -43 00 35 & \multirow{2}{*}{ELAIS\textsuperscript{$\dagger$} S1} \\
~E2 & 00 38 00.0 & -43 59 53 & \\
\hline
~S1 & 02 51 16.8 & ~00 00 00 & \multirow{2}{*}{Stripe 82\textsuperscript{$\ddagger$}} \\
~S2 & 02 44 46.7 & -00 59 18 & \\
\hline
~C1 & 03 37 05.8 & -27 06 42 & \multirow{3}{*}{CDFS\textsuperscript{$\S$}} \\
~C2 & 03 37 05.8 & -29 05 18 & \\
~C3 (deep) & 03 30 35.6 & -28 06 00 & \\
\hline
~X1 & 02 17 54.2 & -04 55 46 & \multirow{3}{*}{XMM-LSS\textsuperscript{$\parallel$}} \\
~X2 & 02 22 39.5 & -06 24 44 & \\
~X3 (deep) & 02 25 48.0 & -04 36 00 & \\
\hline
\\
\multicolumn{4}{l}{\textsuperscript{*}\footnotesize{Overlap with areas covered by other surveys.}} \\
\multicolumn{4}{l}{\textsuperscript{$\dagger$}\footnotesize{European Large-Area ISO Survey.}} \\
\multicolumn{4}{l}{\textsuperscript{$\ddagger$}\footnotesize{An equatorial region repeatedly imaged by SDSS.}} \\
\multicolumn{4}{l}{\textsuperscript{$\S$}\footnotesize{Chandra Deep Field South survey.}} \\
\multicolumn{4}{l}{\textsuperscript{$\parallel$}\footnotesize{X-ray Multi Mirror Large Scale Structure survey.}} \\
\end{tabular*}
\end{table}
OzDES has a total of 100 nights distributed across five
years during the DES observing seasons (August to January). The first
season of OzDES began in 2013B (August 2013 to January 2014) with 12 nights (designated as Y1).
Allocations will progressively increase each year, to
accommodate the increasing number of supernovae host galaxies that DES will
have accumulated in subsequent years.
In the 2012B (August 2012 to January 2013) semester, a year before the start of DES and OzDES, the
DECam was used to execute the DES SN program as part of
its Science Verification (SV) phase of commissioning. In parallel,
AAT/AAOmega-2dF time was awarded for the observation of the
DES SN fields in a precursor program to OzDES.
Supplementary observing time (1 night during SV and 2 nights during Y1) was also obtained through the National Optical Astronomy Observatory (NOAO).
In this paper, we present an overview of OzDES and the
results obtained during the SV and Y1 seasons.
Non-DES fields and targets observed in the SV season are excluded from the analysis.
When appropriate, we also discuss changes and improvements to the OzDES in Y2 (2014B, August 2014 to January 2015).
We organise the paper as follows: In $\S$~\ref{sec:science} we
summarise our science goals and in $\S$~\ref{sec:strategy} we
present the operational details that were current at the end of Y1,
highlighting what was modified to improve the efficiency of the
survey for Y2. Results of redshifts and quality assessments are
presented in $\S$~\ref{sec:results}, followed by discussions of
implications for science in $\S$~\ref{sec:discussion}. Finally, we
conclude in $\S$~\ref{sec:conclusion}.
\begin{figure}
\includegraphics[width=84mm]{ozdes_all_fibres.eps}
\caption{The 2dF spectrograph has 392 optical fibres to be placed at
positions across a field of view 2.1 degree
in diameter (large orange circle). This is well matched to the DECam field
of view (image mosaic in the background), making
spectroscopic follow-up very efficient. The small orange circles represent
the locations of targets selected for one set of AAT observations.}
\label{fig:fov}
\end{figure}
\section{Science Goals}\label{sec:science}
Here we outline the wide range of science goals we aim to achieve with
OzDES. Each science goal may require one or more types of target and
the targeting strategies evolve with time. Details of the target types used
in SV and Y1 are described in $\S$~\ref{sec:targets}.
\subsection{Type Ia Supernova Cosmology}
The main science motivation of OzDES is to obtain host galaxy spectroscopic
redshifts for a sample of 2,500 Type Ia SNe discovered by DES,
with the goal of improving measurements of the Universe's
global expansion history.
This goal is efficiently achieved with AAT's unique combination of
multi-object fibre-fed spectroscopy and wide field of view.
Spectroscopy of live supernovae within a few weeks of maximum light
serves the dual purpose of typing the transient (determining whether
it is a Type Ia SN) and measuring its redshift. However, the spatial
density of SNe Ia that are bright enough to be observed with the AAT
at any instant in time is only a handful per square degree. An 8--10 metre
class telescope is often required for such observations, but can
usually only observe one object at a time. This mode of follow-up is
thus time-consuming and requires an unrealistically large quantity of
resources to acquire a large sample.
Alternatively, well sampled multicolour light curves can be used to
reliably identify a SN Ia \citep{2011ApJ...738..162S}, after which a
spectrum of the host galaxy can be used to measure its precise redshift.
Host galaxy redshifts can also be used to improve photometric
classification \citep{2014AJ....147...75O}.
Host galaxies can be observed at any time, even after the supernova has
faded. Thus one can wait to collect many supernovae in a field before
measuring their host-galaxy redshifts efficiently with multi-object
spectroscopy. This strategy has been tested in the Sloan Digital Sky
Survey \citep[SDSS-II,][]{2014AJ....147...75O} and the SuperNova Legacy
Survey \citep[SNLS,][]{2013PASA...30....1L}.
\subsection{AGN Reverberation Mapping}
The other primary science goal of OzDES is reverberation mapping of
AGN and quasars. Reverberation mapping
\citep{1982ApJ...255..419B,1993PASP..105..247P} is an effective way to
measure supermassive black hole masses in AGN,
over much of the age of the
universe. This is possible because the continuum emission from the AGN
accretion disk is variable, and this continuum emission photoionizes
the clouds of gas at larger scales that give rise to the
characteristic broad emission lines of most AGN. As the continuum
emission varies in intensity, the broad emission lines reverberate in
response with a time delay that depends on the light travel time from
the continuum source.
Measurement of this time delay provides a geometric size for the broad
line region. This size scale
can be used to measure the mass of the supermassive black hole
through application of the virial theorem and measurement of the
velocity width of the broad emission line. Approximately 50 AGN have
reverberation-based black hole masses to date, and the masses of these
black holes agree well with mass measurements from stellar dynamics
\citep{2006ApJ...646..754D, 2014ApJ...791...37O} and yield the same
slope as the $M-\sigma_*$ relation that holds for quiescent galaxies
\citep{2010ApJ...716..269W, 2013ApJ...773...90G}.
The size scale, typically determined from the H$\beta$
emission line, is also very well correlated with the AGN luminosity with
the $R \propto L^{1/2}$ scaling expected from a simple photoionization
model \citep[e.g.][]{2013ApJ...767..149B}. The relatively small
scatter in this relation was used by \citet{2011ApJ...740L..49W} to
demonstrate that AGN could be used as standard candles.
The current sample of about 50 AGN with reverberation-based masses are
all in low to moderate luminosity AGN, and nearly all in the
relatively nearby universe ($z<0.3$). This is because reverberation
mapping requires a substantial amount of telescope time to measure the
time lags, and it has proven most straightforward to get the necessary
allocation to observe bright objects with small telescopes. The
lower-luminosity AGN also have lags of only days to weeks, and thus
can be measured with a single semester of data. It is much more
difficult to measure the year or longer lags of the most luminous AGN
at redshifts $z>1$ and higher \citep[although
see][]{2007ApJ...659..997K}; yet these AGN are arguably the most
interesting as they represent the majority of supermassive black hole
growth in the universe. OzDES is presently monitoring $\sim$1000 AGN
up to $z \approx 4$ and aims to measure reverberation lags and black
hole masses for approximately 40\% of the final sample (King et
al., in prep). This new, multi-object reverberation mapping project, as
well as other similar efforts \citep{2015ApJS..216....4S}, will
provide a wealth of new data on black hole masses out to and beyond
the peak of the AGN era. We will also use new measurements of the
radius-luminosity relation to construct a Hubble diagram out to higher
redshifts than can be reached with supernovae, which provides some
complementary constraints on the time variation in dark energy
\citep{2014MNRAS.441.3454K}.
\subsection{Transients}
Concurrent OzDES and DES observing enables time-critical spectroscopic
observations of transients discovered in imaging.
With a monthly observing cadence, OzDES is expected to target
several hundred active transients, putting these at highest priority.
This sample, supplemented by
observations of fainter events by larger telescopes, will provide
crucial validation of the photometric SN classification
and enable detailed studies of these SNe.
We target all kinds of transient candidates,
including those with uncertain physical nature. A survey of this size and
scope expects to find surprises in the data. With our targeting
strategy we aim to investigate the unexpected and potentially find
as-yet unidentified classes of transients.
\subsection{Photo-$z$ training}
A core requirement of DES is to obtain accurate photometric redshifts
(photo-$z$) for the majority of galaxies in the wide survey. This will
enable key science goals, such as the measurement of baryon acoustic
oscillations (BAO) with millions of galaxies, and the use of weak
lensing for cosmology. Our spectra play an important role in
providing a spectroscopic sample for calibrating and testing the DES
photometric redshifts, and a significant number of our fibres are
allocated to Luminous Red Galaxies (LRG), Emission Line Galaxies
(ELG), and other photo-$z$ targets.
Redshifts from OzDES have already been used in a number of recently
published studies on DES photometric
redshifts. \citet{2014MNRAS.445.1482S} have used the redshifts to
evaluate the performance of various photo-$z$ methods on DES SV data
and found several codes to produce photo-$z$ precisions and outlier
fractions that satisfy DES science requirements.
\citet{2015MNRAS.446.2523B} combines optical data from DES and
near-infrared (NIR) data from VISTA Hemisphere Survey \citep[VHS;][]{2013Msngr.154...35M}
to improve photo-$z$ performance. In particular, selection
criteria based on optical-NIR colors are applied to identify LRG
targets at high redshift ($z\gtrsim0.5$) for OzDES. Spectroscopic
results are used to verify the effectiveness of this selection method.
\subsection{Radio galaxies, cluster galaxies, and strong lenses}
The large number of fibres available to 2dF allows us to pursue
a wide range of supplementary science goals. These include the
following:
\begin{itemize}
\item {Gathering redshifts of galaxies selected from the ATLAS \citep[Australia Telescope Large Area Survey;][Franzen et al. in prep, Banfield et al. in prep]{2006AJ....132.2409N}. ATLAS is a deep, 14/17 uJy/beam rms, 1.4 GHz survey of 3.6/2.7 deg$^2$ of the CDFS/ELAIS-S1 survey fields which have \textgreater90\% overlap with the DES deep imaging. ATLAS is being used to study the astrophysics of radio sources, and is also being used as a pathfinder to develop the science and techniques for the primary radio continuum survey \citep[EMU;][]{2011PASA...28..215N} of the Australian Square Kilometre Array Pathfinder (ASKAP). ATLAS has detected over 5000 radio sources of which at least half will be targeted by OzDES over the course of the survey. These redshifts will be used to determine the evolution of the faint radio population, including both star forming galaxies and radio AGN, up to redshift greater than one. They will also be used to calibrate photometric and statistical redshift algorithms for use with the 70 million EMU sources (for which spectroscopy is impractical). Furthermore, the detected optical emission lines will provide insights into the detailed astrophysics within these galaxies, including distinguishing star forming galaxies from AGN.}
\item {Confirming previously unknown cluster candidates and
gathering redshifts for cluster galaxies, especially
central galaxies for the calibration of the cluster red sequence, as
well as validation of cluster photometric redshifts.
Our repeated returns to the same field allow us to collect
redshifts for multiple galaxies in a single cluster. Usually this is
impossible for all but very nearby clusters because the instrumental
limit of fibre collisions prevents one from measuring closely
neighbouring galaxies in the same exposure.}
\item {Measuring redshifts for both the lens and
the background lensed galaxy or quasar in strong lens candidates.
Some of the lensed quasar targets may be suitable for time-delay experiments.}
\end{itemize}
\subsection{Calibration}
About 10\% of fibres are used for targets that facilitate the
the calibration of the data. These include:
\begin{itemize}
\item Regions that are free of objects (sky fibres). Some of our
targets are 100 times or more fainter than the sky in the $2{\hbox{$^{\prime\prime}$}}$ fibre aperture, so a good
estimate of the sky brightness is crucial.
\item F stars that are used to monitor throughput, which is heavily
dependent on the seeing and the amount of cloud cover. Up to 15
F stars are observed per field and used to derive a mean sensitivity
curve. The variation of the sensitivity curve over each plate allows
for an estimate of the accuracy of the flux calibration and the
mean value allows us to appropriately weight data that are
obtained over multiple occasions.
\item Candidate hot (T$_{\rm eff} \sim 20,000~\rm{K}$) DA (hydrogen
atmosphere) white dwarfs (WDs) that can be used as primary flux
calibrators for the DES deep fields. Stellar atmosphere modeling
uncertainties for hot DA WDs are small so that synthetic
photometry can be compared with DES observations with an expected
accuracy of 2--3\% or better per candidate. A collection of $\sim$100
such candidates over the DES deep fields will allow one to test the
accuracy of the photometric calibration of the DES deep fields. The
number of known DA WDs in the DES deep fields is currently
too small to make this test, so we aim to find new ones.
\end{itemize}
\section{Observing Strategy}\label{sec:strategy}
\subsection{Instrument Setup and Observations}\label{sec:setup}
The 2dF robotic positioner allows up to 392 targets to be observed
simultaneously over a field of view 2.1 degrees in diameter (there are also 8 fibre bundles for guiding).
The projected
fibre diameter is approximately two arcsec. Two sets of fibres are
provided on separate field plates mounted back-to-back on a
tumbler. Configuration of all fibres on a single plate takes about
40 minutes and can be done as the other plate is being observed,
thereby greatly reducing overheads. A minimum separation of 30 to 40
arcsec between fibres is imposed by the physical size of the
rectangular fibre buttons. This constraint and other hardware limits
are respected by the custom fibre configuration software.
The fibres feed AAOmega, which is a bench mounted double beam
spectrograph sitting in one of the Coude rooms of the AAT. The light from
the fibres is first collimated with a mirror before passing through a
dichroic which splits the light at 570~nm into two arms, one red and
one blue. In the blue arm, we used the 580V grating (dispersion of 1~A per pixel). In the red arm,
we used the 385R grating (dispersion of 1.6~A per pixel). The resulting wavelength coverage starts at
370~nm and ends at 880~nm, with a resolution of R$\sim$1400.
Up until the beginning of 2014, the
detectors were two 2k$\times$4k E2V CCDs. These detectors
were replaced with new, cosmetically superior and more efficient 2k$\times$4k E2V CCDs during 2014
\citep{Brough2014}.
The full SV and Y1 observing log is shown in
Table~\ref{tab:observing_log}. The standard observing sequence for a
field configuration is 3 consecutive 40 minute exposures. In practice,
the sequence is constrained by observing conditions and field
observability. Among the ten DES SN fields, the two deep fields (C3
and X3) take priority in the scheduling and have accumulated the
longest integration time, about 16 hours compared to an average of 7
hours for the shallow ones.
For each configuration, we took a single arc and up to two fibre
flats. The arc is used for wavelength calibration, while the flats are
used to define the location of the fibres on the detector (the
so-called tram line map) and to determine the relative chromatic
throughput of the fibres.
At the end of each run the nightly bias and dark frames are combined
to produce a master bias frame and a master dark frame.
During our first observing season,
the blue CCD contained several notable defects. The
master bias and master dark were used to mitigate the impact of these
defects on the science exposures. By comparison, the red CCD was
cosmetically superior, so the corresponding
master bias and master dark for the red CCD were not needed (indeed,
applying the correction just added noise).
\begin{table*}
\centering
\begin{minipage}{\textwidth}
\caption{OzDES first year observing log for DES SN fields.}
\label{tab:observing_log}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}ccccccccccccc}
\hline
\multirow{2}{*}{UT Date} & \multirow{2}{*}{\parbox{1.5cm}{\centering Observing Run}} & \multicolumn{10}{c}{Total exposure time for DES field (minutes)} & \multirow{2}{*}{Note} \\
\cmidrule{3-12}
& & E1 & E2 & S1 & S2 & C1 & C2 & C3(deep) & X1 & X2 & X3(deep)\\
\hline
2012-12-13 & \multirow{4}{*}{001} & -- & -- & -- & -- & 40 & 80 & -- &120& -- & -- & \\
2012-12-14 & & -- & -- & -- & -- &120& 80 & -- & -- & -- & -- \\
2012-12-15 & & -- & -- & -- & -- & -- & 40 &120& -- & -- & -- \\
2012-12-16 & & -- & -- & -- & -- & -- & -- & 80 & -- & -- &160\\
\hline
2013-01-05\textsuperscript{*} & 002 & -- & -- & -- & -- & -- & -- &120& -- & -- & 80 & \\
\hline
2013-09-29\textsuperscript{$\dagger$} & \multirow{3}{*}{003} & -- & -- & -- & -- & -- & -- & 40 & -- & -- & -- & \multirow{3}{*}{\parbox{2.8cm}{About one night lost due to bad weather.}} \\
2013-09-30 & & -- &120& -- & -- & -- &100& 80 & -- & -- &120\\
2013-10-01 & & -- & -- & -- & -- &150& -- & -- & -- & -- & -- \\
\hline
2013-10-30 & \multirow{6}{*}{004} &120 & -- & -- & -- & 80 & -- &120& -- & -- &120& \multirow{6}{*}{\parbox{3cm}{About one night lost due to hardware problem.}}\\
2013-10-31 & & -- &120& -- & -- & -- & -- & -- & -- & -- & -- \\
2013-11-01 & &120 & -- &120& -- & -- & 80 & -- &120& -- & -- \\
2013-11-02 & & -- & -- & -- &120& 80 & -- & -- & -- & 40 & -- \\
2013-11-03 & & -- &120& -- & -- & -- & -- &120& -- & 80 &120\\
2013-11-04 & &120 & -- &120& -- & -- & 80 & -- &120& -- & -- \\
\hline
2013-11-28 & \multirow{4}{*}{005} & -- & -- & -- & -- & -- & -- & 14 & -- & -- &120& \multirow{4}{*}{\parbox{3cm}{About one night lost due to bad weather.}} \\
2013-11-29 & & -- & -- & -- & -- & -- & -- &120& -- & -- &120\\
2013-11-30 & &120 & -- & -- & -- & 40 &120& -- & -- &120& -- \\
2013-12-01 & & -- &120& 40 & -- &107& -- & -- &120& -- & -- \\
\hline
2013-12-25\textsuperscript{*} & \multirow{2}{*}{006} & -- & -- & -- & -- & -- & -- & -- & -- & -- & -- & \multirow{2}{*}{\parbox{3cm}{About one night lost due to bad weather.}} \\
2013-12-26\textsuperscript{*} & & -- & -- & -- & -- & 40 & -- &120& -- & -- &120\\
\hline
{\bf 2012--2013} & {\bf Total (mins)} & 480 & 480 & 280 &120& 657 & 580 & 934 & 480 & 240 & 960 \\
\hline
\\
\multicolumn{13}{l}{\textsuperscript{*}\footnotesize{NOAO time allocation.}}\\
\multicolumn{13}{l}{\textsuperscript{$\dagger$}\footnotesize{C3 field was observed at the end of the night during time allocated to the XMM-XXL collaboration.}} \\
\end{tabular*}
\end{minipage}
\end{table*}
\subsection{Target Selection}
There are three stages to target selection:
\begin{description}
\item[i)] creation of the input catalogues,
\item[ii)] target prioritisation, and
\item[iii)] target allocation.
\end{description}
For each field, the input catalogues contain a large number of potential targets (step
i), from which we select a prioritised set of 800 potential targets
(step ii), which the 2dF configuration software uses to optimally
allocate its fibres for each observation (step iii).
In the first stage, science targets are provided by the science
working groups within DES and OzDES. The working groups are
responsible for updating their input catalogs between observing runs,
e.g. removing objects that have reliable redshifts from earlier runs.
The number of targets greatly exceeds the number of fibres, so not all
targets can be observed. Based on the scientific importance of the
targets, we assign a priority (larger number for higher priority) and
a quota to each type of object, as defined in
Table~\ref{tab:target_type}.
From the initial input catalog, the prioritised target list is
selected based on priority and quota, starting with the highest
priority and ending at priority 4.
Targets of a particular type are randomly selected up to its quota.
If an object cannot be selected
because the quota has been reached,
then the object goes into one of two pools. Objects with priorities
six and above go into a high priority backup pool (priority 3). All
other objects, including objects for photo-$z$ calibration go into a
low priority pool (priority 2). If at the end of this initial
allocation the number of objects is less than 800, then objects from
the high priority backup pool are randomly selected until 800 objects have been
chosen. If the total number of objects is still less than 800, then
objects from the low priority backup pool are selected until 800
objects have been chosen.
In the third stage, targets are allocated to fibres using the custom
2dF fibre configuration software. This software avoids fibre
collisions (see $\S$~\ref{sec:setup}) while optimising the target
distribution so the highest-priority targets are preferentially
observed. A maximum of 392 fibres are allocated in this step including
25 for sky positions. Another 8 fibres are placed on bright sources
for guiding.
The input list size of 800 balances efficiency and performance
of the fibre configuration software.
A sufficient number of objects is required for a good spacial distribution but
the algorithm slows down significantly for a larger number of targets than 800.
If a field is observed a second time during an observing run, the
above three-step allocation process is repeated. Usually the only
change to the input catalogues is the inclusion of just-discovered
transients. During target prioritisation, we deselect targets that
have been given secure redshifts from the previous nights, freeing
those fibres for new targets. The priority of objects that have been
observed in the current run, but do not satisfy the deselection
criteria (defined in Table~\ref{tab:target_type}) are boosted by an
amount that depends on their initial priority.
\subsection{Target Definition}\label{sec:targets}
Here, we define the target types and their related targeting strategy used during SV and Y1, ordered by
object priority from highest to lowest. If more than one priority is available for a type, its location in the list is determined by the higher priority.
In general, we do not expect the
list to evolve significantly, especially for targets that have high
priority. However, it is likely that we will modify the quotas as the
survey progresses so as to maximise the results of the survey.
The number of each type of target we observed by end of Y1 is
given in Table~\ref{tab:target_type}.
Examples of spectra for the main types of targets are shown in Fig.~\ref{fig:sample_specs}.
\begin{table*}
\centering
\begin{minipage}{\textwidth}
\caption{OzDES Target Types and Priorities. Highest priorities are
allocated first, up to their quota.
Objects are removed from the target
pool based on the deselection criteria. In most cases this is when
the target has a successful redshift measurement, but for transients
(which include supernovae) it is after they have faded, and some are
never deselected as they require constant monitoring (AGN and F
stars). The final column shows both the number with successful redshifts, and the redshift success rate as a percentage of the number observed.
}
\label{tab:target_type}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}lcccccc}
\hline
Type & Priority& Quota & Deselection & Number & Average Exposure & Number \\
& (1-9) & & criterion & Observed & (minutes) & With Redshift \\
\hline
Transient & 8& unlimited & faded & 327 & 192& 238 (73\%)\\
AGN reverberation & 7& 150 & redshift or never \textsuperscript{*} & 2103 & 283& 1772 (84\%)\\
SN host & 6& 200 & redshift & 986 & 194& 528 (54\%)\\
WhiteDwarf & 6& 3 & classification & 17 & 139& ... \\
StrongLens & 6& 3 & redshift & 15 & 181& 3 (20\%)\\
ClusterGalaxy I & 6& 10 & redshift & 439 & 165& 232 (53\%)\\
RadioGalaxy I & 6& 25 & redshift & 350 & 259& 161 (46\%)\\
FStar & 5& 15 & never\textsuperscript{$\dagger$} & 48 & 88& ... \\
Sky Fibres & 5& 25 & never & .. & ... & ... \\
ClusterGalaxy II & 4& 50 & redshift & 133 & 249& 89 (67\%)\\
LRG & 4& 50 & redshift & 1208 & 157& 728 (60\%)\\
ELG & 4& 50 & redshift & 2382 & 156& 1326 (56\%)\\
Photo-$z$ & 2& unlimited& redshift & 2192 & 123& 1621 (74\%)\\
\hline
\\
\multicolumn{7}{l}{\textsuperscript{*}\footnotesize{Never deselected if a target is picked to be monitored.}} \\
\multicolumn{7}{l}{\textsuperscript{$\dagger$}\footnotesize{Never deselected if confirmed as an appropriate calibration source.}}\\
\end{tabular*}
\end{minipage}
\end{table*}
\begin{figure*}
\begin{minipage}{\textwidth}
\includegraphics[width=180mm]{sample_specs.eps}
\caption{Sample spectra for targets of typical magnitude and redshift, with flux on a linear scale.
Spectra are truncated at the blue end and smoothed by a gaussian filter of sigma 1 pixel.
Key features used to measure the redshifts are marked.
The spectra are not flux calibrated and the relative variance is plotted in gray. The discontinuity at 570~nm in the variance marks the dichotic split. }
\label{fig:sample_specs}
\end{minipage}
\end{figure*}
\begin{enumerate}
\item \textit{Transient}: transient of any type, including
supernova. Currently, an approximate $r$-band magnitude limit of
22.5, corresponding to the peak r-band magnitude of a Type Ia
supernovae at $z\sim0.5$, is imposed on the brightness of
the transient. This magnitude is the limit at which we can
identify a hostless Type Ia supernova with AAOmega. Because of the
time-sensitive nature of transients, these objects are of the
highest priority. Regardless of whether a redshift or a classification is obtained, a transient remains on the target list until it becomes fainter than the magnitude limit.
There are only 5 to 10 active transients at any
one time in each DES field, so all of these will be assigned a fibre
unless a collision with another transient occurs.
\item \textit{AGN Reverberation}: AGN candidate or previously-identified AGN that we began to monitor in Y1.
The AGN candidates were selected based on photometry from DES, VHS \citep{2013Msngr.154...35M},
and WISE \citep{2010AJ....140.1868W} with a variety of methods \citep{2015MNRAS.446.2523B}.
The selection methods were designed to emphasize completeness over efficiency,
particularly for the brightest candidates, in order to maximize the number of bright AGN.
Besides brightness, we use redshift, emission-line equivalent width, and luminosity to identify targets for monitoring.
For each run in Y1, approximately 150 fibers per field were devoted to AGN targets (Table 2).
As Y1 progressed, a growing fraction of these fibers were devoted to the monitoring program.
\item \textit{SN Host}: host galaxy of a supernova or other
transient detected since the DES SV season. A host galaxy is
defined as the closest galaxy in units of its effective light radius
from the SN \citep[cf.][]{2014arXiv1401.3317S,
2006ApJ...648..868S}. An allocated fibre is placed at the core of the galaxy to maximize the flux input (but see $\S$~\ref{sec:hostless} for a complementary strategy employed in Y2).
Around 20 new hosts are
identified for each DES field for each OzDES observing run. The
total number of targets accumulates as fainter objects remain in the
list until it becomes clear that we are unlikely to get a
redshift. During the first year, no target was dropped. We are
currently using these data to assess when targets should be dropped
in favour of new targets.
\item \textit{White Dwarf}: white dwarf candidate. The list is updated between observing runs based on analysis of the observed spectra,
when successfully classified targets are dropped.
\item \textit{Strong Lens}: strong lens system identified in the DES imaging. Up to five lens systems per field were targeted in Y1.
\item \textit{Cluster Galaxy}: cluster galaxy selected via the
redMaPPer cluster finder on the SVA1 Gold catalog \citep[a galaxy catalog from co-added SV imaging;][Rykoff et
al., in prep]{2014ApJ...785..104R}. High probability cluster galaxies are
selected to be luminous galaxies in moderately rich clusters that
have a luminosity consistent with the cluster richness, as well as
occupying regions with a high local density. An additional
constraint of $m_r<22.5$ mag is put on the galaxies for AAOmega
targeting. Central cluster galaxies with $0.6 < z_\mathrm{photo} <0.9$ are given a
higher priority (ClusterGalaxy I in Table~\ref{tab:target_type}),
with the median target redshift of $0.80$. Another lower
priority list of targets (ClusterGalaxy II) is made for centrals and
bright satellites of clusters at $z<0.6$.
\item \textit{Radio Galaxy}: galaxy selected from the ATLAS survey.
Although these are labelled ``Radio Galaxy'' for convenience, about half the objects are star forming galaxies or radio-quiet AGN.
As for cluster galaxies, we allow two priority settings, for RadioGalaxy I and
RadioGalaxy II. During Y1, category II was rarely used (only two objects), so it
is not listed in Table~\ref{tab:target_type}.
\item \textit{F star}: F star candidate in Y1 (from Nov. 2013), to
obtain spectroscopic classification. The goal is to have
approximately 10 to 15 high signal to noise F stars per field to be
used as flux calibrators. Candidates were selected in the Northern
fields using photometry from SDSS that included u-band.
\item \textit{Sky}: sky regions with no detectable sources in DES
imaging. Approximately 25 fibres are allocated per field.
\item \textit{LRG}: luminous red galaxy used to calibrate DES
photometric redshifts. The high-$z$ (median$\sim$0.7) population is
selected based on DES+VHS photometric data
\citep{2015MNRAS.446.2523B}.
\item \textit{ELG}: emission line galaxy used to calibrate DES
photometric redshifts. Color selections similar to \citet{2013MNRAS.428.1498C} are used.
For bright targets ($19 < m_i < 21.3$), $-0.2 < (m_g - m_r) < 1.1$ and $-0.8 < (m_r - m_i) < 1.4$;
for faint targets ($21.3 < m_i < 22$), $-0.4 < (m_g - m_r) < 0.4$, $-0.2 < (m_r - m_i) < 1.2$ and $m_g - m_r < m_r - m_i$.
\item \textit{photo-$z$}: galaxy target, selected using the DES i-band
magnitude cut $17 \leq m_i < 21$, used to calibrate DES photometric redshifts.
\end{enumerate}
\subsection{Data Reduction}\label{sec:reduction}
We process the data from AAOmega soon after they have been taken so
that we are able to quickly determine redshifts, usually within 48
hours of them being observed and often before the next night's
observations. This gives us the chance to deselect targets if a
secure redshift (see below) has been obtained and therefore free up
fibres to observe other targets later in the run.
All data from AAOmega are processed with \texttt{2dfdr} \citep{Croom2004}. The procedure is
broken into a number of steps, each of which is discussed below.
\begin{itemize}
\item {\it Bias Subtraction and Bad Pixel Masking}. For data taken
with the blue CCD, this consists of using the overscan region to
subtract the bias, subtracting a master bias to remove features that
cannot be removed by using a fit to the overscan region, and
subtracting a scaled version of the master dark. For the red CCD,
this simply consists of using the overscan region to remove the
bias. Cosmic rays are detected with an edge detection filter. The affected pixels
are masked as bad throughout the subsequent analysis and are not
used. Some cosmic rays sneak through this step, but are captured
later when multiple exposures are combined.
\item {\it Tram line mapping and wavelength calibration}. The fibre
flat is used to measure the location of the fibre traces (the tram
line map), while the arc is used to wavelength calibrate the path
along each fibre. In future versions of \texttt{2dfdr}, the fibre flat will
also be used to model the fibre profile.
\item {\it Extraction}. A 2d spline model of the background scattered
light is then subtracted, and the flux from individual fibres is
extracted. The flux perpendicular\footnote{In practice we weight
along the direction that is parallel to the detector columns,
which is almost perpendicular to the fibre traces.} to the fibre
trace is weighted with a Gaussian that has a FWHM that matches the
FWHM of the fibre trace. In future versions of \texttt{2dfdr}, both of these
steps will be replaced with a single step that optimally extracts the
flux and determines the background \citep{2010PASA...27...91S}.
\item {\it Wavelength calibration}. The extracted spectra are
calibrated in units of constant wavelength using the arcs.
\item {\it Sky subtraction}. The relative throughput of the fibres is
normalised, and the sky is removed using the extracted spectrum of
the sky fibres. Usually, there are residuals that remain after this
step. These are removed using a principal component analysis of the
residuals \citep{2010MNRAS.408.2495S}.
\item {\it Combining and Splicing}. If more than one exposure was
obtained, which is usually the case, the reduced spectra are
co-added. Remaining cosmic rays are found as outliers and removed at
this stage. The red and blue halves of the spectra are then
spliced. At this stage, we do not weight the data before we combine it. In future
versions of the OzDES pipeline, we will weight the spectra and both
the splicing and combining will be merged into a single step.
\end{itemize}
\subsection{Redshifting}\label{sec:redshifting}
All spectra are visually inspected by OzDES team members\footnote{A
person who determines redshifts using \texttt{runz} is colloquially referred
to as a redshifter.} using the interactive redshifting software
\texttt{runz}, originally developed by Will Sutherland for the
2dFGRS. \texttt{runz} first attempts an automatic redshift
determination by both cross-correlating to a range of galaxy and
stellar templates and searching for emission line matches. It then
displays the spectrum and marks the locations of the common emission
and absorption features at the best redshift estimate. A redshifter
visually inspects the fit and determines whether the redshift is
correct. A number of interactive tools are provided, e.g. to switch to
a different template, to mark a specific emission line, to add a
comment, or to input the redshift directly.
Uncertainties on the redshifts are calculated by \texttt{runz} based
on the width of the cross-correlation peak or the fit to emission
lines. However, we are able to derive more representative
uncertainties for various classes of objects (e.g.~galaxies and AGN, see~\ref{sec:precision})
using objects that are observed multiple times.
Each redshift is assigned a quality flag. For most objects, we use a
number between 1 and 4, with a larger number meaning a more
secure redshift estimate. We reserve quality flag 6 for stars.
\begin{itemize}
\item Quality 4 is given when there are multiple strong spectral line
matches. With the exception of AGN and transients, these are removed
from the target pool and are not re-observed.
\item Quality 3 is given for multiple weak spectral line matches or single
strong spectral line match (e.g. a bright emission line that is consistent with high-redshift [OII]). These
can be used for science, but for some target types (e.g. SN hosts), may also be
re-observed until the quality is deemed worthy of a 4.
\item Quality 2 is given to targets where there are one or two very weak
features (e.g. a single weak emission line that may be [OII]). The redshift is speculative and
not reliable enough for science.
\item Quality 1 is given to objects where no features can be identified.
\end{itemize}
Targets with quality 1 and quality 2 redshifts are re-observed
until the deselection criteria are met.
In practice, the assignment of quality flags by a redshifter is
subjective to experience and many other factors. We require
independent assessments from two members of the team for each
object. The results from the two redshifters are then compared by a third
``expert'', who chooses the appropriate redshifts and quality flags and provides feedback to the individual
redshifters.
This helps to train redshifters and to homogenise the
redshifting process. The number of cases where there is true
disagreement is small, most disagreements arise from a difference in
quality rating.
At the beginning of Y1, we set the following requirements
for the reliability of the redshift:
\begin{itemize}
\item[] 6: more than 99\% correct (reserved for stars)
\item[] 4: more than 99\% correct (reserved for galaxies)
\item[] 3: more than 95\% correct (for any object)
\end{itemize}
Redshifts with quality of 3 and above are considered
trustworthy for science analysis, so it is important to know the
actual rate of redshift blunders.
By comparing objects that were observed multiple times and redshifted
independently, we find that we are achieving the required level of
reliability for quality flags 4 and 6, but are tracking below 95\% for
quality 3. These results are presented in $\S$~\ref{sec:reliability}.
\subsection{Classification of Active Transients}\label{sec:transient_classification}
Timely identification is critical for transient studies. Discovery of
a supernova at an early phase may trigger observations at other
wavelengths or spectroscopic time series throughout its evolution. The
earlier the follow-up campaign starts, the more information it will
gather to understand the explosion physics. In the exciting case of an
unknown type of transient, early observations may be vital for the
interpretation of its true nature and the only chance to observe it if it is short-lived.
Spectral classification of a SN is usually done by comparison to
templates. The best match type and age are found through either
cross-correlation \citep[e.g. the Supernova Identification,
\texttt{SNID},][]{2007ApJ...666.1024B} or chi-square minimization
\citep[e.g. \texttt{Superfit},][]{2005ApJ...634.1190H}.
During the first season of OzDES, we observed 320 transients and
determined the redshifts (mostly from features of the host galaxies) of
about 73\% of them. Of these, 12 were positively identified as supernovae
\citep{2013ATel.5568....1C,2013ATel.5642....1Y,2014ATel.5757....1Y}. This
is a small fraction of all transients. There are a number or reasons
for this.
First, one of the aims of OzDES is to explore the range of transient
phenomena that exists in the DES deep fields, so a deliberate
decision was made to target objects that were clearly not SN. This
includes objects that turned out to be AGN or variable stars.
Second, the amount of host galaxy light relative to SN light that
enters the $2\hbox{$^{\prime\prime}$}$ 2dF fibre is larger than is normally the case for long slit
observations, which typically use slits that are $1\hbox{$^{\prime\prime}$}$
wide. This, coupled with the typical seeing at the AAT ($2\hbox{$^{\prime\prime}$}$),
makes the SN less clearly visible in spectra from 2dF.
Third, SN typing relies on identifying broad spectral features. At
the beginning of Y1, the pipeline that was used to process the data
imprinted features to the spectra (e.g. a discontinuity between the
red and blue halves) that obfuscated the SN signal. Only the brightest
SN in Y1 could be identified with confidence.
\subsection{Ongoing improvements}\label{sec:y2_upgrades}
While the Y1 spectra are suitable for determining
redshifts, artefacts that come from the processing result in data that
are less suitable for other analyses, such as the spectral typing of
transients. The most common artefact is a flux discontinuity in the
overlap region between the red and blue halves of the spectra.
Since the first year of the OzDES campaign, considerable work has
gone into mitigating these artefacts by improving the algorithms in
\texttt{2dfdr}. These improvements include better tramline tracking,
implementation of optimal extraction (which leads to more accurate
treatment of the background scattered light), better flat fielding,
more effective cosmic ray removal and improved flux
calibration. During Y1, we have used the sensitivity functions provided with \texttt{2dfdr} to do the flux calibration. In future versions of the OzDES pipeline, we will
use the F stars that are observed contemporaneously with the other targets to do the flux calibration. Absolute calibration will be done using the
broad band DES photometry.
Not all of the above algorithms are implemented at once, but better data reduction has already helped us to classify 50\% more SNe in Y2 than in Y1.
Reprocessing of the entire OzDES data set from Y1 and Y2 is planed, after which redshifts and spectra will be publicly released (Childress et al. in prep).
\section{Results}\label{sec:results}
During Y1 and SV, 10482 unique targets were observed and 6727
redshifts (with quality flag 3 and above) were obtained. Figure
\ref{fig:hist_rz} shows the redshift distributions for selected
targets. The fraction of objects with measured redshifts and their
number are listed in Table~\ref{tab:target_type} for each type of
target. For selected groups of extragalactic objects, we also show the
completeness of redshifts as a function of integrated magnitude within
the fibre diameter (Figure~\ref{fig:hist_comp}). For non-transient
objects, photometry measurements are taken from the DES SV Gold
catalog (Rykoff et al., in prep). For transients, the magnitude does not include the
flux of the underlying host and represents luminosity measured in the last epoch when the target was selected.
\begin{figure*}
\begin{minipage}{\textwidth}
\includegraphics[ width=170mm]{hist_rz.eps}
\caption{Redshift distributions for selected target types. Objects that have redshifts consistent with Galactic origins are excluded.
Dark shaded histograms represent objects with redshift quality flag of 4 (most reliable; see $\S$~\ref{sec:redshifting}) and the light shaded histograms represent objects with redshift quality flag of 3 (reliable).
Other galaxies include radio galaxies, cluster galaxies and galaxies that are specifically chosen to calibrate photometric redshifts.
Redshift bins are set to be smaller for ELGs and LRGs to provide higher resolutions around the peak.}
\label{fig:hist_rz}
\end{minipage}
\end{figure*}
\begin{figure*}
\begin{minipage}{\textwidth}
\includegraphics[ width=170mm]{hist_comp.eps}
\caption{Redshift completeness as functions of r-band magnitude measured in a
$2\hbox{$^{\prime\prime}$}$ diameter aperture, for selected groups of extragalactic
objects. Unfilled histograms are for all targets.
Dark shaded histograms represent objects with redshift quality of 4 and above (most reliable; see $\S$~\ref{sec:redshifting}).
Light shaded histograms represent objects with redshift quality of 3 (reliable).
Completeness (as dashed curves) is defined as the fraction of objects for
which redshifts are measured with quality flag 3 and above.
The magnitude range is fixed for all panels for easy comparison.
A few galaxies in the bottom two panels have measured magnitudes fainter than 25. These bins have low completeness and are excluded to emphasize the more typical magnitude range.}
\label{fig:hist_comp}
\end{minipage}
\end{figure*}
\subsection{Efficiency and Completeness}\label{sec:efficiency}
As expected, the probability of a successful redshift measurement
drops for fainter targets. This trend is weak for the transients
because the redshifts of most transients are
inferred from spectral features of their hosts.
Strength of such features depends on the host luminosity, redshift and location of the transient relative to its host.
The redshift efficiency is the likelihood of measuring a redshift
above a certain quality level in a given observation time. Difference
in completeness during comparable integration times reflects difference
in efficiency for different types of targets. In general, emission
lines allow measurements of redshift for sources with faint continuum,
while cross-correlation for a galaxy without identifiable emission
features relies on well measured spectral shape, combined with
absorption features that are usually relatively weak, and therefore
needs brighter continuum. Hence, for ELGs, the efficiency is less
sensitive to the integrated broad-band magnitude than for the LRGs.
In practice, the ELGs are selected to be at $z\sim1$, so only the [OII] $\lambda\lambda$ 3726,3728 doublet is within our wavelength
range. This double line appears blended at the resolution of our
spectra, and although it can often be recognized as a wide or
flat-topped line, it is a single feature that is prone to
misidentification. It is hard for redshifters to say conclusively
that it is [OII] and not another line or an artefact. A high fraction
of the ELGs are therefore rated at a redshift quality of 3.
The nominal target type is known to the redshifter. For ELGs, LRGs and AGN,
this may cause the redshifter to assign a higher redshift quality flag
than would be the case if the target type was not available to the
redshifter.
SN hosts show
lower completeness across a wide magnitude range than either ELGs or LRGs
(see Figure~\ref{fig:hist_comp}). This sample is selected based on criteria
independent of whether the galaxy is of early or late type. The
redshift range may be constrained by the selection criteria, but
knowing the target type does not directly help in determining a
redshift.
At a given magnitude, the redshift completeness grows with integration
time. Many of the faint targets are expected to be observed in
multiple seasons and accumulate significantly more signal than
obtained so far.
The data from Y1 do not have sufficient range of exposure time to determine this trend with much certainty.
In O'Neill et al. (in prep), we use data gathered by the end of Y2 to model how the number of redshifts acquired
increase as a function of number of visits to the field, for various types of targets with different magnitude and redshift distributions.
We then estimate the redshift completeness of the survey
and the number of objects we are likely to observe by extrapolating to a total of 25 visits.
We are on track to obtain
redshifts for 80\% of all SN host galaxies that are targeted.
\subsection{Redshift Precision}\label{sec:precision}
Estimates of the redshift precision based on emission line fitting or
template cross-correlation may suffer different biases. For almost all
of our target groups, a combination of techniques has been used. It
is recognized that for most of our science cases, we exceed the
minimum requirement of precision. Therefore, we do not attempt to
assign accurate uncertainties for individual redshift measurements,
but instead provide an overall statistical error estimate for classes
of targets, based on the dispersion in measured redshifts for
individual objects with multiple independent measurements, either from
different observing runs or from appearing in the overlap regions of
the DES fields. Objects with inconsistent redshift measurements are excluded (see next section).
For each object, we calculate the pair-wise
differences of these redshifts, divided by $1+z_{\rm median}$. The distributions
are examined for several populations of extra-galactic objects. We
quote the 68 percent ($\sigma_{68}$) and 95 percent ($\sigma_{95}$) inclusion regions in Table
\ref{tab:error}, since the distributions have extended tails. Typical uncertainties ($\sigma_{68}$) for AGN and galaxies are 0.0015 and 0.0004 respectively.
We also compare our results against the redshifts from other surveys covering our targeted sky area.
Only redshifts with the most confident quality flag defined by the corresponding survey are considered.
The results are listed in Table \ref{tab:error} as
``external''. No significant systematic offset is found between OzDES and surveys such as GAMA \citep{2011MNRAS.413..971D} or SDSS \citep{2014ApJS..211...17A}.
Distributions of the differences between our redshifts and those
from other surveys are consistent with the internal
comparisons.
A large number of AGN were targeted repeatedly from run to run by
design. This allows us to split the sample and quote the uncertainties
in two redshift bins. The redshifts of AGN are often measured by
cross-correlating with template spectra. Due to intrinsic variation of
the emission profiles from AGN to AGN, the precision of a measurement
depends on the quality of the template, the wavelength region
included in the cross-correlation and/or the line chosen by the redshifter
to centre on. At redshift one and above, high ionization emission
lines with large profile variation, such as CIII] or CIV are often
used, resulting in larger uncertainties in the redshifts.
The number of repeated observations is small for all other galaxy
targets as they are usually removed when secure redshifts are
obtained. To investigate possible type dependence, we examine
separately the dispersions for ELG, LRG and SN host targets. The
smaller dispersion for the ELGs is consistent with the expectation
that better constraints can be obtained from narrow emission lines.
\subsection{Redshift Reliability}\label{sec:reliability}
For a quantitative evaluation of the reliability of our redshifts, we
compare multiple redshifts obtained for the same objects from
independent observations, either from
different observing runs or from appearing in the overlap regions of
the DES fields. For each redshift, we calculate its offset from a redshift
that is deemed to be correct. For simplicity, we call this value the
base redshift. We elaborate on how the base value is determined below.
A redshift is considered wrong if it differs from the base redshift by more than
$0.02 (1+z)$ for a AGN or $0.005 (1+z)$ for a galaxy, which are roughly
ten times the standard deviations measured from the previous section.
The base redshift can be obtained from either internal or external
sources. Internally, if a redshift with a higher quality flag exists,
we consider that redshift as the base redshift. Otherwise, the median
value is used if more than two redshifts are available or a random
selection is made assuming at least one measurement is correct.
The external redshifts are chosen from other surveys to have the highest quality flag defined.
When a conflict between an OzDES redshift and an external redshift is found,
the OzDES spectra are re-examined. In three cases (for quality 3 AGN),
the data quality is not good enough to confirm either redshift. These are excluded in the calculations.
As shown in Table \ref{tab:error}, overall close to 100\% of redshifts
are correct with quality flag 4 but only about 90\% of redshifts are
correct with quality flag 3.
To better understand the sources of error, we examine different galaxy
types separately.
The relatively high error rate for SN hosts (more than 15\% for quality 3)
possibly arises because of the diversity of objects that host
supernovae. The ELG sample appears more homogeneous.
As noted earlier, the error rate for quality 3 objects is higher than
our goal of 5\%. After Y1, we implemented a
number of changes (better training of the human redshifters and more
scrutiny of quality 3 objects by the third person) that have
resulted in fewer errors.
With increasing experience and
better data processing, further
improvements on the reliability and the quality of the
redshifts are expected in the coming seasons and will be closely
monitored.
\begin{table*}
\centering
\begin{minipage}{\textwidth}
\caption{Redshift uncertainties and error rates. Uncertainties are calculated using weighted pair-wise redshift differences, $\Delta z/(1+z)$, for objects observed in multiple overlapping fields or multiple observing runs ($\S$~\ref{sec:precision}). A redshift is considered wrong if it differs from a chosen base redshift ($\S$~\ref{sec:reliability}) by more than
$0.02 (1+z)$ for a AGN or $0.005 (1+z)$ for a galaxy.}
\label{tab:error}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}cccccc}
\hline
Type & Number of redshift pairs & $\sigma_{68}$\textsuperscript{*} & $\sigma_{95}$\textsuperscript{*} & Error rate (Q = 4) & Error rate (Q = 3) \\
\hline
AGN & - & - & - & 1/1647 (0.1\%) & 22/388 (5.7\%) \\
AGN ($z\leq1$) & 521 & 0.0004 & 0.0011 & - & - \\
AGN ($z>1$) & 1568 & 0.0015 & 0.0038 & - & -\\
AGN (external) & 424 & 0.0016 & 0.0048 & 0/387 (0.0\%) \textsuperscript{$\ddagger$}
& 6/46(13.0\%)\textsuperscript{$\S$} \\
Galaxy\textsuperscript{$\dagger$}
& 99 & 0.0004 & 0.0013 & 0/74 (0.0\%) & 10/95 (10.5\%) \\
ELG & 25 & 0.0002 & 0.0006 & 0/8 (0.0\%) & 2/32 (6.3\%) \\
LRG & 21 & 0.0005 & 0.0013 & 0/16 (0.0 \%) & 3/20 (15.0\%) \\
SN host & 36 & 0.0003 & 0.0013 & 0/38 (0.0\%) & 4/25 (16.0\%) \\
Galaxy (external)
& 182 & 0.0004 & 0.0010 & 0/159 (0.0\%)\textsuperscript{$\parallel$} & 0/24 (0.0\%) \\
\hline
\\
\multicolumn{6}{l}{\textsuperscript{*}\footnotesize{68 percent ($\sigma_{68}$) and 95 percent ($\sigma_{95}$) inclusion regions.}} \\
\multicolumn{6}{l}{\textsuperscript{$\dagger$}\footnotesize{All types of galaxy targets that are not AGN.}} \\
\multicolumn{6}{l}{\textsuperscript{$\ddagger$}\footnotesize{In three cases of discrepancy, spectra are inspected and the OzDES measurements are confirmed to be correct.}}\\
\multicolumn{6}{l}{\textsuperscript{$\S$}\footnotesize{All discrepant redshifts are checked by hand. Objects are excluded if the available data is not good enough to confirm either}} \\
\multicolumn{6}{l}{\footnotesize{redshift measurement.}} \\
\multicolumn{6}{l}{\textsuperscript{$\parallel$}\footnotesize{In one case of discrepancy, spectra are inspected and the OzDES measurement is confirmed to be correct.}} \\
\end{tabular*}
\end{minipage}
\end{table*}
\section{Discussion and highlights}\label{sec:discussion}
In this section, we present updates on some of our key science goals in light of the results from Y1. When necessary, complementary data taken in Y2 are included to demonstrate the potential of our observing strategy. Some of the discussions involve changes in Y2 that are inspired by analysis of the Y1 data.
\subsection{SN Ia Cosmology}\label{sec:cos_disc}
The DES SN survey strategy is optimized to detect a large number of SNe Ia with a redshift distribution that extends to redshift 1.2 and peaks around redshift 0.6 \citep[cf. Figure~10 of][]{2012ApJ...753..152B}. Since host galaxy redshifts from OzDES will be the main source of redshifts for Hubble Diagram analyses, the efficiency of obtaining redshifts by OzDES will affect the actual redshift distribution in addition to the discovery efficiency of DES.
In practice, SNe Ia candidates are first selected based on DES light-curves and photo-$z$ of putative host galaxies. The host galaxies of these candidates are then targeted by OzDES. Any redshifts obtained are subsequently used to refine the selection. Only those SNe Ia with a secure host spectroscopic redshift are considered further for cosmology studies. The combined efficiency of this process can be evaluated by comparing the selected SNe Ia sample to a simulated population.
From the DES Y1 data, we select SNe that satisfy light curve quality cuts defined in Table~6 of \citet{2012ApJ...753..152B}, with the exception that only two filters are required to have maximum signal-to-noise above 5 (i.e. 2 filters with SNRMAX $>$ 5). The looser cuts are compensated for by additional selection criteria on peak color and \texttt{SALT2} \citep{2007A&A...466...11G} model fit parameters ($x1$ and color) to achieve $>$ 96\% purity \citep[tuned for simulation, cf.][]{2013APh....42...52G}.
We show in Figure~\ref{fig:comparison_Ia} the redshift distribution of the selected SNe Ia for which host spectroscopic redshifts are available by end of Y2.
The median redshift of this observed sample is 0.52, compared to a median redshift of 0.63 for SNe Ia selected with the same photometric criteria from a \texttt{SNANA} \citep{2009PASP..121.1028K} simulation with realistic survey characteristics (e.g. observing cadence, seeing conditions and photometry zero points) from Y1 as input.
The lower median redshift for the observed sample is mainly due to the difficulty of measuring redshifts in more distant and fainter galaxies.
In the two deep fields (C3 and X3), half of the SNe Ia host galaxies are fainter than 24th magnitude in r-band and the overall redshift completeness is only about half of that for the shallow fields.
With longer exposure times, the redshift completeness will improve at all redshifts. Only for the most distant SNe, spectroscopic redshifts may remain scarce for host galaxies at typical brightness. This bias against faint host galaxies needs to be considered as it has been shown that stretch and color-corrected SN Ia peak magnitude depend on the host galaxy stellar mass \citep[e.g.][]{2010ApJ...715..743K, 2010MNRAS.406..782S, 2010ApJ...722..566L, 2013ApJ...770..108C}.
\begin{figure}
\includegraphics[width=84mm]{sne_Ia_obs.eps}
\caption{Normalized redshift distribution of selected SNe Ia candidates from Y1 that have host spectroscopic redshifts. The selection criteria are tuned to achieve $>$ 96\% purity (see text in $\S$~\ref{sec:cos_disc}). The median redshift of this sample is 0.52.}
\label{fig:comparison_Ia}
\end{figure}
\subsection{Faint SN hosts and Spectra Stacking}
As discussed above, high redshift efficiency across a wide brightness range is critical to maximize the statistics and minimize the bias of a sample of SNe with host galaxy spectroscopic redshifts. Such a goal is achieved by OzDES's unique strategy of repeat targeting, dynamic fibre allocation, and stacking. Multiple observations of the same DES field allows dynamic control of effective exposure times for targets of different brightness. The brighter galaxies are deselected between the observing runs when redshifts are measured; while fainter galaxies remain in the queue. Stacking across many observing runs allows OzDES to go deeper than otherwise possible with the 3.9m AAT. Figure~\ref{fig:faint_spectra} shows an example of stacked spectra, for a SN host target of r-band magnitude 23.7 and redshift 0.732. The growing significance of the emission line, roughly consistent with square root of exposure time, supports the credibility of the feature.
\begin{figure*}
\begin{minipage}{\textwidth}
\includegraphics[width=180mm]{faint_specs_2.eps}
\caption{Selected co-added spectra of a SN host target of r-band magnitude 23.7 and redshift 0.732. The significance of the candidate [OII] feature appears to have increased with exposure time. A quality flag of 3 is assigned so the target will remain in the queue until more data confirms the redshift.}
\label{fig:faint_spectra}
\end{minipage}
\end{figure*}
\subsection{Hostless SNe and Super-Luminous SNe}\label{sec:hostless}
We showed in $\S$~\ref{sec:efficiency} that redshift efficiency
has a different dependence on target brightness for different types of
galaxies. The likelihood of measuring an emission line redshift is
less sensitive to apparent broad-band luminosity. A significant
fraction of SNe (including most SNe Ia and most core-collapse SNe) are
expected to occur in star-forming regions. For some, strong emission
lines may show up in dispersed AAT spectra even though the continuum
is too faint to be detected by DECam imaging in the first few seasons. A new strategy of
targeting hostless SN, by placing a fibre at the position of a SN after
the transient has faded, is implemented after Y1 and has already yielded
redshifts that would otherwise be elusive.
This strategy is particularly interesting for validating Super-Luminous SN (SLSN) candidates. SLSNe are a rare and extreme class of SN discovered in recent years \citep{2012Sci...337..927G}. The origin of SLSNe are unclear, but they play a key role in understanding the evolution of massive stars, chemical enrichment and possibly cosmic re-ionization via their bright UV luminosity. DES will discover many of these intrinsically bright objects out to redshift about 2.5 \citep{2015MNRAS.449.1215P}. At high redshifts, the optical (rest-frame UV) spectra of the SLSNe are poorly understood. Redshifts from host galaxies are thus crucial to constrain their distances and intrinsic luminosities. However, SLSNe preferentially occur in dwarf star-forming galaxies \citep{2014ApJ...787..138L}, many of which are too faint to be detected or to have reliable photo-$z$ estimates from DES multi-band imaging. OzDES provides a cost-effective inspection as the number of SLSN candidates is large with respect to the time available on 8 to 10m class telescope but is small compared to the number of AAT fibres. These SLSNe targets will gain high priority in future AAT observing seasons.
\subsection{AGN}
We obtained almost 6000 spectra of AGN and AGN candidates in Y1 (see $\S$~\ref{sec:targets} for details on the target selection strategy and Figure~\ref{fig:sample_specs} for an example spectrum). After the completion of Y1, we identified 989 AGN to monitor. The number of AGN targets decreases from 150 per field to 100 per field, but the priority is maintained at 7 to ensure that the object is observed as frequently as possible.
The median redshift of these AGN is 1.63 and the distribution extends to z$\sim$4.5 (see Figure~\ref{fig:hist_rz_agn}). The sample will primarily be analyzed using either the Mg~\Rmnum{2} or C\Rmnum{4} emission line, and in some cases both lines. We will also monitor a substantial number of targets using the $H\beta$ emission line, which is the most commonly used line in previous reverberation mapping campaigns \citep[e.g.][]{2004ApJ...613..682P}. We expect to accurately recover the radius-luminosity relationship for all three lines (King et al., in prep).
Based on data obtained from Sept. 2013 through the end of 2014, we presently have 5 (6) or more spectroscopic epochs for 693 (455) AGN (70\% and 46\% of the sample). These numbers are close to the expected 7 epochs (in the first two seasons) for 500 AGN in our survey simulation (King et al., in prep). For more than 10\% of the sample, we have acquired spectra in 8 epochs.
\begin{figure}
\includegraphics[ width=84mm]{hist_rz_agn_monitor_2.eps}
\caption{Redshift distribution of the AGN being monitored after the first year. Shaded area represents objects that have been observed in 5 runs or more by the end of 2014.}
\label{fig:hist_rz_agn}
\end{figure}
\subsection{Radio Galaxies}
By the end of Y1, we had observed 350 targets from the first data release of the ATLAS survey \citep{2006AJ....132.2409N, 2008AJ....135.1276M}, augmenting the earlier work of \citet{2012MNRAS.426.3334M}. Secure redshifts were obtained for 40\% of the targeted sources and the majority currently without redshifts are fainter than $m_r = 22.5$ mag. Sources without a redshift will be re-observed in coming campaign seasons to build up the necessary integration time to determine their redshifts and classifications. As we continue to observe these targets, we expect to increase our completeness.
With the availability of the ATLAS data release 3 (Franzen et al. in prep, Banfield et al. in prep), target selection has been modified slightly after Y1. The additional post-processing provides higher detection reliability at low radio flux levels leading to increased sensitivity to low-level star formation.
The redshifts obtained will be used to further the science goals for ATLAS including:
\begin{itemize}
\item to determine the cosmic evolution of both star forming galaxies and radio AGN, through the measurement of their redshift-dependent radio luminosity functions. For example, we will be able to measure luminosity functions for star-forming galaxies to $\sim L$* (the expected knee of the luminosity function at $z$=1).
\item to calibrate and develop photometric and statistical redshift algorithms for use with the 70 million EMU \citep{2011PASA...28..215N} sources (for which spectroscopy is impractical).
\item to measure line widths and ratios of emission lines to distinguish star forming galaxies from AGN, low-ionization nuclear emission-line regions (LINERs), etc, and identify quasars and broad-line radio galaxies.
\item to measure how radio polarisation evolves with redshift, as a potential measure of cosmic magnetism.
\item to explore how morphology and luminosity of radio-loud AGN evolve with redshift, to understand the evolution of the jets and feedback mechanisms.
\end{itemize}
\subsection{Unusual Objects}
The large number of spectra taken by OzDES almost guarantees discoveries of rare events or objects. Even in the photometrically selected samples, outliers are expected to exist. Although an exhaustive search for unusual objects is beyond the scope of this work, we highlight this potential by noting some objects that clearly stand out when the spectra were visually inspected in \texttt{runz}. These objects fall in the following broad categories:
\begin{enumerate}
\item \textit{Unusual transients}: Time sensitive observations of live transients are of high priority. To maximize the discovery space, we target all transients that satisfy a straightforward magnitude cut. As discussed in $\S$~\ref{sec:transient_classification}, studies of transient spectra have been complicated by host galaxy light contamination and data reduction issues. A more systematic investigation will be carried out when data reduction is refined.
\item \textit{Rare stars}: At least four WD and M-dwarf binaries are recognized as Galactic objects with both a hot and a cool spectral components. We have also found a WD candidate that is likely a rare DQ WD (with carbon bands).
\item \textit{Serendipitous spectroscopic SN}: A SN may be observed unintentionally for two reasons. It happens to occur in a galaxy where a redshift is desired or it is mis-classified into a different target type. During the SV season, one of the photo-$z$ targets turned out to be a Type II SN. The chance for this to happen drops significantly in subsequent seasons as long-term photometry becomes available. However, the first possibility becomes more likely as more galaxies are targeted. Approximately one SN occurs every century in a Milky Way-like galaxy and a SN remains bright for a few weeks to a few months. It is thus expected that every few thousand galaxy spectra at relatively low redshift may contain a visible SN. The concept of spectroscopic SN search has been successfully tested \citep{2003ApJ...599L..33M,2013MNRAS.430.1746G}. However, such a survey method requires more than human eyes because the host galaxy light often dominates and has to be carefully modeled.
\item \textit{Broad Absorption Line (BAL) AGN}: A large number of AGN are observed as reverberation mapping targets, radio sources, galaxies or transients. More than a dozen of these exhibit extraordinary absorption line systems. Selected objects will be monitored throughout our survey.
\item \textit{Multiple redshifts}: If two objects fall in the same fibre, two distinct sets of spectral features may be observed. Searching for double redshifts in galaxy spectra provides a way of finding strong gravitational lens candidates \citep[e.g.][]{2004AJ....127.1860B}, particularly for small Einstein radii and faint background sources that are hard to detect by imaging. Four OzDES targets were noticed to show convincing features at two different extra-galactic redshifts. Inspection of the images reveal that two of these are merely chance alignment between background sources and foreground galaxies. The remaining two systems are lens candidates, both consisting of a foreground early type galaxy and a background emission line galaxy.
\end{enumerate}
\section{Conclusions}\label{sec:conclusion}
OzDES is an innovative spectroscopic program that brings together the
power of multi-fibre spectrograph and time-series observations. In
five years, each DES SN survey field will be targeted in about
25 epochs during the DES observing seasons by the 2dF/AAOmega
spectrograph on the AAT. The 400 fibres are configured nightly to
target a range of objects with the goal of measuring spectroscopic
redshifts of galaxies, monitoring spectral evolution of AGN or
classifying transients. Stacking of multi-epoch spectra allows OzDES
to measure redshifts for galaxies that are as faint as $m_r=25$ mag. Along
with efficient redshifting and recycling of fibres, we expect to
obtain about 2,500 host galaxy redshifts for the DES SN cosmology
study. The long term time series, contemporaneous with DECam imaging,
will enable reverberation mapping of the largest AGN sample to
date. In addition, OzDES is an important source of redshifts for
various DES photo-$z$ programs.
In the above sections, we have summarized the OzDES observing strategy, reported results from the first year of operation and evaluated the outcome in light of various science goals. Overall, our strategy has worked well in the first year and produced a large number of redshifts with good quality. We are on track to achieve our main science goals. Meanwhile, we have identified a number of areas for improvements, including better data reduction procedures to reduce artefacts and more rigorous cross-check to raise redshift reliability.
By the time of writing, the second observing season (Y2) has completed and already a number of updates have been implemented. Additional data and better understanding of the survey yield has allowed us to reassess target selection and de-selection criteria for different target types and science goals. For example, targeting the brightest objects first maximizes the number of measured redshifts for radio galaxies, but such tactics only work when selection bias is not a major consideration. It is also desirable to abandon an unsuccessful target after certain number of exposures while the maximum integration time allowed depends on the total number of redshifts anticipated for this particular target type. Short integration times allow faster recycling of fibres and more redshifts to be measured. During poor observing conditions, a backup program is in place to measure redshifts for bright galaxies, along with high priority targets such as AGN and active transients.
As for all long-term projects, we expect to continue to refine our data quality, actively analyze new data and adapt the observing strategy in the coming years.
\section*{Acknowledgments}
Parts of this research were conducted by the Australian Research Council Centre of Excellence for All-sky Astrophysics (CAASTRO), through project number CE110001020.
ACR acknowledges financial support provided by the PAPDRJ CAPES/FAPERJ Fellowship.
FS acknowledges financial support provided by CAPES under contract No. 3171-13-2
BPS acknowledges support from the Australian Research Council Laureate Fellowship Grant LF0992131.
This work was supported in part by the U.S. Department of Energy contract to SLAC No. DE-AC02-76SF00515.
The data in this paper were based on observations obtained at the Australian Astronomical Observatory (AAO programs A/2012B/11 and A/2013B/12, and NOAO program NOAO/0278). We'd like to thank Marguerite Pierre and the XMM-XXL collaboration for allowing us to use a couple of hours of their time on the AAT to target the DES C3 field.
The authors would like to thank Charles Baltay and the La Silla Quest Supernova Survey to conduct a concurrent transient search during the SV season to help test the targeting strategy.
We are grateful for the extraordinary contributions of our CTIO colleagues and the DECam Construction, Commissioning and Science Verification
teams in achieving the excellent instrument and telescope conditions that have made this work possible. The success of this project also
relies critically on the expertise and dedication of the DES Data Management group.
Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain,
the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing
Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago,
the Center for Cosmology and Astro-Particle Physics at the Ohio State University,
the Mitchell Institute for Fundamental Physics and Astronomy at Texas A\&M University, Financiadora de Estudos e Projetos,
Funda{\c c}{\~a}o Carlos Chagas Filho de Amparo {\`a} Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Cient{\'i}fico e Tecnol{\'o}gico and
the Minist{\'e}rio da Ci{\^e}ncia e Tecnologia, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey.
The DES data management system is supported by the National Science Foundation under Grant Number AST-1138766.
The DES participants from Spanish institutions are partially supported by MINECO under grants AYA2012-39559, ESP2013-48274, FPA2013-47986, and Centro de Excelencia Severo Ochoa SEV-2012-0234,
some of which include ERDF funds from the European Union.
The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energeticas,
Medioambientales y Tecnologicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the Eidgen{\"o}ssische Technische Hochschule (ETH) Z{\"u}rich,
Fermi National Accelerator Laboratory, the University of Edinburgh, the University of Illinois at Urbana-Champaign, the Institut de Ciencies de l'Espai (IEEC/CSIC),
the Institut de Fisica d'Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universit{\"a}t and the associated Excellence Cluster Universe,
the University of Michigan, the National Optical Astronomy Observatory, the University of Nottingham, The Ohio State University, the University of Pennsylvania, the University of Portsmouth,
SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, and Texas A\&M University.
This paper has gone through internal review by the DES collaboration.
|
{
"timestamp": "2015-04-14T02:10:17",
"yymm": "1504",
"arxiv_id": "1504.03039",
"language": "en",
"url": "https://arxiv.org/abs/1504.03039"
}
|
\section{Introduction}
Eruptions in the low solar atmosphere are key elements in generating space weather. Large eruptions can evolve into coronal mass ejections (CMEs) that can plow through the solar wind and ultimately impact the earth's magnetosphere (\cite{Munro}, \cite{Gosling}). The source of many of these CMEs has been associated with prominence eruptions, e.g., \cite{Gopal2003}. \cite{Yan2011} find that approximately half of active region filament eruptions are associated with CMEs and over 90\% are associated with flares. Smaller eruptions may provide the ultimate source for the solar wind (\cite{Tian}). Regardless of their magnitude, eruptions play a significant role in the structure and dynamics of the solar atmosphere.
Identifying eruptions occurring near the solar surface is complicated by the presence of a wide variety of features and scales. Active regions, coronal holes and filaments persist for long periods, while the short-lived eruptions pass through and among them. Flares and other sudden changes in intensity add distractions that can mask or mimic motions that would otherwise be visible. As a result, automated detection of these eruptions has been challenging.
Previous studies have developed automated methods to detect and track filaments primarily in H-alpha images (\cite{Gao},
\cite{2005SoPh..228...97B}, \cite{2005SoPh..228..137Z}, \cite{2005SoPh..228..119Q}, \cite{2008AnGeo..26..243A}, \cite{2008SoPh..248..425S}, \cite{2010SoPh..262..425J}, \cite{2011SoPh..272..101Y}, \cite{2013SoPh..286..385H}, \cite{Schuh}). These methods typically use image-based feature detection followed by a tracking step comparing the results of sequential detections. Measured velocities for prominence eruptions in these studies tend to lie in the range of 10-100 km/s while quiescent prominences show velocities around 4 km/s or less.
We take a different approach by first extracting velocities from a sequence of images and then identifying features within the resulting velocity fields. Here we use an optical flow method to identify regions of significant motion. The use of these velocity fields to define regions of interest, rather than working directly from the images, removes many of the distracting features and permits us to identify and characterize the flows in sufficient detail for further analysis. We note that \cite{Gissot2008} have previously applied an optical flow method to an erupting filament using simultaneous pairs of 30.4nm images acquired by the EUVI instruments on the two STEREO spacecraft (\cite{Kaiser2008}). However their objective was to reconstruct the three-dimensional structure of a filament that had been identified by visual inspection rather than to find the eruption autonomously.
In the following sections we present the underlying method used by Eruption Patrol (EP), assess its performance, survey the statistical properties of the resulting detections and summarize our findings in the conclusion.
\section{Method}
Our approach to identifying solar eruptions is to extract velocity fields from sequences of solar images using the \textsf{opflow3d} method described in \cite{HJ} as applied to images obtained by the Atmospheric Imaging Assembly on the Solar Dynamics Observatory (SDO/AIA, \cite{Lemen}). The Interactive Data Language (IDL) routine, \textsf{opflow3d}, creates an estimate for an unknown, steady, velocity field ${\bf v}(x,y)$ from a time sequence of images $I(x,y,t)$ with the assumption that it satisfies the advection equation
\begin{equation}
\ddt{I}+ {\bf v} \cdot \nabla I = 0. \label{eq:advect}
\end{equation}
The best fit velocity field ${\bf v}_f$ can then be found using least squares applied over
a set of images $I(x,y,t)$ over some time interval, $\Delta t$. \textsf{opflow3d} expands the fit velocity ${\bf v_f}$ in a truncated Fourier series
\begin{equation}
{\bf v}_f= \Sigma_{i=-N_x}^{i=N_x}\Sigma_{j=-N_y}^{j=N_y}(\alpha_{ij}{\bf\hat x} +
\beta_{ij}{\bf\hat y}) e^{-2\pi{J( ix/X + jy/Y)}}, \label{eq:vel}
\end{equation}
where $J=\sqrt{-1}$, and $X$ and $Y$ are the spatial dimensions of the image set. It then determines the optimal values of the complex amplitudes $\alpha_{ij}$
and $\beta_{ij}$, where {$\bf\hat x$} and {$\bf\hat y$} are unit vectors, and $ N_x$ and $N_y$ are the
number of Fourier modes retained in the expansion.
The primary goal of EP is to identify regions of high velocity while operating within the near-real time AIA data pipeline. Hence we choose parameters for EP that can keep up with the data flow. By setting $\Delta t=2$ min, $N_x, N_y =8$ and $X,Y=4096$ pixels, \textsf{opflow3d} can extract a velocity fit from the full-resolution AIA images in approximately one minute of computation on a 2013-vintage Apple iMac. The effective resolution element of this fit is one quarter of the shortest wavelength captured in equation \ref{eq:vel}, which is $0.25 X/N_x=128$ pixels, or about 80 arcsec.
Systematic errors are generally negligible given the quality and consistency of AIA data, but there may be some ringing due to Gibbs phenomena when strong motions are present at the edge of the field of view. Additional sources of error are discussed below.
A two-minute sequence of ten, full-resolution Level 1 images taken in the He II line at 30.4 nm is processed to create one velocity map. These images have had dark current and flat-field corrections and spikes caused by bad pixels and radiation hits removed. The intensity in AIA images is expected to scale as the square of the density of the emitting plasma multiplied by a function temperature (\cite{1983ApJS...52..155G}). Over the two minutes used in determining the fit, we expect that changes in intensity are mostly due to changes in density since temperature changes are dampened relatively quickly (\cite{1982soma.book.....P}) except in actively flaring regions. Hence, the square-root of these images is a proxy for the density, which we expect to approximately obey equation \ref{eq:advect}. This is consistent with the visual impression given by viewing the image sequences. We therefore apply a square-root to the images before passing them into \textsf{opflow3d}.
Previous studies found velocities exceeding 100 km/s, which translates to about 17 arcsec over the two minute sampling period used for a velocity fit. This is well within our effective resolution so systematic error due to smearing should be small; it also suggests that the maximum reliable velocity estimate we can expect for our sampling choice is about 500 km/s, i.e., a displacement of 80 arcsec over the interval $\Delta t=2$ min. Fitting for higher speeds would require either smaller $\Delta t$ or larger spatial windows. For instance, we could detect speeds approaching 5 Mm/s over 80 arcsec by setting $\Delta t = 12$s, the maximum AIA cadence. The minimum speed of the \textsf{opflow3d} method itself is only limited by roundoff errors. Note that if we were to apply this method to the coronal images collected by AIA, say 19.3 nm where velocities are expected to reach these ranges, we would need to adjust our parameters accordingly. In that case, the more complex variations in intensity may require an explicit treatment, such as the multi-scale optical flow method described by \cite{2007A&A...464.1107G} which was used by \cite{Gissot2008}.
The derived velocity fields are composed of multiple components, some of which are sources of error for our application: these include solar rotation, super-granulation and other quasi-static motions. Most of these motions are small and reasonably isotropic. The solar rotation profile is neither, with a peak value of about 2.2 km/s. This can introduce a bias when using a thresholding technique to identify eruption sites. Hence, EP subtracts a background velocity corresponding to that of solid-body rotation. We do not attempt to remove smaller effects such as differential rotation and meridional circulations.
An example of the resulting flow field is displayed in Figure \ref{fig:Aug2010}. The spatial resolution of our velocity fit was doubled to 30 arcsec here to better define the regions. The flow associated with the large filament eruption near the northern pole is clearly captured, as well as a few smaller-scale flows around the limb.
\begin{figure*}
\centering
\includegraphics[width=\columnwidth]{Figure1.png}
\caption{\small The velocity field obtained by applying our method to two minutes of SDO/AIA 30.4 nm data starting at 2010-Aug-01T21:20. The arrows are aligned with the local velocity with areas proportional to the speed. The corresponding image is shown in the color background. Two regions are seen to be erupting: a long filament on disk is ascending into the corona; and another region in the upper left that may be part of the same eruption, or a sympathetic response. Only velocities over 1 km/s are displayed and solar rotation is not removed. The peak velocity here is 3.8 km/s.}
\label{fig:Aug2010}
\end{figure*}
EP samples velocities every 20 minutes and records the time, location and velocity at the point of maximum speed within each sample. As described above, this velocity corresponds to the best-fit over a region of about 80 arcsec in a two minute interval. Hence the precise position of the peak is only known to that resolution. Figure \ref{fig:raw} displays the raw output of the patrol over a seven week period starting on 29 March 2014. The effect of the rotation removal can be seen as a drop in the floor of the velocity measurements to values consistent with those expected from super-granulation (e.g., \cite{Shine}) and other sources. Peaks corresponding to eruptions and spacecraft motions are also clearly visible. The latter are excluded in the production version of the method.
The decision to only sample 10\% of the images, i.e., 2 out of 20 minutes, risks missing short-lived events. However eruptions with lifetimes shorter than 20 minutes probably have little impact on their surroundings. Our goal here is to identify eruptions that may have significant impacts, so the computational cost savings outweigh the loss of information. Our intent is to return to periods of significant motion for more detailed analysis in the future.
The results of this first pass are then processed to identify time periods where velocity exceeds the threshold of 3.6 km/s. This value was settled upon by the need to exclude the background motions seen in Figure \ref{fig:raw} while generating a moderate detection rate. It also corresponds to the level of motion found by \cite{Wang} in quiescent prominences. These periods, along with the largest velocity and its position, are then recorded to the Heliophysics Events Knowledgebase (HEK, \cite{Hurlburt}) as preliminary reports of eruptions. Our intent is to analyze these more carefully in a second "characterization" routine and then replace or update these entries with more details.
\begin{figure}
\includegraphics[width=\columnwidth]{Figure2.png}
\caption{\small The peak speeds reported by EP in 3000 samples taken between 2014-03-29 to 2014-05-16. The preliminary results for April did not remove the effects of solar rotation, and hence have a floor of about 2 km/s. Data in May have that correction applied, and the remaining floor is a combination of differential rotation, meridional flow, super-granulation and other ubiquitous sources. Spacecraft maneuvers and other calibrations are also present in the April results, most notably during April 23.}
\label{fig:raw}
\end{figure}
\section{Results}
\subsection{Comparison with manual selection}
We assess the performance of our method by comparing it to eruptions recorded manually to the HEK. These entries are primarily provided by members of the SDO/AIA science team who monitor data as they arrive at the AIA Validation Center (see \cite{Hurlburt} for details). The volunteer annotators regularly sign up for three-day shifts. Thus all the datasets used by EP, along with other AIA channels, have also been reviewed by this team. Over the interval from 18 April 2014 to 17 July 2014, a total of 43 filament eruptions and 44 eruptions were recorded by the team. For this case we consider an eruption to be any of two classes accepted by the HEK: eruptions and filament eruptions. The first is a catch-all category that may or may not be associated with a filament; the later is associated with a filament that the observer considered to have ejected material into the corona.
As a first test, we queried the HEK for both classes of eruptions using iSolsearch\footnote{\url{http://www.lmsal.com/isolsearch}} to select the events and then exporting them into SolarSoft (\cite{Freeland}) and using the IDL routine \textsf{hek\_match\_events}. For this study we considered events that overlapped within an hour in time. The results are displayed in Table 1. Of the 43 filament eruptions reported by humans, 37 (79\%) matched times reported by EP. The success drops to 24 (44\%) when we also require a separation of less than 120 arcsec. Human reports of eruptions displayed a similar behavior, which is partly due to some observers selecting both when they generate their reports. We will discuss this further below.
\begin{table*}
\label{table:1}
\centering
\begin{tabular}{c c c c c c}
\hline\hline
Comparison & Count& Hit (t) & Miss (t) & Hit (x,t) & Miss (x,t) \\
\hline
EP from FE & 43 & 79\% & 21\% &44\%&56\% \\
EP from ER & 44& 84\% & 16\%&55\%&45\% \\
FE+ER from EP & 29 & 31\% & 69\%&24\%&86\% \\
\hline
\end{tabular}
\caption{A bi-directional comparison between filament eruptions (FE) and generic eruptions (ER) reported manually and EP reports over the time span from 4/18/2014 to 7/17/2014. The first two rows show the success of EP in matching times recorded by FE and ER. The third row shows the success of the manual reports in matching times reported by EP. The columns show the total number of reports in each set, the percentage of hits and misses for the reported times, and hits and misses for both time and position.}
\end{table*}
\begin{table*}
\label{table:2}
\centering
\begin{tabular}{c c c c c c}
\hline\hline
Comparison & Count& Hit (t) & Miss (t) & Hit (x,t) & Miss (x,t) \\
\hline
EP from FE & 813 & 68\% & 32\% &27\%&73\% \\
EP from ER& 327& 70\% & 30\%&53\%&47\% \\
FE+ER from EP& 289 & 24\% & 76\%&11\%&89\% \\
\hline
\end{tabular}
\caption{A similar comparison as in Table 1, but over the large time span from 5/15/2010 to 11/17/2014.}
\end{table*}
As a second test, we selected the 29 events with speeds exceeding 30 km/s from EP over the same interval and compared them to entries reported by human annotators. Nine (31\%) match the human annotations in time, while the remaining 20 did not. Only 7 of those 9 also overlapped spatially. All of these missed events were reviewed visually using the daily movies posted by the AIA team\footnote{\url{http://sdowww.lmsal.com}} and were found to be associated with significant eruptions.
These patterns persist over the entire AIA dataset, as can be seen in Table 2. Overall EP finds about 70\% of all time periods manually reported as erupting in either category. This moderate drop for Hit(t) in Table 2, as compared to Table 1, may be due to the fact that humans are particularly good at catching slow, long-duration eruptions. Comparisons of EP detections exceeding 30 km/s to manual ones shows similar drop from 31\% in Table 1 to 24\% in Table 2. This might indicate that the converse is also true: fast, short-duration eruptions may be overlooked by human annotators. This is consistent with what we found in the smaller sample, but we have not manually reviewed the 289 events to confirm this.
The success at matching both the time and location of filament eruptions drops to 27\% in the larger sample. This may again be due to large-scale filament eruptions. In that case, the position selected by annotators will tend to be the geometric center of the filament, while the position reported by EP will be the fastest moving element, or perhaps that of a separate fast, short-lived eruption. The two eruptions seen in Figure \ref{fig:Aug2010} provides an example of this situation.
The matches to generic eruption continues to be slightly over 50\%. In this larger time span, the two sets (filament eruptions and generic eruptions) are reasonably independent, with the latter catching a larger range of behavior which is left to the discretion of the annotator. The fact that there is not higher success rate here is to be expected, since EP reports only a single location for a given time sample. If multiple eruptions are occurring, all but one of them will be missed. This is another issue that will be addressed in the characterization module.
The overall accuracy of manual entries, finding less than a quarter of significant time periods identified by EP and with only 11\% overlapping spatially is noteworthy. Their accuracy appears to be uncorrelated with the eruption's duration or magnitude. There may be some relation to spatial extent, but in reviewing a smaller sample with members of the annotation team, the main source of discrepancy is most likely a lack of attention or interest. Each annotator has particular interests related to their individual research, and there is a clear correlation between their success at this task and their interests.
\begin{figure*}
\centering
\includegraphics[width=\columnwidth]{Figure3.png}
\caption{\small(Left) Histogram of speeds (black dots) of all eruptions from 5/15/2010 to 7/12/2014. Eruptions appear to possess a power law distribution near that of an inverse square (solid line). (Right) A polar plot of the velocity vectors show them to be reasonably isotropic with a maximum speed of 96.2 km/s. For clarity, only speeds over 10 km/s are displayed).}
\label{fig:speeds}
\end{figure*}
\subsection{Statistical properties}
The EP module described above was run over the entire SDO mission up to July 12, 2014, thus spanning just over four years. Here we give an overview of the statistical properties of this sample. The left panel in Figure \ref{fig:speeds} displays the histogram of peak speeds detected for each recorded eruption. The distribution appears to have power law dependence on the speed, with the largest event having a speed of 96 km/s. These are consistent with previous studies, such as \cite{Gopal2003} and \cite{Wang}. The right panel in Figure \ref{fig:speeds} displays the distribution of velocities as a polar plot. There does not appear to be a significant directional bias in the sample.
\begin{figure*}
\centering
\includegraphics[width=\columnwidth]{Figure4.png}
\caption{\small (Left) The distribution of all eruptions from 5/15/2010 to 7/12/2014 show clustering near activity belts and the limb. Black, blue and red circles display weak ($\langle10$ km/s), medium ($\langle30$ km/s) and strong ($\rangle30$ km/s) eruptions respectively. (Right) Histogram of radial positions. Dots represent number of events in 10 arc second bins. The solid line is a theoretical distribution assuming the formation layer is a uniform 15 Mm thick, while the dashed is same for a 3.5 Mm layer. Both lines are scaled to match the maximum. The distribution is compatible with a layer thickness somewhere within the range of 3-15 Mm.}
\label{fig:dist}
\end{figure*}
Figure \ref{fig:dist} displays the spatial distribution of all events over this period. Eruptions are detected almost everywhere on the disk, as seen in the left panel, but are clearly clustered near the activity belts and the limb. This distribution appears to be independent of the magnitude of the events (as indicated by the color of the dots). There is a clear lack of eruptions reported near the poles which may be due to relatively slow-moving polar crown filament eruptions being masked by more dynamic regions as described in the last section.
The apparent clustering near the limb is examined in the right panel, where the histogram as a function of radius (r) for these events is displayed. The distribution rises from zero near disk center ($r=0$) until the active region bands begin to contribute at $r \approx 0.4 R_{sun},$ where $R_{sun}$ is the solar radius. The distribution remains relatively constant between that point until near the limb ($r \approx R_{sun}$), where the number of eruptions climb rapidly before falling back to zero. This distribution is consistent with that expected in the case of a shallow, optically-thin, formation layer (between 3-15 Mm) containing a uniformly random distribution of eruptions. \cite{Gopal2003} report that (relatively large) eruptive prominences have heights between 1.1 to 1.5 $R_{sun}$. This suggests that there may be some scale dependence in the distribution that is neglected in our simple model, which might also explain the deviations from our model for $r > R_{sun}$. Some level of scale dependence is expected in the structuring of the solar atmosphere by magnetic fields, as in the magnetic carpet model of \cite{Title}.
Another source of systematic error may result from projection effects. If all eruptions were predominantly radial, we would expect a radial dependence in the magnitude and direction of the reported velocities. \cite{Gopal2003} found this to be the case overall, but with many eruptions possessing tangential motions on the order of 10 km/s. In contrast, the motions we detect are randomly oriented and show no significant projection effects. We may resolve this discrepancy by noting that the former study was effectively tracking the centroid of a prominence while our method is measuring local velocities, which includes twisting, writhing and streaming motions that frequently accompany eruptions. Hence we can identify times and location of eruptions based on these associated motions regardless of where they appear on the disk. A more complete reckoning of these velocity contributions falls to the future characterization module.
\section{Conclusion}
We have developed an automated method for finding eruptions in the lower solar atmosphere and have deployed it within the SDO/AIA Event Detection System which operates on the data as they arrive. The method has been found to measure velocities with statistical properties consistent with previous studies. The reported eruptions also appear to be consistent with those reported by human reviewers. The automated detections are less prone to lapses in attention or skewed by personal interests, but may miss slow, long-duration eruptions. They also provide a more complete characterization of eruptions by reporting both the location and plane-of-sky velocity. The reported events are found to be distributed in a layer near the solar surface and possess a power law distribution in peak speed. Details of these events, including summary movies, can be found using a variety of tools including Helioviewer\footnote{\url{http://helioviewer.org}}, iSolsearch\footnote{\url{http://www.lmsal.com/isolsearch}} and SolarSoft. As part of the HEK, they are automatically cross-referenced with solar datasets obtained by the Hinode (\cite{Kosugi}) and Interface Region Imaging Spectrograph (IRIS, \cite{BDP}) missions.
Subsequent papers will explore how these eruptions compare with those found with other automated processes recording in the HEK and will describe a characterization module that confirms and extracts more detailed information on the eruptions reported by the EP.
\begin{acknowledgements}
Thanks to Dr Paul Higgins for his helpful discussions and to Ryan Timmons and Sam Freeland for help implementing the module. This work has been supported by NASA under contract NNG04EA00C and Lockheed Martin Internal Research Funds. The editor thanks Jean-Fran\c cois Hochedez and two anonymous referees for their assistance in evaluating this paper.
\end{acknowledgements}
|
{
"timestamp": "2015-12-02T02:16:37",
"yymm": "1504",
"arxiv_id": "1504.03395",
"language": "en",
"url": "https://arxiv.org/abs/1504.03395"
}
|
\section{Introduction}
Feedback processes associated with the collapse of protostellar envelopes
at $10^3-10^4$ AU scales limit the accretion onto the protostar and contribute
to the overall low efficiency of transferring gas into stars on global
scales \citep{Of09,Krum14}. Calculating the physical conditions
help to identify the most relevant phenomena
and constrain their role in the star formation process \citep{Ev99}.
Since gas in protostellar envelopes is heated to temperatures much higher than the dust temperatures,
molecular transitions are the suitable tracers of physical conditions of hot ($T\gtrsim100$ K) gas
around protostars. In particular, the far-infrared (IR)
lines of CO and H$_2$O dominate the cooling of hot and dense gas \citep{GL78}.
The excitation of CO and H$_2$O depends on the local physical conditions (temperature, density) and thus
is crucial to determine which physical mechanisms are responsible for the gas heating and
to study whether the energetics involved in the feedback
scale from low- to intermediate- to high-mass young stellar objects (YSOs).
Recent observations of CO and H$_2$O lines with the Photodetector
Array Camera and Spectrometer \citep[PACS,][]{Po10} on board \textit{Herschel}
found evidence for large columns of dense ($\gtrsim 10^4$ cm$^{-3}$) and hot ($\gtrsim 300$ K)
gas towards low-mass ($L_\mathrm{bol}\lesssim10^2 L_\odot$) protostars \citep{vK10,He12,Ma12,Ka13,Gr13,Li14}, originate largely
from UV-irradiated shocks associated with jets and winds \citep{Ka14b}.
CO and H$_2$O line luminosities of the high-mass protostars (with $L_\mathrm{bol}\sim10^{4}-10^{6}$ $L_{\odot}$)
follow the correlations with bolometric luminosities found in the low-mass protostars \citep{Ka14} and
show similar velocity-resolved line profiles regardless of the mass of the protostar \citep{Yi13,IreneCO,vT13}.
In contrast, rotational temperatures of H$_2$O are lower and the H$_2$O fraction
contributed to the total cooling in lines with respect to CO is higher for the low-mass protostars \citep{Ka14,Go15},
suggesting that the physical mechanism causing the excitation in low- and high-mass protostars are different.
\begin{figure*}
\sidecaption
\includegraphics[angle=90,width=12cm]{codiags.eps}
\caption{\label{coex} Rotational diagrams of CO.
The base-10 logarithm of the number of
emitting molecules from the upper level,
$\mathcal{N}_\mathrm{up}$, divided by the degeneracy of the level,
$g_\mathrm{up}$, is shown as a function of energy of the upper level
in kelvins, $E_\mathrm{up}$. Detections are shown as filled
circles, whereas three-sigma upper limits are shown as empty
circles. Empty upper triangle corresponds to the line flux calculated using
a smaller area on the map than the rest of the lines. Blue lines show linear fits to the data and the
corresponding rotational temperatures. Errors associated with the least-square linear fit are shown in brackets.}
\end{figure*}
\begin{figure*}[!tb]
\sidecaption
\includegraphics[angle=90,width=12cm]{h2odiags.eps}
\caption{\label{wex} Similar to Figure \ref{coex} but for H$_2$O.}
\end{figure*}
Intermediate-mass YSOs (with $L_\mathrm{bol}\sim10^{2}-10^{3}$ $L_{\odot}$\footnote{
We use bolometric luminosity as a proxy of the
protostellar mass for practical and historical reasons, but we note that $L_\mathrm{bol}$ changes
significantly during the protostellar phase if the accretion is episodic \citep{Yo05,MD10}. Moreover,
some of our intermediate-mass sources may be in fact a collection of unresolved low-mass protostars.})
provide a natural link between low- and high mass protostars, but their far-IR CO and H$_2$O emission has only
been studied for a single protostar position of NGC 7129 FIRS 2 \citep{Fi10} and the outflow position of
NGC 2071 \citep{Ne14}. CO emission alone was analyzed for two intermediate-mass protostars in Orion,
HOPS 288 and 370 \citep{Ma12}. In this paper, we present the analysis of PACS spectra for the
full sample of intermediate-mass protostars from the `Water in star forming regions with Herschel'
(WISH) key program \citep{WISH}, including the maps of NGC 7129 and NGC 2071 centered on the
YSO position. These results complement the work by \citet{Wa13}, which describes the OH excitation in our source sample
and the sample of low- and high-mass protostars for which CO and H$_2$O emission is discussed in \citet{Ka13,Ka14}.
The main question addressed is whether CO and H$_2$O rotational temperatures differ
from low- to high-mass protostars.
The paper is organized as follows. \S 2 briefly introduces the observations,
\S 3 the excitation analysis using rotational diagrams, and \S 4 discusses the results.
\section{Observations}
\begin{table*}[tb!]
\caption{\label{tab:exc} CO and H$_2$O rotational excitation}
\centering
\renewcommand{\footnoterule}{}
\begin{tabular}{lcccccccccccccc}
\hline \hline
Source & $D$ & $L_\mathrm{bol}$ & \multicolumn{2}{c}{Warm CO} & \multicolumn{2}{c}{Hot CO} & \multicolumn{2}{c}{H$_2$O} \\
~ & (pc) & ($L_{\odot}$) & $T_\mathrm{rot}$(K) & $\mathrm{log}_\mathrm{10}\mathcal{N}$ & $T_\mathrm{rot}$(K) &
$\mathrm{log}_\mathrm{10}\mathcal{N}$ & $T_\mathrm{rot}$(K) & $\mathrm{log}_\mathrm{10}\mathcal{N}$ \\
\hline
AFGL 490 & 1000 & 2000 & 300(20) & 51.6(0.1) & \ldots & \ldots & 90(100) & 48.3(1.2) \\
L 1641 S3 MMS1 & 465 & 70 & 325(30) & 49.9(0.1) & \ldots & \ldots & 150(100) & 46.8(0.1) \\
NGC 2071 & 422 & 520 & 310(30) & 51.2(0.1) & 700(50) & 50.1(0.1) & 135(100) & 47.9(0.1) \\
Vela 17 & 700 & 715 & 265(25) & 50.8(0.2) & \ldots & \ldots & 90(40) & 47.8(0.5) \\
Vela 19 & 700 & 776 & 335(50) & 50.4(0.2) & \ldots & \ldots & 130(80) & 47.2(0.4) \\
NGC 7129 FIRS 2 & 1250 & 430 & 370(45) & 50.7(0.2) & 710(70) & 49.9(0.2) & 130(50) & 47.8(0.1) \\
\hline
\end{tabular}
\tablefoot{Distances and bolometric luminosities are taken from \citet{Wa13} and references therein.}
\end{table*}
Our sample includes 6 YSOs with bolometric luminosities from 70 to 2000 $L_\odot$
and located at an average distance of 700 pc (see Table \ref{tab:exc}).
The sources were selected based on their small distances ($\lesssim$ 1 kpc)
and location accessible for follow-up observations from the southern hemisphere
\citep[for more details see \S 4.4.2. in][]{WISH}. Spectroscopy for all
sources was obtained with PACS
as part of the WISH program. For observing details see Table \ref{log} in the Appendix.
With PACS, we obtained single footprint spectral maps covering a field of view of $\sim47''\times47''$ and resolved
into 5$\times$5 array of spatial pixels (spaxels) of $\sim9.4''\times9.4''$ each. At the distance to the sources,
the full array corresponds to spatial scales of $\sim 2-6\times 10^4$ AU in diameter, of the order of
full maps of low-mass YSOs ($\sim 10^4$ AU) in \citet{Ka13} and the central spaxel spectra of more distant high-mass YSOs
($\sim 3\times 10^4$ AU) in \citet{Ka14}.
The observations were taken in the line spectroscopy mode, which provide deep integrations of
0.5--2 $\mu$m wide spectral regions within the $\sim$55--210 $\mu$m PACS range. Two nod positions
were used for chopping 3$^\prime$ on each side of the source \citep[for
details of our observing strategy and basic reduction methods, see][]{Ka13}. The full list of targeted
CO and H$_2$O lines and the calculated line fluxes are shown in Table \ref{lines} in the Appendix.
The quality of the spectra is illustrated in Figure \ref{spec} in the Appendix.
We note that the simultaneous non-detections of the H$_2$O 7$_{16}$-7$_{07}$ line at 84.7 $\mu$m
and detections of the H$_2$O 8$_{18}$-7$_{07}$ line at 63.3 $\mu$m in L 1641, NGC 2071, and
NGC 7129 FIRS 2 are related to the structure of the H$_2$O energy levels and not
the differences in the sensitivities of the instrument in the 2nd and 3rd order observations.
The 8$_{18}$-7$_{07}$ line is a backbone transition, with
the Einstein A coefficient higher than the one for the 7$_{16}$-7$_{07}$ line.
The data reduction was done using HIPE v.13 with Calibration Tree 65 and subsequent
analysis with customized IDL programs \citep[see e.g.][]{Ka14b}. The fluxes were calculated
using the emission from the entire maps. Figure \ref{maps} in the Appendix illustrates
that both the line and continuum emission peaks approximately at the source position, with
small shifts in continuum due to mispointing. The extent of the line emission follows typically
the continuum pattern with the exception of Vela 17 where the line emission extends from NE to SW
direction, while the continuum is centrally peaked. There is no contamination
detected from the nearby sources or their outflows in the targeted regions.
\section{Rotational diagrams}
\subsection{Results}
\begin{figure}[!tb]
\begin{center}
\includegraphics[angle=90,height=7cm]{imtrot.eps}
\vspace{-0.4cm}
\caption{\label{trot} Rotational temperatures of \lq warm' ($E_\mathrm{up}=14-24$) CO (top) and H$_2$O (bottom) as function
of bolometric luminosity. Orange circles show YSOs from
\citet{Gr13}, \citet{Ka13}, and \citet{Ma12}. Dark blue crosses show intermediate-mass
YSOs from the WISH program and light blue diamonds high-mass YSOs from \citet{Ka14}.
A few sources with luminosities $\sim10^2$ $L_\odot$ shown in orange crosses are from
\citet{Ma12} and can be also regarded as intermediate-mass YSOs. Uncertainties
of the CO and H$_2$O rotational temperatures are shown for L1641 S3 MMS1, which
is representative of the sample.}
\end{center}
\end{figure}
Figure \ref{coex} shows rotational diagrams of CO calculated for all sources in the
same way as in \citet{Ka14}. The corresponding rotational temperatures, $T_\mathrm{rot}$,
and total numbers of emitting molecules, $\mathcal{N}$, are shown in Table \ref{tab:exc}.
All sources show a 300 K, \lq warm' CO component \citep{Ma12,Ka13,Gr13}, with a mean temperature
of $\sim 320$ K for the range of total numbers of emitting molecules, $\mathcal{N}\sim 10^{50} - 10^{51}$.
In addition, NGC2071 and NGC7129 show a \lq hot' CO component with temperatures of $\sim 700$ K
and $\mathcal{N}\sim 10^{50}$.
The rotational diagrams for H$_2$O, presented in Figure \ref{wex}, show a single component with
a mean temperature of $\sim 120$ K. The scatter due to subthermal excitation and
high opacities exceeds the uncertainties in the observed fluxes, similar to diagrams of low- and high-mass
YSOs \citep[for the discussion of both effects see \S 4.2.2 of][]{Ka14}.
The corresponding numbers of emitting molecules are about 3 orders of magnitude lower than
for CO, $\mathcal{N}\sim 10^{47} - 10^{48}$.
\subsection{Comparison to low- and high-mass sources}
Figure \ref{trot} shows a comparison between rotational temperatures obtained for CO
and H$_2$O for the intermediate-mass sources presented here and these quantities
determined in the same way for low- and high-mass YSOs.
The comparison is restricted only to the \lq warm' component
seen in CO rotational diagrams due to the low number of sources with detections of the \lq hot' CO component;
those will be discussed in detail in Karska et al. (in prep.). A bolometric luminosity
is used here as a proxy for the protostellar mass, but in fact some of
our intermediate-mass sources may be a collection of unresolved low-mass protostars.
The median $T_\mathrm{rot}$ of CO in low-mass protostars is 325 K, using the results from the WISH
\citep{Ka13}, \lq Dust, Ice, and Gas in Time' \citep{Gr13}, and \lq Herschel Orion Protostar Survey'
\citep{Ma12} programs for a total of about 50 sources. A comparable value of $\sim$290 K was
found for 10 high-mass sources ($L_\mathrm{bol}\sim10^4 - 10^6$ $L_\odot$) in \citet{Ka14}.
The range of CO $T_\mathrm{rot}$ of 265-370 K (see Table \ref{tab:exc}) determined
here for intermediate-mass YSOs is thus fully consistent with previous results. The fact that the
300 K CO component does not depend on the source bolometric luminosity
over 6 orders of magnitude suggests the origin in a shock associated with the jet / winds
impact on the envelope rather than in a photodissociation region where $T_\mathrm{rot}$
should scale with the UV flux and luminosity \citep{vK10,Vi12,Ma12,FP13,Kr13}.
The median H$_2$O rotational temperatures (see the bottom panel of Figure \ref{trot})
for low- and high-mass YSOs are $\sim130$ K and $\sim230$ K, respectively \citep{Ka13,Ka14}.
The range of temperatures obtained for the intermediate-mass YSOs (90-150 K, Table \ref{tab:exc})
is thus comparable to the low-mass sources. However, the uncertainties in the rotational
temperatures are high, of the order of $\sim100$ K, and do not account for the optical depth
effects, the density effects, and the possible complexity of the line profiles (absorption and emission
components). It is therefore unclear if there is
a true jump in rotational temperatures at $L_\mathrm{bol}\sim10^4$ $L_\odot$, or a smooth trend
toward higher values of $T_\mathrm{rot}$. In either case, these higher excitation
temperatures could be due to higher densities in the more massive envelopes.
\section{Summary}
We analyze the excitation of far-infrared CO and H$_2$O lines in 6 intermediate-mass
YSOs observed in the WISH survey and compare the results to low- and high-mass protostars.
Rotational temperatures of CO and H$_2$O are found to be $\sim320$ K and $\sim120$ K, respectively,
and are consistent with low-mass and high-mass YSOs within the uncertainties.
The large uncertainties in the H$_2$O rotational temperatures, of the order of 100 K,
and the order of magnitude gap in the bolometric luminosity between intermediate-
and high-mass protostars does not allow us to conclude whether the changes are a smooth
function of luminosity. Still, the similarities in rotational
temperatures seen for sources with luminosities spanning 6 orders of magnitude and
probed at different spatial scales strongly suggest
the same excitation mechanism, the UV-irradiated shocks associated with jets and winds
for all sources across the luminosity range \citep{Kr13,Ka14b,Mo14}.
\begin{acknowledgements}
The authors would like to thank the referee for the valuable comments
which helped to improve the manuscript. Herschel is an ESA space observatory with science instruments provided
by European-led Principal Investigator consortia and with important
participation from NASA. AK acknowledges support from
the Polish National Science Center grant 2013/11/N/ST9/00400.
Research conducted within the scope of the HECOLS International Associated
Laboratory, supported in part by the Polish NCN grant DEC-2013/08/M/ST9/00664.
\end{acknowledgements}
\bibliographystyle{aa}
|
{
"timestamp": "2015-05-05T02:14:19",
"yymm": "1504",
"arxiv_id": "1504.03347",
"language": "en",
"url": "https://arxiv.org/abs/1504.03347"
}
|
\section{Introduction}
The importance of (small) categories enriched in a (unital) quantale rather than in an arbitrary monoidal category was discovered by Lawvere \cite{Lawvere1973} who enabled us to look at individual mathematical objects, such as metric spaces, as small categories. Through the study of lax algebras \cite{Hofmann2014}, quantale-enriched categories have become the backbone of a larger array of objects that may be viewed as individual generalized categories. Prior to this development, Walters \cite{Walters1981} had extended Lawvere's viewpoint in a different manner, replacing the quantale at work by a \emph{quantaloid} (a term proposed later by Rosenthal \cite{Rosenthal1996}), thus by a bicategory with the particular property that its hom-objects are given by complete lattices such that composition from either side preserves suprema; quantales are thus simply one-object quantaloids.
Based on the theory of quantaloid-enriched categories developed by Stubbe \cite{Stubbe2005,Stubbe2006}, recent works \cite{Hohle2011,Pu2012,Stubbe2014,Tao2014} have considered in particular the case when the quantaloid in question arises from a given quantaloid by a ``diagonal construction'' whose roots go far beyond its use in this paper; see \cite{Grandis2000,Grandis2002}. Specifically for the one-object quantaloids (i.e., quantales) whose enriched categories give (pre)ordered sets and (generalized) metric spaces, the corresponding small quantaloids of diagonals lead to truly partial structures, in the sense that the full structure is available only on a subset of the ambient underlying set of objects.
In the first instance then, this paper aims at exploring the categorical properties of the category $\QCat$ of small $\CQ$-enriched categories and their $\CQ$-functors for a small quantaloid $\CQ$. By showing that $\QCat$ is topological \cite{Adamek1990} over the comma category $\Set/\ob\CQ$ (Proposition \ref{QCat_topological}) one easily describes small limits and colimits in this category, and beyond. In fact, one concludes that categories of this type are total \cite{Street1978} and cototal, hence possess even those limits and colimits of large diagrams whose existence is not made impossible by the size of the small hom-sets of $\QCat$ (see \cite{Borger1990}).
Our greater interest, however, is in the category $\QChu$ whose objects are often called $\CQ$-Chu spaces, the prototypes of which go back to \cite{Barr1991,Pratt1995} and many others (see \cite{Barwise1997,Ganter2007}). Its objects are \emph{$\CQ$-distributors} of $\CQ$-categories (also called $\CQ$-(bi)modules or $\CQ$-profunctors), hence they are compatible $\CQ$-relations (or $\CQ$-matrices) that have been investigated intensively ever since B{\'e}nabou \cite{Benabou1973} introduced them (see \cite{Benabou2000,Borceux1994a}). While when taken as the morphisms of the category whose objects are $\CQ$-categories, they make for a in many ways poorly performing category (as already the case $\CQ={\bf 2}$ shows), when taken as objects of $\QChu$ with morphisms given by so-called \emph{$\CQ$-Chu transforms}, i.e., by pairs of $\CQ$-functors that behave like adjoint operators, we obtain a category that in terms of the existence of limits and colimits behaves as strongly as $\QCat$ itself. In analogy to the property shown in \cite{Giuli2007} in a different categorical context, we first prove that the domain functor $\QChu\to\QCat$ allows for initial liftings \cite{Adamek1990} of structured cones over small diagrams (Theorem \ref{dom_initial_lifting}), which then allows for an explicit description of all limits and colimits in $\QChu$ over small diagrams. But although the domain functor fails to be topological, just as for $\QCat$ we are able to show totality (and, consequently, cototality) of $\QChu$. A key ingredient for this result is the existence proof of a generating set in $\QChu$, and therefore also of a cogenerating set (Theorem \ref{QChu_generator}).
\section{Limits and colimits in $\QCat$}
Throughout, let $\CQ$ be a small \emph{quantaloid}, i.e., a small category enriched in the category $\Sup$ of complete lattices and sup-preserving maps. A small \emph{$\CQ$-category} is given by a set $X$ (its set of objects) and a lax functor
$$a:X\to\CQ,$$
where the set $X$ is regarded as a quantaloid carrying the chaotic structure, so that for all $x,y\in X$ there is precisely one arrow $x\to y$, called $(x,y)$. Explicitly then, the $\CQ$-category structure on $X$ is given by
\begin{itemize}
\item a family of objects $|x|_X:=ax$ in $\CQ$ $(x\in X)$,
\item a family of morphisms $a(x,y):|x|\to|y|$ in $\CQ$ $(x,y\in X)$, subject to
$$1_{|x|}\leq a(x,x)\quad\text{and}\quad a(y,z)\circ a(x,y)\leq a(x,z)$$
\end{itemize}
$(x,y,z\in X)$. When one calls $|x|=|x|_X$ the \emph{extent} (or \emph{type}) of $x\in X$, a \emph{$\CQ$-functor} $f:(X,a)\to(Y,b)$ of $\CQ$-categories $(X,a)$, $(Y,b)$ is an extent-preserving map $f:X\to Y$ such that there is a lax natural transformation $a\to bf$ given by identity morphisms in $\CQ$; explicitly,
$$|x|_X=|f(x)|_Y\quad\text{and}\quad a(x,y)\leq b(f(x),f(y))$$
for all $x,y\in X$. Denoting the resulting ordinary category by $\QCat$, we have a forgetful functor
$$\bfig
\morphism<1200,0>[\QCat`\Set/\ob\CQ;\ob]
\Vtriangle(-300,-500)<300,300>[X`Y`\CQ;f`a`b]
\Vtriangle(900,-500)<300,300>[X`Y`\ob\CQ;f`|\text{-}|`|\text{-}|]
\place(600,-350)[\mapsto] \place(0,-320)[\leq]
\efig$$
\begin{exmp} \phantomsection \label{QCat_exmp}
\begin{itemize}
\item[\rm (1)] If $\CQ$ is a \emph{quantale}, i.e., a one-object quantaloid, then the extent functions of $\CQ$-categories are trivial and $\QCat$ assumes its classical meaning (as in \cite{Kelly1982}, where $\CQ$ is considered as a monoidal (closed) category). Prominent examples are $\CQ={\bf 2}=\{0<1\}$ and $\CQ=([0,\infty],\geq,+)$, where then $\QCat$ is the category $\Ord$ (sets carrying a reflexive and transitive relation, called \emph{order} here but commonly known as preorder, with monotone maps) and, respectively, the category $\Met$ (sets $X$ carrying a distance function $a:X\times X\to[0,\infty]$ required to satisfy only $a(x,x)=0$ and $a(x,z)\leq a(x,y)+a(y,z)$ but, in accordance with the terminology introduced by Lawvere \cite{Lawvere1973} and used in \cite{Hofmann2014}, nevertheless called \emph{metric} here, with non-expanding maps $f:X\to Y$, so that $b(f(x),f(y))\leq a(x,y)$ for all $x,y\in X$).
\item[\rm (2)] (Stubbe \cite{Stubbe2014}) Every quantaloid $\CQ$ gives rise to a new quantaloid $\DQ$ whose objects are the morphisms of $\CQ$, and for morphisms $u,v$ in $\CQ$, a morphism $(u,d,v):u\to/~>/v$ in $\DQ$, normally written just as $d$, is a $\CQ$-morphism $d:\dom u\to\cod v$ satisfying
$$(d\lda u)\circ u=d=v\circ(v\rda d),$$
also called a \emph{diagonal} from $u$ to $v$:
$$\bfig
\square[\bullet`\bullet`\bullet`\bullet;v\rda d`u`v`d\lda u]
\morphism(0,500)<500,-500>[\bullet`\bullet;d]
\efig$$
(Here $d\lda u$, $v\rda d$ denote the \emph{internal homs} of $\CQ$, determined by
$$z\leq d\lda u\iff z\circ u\leq d,\quad t\leq v\rda d\iff v\circ t\leq d$$
for all $z:\cod u\to\cod d$, $t:\dom d\to\dom v$.) With the composition of $d:u\to/~>/v$ with $e:v\to/~>/w$ in $\DQ$ defined by
$$e\diamond d=(e\lda v)\circ d=e\circ(v\rda d),$$
and with identity morphisms $u:u\to/~>/u$, $\DQ$ becomes a quantaloid whose local order is inherited from $\CQ$. In fact, there is a full embedding
$$\CQ\to\DQ,\quad(u:t\to s)\mapsto(u:1_t\to/~>/1_s)$$
of quantaloids.
We remark that the construction of $\sD$ works for ordinary categories; indeed it is part of the proper factorization monad on $\CAT$ \cite{Grandis2002}.
\item[\rm (2a)] For $\CQ={\bf 2}$, the quantaloid $\DQ$ has object set $\{0,1\}$. There are exactly two $\DQ$-arrows $1\to/~>/1$, given by $0,1$, and $0$ is the only arrow in every other hom-set of $\DQ$; composition is given by infimum. A $\DQ$-category is given by a set $X$, a distinguished subset $A\subseteq X$ (those elements of $X$ with extent $1$) and a (pre)order on $A$. Hence, a $\DQ$-category structure on $X$ is a (truly!) \emph{partial order} on $X$. With morphisms $f:(X,A)\to(Y,B)$ given by maps $f:X\to Y$ monotone on $A=f^{-1}B$ we obtain the category
$$\ParOrd=\DQCat,$$
which contains $\Ord$ as a full coreflective subcategory.
\item[\rm (2b)] For $\CQ=([0,\infty],\geq,+)$, the hom-sets of $\DQ$ are easily described by
$$\DQ(u,v)=\{s\in[0,\infty]\mid u,v\leq s\},$$
with composition given by $t\diamond s=s-v+t$ (for $t:v\to/~>/w$). A $\DQ$-category structure on a set $X$ consists of functions $|\text{-}|:X\to[0,\infty]$, $a:X\times X\to[0,\infty]$ satisfying
$$|x|,|y|\leq a(x,y),\quad a(x,x)\leq|x|,\quad a(x,z)\leq a(x,y)-|y|+a(y,z)$$
$(x,y,z\in X)$. Obviously, since necessarily $|x|=a(x,x)$, these conditions simplify to
$$a(x,x)\leq a(x,y),\quad a(x,z)\leq a(x,y)-a(y,y)+a(y,z)$$
$(x,y,z\in X)$, describing $a$ as a \emph{partial metric} on $X$ (see \cite{Hohle2011,Matthews1994,Pu2012}\footnote{Here our terminology naturally extends Lawvere's notion of metric and is synonymous with ``generalized partial metric'' as used by Pu-Zhang \cite{Pu2012} who dropped finiteness ($a(x,y)<\infty$), symmetry ($a(x,y)=a(y,x)$) and separation ($a(x,x)=a(x,y)=a(y,y)\iff x=y$) from the requirement for the notion of ``partial metric'' as originally introduced by Matthews \cite{Matthews1994}.}). With non-expanding maps $f:(X,a)\to(Y,b)$ satisfying $a(x,x)=b(f(x),f(x))$ for all $x\in X$ one obtains the category
$$\ParMet=\DQCat,$$
which contains $\Met$ as a full coreflective subcategory: the coreflector restricts the partial metric $a$ on $X$ to those elements $x\in X$ with $a(x,x)=0$.
\end{itemize}
\end{exmp}
To see how limits and colimits in the (ordinary) category $\QCat$ are to be formed, it is best to first prove its topologicity over $\Set/\ob\CQ$. Recall that, for any functor $U:\CA\to\CX$, a \emph{$U$-structured cone} over a diagram $D:\CJ\to\CA$ is given by an object $X\in\CX$ and a natural transformation $\xi:\De X\to UD$. A \emph{lifting} of $(X,\xi)$ is given by an object $A$ in $\CA$ and a cone $\al:\De A\to D$ over $D$ with $UA=X$, $U\al=\xi$. Such lifting $(A,\al)$ is \emph{$U$-initial} if, for all cones $\be:\De B\to D$ over $D$ and morphisms $t:UB\to UA$ in $\CX$, there is exactly one morphism $h:B\to A$ in $\CA$ with $Uh=t$ and $\al\cdot\De h=\be$. We call $U$ \emph{small-topological} \cite{Giuli2007} if all $U$-structured cones over small diagrams admit $U$-initial liftings, and $U$ is \emph{topological} when this condition holds without the size restriction on diagrams. Recall also the following well-known facts:
\begin{itemize}
\item Topological functors are necessarily faithful \cite{Borger1978}, and for faithful functors it suffices to consider discrete cones to guarantee topologicity.
\item $U:\CA\to\CX$ is topological if, and only if, $U^{\op}:\CA^{\op}\to\CX^{\op}$ is topological.
\item The two properties above generally fail to hold for small-topological functors. However, for any functor $U$, a $U$-initial lifting of a $U$-structured cone that is a limit cone in $\CX$ gives also a limit cone in $\CA$.
\item Every small-topological functor is a fibration (consider singleton diagrams) and has a fully faithful right adjoint (consider the empty diagram).
\end{itemize}
\begin{prop} \label{QCat_topological}
For every (small) quantaloid $\CQ$, the ``object functor'' $\QCat\to\Set/\ob\CQ$ is topological.
\end{prop}
\begin{proof}
Given a (possibly large) family $f_i:(X,|\text{-}|)\to(Y_i,|\text{-}|_i)$ $(i\in I)$ of maps over $\ob\CQ$, where every $Y_i$ carries a $\CQ$-category structure $b_i$ with extent function $|\text{-}|_i$, we must find a $\CQ$-category structure $a$ on $X$ with extent function $|\text{-}|$ such that (1) every $f_i:(X,a)\to(Y,b)$ is a $\CQ$-functor, and (2) for every $\CQ$-category $(Z,c)$, any extent preserving map $g:Z\to X$ becomes a $\CQ$-functor $(Z,c)\to(X,a)$ whenever all maps $f_i g$ are $\CQ$-functors $(Z,c)\to(Y_i,b_i)$ $(i\in I)$. But this is easy: simply define
$$a(x,y):=\bw_{i\in I}b_i(f_i(x),f_i(y))$$
for all $x,y\in X$. Hence, $a$ is the $\ob$-initial structure on $X$ with respect to the structured source $(f_i:X\to(Y_i,b_i))_{i\in I}$.
\end{proof}
\begin{cor} \label{QCat_complete}
$\QCat$ is complete and cocomplete, and the object functor has both a fully faithful left adjoint and a fully faithful right adjoint.
\end{cor}
\begin{rem} \phantomsection \label{QCat_limit}
\begin{itemize}
\item[\rm (1)] The set of objects of the product $(X,a)$ of a small family of $\CQ$-categories $(X_i,a_i)$ $(i\in I)$ is given by the fibred product of $(X_i,|\text{-}|_i)$ $(i\in I)$, i.e.,
$$X=\{((x_i)_{i\in I},q)\mid q\in\ob\CQ,\ \forall i\in I(x_i\in X_i,|x_i|=q)\},$$
and (when writing $(x_i)_{i\in I}$ instead of $((x_i)_{i\in I},q)$ and putting $|(x_i)_{i\in I}|=q$) we have
$$a((x_i)_{i\in I},(y_i)_{i\in I})=\bw_{i\in I}a_i(x_i,y_i):|(x_i)_{i\in I}|\to|(y_i)_{i\in I}|$$
for its hom-arrows. In particular, $(\ob\CQ,\top)$ with
$$\top(q,r)=\top:q\to r$$
the top element in $\CQ(q,r)$ (for all $q,r\in\ob\CQ$), is the terminal object in $\QCat$.
\item[\rm (2)] The coproduct $(X,a)$ of $\CQ$-categories $(X_i,a_i)$ $(i\in I)$ is simply formed by the coproduct in $\Set$, with all structure to be obtained by restriction:
$$X=\coprod_{i\in I}X_i,\quad |x|_X=|x|_{X_i}\ \text{if}\ x\in X_i,\quad a(x,y)=\begin{cases}
a_i(x,y) & \text{if}\ x,y\in X_i,\\
\bot:|x|\to|y| & \text{else}.
\end{cases}$$
In particular, $\varnothing$ with its unique $\CQ$-category structure is an initial object in $\QCat$.
\item[\rm (3)] The equalizer of $\CQ$-functors $f,g:(X,a)\to(Y,b)$ is formed as in $\Set$, by restriction of the structure of $(X,a)$. The object set of their coequalizer $(Z,c)$ in $\QCat$ is also formed as in $\Set$, so that $Z=Y/\sim$, with $\sim$ the least equivalence relation on $Y$ with $f(x)\sim g(x)$, $x\in X$. With $\pi:Y\to Z$ the projection, necessarily $|\pi(y)|_Z=|y|_Y$, and $c(\pi(y),\pi(y'))$ is the join of all
$$b(y_n,y'_n)\circ b(y_{n-1},y'_{n-1})\circ\dots\circ b(y_2,y'_2)\circ b(y_1,y'_1),$$
where $|y|=|y_1|,|y'_1|=|y_2|,\dots,|y'_{n-1}|=|y_n|,|y'_n|=|y'|$ $(y_i,y'_i\in Y, n\geq 1)$.
\item[\rm (4)] The fully faithful left adjoint of $\QCat\to\Set/\ob\CQ$ provides a set $(X,|\text{-}|)$ over $\ob\CQ$ with the discrete $\CQ$-structure, given by
$$a(x,y)=\begin{cases}
1_{|x|} & \text{if}\ x=y,\\
\bot:|x|\to|y| & \text{else};
\end{cases}$$
while the fully faithful right adjoint always takes $\top:|x|\to|y|$ as the hom-arrow, i.e., it chooses the indiscrete $\CQ$-structure.
\end{itemize}
\end{rem}
\begin{exmp}
The product of partial metric spaces $(X_i,a_i)$ $(i\in I)$ provides its carrier set
$$X=\{((x_i)_{i\in I},s)\mid s\in[0,\infty],\ \forall i\in I(x_i\in X_i,|x_i|=s)\}$$
with the ``sup metric'':
$$a((x_i)_{i\in I},(y_i)_{i\in I})=\sup_{i\in I}a_i(x_i,y_i).$$
$[0,\infty]$ is terminal in $\ParMet$ when provided with the chaotic metric that makes all distances $0$, and it is a generator when provided with the discrete metric $d$:
$$d(s,t)=\begin{cases}
0 & \text{if}\ s=t,\\
\infty & \text{else}.
\end{cases}$$
\end{exmp}
Beyond small limits and colimits, $\QCat$ actually has all large-indexed limits and colimits that one can reasonably expect to exist. More precisely, recall that an ordinary category $\CC$ with small hom-sets is (see \cite{Borger1990})
\begin{itemize}
\item \emph{hypercomplete} if a diagram $D:\CJ\to\CC$ has a limit in $\CC$ whenever the limit of $\CC(A,D-)$ exists in $\Set$ for all $A\in\ob\CC$; equivalently: whenever, for every $A\in\ob\CC$, the cones $\De A\to D$ in $\CC$ may be labeled by a set;
\item \emph{totally cocomplete} if a diagram $D:\CJ\to\CC$ has a colimit in $\CC$ whenever the colimit of $\CC(A,D-)$ exists in $\Set$ for all $A\in\ob\CC$; equivalently: whenever, for every $A\in\ob\CC$, the connected components of $(A\da D)$ may be labelled by a set.
\end{itemize}
The dual notions are \emph{hypercocomplete} and \emph{totally complete}. It is well known (see \cite{Borger1990}) that
\begin{itemize}
\item $\CC$ is totally cocomplete if, and only if, $\CC$ is \emph{total}, i.e., if the Yoneda embedding $\CC\to\Set^{\CC^{\op}}$ has a left adjoint;
\item total cocompleteness implies hypercompleteness but not vice versa (with Ad{\'a}mek's monadic category over graphs \cite{Adamek1977} providing a counterexample);
\item for a \emph{solid} (=\emph{semi-topological} \cite{Tholen1979}) functor $\CA\to\CX$, if $\CX$ is hypercomplete or totally cocomplete, $\CA$ has the corresponding property \cite{Tholen1980};
\item in particular, every topological functor, every monadic functor over $\Set$, and every full reflective embedding is solid.
\end{itemize}
It is also useful for us to recall \cite[Corollary 3.5]{Borger1990}:
\begin{prop} \label{total_condition}
A cocomplete and cowellpowered category with small hom-sets and a generating set of objects is total.
\end{prop}
Since $\QCat$ is topological over $\Set/\ob\CQ$ which, as a complete, cocomplete, wellpowered and cowellpowered category with a generating and a cogenerating set, is totally complete and totally cocomplete, we conclude:
\begin{thm} \label{QCat_total}
$\QCat$ is totally complete and totally cocomplete and, in particular, hypercocomplete and hypercomplete.
\end{thm}
\begin{rem} \label{QCat_generator}
Of course, we may also apply Proposition \ref{total_condition} directly to obtain Theorem \ref{QCat_total} since the left adjoint of $\QCat\to\Set/\ob\CQ$ sends a generating set of $\Set/\ob\CQ$ to a generating set of $\QCat$, and the right adjoint has the dual property. Explicitly then, denoting for every $s\in\ob\CQ$ by $\{s\}$ the discrete $\CQ$-category whose only object has extent $s$, we obtain the generating set $\{\{s\}\mid s\in\ob\CQ\}$ for $\QCat$. Similarly, providing the disjoint unions $D_s=\{s\}+\ob\CQ$ $(s\in\ob\CQ)$ with the identical extent functions and the indiscrete $\CQ$-category structures, one obtains a cogenerating set in $\QCat$.
\end{rem}
\section{Limits and colimits in $\QChu$}
For $\CQ$-categories $X=(X,a)$, $Y=(Y,b)$, a \emph{$\CQ$-distributor} \cite{Benabou1973} $\phi:X\oto Y$ (also called \emph{$\CQ$-(bi)module} \cite{Lawvere1973}, \emph{$\CQ$-profunctor}) is a family of arrows $\phi(x,y):|x|\to|y|$ $(x\in X,y\in Y)$ in $\CQ$ such that
$$b(y,y')\circ\phi(x,y)\circ a(x',x)\leq\phi(x',y')$$
for all $x,x'\in X$, $y,y'\in Y$. Its composite with $\psi:Y\oto Z$ is given by
$$(\psi\circ\phi)(x,z)=\bv_{y\in Y}\psi(y,z)\circ\phi(x,y).$$
Since the structure $a$ of a $\CQ$-category $(X,a)$ is neutral with respect to this composition, we obtain the category
$$\QDis$$
of $\CQ$-categories and their $\CQ$-distributors which, with the local pointwise order
$$\phi\leq\phi'\iff\forall x,y:\ \phi(x,y)\leq\phi'(x,y),$$
is actually a quantaloid. Every $\CQ$-functor $f:X\to Y$ gives rise to the $\CQ$-distributors $f_{\nat}:X\oto Y$ and $f^{\nat}:Y\oto X$ with
$$f_{\nat}(x,y)=b(f(x),y)\quad\text{and}\quad f^{\nat}(y,x)=b(y,f(x))$$
$(x\in X,y\in Y)$. One has $f_{\nat}\dv f^{\nat}$ in the 2-category $\QDis$, and if one lets $\QCat$ inherit the order of $\QDis$ via
$$f\leq g\iff f^{\nat}\leq g^{\nat}\iff g_{\nat}\leq f_{\nat}\iff 1_{|x|}\leq b(f(x),g(x))\quad (x\in X),$$
then one obtains 2-functors
$$(-)_{\nat}:(\QCat)^{\co}\to\QDis,\quad (-)^{\nat}:(\QCat)^{\op}\to\QDis$$
which map objects identically; here ``$\op$'' refers to the dualization of 1-cells and ``$\co$'' to the dualization of 2-cells.
\begin{exmp}[See Example \ref{QCat_exmp}]
\begin{itemize}
\item[\rm (1)] A ${\bf 2}$-distributor is an \emph{order ideal relation}; that is, a relation $\phi:X\oto Y$ of ordered sets that behaves like a two-sided ideal w.r.t. the order:
$$x'\leq x\ \&\ x\phi y\ \&\ y\leq y'{}\Lra{}x'\phi y'.$$
A $[0,\infty]$-distributor $\phi:X\oto Y$ introduces a distance function between metric spaces $(X,a)$, $(Y,b)$ that must satisfy
$$\phi(x',y')\leq a(x',x)+\phi(x,y)+a(y,y')$$
for all $x,x'\in X$, $y,y'\in Y$.
\item[\rm (2)] A $\sD{\bf 2}$-distributor $\phi:X\oto Y$ is given by a ${\bf 2}$-distributor $A\oto B$ where $A=\{x\in X\mid x\leq x\}$, $B=\{y\in Y\mid y\leq y\}$ are the coreflections of $X$, $Y$, respectively. Likewise, a $\sD[0,\infty]$-distributor $\phi:X\oto Y$ is given by a distributor of the metric coreflections of the partial metric spaces $X$ and $Y$.
\end{itemize}
\end{exmp}
In our context $\QDis$ plays only an auxiliary role for us in setting up the category
$$\QChu$$
whose objects are $\CQ$-distributors and whose morphisms $(f,g):\phi\to\psi$ are given by $\CQ$-functors $f:(X,a)\to(Y,b)$, $g:(Z,c)\to(W,d)$ such that the diagram
\begin{equation} \label{Chu_transform_def_diagram}
\bfig
\square<700,500>[X`Y`W`Z;f_{\nat}`\phi`\psi`g^{\nat}]
\place(350,0)[\circ] \place(350,500)[\circ] \place(0,250)[\circ] \place(700,250)[\circ]
\efig
\end{equation}
commutes in $\QDis$:
\begin{equation} \label{Chu_transform_def}
\psi(f(x),z)=\phi(x,g(z))
\end{equation}
for all $x\in X$, $z\in Z$. In particular, with $\phi=a$, $\psi=b$ one obtains that the morphisms $(f,g):1_{(X,a)}\to 1_{(Y,b)}$ in $\QChu$ are precisely the adjunctions $f\dv g:(Y,b)\to(X,a)$ in the 2-category $\QCat$. With the order inherited from $\QCat$, $\QChu$ is in fact a 2-category, and one has 2-functors
\renewcommand\arraystretch{1.5}
$$\begin{array}{lll}
\dom: & \QChu\to\QCat, & (f,g)\mapsto f,\\
\cod: & \QChu\to(\QCat)^{\op}, & (f,g)\mapsto g.
\end{array}$$
In order for us to exhibit properties of $\QChu$, it is convenient to describe \emph{$\CQ$-Chu transforms}, i.e., morphisms in $\QChu$, alternatively, with the help of \emph{presheaves}, as follows. For every $s\in\ob\CQ$, let $\{s\}$ denote the discrete $\CQ$-category whose only object has extent $s$. For a $\CQ$-category $X=(X,a)$, a \emph{$\CQ$-presheaf $\phi$ on $X$} of extent $|\phi|=s$ is a $\CQ$-distributor $\phi:X\oto\{s\}$. Hence, $\phi$ is given by a family of $\CQ$-morphisms $\phi_x:|x|\to|\phi|$ $(x\in X)$ with $\phi_y\circ a(x,y)\leq\phi_x$ $(x,y\in X)$. With
$$[\phi,\psi]=\bw_{x\in X}\psi_x\lda\phi_x,$$
$\PX$ becomes a $\CQ$-category, and one has the \emph{Yoneda $\CQ$-functor}
$$\sy_X=\sy:X\to\PX,\quad x\mapsto(a(-,x):X\oto\{|x|\}).$$
$\sy$ is fully faithful, i.e., $[\sy(x),\sy(y)]=a(x,y)$ $(x,y\in X)$. The point of the formation of $\PX$ for us is as follows (see \cite{Heymans2010,Shen2015}):
\begin{prop} \label{cograph_Kan_adjunction}
The 2-functor $(-)^{\nat}:(\QCat)^{\op}\to\QDis$ has a left adjoint $\sP$ which maps a $\CQ$-distributor $\phi:X\oto Y$ to the $\CQ$-functor
$$\phi^*:\PY\to\PX,\quad\psi\mapsto\psi\circ\phi;$$
hence,
$$(\phi^*(\psi))_x=\bv_{y\in Y}\psi_y\circ\phi(x,y)$$
for all $\psi\in\PY$, $x\in X$. In particular, for a $\CQ$-functor $f:X\to Y$ one has
$$f^*:=(f_{\nat})^*:\PY\to\PX,\quad (f^*(\psi))_x=\psi_{f(x)}.$$
\end{prop}
Denoting by $\tphi:Y\to\PX$ the transpose of $\phi:X\oto Y$ under the adjunction, determined by $\tphi^{\nat}\circ(\sy_X)_{\nat}=\phi$, so that $(\tphi(y))_x=\phi(x,y)$ for all $x\in X$, $y\in Y$, we can now present $\CQ$-Chu transforms, as follows:
\begin{cor} \label{Chu_transform_presheaf}
A morphism $(f,g):\phi\to\psi$ in $\QChu$ (as in {\rm(\ref{Chu_transform_def_diagram})}) may be equivalently presented as a commutative diagram
\begin{equation}
\bfig
\square/<-`<-`<-`<-/<700,500>[\PX`\PY`W`Z;f^*`\tphi`\tpsi`g]
\efig
\end{equation}
in $\QCat$. Condition {\rm(\ref{Chu_transform_def})} then reads as
\begin{equation}
(\tphi(g(z)))_x=(\tpsi(z))_{f(x)}
\end{equation}
for all $x\in X$, $z\in Z$.
\end{cor}
\begin{proof}
For all $z\in Z$,
$$f^*(\tpsi(z))=\tpsi(z)\circ f_{\nat}=\sy_Z(z)\circ\psi\circ f_{\nat}=\sy_Z(z)\circ g^{\nat}\circ\phi=\sy_Y(g(z))\circ\phi=\tphi(g(z)).$$
\end{proof}
\begin{thm} \label{dom_initial_lifting}
Let $D:\CJ\to\QChu$ be a diagram such that the colimit $W=\colim\cod D$ exists in $\QCat$. Then any cone $\ga:\De X\to\dom D$ in $\QCat$ has a $\dom$-initial lifting $\Ga:\De\phi\to D$ in $\QChu$ with $\phi:X\oto W$, $\dom\Ga=\ga$. In particular, if $\ga$ is a limit cone in $\QCat$, $\Ga$ is a limit cone in $\QChu$.
\end{thm}
\begin{proof}
Considering the functors
$$\QChu\to^{\dom}\QCat\to^{(-)_{\nat}}\QDis,\quad\QChu\to^{\cod}(\QCat)^{\op}\to^{(-)^{\nat}}\QDis,$$
one has the natural transformation
$$\ka:(\dom(-))_{\nat}\oto(\cod(-))^{\nat},\quad\ka_{\phi}:=\phi\ (\phi\in\ob\QChu).$$
By the adjunction of Proposition \ref{cograph_Kan_adjunction}, $\ka D:(\dom D)_{\nat}\oto(\cod D)^{\nat}$ corresponds to a natural transformation $\tkD:\cod D\to\sP(\dom D)_{\nat}$, and the given cone $\ga$ gives a cocone $\ga^*:\sP(\dom D)_{\nat}\to\De\PX$. Forming the colimit cocone $\de:\cod D\to\De W$ one now obtains a unique $\CQ$-functor $\tphi:W\to\PX$ making
$$\bfig
\square/<-`<--`<-`<-/<1000,500>[\De\PX`\sP(\dom D)_{\nat}`\De W`\cod D;\ga^*`\De\tphi`\tkD`\de]
\efig$$
commute in $\QCat$ or, equivalently, making
$$\bfig
\square<1000,500>[\De X`(\dom D)_{\nat}`\De W`(\cod D)^{\nat};\ga_{\nat}`\De\phi`\ka D`\de^{\nat}]
\place(500,0)[\circ] \place(500,500)[\circ] \place(0,250)[\circ] \place(1000,250)[\circ]
\efig$$
commute in $\QDis$, with $\phi:X\oto W$ corresponding to $\widetilde{\phi}$. In other words, we have a cone $\Ga:\De\phi\to D$ with $\dom\phi=X$, $\dom\Ga=\ga$, namely $\Ga=(\ga,\de)$.
Given a cone $\Theta:\De\psi\to D$ with $\psi:Y\oto Z$ in $\QDis$ and a $\CQ$-functor $f:Y\to X$ with $\ga\cdot\De f=\ep:=\dom\Theta$, the cocone $\vartheta:=\cod\Theta:\cod D\to\De Z$ corresponds to a unique $\CQ$-functor $g:W\to Z$ with $\De g\cdot\de=\vartheta$ by the colimit property. As the diagram
$$\bfig
\square|arra|/<-`<-`<-`<-/<1200,600>[\De\PX`\sP(\dom D)_{\nat}`\De W`\cod D;\ga^*``\tkD`\de]
\place(70,350)[\mbox{\scriptsize$\De\tphi$}]
\morphism(0,600)<-500,-500>[\De\PX`\De\PY;]
\place(-300,400)[\mbox{\scriptsize$\De f^*$}]
\morphism(1200,600)|b|<-1700,-500>[\sP(\dom D)_{\nat}`\De\PY;\ep^*]
\morphism<-500,-500>[\De W`\De Z;]
\place(-300,-200)[\mbox{\scriptsize$\De g$}]
\morphism(1200,0)|b|<-1700,-500>[\cod D`\De Z;\vartheta]
\morphism(-500,-500)|l|<0,600>[\De Z`\De\PY;\De\tpsi]
\efig$$
shows, the colimit property of $W$ also guarantees $f^*\tphi=\tpsi g$ (with $\tpsi$ corresponding to $\psi$) which, by Corollary \ref{Chu_transform_presheaf}, means that $(f,g):\psi\to\phi$ is the only morphism in $\QChu$ with $\dom(f,g)=f$ and $\Ga\cdot\De(f,g)=\Theta$.
\end{proof}
\begin{cor}
$\dom:\QChu\to\QCat$ is small-topological; in particular, $\dom$ is a fibration with a fully faithful right adjoint which embeds $\QCat$ into $\QChu$ as a full reflective subcategory. $\cod:\QChu\to(\QCat)^{\op}$ has the dual properties.
\end{cor}
\begin{proof}
With the existence of small colimits guaranteed by Corollary \ref{QCat_complete}, $\dom$-initial liftings to small $\dom$-structured cones exist by Theorem \ref{dom_initial_lifting}.
For the assertion on $\cod$, first observe that every $\CQ$-category $X=(X,a)$ gives rise to the $\CQ^{\op}$-category $X^{\op}=(X,a^{\circ})$, where $a^{\circ}(x,y)=a(y,x)$ $(x,y\in X)$. With the commutative diagram
$$\bfig
\square<1200,500>[(\QChu)^{\op}`\CQ^{\op}\text{-}\Chu`\QCat`\CQ^{\op}\text{-}\Cat;(-)^{\op}`\cod^{\op}`\dom`(-)^{\op}]
\efig$$
one sees that, up to functorial isomorphisms, $\cod^{\op}:(\QChu)^{\op}\to\QCat$ coincides with the small-topological functor $\dom:\CQ^{\op}\text{-}\Chu\to\CQ^{\op}\text{-}\Cat$.
\end{proof}
\begin{cor} \label{QChu_complete}
$\QChu$ is complete and cocomplete, all small limits and colimits in $\QChu$ are preserved by both $\dom$ and $\cod$.
\end{cor}
\begin{proof}
The $\dom$-initial lifting of a $\dom$-structured limit cone in $\QCat$ is a limit cone in $\QChu$, which is trivially preserved. Having a right adjoint, $\dom$ also preserves all colimits.
\end{proof}
\begin{rem} \phantomsection
\begin{itemize}
\item[\rm (1)] Let us describe (small) products in $\QChu$ explicitly: Given a family of $\CQ$-distributors $\phi_i:X_i\oto Y_i$ $(i\in I)$, one first forms the product $X$ of the $\CQ$-categories $X_i=(X_i,a_i)$ as in Remark \ref{QCat_limit}(1) with projections $p_i$ and the coproduct of the $Y_i=(Y_i,b_i)$ as in Remark \ref{QCat_limit}(2) with injections $s_i$ $(i\in I)$. The transposes $\tphi_i$ then determines a $\CQ$-functor $\tphi$ making the left square of
$$\bfig
\square/<-`<-`<-`<-/<700,500>[\PX`\PX_i`Y`Y_i;p_i^*`\tphi`\tphi_i`s_i]
\square(1500,0)<700,500>[X`X_i`Y`Y_i;(p_i)_{\nat}`\phi`\phi_i`(s_i)^{\nat}]
\place(1500,250)[\circ] \place(2200,250)[\circ] \place(1850,0)[\circ] \place(1850,500)[\circ]
\efig$$
commutative, while the right square exhibits $\phi$ as a product of $(\phi_i)_{i\in I}$ in $\QChu$ with projections $(p_i,s_i)$, $(i\in I)$; explicitly,
$$\phi(x,y)=(\tphi(y))_x=(\tphi_i(y)\circ(p_i)_{\nat})_x=(\tphi_i(y))_{p_i(x)}=\phi_i(x_i,y)$$
for $x=((x_i)_{i\in I},q)$ in $X$ and $y=s_i(y)$ in $Y_i$, $i\in I$.
\item[\rm (2)] The coproduct of $\phi_i:X_i\oto Y_i$ $(i\in I)$ in $\QChu$ is formed like the product, except that the roles of domain and codomain need to be interchanged. Hence, one forms the coproduct $X$ of $(X_i)_{i\in I}$ and the product $Y$ of $(Y_i)_{i\in I}$ in $\QCat$ and obtains the coproduct $\phi:X\oto Y$ in $\QChu$ as in
$$\bfig
\square<700,500>[X_i`X`Y_i`Y;(s_i)_{\nat}`\phi_i`\phi`(\phi_i)^{\nat}]
\place(0,250)[\circ] \place(700,250)[\circ] \place(350,0)[\circ] \place(350,500)[\circ]
\efig$$
so that $\phi(x,y)=\phi_i(x,y_i)$ for $y=((y_i)_{i\in I},q)$ in $Y$ and $x=s_i(x)$ in $X_i$, $i\in I$.
\item The equalizer of $(f,g),(\of,\og):\phi\to\psi$ in $\QChu$ is obtained by forming the equalizer and coequalizer
$$U\to^i X\two^f_{\of}Y\quad\text{and}\quad W\two^g_{\og}Z\to^p V$$
in $\QCat$, respectively. With $\tchi:V\to\PU$ obtained from the coequalizer property making
$$\bfig
\square/<-`<-`<-`<-/<700,500>[\PU`\PX`V`W;i^*`\tchi`\tphi`p]
\efig$$
commutative, Theorem \ref{dom_initial_lifting} guarantees that
$$\chi\to^{(i,p)}\phi\two^{(f,g)}_{(\of,\og)}\psi$$
is an equalizer diagram in $\QChu$, where
$$\chi(x,p(w))=(\tchi(p(w))_x=(i^*(\tphi(w)))_x=(\tphi(w)\circ i_{\nat})_x=\phi(i(x),w)$$
for all $x\in U$, $w\in W$.
\item Coequalizers in $\QChu$ are formed like equalizers, except that the roles of domain and codomain need to be interchanged.
\end{itemize}
\end{rem}
We will now strengthen Corollary \ref{QChu_complete} and show total completeness and total cocompleteness of $\QChu$ with the help of Proposition \ref{total_condition}. To that end, let us observe that, since the limit and colimit preserving functors $\dom$ and $\cod$ must in particular preserve both monomorphisms and epimorphisms, a monomorphism $(f,g):\phi\to\psi$ in $\QChu$ must be given by a monomorphism $f$ and an epimorphism $g$ in $\QCat$, i.e., by an injective $\CQ$-functor $f$ and a surjective $\CQ$-functor $g$. Consequently, $\QChu$ is wellpowered, and so is its dual $(\QChu)^{\op}\cong\CQ^{\op}\text{-}\Chu$. The main point is therefore for us to prove:
\begin{thm} \label{QChu_generator}
$\QChu$ contains a generating set of objects and, consequently, also a cogenerating set.
\end{thm}
\begin{proof}
With the notations explained below, we show that
$$\{\eta_s:\varnothing\oto D_s\mid s\in\ob\CQ\}\cup\{\lam_t:\{t\}\oto\hC\mid t\in\ob\CQ\}$$
is generating in $\QChu$. Here $D_s$ belongs to a generating set of $\QCat$ (see Remark \ref{QCat_generator}), and
$$C=\coprod\limits_{t\in\ob\CQ}\sP\{t\}$$
is a coproduct in $\QCat$ (see Remark \ref{QCat_limit}(2)) of the presheaf $\CQ$-categories of the singleton $\CQ$-categories $\{t\}$ (see Proposition \ref{cograph_Kan_adjunction}). From $C$ one obtains $\hC$ by adding an isomorphic copy of each object in $C$, which may be easily explained for a $\CQ$-category $(X,a)$: simply provide the set $\hX:=X\times\{1,2\}$ with the structure
$$|(x,i)|_{\hX}=|x|_{X}\quad\text{and}\quad\ha((x,i),(y,j))=a(x,y)$$
for all $x,y\in X$, $i,j\in\{1,2\}$. Noting that the objects of $\sP\{t\}$ are simply $\CQ$-arrows with domain $t$, we now define $\lam_t:\{t\}\oto\hC$ by
$$\lam_t(u,i)=\begin{cases}
u & \text{if}\ \dom u=t,\\
\bot & \text{else}
\end{cases}$$
for $i\in\{1,2\}$ and every object $t$ and arrow $u$ in $\CQ$. For another element $(v,j)$ in $\hC$, if $\dom v=\dom u=t$ one then has
$$[(u,i),(v,j)]\circ\lam_t(u,i)=(v\lda u)\circ u\leq v=\lam_t(v,j),$$
and in other cases this inequality holds trivially. Hence, $\lam_t$ is indeed a $\CQ$-distributor.
Let us now consider $\CQ$-Chu transforms $(f,g)\neq(\of,\og):\phi\to\psi$ as in
$$\bfig
\square|alra|/@{->}@<3pt>`->`->`@{->}@<3pt>/<800,500>[(X,a)`(Y,b)`(W,d)`(Z,c);f_{\nat}`\phi`\psi`g^{\nat}]
\morphism(0,500)|b|/@{->}@<-3pt>/<800,0>[(X,a)`(Y,b);\of_{\nat}]
\morphism(0,0)|b|/@{->}@<-3pt>/<800,0>[(W,d)`(Z,c);\og^{\nat}]
\place(0,250)[\circ] \place(800,250)[\circ] \place(400,-30)[\circ] \place(400,30)[\circ] \place(400,470)[\circ] \place(400,530)[\circ]
\efig$$
{\bf Case 1}: $X=\varnothing$ is the initial object of $\QCat$ (and $\QDis$). Then $g\neq\og$, and we find $s\in\ob\CQ$ and $h:W\to D_s$ with $hg\neq h\og$ in $\QCat$. Consequently, $(1_{\varnothing},h):\eta_s\to\phi$ satisfies $(f,g)(1_{\varnothing},h)\neq(\of,\og)(1_{\varnothing},h)$.
{\bf Case 2}: $f\neq\of$, so that $f(x_0)\neq\of(x_0)$ for some $x_0\in X$. Then, for $t:=|x_0|$, $e:\{t\}\to X$, $|x_0|\mapsto x_0$, is a $\CQ$-functor with $fe\neq\of e$, and it suffices to show that
$$h:W\to\hC,\quad w\mapsto(\phi(x_0,w),1)$$
is a $\CQ$-functor making $(e,h):\lam_t\to\phi$ a $\CQ$-Chu transform. Indeed,
\begin{align*}
&d(w,w')\leq\phi(x_0,w')\lda\phi(x_0,w)=[h(w),h(w')],\\
&\lam_t(h(w))=\phi(x_0,w)=\phi(e(t),w)
\end{align*}
for all $w,w'\in W$.
{\bf Case 3}: $X\neq\varnothing$ and $g\neq\og$. Then $g(z_0)\neq\og(z_0)$ for some $z_0\in Z$, and with any fixed $x_0\in X$ we may alter the previous definition of $h:W\to\hC$ by
$$h(w):=\begin{cases}
(\phi(x_0,w),2) & \text{if}\ w=\og(z_0),\\
(\phi(x_0,w),1) & \text{else}.
\end{cases}$$
The verification for $h$ to be a $\CQ$-functor and $(e,h):\lam_t\to\phi$ a $\CQ$-Chu transform remain intact, and since $hg\neq h\og$, the proof is complete.
\end{proof}
\begin{rem}
A generating set in $\QChu$ may be alternatively given by
$$\{\lam_{\varnothing}:\varnothing\oto\hC\}\cup\{\lam_t:\{t\}\oto\hC\mid t\in\ob\CQ\},$$
so that in Case 1 one may proceed exactly as in Case 3 only by replacing $\phi(x_0,w)$ with $\top:q\to|w|$ for any fixed $q\in\ob\CQ$.
\end{rem}
With Theorem \ref{QChu_generator} we obtain:
\begin{cor}
$\QChu$ is totally complete and totally cocomplete and, in particular, hypercocomplete and hypercomplete.
\end{cor}
|
{
"timestamp": "2015-10-13T02:22:12",
"yymm": "1504",
"arxiv_id": "1504.03348",
"language": "en",
"url": "https://arxiv.org/abs/1504.03348"
}
|
\section{Introduction}
The study of the parsec-scale properties of radio sources is crucial to obtain information on the
nature of the central engine and provide the foundations for the current unified theories, which
suggest that the appearance of active galactic nuclei (AGN) depends strongly on orientation.
The standard unified scheme is nowadays generally accepted as a consequence of relativistic
beaming effects and obscuration related to the orientation with respect to the line of sight. However, many
recent findings are posing challenges for the standard unified scheme. The discovery of
quasars with a FR I morphology, BL Lac objects with broad emission lines and high radio power (in the
FR II range) suggests that we still have to improve our knowledge in this field. Recently,
Kharb et al. 2009 found that among MOJAVE Blazars many BL Lacs exhibit radio
power and kpc scale morphology typical of FR II sources, while a substantial number of quasars
show a radio power intermediate between FR I and FR II. Moreover, the Fermi satellite has detected
gamma-ray emission in FR I radio galaxies and in steep spectrum radio sources. The origin and
nature of this high frequency emission is still under discussion. At the same time, the number of
radio sources with a very low core radio power in complete samples is high, suggesting the presence
of both high and low activity regimes in nuclear sources, and the possibility of restarted activity.
However, the duty cycle, and the triggering mechanisms, are not yet understood.
To properly discuss these points and to improve our knowledge of AGN, observations of statistical properties a large sample of radio sources unbiased
with respect to orientation effects are
needed. Large surveys such as the Caltech-Jodrell Bank survey (e.g., Taylor et al. 1994) observed
sources selected at high frequency and therefore biased towards objects at small angles to the line
of sight. Large samples of well studied sources such as the MOJAVE sample collect only sources
with a bright radio core or gamma-ray emission, and are also likely biased to sources at small angles
with respect to the line of sight.
At present the largest complete sample with a large amount of available data on the kpc and
pc scale is the Bologna Complete Sample (BCS) which we are studying in detail over several years
(Giovannini et al. 2001, 2005, and Liuzzo et al. 2009a, 2009b). It consists of 94 sources selected at
low frequency including all the B2 and 3CR radio sources present with z < 0.1, regardless of the core
flux density. Therefore, it is a sample not affected by any selection effect on the jet velocity and orientation
with respect to the line of sight.
For all the galaxies in our sample good arcsecond scale radio maps, as well X-ray data and HST optical images are available, allowing the necessary the comparison between the parsec and kiloparsec-scale radio structure and multiband emission properties. Up to now, we published results obtained at parsec scale for 77 of the 94 sources using VLBI observations (Giovannini et al. 2001, 2005, and Liuzzo et al. 2009a, 2009b).
\section{New VLBI data}
To complete the parsec scale analysis of the BCS, we asked and obtained new multifrequency VLBI data of the 17 remaining sources with very faint radio core ($<$ 5 mJy in 5 GHz VLA data) not yet observed in radio band at this high resolution. Thanks to these data, we would like to clarify some points as: a) how many two-sided sources are present in this complete sample? Are there non-relativistic or only mildly relativistic parsec-scale jets? Is the jet
velocity related to the core radio power? b) how to explain the faint radio core with the extended kpc scale structure and LogP(tot,408MHz)$>$ 25.0 W/Hz
observed in 7/17 sources of these very faint objects? c) Could we give constraints on the duty cycle of radio loud AGN on the base of statistical considerations of the BCS? d) what can we understand of these faint peculiar sources compared to the high-power ones?\\
In particular, we asked and obtained phase referencing observations with: \\
1) {\bf VLBA at 5 GHz} ($\sim$ 2h on source at 512 Mbps) for homogeneity with observations of the others BCS sources which also show
that many objects have a nuclear emission self-absorbed at 5-8GHz.\\
2) {\bf EVN at 18 cm} ($\sim$ 2h on source at 1 Gbps) to compare with 5 GHz VLBA data and derive spectral index information for these sources showing a non dominant core emission. Moreover, the good (u,v) coverage of these data at the short baselines could allow in principle to map the possible intermediate scale structures poorly known in active radiogalaxies.\\
The achieved angular resolution in the final images is $\sim$ 15 $\times$ 5 mas$^{2}$ for EVN data while $\sim$ 3 $\times$ 2 mas$^{2}$ for VLBA data. The noise level is $\sim$ 0.03-0.12 mJy/beam for EVN data and $\sim$ 0.05-0.13 mJy/beam for VLBA data . The detection rate is 14/17 detected with S(tot) $<$ 4 mJy both at 1.7 GHz (EVN) and 5 GHz (VLBA). Most of these very faint sources are point-like, i.e. no jet clearly visible from modelfit results (e.g. see Fig.1). The core dominance (CD) distribution for the 17 very faint BCS sources shows that 12/17 radiogalaxies have CD between 0.25 and 1, while 5/17 objects have CD values $<$ 0.25 suggesting that nuclear variability and greater radio activity are present in the past.
\section{Preliminary results on the complete sample}
The new VLBI data of these 17 very faint objects allow to complete the pc-scale analysis of the BCS sample. In the following, we report our preliminary findings considering all the 94 sources of the sample.\\
a) The VLBI detection rate is high (93\%), even though we observed sources with an arcsecond core flux density as low as 5 mJy at 5 GHz. This result
confirms the presence of compact nuclei at the center of radio galaxies with very low power radio core.\\
b) The one-sided jet morphology is the predominant structure on the parsec-scale (in $\sim$ 80\% of sources).
This is in agreement with a random orientation of radio sources and a high jet velocity ($\beta\sim$ 0.9). No intrinsic difference in jet velocity and morphology has been found between high and
low power radio galaxies.\\
c) With very few exceptions, the parsec and kiloparsec-scale radio structures are aligned confirming that the large bends present in some BL Lacs are likely amplified by the small jet
orientation angle with respect to the line-of-sight. In sources with aligned pc and kpc scale structure,
the main jet is always on the same side with respect to the nuclear emission.\\
d) We find two sources with a Z-shaped structure on the pc-scale (4c26.42, Liuzzo et al. 2009a, and 3C310, Liuzzo et al. 2009b) suggesting the presence of low velocity jets in these objects.\\
e) In $\sim$ 40\% of the sources, there is evidence of nuclear variability and/or of a significant
sub-kpc-scale structure which will be better investigated with the EVLA at high frequency or with
the E-MERLIN array. \\
f) $\sim$3\% sources present very high core dominance values ($>$10) which suggest the presence of recurrent or re-starting activity. \\
g) In 20/94 objects (mainly among these 17 recently observed very faint BCS sources), there is evidence of low activity state, but not always completely quiescent. These sources show in fact low core
dominance values ($<$0.25) but the presence of pc-scale core and in some cases even the jet.
\section{Future work}
The analysis of these new VLBI data and the statistical study of the BCS are still in progress. In particular, we have to deeply investigate our preliminary results to properly address the points discussed in Sect. 2. Our final aim is to discuss more in general the BCS nuclear multiband properties in comparison with those of low-z BL Lac objects (Giovannini et al. 2014, Liuzzo et al. 2013) for a better understanding of unified models of radio loud AGN.\\
{\bf Acknowledgments:} We sincerely thank the organizers for this very interesting meeting.
\\\\
{\bf References.}\\
Giovannini G. et al. 2001: ApJ 552, 508 \\
Giovannini G. et al. 2005: ApJ 618, 635 \\
Giovannini G. et al. 2014: 2014cosp...40E.998G \\
Kharb P. et al. 2010: ApJ 710, 764K\\
Liuzzo E. et al. 2009a: A\&A 501, 933\\
Liuzzo E. et al. 2009b: A\&A 505, 509\\
Liuzzo et al. 2013: A\&A 560, 23L\\
Taylor, G.B. et al. 1994: ApJSS 95, 345.
\begin{figure}
\centering
\includegraphics[width=16cm, angle=0]{fig1.eps}
\caption{\footnotesize {18 cm EVN and 6 cm VLBA images of 3 of the faintest BCS sources recently observed.}}
\label{fig_images}
\end{figure}
\end{document}
|
{
"timestamp": "2015-04-14T02:16:01",
"yymm": "1504",
"arxiv_id": "1504.03235",
"language": "en",
"url": "https://arxiv.org/abs/1504.03235"
}
|
\section{ INTRODUCTION }
Since magnetic reconnection processes are believed to be the primary
source of energy for producing solar flares, filament eruptions,
and coronal mass ejections (e.g., Priest 1982; 2014),
the measurement of the dissipated
magnetic energy provides a key parameter in the understanding of
the underlying physics. The dissipated magnetic energy is thought to
represent an absolute upper limit to all secondary energy conversions,
such as thermal, nonthermal, radiative, and kinetic energies. The
measurement of the dissipated magnetic energy requires a reliable method
of calculating the evolution of the nonpotential magnetic field
during a flare. The difference between the
nonpotential $E_{np}(t)$ and the potential energy $E_p(t)$
is the maximum free energy, i.e., $E_{free}(t)=E_{np}(t)-E_{p}(t)$, which
provides an upper limit on the total dissipated energy in a flare.
There are two fundamentally different methods to calculate the
nonpotential energy: (i) using a nonlinear force-free field (NLFFF)
code that extrapolates from the 3D vector magnetic field at the
photospheric boundary (which we call the PHOT-NLFFF method;
e.g., Wiegelmann 2004), and
(ii) by forward-fitting of a NLFFF approximation
to the observed geometry of coronal loops, using a line-of-sight (LOS)
magnetogram to constrain the potential field (which we call the
COR-NLFFF method; Aschwanden 2013a).
For the first method exist about a dozen of
NLFFF codes, which have been compared and showed a large scatter
of the free energy (Schrijver et al.~2006, 2008). Moreover, the most
severe problem of PHOT-NLFFF codes is their underlying assumption that
the photospheric boundary is force-free (DeRosa et al.~2009),
although attempts have been made to bootstrap the force-freeness
of the photospheric boundary by a ``pre-processing method''
(Wiegelmann et al.~2006). The question arised:
{\sl Can we improve the pre-processing of photospheric vector
magnetograms by the inclusion of chromospheric observations?}
(Wiegelmann et al.~2008). In contrast, the COR-NLFFF
code circumvents the non-force-free photosphere by fitting a
quasi-force-free solution to loops in force-free regions of the corona.
Using this second (COR-NLFFF) method, the dissipated magnetic energies
could be determined for 172 major (GOES M and X-class) flare events,
yielding dissipated energies that amount to a fraction $E_{diss}/E_p
\approx 1\%-25\%$ of the potential energy, where the potential
field covers a range of $E_p \approx 10^{31}-10^{33}$ erg for large
(M and X-class) flares (Aschwanden, Xu, and Jing 2014;
Emslie et al.~2012).
The accuracy in the calculation of free energy crucially depends
on the force-freeness of the boundary field or fitted loops.
While the solar corona is believed to be force-free in most places,
major parts of the transition region,
the chromosphere, and the photosphere are dominated by regions with
a high plasma-$\beta$ (i.e., the ratio of the thermal to magnetic energy)
that is larger than unity (e.g., Gary et al.~2001), which can enable
cross-field electric currents that disturb the force-freeness
condition. Measurements of
the chromospheric vector field and application of the virial theorem
demonstrated that the photosphere and lower chromosphere is not
force-free, while it becomes force-free at an altitude of
$h \lower.4ex\hbox{$\;\buildrel >\over{\scriptstyle\sim}\;$} 400$ km (Metcalf et al.~1995). Attempts to improve the
accuracy of NLFFF solutions have been made by using H$\alpha$ observations
(Wiegelmann et al.~2008), which outline loop-like or ribbon-like structures
in the chromosphere. Here we apply the COR-NLFFF
method to images obtained in coronal EUV wavelengths (with AIA),
as well as (for the first time) to images obtained in the chromosphere and
transition region in UV wavelengths (with IRIS and AIA), and we compare the
evolution of the free energy inferred in both height regimes.
\section{ OBSERVATIONS AND MEASUREMENTS }
\subsection{ AIA, HMI, and IRIS Observations }
We perform modeling of the nonpotential magnetic field for the
flare on 2014 March 29, 17:35-17:54 UT, classified as a GOES
X1.0-class event, which occurred in the NOAA active region 12017,
located at heliographic position N11W32. It was the first X-class
flare observed by IRIS and has been declared as the
{\sl ``best-ever observed flare''} (NASA press release of 2014 May 7).
Recent studies on this flare deal with the origin of a sunquake
(Judge et al.~2014), the hydrogen Balmer continuum emission during
the flare (Heinzel and Kleint 2014), and spectroscopy at subarcsecond
resolution (Young et al.~2015).
We are using images from the {\sl Atmospheric
Imager Assembly (AIA)} (Lemen et al.~2012) onboard the {\sl Solar
Dynamics Observatory (SDO)} (Pesnell et al.~2011), and slit-jaw
images (SJI) from the {\sl Interface Region Imaging Spectrograph
(IRIS)} (De Pontieu et al.~2014). A list of the analyzed
wavelengths is given in Table 1, which we group into three
wavelength sets used in three independent data analysis runs:
(i) IRIS-UV with wavelengths 1400 and 2796 \AA\
that probe the chromosphere and transition region,
(ii) AIA-UV wavelengths 304 and
1600 \AA\ that probe the transition region also, and (iii) AIA-EUV
wavelengths 94, 131, 171, 193, 211, 335 \AA , that probe the corona.
In addition we use magnetograms from the {\sl Helioseismic and
Magnetic Imager (HMI)} (Scherrer et al.~2012) onboard SDO. For our
analysis we use a cadence of 3 minute in all wavelengths, which
yields 26 time steps during an interval that covers the entire
flare duration plus 0.5 hrs margins before and after (i.e.,
17:05-18:24 UT). The AIA images have a pixel size of $0.6\hbox{$^{\prime\prime}$}$,
the HMI magnetograms have a pixel size of $0.5\hbox{$^{\prime\prime}$}$, and
IRIS SJI images have a pixel size of $0.166\hbox{$^{\prime\prime}$}$.
For the field-of-view of the analyzed subimages we use a
square with a length of 0.2 solar radii, centered at location
N11W32, adjusted to solar rotation tracking during the observed
time interval.
\subsection{ Data Analysis Method }
We determine the evolution of the free energy $E_{free}(t)$ during
the flare episode by applying the COR-NLFFF code, for which
the theoretical framework is given in Aschwanden (2013a), numerical
tests in Aschwanden and Malanushenko (2013), and the determination
of the free energy in Aschwanden (2013b). The theoretical concept
of the COR-NLFFF code is based on an analytical NLFFF solution of
buried magnetic charges, each one having a variable vertical current
and an associated helical twist of the field lines. The analytical
solution is divergence-free and force-free with an accuracy of
second order of the azimuthal (non-potential) magnetic field component
${\bf B}_{\varphi}({\bf x})$.
The performance of the COR-NLFFF code includes three tasks:
(1) A decomposition of the LOS component of the HMI magnetogram into
a finite number of magnetic charges that yield the potential
field solution ${\bf B}_p({\bf x})$ of an active region,
(ii) automated loop tracing in AIA and IRIS images with the
OCCULT-2 code (Aschwanden et al.~2008, 2013b; Aschwanden 2010),
separately executed for wavelength sets with
coronal and transition-region temperatures,
and (iii) forward-fitting of the non-potential (force-free)
$\alpha$-parameters to the coronal magnetic field by optimizing
the 2D-misalignment angles between the theoretical model of the
nonpotential field ${\bf B}_{np}({\bf x})$
and the observed geometry of coronal loops. The NLFFF fitting procedure
follows closely the code version applied in the most recent statistical
study of 172 major flares (Aschwanden et al.~2014), while the
application to loop structures observed in the transition region,
imaged by IRIS and AIA, represents a new experimental step explored
for the first time here.
The standard control parameters of the COR-NLFFF code are given
in Table 2 of Aschwanden et al.~(2014). In the present experiment
we used slightly different settings to optimize the results
obtained from the IRIS images, in the sense of maximizing the
number of detected structures and minimizing the number of non-loop
features:
curvature radius $r_{min}=15$ (or 10) pixels for EUV (or UV) images,
loop length limit $l_{min}=r_{min}$,
no gaps in the loop structures $n_{gap}=0$,
flux profile rippledness $q_{ripple} \le 0.4, 0.6, 0.8$,
flux threshold $q_{thr} \ge 3.0$ standard deviations,
number of magnetic sources $n_{mag}=100$,
proximity for the separation of loop footpoints $d_{prox} \le 3$ FWHM,
number of iterations $10 \le n_{iter} \ge 20$,
limit of loop structures per wavelength $n_{loop} \le 200$,
number of loop segments $n_{seg}=7$,
and maximum altitude of $h=0.05$ (0.20) solar radii for IRIS (AIA).
Differences between AIA and IRIS images are mostly the spatial
resolution, the flux contrast, and the morphology of wavelength-specific
features.
\section{ RESULTS }
A snapshot of the results of the magnetic modeling of the 2014 Mar 29
flare is shown in Figs.~1 and 2, at 17:53 UT, at the
time of the flare peak. Original images
of the flare region are shown on a logarithmic flux scale in Fig.~1
(left), juxtaposed to the highpass-filtered images (Fig.~1, right)
that have been used for automated loop detection (red curves in
Fig.~1 right panels). Fig.~2 shows an AIA EUV image sensitive to
coronal temperatures (211 \AA\ ), an AIA 304 \AA\ (Fig.~2, top left)
and an IRIS SJI image at 2796 \AA\ (Fig.~2 middle left), both being
sensitive to chromospheric and transition region temperatures. The
loop structures traced in these images, found with the automated
pattern recognition code OCCULT-2, are indicated in red. The structures
detected in IRIS images are generally shorter segments than those
detected in AIA images, which is partly due to a different morphology,
and partly due to the 4 times higher spatial resolution of IRIS.
The geometric 2D coordinates of the automated
loop tracings constrain the best fit of the NLFFF solution,
which is shown together with the HMI magnetogram
in Fig.~2 (middle right panel), (from which the NLFFF code uses
only the LOS component). We see that the active region contains
closed loops (blue curves) in the eastern dipolar part, while there
are mostly open field lines in the western dipolar spot group,
except for a small bipolar arcade in the core, where most of the
flare action occurs. Integrating the free magnetic energy in every
LOS, we created a free energy map (Fig.~2 bottom right),
which reveals that the largest amount of free energy is concentrated
in a semi-circular configuration surrounding the penumbra of the main
western sunspot. We show the 50\% contours of the free
energy map $E_{free}(x,y,t)$ at 17:53 UT, shortly after the flare peak,
overlaid on the magnetogram (Fig.~2 middle right panel, red contour),
which is essentially identical with
the location of the dissipated magnetic energy (i.e., the difference
of the free energy between flare start and end). The eastern part
of the semi-circular energy dissipation structure is cospatial with
the location where 30-70 keV non-thermal hard X-ray emission was
detected with RHESSI, as well as enhanced (hydrogen Balmer) white
light continuum with IRIS 1400 \AA\ (see Fig.~1 of Heinzel and
Kleint 2014). In any case, the spatial map of the free energy
localizes the footpoint areas of the flare loops that undergo
magnetic reconnection. We find essentially identical free energy
maps for coronal (AIA EUV wavelengths) and chromospheric or
transition region (AIA UV and IRIS wavelengths) loop tracers,
which implies that the magnetic information of the nonpotential
field $B_{np}(x,y,t)$ and free energy $E_{free}(x,y,t)$
is imprinted in field-aligned structures that can be observed
in the chromosphere, the transition region, and in the corona.
A new result of this study is that we extended nonpotential
magnetic modeling from the corona down to the transition region
and chromosphere, by calculating NLFFF solutions from three
different wavelength sets, i.e., AIA-EUV, AIA-UV, and IRIS-UV.
We show the resulting evolution of the magnetic energies calculated
from these three wavelength sets in Fig.~3, performed in time steps
of $\Delta t=0.05$ hrs (3 min) during the flare time interval
(with 0.5 hrs margin before and after). Error bars of the free
energy measurements are estimated from 3 different loop selection
criteria (defined by the level of flux variability along each
detected loop segment, $q_{ripple}=0.4, 0.6, 0.8$).
We find that the free energy peaks at a value of
$E_{free} \approx (45 \pm 2) \times 10^{30}$ erg
for all three wavelength sets. All three evolutions of free
energies peak consistently within a few minutes after the
GOES-based flare peak time (Fig.~3, bottom panel). Both the
AIA-EUV and AIA-UV wavelength sets exhibit a step-wise drop
of the free energy within 10-20 min after the flare peak, while
this step could be verified in the IRIS data partially only, due
to an observing sequence that unfortunately ends at 17:54:19 UT.
Nevertheless, both the coronal and the chromospheric
data exhibit a consistent drop in free energy, by an average
amount of $\Delta E_{free} \approx (29 \pm 3) \times 10^{30}$ erg,
which indicates that the helically twisted magnetic field lines
relax to a state of lower twist in both the corona and the
chromosphere, because the COR-NLFFF solution is most sensitive
to vertical currents and the associated helical twist around
vertical twist axes. This is the first quantitative measurement
that demonstrates in both the chromosphere and corona that the
dissipated magnetic energy in a flare is caused by untwisting of
(helically-twisted) stressed field lines.
\section{ DISCUSSION }
\subsection{ Chromospheric Magnetic Field Tracers }
Coronal loops are most conspicuously seen in EUV wavelength
images with a temperature sensitivity in the $T_e \approx 1-2$ MK
temperature range, such as in the Fe IX (171 \AA ) and in the
Fe XII (193 \AA ) line, and to a lesser extent in the weaker
Fe XIV (211 \AA ) and Fe XVI (355 \AA ) lines (Table 1),
which all are most useful in constraining theoretical magnetic field
models, such as calculated with the COR-NLFFF code (Aschwanden 2013a)
or with a Grad-Rubin NLFFF method (Malanushenko et al.~2014).
In the temperature regime of the photosphere, chromosphere, and
transition region, which ranges from $T_e=5000$ K to $T_e \lower.4ex\hbox{$\;\buildrel <\over{\scriptstyle\sim}\;$}
1$ MK, we expect to see cooler loops or the footpoints of hotter
coronal loops, which manifest themselves as a reticulated ``moss''
structure (Berger et al.~1999; De Pontieu et al.~1999).
These chromospheric structures appear
to be more irregular and fragmented than the smooth curvi-linear loop
structures seen in the corona. Experiments to use the directions
of H$\alpha$ fibrils to constrain NLFFF solutions have been undertaken
by Wiegelmann et al.~(2008), a method that became to be known as
{\sl pre-processing}, which supposedly makes the photospheric 3D
vector field more force-free and provides then a more suitable
near force-free boundary for NLFFF extrapolation methods.
Due to regions with high plasma $\beta$-parameters and
non-magnetic forces in the chromosphere, one cannot expect to model
the chromospheric field correctly under the force-free assumption,
neither with PHOT-NLFFF nor with COR-NLFFF codes.
Alternatively, we explore here for the first time the method of
using (automatically detected) chromospheric structures to measure
the local directivity of the magnetic field, which is then used
to constrain a nonpotential magnetic field (COR-NLFFF) solution in
an active region during a major flare. We use UV images in several
wavelengths, such as the He II (304 \AA ) and C IV (1600 \AA ) lines
from AIA/SDO, and the Mg II (2796 \AA ) and
Si IV (1400 \AA ) lines from IRIS, which all exhibit some segments of
loop-like features in the chromosphere and transition
region, and thus may contain some information on the local magnetic field
direction. Since the OCCULT-2 code is designed to detect
curvi-linear loop structures with large curvature radii, the
more irregular and inhomogeneous structures detected in the
transition region and chromosphere are more challenging for
automated detection of magnetic field directions, and thus
it is not clear how useful they are for magnetic field modeling.
However, our experiment demostrates that we detect a consistent
evolution of the free energy $E_{free}(t)$ during the investigated
flare in both coronal EUV and chromospheric UV wavelengths
(Fig.~3), within the uncertainties of the measurements. The biggest
challenge is to distinguish between contiguous field-aligned structures
(segments of loops or filaments), curvi-linear aggregations
of ``moss''-like structures, and curved flare ribbons.
\subsection{ Energy Dissipation in Flares }
The coronal and/or chromospheric feature tracing provides field
directions, which can be used to calculate the nonpotential magnetic
field ${\bf B}({\bf x})$ in a computation box that encompasses an
active region or flare region. The volume integral of the free
energy, $E_{free}(t) = (1/8\pi ) \int B_{np}(x,y,t)^2-B_p(x,y,t)^2
\ dx \ dy$, is ideally expected to have a higher level of free energy
before the flare, and to drop to a lower level after the flare
(e.g., Jing et al.~2009). We indeed find
a step-wise decrease of the free energy after the flare, which defines
the total dissipated magnetic energy during the flare, and is found
to have a value of $\Delta E_{free} = E_{free}(t_{start})-E_{free}(t_{end})
\approx (29 \pm 3) \times 10^{30}$ erg here. This value is indeed typical for an
X-class flare (Aschwanden et al.~2014). The apparent increase
of the free energy before the flare peak time has been interpreted as
an illumination effect, caused by chromospheric evaporation that fills up
flare loops, whereby their helical twist becomes visible
(Aschwanden et al.~2014). The dissipated magnetic energy is an
important physical parameter for flare models, because it sets rigorous
upper limits on other secondary flare energy conversions
(thermal and nonthermal energy, kinetic energy of CMEs, etc.), it
constrains the number problem for particle acceleration, and
constrains physical scaling laws for magnetic reconnection processes.
It is therefore imperative to establish a reliable method for the
determination of nonpotential magnetic energies. A comparison
of potential, nonpotential, and free energies between PHOT-NLFFF
and COR-NLFFF codes has been conducted in a recent statistical study,
where it was found that the dissipated flare energies determined with
PHOT-NLFFF and COR-NLFFF methods disagree up to a factor of 10
(Aschwanden et al.~2014). At this point it is not clear what
the largest source of uncertainties is. It is possible that the
pre-processing of photospheric vector field data in the PHOT-NLFFF
method introduces an over-smoothing and leads to an underestimation
of magnetic energies (Sun et al.~2012). Alternatively, mis-detections
or sparseness of field-aligned structures with the OCCULT-2 code could
spoil the optimization of force-free $\alpha$-parameters in the
COR-NLFFF code.
However, the consistent result of the evolution of the free energy
found in this study from both coronal and chromospheric images yields
an independent test and corroboration of the COR-NLFFF method.
\section{ CONCLUSIONS }
We calculated the time evolution of the free energy $E_{free}(t)$
during the 2014-Mar-29 flare, the first X-class flare detected by IRIS.
We used HMI/SDO data to compute the potential field,
and AIA/SDO and IRIS images to delineate the geometry of coronal loop
segments in EUV wavelengths, as well as to delineate magnetic field
directions from automatically traced structures seen in the transition
region and chromosphere in UV wavelengths.
The major result of this study is that we find a similar
evolution of the free energy for both coronal and chromospheric
structures, peaking at a free energy of $E_{free}(t_{peak})
\approx (45 \pm 2) \times 10^{30}$ erg, which decreases by an amount of
$\Delta E_{free} \approx (29 \pm 3) \times 10^{30}$ erg in the flare
decay phase, and thus represents the total magnetic energy dissipated
during this X-class flare. The consistency of magnetic energy measurements
from both coronal and chromospheric tracers represents an independent
test and corroboration of the COR-NLFFF method. The COR-NLFFF code
provides also maps of dissipated flare energies, which enables us
to localize and map out magnetic reconnection regions during solar
flares in great detail, which is the subject of future studies.
\bigskip
\acknowledgements
We acknowledge helpful comments from the referee, the LMSAL Team,
and guidance for using IRIS data by Bart De Pontieu.
Part of the work was supported by the
NASA contracts NNG04EA00C of the SDO/AIA instrument and
NNG09FA40C of the IRIS mission.
|
{
"timestamp": "2015-04-14T02:17:54",
"yymm": "1504",
"arxiv_id": "1504.03301",
"language": "en",
"url": "https://arxiv.org/abs/1504.03301"
}
|
\section{Introduction}
Let $N>1$ be an integer, $A_1,\ldots,A_N,B$ real valued
random variables such that $A_i$ are independent and identically
distributed (i.i.d.). On the set $P (\R)$ of
probability measures on the real line the smoothing transform is
defined as follows
\[
\mu\mapsto{\mathcal L} \Biggl(\sum_{i=1}^NA_iR_i
+ B \Biggr),
\]
where $R_1,\ldots,R_N$, are i.i.d. random variables with common
distribution $\mu$, independent of $(B, A_1,\ldots, A_N)$ and
${\mathcal L} (R)$ denotes the law of the random variable $R$. A
fixed point of the smoothing transform is given by any $\mu\in P
(\R)$ such that, if $R$ has distribution $\mu$, the equation
\begin{equation}
\label{equation} R=_d \sum_{i=1}^N
A_i R_i + B,
\end{equation}
holds true. We are going to distinguish between the case of $B=0$ a.s.
(the homogeneous smoothing transform) and the
other one called the nonhomogeneous smoothing transform.
The homogeneous equation \eqref{equation} is used for example, to study
interacting particle systems \cite{DL} or the branching random walk
\cite{HS,AR}.
In recent years, from practical reasons, the inhomogeneous
equation has gained importance. It appears for example, in the stochastic
analysis of the Pagerank algorithm (which is the heart of the
Google engine) \cite{JC1,JC2,VL} as well as in the analysis of a
large class of divide and conquer algorithms including the
Quicksort algorithm \cite{NR,R}. Both the homogeneous and the
inhomogeneous equation were recently used to describe equilibrium
distribution of a class of kinetic models and used for example, to
study the distribution of particle velocity in Maxwell gas (see, e.g.,
\cite{bassetti}).
Properties of the fixed points of equation \eqref{equation} are
governed by the function
\[
m(s) = \E \Biggl[ \sum_{i=1}^N
|A_i|^s \Biggr] = N \E\bigl[ |A_1|^s
\bigr].
\]
Suppose that $s_1 = \sup\{s\dvt m(s)<\infty \}$ is strictly positive.
Clearly $m$ is convex and differentiable on $(0,s_1)$. We
assume that there are $0<\g<\a<s_1$ such that
\[
m(\g)=m (\a)=1.
\]
Then
\[
0<m'(\a)=\E \Biggl[\sum_{i=1}^N
|A_i|^{\a}\log|A_i| \Biggr]
\]
and the latter
quantity is finite. The main result of this paper is
the following theorem.
\begin{theorem}
\label{mthm} Suppose that
\begin{itemize}[$\bullet$]
\item[$\bullet$]$\log|A_1|$ is nonlattice;
\item[$\bullet$]$\P[A_1 > 0] > 0$ and $\P[A_1<0]>0$;
\item[$\bullet$]$s_1>0$;
\item[$\bullet$] there are $0<\g<\a<s_1$ such that $m(\g)=m (\a)=1$;
\item[$\bullet$] there is $\eps>0$ such that $\E|B|^{\g+\eps}<\infty $.
\end{itemize}
Suppose that $R$ is a nontrivial solution to \eqref{equation}
such that $\E|R|^{\g+\eps}<\infty $. Then
\[
\liminf_{t\to\infty } t^\a\P[R>t] >0 \quad \mbox{and}\quad
\liminf_{t\to\infty } t^\a\P[R<-t] >0.
\]
\end{theorem}
\begin{rem}
Under the assumptions of Theorem~\ref{mthm} the random variable $R$ is
real valued and it attains both positive and negative values. If $\P
[A_1>0] = \P[B>0]=1$ then $R$ is a positive random variable and
exactly the same proof shows that
\[
\liminf_{t\to\infty } t^\a\P[R>t] >0.
\]
\end{rem}
Existence of such a solution implies $\g<2$
for the nonhomogeneous case and $1\leq\g<2$ for the homogeneous
one (see \cite{ADM}). Then the solution is basically unique (given the
mean of it
exists) and, if $\E|B| ^{\a}<\infty $ then for every $s<\a$
\begin{equation}
\label{moment} \E|R|^s <\infty .
\end{equation}
In view of the result of Jelenkovic
and Olvera-Cravioto (Theorem~4.6 in \cite{JC3}), Theorem~\ref{mthm}
implies.
\begin{cor}
Suppose that the assumptions of Theorem~\ref{mthm} are satisfied
and additionally let $\E|B|^{\a}<\infty $.
Then
\begin{equation}
\label{asym} \lim_{t\to\infty } t^\a\P[R>t]=\lim
_{t\to\infty } t^\a\P [R<-t]=K >0.
\end{equation}
\end{cor}
The existence of the limit in \eqref{asym} for such $R$, in a more
general case
of random $N$, was proved by Jelenkovic and Olvera-Cravioto
\cite{JC3}, Theorem~4.6, but from the expression for $K$, given by
their renewal theorem, it is not possible to conclude its strict
positivity except of the very particular case when $A_1,\ldots, A_N,B$ are
positive and $\a
\geq1$. There are other solutions to \eqref{equation} than those
mentioned in the above corollary. For the full description of them
see \cite{ABM,AM,AM2}. Clearly, Theorem~\ref{mthm} matters only
for solutions satisfying \eqref{moment}.
Some partial results concerning positivity of $K$ are contained in
\cite{BDMM} and \cite{ADM}. The paper \cite{BDMM} deals with
matrices but Theorem~2.12 and Proposition~2.13 there can be
specified to our case. Under additional assumption that $\E
|B|^{s_0}<\infty $ for some $\a<s_0<s_1$ they say that either $K>0$ or
$\E|R|^s<\infty $ for all $s<s_0$. If $R$ is not constant, the latter
is not possible when there is $\b\leq s_0$ such that $\E
|A_1|^{\b}=1$. Indeed, then $R$ becomes the solution of
\[
R=AR+Q
\]
with $Q=\sum_{i=2}^NA_iR_i+B$ and the conclusion of
Goldie's theorem \cite{Go} would be violated. It is interesting that
for the
asymptotics in \eqref{asym} in the case of $N$ being constant the
implicit renewal theorem of Jelenkovic and Olvera-Cravioto is not
needed. The usual one on $R$ is sufficient \cite{BDMM}, Theorem~2.8. For positivity of $K$ in the general case of random $N$ see
\cite{ADM}, Theorem~9.
Clearly, Theorem~\ref{mthm} improves considerably the results of
\cite{BDMM} specialised to the one dimensional case. Also, the technique
is purely probabilistic while in \cite{BDMM} holomorphicity of $\E
|R|^z $ and the Landau theorem is used.
Let $\mu_A$ be the law of $A_i$. In Section~\ref{sec2}, we show some
necessary properties of the random walks with the transition
probability $\mu_A$. A version of the Bahadur, Rao theorem (\cite
{DZ}, Theorem
3.7.4) is needed and its proof is included in the \hyperref[appendix]{Appendix}.
Section~\ref{sec3} is devoted to the proof of Theorem~\ref{mthm}.
\section{Random walk generated by the measure \texorpdfstring{$\mu_A$}{$mu_A$}}\label{sec2}
In this section, we will study
properties of the random walk $\{|\wt A_1\cdots \wt A_n|\}_{n\in\N}$,
where $\wt A_i$ are independent and distributed according to the
measure $\mu_A$ (it is convenient for our purpose to use the
multiplicative notation). Since $\E\log|\wt A_1|<0$, by the
strong law of large numbers, this random walk converges to 0 a.s.
Nevertheless, our aim is to describe a sufficiently large set on
which trajectories of the process exceed an arbitrary large, but
fixed number $t$. Given $n$, one can prove that the probability of the
event $ \{|\wt A_1\cdots \wt A_n|>t \}$ is largest when $n$ is
comparable with $n_0$ defined by
\begin{equation}
\label{n0} n_0 = \biggl\lfloor\frac{\log t}{N\rho} \biggr\rfloor,
\end{equation}
where $\rho= \E [ |\wt A_1|^\a\log|\wt A_1| ]$.
Notice that $n_0$ depends on $t$. However, since we are interested only
in estimates from below we need less and for our purpose it is
sufficient to consider
sets
\begin{equation}
\label{vn} {V}_{n} = \bigl\{ |\wt A_1\cdots \wt
A_n|\ge t \mbox{ and } |\wt A_1\cdots \wt A_s|\le
\mathrm{e}^{-(n-s)\d}tC_0 \mbox{ for every }s\le n-1 \bigr\},
\end{equation}
where $C_0$ is a large constant and $\d$ is a small constant (both
will be defined later).
Our main result of this section is the following
theorem.
\begin{theorem}
\label{thm_Vn} Assume $\E[|\wt A_1|^{\a+\d}]<\infty $, $\E[|\wt
A_1|^{\a}]=\frac{1}N$ and $0<\rho<\infty $.
There are constants $C_0, C_1, C_2$ such that for
sufficiently large $t$ and
for $n_0-\sqrt{n_0}\leq n \leq n_0$
\[
\frac{C_1}{\sqrt n t^\a N^n} < \P [ V_n ] \le \frac{C_2}{\sqrt n t^\a N^n}.
\]
\end{theorem}
In order to prove the theorem above we will need precise estimates
of $\P[|\wt A_1\cdots \wt A_s|>t]$. We will use the following extension
of the Bahadur, Rao theorem (\cite{DZ}, Theorem 3.7.4, see also Example
3.7.10).
\begin{prop}
\label{prop_br}
Assume $\E[|\wt A_1|^{\a+\d}]<\infty $, $\E[|\wt A_1|^{\a}]=\frac{1}N$ and $0<\rho<\infty $.
There is $C$ such that for every $d\geq0$ and every
$n\in\N$
\begin{equation}
\label{eq:p1} \P \bigl\{ |\wt A_1\cdots \wt A_n|>\mathrm{e}^d
\mathrm{e}^{ \rho nN} \bigr\}\leq \frac{C}{\sqrt{2\pi}\a\lambda\sqrt n \mathrm{e}^{\rho\a nN}N^{n}
\mathrm{e}^{\a d}} ,
\end{equation}
where $\lam=\sqrt{\Lambda''(\a)}$ for
$
\Lambda(s) = \log\E [|\wt A_1|^s ]$.
Moreover, let $\theta\geq0$ and
\begin{equation}
\label{eq: p222} 0\leq\frac{d}{\sqrt n}\leq\theta
\end{equation}
for sufficiently large $n$.
Then there is $C=C(\theta)$ such that for large $n$:
\begin{equation}
\label{eq:p2} \sqrt{2\pi}\a\lambda\sqrt n \mathrm{e}^{\rho\a n N}N^n
\mathrm{e}^{\a d} \mathrm{e}^{\afrac{d^2}{2\lambda^2 n}}\cdot\P \bigl\{ |\wt A_1\cdots \wt
A_n|>\mathrm{e}^d \mathrm{e}^{\rho nN} \bigr\} = 1 + C(\theta)\mathrm{o}(1),
\end{equation}
where as usual $\lim_{n\to\infty } \mathrm{o}(1)=0$ uniformly for $d$
satisfying \eqref{eq: p222}.
\end{prop}
The proof is a slight modification of the proof of Theorem~3.7.4
in \cite{DZ}. For reader's convenience we give all the details but
we postpone the proof to the \hyperref[appendix]{Appendix}.
We will also use the following. Since $\E [|\wt A_1|^\b
]<\frac{1}N$ for $\b<\a$ and sufficiently close to $\a$, one can find
$\b<\a$ and $\g>0$ such that
\begin{equation}
\label{eq:2.4} \E \bigl[|\wt A_1|^\b \bigr] =
\frac{1}{N^{1+\g}}.
\end{equation}
\begin{pf*}{Proof of Theorem~\ref{thm_Vn}}
Denote
\begin{eqnarray*}
U_n &=& \bigl\{ |\wt A_1\cdots \wt A_n|>t \bigr\},
\\
W_{s,n} &=& \bigl\{ |\wt A_1\cdots \wt A_s| >
\mathrm{e}^{-\d(n-s)} C_0 t \bigr\}.
\end{eqnarray*}
We have
\begin{eqnarray*}
\P [ V_n ] &=& \P \biggl[ U_n \cap\bigcap
_{s<n} W_{s,n}^c \biggr]
\\
&=& \P [ U_n ] - \P \biggl[ U_n \cap \biggl(\bigcap
_{s<n} W_{s,n}^c
\biggr)^c \biggr]
\\
&=& \P [ U_n ] - \P \biggl[\bigcup_{s<n}
( U_n\cap W_{s,n} ) \biggr].
\end{eqnarray*}
By Proposition~\ref{prop_br} ($s=n, d = N\rho(n_0-n)$,
$\theta=N\rho+1$)
\begin{eqnarray*}
\P [ U_n ] &=& \P \bigl[ |\wt A_1\cdots \wt A_n|>t \bigr]
= \P \bigl[ |\wt A_1\cdots \wt A_n| > \mathrm{e}^{N\rho n}\mathrm{e}^{N\rho(n_0-n)}
\bigr]
\\
&\geq& \frac{C_1 \mathrm{e}^{-N\rho\a(n_0-n)}}{\sqrt n \mathrm{e}^{N\rho\a n}N^n} =
\frac{C_1}{ \sqrt n \mathrm{e}^{N\rho\a n_0}N^n} = \frac{C_1}{\sqrt n t^\a
N^n}
\end{eqnarray*}
for sufficiently large $t$ and $C_1 = \frac{1+C(N\rho
+1)\mathrm{o}(1)} {\sqrt{2\pi}\a\lam}\exp ({-\frac{(N\rho
+1)^2}{2\lambda
^2}} )$. Exactly in the same way \eqref{eq:p2} gives estimates from
above with $C_2 = \frac{1+C(N\rho+1)\mathrm{o}(1)} {\sqrt{2\pi}\a\lam}$.
Therefore to prove the theorem, it is sufficient to justify that
\begin{equation}
\label{square} \P \biggl[\bigcup_{s<n} (
U_n\cap W_{s,n} ) \biggr] \le \frac{\eps}{\sqrt n t^\a N^n}.
\end{equation}
We fix $t$, $n_0$ and $n$ such that
$n_0-\sqrt{n_0}\le n \le n_0$.
First we estimate $\P[ U_n\cap W_{s,n} ]$ for $s< n-D\log n$, where
the constant $D$ will be defined later. By the Chebyshev inequality and
\eqref{eq:2.4}, we have
\begin{eqnarray*}
\P[ U_n\cap W_{s,n} ] &=& \sum
_{m=0}^\infty \P \bigl[\mathrm{e}^m
\mathrm{e}^{-\d
(n-s)}C_0 t < |\wt A_1\cdots \wt A_s|
\le \mathrm{e}^{m+1} \mathrm{e}^{-\d(n-s)}C_0 t \mbox{ and } |\wt
A_1\cdots \wt A_n| > t \bigr]
\\
&\le& \sum_{m=0}^\infty \P \bigl[ |\wt
A_1\cdots \wt A_s| > \mathrm{e}^{m} \mathrm{e}^{-\d
(n-s)}C_0
t \bigr] \P \bigl[ |\wt A_{s+1}\cdots \wt A_n| >
\mathrm{e}^{-(m+1)}\mathrm{e}^{\d(n-s)}C_0^{-1} \bigr]
\\
&\le& \sum_{m=0}^\infty \frac{ \mathrm{e}^{\d\a(n-s)} }{\mathrm{e}^{m\a}C_0^\a
t^\a}
\bigl(\E|\wt A_1|^\a \bigr)^s \cdot
\frac{\mathrm{e}^{\b(m+1)}C_0^\b
}{\mathrm{e}^{\d\b(n-s)}} \bigl(\E|\wt A_1|^\b
\bigr)^{n-s}
\\
&\le& \frac{\mathrm{e}^{\d\a(n-s)}}{C_0^{\a-\b}t^\a} \cdot\frac{1}{N^s} \cdot\frac{1}{ \mathrm{e}^{\d\b(n-s)} N^{n-s} N^{\g(n-s)} } \cdot
\sum_{m=0}^\infty \frac{\mathrm{e}^\b}{\mathrm{e}^{m(\a-\b)}}
\\
&\le& \frac{C}{C_0^{\a-\b}t^\a N^n \mathrm{e}^{(\g\log N+\d(\b-\a
))(n-s)}} = \frac{C}{C_0^{\a-\b}t^\a N^n \mathrm{e}^{\g_1(n-s)}},
\end{eqnarray*}
where
$
\g_1:=\g\log N + (\b-\a)\d
$ and choosing appropriately small $\d$ we can assume that $\g_1>0$.
Hence, for $s<n-D\log n$
\begin{equation}
\label{star} \P[ U_n\cap W_{s,n} ] \le
\frac{C}{C_0^{\a-\b}t^\a N^n \mathrm{e}^{\g_1(n-s)}}.
\end{equation}
For $s> n-D\log n$, we estimate
\begin{eqnarray*}
\P[ U_n\cap W_{s,n} ] &=& \sum_{m=0}^\infty
\P \bigl[\mathrm{e}^m \mathrm{e}^{-\d
(n-s)}C_0 t < |\wt
A_1\cdots \wt A_s| \le \mathrm{e}^{m+1} \mathrm{e}^{-\d(n-s)}C_0
t \mbox{ and } |a_1\cdots a_n| > t \bigr]
\\
&\le&\sum_{m=0}^\infty \P \bigl[ |\wt
A_1\cdots \wt A_s| > \mathrm{e}^m \mathrm{e}^{-\d
(n-s)}C_0
t \bigr] \P \bigl[ |\wt A_{s+1}\cdots \wt A_n| >
\mathrm{e}^{-(m+1)}\mathrm{e}^{\d(n-s)}C_0^{-1} \bigr].
\end{eqnarray*}
We denote the first factor of the sum by $I_m$. To estimate it, we will
use Proposition~\ref{prop_br}. Namely let
\begin{eqnarray*}
k&=& n-s, \qquad k_0 = n_0 - s,
\\
d_1 &=& -\d k + m + \log C_0 + N\rho
k_0,
\\
d_2 &=& d_1 + 1,
\end{eqnarray*}
then (recall $\log t = (s+k_0)N\rho$)
\[
\mathrm{e}^m \mathrm{e}^{-\d(n-s)}C_0 t = \mathrm{e}^{d_1}
\mathrm{e}^{N\rho s}.
\]
So, by Proposition~\ref{prop_br}:
\[
\P \bigl[ |\wt A_1\cdots \wt A_s| > \mathrm{e}^{d_1+N\rho s} \bigr]
\le\frac
{C}{\sqrt s} N^{-s}\mathrm{e}^{-N\rho\a s-\a d_1} \leq \frac{C\mathrm{e}^{\d\a k}}{ C_0^\a
\mathrm{e}^{\a m} t^\a N^s \sqrt s}.
\]
The second factor we estimate exactly in the same way as previously and
we obtain
\begin{eqnarray}
\label{2star}
\P[ U_n\cap W_{s,n} ] &=& \sum
_{m=0}^\infty \frac{C\mathrm{e}^{\d\a(n-s)}}{
C_0^\a \mathrm{e}^{\a m} t^\a N^s \sqrt s} \cdot
\frac{ \mathrm{e}^{\b(m+1)} C_0^\b}{ \mathrm{e}^{\d\b(n-s)} N^{(1+\g)(n-s)}
}\nonumber
\\[-8pt]\\[-8pt]
&\le&\frac{C}{ C_0^{\a-\b} t^\a N^n \sqrt n \mathrm{e}^{\g_1(n-s)} }. \nonumbe
\end{eqnarray}
Next, in view of \eqref{star} and \eqref{2star}
\begin{eqnarray*}
\P \biggl[\bigcup_{s<n} ( U_n\cap
W_{s,n} ) \biggr] &\le& \sum_{s< n-D\log n} \P [
U_n\cap W_{s,n} ] + \sum_{ n-D\log n\le s < n}
\P [ U_n\cap W_{s,n} ]
\\
&\le& \sum_{s< n-D\log n} \frac{C}{C_0^{\a-\b}t^\a N^n \mathrm{e}^{\g_1(n-s)}} + \sum
_{ n-D\log n\le s < n} \frac{C}{C_0^{\a-\b}t^\a N^n \sqrt n
\mathrm{e}^{\g_1(n-s)}}
\\
&\le& \frac{C}{C_0^{\a-\b}t^\a N^n} \biggl( \frac{n}{n^{\g_1 D }} +\frac{1}{\sqrt n} \sum
_{ s < D\log n} \frac{1}{\mathrm{e}^{\g_1 s}} \biggr)
\\
&\le& \frac{C}{C_0^{\a-\b}t^\a N^n} \biggl( \frac{1}{n^{\g_1 D -1}} +\frac{1}{\sqrt n} \biggr)
\\
&\le& \frac{\eps}{\sqrt n t^\a N^n }
\end{eqnarray*}
assuming that $\frac{C}{C_0^{\a-\b}}<\eps$ and $\g_1 D \ge\frac{3}2$. Hence, \eqref{square} and the proof is finished.
\end{pf*}
\section{Proof of Theorem \texorpdfstring{\protect\ref{mthm}}{1.1}}\label{sec3}
We start with the following lemma.
\begin{lem} If $\E [|A_1|^\b\log|A_1| ] >0$ for some $\b
>0$, $\P[A_1>0]>0$ and $\P[A_1<0]>0$, then any nontrivial solution of
\eqref{equation} is unbounded at $+\infty $ and $-\infty $.
\end{lem}
\begin{pf}
Suppose that $R$ is a bounded solution of \eqref{equation} and $R\neq C$ a.s. for any $C$. Assume first that $R$ is bounded a.s. from below
and from above. Let $[r,s]$ be the smallest interval containing the
support of $R$ for some finite numbers $r$ and $s$. Of
course $r\neq s$.
Denote $\wt B = \sum_{i=2}^N A_i R_i+B$, then
\begin{equation}
\label{eqaffine} R =_d A_1R_1+\wt B.
\end{equation}
Since $\E[|A_1|^\b\log|A_1|]>0$, the probability of the set $ U =
\{ (A_1,\wt B)\dvt |A_1| > 1 \} $ is strictly positive.
Then by \eqref{eqaffine}
we must have
\[
A_1 r +\wt B \ge r \quad \mbox{and}\quad A_1 s +\wt B \le s\qquad
\mbox{a.s.}
\]
But if we take a random pair $(A_1,\wt B)\in U$, then
\[
\bigl|(A_1 r + \wt B) - (A_1 s + \wt B)\bigr| =
|A_1||r-s| > |r-s|.
\]
Thus, we are led to a contradiction and at least one constant $r$ or
$s$ must be infinite. Without loss of generality, we can assume that
$s=+\infty $. In view of our assumptions, we can choose a large
constant $M$ and a small constant $\eps$ such that the probability of
the set $ V =
\{ (A_1,\wt B)\dvt A_1<-\eps, \wt B < M \} $ is strictly
positive. Now, take any $x> (r-M)/(-\eps)$ belonging to the support of
$R$. Then for any $(A_1,\wt B)\in V$ we have
\[
A_1 x+\wt B < -\eps x+M < r.
\]
Thus, by \eqref{eqaffine}, $r$ cannot be a lower bound of the support
of $R$ and must be equal $-\infty $.
\end{pf}
Let $\T$ be an $N$-ary rooted tree, that is, the tree with a
distinguished vertex $o$ called root, such that every vertex has $N$
daughters and one mother
(except the root). The tree $\T$ can be identified with the set of
finite words over the alphabet $\{1,2,\ldots ,N\}$:
\[
\T= \bigcup_{k=0}^\infty \{1,2,\ldots ,N
\}^k,
\]
where the empty word $\emptyset$ is the root and given $i_1i_2\cdots i_n\in
\T$ its daughters are the words of the form $i_1i_2\cdots i_nj$ for $j=1,\ldots ,N$.
We denote a typical vertex of the tree by $\g= i_1i_2\cdots i_n$ and we
identify it with the shortest path connecting $\g$ with $o$. We write
$|\g|=n$
for the length of $\g$ and $\g_{|_k} = i_1\cdots i_k$ for the curtailment
of $\g$ after $k$ steps. Conventionally, $|\emptyset|=0$ and $\g
_{|_0} = \emptyset$. If $\g_1 = i_1^1i_2^1\cdots i_{n_1}^1\in\T$ and $\g
_2 = i_1^2i_2^2\cdots i_{n_2}^2\in\T$ then we write $\g_1\g_2 =
i_1^1i_2^1\cdots i_{n_1}^1i_1^2i_2^2\cdots i_{n_2}^2 $
for the element of $\T$ obtained by juxtaposition. In particular, $\g
\emptyset= \emptyset\g= \g$. We partially order $\T$ by writing
$\g_1\le\g_2$ if there exists $\g_0\in\T$ such that $\g_2 = \g
_1\g_0$. For two vertices $\g_1$ and $\g_2$, we denote by $\g_0 =
\g_1\wedge\g_2$ the longest common subsequence of $\g_1$ and $\g
_2$ that is, the maximal $\g_0$ such that both $\g_0\le\g_1$ and
$\g_0\le\g_2$.
To every vertex $\g\in\T$ we associate random variables $(A_{\g
1},\ldots ,A_{\g N},B_\g,R_{\g1},\ldots ,R_{\g N})$ which are independent
copies of
$(A_{1},\ldots ,A_{N},B,R_{1},\ldots ,R_{N})$ defined in \eqref{equation}. It is
more convenient to think that $A_{\g i}$ and $R_{\g i}$ are indeed
attached not to the vertex $\g$ but to the edge connecting $\g$ with
$\g i$.
We write $\Pi_\g= A_{\g_{|_1}}A_{\g_{|_2}}\cdots A_{\g}$, then $\Pi_\g
$ is just the product of random variables $A_{\g_{|_k}}$ which are
associated with consecutive edges connecting the root $o$ with $\g$.
We fix $\g= i_1\cdots i_n$ and we apply $n$ times the stochastic equation
\eqref{equation} in such a way that in $k$th step we apply recursively
this equation to $R_{\g_{|_k}}$:
\begin{eqnarray}
\label{eq:2star}
R &=_d& \sum_{i=1}^N
A_i R_i + B_0\nonumber
\\
&=_d& A_{i_1} \Biggl( \sum_{j=1}^N
A_{i_1j}R_{i_1j} + B_{i_1} \Biggr) + \sum
_{i\neq i_1} A_i R_i + B_0\nonumber
\\
&=_d& A_{i_1}A_{i_1i_2}R_{i_1i_2} +
A_{i_1} \Biggl( \sum_{j\neq i_2}^N
A_{i_1j}R_{i_1j} + B_{i_1} \Biggr) + \sum
_{i\neq i_1} A_i R_i + B_0\nonumber
\\
&=_d& \Pi_{\g_{|_2}} R_{\g_{|_2}} + \sum
_{j\neq i_2}^N \Pi_{(\g
_{|_1}j)}R_{(\g_{|_1}j)} +
A_{i_1}B_{i_1} + \sum_{i\neq i_1}
A_i R_i + B_0
\\
&=_d& \Pi_{\g_{|_2}} \Biggl( \sum_{i=1}^N
A_{(\g_{|_2}i)}R_{(\g
_{|_2}i)} + B_{\g_{|_2}} \Biggr) + \sum
_{i\neq i_2}^N \Pi_{(\g
_{|_1}i)}R_{(\g_{|_1}i)} +
\sum_{i\neq i_1} A_i R_i +
A_{i_1}B_{i_1} + B_0\nonumber
\\
&=_d& \cdots\nonumber
\\
&=_d& \Pi_\g R_\g+ \sum
_{k<n} \sum_{i\neq i_k}
\Pi_{(\g
_{|_k}i)}R_{(\g_{|_k}i)} + \sum_{k<n}
\Pi_{\g_{|_k}}B_{\g_{|_k}}.\nonumber
\end{eqnarray}
We define
\[
V_{\g} = \bigl\{ |\Pi_\g |\ge t \mbox{ and } |\Pi
_{\g_{|_s}} |\le \mathrm{e}^{-(|\g|-s)\d}C_0 t \mbox{ for every }s<|
\g| \bigr\}.
\]
Notice that if we denote $\wt A_k = A_{\g_{|_k}}$, then the set $V_\g
$ coincides with the set $V_{|\g|}$ defined in \eqref{vn}. Thus, by
Theorem~\ref{thm_Vn} we can choose large $C_0$ such that if $n=|\g|$
and $n_0 - \sqrt{n_0} < n < n_0$, then
\[
\P[V_\g] \ge\frac{C}{\sqrt n t^\a N^n}.
\]
For a sufficiently large constant $d$ (defined later) and
$D=\frac{Nd^2 + d}{1-\mathrm{e}^{-\sfrac{\d}{2}}}$, we define sets
\begin{eqnarray*}
W_\g&=& \bigl\{ |R_{(\g_{|_s}i)} | < d \mathrm{e}^{{(|\g|-s)\d
}/{4}},
|A_{(\g_{|_s}i)} | < d \mathrm{e}^{{(|\g|-s)\d}/{4}}, |B_{\g
_{|_s}}| < d
\mathrm{e}^{{(|\g|-s)\d}/{2}}, \\
&&\hphantom{\bigl\{}s=0,\dots,|\g|-1; i\neq i_{s+1} \bigr\};
\\
W^+_\g&=& W_\g\cap \{ R_\g> 2D \};
\\
W^-_\g&=& W_\g\cap \{ R_\g< -2D \};
\\
V^+_\g&=& V_\g\cap \{ \Pi_\g>0 \};
\\
V^-_\g&=& V_\g\cap \{ \Pi_\g<0 \}.
\end{eqnarray*}
Finally we define
\[
\wt V_\g= \bigl(V_\g^+ \cap W_\g^+ \bigr)
\cup \bigl(V_\g^- \cap W_\g^- \bigr).
\]
\begin{lem}
Assume $\g\in\T$. Then on the set $\wt V_\g$ we have
\[
R > At.
\]
\end{lem}
\begin{pf}
Let $n=|\g|$, then by \eqref{eq:2star} on $\wt V_{\g}$ we have
\begin{eqnarray*}
R &\ge& \Pi_\g R_\g- \biggl| \sum
_{k<n}\sum_{i\neq i_k} \Pi
_{(\g_{|_k}i)} R_{(\g_{|_k}i)} + \sum_{k<n}
\Pi_{\g_{|_k}} B_{\g_{|_k}} \biggr|
\\
&\ge& 2Dt - \sum_{k<n} \bigl(Nd^2+d
\bigr) \mathrm{e}^{-{(n-k)\d}/{2}}C_0 t
\\
&\ge& Dt.
\end{eqnarray*}
\upqed
\end{pf}
We are going to prove that for some $\eta>0$
\begin{equation}
\label{eq:3star} \P \biggl[\bigcup_{\{\g\in\T: n_0-\sqrt{n_0} < |\g| <n_0\}} \wt
V_\g \biggr]\ge\eta t^{-\a},
\end{equation}
which immediately
implies that
\[
\liminf_{t \to\infty }\P\{ R>t\}t^{\a}>0.
\]
\begin{lem}
Let $X_i$ be a sequence of i.i.d. random variables such that $\E
|X_1|^{\eps}<\infty $ for some $\eps>0$. Let $\d_0>0$. Then there
exist constants $d_0$ and $p_0>0$
such that for every $n$
\[
\P \bigl[ |X_i| < d_0 \mathrm{e}^{(n-i)\d_0}, i=1,2,\ldots ,n-1
\bigr] \ge p_0.
\]
\end{lem}
\begin{pf}
By the Chebyshev inequality, we have
\[
\P \bigl[ |X_i| \ge d_0 \mathrm{e}^{(n-i)\d_0} \bigr] \le
\frac{\E|X_i|^{\eps
}}{d_0^{\eps}} \mathrm{e}^{-(n-i)\d_0\eps}.
\]
Take $d_0$ such that $d_0^{\eps} > 3 \E|X_i|^{\eps}$. Then,
since $1-\frac{x}{3} > \mathrm{e}^{-x}$ for $x\in[0,1]$ we have
\[
\P \bigl[ |X_i| < d_0 \mathrm{e}^{(n-i)\d_0} \bigr] \ge1 -
\tfrac{1}{3} \mathrm{e}^{-(n-i)\d_0 \eps} \ge\exp \bigl({- \bigl(\mathrm{e}^{-\d_0 \eps}
\bigr)^{n-i}} \bigr).
\]
Therefore,
\begin{eqnarray*}
&&\P \bigl[ |X_i| < d_0 \mathrm{e}^{(n-i)\d_0}, i=1,2,\ldots ,n-1
\bigr] \\
&&\quad = \prod_{i=1}^{n-1} \P \bigl[
|X_i| < d_0 \mathrm{e}^{(n-i)\d_0} \bigr]\ge\prod
_{i=1}^{n-1} \mathrm{e}^{- (\mathrm{e}^{-\d_0 \eps})^{n-i}}
\\
&&\quad = \exp \Biggl({-\sum_{i=1}^{n-1}
\bigl(\mathrm{e}^{-\d_0\eps}\bigr)^i} \Biggr)\ge\exp \bigl({-
\bigl(1-\mathrm{e}^{-\d_0 \eps}\bigr)^{-1}} \bigr) =:p_0.
\end{eqnarray*}
\upqed
\end{pf}
Since $B$ and $R$ have absolute moments of order bigger then $\g$ we
obtain the following corollary.
\begin{cor}
There are constants $d$ and $p>0$ such that for every $\g\in\T$
\[
\P\bigl[W_\g^+\bigr] \ge p \quad \mbox{and}\quad \P\bigl[W_\g^-
\bigr]\ge p.
\]
\end{cor}
In view of the last result to obtain \eqref{eq:3star}, it is
sufficient to prove
\[
\P \biggl[\bigcup_{\{\g\in\T: n_0-\sqrt{n_0} < |\g| <n_0\}} V_\g \biggr]
\ge\eta_1 t^{-\a},
\]
for some $\eta_1>0$.
In fact, we will estimate from below much smaller sum over a sparse
subset of $\T$. The details are as follows.
We fix a large integer $C_1$ (determined later) and an arbitrary
element $\ov\g$ of $\T$ such that $|\ov\g| = C_1$ (e.g., $\ov\g$
can be chosen as the word consisting of $n$ one's). We define a sparse
subset of vertices of $\T$:
\[
\ov\T= \bigl\{ \g\in\T\dvt \bigl(|\g|\quad \mathrm{mod}\ C_1\bigr) = 0, \g= \g_{|_{|\g
|-C_1}}
\ov\g, n_0 - \sqrt{n_0} < |\g| < n_0
\bigr\},
\]
that is, $\ov\T$ is the set of vertices of $\T$ located on the level
$kC_1$ (for some integer $k$) such that
$n_0-\sqrt{n_0} < kC_1 < n_0$ and such that the last $n$ letters of
$\g$ form the word $\ov\g$. Notice that for every $\g$ such that
$|\g|=kC_1$ the set
\[
\Biggl\{ \g\g_1, \g_1\in\bigcup
_{i=1}^{C_1}\{1,\ldots ,N\}^i \Biggr\}
\]
contains exactly one element of $\T$. Thus there are exactly
$N^{kC_1}$ elements of $\ov\T$ of length $(k+1)C_1$. Moreover, the
crucial property of the set $\ov\T$, that will be strongly used
below, is that the distance between two different elements of $\ov\T$
is at least $C_1$ (by ``distance'' we mean the usual distance on graphs,
that is, the minimal number of edges connecting two vertices).
\begin{theorem} There is $\eta>0$ such that
\[
\P \biggl(\bigcup_{\gamma\in \ov\T} V_\g \biggr)
\geq\frac{C\eta
}{N^{C_1}C_1t^{\a}}.
\]
\end{theorem}
\begin{pf}
By the inclusion--exclusion principle, we have
\begin{equation}
\label{eq:1} \P \biggl(\bigcup_{\gamma\in\ov\T}
V_\g \biggr) \ge \sum_{\g\in\ov\T}\P (
V_\g ) - \sum_{\g\in\ov\T
}\sum
_{U_\g} \P ( V_\g\cap V_{\g'} ),
\end{equation}
where $U_\g= \{ \g'\in\ov\T\setminus\{\g\}\dvt |\g'|\le|\g| \}$.
Therefore, we have to estimate
\[
\sum_{\g\in\ov\T}\P ( V_\g ) \quad \mbox{and}\quad
\sum_{\g\in
\ov\T}\sum_{ U_\g} \P
( V_\g\cap V_{\g'} ).
\]
Let $K$ be the set of levels on which there are some elements of $\ov
\T$, that is,
\[
K= \{ kC_1\dvt n_0 - \sqrt{n_0} <
kC_1 < n_0 \}.
\]
Let $L=|K|$ be the number of elements of the set $K$ and let $n_j$ be
the $j$th element of $K$.
Observe that for given $n\in K$ there are
exactly $N^{n-C_1}$ elements of $\ov\T$ located on the level $n$ and
for every such $\g$, by Theorem~\ref{thm_Vn}, we have
$\P( V_\g)\geq
\frac{C}{\sqrt{n}N^nt^{\a}}$. Hence,
\begin{equation}
\label{eq:2} \sum_{\g\in\ov\T}\P(V_\g)\geq
\sum_{j=1}^L \frac{C}{\sqrt{n_j}N^{n_j}t^{\a}}N^{n_j-C_1}
\geq \frac{C }{N^{C_1}C_1t^{\a}}.
\end{equation}
Now, let us estimate the sum of intersections. We fix $\g\in\ov\T$
and $\g'\in U_\g$. Let $\g_0 = \g\wedge\g'$ and let $s$ be the
length of $\g_0$. We have
\begin{eqnarray}
\label{int}
\P [ V_\g\cap V_{\g'} ] &\le&\P
\bigl[ V_\g\cap \bigl\{ |\Pi_{\g_0}| < \mathrm{e}^{-\d(|\g|-s)}C_0
t, |\Pi_{\g'}| > t \bigr\} \bigr]\nonumber
\\
&\le&\P [ V_{\g} ]\P \bigl[ | A_{\g'_{|_{s+1}}}A_{\g
'_{|_{s+2}}}\cdots A_{\g'}
| > \mathrm{e}^{\d(|\g|-s)} C_0^{-1} \bigr]
\\
&\le&\P [ V_{\g} ]\cdot\frac{C_0^\a}{\mathrm{e}^{\a\d(|\g|-s)}
N^{|\g'|-s}},\nonumber
\end{eqnarray}
where for the last inequality we have used the Chebyshev
inequality. We fix $\g\in\ov\T$ and we consider $\g'\in U_\g$.
Notice that if $\g$ and $\g'$ connect on the level $s$, that is, $\g
_{|_s} = \g\wedge\g'$, then $s$ must be smaller than
$|\g|-C_1$. Given $s$ let us estimate the number of elements
$\g'\in U_\g$ such that $\g_{|_s} = \g\wedge\g'$. All these
elements must be located on levels $|\g|, |\g|-C_1,\ldots , |\g|-k
C_1$, where $k$ is the largest number such that $|\g|-kC_1 \ge
\max\{ s,n_0-\sqrt{n_0}\}$, that is,
\[
k \le\frac{1}{C_1} \min \bigl\{ |\g|-s , |\g|-n_0 +
\sqrt{n_0} \bigr\} \le\frac{1}{C_1} \bigl(|\g|-s\bigr).
\]
Moreover on the level $|\g|-jC_1$ ($j<k$), there are exactly $N^{|\g
|-jC_1-s-C_1}$ elements of $U_\g$. Thus for $C_1$ sufficiently large,
by \eqref{int}, we have
\begin{eqnarray*}
&&\sum_{\g\in\ov\T}\sum_{\g' \in U_\g}
\P [ V_\g\cap V_{\g
'} ] \\
&&\quad \le \sum
_{\g\in\ov\T} \sum_{s\le|\g|-C_1} \sum
_{\{
\g'\in U_\g: \g_{|_s} = \g\wedge\g'\}} \P[V_\g] \cdot\frac{C_0^\a}{\mathrm{e}^{\a\d(|\g|-s)} N^{|\g'|-s}}
\\
&&\quad \le \sum_{\g\in\ov\T} \P[V_\g] \sum
_{s\le|\g|-C_1} \sum_{0\le j\le\sklfrac{1}{C_1}(|\g|-s)} \sum
_{\{\g'\in U_\g: \g_{|_s} = \g\wedge\g', |\g'| = |\g
|-jC_1\}} \frac{C_0^\a}{\mathrm{e}^{\a\d(|\g|-s)} N^{|\g'|-s}}
\\
&&\quad \le \sum_{\g\in\ov\T} \P[V_\g] \sum
_{s\le|\g|-C_1} \sum_{0\le j\le\sklfrac{1}{C_1}(|\g|-s)}
\frac{C_0^\a}{\mathrm{e}^{\a\d(|\g|-s)} N^{|\g|-jC_1-s}} \cdot N^{|\g| -
jC_1 -s - C_1}
\\
&&\quad \le \sum_{\g\in\ov\T} \P[V_\g] \sum
_{s\le|\g|-C_1} \frac{C_0^\a(|\g|-s)}{ C_1 N^{C_1} \mathrm{e}^{\a\d(|\g|-s)} }
\\
&&\quad \le \sum_{\g\in\ov\T} \P[V_\g]
\frac{C_0^\a}{ C_1 N^{C_1} \mathrm{e}^{{\a\d C_1}/2 }} \le\frac{1}2 \sum_{\g
\in\ov\T}
\P[V_\g].
\end{eqnarray*}
Finally, combining the above estimates with \eqref{eq:1} and \eqref
{eq:2}, we obtain
\[
\P \biggl[ \bigcup_{\g\in\ov\T} V_\g \biggr]
\ge\frac{1}2 \frac
{C}{N^{C_1} C_1 t^\a}.
\]
\upqed
\end{pf}
\begin{appendix}
\section*{Appendix: Proof of Proposition \texorpdfstring{\lowercase{\protect\ref{prop_br}}}{2.2}}
\label{appendix}
\setcounter{equation}{0}
\begin{pf}
We proceed as in \cite{DZ} and for reader's convenience we use the
same notation. Define
$X_i = \log|\wt A_i|$ and $\wh S_n = \frac{1}n\sum_{i=1}^n X_i$.
We introduce a new probability measure: $\wt\mu(dx) = N \mathrm{e}^{\a x
}\mu(dx)$, where $\mu$ is the law of $X_i$. Next, we normalize
$X_i$ and we define
new random variables: $Y_i =
\frac{X_i-N\rho}{\sqrt{\Lambda''(\a)}}$ and $W_n= \frac{1}{\sqrt
n}\sum_{i=1}^n Y_i$. Then $\E_{\wt\mu} Y_i=0$
and
\[
\wh S_n - N\rho=\frac{\lam}{\sqrt{n}}\frac{1}{\sqrt{n}}\sum
_{i=1}^n Y_i= \frac{\lam}{\sqrt{n}}W_n,
\]
where $\lam= \sqrt{\Lambda''(\a)}$ and $\Lambda(s) = \log (
\E [ |\wt A_1|^s ] )$.
Let $F_n$
be the distribution of $W_n$ with respect to the changed measure $\wt
\mu$.
Let $\psi_n = \a\lambda\sqrt n$.
Then,
\begin{eqnarray*}
\P \bigl\{ |\wt A_1\cdots \wt A_n|>\mathrm{e}^d
\mathrm{e}^{N\rho n} \bigr\} &=& \P \{ \wh S_n > N\rho+ d/n \}
\\
&=& \P \biggl\{ W_n> \frac{d}{\lam\sqrt {n}} \biggr\} =\E_{\wt
\mu}
\bigl[ N^{-n} |\wt A_1\cdots\wt A_n|^{-\a}
\mathbf{1}_{\{W_n >
\afrac{d}{\lambda
\sqrt n}\}} \bigr]
\\
&=& \mathrm{e}^{-\a n\rho N} N^{-n} \E_{\wt\mu} \bigl[\mathrm{ e}^{-\psi_n W_n}
\mathbf{1}_{\{W_n > \afrac{d}{\lambda\sqrt n}\}} \bigr]
\\
&=& \mathrm{e}^{-\a n\rho N} N^{-n} \int_{\afrac{d}{\lambda\sqrt
n}}^\infty
\mathrm{e}^{-\psi_n x}\,\mathrm{d}F_n(x).
\end{eqnarray*}
We will use here the Berry--Esseen expansion for nonlattice
distributions of $F_n$ (see \cite{F}, page 538):
\begin{equation}
\label{Berry-Essen} \lim_{n\to\infty } \biggl( \sqrt n \sup
_x \biggl| F_n(x) - \Phi(x) - \frac{m_3}{6\sqrt n}
\bigl(1-x^2\bigr)\phi(x) \biggr| \biggr) = 0,
\end{equation}
where $m_3 = \E_{\wt\mu}[Y_1^3]<\infty $, $\phi(x)=\frac{1}{\sqrt {2\pi}} \mathrm{e}^{-\sfrac{x^2}2}$ is the standard normal density, and $\Phi
(x)=\int_{-\infty }^x \phi(y)\,\mathrm{d}y$ is its distribution function.
First, we integrate by parts and then we use the above result
\begin{eqnarray*}
J&=&\a\lambda\sqrt n \mathrm{e}^{N\rho\a n}N^n \P \bigl\{ |\wt
A_1\cdots \wt A_n|>\mathrm{e}^d \mathrm{e}^{N\rho n} \bigr\}
\\
&=& \int_{\afrac{d}{\lambda\sqrt n}}^\infty
\psi_n \mathrm{e}^{-\psi_n
x}\,\mathrm{d}F_n(x)
\\
&=& \psi_n \mathrm{e}^{-\psi_n x} F_n(x) \biggl|_{\afrac{d}{\lambda\sqrt
n}}^\infty
+ \int_{\afrac{d}{\lambda\sqrt n}}^\infty \psi _n^2\mathrm{e}^{-\psi_n x}F_n(x)\,\mathrm{d}x
\\
&=& - \psi_n \mathrm{e}^{-\a d} F_n \biggl(
\frac{d}{\lambda\sqrt n} \biggr) + \int_{\a d}^\infty
\psi_n \mathrm{e}^{- x}F_n \biggl(\frac{x}{\psi_n}
\biggr)\,\mathrm{d}x
\\
&=& \int_{\a d}^\infty \psi_n
\mathrm{e}^{- x} \biggl[ F_n \biggl(\frac{x}{\psi
_n} \biggr) -
F_n \biggl(\frac{d}{\lambda\sqrt n} \biggr) \biggr]\, \mathrm{d}x
\\
&=& \mathrm{o}(1) \mathrm{e}^{-\a d} + \int_{\a d}^\infty
\psi_n \mathrm{e}^{- x} \biggl[ \Phi \biggl(\frac{x}{\psi_n}
\biggr) - \Phi \biggl(\frac{d}{\lambda\sqrt n} \biggr) \biggr] \,\mathrm{d}x
\\
&&{} + \frac{m_3 }{6\sqrt n} \int_{\a d}^\infty
\psi_n \mathrm{e}^{- x} \biggl[ \biggl(1- \biggl(\frac{x}{\psi_n}
\biggr)^2 \biggr)\phi \biggl(\frac{x}{\psi
_n} \biggr) - \biggl(1-
\biggl(\frac{d}{\lambda\sqrt n} \biggr)^2 \biggr) \phi \biggl(
\frac{d}{\lambda\sqrt n} \biggr) \biggr] \,\mathrm{d}x.
\end{eqnarray*}
We denote the second term by $I(n)$ and the third one by $\mathit{II}(n)$. Thus,
\[
J(n) = \mathrm{o}(1)\mathrm{e}^{-\a d} + I(n) + \mathit{II}(n).
\]
We estimate first $I$:
\begin{eqnarray*}
\sqrt{2\pi} I(n) &=& \int_{\a d}^\infty
\psi_n \mathrm{e}^{-x}\int_{\afrac{d}{\lambda\sqrt n}}^{\afrac{x}{\psi_n}}
\mathrm{e}^{-\sfrac{y^2}{2}}\,\mathrm{d}y\,\mathrm{d}x = \int_{\afrac{d}{\lambda\sqrt n}}^\infty
\psi_n \mathrm{e}^{-\sfrac
{y^2}{2}}\int_{ \psi_n y}^{\infty }
\mathrm{e}^{-x}\,\mathrm{d}x\,\mathrm{d}y
\\
&=& \int_{\afrac{d}{\lambda\sqrt n}}^\infty \psi_n
\mathrm{e}^{-\psi_n
y}\mathrm{e}^{-\sfrac{y^2}{2}}\,\mathrm{d}y = - \mathrm{e}^{-\psi_n y} \mathrm{e}^{-\sfrac{y^2}{2}}
\biggl|_{\afrac{d}{\lambda\sqrt
n}}^\infty - \int_{\afrac{d}{\lambda\sqrt n}}^\infty
y \mathrm{e}^{-\psi_n
y}\mathrm{e}^{-\sfrac{y^2}{2}}\,\mathrm{d}y
\\
&=& \mathrm{e}^{-\a d} \mathrm{e}^{-\afrac{d^2}{2\lambda^2 n}} - \int_{\afrac{d}{\lambda
\sqrt n}}^\infty
y \mathrm{e}^{-\psi_n y}\mathrm{e}^{-\sfrac{y^2}{2}}\,\mathrm{d}y.
\end{eqnarray*}
Let $\d>0$. We divide the last integral into two parts
\[
\int_{\afrac{d}{\lambda\sqrt n}}^\infty y \mathrm{e}^{-\psi_n y}\mathrm{e}^{-\sfrac
{y^2}2}\,\mathrm{d}y
= \int_{\afrac{d}{\lambda\sqrt n}}^{{\afrac{d}{\lambda\sqrt n}} + \sfrac\d
{\lambda}} y \mathrm{e}^{-\psi_n y}\mathrm{e}^{-\sfrac{y^2}{2}}\,\mathrm{d}y
+ \int_{{\afrac{d}{\lambda\sqrt n}} + \sfrac\d{\lambda}}^\infty y \mathrm{e}^{-\psi_n y}\mathrm{e}^{-\sfrac{y^2}{2}}\,\mathrm{d}y
\]
and denote the first one by $I_1(n)$ and the second one by $I_2(n)$. Then
\[
\mathrm{e}^{\a d} I_1(n) = \int_{\afrac{d}{\lambda\sqrt n}}^{\afrac{d}{\lambda
\sqrt n}+ \sfrac\d{\lambda}}
y \mathrm{e}^{\a d-\psi_n y}\mathrm{e}^{-\sfrac
{y^2}{2}}\,\mathrm{d}y \le\frac{\d}{\lambda}\frac{\theta+{\d}}{\lambda}
\cdot \mathrm{e}^{-\afrac{d^2}{2\lambda^2 n}}
\]
and large $n$ we have
\[
\mathrm{e}^{\a d}I_2(n) = \int_{\afrac{d}{\lambda\sqrt n}+ \sfrac\d{\lambda
}}^\infty
y \mathrm{e}^{\a d-\psi_n y}\mathrm{e}^{-\sfrac{y^2}{2}}\,\mathrm{d}y \le \mathrm{e}^{-\a\d\sqrt n} \mathrm{e}^{-\afrac{d^2}{2\lambda^2 n}}
\le\d \mathrm{e}^{-\afrac{d^2}{2\lambda^2 n}}.
\]
Thus, we have proved that for large $n$
\[
\sqrt{2\pi} \mathrm{e}^{\a d} I(n) = \mathrm{e}^{-\afrac{d^2}{2\lambda^2 n}}\bigl(1+ C(\theta)\d\bigr).
\]
We may also write for any {$d \geq0$}
\[
\int_{\afrac{d}{\lambda\sqrt n}}^\infty y \mathrm{e}^{\a d-\psi_n y}\mathrm{e}^{-\sfrac
{y^2}{2}}\,\mathrm{d}y
\leq\int_{\afrac{d}{\lambda\sqrt n}}^\infty y \mathrm{e}^{-\sfrac
{y^2}2}\,\mathrm{d}y\leq
\mathrm{e}^{-\afrac{d^2}{2\lambda^2 n}}.
\]
Hence,
\[
\sqrt{2\pi} \mathrm{e}^{\a d} \bigl|I(n)\bigr| \leq2\mathrm{e}^{-\afrac{d^2}{2\lambda^2 n}}.
\]
Now we compute the second term $\mathit{II}(n)$. Denote $g(x) =\sqrt{2\pi}
(1-x^2) \phi(x)$. Then
\begin{eqnarray*}
\sqrt{2\pi} \mathit{II}(n) &=& \frac{m_3 \a\lambda}{6} \int_{\a d}^\infty
\mathrm{e}^{-x} \biggl[ g \biggl( \frac{x}{\psi_n} \biggr)- g \biggl(
\frac
{d}{\lambda\sqrt n} \biggr) \biggr]\,\mathrm{d}x
\\
&=& C\int_{\a d}^\infty \mathrm{e}^{-x} \int
_{\afrac{d}{\lambda\sqrt
n}}^{\sfrac{x}{\psi_n}} g'(y)\,\mathrm{d}y\,\mathrm{d}x
\\
&=& C\int_{\afrac{d}{\lambda\sqrt n}}^\infty g'(y) \int
_{\psi_n
y}^{\infty } \mathrm{e}^{-x}\,\mathrm{d}x \,\mathrm{d}y
\\
&=& C\int_{\afrac{d}{\lambda\sqrt n}}^\infty \mathrm{e}^{-\psi_n y}
g'(y) \,\mathrm{d}y.
\end{eqnarray*}
Hence,
\[
\sqrt{2\pi}\bigl|\mathit{II} (n)\bigr|\leq C \int_{\afrac{d}{\lambda\sqrt n}}^\infty
\mathrm{e}^{-\psi_n y} \,\mathrm{d}y= -\frac{C}{\psi_n} \mathrm{e}^{-\psi_n y} \biggl|_{\afrac{d}{\lambda\sqrt n}}^\infty
= \frac{C}{\lam\a\sqrt n} \mathrm{e}^{-\a d},
\]
and so
\[
\mathrm{e}^{\a d}\bigl|\mathit{II}(n)\bigr| = \mathrm{O} \biggl(\frac{1}{\sqrt n} \biggr).
\]
Finally,
\[
\sqrt{2\pi}\mathrm{e}^{\a d}J\leq \mathrm{o}(1)+2\mathrm{e}^{-\afrac{d^2}{2\lambda^2
n}}+\mathrm{O}\biggl(
\frac{1}{\sqrt{n}}\biggr),
\]
which shows \eqref{eq:p1} and
\[
\sqrt{2\pi}\mathrm{e}^{\a d}J= \mathrm{o}(1)+\mathrm{e}^{-\afrac{d^2}{2\lambda^2n}}\bigl(1+C(\theta)\d\bigr) +\mathrm{O}
\biggl(\frac{1}{\sqrt{n}}\biggr).
\]
{We may always take $\d=\d(n) =
\mathrm{o}(1)$. Hence \eqref{eq:p2} follows.}
\end{pf}
\end{appendix}
\section*{Acknowledgements}
The authors were supported in part by NCN Grant DEC-2012/05/B/ST1/00692.
|
{
"timestamp": "2015-04-14T02:13:46",
"yymm": "1504",
"arxiv_id": "1504.03144",
"language": "en",
"url": "https://arxiv.org/abs/1504.03144"
}
|
\section{Introduction}
This paper extends a body of classification results for countably infinite ordered structures,
under various homogeneity assumptions. As background we mention that
Morel \cite{Morel} classified the countable 1-transitive linear orders, of which there are $\aleph_1$,
Campero-Arena and Truss \cite{TrussGabi} extended this classification to
{\em coloured} countable 1-transitive linear orders, and Droste \cite{Droste} classified the
countable 2-transitive trees. The work of Droste was later generalised by Droste, Holland and Macpherson
\cite{Drost&Holl&Macph}
to give a classification of all countable `weakly 2-transitive' trees; there are $2^{\aleph_0}$ non-isomorphic such trees.
The goal of this paper, and of \cite{me}, is to extend this last classification result to a considerably richer class,
by working under a much weaker symmetry hypothesis, namely 1-transitivity.
We first define the terminology used above and later. A \textbf{tree} is a partial order
in which any two elements have a common lower bound and the lower bounds of any element are linearly ordered.
A relational structure is said to be \textbf{$k$-transitive} if for any two isomorphic $k$-element substructures there is an automorphism
taking the first to the second. For partial orders, there is a notion, called \textit{weak 2-transitivity}, that generalises that of 2-transitivity: a partial order is weakly 2-transitive if for any two 2-element chains there is an automorphism taking
the first to the second (but not necessarily for 2-element antichains).
A weaker notion still
is that of \textit{1-transitivity}.
The classification of countable 1-transitive trees is considerably more involved than that of
the weakly-2-transitive trees, and it rests on the classification of countable lower 1-transitive linear orders --- the subject of this paper.
\begin{definition}
\noindent A linear order $(X , \leqslant)$ is \textbf{lower 1-transitive} if
$$(\forall a, b \in X) \ \{ x \in X : x \leqslant a\} \cong \{x \in X : x \leqslant b\}.$$
\end{definition}
An example of a lower 1-transitive, not 1-transitive linear order
is $\omega^{\ast}$, (that is, $\omega$ reversed).
It is easy to see that any branch (that is, maximal chain) of a 1-transitive tree must be lower 1-transitive, though it is not necessarily 1-transitive.
The natural relation of equivalence between lower 1-transitive linear
orders is \textit{lower isomorphism}, rather than isomorphism.
\begin{definition}
Two linear orders, $(X , \leqslant)$ and $(Y , \leqslant)$ are \textbf{lower isomorphic} if
$$(\forall a \in X)(\forall b \in Y) \ \{ x \in X : x \leqslant a\} \cong \{y \in Y : y \leqslant b\}.$$
When this happens, we write $(X , \leqslant) \cong_l (Y , \leqslant)$.
\end{definition}
We shall use interval notation from now on where appropriate, that is,
$$(-\infty,a]:=\{ x \in X : x \leqslant a\}\, .$$
With this notation, the isomorphisms in the above definitions may then be written more succinctly as $(- \infty , a] \cong (- \infty , b]$.
The classification of countable lower 1-transitive linear orders is rather involved and so the current paper
is devoted entirely to this, and the resultant classification of countable 1-transitive trees is deferred to \cite{me}.
A principal feature of the classification of coloured 1-transitive countable linear orders,
given in \cite{gabithesis} and \cite{TrussGabi},
is the use of \textit{coding trees} to describe the construction of the orderings. In these papers, coding trees play a totally different role from that of
the 1-transitive trees which we aim to classify: they are classifiers, rather than structures being classified. In order to emphasise this distinction, and to be consistent
with previous references such as \cite{Droste} and \cite{gabithesis}, we adopt the convention that coding trees
`grow downwards',
that is, any two elements have a common upper bound, and the upper bounds of any element are linearly ordered.
Section~2 of this paper contains the definition of coding tree and related notions. Section~3 describes how to recover a linear order from a coding tree. The main work of the paper is in Section~4, where we show how to construct a coding tree from a linear order.
The main theorem is Theorem~\ref{T4}, which, in conjunction with Theorem~\ref{T3}, gives our classification.
In order to give the flavour of the classification, we conclude this introduction with some examples of lower 1-transitive linear orders. First, some notation and terminology are needed.
Let $(A, \leqslant)$,$(B, \leqslant)$ be linear orders; for convenience, we often omit the order symbol. Then $A.B$ denotes the lexicographic product of $A$ and $B$, where for $(a,b), (a^\prime, b^\prime) \in A \times B$,
$(a, b) \leqslant (a^\prime, b^\prime)$ if and only if $a < a^\prime$, or $a = a^\prime$ and $b \leqslant b^\prime$. Also, $A+B$ denotes $A$ followed by $B$, that is, the disjoint union of $A$ and $B$ with $a<b$ for all $a\in A$ and $b\in B$. We write $\dot{\ensuremath{\mathbb{Q}}}$ for ${\ensuremath{\mathbb{Q}}}+\{+\infty\}$. If $A$ is a linear order, then $A^*$ denotes the ordering with the same domain and the reverse order. If $n \in \mathbb{N} \cup \{ \aleph_0 \}$, then ${\ensuremath{\mathbb{Q}}}_n$ is a countable dense linear order coloured by $n$ colours $c_0,\ldots,c_{n-1}$ and such that between any two distinct points there is a point of each colour. Likewise, $\dot{\ensuremath{\mathbb{Q}}}_n$ is ${\ensuremath{\mathbb{Q}}}_n +\{+\infty\}$, where the point $+\infty$ is also coloured by any of the $c_i$, or indeed any other colour. If $Y_0,\ldots,Y_{n-1}$ are linear orders, ${\ensuremath{\mathbb{Q}}}_n(Y_0,\ldots,Y_{n-1})$ denotes the ordering
obtained by replacing each point of ${\ensuremath{\mathbb{Q}}}_n$ coloured $c_i$ by a convex copy of $Y_i$ (with the natural induced ordering).
If $n=\aleph_0$, we write
${\ensuremath{\mathbb{Q}}}_{\aleph_0}(Y_0,Y_1,\ldots)$.
The simplest countable lower 1-transitive linear orders are singletons, then ${\omega}^{\ast}$ and $\ensuremath{\mathbb{Z}}$ (which are lower isomorphic), and $\ensuremath{\mathbb{Q}}$ and $\dot{\ensuremath{\mathbb{Q}}} $ (which are also lower isomorphic). These orders are the basic building blocks for our constructions. We obtain new lower 1-transitive linear orders by concatenating and taking lexicographic products of existing ones. More precisely, Theorem~\ref{T3} implies that if $A$ and $B$ are any lower 1-transitive
linear orders which are lower isomorphic, then ${\omega}^{\ast}.A + B$ is lower 1-transitive.
For example, a lower isomorphism class (that is, a class of lower-isomorphic linear orders) consists of ${\ensuremath{\mathbb{Z}}}.{\ensuremath{\mathbb{Z}}}$, which by convention we write as ${\ensuremath{\mathbb{Z}}}^{2}$, ${\omega}^{\ast}.{\ensuremath{\mathbb{Z}}} + {\ensuremath{\mathbb{Z}}}$ and ${\omega}^{\ast}.{\ensuremath{\mathbb{Z}}} + {\omega}^{\ast}$.
Note that we can concatenate ${\omega}^{\ast}.{\ensuremath{\mathbb{Z}}}$ with either ${\ensuremath{\mathbb{Z}}}$ or ${\omega}^{\ast}$ and the resulting linear order will still be lower 1-transitive.
This is because ${\omega}^{\ast}$ has a right-hand endpoint and because ${\ensuremath{\mathbb{Z}}}$ and ${\omega}^{\ast}$ are lower isomorphic.
Notice that ${\omega}^{\ast}.A + A \cong {\omega}^{\ast}.A$. We use the former form to streamline subsequent definitions in the paper. A yet more complex lower isomorphism class is that of ${\ensuremath{\mathbb{Z}}}^{3}$, which includes
${\omega}^{\ast}.{\ensuremath{\mathbb{Z}}}^{2} + {\ensuremath{\mathbb{Z}}}^{2}$, $ {\omega}^{\ast}.{\ensuremath{\mathbb{Z}}}^{2} + {\omega}^{\ast}.{\ensuremath{\mathbb{Z}}} + {\ensuremath{\mathbb{Z}}}$ and
$ {\omega}^{\ast}.{\ensuremath{\mathbb{Z}}}^{2} + {\omega}^{\ast}.{\ensuremath{\mathbb{Z}}} + {\omega}^{\ast}$.
Theorem~\ref{T3} gives another construction of lower 1-transitive linear orders from existing ones. This construction involves the building block $\ensuremath{\mathbb{Q}}$: the linear order ${\ensuremath{\mathbb{Q}}}_n(Y_0,\ldots,Y_{n-1})$ (possibly with $n=\aleph_0$) is lower 1-transitive provided the $Y_{i}$ are lower isomorphic to each other. Moreover, as above, ${\ensuremath{\mathbb{Q}}}_n(Y_0,\ldots,Y_{n-1})+ Y$ is lower 1-transitive provided $Y$ and the $Y_{i}$ are all lower isomorphic to each other.
A simple example is $X = {\ensuremath{\mathbb{Q}}}_{2}({\omega^{\ast}}, {\ensuremath{\mathbb{Z}}})$, which is
countable and lower 1-transitive. Its lower isomorphism class also includes
${\ensuremath{\mathbb{Q}}}_{2}( {\omega^{\ast}}, {\ensuremath{\mathbb{Z}}}) + {\ensuremath{\mathbb{Z}}}$ and ${\ensuremath{\mathbb{Q}}}_{2}( {\omega^{\ast}}, {\ensuremath{\mathbb{Z}}}) + {\omega}^{\ast}$.
\section{Coding Trees}
This section introduces coding trees, which carry all the relevant information about lower 1-transitive linear orders.
First, recall that a downwards growing tree $(T,\leqslant)$ is \textbf{Dedekind-MacNeille complete} if its maximal chains are Dedekind-complete in the usual sense,
and if any two incomparable elements have a least upper bound. In fact, this is a special case of a general notion for partial
orders, and the basics are given, for example, in Chapter~7 of \cite{Davey}. Any tree $(T,\leq)$ has a unique (up to isomorphism over $T$) Dedekind-MacNeille completion, that is, a minimal Dedekind-MacNeille complete tree containing it, which is obtained as follows. If $A \subseteq T$ then $A^\mathrm{u}$ denotes the set of upper bounds of $A$ and $A^\mathrm{l}$ the set of lower bounds, that is,
$$A^\mathrm{u}:=\{x \in T : (\forall a \in A) \ (x \geqslant a) \},\ \mbox{ and}$$
$$A^\mathrm{l}:=\{x \in T : (\forall a \in A) \ (x \leqslant a) \}. \quad$$
A subset $A \neq \emptyset$ is an \textbf{ideal} of $T$ if
$(A^\mathrm{u})^\mathrm{l}=A$. If $x$ is any vertex of $T$, then the set $\mathrm{I}(x):= \{y \in T : y \leqslant x \}$ is an ideal of $T$. The Dedekind-MacNeille completion of $T$ is the set $\mathrm{I}^D(T)$ of the ideals of $T$ ordered by inclusion. It is easy to see that $T$ embeds in $\mathrm{I}^D(T)$ via the map which takes $x \in T$ to $\mathrm{I}(x) \in \mathrm{I}^D(T)$.
\begin{definition}
If $(T,\leqslant)$ is a downward growing tree, and $x\in T$, then a \textbf{child} of $x$ is some $y$ such
that $y<x$ and there is no $z\in T$ with $y<z<x$. If $x$ is a child of $y$ then $y$ is a \textbf{parent} of $x$.
We write ${\rm child}(x)$ for the set of children of $x$.
A \textbf{leaf} of $(T,\leqslant)$ is some $x\in T$ such that there is no $y \in T$ with $y < x$. We write ${\rm leaf}(T)$ for the set of leaves of $(T,\leqslant)$.
A \textbf{levelled tree} is a downward growing tree $(T , \leqslant)$ together with a partition, $\pi$, of $T$ into
maximal antichains, called \textbf{levels}, such that $\pi$ is linearly ordered by $\ll$ so that $x \leqslant y$ in $T$ implies
that the level containing $x$ is below the level containing $y$ in the $\ll$ ordering.
A \textbf{leaf-branch} $B$ of a (levelled) $(T,\leqslant)$ is a maximal chain of $T$ which contains a leaf.
The supremum of two incomparable points (which exists in the Dedekind-MacNeille completion of $T$,
even if not in $T$ itself) is called a \textbf{ramification point}.
If $x \in T$ then the relation $\asymp_{x}$ on
$\{ y \in T : y < x \}$ given by
\[a \asymp_{x} b\ \text{ if there is } c \in T \, \text{ such that } a, b \leqslant c < x\]
is an equivalence
relation. The equivalence classes are called \textbf{cones} at $x$.
\end{definition}
In the definitions of \textit{right} and \textit{left children} and \textit{coding trees} below, a tree $(T,\leqslant)$ is equipped with a labelling, that is, each vertex is labelled
by one of the symbols $\ensuremath{\mathbb{Z}}$, ${\omega}^{\ast}$,
$\ensuremath{\mathbb{Q}}$, $\dot{\ensuremath{\mathbb{Q}}}$, ${\ensuremath{\mathbb{Q}}}_{n}$, ${\dot{\ensuremath{\mathbb{Q}}}}_n$ ( for $2 \leqslant n \leqslant {\aleph}_{0}$), $\{ 1 \}$ (singleton), or $\mathrm{lim}$.
Isomorphisms between such trees are required to preserve the labelling.
\begin{definition}\label{A2} Let $x$ be a vertex of $T$ and $\triangleleft$ a linear order on ${\rm child}(x)$. If a vertex is labelled by one of ${\omega}^{\ast}$, ${\dot{\ensuremath{\mathbb{Q}}}}$ and ${\dot{\ensuremath{\mathbb{Q}}}}_{n}$, the \textbf{right} child of that vertex is the child which is greatest under the $\triangleleft$ ordering. All the remaining children are \textbf{left} children. If a vertex is labelled by one of $\ensuremath{\mathbb{Z}}$, $\ensuremath{\mathbb{Q}}$ or ${\ensuremath{\mathbb{Q}}}_{n}$, we consider all its
children to be \textbf{left} children.
The \textbf{left forest} of a vertex is defined to be the partially ordered set consisting of the
left children of the given vertex together with their descendants, with the induced structure of levels and labels.
Two forests are \textbf{isomorphic} provided the subtrees rooted at the
greatest elements in each forest can be put into one-to-one correspondence in such a way that they are isomorphic as trees.
\end{definition}
Thus, an isomorphism between two forests preserves the levelling and the labelling, but it is not required to preserve the $\triangleleft$ ordering among children.
\begin{definition}
\label{A1}
A \textbf{coding tree} has the form $( T, \leqslant , \triangleleft, \varsigma , \ll)$ where\\
1. $T$ is a levelled tree with a greatest element, the root. The tree ordering is $\leqslant$, $\triangleleft$ is a linear ordering on the set of children of each parent and $\ll$ is the ordering of the levels.\\
2. There are countably many leaves.\\
3. Every vertex is a leaf or is above a leaf.\\
4. $T$ is Dedekind-MacNeille complete.\\
5. The vertices are labelled by $\varsigma$, the labelling function, which assigns to the vertices one of the following labels: $\ensuremath{\mathbb{Z}}$, ${\omega}^{\ast}$,
$\ensuremath{\mathbb{Q}}$, $\dot{\ensuremath{\mathbb{Q}}}$, ${\ensuremath{\mathbb{Q}}}_{n}$, ${\dot{\ensuremath{\mathbb{Q}}}}_n$ ( for $2 \leqslant n \leqslant {\aleph}_{0}$), $\{ 1 \}$ (singleton), or $\mathrm{lim}$.\\
6. For any two vertices $x_{i}$ and $x_{j}$ on the same level, $\varsigma (x_{i}) \ {\cong}_{l} \ \varsigma (x_{j})$ or $\varsigma (x_{i}) \ = \ \varsigma (x_{j})$.\\
7. For any vertex $x$ of the tree:\\
if $\varsigma (x) = {\ensuremath{\mathbb{Z}}}$ or $\ensuremath{\mathbb{Q}}$ then $x$ has one child;\\
if $\varsigma (x) = {\omega}^{\ast}$ or $\dot{\ensuremath{\mathbb{Q}}}$ then $x$ has two children;\\
if $\varsigma (x) = {\ensuremath{\mathbb{Q}}}_{n}$ then $x$ has $n$ children;\\
if $\varsigma (x) = {\dot{\ensuremath{\mathbb{Q}}}}_n$ then $x$ has $n +1$ children;\\
if $\varsigma (x) = \{ 1 \} $ then $x$ is a leaf and has no children;\\
if $\varsigma (x) = \mathrm{lim}$ then there is only one cone at $x$ ($x$ is not a leaf and has no children).\\
8. At each given level of $T$, the left forests of vertices at that level are all isomorphic in the sense of Definition~\ref{A2}.\\
9. If $x$ is a parent vertex and $y_{0}, y_{1}$ are two of its left children, then the subtrees with roots $y_{0}, y_{1}$ are not isomorphic.
\end{definition}
We will not explain how to define a linear order from a coding tree until Section~\ref{sec3}, but we illustrate Definition~\ref{A1} in Figure~\ref{Fig1}, where we give the coding trees for the lower 1-transitive linear orders in the lower isomorphism class of $\ensuremath{\mathbb{Z}}^3$, that is, $\ensuremath{\mathbb{Z}}^3$, $\omega^*.\ensuremath{\mathbb{Z}}^2 + \ensuremath{\mathbb{Z}}^2$, $\omega^*.\ensuremath{\mathbb{Z}}^2 + \omega^*.\ensuremath{\mathbb{Z}} +\ensuremath{\mathbb{Z}}$ and $\omega^*.\ensuremath{\mathbb{Z}}^2 + \omega^*.\ensuremath{\mathbb{Z}} +\omega^*$.
\newpage
\begin{figure}
\begin{tikzpicture}
\node [draw,circle,inner sep = 1pt,fill, label=right:$\ensuremath{\mathbb{Z}}$] (0) at (-6, 2) {};
\node [draw,circle,inner sep = 1pt,fill, label=right:$\ensuremath{\mathbb{Z}}$] (1) at (-6, -0) {};
\node [draw,circle,inner sep = 1pt,fill, label=right:$\ensuremath{\mathbb{Z}}$] (2) at (-4, -0) {};
\node [draw,circle,inner sep = 1pt,fill, label=right:$\ensuremath{\mathbb{Z}}$] (3) at (-4, 1) {};
\node [draw,circle,inner sep = 1pt,fill, label=right:$\ensuremath{\mathbb{Z}}$] (4) at (-2, 1) {};
\node [draw,circle,inner sep = 1pt,fill, label=right:$\ensuremath{\mathbb{Z}}$] (5) at (-2, -0) {};
\node [draw,circle,inner sep = 1pt,fill, label=right:${\omega}^{\ast}$] (6) at (-3, 2) {};
\node [draw,circle,inner sep = 1pt,fill, label=right:$\ensuremath{\mathbb{Z}}$] (7) at (0, 1) {};
\node [draw,circle,inner sep = 1pt,fill, label=right:$\ensuremath{\mathbb{Z}}$] (8) at (0, -0) {};
\node [draw,circle,inner sep = 1pt,fill, label=right:${\omega}^{\ast}$] (9) at (1, 2) {};
\node [draw,circle,inner sep = 1pt,fill, label=right:${\omega}^{\ast}$] (10) at (2, 1) {};
\node [draw,circle,inner sep = 1pt,fill, label=right:$\ensuremath{\mathbb{Z}}$] (11) at (1.5, -0) {};
\node [draw,circle,inner sep = 1pt,fill, label=right:$\ensuremath{\mathbb{Z}}$] (12) at (2.5, -0) {};
\node [draw,circle,inner sep = 1pt,fill, label=right:$\ensuremath{\mathbb{Z}}$] (13) at (4, -0) {};
\node [draw,circle,inner sep = 1pt,fill, label=right:$\ensuremath{\mathbb{Z}}$] (14) at (4, 1) {};
\node [draw,circle,inner sep = 1pt,fill, label=right:${\omega}^{\ast}$] (15) at (5, 2) {};
\node [draw,circle,inner sep = 1pt,fill, label=right:${\omega}^{\ast}$] (16) at (6, 1) {};
\node [draw,circle,inner sep = 1pt,fill, label=right:$\ensuremath{\mathbb{Z}}$] (17) at (5.5, -0) {};
\node [draw,circle,inner sep = 1pt,fill, label=right:${\omega}^{\ast}$] (18) at (6.5, -0) {};
\node [draw,circle,inner sep = 1pt,fill, label=right:$\ensuremath{\mathbb{Z}}$] (19) at (-6, 1) {};
\node [draw,circle,inner sep = 1pt,fill, label=below:$\{1\}$] (21) at (-6, -1) {};
\node [draw,circle,inner sep = 1pt,fill, label=below:$\{1\}$] (22) at (-4, -1) {};
\node [draw,circle,inner sep = 1pt,fill, label=below:$\{1\}$] (23) at (-2, -1) {};
\node [draw,circle,inner sep = 1pt,fill, label=below:$\{1\}$] (24) at (0, -1) {};
\node [draw,circle,inner sep = 1pt,fill, label=below:$\{1\}$] (25) at (4, -1) {};
\node [draw,circle,inner sep = 1pt,fill, label=below:$\{1\}$] (26) at (1.5, -1) {};
\node [draw,circle,inner sep = 1pt,fill, label=below:$\{1\}$] (27) at (2.5, -1) {};
\node [draw,circle,inner sep = 1pt,fill, label=below:$\{1\}$] (28) at (5.5, -1) {};
\node [draw,circle,inner sep = 1pt,fill, label=below:$\{1\}$] (29) at (6.25, -1) {};
\node [draw,circle,inner sep = 1pt,fill, label=below:$\{1\}$] (30) at (6.75, -1) {};
\node at (-6, -2.5) {$\ensuremath{\mathbb{Z}}^3$};
\node at (-3, -2.5) {$\omega^*.\ensuremath{\mathbb{Z}}^2 + \ensuremath{\mathbb{Z}}^2$};
\node at (1.5, -2.5) {$\omega^*.\ensuremath{\mathbb{Z}}^2 + \omega^*.\ensuremath{\mathbb{Z}} +\ensuremath{\mathbb{Z}}\quad$};
\node at (5.5, -2.5) {$\quad \omega^*.\ensuremath{\mathbb{Z}}^2 + \omega^*.\ensuremath{\mathbb{Z}} +\omega^*$};
\draw (0) to (1);
\draw (6) to (3);
\draw (3) to (2);
\draw (6) to (4);
\draw (4) to (5);
\draw (9) to (7);
\draw (7) to (8);
\draw (9) to (10);
\draw (10) to (11);
\draw (10) to (12);
\draw (15) to (14);
\draw (14) to (13);
\draw (15) to (16);
\draw (16) to (17);
\draw (16) to (18);
\draw (1) to (21);
\draw (2) to (22);
\draw (5) to (23);
\draw (8) to (24);
\draw (11) to (26);
\draw (12) to (27);
\draw (13) to (25);
\draw (17) to (28);
\draw (18) to (29);
\draw (18) to (30);
\end{tikzpicture}
\caption{Coding trees for lower 1-transitive linear orders in the lower isomorphism class of $\ensuremath{\mathbb{Z}}^3$}
\label{Fig1}
\end{figure}
In order to recover a lower 1-transitive linear order from a coding tree, we will need \textit{expanded coding
trees}, which are closely related to coding trees and are defined next. The reason why we need expanded coding trees should become clear in Section~\ref{sec3}. In place of a labelling function on vertices, expanded coding trees carry, as part of the structure,
a total ordering on the set of children of each vertex, induced by a binary relation~$\triangleleft$. In general, a coding tree and the corresponding expanded coding tree do not have the same vertex set.
For instance, a point of the expanded
coding tree corresponding to a point labelled $\dot{\ensuremath{\mathbb{Q}}}$ in the coding tree will have infinitely many children in the expanded coding tree. All the children but the last one are associated with the left child in the coding tree.
The idea is that a lower 1-transitive linear order $(X,\leqslant)$ lives on the set of leaves of the expanded coding tree, so the expanded coding tree facilitates the transition
between coding tree and encoded order.
\begin{definition}
\label{A3} An \textbf{expanded coding tree} is a structure of the form $( E, \leqslant , \ll, \triangleleft)$ where:\\
1. $E$ is a levelled tree with a greatest element, the root, denoted by $r$. The tree ordering is $\leqslant$, $\ll$ is the ordering of the levels and $ \triangleleft$ is the ordering of the children of each parent vertex.\\
2. $(E,\triangleleft)$ is a partial ordering consisting of a disjoint union of antichains whose elements are exactly the levels of $(E,\leqslant,\ll)$.\\
3. $(E,\leqslant)$ has at most countably many leaves.\\
4. Every vertex of $(E,\leqslant)$ is a leaf or is above a leaf.\\
5. $(E,\leqslant)$ is Dedekind-MacNeille complete.\\
6. If a vertex has any children, then their $\triangleleft$-order type is one of the following; $\ensuremath{\mathbb{Z}}$, ${\omega}^{\ast}$,
$\ensuremath{\mathbb{Q}}$, $\dot{\ensuremath{\mathbb{Q}}}$, ${\ensuremath{\mathbb{Q}}}_{n}$ or ${\dot{\ensuremath{\mathbb{Q}}}}_n$ for $2 \leqslant n \leqslant {\aleph}_{0}$.\\
7. Any two vertices $x$ and $x'$ on the same level are either both parent vertices, or they are both leaves, or
they both have exactly one cone below them. If $x$ and $x'$ are both parent vertices, then
$({\rm child}(x),\triangleleft) \cong_l ({\rm child}(x'),\triangleleft)$. \\
8. For any parent vertex $x$ of the tree, one of the following holds:
\begin{enumerate}
\item[(i)] the $\triangleleft$-order type of ${\rm child}(x)$ is ${\ensuremath{\mathbb{Z}}}$, $\ensuremath{\mathbb{Q}}$, $ {\omega}^{\ast}$ or $\dot{\ensuremath{\mathbb{Q}}}$ and the left trees rooted at the children of $x$ are all isomorphic, or
\item[(ii)] the children of $x$ are densely ordered by $\triangleleft$ and the trees rooted at the children of $x$
fall into $n \geqslant 2$ isomorphism classes and this makes them isomorphic to $ {\ensuremath{\mathbb{Q}}}_{n}$
(for $2 \leqslant n \leqslant {\aleph}_{0}$), or
\item[(iii)] the left children are as in (ii) above, and $x$ has a right child and this makes $\mathrm{child}(x)$ order-isomorphic to ${\dot{\ensuremath{\mathbb{Q}}}}_n$.
\end{enumerate}
9. At each given level of $E$ the left forests (see Definition~\ref{A2}) from that level are order-isomorphic (meaning that $\leqslant , \ll, \triangleleft$ are preserved).
\end{definition}
In 8(ii), we mean that if the elements of ${\rm child}(x)$ are coloured according to the isomorphism type of the trees below
them, then the corresponding coloured linear order
(with respect to $\triangleleft$) is isomorphic to ${\ensuremath{\mathbb{Q}}}_n$; likewise in 8(iii).
\section{Construction of a Linear Order from a Coding Tree} \label{sec3}
In this section we describe the relationship between a coding tree and expanded coding tree, and explain how the latter determines a lower 1-transitive linear order. For the coding trees in Figure~\ref{Fig1} it is possible to start either at the root, or at the leaves, and inductively proceed through the tree to determine
the linear order encoded by it. However, Definition~\ref{A1} does not imply that the levels of a coding tree are well ordered or conversely well ordered. Consider the example in Figure~\ref{Fig2}.
\newpage
\begin{figure}
\begin{tikzpicture}[scale=0.8]
\node [draw,circle,inner sep = 1pt,fill] (0) at (0, 3) {};
\node [draw,circle,inner sep = 1pt,fill] (1) at (0, 2) {};
\node [draw,circle,inner sep = 1pt,fill] (2) at (0, 1) {};
\node [draw,circle,inner sep = 1pt,fill] (3) at (1.5, 1) {};
\node [draw,circle,inner sep = 1pt,fill] (4) at (-1.5, 1) {};
\node [draw,circle,inner sep = 1pt,fill] (5) at (-0.5, -0) {};
\node [draw,circle,inner sep = 1pt,fill] (6) at (0, -0) {};
\node [draw,circle,inner sep = 1pt,fill] (7) at (0.5, -0) {};
\node [draw,circle,inner sep = 1pt,fill] (8) at (1, -0) {};
\node [draw,circle,inner sep = 1pt,fill] (9) at (1.5, -0) {};
\node [draw,circle,inner sep = 1pt,fill] (10) at (2, -0) {};
\node [draw,circle,inner sep = 1pt,fill] (11) at (-1, -0) {};
\node [draw,circle,inner sep = 1pt,fill] (12) at (-1.5, -0) {};
\node [draw,circle,inner sep = 1pt,fill] (13) at (-2, -0) {};
\node [draw,circle,inner sep = 1pt,fill] (14) at (3, -0) {};
\node [draw,circle,inner sep = 1pt,fill] (15) at (3.5, -0) {};
\node [draw,circle,inner sep = 1pt,fill] (16) at (4, -0) {};
\node [draw,circle,inner sep = 1pt,fill] (17) at (4.5, -0) {};
\node [draw,circle,inner sep = 1pt,fill] (18) at (5, -0) {};
\node [draw,circle,inner sep = 1pt,fill] (19) at (5.5, -0) {};
\node [draw,circle,inner sep = 1pt,fill] (20) at (6, -0) {};
\node [draw,circle,inner sep = 1pt,fill] (21) at (6.5, -0) {};
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\node [draw,circle,inner sep = 1pt,fill] (57) at (-6, -2) {};
\node at (11, 2) {\small The labels on these levels are all ${\dot{\ensuremath{\mathbb{Q}}}}_2$};
\node at (11, -1) {\small The labels on this level are all $\mathrm{lim}$};
\node at (11,-2) {\small The labels on this level are all $\{1\}$};
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\end{tikzpicture}
\caption{A coding tree that is neither well founded nor conversely well founded}
\label{Fig2}
\end{figure}
In this tree, there are infinitely many levels of vertices labelled $\dot{\ensuremath{\mathbb{Q}}}_2$. The branches are maximal chains which will eventually constantly descend through the right children of $\dot{\ensuremath{\mathbb{Q}}}_2$. This tree is a coding tree, yet it is neither well founded nor conversely well founded. Examples of this kind are the reason why expanded coding tree are necessary to recover a lower 1-transitive linear order from a coding tree.
As in \cite{TrussGabi}, we start by defining a map which
associates an expanded coding tree to a coding tree.
\begin{definition}
\label{d1} Let $( T, \leqslant , \varsigma , \ll, \triangleleft)$ be a coding tree, and $( E, \leqslant , \ll, \triangleleft)$ be
an expanded coding tree. We say that $E$ is \textbf{associated} with $T$ via $\phi$ if there is a function $\phi: E \to T$ which
takes the root $r$ of $E$ to the root of $T$, each leaf of $E$ to some leaf of $T$, and\\
(i) $v_{1} \leqslant v_{2}$ $\implies$ $\phi(v_{1}) \leqslant \phi(v_{2})$,\\
(ii) $\phi$ induces an order-isomorphism from the set of levels of $E$ (ordered by $\ll$) to the set of levels of $T$.
\\
(iii) for each vertex $v$ of $E$, $\phi$ maps $\{ u \in E : u \leqslant v \}$ onto $\{ u \in T : u \leqslant \phi(v) \}$, and
for any leaf $l$ of $E$, $\phi$ maps $[l, r]$ onto $[\phi(l), \phi(r)]$,\\
(iv) for each parent vertex $v$ of $E$, one of the following holds:\\
- $\varsigma(\phi(v)) =$ $\ensuremath{\mathbb{Z}}$, ${\omega}^{\ast}$, $\ensuremath{\mathbb{Q}}$, $\dot{\ensuremath{\mathbb{Q}}}$, and this is
the order type of the children of $v$ under $\triangleleft$; \\
- $\varsigma(\phi(v)) =$ ${\ensuremath{\mathbb{Q}}}_{n}$, ${\dot{\ensuremath{\mathbb{Q}}}}_n$ (for
$2 \leqslant n \leqslant {\aleph}_{0}$) and for any left children $u$, $u^\prime$ of $v$, if the trees rooted at $u$ and $u^\prime$ are isomorphic then $\phi(u)=\phi(u^\prime)$;\\
- $\varsigma(\phi(v)) = \mathrm{lim}$ if $v$ is
neither a parent nor a leaf (in which case $v$ has just one cone); \\
- $\varsigma(\phi(v)) = \{ 1 \}$ if $v$ is a leaf.
The map $\phi$ is said to be an \textbf{association map} between $T$ and $E$.
\end{definition}
We are now in a position to say explicitly how a tree encodes a linear order. First note
that if $E$ is an expanded coding tree, then there is a natural linear order on ${\rm leaf}(E)$ which we denote by $\triangleleft^*$ and call the \textbf{leaf order}.
If $x,y$ are leaves, we write $x\triangleleft^*y$ if there are $x',y'\in E$ with $x\leqslant x'$, $y\leqslant y'$,
and $x' \triangleleft y'$.
\begin{definition}
\label{d2}
The coding tree $( T, \leqslant , \varsigma , \ll, \triangleleft)$ \textbf{encodes} the linear order $(X, \leqslant)$ if there is an expanded coding tree associated with $T$ such that $X$ is (order) isomorphic to the set of leaves of $E$ under the leaf order induced by $\triangleleft$.
\end{definition}
In Theorem~\ref{T1} below, we show how to recover a linear order from a coding tree. In order to do this, we need to define certain functions called \textit{decoding functions}, whose domains are the leaf--branches of a given coding tree and which take a vertex $x$ to an element of the ordered set $\varsigma(x)$. To cut down to a countable set of functions, even when the coding tree is not well founded or conversely well founded, we begin by choosing arbitrary default values for each of the labels. For each of $\ensuremath{\mathbb{Z}}$ and $\ensuremath{\mathbb{Q}}$, there is one default value. In the cases of ${\omega}^{\ast}$, $\dot{\ensuremath{\mathbb{Q}}}$ and ${\dot{\ensuremath{\mathbb{Q}}}}_n$, there are two default values, one for the end points and one other. In the case of ${\ensuremath{\mathbb{Q}}}_{n}$, there are $n$ default values, one of each `colour', whereas $\dot{\ensuremath{\mathbb{Q}}}_n$ has the same default values as $\ensuremath{\mathbb{Q}}_n$ plus an additional one for the endpoint.
\begin{definition}
\label{d3} A \textbf{decoding function} is a function $f$ defined on a leaf-branch $B$ of $T$ and such that \\[3pt]
-- the set of non-default values taken by $f$ is finite \\[1pt]
-- for each $x \in B$ with $\varsigma (x) \neq$ $\mathrm{lim}$, $f(x) \in \varsigma (x)$ \\[1pt]
-- if $x$ is a parent vertex and a left child of $x$ is in $\mathrm{dom}\, f$, then $f(x) \neq d_e$, where $d_e$ is the default value for the endpoint \\[1pt]
-- if $x$ is a parent vertex and the right child of $x$ is in $\mathrm{dom}\, f$, then $f(x) = d_e$, \\[1pt]
-- if $\varsigma(x) = \ensuremath{\mathbb{Q}}_n\,$ or $ \dot{\ensuremath{\mathbb{Q}}}_n \,$ and $\mathrm{dom}\, f$ contains a left child of $x$ with `colour' $m$, then $f(x)$ has the colour $m$\\[3pt]
If $\varsigma (x) = \, \mathrm{lim}$, we consider $f$ to be undefined at $x$.
\end{definition}
\begin{theorem}
\label{T1}
Any coding tree encodes some linear order, and any two linear orders encoded by the same coding tree are isomorphic.
\end{theorem}
\begin{proof}
We proceed as in \cite{TrussGabi}. Given a coding tree $T$, we construct an expanded coding tree which is associated with $T$ as in Definition~\ref{d1}.
Let $\Sigma_{T}$ be the set of decoding functions on $T$ ordered by $<$, where for $f, g \in \Sigma_T$, $f < g$ if $f(x_0) < g(x_0)$ with $x_0$ the greatest point for which $f(x) \neq g(x)$. We show that $<$ is a linear ordering. Let $f$ and $g$ be decoding functions such that $f \neq g$. Consider the greatest vertex $x$ such that $f(x) \neq g(x)$. Such a vertex exists since for $\mathrm{dom}\,f = \mathrm{dom} g$, $f$ and $g$ differ only finitely often. In the case where $\mathrm{dom\,}f \neq \mathrm{dom}\, g$, we appeal to the fact that $T$ is Dedekind-MacNeille complete and therefore it contains all its ramification points, hence there is a vertex $x \in\mathrm{dom}\, f \cap \mathrm{dom}\, g$ such that $f(x) \neq g(x)$. Then $f(x) < g(x) \Rightarrow f < g$ and $f(x) > g(x) \Rightarrow f > g$. It is clear this relation is irreflexive and transitive, hence $(\Sigma_{T}, <)$ is a linear order.
In order for $T$ to encode $(\Sigma_{T}, <)$ according to Definition~\ref{d2}, we must produce an expanded coding tree associated with $T$. Such a tree is given by
$$ E = \{ (x, f\upharpoonright(x, r]) : f \in \Sigma_{T}, x \in \mathrm{dom}\, f \}.$$
The tree ordering is given by letting $(x, f\upharpoonright(x, r]) \leqslant (y, f\upharpoonright(y, r])$ if $x \leqslant y \in \mathrm{dom}\, f$. In addition $(v_{1}, f\upharpoonright(v_{1}, r]) $ is level with $ (v_{2}, g\upharpoonright(v_{2}, r])$ if and only if $v_{1}$ is level with $v_{2}$. It is now clear that $E$ is a levelled tree. Its root is $(r, \varnothing)$. Also, any $(x, f\upharpoonright(x, r])$ lies above a leaf $(l, f\upharpoonright(l, r])$ where $l$ is a leaf in $\mathrm{dom}\, f$.
Each leaf-branch of $E$ is isomorphic to a leaf-branch of $T$, and so it is Dedekind complete. Furthermore, since $T$ contains all its ramification points, so does $E$, and therefore $E$ is Dedekind-MacNeille complete.
Now we consider the possible order types of sets of children of vertices of $E$. Let $(x, f\upharpoonright(x, r])$ be a parent vertex in $E$. Then $x$ is a parent vertex in $T$. The order type of the children of $(x, f\upharpoonright(x, r])$ is determined by
\[\{(x, g\upharpoonright[x, r]) : g \in \Sigma_{T}, x \in \mathrm{dom}\,g, f\upharpoonright(x, r] = g\upharpoonright(x, r] \}\, . \]
Since $x$ is a parent vertex, $f(x) \in \varsigma(x)$. Hence the order type of the children of $(x, f\upharpoonright(x, r])$ is equal to the label of $x$ in $T$. In the case of $\varsigma(x) = {\ensuremath{\mathbb{Q}}}_{n}$ or ${\dot{\ensuremath{\mathbb{Q}}}}_n$ we may say that the `coloured' order type of the children in $E$ is ${\ensuremath{\mathbb{Q}}}_{n}$ or ${\dot{\ensuremath{\mathbb{Q}}}}_n$.\\
If $(x, f\upharpoonright(x, r])$ is neither a parent vertex nor a leaf, then $x$ is neither a parent nor a leaf, and so $x$ is labelled $\mathrm{lim}$.
The mapping $\phi$ is given by $\phi((x, f\upharpoonright(x, r])) = x$. This preserves root, leaves and, as we have just seen, it preserves the relation between labels of vertices in $T$ and the (coloured) order type of the children of those vertices in $E$. Also $x \leqslant y \Rightarrow \phi(x) \leqslant \phi(y)$ and it is clear that for each vertex $x$ of $E$, $\phi$ maps $\{ u \in E : u \leqslant x \}$ onto $\{ u \in T : u \leqslant \phi(x) \}$ and for any leaf $l$ of $E$, $\phi$ maps $[l, r]$ onto $[\phi(l), \phi(r)]$. Therefore $E$ is associated with $T$ and $\Sigma_{T}$ is order isomorphic to the set of leaves of $E$. Hence $T$ encodes $\Sigma_{T}$.
A back-and-forth argument shows that any two countable linear orders encoded by the same coding tree $( T, \leqslant , \varsigma , \ll, \triangleleft)$ are isomorphic.
\end{proof}
\begin{theorem}
\label{T3}
The ordering $\Sigma_{T}$ encoded by the coding tree $( T, \leqslant , \varsigma , \ll, \triangleleft)$ is countable and lower 1-transitive.
\end{theorem}
\begin{proof}
The way in which $\Sigma_{T}$ has been defined ensures that it is countable.
We now show that $\Sigma_{T}$ is lower 1-transitive. Take any $f, g \in \Sigma_{T}$ and consider the initial segments $( -\infty , f]$ and $( -\infty , g]$. Now $\Sigma_{T}$ is defined to be the set of all functions on the leaf-branches of $T$ which take a default value at all but finitely many points. By definition of the ordering on $\Sigma_{T}$, an initial segment of $\Sigma_{T}$ at $f$ can be written as
$$( -\infty , f] = \{ f \} \cup \{ p \in \Sigma_{T} : (\exists x \in \mathrm{dom}\, f )( p(x) < f(x)) \wedge (\forall y > x)( p(y) = f(y)) \}.$$
Let $L_{i}$ be the $i$th level of the tree, and let
$$\Gamma^{f}_{i} = \{ p \in ( -\infty , f] : p(x^{f}_{i}) < f(x^{f}_{i}) \wedge (\forall y > x^{f}_{i})( p(y) = f(y)) \},$$
where $x^{f}_{i}$ denotes the element of $\mathrm{dom}\, f$ on the level $L_{i}$, and $x_{i}$ denotes the element of $\mathrm{dom}\, p$ in $L_{i}$, where $p$ is a typical member of $\Sigma_{T}$.)
Then, by definition of the $\ll$ - ordering of the levels, it is clear that $( -\infty , f]$ is the disjoint union of all the $\Gamma^{f}_{i}$, and furthermore that $i \ll j \Rightarrow \Gamma^{f}_{i} > \Gamma^{f}_{j}$ (where this means that every element of $\Gamma^{f}_{i}$ is greater than every element of $\Gamma^{f}_{j}$). Since the same is true of the $\Gamma^{g}_{i}$, to show that $( -\infty , f] \cong ( -\infty , g]$, it suffices to show that $\Gamma^{f}_{i} \cong \Gamma^{g}_{i}$ for each $i$, and the desired isomorphism from $( -\infty , f]$ to $ ( -\infty , g]$ is obtained by patching together all the individual isomorphisms.
We remark that for $\varsigma (x^{f}_{i}) = \, \mathrm{lim}$, we have $\Gamma^{f}_{i} = \varnothing$. The label $\mathrm{lim}$ is not a linear order, so by condition 3 of Definition~\ref{A1}, if a vertex on level $i$ is labelled $\mathrm{lim}$, then all vertices on level $i$ are labelled $\mathrm{lim}$. This shows that when $i$ is a level with vertices labelled $\mathrm{lim}$, we have $\Gamma^{f}_{i} \cong \Gamma^{g}_{i}$.
We now consider the cases where the vertices on level $i$ are not labelled $\mathrm{lim}$. There is an isomorphism $\varphi$ from $( - \infty , f(x^{f}_{i})] \cap \varsigma (x^{f}_{i}) $ to $( - \infty , g(x^{g}_{i})] \cap \varsigma (x^{g}_{i})$. Moreover, there is an isomorphism $\psi$ between the left forests at the points $x_i^f$ and $x_i^g$. Now let $x_j$ be the member of $\mathrm{dom}\,p$ at the level $L_j$ for a typical $p$. We now define
\begin{displaymath}
\Phi_{i} (p) (\psi (x_j))= \left\{ \begin{array} {ll}
g(x^{g}_{j}) & \textrm{ if } j > i\\
\varphi (p(x_{j})) & \textrm{ if } j = i\\
p(x_{j}) & \textrm{ if } j < i\\
\end{array} \right.
\end{displaymath}
where $p \in \Gamma^{f}_{i}$.
We must now show that $\Gamma^{f}_{i}$ is mapped 1-1 into $\Gamma^{g}_{i}$ by $ \Phi_{i}$. This gives our result. We have that $ \Phi_{i}(p) \in \Sigma_{T}$, because all such $ \Phi_{i}(p)$ are defined on leaf-branches of $T$ and they take a default value at all but finitely many points, since both $ g(x^{g}_{j})$ and $p(x_{j})$ take the default value at all but finitely many points (possibly with $\varphi (p(x_{j}))$ in addition).
It is easy to see that $\Phi_i$ is surjective. For injectivity, suppose $ \Phi_{i}(p_{1}) = \Phi_{i}(p_{2})$. Then since $p_{1} , p_{2} \in \Gamma^{f}_{i}$, $p_{1} (x_{j}) = p_{2} (x_{j}) = f(x^{f}_{j})$ for all $j > i$ and $p_{1} (x_{j}) = p_{2} (x_{j})$ for $j < i$ by the third clause. Since $\varphi$ is an isomorphism, $\varphi (p_{1} (x_{i})) = \varphi (p_{2} (x_{i}))$ implies that $p_{1} (x_{j}) = p_{2} (x_{j})$. Hence $p_{1} (x_{j}) = p_{2} (x_{j})$ for all $j$.\\
\end{proof}
\section{Construction of a Coding Tree from a Linear Order}
In this section we complete the classification by showing that any countable and lower 1-transitive $(X, \leqslant)$ is encoded by a suitable coding tree. We first find the associated expanded coding tree for $(X, \leqslant)$. This is done by building a tree of \textit{invariant partitions} of $(X, \leqslant)$ in the sense of Definition~\ref{D2} below. We show this is in fact an expanded coding tree for $(X, \leqslant)$ and then give the association map between them.
\begin{definition}
\label{D2}
An \textbf{invariant partition} of $X$ is a partition $\pi$ into convex subsets, called \textbf{parts}, which is invariant under lower isomorphims of $(X, \le)$ into itself. That is, for any $a, b \in X$, any order isomorphism $f : ( -\infty , a] \to ( - \infty , b]$, and any $x, y \leqslant a$,
$$x \sim_{\pi} y \iff f(x) \sim_{\pi} f(y).$$
\end{definition}
The family $I$ of all parts of invariant partitions of $X$ is partially ordered by inclusion. This allows us to define a levelled tree structure on $I$.
\begin{definition}
\label{D3}
For a lower 1-transitive linear order $(X, \le)$, the \textbf{invariant tree} associated with $X$ is the levelled tree $I$ whose vertices are parts in the invariant partitions of $X$ ordered by $\subseteq$ in such a way that
\begin{itemize}
\item[(i)] $X \in I$ is the root
\item[(ii)] each level is an invariant partition of $X$
\item[(iii)] the leaves are the singletons $\{ x \}$ for $x \in X$
\item[(iv)] every invariant partition of $X$ into convex subsets of $X$ is represented by a level of vertices in $I$.
\end{itemize}
\end{definition}
We remark that $I$ has a root since $X$ is itself lower 1-transitive and a convex subset of $X$. Moreover, the parts of any invariant partition of $X$ are lower isomorphic and lower 1-transitive. Lemmas~\ref{F2} and~\ref{F3} show that for any countable, lower 1-transitive linear order, the family $I$ is a levelled tree, thereby justifying the description \emph{the} invariant tree. The proof of Lemma~\ref{F2} is left to the reader.
\begin{lemma}
\label{F2}
If $(X, \leqslant)$ is a countable lower 1-transitive linear order and $\pi$ is an invariant partition of $X$, then $X/_{\sim_{\pi}}$ is also a countable lower 1-transitive linear order with the ordering induced by $(X, \leqslant)$.
\end{lemma}
\begin{definition} Let $\pi_i, \pi_j$ be invariant partitions of $(X, \leqslant)$. We say that $\pi_{i} $ is a \textbf{refinement} of $\pi_{j}$ if every element of $\pi_{j}$ is a union of members of $\pi_{i}$.\end{definition}
\begin{lemma}
\label{F3}
Given any two nontrivial invariant partitions $\pi_{1} , \pi_{2}$ of $X$ into convex subsets of $X$, one is a refinement of the other, and moreover $\pi_1$ and $\pi_2$ have no part in common.
\end{lemma}
\begin{proof}
Let $\sim_1, \sim_2$ be the equivalence relations defining $\pi_{1} , \pi_{2}$ respectively.
We want to show that
$$ (\forall x , y \in X)(x \sim_1 y \Rightarrow x \sim_2 y) \vee (\forall x , y \in X)(x \sim_2 y \Rightarrow x \sim_1 y). $$
Suppose both disjuncts are false. Then there are $x, y, u, v$ such that
\begin{itemize}
\item $x \sim_{1} y$ and $x \nsim_{2} y$, and
\item $u \nsim_{1} v$ and $u \sim_{2} v$.
\end{itemize}
We may assume that $x < y$ and $u < v$. Let $f : (- \infty , y] \to (- \infty , v]$ be an isomorphism. Then
$f(x) < v$ and $f(x) \sim_1 v$. Moreover, we must have $u < f(x)$, otherwise $u \sim_1 v$ by convexity. So
$u < f(x) < v$, and therefore $f(x) \sim_2 v$ by convexity. However, $x \nsim_2y$ implies $f(x) \nsim_2 v$, which is a contradiction.
Without loss of generality, assume that $\pi_1$ is a refinement of $\pi_2$. We want to show that $\pi_1 \cap \pi_2 = \varnothing$. So suppose for a contradiction that there is $p \in \pi_1 \cap \pi_2$, and let $x, y \in X$ be such that $x \nsim_1y$ and $x \sim_2y$. Pick $z \in p$ and let
$g : (- \infty , y] \to (- \infty , z]$ be an isomorphism. Then $g(x) \sim_2 z$, and since $p \in \pi_1 \cap \pi_2$, we have $g(x) \sim_1 g(y)$, contradicting our choice of $x$ and $y$.
\end{proof}
The next lemma proves the Dedekind-MacNeille completeness of the invariant tree.
\begin{lemma}
\label{F4}
The invariant tree $I$ of a lower 1-transitive linear order $(X, \leqslant)$ is Dedekind-MacNeille complete.
\end{lemma}
\begin{proof} We need to show that
\begin{itemize}
\item[(i)] the supremum of any two vertices in $I$ is also a vertex in $I$, and
\item[(ii)] every descending chain of vertices in the tree which is bounded below has an infimum in the tree.
\end{itemize}
To show (i), consider two vertices $p_1, p_2 \in I$ that are parts of two partitions $\pi_{1}, \pi_{2}$, respectively. Without loss of generality, assume that $\pi_{1}$ refines $\pi_{2}$. Then either $p_{1} \subseteq p_{2}$ (and $p_{2}$ is the supremum of $p_{1}$ and $p_{2}$) or $p_{1} \subseteq p'_{2} \in \pi_{2}$. So this problem reduces to showing that the supremum of any two vertices on the same level is in $I$.
We know that $p_{2} , p'_{2} \subseteq p$ with $p \in \pi$, for some $\pi \in I$ which coarsens $\pi_{2}$ --- for instance $\{ X \}$ itself. Let $\sim_\pi$ be the equivalence relation corresponding to $\pi$. Then $a \sim_\pi b$ for $a, b \in p_{2} , p'_{2}$ respectively.
Consider the set of partitions $\pi^\prime$ that refine $\pi$ for which $a \sim_{\pi^\prime} b$, where $\sim_{\pi^\prime}$ is the corresponding equivalence relation. By Lemma~\ref{F3} this is a descending chain of partitions. If the set of parts containing both $a$ and $b$ has an infimum, then $p_{2} , p'_{2}$ have a supremum. So the verification of (i) reduces to that of (ii).
For (ii), consider a descending chain of vertices $p_{\gamma}$ that are parts of a descending chain of partitions $\pi_{\gamma}$ bounded below by $p$, say, where $p \neq \varnothing$. Let $\sim_{\gamma}$ be the equivalence relation corresponding to $\pi_{\gamma}$. Then define $x \sim y$ if $ x \sim_{\gamma} y$ for all $\gamma$. Let $f$ be a lower isomorphism of $(X, \leqslant)$. Then $x \sim y$ implies $f(x) \sim_{\gamma} f(y)$ for all $\gamma$ because each of the $\sim_{\gamma}$ is an invariant relation. Hence $f(x) \sim f(y)$ and so $\sim $ is an invariant relation. If $\pi$ is the corresponding partition, then $\pi$ is a partition into lower 1-transitive, lower isomorphic convex subsets of $X$, and so its parts are vertices in $I$. Then $p$ is contained in some member of $p^\prime$ of $\pi$, and $\pi^\prime$ is the infimum of the $p_{\gamma}$.
\end{proof}
\begin{theorem}
\label{T4}
The invariant tree $I$ of a lower 1-transitive linear order $(X, \leqslant)$ is an expanded coding tree whose leaves are order-isomorphic to $(X, \leqslant)$.
\end{theorem}
\begin{proof}
Firstly, the leaves of $I$ are singletons containing the elements of $X$, and so they are isomorphic to $X$.
Definition~\ref{D3} ensures that $I$ is a levelled tree whose root is $X$. The tree ordering is containment, the ordering of the levels is the one induced by $\subseteq$ on the set of invariant partitions of $X$, and the ordering of the children of a parent vertex is the one induced by the linear order on $X$. Since $X$ is countable, $I$ has countably many leaves. It is clear that every vertex of $I$ is a leaf or is above a leaf. So conditions 1 to 4 of Definition~\ref{A3} are satisfied. Moreover, $I$ is Dedekind-MacNeille complete by Lemma~\ref{F4}.
In order to verify condition 6 of Definition~\ref{A3}, we need to show that the order type of the children of a parent vertex in $I$ is one of $\ensuremath{\mathbb{Z}}$, ${\omega}^{\ast}$, $\ensuremath{\mathbb{Q}}$, $\dot{\ensuremath{\mathbb{Q}}}$, ${\ensuremath{\mathbb{Q}}}_{n}$ or ${\dot{\ensuremath{\mathbb{Q}}}}_n$ ( for $2 \leqslant n \leqslant {\aleph}_{0}$). Consider a successor level $\pi_{i+1}$ of $I$, so $\pi_{i}$ is the predecessor. Let $p \in \pi_{i + 1}$. Then $p$ is lower 1-transitive, and the children of $p$ are those elements of $\pi_{i}$ which are convex subsets of $p$. These children are lower 1-transitive linear orders and are lower isomorphic to each other. Let $\sim_{\pi_i}$ be the equivalence relation that defines $\pi_{i}$. Then, by Lemma~\ref{F2}, $p / \sim_{\pi_i}$ is also lower 1-transitive, and the order type of $p / \sim_{\pi_{i}}$ tells us how the children of $p$ are ordered. In order to describe the possible order types, we look at the structure forced by a specific invariant equivalence relation, namely, the relation $\sim_{\mathrm{fin}}$ that identifies points that are finitely far apart, defined by
\[ x \sim_{\mathrm{fin} } y \ \mbox{ iff } \ x \leqslant y \ \mbox{ and } [x, y] \ \mbox{ is finite, or } \ y \leqslant x \ \mbox{ and } \ [y, x] \ \mbox{ is finite. }\]
For any linear order, the equivalence classes of $\sim_{\mathrm{fin}}$ must be either finite, $\omega$, ${\omega}^{\ast}$ or ${\ensuremath{\mathbb{Z}}}$. If $(X, <)$ is lower 1-transitive, the equivalence classes of this form are either singletons, ${\omega}^{\ast}$, or ${\ensuremath{\mathbb{Z}}}$. If one equivalence class is a singleton, then they all are, and then the ordering is dense with no least endpoint. Hence it is isomorphic to $\ensuremath{\mathbb{Q}}$ or $\dot{\ensuremath{\mathbb{Q}}}$.
Since $p / \sim_{\pi_{i}}$ is a lower 1-transitive linear order, we can take its quotient by $\sim_{\mathrm{fin}}$. There are two cases.
Case 1: the equivalence classes of $(p / \sim_{\pi_{i}}) / \sim_{\mathrm{fin} }$ are non-trivial. Then, by the maximality of $I$, there can be only one equivalence class, that is, $p / \sim_{\pi_{i}}$ itself. If there is no last child, then $p$ is equal to $\ensuremath{\mathbb{Z}}$ copies of its children; otherwise, the order type of $p / \sim_{\pi_{i}}$ is ${\omega}^{\ast}$.
Case 2: the equivalence classes of $(p / \sim_{\pi_{i}}) / \sim_{\mathrm{fin} }$ are trivial.
Then the parts of $\pi_{i}$ are dense within $p$. We aim to show that $p / \sim_{\pi_{i}}$ is a ${\ensuremath{\mathbb{Q}}}$, $\dot{\ensuremath{\mathbb{Q}}}$, ${\ensuremath{\mathbb{Q}}}_{n}$ or ${\dot{\ensuremath{\mathbb{Q}}}}_{n}$ combination of its children.
If all the left children of $p$ are isomorphic, then $\mathrm{child}(p)$ is isomorphic to $ {\ensuremath{\mathbb{Q}}}$, or $\dot{\ensuremath{\mathbb{Q}}}$ if the right child exists.
If not all the left children of $p$ are isomorphic, then we show that $\mathrm{child}(p)$ is isomorphic to $ {\mathbb Q}_n$, or ${\dot{\mathbb Q}}_n$ if $p$ has a
right child, where the set $\Gamma$ of (colour, order-)isomorphism types of the left children of $p$ has size $n$. Suppose, for a contradiction,
that $p$ is not the ${\mathbb Q}_n$ mixture of its children.
Then there are two elements of $\Gamma$ such that not all other elements of $\Gamma$ occur between them in $p$. Let $\gamma$ be a member of $\Gamma$ which does not occur
between all pairs, and let us define $\sim$ on $\pi$ by $y \sim z$ if $y = z$, or if no point of $[y, z]$ (or $[z, y]$ if $z < y$) has isomorphism type
$\gamma$. This is an invariant partition of $\pi$ into convex pieces, and is proper and non-trivial, which contradicts $\pi_i$ and $\pi_{i+1}$ being on consecutive levels.
This verifies condition 6 of Definition~\ref{A3} for a parent vertex on a successor level of the invariant tree $I$.
Now consider the levels which are not successor levels. Firstly, this includes the trivial partition, $\pi_{0}$, given by the relation $ x \sim_{\pi_{0}} y \iff x = y$. These vertices are leaves.
There remains the case of vertices which do not have children in $I$. If one part of an invariant partition does not have a child then, clearly, none of them do. Dedekind-MacNeille completeness implies that these vertices have one cone below them.
For condition 7, let $x$ and $ x'$ be two vertices of $I$ on the same level. Then $x, x^\prime$ are parts of an invariant partition, so either they are both parents, or they are both leaves, or both are neither of these, in which case they have a single cone below them. Moreover, if $x, x'$ are both parent vertices, then $({\rm child}(x),\triangleleft)$ is lower-isomorphic to $({\rm child}(x'),\triangleleft)$, since $(X, \leqslant)$ is lower 1-transitive and $x$ and $ x'$ are parts of an invariant partition.
For condition 8, let $x \in I$ be a parent vertex. Suppose that $({\rm child}(x), \triangleleft) \cong {\ensuremath{\mathbb{Q}}}$, $\dot{\ensuremath{\mathbb{Q}}}$, ${\ensuremath{\mathbb{Q}}}_{n}$ or ${\dot{\ensuremath{\mathbb{Q}}}}_n$. Here two children vertices $a, b$ have the same colour when they are isomorphic. This isomorphism induces an isomorphism on the trees rooted at $a, b$.
If $({\rm child}(x), \triangleleft) \cong {\ensuremath{\mathbb{Z}}}$, we wish to show the children of $x$ are all isomorphic, and hence the trees below the children are isomorphic. Now, the children of $x$ are all a finite distance apart. In particular, each child has a successor and a predecessor. If $a$ and $b$ are children of $x$, the existence of an isomorphism from the successor of $a$ to the successor of $b$ implies that $a$ and $b$ are isomorphic. The argument in the case $({\rm child}(x), \triangleleft) \cong {\omega}^{\ast}$ is similar.
Finally we show $I$ satisfies condition 9 of~Definition~\ref{A3}.
Let $x$ and $y$ be distinct vertices on the same level. If $x$ and $y$ have no children, the condition holds trivially. So suppose that $x$ and $y$ are parent vertices and let $a \in x$ and $b \in y$. By lower 1-transitivity, there is an isomorphism $\varphi : (- \infty , a] \to (- \infty , b]$
which induces an isomorphism between $(- \infty , a] \cap x$ and $(- \infty , b] \cap y$. Let $x_{a}, y_{b}$ be the children of $x, y$ containing $a, b$ respectively. Then $(- \infty , a] \cap x_{a}$ and $(- \infty , b] \cap y_{b}$ are isomorphic. Consider the sets $\Gamma_{a}$, $\Gamma_{b}$ of children of $x, y$ to the left of $x_{a}, y_{b}$ respectively. Since $\varphi (\Gamma_{a}) = \Gamma_{b}$, the left forests of $x$ and $y$ are isomorphic. Since $a, b$ are arbitrary, $\Gamma_{a}$ and $ \Gamma_{b}$ can contain any particular left children of $x$ and $y$.
\end{proof}
We now must show how to construct a coding tree for $(X, \leqslant)$ given the invariant tree $I$, and give an inverse association map between them.\\
Informally, the coding tree is obtained from $I$ by amalgamating left children who are siblings and whose trees of descendants are isomorphic. The parent vertex is then labelled according to the order type of its children in $I$.\\
For each level $s$ of $I$ we define a relation $\simeq_{s}$ on $I$ that tells us which vertices to amalgamate:\\[3pt]
$x \simeq_{s} y \text{ if there are } x' \supseteq x, \ y' \supseteq y$ such that
\begin{itemize}
\item[(i)] the tree of descendants of $x'$ is isomorphic to the tree of descendants $y'$ under $\theta$, a suitable order isomorphism (respecting both the $\leqslant$ order and the $\vartriangleleft$ order)
\item[(ii)] $x', y'$ are left children of a vertex $z$ and lie on level $s$, or $x' = y'$
\item[(iii)] $\theta(x) = y$.
\end{itemize}
Note that the clauses guarantee that $x, y$ are level. Now we define a relation $\simeq$ on the whole of $E$ as follows:
\[x \simeq y \iff \exists x = x_{0}, \ldots , x_{n} = y, \text{ where for each } i=0, \ldots, n-1 \text{ there is } s_i \text{ with } x_{i} \simeq_{s_i} x_{i + 1}.\]
The relation $\simeq$ is an equivalence relation on $I$, and $T$ is then the set of equivalence classes on $I$, labelled as described above. We denote an element of $T$ by $[x]$, where $x \in I$. The next lemma ensures that the ordering on $I$ induces one on $T$.
\begin{lemma}\label{lemma48}
Let $[x],\,[y]\in T$ be such that $x \leqslant y$ (in $I$), and let $x^\prime \in [x]$. Then there is $y^\prime \in [y]$ such that $x^\prime \leqslant y^\prime$.
\end{lemma}
\begin{proof}
Let $x, y$ and $x^\prime$ be as in the statement. Since $x^\prime \in [x]$, there are $u, v$ and $w$ in $I$ such that $u, v$ are left children of $w$ and $x \leq u$, $x^\prime \leqslant v$. Moreover, the tree of descendants of $u$ is isomorphic to the tree of descendants of $v$ by an isomorphism $\theta$ such that $\theta(x)=x^\prime$. Now, either $y \geqslant w$ or $y <w$. If $y \geqslant w$ then $x^\prime < w \leqslant y$, so $y$ is the required $y^\prime$. If $y<w$, then $y \leqslant u$. Then $x^\prime = \theta(x) \leqslant \theta(y)$ and, since $\theta(y) \leqslant v$, this implies that $x^\prime \leqslant v$. But $\theta(y) \in [y]$ because of the way $\simeq$ is defined, so $\theta(y)$ is the required $y^\prime$.
\end{proof}
\begin{theorem}\label{t49}
The set of $\simeq$-classes on the invariant tree of $(X, \leqslant)$ is a coding tree for $(X, \leqslant)$.
\end{theorem}
\begin{proof}
Let $T$ be the family of $\simeq$-equivalence classes on $I$. Let $[x], [y] \in T$ and define
\[ [x] \leqslant [y] \iff (\exists x^\prime \in [x]) (\exists y^\prime \in [y]) (x^\prime \leqslant y^\prime) \ \text{ (in } I).\]
Lemma~\ref{lemma48} ensures that $\leqslant$ is well defined and transitive, so $\leqslant$ is an order.
Since $\leqslant$ is the order induced by that on $I$, $T$ is a tree with root $[r]$ and, since $\simeq$ is level preserving, $T$ is a levelled tree. Moreover, $T$ is countable, and every vertex of $T$ is a leaf or is above a leaf. We verify Dedekind-MacNeille completeness. Firstly note that all leaf-branches of $T$ are isomorphic to some leaf-branch of $I$ and so the leaf-branches of $T$ are Dedekind complete. We must now show that the least upper bound of any two vertices $[x], [y] \in T$ is in $T$. Since $I$ is Dedekind-MacNeille complete, any $x^\prime \in [x]$ and $y^\prime \in [y]$ have a least upper bound in $I$. Let
\[ \Gamma = \{ z \in I : z \text{ is the least upper bound of } x^\prime \text{ and } y^\prime \text{ for some } x^\prime \in [x], y^\prime \in [y] \}. \]
If $z \in \Gamma$, then $[x^\prime]=[x] \leqslant [z]$ and $[y^\prime] = [y] \leqslant [z]$ for some $x^\prime, y^\prime$, and so $[z]$ is an upper bound for $[x]$ and $[y]$. Now let
$ \Gamma^\prime = \{ [z] : z \in \Gamma \}$.
Since $\Gamma^\prime$ contains the upper bounds of $[x]$ and $[y]$, it is linearly ordered. Moreover, it is bounded above by $[r]$ and below by $[x]$. Let $\Gamma$ be the chain
\[ \ldots [z_{-n}] \geqslant \ldots \geqslant [z_{0}] \geqslant [z_{1}] \geqslant \ldots\ [z_n] \geqslant \ldots \]
By Lemma~\ref{lemma48}, for any $u \in [z_{i+1}]$ there is $v \in [z_i]$ such that $u \leqslant v $. Hence we can construct a corresponding chain of vertices in $I$.
If the $[z_{i}]$ do not have an infimum, then there is a chain of vertices in $I$ bounded below by $x \in [x]$ and without an infimum. This contradicts the Dedekind-MacNeille completeness of $I$. Then the infimum of $\Gamma^\prime$ is the least upper bound of $[x]$ and $[y]$.
Next we examine the labelling. Suppose $x \in I$ is a parent vertex. Then $[x] \in T$ is also a parent vertex and we let $\varsigma([x]) = ({\rm child}(x), \triangleleft)$, the order type of the children of $x$ in $I$. This is well defined, as $x \simeq y$ implies that $x$ and $y$ are isomorphic and hence the sets of their children have the same order type. Now, since $({\rm child}(x), \triangleleft)$ is one of $\ensuremath{\mathbb{Z}}$, ${\omega}^{\ast}$,
$\ensuremath{\mathbb{Q}}$, $\dot{\ensuremath{\mathbb{Q}}}$, ${\ensuremath{\mathbb{Q}}}_{n}$, ${\dot{\ensuremath{\mathbb{Q}}}}_n$ ( for $2 \leqslant n \leqslant {\aleph}_{0}$), it follows that $\varsigma([x])$ is also one of the above.\\
If $x$ is neither a parent nor a leaf, then neither is $[x]$. Hence we label $[x]$ by $\mathrm{lim}$. The leaves are labelled $\{ 1 \}$.\\
Let $[x], [y] \in T$ be level parent vertices and let $x, y \in I$ be representatives. Then $\varsigma ([x]) \ {\cong}_{l} \ \varsigma ([y])$ follows from the fact that $({\rm child}(x), \triangleleft) \ {\cong}_{l} \ ({\rm child}(y), \triangleleft)$.\\
When $[x], [y] \in T$ are level but neither parent vertices nor leaves (if $[x]$ is not a parent vertex and $[x]$ are $[y]$ level, then $[y]$ is not a parent vertex), both are labelled $\mathrm{lim}$, as remarked earlier. Hence $\varsigma ([x]) = \varsigma ([y])$ as required. The case when $[x], [y] \in T$ are leaves is similar.\\
We now show that $T$ fulfils condition 7 of Definiton~\ref{A1}. The number of children of $[x] \in T$ is the number of equivalence classes of the children of vertices $x^\prime \in [x]$ in $I$
We consider various cases.\\
Case 1: $({\rm child}(x), \triangleleft) \cong {\ensuremath{\mathbb{Z}}}, {\ensuremath{\mathbb{Q}}}$\\
All the children of $x$ are left children. We have also seen that they are all isomorphic and hence they are all $\simeq$-equivalent. Therefore there is one equivalence class below $[x]$.\\
Case 2: $({\rm child}(x), \triangleleft) \cong {\omega}^{\ast}, \dot{\ensuremath{\mathbb{Q}}}$\\
Again all the left children of $x$ are isomorphic and hence they are all $\simeq$-equivalent. A right child of $x$ forms its own equivalence class under $\simeq$. In these cases $[x]$ has two children.\\
Case 3. $({\rm child}(x), \triangleleft)\cong{\ensuremath{\mathbb{Q}}}_{n}$, ${\dot{\ensuremath{\mathbb{Q}}}}_n$.\\
The `colours' are the isomorphism types of the children of $x$ in $I$. There are $n$ isomorphism types amongst the left children. The left children which are isomorphic are also $\simeq$-equivalent. Hence there are $n$ ($n + 1$ in the case of ${\dot{\ensuremath{\mathbb{Q}}}}_n$) $\simeq$-classes below $[x]$.\\
Clause 8 of Definition~\ref{A1} follows from the corresponding fact about the expanded coding tree. Given two order isomorphic forests in the expanded coding tree, clearly the $\simeq$-classes on two such forests are also isomorphic.\\
Finally, since $\simeq$ amalgamates isomorphic trees of descendants of sibling left vertices, the tree of descendants of two sibling vertices in the resulting $T$ will not be isomorphic.
\end{proof}
\noindent We have obtained a coding tree from $(X, \leqslant)$. We have now to show that this tree does encode $(X, \leqslant)$.
\begin{theorem}
\label{C2}
The coding tree, $(T, \leqslant, \vartriangleleft, \varsigma, \ll)$ obtained from $(X, \leqslant)$ encodes $(X, \leqslant)$ in the sense of Definition~\ref{A3}.
\end{theorem}
\begin{proof}
Firstly we show that the expanded coding tree $I$ of invariant partitions of $X$ is associated with $T$ in the sense of Definition~\ref{d1}. The association function $\phi \rightarrow T$ is defined by $\phi(x)=[x]$, and the labelling function on $T$ is defined as follows:
\begin{itemize}
\item[(i)] if $x$ is a parent vertex, the label of $\phi(x)$ is equal to $({\rm child}(x), \triangleleft)$, the (coloured) order type of the children of $x$ in $I$,
\item[(ii)] if $x$ is neither a parent nor a leaf, the label of $\phi(x)$ is $\mathrm{lim}$,
\item[(iii)] if $x$ is a leaf, the label of $\phi(x)$ is $\{ 1 \}$.
\end{itemize}
As remarked in the proof of Theorem~\ref{t49}, this labelling is well defined. Moreover, the labels satisfy condition (iv) of Definition~\ref{d1}. By the way $T$ is constructed, it is clear that $\phi$ preserves levels. Moreover, the ordering on $T$ is such that $x \leqslant y$ in $I$ implies that $\phi(x) \leqslant \phi(y)$ in $T$. This ensures that conditions (i), (ii) and (iii) of Definition~\ref{d1} are satisfied.
The construction of $I$ ensures that $X$ is order isomorphic to the set of leaves of $I$. Therefore $T$ encodes the linear order $X$ in the sense of Definition~\ref{d2}, as required.
\end{proof}
Therefore $(T, \leqslant, \varsigma, \ll, \triangleleft)$ encodes $(X, \leqslant)$. Furthermore the set of $\simeq$-classes on the expanded coding tree $( E, \leqslant , \ll, \triangleleft)$ constructed from the coding tree $(T, \leqslant, \varsigma, \ll, \triangleleft)$ is isomorphic to $(T, \leqslant, \varsigma, \ll, \triangleleft)$, so the two procedures, from coding tree to encoded order, back to coding tree are converse operations.\\
Theorem~\ref{C2} concludes our classification of countable lower 1-transitive linear orders. The companion paper \cite{me} is a major extension of this work, since it classifies countable 1-transitive trees. The branches of these trees are countable lower 1-transitive linear orders. However, two non-isomorphic trees can have branch sets where the branches are isomorphic as linear orders. In order to consider the way lower 1-transitive linear orders embed in the trees, it is necessary to consider the ramification points of the trees. These points might not be vertices of the tree, and different types of ramification points give rise to the notion of \textit{colour lower 1-transitivity}. The starting point in \cite{me} is the classification of coloured 1-transitive linear orders. In order to give a complete description of each countable 1-transitive tree, it is then necessary to consider the number and type of cones at each ramification point.
\bibliographystyle{plain}
|
{
"timestamp": "2015-10-22T02:10:53",
"yymm": "1504",
"arxiv_id": "1504.03372",
"language": "en",
"url": "https://arxiv.org/abs/1504.03372"
}
|
\section{Introduction}\label{sec:intro}
The possibilities to obtain kinematic information for the hot phase of the
interstellar medium (ISM) are generally very limited. While multimillion-degree
gas is common, and metal lines are observed \citep[e.g.,][]{HS12}, the spectral
resolution typically does not allow \rv{one} to meaningfully constrain flows of hot gas in galaxy clusters
\citep{BDB13}, and more so for the smaller velocities in the ISM.
The gamma-ray spectrometer aboard {\it INTEGRAL~} \citep[SPI,][]{Vedrea03,Winklea03},
has a spectral resolution of $\approx$~3~keV at
1.8~MeV, where {$^{26}$Al~} can be observed through its characteristic
gamma-ray decay line. With increasing exposure times
\citep[Paper~I in the following]{Diehlea06,Kretschea13}, it has become
possible to measure the centroid
position of the line with an accuracy of tens of {km~s$^{-1}$}, sufficient to clearly observe
the Doppler shift due to large-scale rotation along the ridge of the Galaxy
within longitudes $|l|<35$~deg.
Towards the Galactic
centre ($l=0$), the apparent {$^{26}$Al~} velocity is zero with a hint for a small blue-shift. For
greater positive (negative) longitudes, the projected velocity
rises beyond 200 (-200)~{km~s$^{-1}$}. The direction of the line shift corresponds
to
Galactic rotation, but its magnitude is significantly larger
than what is expected from CO and \mbox{H{\hspace{1pt}}I}~. \citetalias{Kretschea13} also showed that an ad hoc model
assuming forward blowout at 200~{km~s$^{-1}$}~from the spiral arms of the inner Galaxy can well explain the data.
The physical interpretation
would be that {$^{26}$Al~} is ejected into \rv{the hot phase of the ISM in}
superbubbles at the leading edges of the gaseous
spiral arms. Hydrodynamic interaction with the locally anisotropic ISM
would then lead to a preferential expansion of the superbubbles into the
direction of Galactic rotation (in addition to out-of-plane-blowout).
The sources of \rv{diffuse, interstellar} {$^{26}$Al~} are massive star winds and supernovae \citep{PD96}.
These are energetic events, which lead to the formation of bubbles
\rv{(one massive star)}
and, because massive stars \rv{often} occur
\rv{together with other massive stars in associations and bound clusters, }
\citep[e.g.,][]{ZY07,Kroupea13,Krumholz14},
superbubbles. Superbubbles are observed in many different wavelengths
\citep[e.g.,][]{Krausea14a}. Statistical information is, however, mainly restricted
to sizes and kinematics of the cavities seen in \mbox{H{\hspace{1pt}}I}.
\rv{\citet{Bagea11} analysed 20~nearby spiral galaxies, whose properties are thought to be similar to those of the Milky Way, and found more than 1000 "\mbox{H{\hspace{1pt}}I}~ holes". We use their data
as reference below.}
\citet{OC97} have connected the statistics of \mbox{H{\hspace{1pt}}I}~ holes to the star cluster mass function,
finding that the sizes and velocities of \mbox{H{\hspace{1pt}}I}~ holes may be explained
by massive star activity in star clusters (compare below, however).
\rv{Because this association is established now, we will in the following
use the term "HI supershells" instead of "HI holes", for clarity.}
The {$^{26}$Al~} measurement constitutes another piece of statistical information for
\rv{bubbles and} superbubbles.
{$^{26}$Al~} decays on a timescale of 1~Myr\rv{, much less than typical superbubble lifetimes
\citep[e.g.,][]{OG04,Bagea11,Heesea15}}. Hence, we may expect it to reflect internal dynamics.
Here, we connect the observed {$^{26}$Al~}
kinematics to the statistics of star clusters (Sect.~\ref{sec:sc-al}) and superbubbles
(HI supershells, Sect.~\ref{sec:sc-sb})\rv{, in order to better understand the large-scale
gas flows traced by {$^{26}$Al}}.
\rv{In particular, we are interested to constrain superbubble merging, because superbubble
merging may lead to asymmetric motions relative to the parent star clusters, when gas
from a high pressure superbubble streams into a low-pressure cavity.}
We find that star clusters of all masses contribute
to the {$^{26}$Al~} signal.
\rv{\citet{OC97} investigate superbubble merging in the Milky Way with
inconclusive results.
With updated models and star-formation rate we find frequent merging.
Hence, we expect the {$^{26}$Al}-traced
hot outflows} to be injected into pre-existing superbubbles.
We then argue in Sect.~\ref{sec:concmod} that the spatial co-ordination of star formation in the Milky Way by
the spiral arms may lead to the observed {$^{26}$Al~} kinematics.
\section{Which star clusters produce how much {$^{26}$Al}?}
\label{sec:sc-al}
\rv{Star formation generally takes place in clusters and associations,
the majority of which disperse after some time
\citep{LL03,Kruijssen12}. For the case of bound star clusters
it is debated if the dispersal is
due to gas expulsion \citep[e.g.,][]{GiBa08}. Recent observations did
not find the expansion velocities expected if gas expulsion was important
\citep[e.g.,][]{HenBruea12}. Hence,
the dispersal is probably related to tidal effects \citep[e.g.,][]{Kruijea12a},
which could take as long as 200~Myr \citep{Kruijea12b}. It is therefore
reasonable to assume that for the timescales of interest here,
the great majority of massive stars are grouped \citep[compare also][]{ZY07}.
The mass function of embedded star clusters (mostly unbound), which is where
most star formation takes place locally \citep{LL03} has a very similar slope
than the one of star clusters in external galaxies (compare below). Therefore,
we assume just one mass function for star forming regions in the following,
and generally use the term "star cluster" without qualifying adjective
to subsume bound and unbound star forming regions. }
For spiral galaxies like the Milky Way, the initial cluster mass function (ICMF) is given
by \citep[e.g.,][]{Larsen09,Bastea12}
\begin{equation}\label{eq:ICMF}
\frac{\mathrm{d}N}{\mathrm{d}M} = a \nbr{M/M_\mathrm{c}}^\alpha
\exp\nbr{-M/M_\mathrm{c}} \, ,
\end{equation}
where $a$ is the normalisation, the cutoff mass
$M_\mathrm{c}= 2 \times 10^{5}$~{M$_\odot$}, and we take the
power-law index $\alpha$ to be $-2$;
compare also the reviews
by \citet{LL03,Kroupea13,Krumholz14}. Following \citet{LL03}, we
adopt a lower limit for star cluster masses of 50~{M$_\odot$}.
\rv{Embedded star clusters have not been shown to possess the exponential cutoff.}
We have\rv{, therefore,} checked that the \rv{presence} of the high-mass cutoff only marginally influences our results.
Since only massive stars produce {$^{26}$Al}, we have to relate the occurrence of massive
stars to the masses of star clusters.
We \rv{carry out the entire analysis for both, optimal sampling \citep{Kroupea13},
where the masses of massive stars are fixed for given star cluster mass,
and random sampling \citep[e.g.,][]{Krumholz14}. For random sampling, we
fix the stellar mass above 6~{M$_\odot$}~to the corresponding fraction of the IMF from \citep{Kroupea13}.
While the extreme assumption of
optimal sampling has been challenged recently \citep{Andrewsea14}, we use it here
to demonstrate that even such a strong truncation of the IMFs would not affect
the conclusions.}
\rv{For such groups of massive stars, w}e use the population synthesis results from \citet{Vossea09} (stellar evolutionary tracks of rotating
stars of \citet{MM05} and wind velocities from \citet{Lamea95} and \citet{NiSk02}
for the Wolf-Rayet phase) to obtain the {$^{26}$Al~} mass as well as the energy
injected into the ISM by massive stars as a function of time and stellar
mass. The release of mass, energy and {$^{26}$Al~}
is largely completed after about 48~Myr, the lifetime of stars of about 8~{M$_\odot$},
also broadly consistent with the age estimates
for \mbox{H{\hspace{1pt}}I}~ supershells given by \citet{Bagea11}.
Not all the stars in a cluster might form at the same time. However typical age spreads within clusters are of order 1~Myr or below \citep[e.g.,][]{Niedea15},
which is much shorter than the timescales of interest.
We therefore use the star cluster population up to 48~Myr for our model.
Following \citet{CP11}, we take 1.9~{M$_\odot$~yr$^{-1}$}~
for the star formation rate of the Milky Way.
This \rv{sets} the constant in eq.~(\ref{eq:ICMF}) to $a=3\times10^{-4}$~{M$_\odot$}$^{-1}$.
Uncertainties in this parameter are substantial
\citep[compare also][]{KE12}.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{pfp_Alcomb_v04.eps}
\caption{\tiny {$^{26}$Al~} mass for individual star clusters of given mass (dotted blue, right vertical scale) and
cumulative {$^{26}$Al~} mass for the Milky Way as a function of star cluster mass,
assuming a star formation rate of 1.9~{M$_\odot$~yr$^{-1}$} (solid black, left vertical scale).
\rv{Thin (thick) lines are for the case of truncated IMFs (random sampling).
In the limit of high star cluster masses, $2.6 \times 10^{-8}$~{M$_\odot$}~of {$^{26}$Al~} is
produced per unit stellar mass formed. The blue dashed curves are therefore
linear, down to about 1000~{M$_\odot$}, where sampling effects become important.}}
\label{f:al_per_sc}%
\end{figure}
With these assumptions, we calculate the time-averaged {$^{26}$Al~} mass for a given star cluster.
For each star cluster, we first determine the masses of its
stars above 8~{M$_\odot$}~by the optimal sampling method, the amount of released
{$^{26}$Al~} from \citet{Vossea09}, taking into account radioactive decay, and finally average over time (48~Myr).
The result is shown in Fig.~\ref{f:al_per_sc}. Apart from small
features towards lower masses, the {$^{26}$Al~} yield is almost linear
\rv{even for optimal sampling. For star clusters below about 1000~{M$_\odot$},
the sampling method matters. For both, random and optimal sampling, the
{$^{26}$Al~} mass per cluster drops below the linear relation, because the IMF can no longer
be fully sampled (e.g. a 120~{M$_\odot$}~star may not live in a 50~{M$_\odot$}~star cluster).}
\rv{We note that using the IMF directly, without dividing the mass of young stars into star clusters, to predict the Galactic {$^{26}$Al~} mass yields a
higher value by about 20 per cent.}
The ICMF has roughly equal mass in each decade of star cluster mass
(within the cutoffs). This remains true with the {$^{26}$Al~} mass folded in, because
the latter is roughly proportional to the star cluster mass: star clusters of
each decade in mass\rv{, from a few hundred to about $10^5$~{M$_\odot$},}
contribute about equally to the observed {$^{26}$Al~} signal (Fig.~\ref{f:al_per_sc}).
\section{Superbubble size distributions and merging}
\label{sec:sc-sb}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{MW_cls_bubcomp_nd_v4.eps}
\includegraphics[width=0.48\textwidth]{MW_cls_bubcomp_wd_v4.eps}
\caption{\tiny Superbubble diameter distributions for the weakly (top) and the strongly (bottom)
dissipative model for three different choices of the background density (bg-den. in the legends). The size of the bins is 200~pc. \rv{Thick lines are for random sampling,
thinner ones for optimal sampling, and the thinnest ones in the top panel
are for optimal sampling where the background pressure and superbubble destruction by ISM turbulence are taken into account.}
The minimum near 400~pc for the solid curves is due to the strong acceleration after the first supernova in a superbubble. It is below the data range for the other curves. Large superbubbles are better explained by the weakly dissipative model.}
\label{f:hi_hole_size}%
\end{figure}
\rv{Here, we investigate if merging of superbubbles is common in the Milky Way.}
\rv{We follow the overall procedure described in \citet{OC97}, but update
the expansion models
\rv{from our own 3D hydrodynamics simulation studies}
(compare below). While towards the low mass end, the ICMF includes many
objects with only one massive star, which will produce a single-star bubble,
we use the term
"superbubble", below for simplicity for all bubbles produced by the star clusters.}
\citet{McLMcC88}
present a self-similar model for superbubble expansion, where the superbubble expands
steadily with radius $r$ proportional to a power law in time $t$. About 35 per cent of
the injected energy, $E(t)$, is dissipated radiatively in this model. This model should be increasingly
adequate for larger superbubbles, with more frequent explosions, and at later times.
In \citet{Krausea13a} and
\citet{KD14}, we have developed a more strongly dissipative model from 3D
hydrodynamics simulations. The reason for the stronger dissipation is the
more realistic, non-steady energy input and the emergence of a highly radiative mixing layer due to 3D instabilities.
Our results are well approximated by 90 per cent dissipation in the
steady energy input phase before the first supernova and a decline of the current energy, $E(t)$, after each supernova with time $t$ as $t^{-3/4}$
\rv{(momentum-conserving snowplough)}. Both are an upper limit on the
energy dissipation, as in the pre-supernova phase we still observed a slight
dependence on numerical resolution ($\approx 88$ per cent dissipation at the highest resolution) and, as the superbubble expands, the
density around star clusters will drop below the 10~{cm$^{-3}$}~we assumed
in the simulations.
The strongly dissipative model should be more adequate for superbubbles
with few supernovae,
and indeed explains, e.g., the X-ray-luminosity--kinematics relation well \citep{KD14}.
We use the evolution of the superbubble energy $E(t)$ from both models and
predict the radius in the thin shell approximation
following \citet{KD14}. Their eq.~(3) for constant \rv{ambient} density
$\rho_0$ evaluates to
\rv{$r^5 = 15/(2\pi\rho_0) \int_0^t dt^\prime \int_0^{t^\prime} dt^{\prime\prime} E(t^{\prime\prime}).$}
\rv{We calculate models for both, random sampling and optimal sampling.
For the weakly dissipative models, we also add models where we take a constant
ISM pressure of $P_0=3800 k_\mathrm{B} \mathrm{K} \mathrm{cm}^{-3}$
\citep{JT11} into account which limits the expansion. The momentum
equation may then be written as \citep{Krause2005a}:
$\partial^2Y(r)/\partial t^2 = E(t) - 2 \pi r^3 P_0$, with
$Y(r) = 2\pi \rho_0 r^5 / 15$, which we solve numerically.
For this model, we also
regard a superbubble as dissolved when the expansion velocity has dropped
to 10~km~s$^{-1}$ and perturbations with this velocity had time to grow
to the size of the superbubble, similar to the "stalled and surviving" mode
in \citep{OC97}. We do not investigate this option for the strong dissipation models,
because the assumption of momentum conservation after each supernova explosion
implies a total pressure force of zero.}
For the following analysis, we neglect the shear gradient from galactic rotation.
It is typically 10-50~{km~s$^{-1}$}~kpc$^{-1}$ \citep{Bagea11},
and therefore has a small effect on active superbubbles, in agreement with the moderate
asymmetries found by \citet{Bagea11}, but will eventually destroy old ones.
The finite exponential scaleheight $H$ of the ISM introduces a cutoff in the superbubble radii
in the Galactic plane
due to blow-out related pressure loss at $\approx 3H$ \citep{BB13}. We set this cutoff superbubble radius to 1~kpc for the whole sample, and 0.5~kpc for the Milky Way modelling below due to the lower \mbox{H{\hspace{1pt}}I}~ scaleheight \rv{\citep{NJ02,LPV14}}.
\rv{We first calculate the fractional distributions of superbubble diameters
for three different
assumptions for the background density
for the sample of star-forming galaxies from \citet{Bagea11}, i.e. $H=1/3$~kpc,
and compare this to the observations in Fig.~\ref{f:hi_hole_size}.}
Generally, models with lower background density provide a better match to the observations.
As expected, the weakly dissipative model more closely represents the large superbubbles.
The model is \rv{not quite} satisfactory, \rv{because}
the density \rv{required to reach the larger diameters}, 0.1~{cm$^{-3}$}, is on the low side of values suggested by observations,
0.1-0.7~{cm$^{-3}$} \citep{Bagea11}. One might be able to interpret this finding by
shear effects, adopting a higher density, i.e. choosing
a curve between the dotted blue and solid black lines in Fig.~\ref{f:hi_hole_size}.
\rv{The strongly dissipative models may produce
a significant population at around 1 kpc diameter, but, on the other hand,
cannot account for large \mbox{H{\hspace{1pt}}I}~ supershells.}
\rv{ISM pressure becomes most important for intermediate-size \mbox{H{\hspace{1pt}}I}~ supershells
and for low ISM density
($\approx 1$~kpc for $\rho_0=0.1$~cm$^{-3}$). At high ISM densities, ISM pressure is negligible, but in these models
many slower and smaller superbubbles are destroyed when considering ISM turbulence,
which increases the fraction of larger superbubbles.}
\rv{The IMF sampling method has a minor effect on the results
(compare Fig.~\ref{f:hi_hole_size}).}
\rv{We can now predict the superbubble distribution for the Milky Way from the star formation rate using the procedure outlined above, now with $H=1/6$~kpc.
The fractional distributions are identical to Fig.~\ref{f:hi_hole_size}, but cut
at 1~kpc due to the reduced scaleheight. The observed fractional \mbox{H{\hspace{1pt}}I}~ supershell
diameter distribution for the Milky Way \citep{EP13} is consistent with the one
of external star-forming galaxies from \citep{Bagea11}, which we used here.}
\rv{Because for the Milky Way, the total number of superbubbles is constrained by the star-formation rate, we
may now check for superbubble merging by calculating the total volume predicted by our model to be occupied by superbubbles and comparing it to the volume of the Milky Way ISM.
The} total
occupied volume for the given star formation rate exceeds the one of the Milky Way ISM
(cylinder: 10~kpc radius, 1~kpc thickness) for all assumptions (Table~\ref{t:vols}).
\rv{This indicates that the superbubbles merge frequently. In the case of merging superbubbles,
the total volume is not simply the sum of the individual volumes, but much smaller.
The observed volume fractions of \mbox{H{\hspace{1pt}}I}~ supershells (3D porosity) are typically below 10 per cent
and may reach 20 per cent in later Hubble types \citep{Bagea11}.
A superbubble volume fraction around 20 per cent is expected from the hot gas fraction
in the ISM simulations of \citet{dAB05}. Combined
with our analysis, this strengthens the point about superbubble merging. A consistent
interpretation would be that the smaller superbubbles in the diameter distribution
(Fig.~\ref{f:hi_hole_size}) merge to obtain more HI supershells at large diameters. This would
also alleviate the requirement for low ambient density (compare above).
}
\rv{There is a lot of direct evidence for superbubble merging in the Milky Way:
29 per cent of the bubbles identified in "The Milky Way Project", a citizen science project
that identified 5106 bubbles in the Milky Way (many of which are single star bubbles),
showed signs of merging \citep{Simpsea12}. Often, secondary bubbles are found on the edge of larger bubbles. \citet{EP13} calculate the porosity for the Milky Way as a function
of radius from 333 identified HI supershells.
They find porosities above unity inside of the solar circle, and thus strong overlap
of superbubbles. The closest massive star group, \object{Scorpius-Centaurus OB2},
is an excellent example for superbubble merging \citep{Poepea10,PreibMam08}:
the different subgroups of the OB association appear to have been triggered by expanding
shells from the older parts, and the shell around Upper Scorpius is half merged into an older supershell. The whole structure is expected to merge within a few Myr with the Local Bubble \citep{BdA06}. Evidence for superbubble merging from extragalactic studies
is, however, scarce, probably because of the low resolution (typically around 200~pc).
H$\alpha$ bubbles are however found at the rims of HI supershells \citep{Egorea14}. }
\rv{Superbubble merging may produce significant net velocities in ejecta flows
with respect to the driving massive-star group. Because the {$^{26}$Al~} content
is correlated with the energy content of a superbubble (Fig.\ref{f:al-en}),
we expect overpressured {$^{26}$Al}-rich material to often stream into lower pressured
superbubbles, once the interface is eroded. The situation is similar, if the {$^{26}$Al~}
production site is located towards one end of an already merged larger superbubble.}
\begin{table}
\caption{\rv{Galaxy integrated superbubble volumes in units of the Milky Way volume, assuming a maximum superbubble diameter of 1~kpc due to blowout. For each entry, the first (second) number is for random sampling (truncated IMFs). For weak-dissipation
models, we also give the numbers for the models that take into account the ISM background pressure and turbulence as the third number.}}
\label{t:vols}
\centering
\begin{tabular}{l r r r }
\hline\hline
Dissipation & $\rho_0=0.1$~{cm$^{-3}$}& $\rho_0=1$~{cm$^{-3}$}& $\rho_0=10$~{cm$^{-3}$}\\
\hline
weak & 115/115/33 & 40/38/22 & 11/11/6.1 \\
strong & 47/49 & 12/13 & 3.1/3.3 \\
\hline
\end{tabular}
\end{table}
\section{A model for the {$^{26}$Al~} kinematics}
\label{sec:concmod}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{al-e_v4.eps}
\caption{\tiny Cumulative {$^{26}$Al~} mass over current superbubble energy for weakly (dotted) and strongly (solid)
dissipative models for a star cluster population representative of the Milky Way. 1 Bethe = $10^{51}$~erg.}
\label{f:al-en}%
\end{figure}
In the preceding sections, we have demonstrated that star clusters of all masses
are equally important as {$^{26}$Al~} producers, and that, on Galactic scales, star clusters
cannot be assigned to individual superbubbles due to frequent superbubble merging.
Our model also shows that {$^{26}$Al~} injection from star clusters is strongly correlated
to energy injection (Fig.\ref{f:al-en}). It follows that {$^{26}$Al~} is likely to be observed
in motion, and in particular it is likely that it traces gas involved in superbubble merging.
Based on these findings, we suggest the following model (Fig.\ref{f:sketch}) to explain
the {$^{26}$Al~} kinematics.
When spiral arms sweep through
the Galactic disc, they trigger the formation of young star clusters that produce
large superbubbles, \rv{traced as} \mbox{H{\hspace{1pt}}I}~ supershells. During the observed lifetimes of \mbox{H{\hspace{1pt}}I}~ supershells,
$\lesssim 100$~Myr \citep{Bagea11}, a spiral arm may lag behind stars and gas by
as much as a few kpc, due to the pattern speed
\rv{of the arm which is lower within corotation than the rotational speed of the stars and gas}.
The current young star clusters in a spiral arm
therefore feed {$^{26}$Al}-carrying ejecta into the \mbox{H{\hspace{1pt}}I}~ supershells left behind by the receding spiral arm
(sketch in Fig.~\ref{f:sketch}).
\rv{Despite uncertainties regarding wind clumping \citep[e.g.,][]{Bestenlea14} and
dust production and clumping \citep[e.g.][]{Indea14,Williams14}, the bulk of {$^{26}$Al~} is
likely mixed into the diffuse gaseous ejecta, expelled into the hot immediate surroundings of the stars. The ejecta} do not keep their initial velocity ($\approx 1000$~{km~s$^{-1}$}) for long:
for supernovae, they are
shocked on timescales of $10^3$~yr \citep{Tenea90}. For Wolf-Rayet winds inside
superbubbles, the free expansion phase can be up to $10^4$~yr, or
$\approx 10$~pc \citep{Krausea13a}.
The ejecta then travel at
\rv{a reasonable fraction of}
the sound speed in superbubbles,
$c_\mathrm{s} = \sqrt{1.62 kT/m_\mathrm{p}} = 279\,T_{0.5}^{1/2}$~{km~s$^{-1}$}.
Here, $k$ is Boltzmann's constant, $m_\mathrm{p}$ the proton mass, $T$ ($T_{0.5}$)
the temperature (in units of 0.5~keV), and the numerical factor is calculated for a
fully ionised plasma of 90~per cent hydrogen and 10~per cent helium by volume.
Measurements of superbubble temperatures range from 0.1~keV to about 1~keV
\citep[e.g.,][]{DPC01,Jaskea11,Sasea11,KSP12,Warthea14}, in good agreement with expectations, if instabilities and mixing are taken into account \citep{Krausea14a}.
\rv{In simulations of merging bubbles \citep{Krausea13a}, we find such kinematics for gas flooding
the cavities at lower pressure
shortly after merging. T}he ejecta travel about 300~pc during one
decay time ($\tau = 1$~Myr), which
corresponds to the size of the smaller \mbox{H{\hspace{1pt}}I}~ supershells (Fig.~\ref{f:hi_hole_size}),
i.e. the decay is expected to happen during the first crossing of the \mbox{H{\hspace{1pt}}I}~ supershell.
Hence, we expect a one-sided {$^{26}$Al~} outflow at the superbubble sound speed, $\approx 300$~{km~s$^{-1}$},
in excellent agreement with the observations and their analysis presented in \citetalias{Kretschea13}.
\rv{This} model
predicts a change in \rv{relative} outflow direction near the corotation radius.
\rv{But, c}orotation \rv{in the Galaxy} is unfortunately too far out
\citep[8.4-12 kpc, e.g.,][]{MBP15}
to check for direction reversals in the data set of \citetalias{Kretschea13}.
At such galactocentric distances,
individual {$^{26}$Al}-emission regions are \rv{only a few, faint, and} not associated with
spiral arms. Thus, we do not expect large {$^{26}$Al~} velocity asymmetries,
in good agreement with the measurements in Cygnus \citep{Martinea09}
and Scorpius-Centaurus \citep{Diehlea10}.
\begin{figure}
\centering
\includegraphics[width=.4\textwidth]{sketch_v2.eps}
\caption{\tiny Sketch of the proposed model to explain the {$^{26}$Al~} kinematics. In the co-rotating
frame chosen here, a spiral arm (solid line) moves anti-clockwise. At its
previous location (dashed line), it created large superbubbles (ellipses), blowing out of the disc. The young
star clusters (blue stars) at the current spiral arm location feed {$^{26}$Al~} (colour gradient in ellipses)
into the old superbubbles.}
\label{f:sketch}%
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{N628-HI-HII-in.eps}
\caption{\tiny
The grand-design spiral galaxy \object{NGC~628}. The background
image is the 21~cm map from The \mbox{H{\hspace{1pt}}I}~ Nearby Galaxy Survey
\citep[THINGS, ][]{Waltea08}. Red ellipses denote \mbox{H{\hspace{1pt}}I}~ supershells from
\citet{Bagea11}. Blue 'plus'-signs denote the 650~\mbox{H{\hspace{1pt}}II}~ regions identified
by \citet{HR15}. Their spiral arm designations, 'A' and 'B', are also indicated.
The large green circle indicates the median corotation radius of
$4.6\pm1.2$~kpc
from a number of studies as compiled by \citet{SL13}.
For the first half-turn, arm 'A' has no \mbox{H{\hspace{1pt}}I}~ supershell on its trailing edge, but four
are close to or even overlapping the leading edge in the way envisaged by our model.
Arm 'B' begins just inside of corotation and has three prominent \mbox{H{\hspace{1pt}}I}~ supershells
at its leading edge, with only a minor one towards the trailing edge.
From about the corotation radius outwards, \mbox{H{\hspace{1pt}}I}~ supershells are no longer
at the edges of the \mbox{H{\hspace{1pt}}II}~ arm, but appear all over it.}
\label{f:NGC628}%
\end{figure}
\rv{We might, however, expect to find \mbox{H{\hspace{1pt}}I}~ supershells associated with the
leading-edge of spiral-arm star-formation regions
in nearby face-on spiral galaxies, inside their corotation radii. We have
investigated this for a few objects by combining \mbox{H{\hspace{1pt}}II}~ regions from
\citet{HR15} to \mbox{H{\hspace{1pt}}I}~ images with \mbox{H{\hspace{1pt}}I}~ supershells using corotation radii from
\citet{Tambea08} and \citet{SL13}.
For \object{NGC~3184} and \object{NGC~5194} we find evidence for \mbox{\mbox{H{\hspace{1pt}}I}~} supershells
close to \mbox{H{\hspace{1pt}}II}~
regions in the spiral arms. There is no clear trend where the \mbox{H{\hspace{1pt}}I}~ supershells
are located with respect to the \mbox{H{\hspace{1pt}}II}~ regions in \object{NGC~5194}, whereas more supershells
appear on the trailing edge for \object{NGC~3184}.
In the case of \object{NGC~628} (Fig.~\ref{f:NGC628}), \citet{HR15}
map \mbox{H{\hspace{1pt}}II}~ regions for two arms, 'A' and 'B', and inside corotation, \mbox{H{\hspace{1pt}}I}~ supershells
are indeed found close to and overlapping with the \mbox{H{\hspace{1pt}}II}~ regions, preferentially at their leading edges.
Especially for
arm 'B', which is located in an \mbox{H{\hspace{1pt}}I}~ rich part of the galaxy, the \mbox{H{\hspace{1pt}}I}~ supershell locations
relative to the \mbox{H{\hspace{1pt}}II}~ regions change strikingly near the corotation radius:
Inside, three prominent \mbox{H{\hspace{1pt}}I}~ supershells lie towards the leading edge of the \mbox{H{\hspace{1pt}}II}~ arm,
extending over about a quarter of a turn. Only one small supershell is located
at the trailing edge. From about the corotation radius outwards, the \mbox{H{\hspace{1pt}}I}~ supershells
are spread over the widening \mbox{H{\hspace{1pt}}II}~ arm. None is clearly associated with the leading
or trailing edges.
It is beyond the scope of this article to explain the differences between these
galaxies. The fact that the effect we postulate is consistent with the data in
NGC~628 is, however, encouraging.
}
The {$^{26}$Al~} decay time is comparable to the crossing time through
the \mbox{H{\hspace{1pt}}I}~ supershell, and thus we \rv{expect to} observe it \rv{while it crosses the HI supershells}.
A few Myr later, \rv{{$^{26}$Al~} should isotropise, advect} "vertically" into the halo \citep[e.g.,][]{dAB05}, or mix due to interaction with the cavity walls. Most of the
{$^{26}$Al~} has then decayed, and the contribution to the observed $\gamma$-ray
signal is small.
\section{Conclusions}
We interpret the observed {$^{26}$Al~} kinematics in the Galaxy
as a consequence of superbubble formation propagating
with the spiral arms and merging of young superbubbles into older \mbox{H{\hspace{1pt}}I}~ supershells,
\rv{with outflows from currently star-forming
regions into the pre-shaped cavities from preceding star-formation towards the
leading edges of spiral arms}.
The model does not rely on independent offsets between young stars and gaseous
spiral arms, which
\rv{might be created by other -- not feedback related -- processes and which}
are a matter of ongoing research \citep[compare, e.g., the review by][]{DB14}.
We conclude that {$^{26}$Al~} mainly decays during the first crossing of superbubbles while in the hot phase.
The bulk of {$^{26}$Al~} is therefore not mixing with cold gas on its decay timescale.
{$^{26}$Al~} has however been found in meteorites indicating its presence in the gas
that formed the Sun
\citep[e.g.,][]{GM12}. The corresponding fraction of {$^{26}$Al~} required to mix into a
star-forming cloud during the decay timescale is, however, small
\citep{Vasilea13}, and would hardly affect our model.
\begin{acknowledgements}
This research was supported by the cluster of excellence ``Origin
and Structure of the Universe''
and by
Deutsche Forschungsgemeinschaft
under DFG project number PR 569/10-1 in the context of the
Priority Program 1573 “Physics of the Interstellar Medium.
K.\ K.\ was supported by CNES.
\end{acknowledgements}
\bibliographystyle{aa}
|
{
"timestamp": "2015-04-14T02:13:21",
"yymm": "1504",
"arxiv_id": "1504.03120",
"language": "en",
"url": "https://arxiv.org/abs/1504.03120"
}
|
\section{Hurwitz numbers }
The study of Hurwitz numbers, which enumerate branched covers of the Riemann sphere with specified ramification profiles, began with the pioneering work of Hurwitz \cite{Hu1, Hu2}. Their relation to enumerative factorization problems in the symmetric group and irreducible characters was developed by Frobenius \cite{Frob1, Frob2} and Schur \cite{Sch}. In recent years, following the discovery by Pandharipande \cite{Pa} and Okounkov \cite{Ok} that certain KP and 2D Toda $\tau$-functions \cite{Ta, UTa, Takeb}, fundamental to the modern theory of integrable systems \cite{Sa, SS}, could serve as generating functions for weighted Hurwitz numbers, there has been a flurry of activity \cite{GGN1, GGN2, GH1, GH2, AMMN, HO3, AC1, AC2, BEMS, Z, KZ, H1, NOr1, NOr2, H2} concerned with finding new classes of $\tau$-functions that can similarly serve as generating functions for various types of weighted Hurwitz numbers.
Two closely related interpretations of these weighted Hurwitz numbers exist. The enumerative geometrical one consists of weighted sums of Hurwitz numbers for $n$-sheeted branched coverings of the Riemann sphere. The other consists of weighted enumeration of factorizations of elements of the symmetric group $S_n$ in which the factors are in specified conjugacy classes. This may equivalently be interpreted as a weighted counting of paths in the Cayley graph generated by transpositions, starting and ending in specified classes. The two approaches are related by the monodromy representation of the fundamental group of the sphere punctured at the branch points obtained by lifting closed paths to the covering surface. Variants of this also exist for branched coverings of higher genus surfaces \cite{LZ} and other groups.
Some generating functions of enumerative invariants are known to also have representations as matrix integrals \cite{GGN1, GGN2, GH1, AC1, AC2, BEMS, NOr1, NOr2}. These include, in particular, the well-known Harish-Chandra-Itzykson-Zuber (HCIZ) integral \cite{HC, IZ}, which plays a fundamental r\^ole both in representation theory and in coupled matrix models. In \cite{GGN1, GGN2, GH1}, it was shown that when the Toda flow parameters
are equated to the trace invariants of a pair of $N \times N$ hermitian matrices, and the expansion parameter is equated to $-1/N$, this gives the generating function for the enumeration of weakly monotonic paths in the Cayley graph with a fixed number of steps while, geometrically, it coincides with signed enumeration of branched coverings of fixed genus and variable numbers of branch points \cite{HO3, GH2}. Other matrix integrals
give ``hybrid'' paths consisting of both weakly and strongly monotonic segments or, equivalently, enumeration of coverings with multispecies ``coloured'' branch points \cite{HO3, GH1}. Certain of these may also be shown to satisfy differential constraints, the so-called {\em Virasoro constraints} \cite{MM, Z, KZ}, due to reparametrization invariance, and {\em loop equations} \cite{BM, BEMS, AC1, AC2} following from the structure of the underlying matrix integrals.
These, and other generating functions for various enumerative, topological, combinatorial and geometrical invariants related to Riemann surfaces, such as intersection numbers \cite{Ko}, higher Gromov-Witten invariants, Hodge numbers \cite{KL, K}, knot invariants \cite{BE, MMN}, and a growing number of other cases, can be placed into the {\em topological recursion} scheme \cite{GJ2, EO1, EO2, EO3}, which aims at determining the generating functions through algorithmic recursion sequences stemming from an underlying {\em spectral curve} \cite{BEMS, KZ}. This has turned out to be a very effective approach to a broad class of examples.
However not all such generating functions are known to be $\tau$-functions in the usual sense of integrable systems, nor partition functions or correlaters for matrix models. It remains something of a mystery exactly which class of invariants is amenable to such a representation. A further remarkable fact is that, in some cases, different generating functions corresponding to distinct enumerative problems, such as Hurwitz numbers and Hodge integrals, may be $\tau$-functions that are related through algebraic transformations that themselves involve the spectral curve \cite{K, KL}.
The present work is concerned solely with the case of Hurwitz numbers, but in the generalized
sense, allowing infinite parametric families of weightings. It provides a unified approach
encompassing all cases of weighted Hurwitz numbers that have appeared to date,
interpreting these as special cases of an infinite parametric family of weighting functions
determining mKP or 2D-Toda $\tau$-functions of generalized hypergeometric type.
The parameters serve to specify the particular weighting used when summing over the various
configurations. Their values are determined by a ``weight generating'' function $G(z)$, and define the weighting by
evaluation of the standard bases $(e_\lambda, h_\lambda, m_\lambda, f_\lambda)$, for the space
$\Lambda$ of symmetric functions in an infinite number of indeterminates \cite{Mac}
consisting of {\em elementary}, {\em complete}, {\em monomial} and {\em forgotten} symmetric functions, respectively, at the given set of parameters $(c_1, c_2, \dots)$ determined by $G(z)$. The other two standard bases, the Schur functions $\{s_\lambda\}$ and the power sum symmetric functions $\{p_\mu\}$, serve as bases for expansions of the $\tau$-function, in which the coefficients in the first are diagonal and of {\em content product} form, guaranteeing that the Hirota bilinear equations of the integrable hierarchy are satisfied, while those in the second provide the weighted Hurwitz numbers.
Besides the various ``classically weighted'' cases, arising through different choices of the parameters
$(c_1, c_2, \dots)$, there are also ``quantum deformations'', depending on an additional pair $(q,t)$ that
are closely linked to the MacDonald symmetric functions \cite{Mac}. This leads to the notion of `quantum weighted'' Hurwitz numbers, of various types \cite{GH2, H2}, which may depend both on the infinity of classical weighting parameters $(c_1, c_2, \dots)$, and the further pair $(q,t)$, in a specific way, involving $q$-deformations.
Another generalization consists of introducing multiple expansion parameters $(z_1, z_2, \dots)$, leading to
generating functions for weighted ``multispecies'' weighted Hurwitz numbers \cite{H1},
which are counted with different weighting factors, depending on the species type, or ``colour''.
In \autoref{tau_functions}, a quick review is given of the fermionic approach to $\tau$-functions
for the KP hierarchy and modified KP sequence of $\tau$-functions as introduced by Sato,
\cite{Sa, SS} as well as the 2D Toda case introduced in \cite{Ta, UTa, Takeb}.
\autoref{center_group_algebra_hurwitz} recalls basic notions regarding the $S_n$ group algebra,
including the commuting {\em Jucys-Murphy elements}\cite{Ju, Mu}, Frobenius' characteristic map from the center
$\mathbf{Z}(\mathbf{C}[S_n])$ to the algebra $\Lambda$ of symmetric functions, and the abelian group within $\mathbf{Z}(\mathbf{C}[S_n])$ that is generated through a combination of these. \autoref{hypergeom_hurwitz} gives a summary of the new approach to the construction of $\tau$-functions of hypergeometric type interpretable as generating functions for infinite parametric families of weighted Hurwitz numbers developed in \cite{GH1, GH2, H1, H2, HO3}. The weightings are interpreted both geometrically, as weighted enumeration of $n$-sheeted branched covers of the Riemann sphere, and combinatorially, as weighted enumeration of paths in the Cayley graph of $S_n$ generated by transpositions. The relation between these is easily seen algebraically through the Cauchy- Littlewood generating functions for dual pairs of bases for $\Lambda$.
\autoref{examples_classical} is devoted to the various examples that have so far been
considered in the literature. These include: the original case of single and double Hurwitz numbers, generated by the special KP and 2D Toda $\tau$-functions studied by Pandharipande and Okounkov \cite{Pa, Ok}; the case of the HCIZ integral \cite{IZ, HC, GGN1, GGN2, GH1}, which is known to have the combinatorial interpretation of counting weakly monotonically increasing paths of transpositions in the Cayley graph, to which is added the geometrical one of signed enumeration of branched coverings with an arbitrary number of branch points with arbitrary branching profiles, at fixed genus; another case \cite{GH1}, which counts strongly monotonic such paths, and can be related to the special case of counting Belyi curves \cite{AC1, AC2, Z, KZ} (with three branch points) or ``Dessins d'enfants''; and
a hybrid case \cite{GH1}, which combines the two, and counts branching configurations of multiple ``colour'' type and, moreover also has a matrix model representation. More general ``multispecies'' branched coverings, with their associated combinatorial equivalents \cite{HO3, H1}, and other, more general parametric families of weighted Hurwitz numbers are considered in \autoref{multispecies}.
Already in the ``classical'' setting, it is possible to select the parameters $(c_1, c_2, \dots)$ appearing in the associated weight generating functions in such a way that the resulting weightings, both for branched coverings and for paths, involve what may be interpreted as a quantum deformation parameter $q$ . When suitably interpreted in terms of Planck's constant $\hbar$ and temperature, the resulting distributions can be related to the energy distribution law for a Bose gas with linear energy spectrum. In \autoref{macdonald_quantum_hurwitz}, we extend the family of weight generating functions by introducing a further pair $(q,t)$ of deformation parameters that play the same r\^ole as those appearing in the MacDonald symmetric functions \cite{Mac}, with the Cauchy-Littlewood generating functions replaced by the corresponding one for Macdonald functions \cite{H2}. The resulting weighted Hurwitz numbers are interpretable as multispecies quantum Hurwitz numbers, whose distributions are again related to those for a Bosonic gas. Various specializations are obtained by choosing specific values for the parameters $q$ and $t$, or relations between them, or various limits. Besides recovering the ``classical'' weighting, for $q=t$,
this leads to various other specializations, such as weightings involving the quantum analog of the elementary and complete symmetric functions, the Hall-Littlewood polynomials and the Jack polynomials.
\subsection{Enumerative geometrical Hurwitz numbers}
For any set of partitions $\{\mu^{(1)}, \dots, \mu^{(k)}\}$ of $n \in \mathbf{N}^+$, we define the geometrical Hurwitz number
$H(\mu^{(1)}, \dots, \mu^{(k)})$ to be the number of $n$-sheeted branched coverings of the Riemann
sphere having no more than $k$ branch points $\{q_1, \dots , q_k\}$, with ramification profiles
of type $\{\mu^{(i)}\}$, weighted by the inverse of the order of their automorphism groups.
The Frobenius-Schur formula \cite{Frob1, Frob2, Sch, LZ} expresses these in terms of the irreducible characters $\chi_\lambda(\mu^{(i)})$ of the symmetric group $S_n$
\begin{equation}
H(\mu^{(1)}, \dots, \mu^{(k)}) = \sum_{\lambda, |\lambda|=n} h_\lambda^{k-2} \prod_{i=1}^k z_{\mu^{(i)}}^{-1} \chi_\lambda(\mu^{(i)})
\label{frob_schur}
\end{equation}
where $\lambda$ is the partition corresponding to the irreducible representation with Young symmetrizer of type $\lambda$, and the parts of the partitions $\{\mu^{(i)}\}$ are the cycle lengths defining the ramifications profiles that determine the conjugacy classes $\cyc(\mu^{(i)})$ on which $\chi_\lambda$ is evaluated. Here
\begin{equation}
z_\mu = \prod_{i=1}^{\ell(\mu^{(i)})} i^{m_i(\mu)} (m_i(\mu))!
\end{equation}
is the order of the stabilizer of any element in $\cyc(\mu)$ under conjugation,
where $m_i(\mu)$ is the number of parts of $\mu$ equal to $i$ and
\begin{equation}
h_\lambda := \mathrm {det}\left({1\over (\lambda_i - i+ j)!}\right)^{-1}
\end{equation}
is the product of the hook lengths of the partition $\lambda$.
\subsection{Combinatorial Hurwitz numbers}
The combinatorial definition of the Hurwitz number, denoted $F(\mu^{(1)}, \dots, \mu^{(k)})$ (and perhaps more
aptly called the {\em Frobenius number}, although the two turn out to be equal!) is the following:
$n! F(\mu^{(1)}, \dots, \mu^{(k)})$ is the number of ways the identity element $\mathbf{I} \in S_n$ may be factorized into a product
\begin{equation}
\mathbf{I} = g_1 \cdots g_k,
\end{equation}
in which the $i$th factor $g_i\in S_n$ is in the conjugacy class $\cyc(\mu^{(i)})$.
The equality of these two quantities
\begin{equation}
F(\mu^{(1)}, \dots, \mu^{(k)}) =H(\mu^{(1)}, \dots, \mu^{(k)})
\label{hurwitz_frobenius}
\end{equation}
follows from the monodromy representation of the fundamental group $\pi_1(\mathbf{C}\mathbf{P}^1/\{q_1, \dots, q_k\})$ of the punctured sphere with the branch points removed \cite[Appendix~A]{LZ}.
As shown in \autoref{hurwitz_numbers}, relation (\ref{frob_schur}) follows from (\ref{hurwitz_frobenius})
and the Frobenius character formula. Avatars of this equality will be seen to recur repeatedly in the
various versions of weighted Hurwitz numbers studied below.
\section{mKP and 2D Toda $\tau$-functions}
\label{tau_functions}
\subsection{Fermionic Fock space}
\label{fermionic_fack_space}
The fermionic Fock space $\mathcal{F}$ is defined \cite{SS} as the semi-infinite wedge product space
\begin{equation}
\mathcal{F} := \Lambda^{\infty/2} \mathcal{H}
\end{equation}
constructed from a separable Hilbert space $\mathcal{H}$ with orthonormal basis $\{e_i\}_{i\in \mathbf{Z}}$,
that is split into an orthogonal direct sum of two subspaces
\begin{equation}
\mathcal{H} = \overline{\mathcal{H}_+ \oplus \mathcal{H}_-},
\end{equation}
where
\begin{equation}
\mathcal{H}_- = \mathrm {span} \{e_i\}_{i\in \mathbf{N}}, \quad \mathcal{H}_+=\mathrm {span}\{e_{-i}\}_{i\in \mathbf{N}^+}.
\end{equation}
and $\{e_i\}_{i\in \mathbf{Z}}$ is an orthonormal basis.
\begin{remark}\small \rm
The curious convention of using negative $i$'s to label the basis for $\mathcal{H}_+$ and
positive ones for $\mathcal{H}_-$ stems from the notion of the ``Dirac sea'', in which
all negative energy levels are filled and all positive ones empty, where the
integer lattice is identified with the energies. If we take Segal and Wilson's \cite{SW} model
for $\mathcal{H}$
\begin{equation}
\mathcal{H} := L^2(S^1) = \overline{\mathrm {span}\{z_i\}}_{i\in \mathbf{Z}} \text{ with } e_i:= z^{-i-1},
\end{equation}
we may view $\mathcal{H}_+$ and $\mathcal{H}_-$ either as the subspaces of positive and negative Fourier series
on the circle $S^1$ or, equivalently, the Hardy spaces of square integrable functions admitting a
holographic extension to inside and outside the unit circle, with the latter vanishing at $z= \infty$.
\end{remark}
$\mathcal{F}$ is the graded sum
\begin{equation}
\mathcal{F} = \oplus_{N\in \mathbf{Z}} \mathcal{F}_N
\end{equation}
of the subspaces $\mathcal{F}_N$ with fermionic charge $N\in \mathbf{Z}$.
An orthonormal basis $\{|\lambda; N\rangle\}$ for these is provided by the
semi-infinite wedge product states
\begin{equation}
|\lambda; N\rangle := e_{\ell_1} \wedge e_{\ell_2} \wedge \cdots
\end{equation}
labeled by pairs of partitions $\lambda$ and integers $N\in \mathbf{Z}$, where
\begin{equation}
\{\ell_i := \lambda_i - i +N\}
\end{equation}
are the ``particle coordinates'', indicating the occupied points on the integer lattice, corresponding to
the parts of the partition $\lambda$, with the usual convention that, for $i$ greater
than the length $\ell(\lambda)$ of the partition, $\lambda_i := 0$. The vacuum state in the
charge $N$ sector $\mathcal{F}_N$ of the Fock space is denoted
\begin{equation}
|N\rangle := | 0; N\rangle.
\end{equation}
In Segal and Wilson's \cite{SW} sense, the image $\mathcal{P}(W)$, under the Pl\"ucker map
\begin{eqnarray}
\mathcal{P}: Gr_{\mathcal{H}_+}(\mathcal{H}) &\&\rightarrow \mathbf{P}(\mathcal{F}) \cr
\mathcal{P}: W\ &\&\ {\mapsto} \ \mathbf{P}(W) \cr
\mathcal{P}: \Span\{w_i \in \mathcal{H}\}_{i\in \mathbf{N}^+} &\&\mapsto [w_1 \wedge w_2 \wedge \cdots],
\end{eqnarray}
of an element $W\in Gr_{\mathcal{H}_+}(\mathcal{H})$ of the infinite Grassmannian modeled
on $\mathcal{H}_+ \subset \mathcal{H}$, having {\em virtual dimension} $N$ (i.e., such that the Fredholm
index of the orthogonal projection map $\pi^{\perp}: W \rightarrow \mathcal{H}_+$ is $N$)
is in the charge $N$ sector $\mathcal{P}(W)\in \mathcal{F}_N \subset \mathcal{F}$, and the entire image consists of all decomposable elements of $\mathcal{F}$.
In particular, $\mathcal{H}_+$ is mapped to the projectivization of the vacuum element
\begin{equation}
\mathcal{P}: \mathcal{H}_+ {\mapsto} [|0 \rangle]:=[ |0;0\rangle] = [e_{-1}\wedge e_{-2}\wedge \cdots ] .
\end{equation}
The Fermi creation and annihilation operators $\psi_i$, $\psi_i^\dag$ are defined as exterior multiplication
by the basis element $e_i$ and interior multiplication by the dual basis element $\tilde e^i$, respectively.
\begin{equation}
\psi := e_i \wedge \quad \psi^\dag := i(\tilde{e}^i ).
\end{equation}
These satisfy the usual anticommutation relations
\begin{equation}
[\psi_i, \, \psi_j^\dag]_+ = \delta_{ij}
\end{equation}
defining the corresponding Clifford algebra on $\mathcal{H} + \mathcal{H}^*$ with respect to the natural quadratic form
in which both $\mathcal{H}$ and $\mathcal{H}^*$ are totally isotropic.
The infinite general linear algebra $\grg\grl(\mathcal{H}) \subset \Lambda^2(\mathcal{H} +\mathcal{H}^*)$, in the standard Clifford representation,
is spanned by the elements $:\psi_i \psi_j^\dag:$, with the usual convention for normal ordering
\begin{equation}
:\psi_i \psi_j^\dag: = :\psi_i \psi_j^\dag: - \langle \psi_i \psi_j^\dag \rangle,
\end{equation}
where $\langle \mathcal{O}\rangle$ denotes the vacuum expectation value
\begin{equation}
\langle \mathcal{O}\rangle := \langle 0 |\mathcal{O} | 0\rangle.
\end{equation}
The corresponding group $\mathfrak{G}} \def\grg{\mathfrak{g}\grl(\mathcal{H})$ consists of invertible endomorphisms,
having well defined determinants. (See \cite{SS, Sa, SW} for more detailed definitions.)
A typical exponentiated element in the Clifford representation is of the form
\begin{equation}
\hat{g} = e^{\sum_{ij \in \mathbf{Z}} A_{ij} :\psi_i \psi_j^\dag:},
\end{equation}
where the doubly infinite square matrix with elements $A_{ij}$ satisfies suitable
convergence conditions \cite{Sa, SS, Sa, SW} that will not be detailed here.
\subsection{Abelian group actions, mKP and 2D Toda lattice $\tau$-functions and Hirota relations}
The KP and 2D-Toda flows are generated by the multiplicative action on $\mathcal{H}$ of the two infinite abelian subgroups
$\Gamma_{\pm} \subset \mathfrak{G}} \def\grg{\mathfrak{g}\grl_0(\mathcal{H})$ of the identity component $\mathfrak{G}} \def\grg{\mathfrak{g}\grl_0(\mathcal{H})$ of the general linear group $\mathfrak{G}} \def\grg{\mathfrak{g}\grl(\mathcal{H})$,
defined by:
\begin{equation}
\Gamma_+:=\{\gamma_+({\bf t}):=e^{\sum_{i=1}^\infty t_i z^i}\},
\quad {\rm and \quad} \Gamma_- :=\{\gamma_-({\bf s}):=e^{\sum_{i=1}^\infty s_i z^{-i}}\},
\end{equation}
where ${\bf t}= (t_1, t_2, \dots)$ is an infinite sequence of (complex) flow parameters corresponding to one-parameter
subgroups, and ${\bf s}= (s_1, s_2, \dots)$ is a second such sequence. These in turn have the following Clifford group
representations on $\mathcal{F}$
\begin{equation}
\hat{ \Gamma}_+:=\{\hat{\gamma}_+({\bf t}):=e^{\sum_{i=1}^\infty t_i J_i}\},
\quad {\rm and \quad} \hat{\Gamma}_- :=\{\hat{\gamma}_-({\bf s}):=e^{\sum_{i=1}^\infty s_i J_{-i}}\},
\end{equation}
where
\begin{equation}
J_i:= \sum_{k \in \mathbf{Z}}: \psi_i \psi_{i+k}^\dag :, \quad \pm i \in \mathbf{N}^+
\end{equation}
are referred to as the ``current components''. In this infinite dimensional setting, whereas the abelian
groups $\Gamma_\pm$ commute, their Clifford representations $\hat{\Gamma}_\pm$ involve
a central extension, so that
\begin{equation}
\hat{\gamma}_+({\bf t})\hat{\gamma}_-({\bf s}) = \hat{\gamma}_-({\bf s}) \hat{\gamma}_+({\bf t}) e^{\sum_{i\in \mathbf{Z}} i t_i s_i}.
\end{equation}
The mKP-chain and 2D-Toda $\tau$-functions corresponding to the element
$g \in \mathfrak{G}} \def\grg{\mathfrak{g}\grl_0(\mathcal{H}$ are given, within a nonzero multiplicative constant, by the vacuum expectations values (VEV's)
\begin{eqnarray}
\tau_g^{mKP} (N, {\bf t})&\& := \langle N \vert \hat{\gamma}_+({\bf t}) \hat{g} \vert N \rangle,
\label{KPtau} \\
\tau^{(2Toda)}_g(N, {\bf t}, {\bf s}) &\&: =
\langle N \vert \hat{\gamma}_+({\bf t})\hat{g}\hat{\gamma}_-({\bf s}) \vert N\rangle.
\label{2KPtau}
\end{eqnarray}
If the group element $g\in \mathfrak{G}} \def\grg{\mathfrak{g}\grl_0(\mathcal{H} $ is interpreted, relative to the $\{e_i\}_{i\in \mathbb{Z}}$ basis, as a
matrix exponential $g=e^A$, where the algebra element $A\in \grgl(\mathcal{H})$ is represented by the infinite matrix with
elements $\{A_{ij}\}_{i,j \in \mathbb{Z}}$, then the corresponding representation of $GL(\mathcal{H})$ on $\mathcal{F}$ is given by
\begin{equation}
\hat{g} := e^{\sum_{i,j \in \mathbb{Z}} A_{ij} : \psi_i \psi^\dag_j : } ,
\label{ghat}
\end{equation}
These satisfy the Hirota bilinear relations
\begin{eqnarray}
&\& \oint_{z=\infty} z^{N'-N}e^{-\xi(\delta{\bf t}, z)} \tau^{mKP}_g(N, { \bf t}+ \delta{\bf t} + [z^{-1}])
\tau^{mKP}_g(N',{ \bf t} -[z^{-1}] ) =0
\label{hirotaKP}
\\
&\& \oint_{z=\infty} z^{N'-N}e^{-\xi(\delta{\bf t}, z)} \tau^{2D\text{T}}(N,{ \bf t} +[z^{-1}] , {\bf s})
\tau^{2D\text{T}}(N', { \bf t}+ \delta{\bf t} - [z^{-1}], {\bf s} + \delta{\bf s}) =
\cr
&\& \oint_{z=0} z^{N'-N} e^{-\xi(\delta{\bf s}, z^{-1})} \tau^{2D\text{T}}(N-1, {\bf t}, { \bf s} +[z])
\tau^{2D\text{T}}(N'+1, {\bf t} + \delta{\bf t}, {\bf s}+ \delta{\bf s} - [z] )
\label{hirotaToda}
\end{eqnarray}
understood to hold identically in $\delta{\bf t} = (\delta t_1, \delta t_2, \dots),\ \delta{\bf s} := (\delta s_1, \delta s_2, \dots)$,
where
\begin{equation}
\quad [z]_i := {1 \over i }z^i.
\end{equation}
\subsection{Bose-Fermi equivalence and Schur function expansions}
\label{bose_fermi_schur_expansion}
It follows from the identities \cite{Sa, SS}
\begin{equation}
\langle N | \hat{\gamma}_+({\bf t}) | \lambda; N\rangle = \langle\lambda; N | \hat{\gamma}_-({\bf t}) | N\rangle = s_\lambda({\bf t}),
\end{equation}
where $s_\lambda$ is the Schur function corresponding to partition $\lambda$, viewed
as function of the parameters
\begin{equation}
t_i := {p_i \over i},
\end{equation}
where the $p_i$'s are the power sums, that the $\tau$-functions may be expressed,
at least formally, as single and double Schur functions expansions
\begin{eqnarray}
\tau_g^{mKP}(N, {\bf t}) &\& = \sum_{\lambda} \pi_\lambda(N, g) s_\lambda ({\bf t}) \\
\tau_g^{2Toda}(N, {\bf t}, {\bf s})) &\& = \sum_{\lambda} \sum_\mu B_{\lambda \mu} (N, g) s_\lambda ({\bf t}) s_\mu ({\bf t})
\end{eqnarray}
where
\begin{equation}
\pi_\lambda(N,g) := \langle \lambda; N | \hat{g} |N \rangle, \quad B_{\lambda \mu}(N, g) := \langle \lambda; N | \hat{g} |\mu; N \rangle,
\end{equation}
are the Pl\"ucker coordinates of the elements $ \hat{g} |N \rangle$ and $\hat{g} |\mu; N\rangle$ when $g\in \mathfrak{G}} \def\grg{\mathfrak{g}\grl_0(\mathcal{H})$ is in the
identity component of $\mathfrak{G}} \def\grg{\mathfrak{g}\grl(\mathcal{H})$. The Hirota bilinear relations (\ref{hirotaKP}), (\ref{hirotaToda}) are then equivalent to the Pl\"ucker relations satisfied by these coefficients.
The ``Bose-Fermi equivalence'' gives an isomorphism between a completion $\mathcal{B}_0$ of the space of symmetric functions $\Lambda$ of an infinite number of ``bosonic'' variables $\{x_i \}_{i\in \mathbf{N}}$, labelled by the natural numbers and the $N=0$ (zero charge) sector of the Fermionic Fock space $\mathcal{F}_0 \subset \mathcal{F}$ which identifies the basis states $\{|\lambda; 0 \rangle\}$ with the basis of Schur functions $\{s_\lambda \in \Lambda\}$ through the ``bosonization'' map:
\begin{eqnarray}
\mathbf{B} : \mathcal{F}_0 \, &\& \rightarrow \, \mathcal{B}_0 \cr
\mathbf{B}: | v \rangle \ &\& \ {\mapsto} \ \langle 0 | \hat{\gamma}_+ |v\rangle \cr
\mathbf{B}: |\lambda; 0 \rangle \ &\& \ {\mapsto} \ \ s_\lambda.
\end{eqnarray}
More generally, this can be extended to the full (graded) fermonic Fock space $\mathcal{F}= \oplus_{N \in \mathbf{Z}}\mathcal{F}_N$
by adding a parameter $\zeta$ to the Bosonic Fock space, taking formal Laurent expansions in this
\begin{equation}
\mathcal{B}:= \mathcal{B}_0 [[\zeta]],
\end{equation}
and defining
\begin{eqnarray}
\mathbf{B} : \mathcal{F}_N &\& \rightarrow \, \mathcal{B}_N \cr
\mathbf{B}: | v \rangle &\& \ {\mapsto} \ \langle N | \hat{\gamma}_+ |v\rangle \zeta^N
\end{eqnarray}
Using $\mathbf{B}$ as an intertwining map, this defines identifications between
operators in $\mathrm {End}(\mathcal{F})$ and those in $\mathrm {End}(\mathcal{B})$. However, what appears
in the Fermonic representation as a ``locally'' defined element of the Clifford algebra
or group is in general a nonlocal operator in the Bosonic representation (involving exponentiated
differential operators in terms of the ${\bf t}$ coordinates), as is the case, e.g. , for the
Bosonic representations of the operators $\psi, \psi_i^\dag $, which are special types
of ``vertex operators''. In particular, the Bosonization of fermionic states of the type
$\hat{g} \hat{\gamma}_- | 0\rangle$ is given by application of nonlocal operators of the type
that were interpreted in \cite{GJ1} as ``cut-and-join'' operators, to the gauge transform
of the vacuum state, defined by $ \hat{\gamma}_- | 0\rangle$
\subsection{Hypergeometric $\tau$-functions and convolution symmetries}
\label{hypergeometric_convolution_sym}
\label{convolution_symmetries}
A special subfamily of the above consists of those $\tau$-functions for which the group
element $\hat{g}$ is diagonal
\begin{equation}
\hat{g} = e^{\sum_{i \in \mathbf{Z}} T_i : \psi_i \psi_i^\dag}, \quad A_{ij} = T_i \delta_{ij}
\label{conv_symm}
\end{equation}
in the basis $|\lambda;N\rangle$. These were named {\em convolutions symmetries} in \cite{HO2}, since in the Segal-Wilson representations of $\mathfrak{G}} \def\grg{\mathfrak{g}\grl(\mathcal{H})$ they may be interpreted as (generalized) convolution products
on $\mathcal{H} \sim L^2(S_1)$. Their eigenvalues $ r_\lambda(N, g)$ in the basis $|\lambda;N\rangle$
\begin{equation}
e^{\sum_{ i \in \mathbf{Z}} T_i : \psi_i \psi_i^\dag } |\lambda; N\rangle = r_\lambda(N, g) |\lambda; N\rangle
\end{equation}
can be written in the form of a {\em content product} \cite{OrSc, HO2}:
\begin{equation}
r_\lambda(N, g) := r_0(N,g) \prod_{(i,j) \in \lambda} r_{N+j-i}( g), \quad r_i(g) := e^{T_{i} - T_{i-1}}
\label{content_product_fermion}
\end{equation}
where
\begin{equation}
r_0(N,g) := \begin{cases} \prod_{i=0}^{N-1}e^{T_i} \quad {\rm if} \quad N > 0 \cr
\quad \ 1 \qquad \ \ {\rm if} \quad N=0 \cr
\prod _{i=N}^{-1} e^{-T_i} \quad \ \ {\rm if } \quad N < 0.
\end{cases}
\end{equation}
The double Schur function expansion (\ref{2KPtau}) in this case reduces to the diagonal form
\begin{equation}
\tau_g^{2Toda}(N, {\bf t}, {\bf s})) = \sum_{\lambda} r_\lambda(N,g) s_\lambda ({\bf t}) s_\lambda ({\bf s}).
\label{2DT_hypergeom}
\end{equation}
If we view the second set of parameters $(c_1, c_2, \dots)$ as fixed, and consider only the first set $(1_1, 1_2,, \dots)$ as KP flow parameters, we may interpret (\ref{2DT_hypergeom}) as defining a chain of mKP $\tau$ functions.
A specific value of special interest is $(c_1, c_2, \dots )$ = $(1, 0, 0 \dots)$, for which the Schur function
evaluates to
\begin{equation}
s_\lambda(1,0, \dots) = h_\lambda^{-1}
\end{equation}
and (\ref{2DT_hypergeom}) reduces to
\begin{equation}
\tau_g^{mKP}(N, {\bf t}, {\bf s})) = \sum_{\lambda} r_\lambda(N,g) h_\lambda^{-1}s_\lambda ({\bf t}).
\label{mKP_hypergeom}
\end{equation}
In the following, only such {\em hypergeometric} $\tau$-functions will be needed.
By defining suitable parametric families of the latter, and expanding these in powers
of some auxiliary parameters, while leaving the others to define the weightings, It will be seen that
we can interpret them as generating functions for finite or infinite parametric families of weighted Hurwitz numbers, both classical and quantum, obtaining both a natural enumerative geometric and combinatorial interpretation in all cases.
\section{The center $\mathbf{Z}(\mathbf{C}[S_n])$ of the $S_n$ group algebra and symmetric functions}
\label{center_group_algebra_hurwitz}
\subsection{The $\{C_\mu\}$ and $\{F_\lambda\}$ bases}
There are two natural bases for the center $\mathbf{Z}(\mathbf{C}[S_n])$ of the group algebra of
the symmetric group $S_n$, both labelled by partitions of $n$. The first is
the basis of cycle sums $\{C_\mu\}|_{|\mu|=n}$, defined by
\begin{equation}
C_\mu := \sum_{h\in \cyc(\mu)} h.
\end{equation}
The second is the basis of orthogonal idempotents $\{F_\lambda\}_{|\lambda|=n}$,
which project onto the irreducible representations of type $\lambda$ and satisfy
\begin{equation}
F_\lambda F_\mu = F_\lambda \delta_{\lambda \mu}.
\label{FF_F}
\end{equation}
These are related by
\begin{eqnarray}
F_\lambda &\&= h_\lambda^{-1}\sum_{\mu, \, |\mu|=|\lambda| =n} \chi_\lambda(\mu) C_\mu
\label{F_lambda_C_mu}
\\
C_\mu &\&= z_\mu^{-1}\sum_{\lambda, \, |\lambda| = |\mu| =n} h_\lambda \chi_\lambda(\mu) F_\lambda
\label{C_mu_F_lambda}
\end{eqnarray}
which is equivalent to the Frobenius character formula (see below).
The main property of the $\{F_\lambda\}$ basis is that multiplication by any element
of the center $\mathbf{Z}(\mathbf{C}[S_n])$ is diagonal in this basis (as follows immediately from
(\ref{FF_F})).
\subsection{The characteristic map}
Frobenius' characteristic map defines a linear isomorphism between the characters
of $S_n$ and the characters of tensor representations of $GL(k)$, of total tensor
weight $n$, for $k$ sufficiently large. It maps the irreducible character $\chi_\lambda$ to the
Schur function $s_\lambda$, viewed as the corresponding $GL(k)$ character through the
Weyl character formula for any $k\ge \ell(\lambda).$ Equivalently, it defines a linear endomorphism
\begin{eqnarray}
\ch: \mathbf{Z}(\mathbf{C}[S_n]) &\& \, \rightarrow \, \Lambda \cr
\ch: F_\lambda &\& \ \, {\mapsto} \ {s_\lambda \over h_\lambda}
\end{eqnarray}
from the center $\mathbf{Z}(\mathbf{C}[S_n])$ of the group algebra to the algebra $\Lambda$ of symmetric functions \cite{Mac}.
The change of basis formulae (\ref{F_lambda_C_mu}), (\ref{C_mu_F_lambda}), together
with the Frobenius character formula
\begin{equation}
s_\lambda = \sum_{\mu, \, |\mu|= |\lambda| =n} z_\mu^{-1} \chi_\lambda(\mu) p_\mu,
\end{equation}
where
\begin{equation}
p_\mu := \prod_{i=1}^{\ell(\mu)} p_{\mu_i}
\end{equation}
is the power sum symmetric function, then imply that the characteristic map
takes the cycle sum basis into the $\{p_\mu\}$ basis for $\Lambda$
\begin{equation}
\ch: C_\mu \, {\mapsto} \, {p_\mu \over z_\mu}.
\end{equation}
\subsection{Combinatorics of Hurwitz numbers and the Frobenius-Schur formula}
\label{hurwitz_numbers}
The two bases $\{C_\mu\}$, $\{F_\lambda\}$ can be used to deduce
the Frobenius-Schur formula (\ref{frob_schur}), expressing $H(\mu^{(1)}, \dots, \mu^{(k)})$
in terms of the irreducible group characters $\chi_\lambda(\mu)$.
The product $\prod_{i=1}^k C_{\mu^{(i)}}$ of elements of the cycle sum basis
is central and hence can be expressed relative to the same basis:
\begin{equation}
\prod_{i=1}^k C_{\mu^{(i)}} = \sum_{\nu, |nu|= n} H(\mu^{(1)}, \dots, \mu^{(i)}, \nu) z_\nu C_\nu,
\label{C_mu_prod}
\end{equation}
and, in particular, the coefficient of the identity class, for which $\mu = (1)^n$ is $n!$ times
the Hurwitz number
\begin{equation}
[\mathbf{I} = C_{ (1)^n}]\prod_{i=1}^k C_{\mu^{(i)}}= n! H(\mu^{(1)}, \dots , \mu^{(k)}),
\label{C_mu_prod_Id}
\end{equation}
giving the number of factorizations of the identity element into a product of
$k$ elements within the conjugacy classes $\{\cyc(\mu^{(i)}\}_{i=1 , \dots, k}$.
Substituting the change of basis formula (\ref{C_mu_F_lambda}) into (\ref{C_mu_prod_Id}),
applying both sides to the basis element $\{F_\lambda\}$ and equating the eigenvalues
that result gives the Frobenius-Schur formula:
\begin{equation}
H(\mu^{(1)}, \dots , \mu^{(k)}) = \sum_{\lambda, \, |\lambda|=|\mu|= n} h_\lambda^{k-2} \prod_{i=1}^k {\chi_\lambda(\mu^{(i)} )\over z_\mu^{(i)}}.
\end{equation}
\subsection{Jucys-Murphy elements, central elements and weight generating functions }
We now recall the special commuting elements $(\mathcal{J}_1, \dots, \mathcal{J}_n\}$ of the group algebra $\mathbf{C}[S_n]$ introduced
by Jucys \cite{Ju} and Murphy \cite{Mu}. (See also \cite{DG}). These are defined by
\begin{equation}
\mathcal{J}_b := \sum_{a=1}^{n-1} (ab) \ \text{ for } b>1, \text{ and } J_1 :=0.
\end{equation}
where $(ab) \in S_n$ is the transposition that interchanges $a$ with $b$.
Although these are not central elements, they have two remarkable properties:
Any symmetric function $f(\mathcal{J}_1, \dots , \mathcal{J}_n)$, $f \in \Lambda_n$ formed from them is
central, and this central element has eigenvalues in the $F_\lambda$ basis that
are equal to the evaluation on the {\em content} of the partition $\lambda$; i.e. the
set of number $j-i$, where $\{(i,j) \in \lambda\}$ are the set of positions (in the English
convention) in the Young diagram of $\lambda$:
\begin{equation}
f(\mathcal{J}_1, \dots, \mathcal{J}_n) F_\lambda = f(\{j-i\}_{(ij) \in \lambda}) F_\lambda.
\end{equation}
A particular case of symmetric functions of $n$ variables consists of taking
a single generating function $G(z)$, expressed formally either as an infinite product
\begin{equation}
G(z) = \prod_{i=1}^\infty (1+ c_i z)
\label{G_inf_prod}
\end{equation}
or an infinite sum
\begin{equation}
G(z) = 1 +\sum_{i=1}^\infty G_i z^i
\end{equation}
or some limit thereof, and defining the central element as a product
\begin{equation}
G_n(z, \mathcal{J}) :=\prod_{b=1}^n G(z\mathcal{J}_a).
\end{equation}
(For the present, we are not concerned with whether $G(z)$
is polynomial, rational, a convergent series, in some field extension or just a formal
infinite series or infinite product; the considerations that follow are mainly algebraic, but are easily extended
to include either convergent series, through suitable completions, or formal series and products,
as in the generating functions for symmetric functions.)
When applied multiplicatively to the $\{F_\lambda\}$ basis, the central element $G_n(z, \mathcal{J})$ has eigenvalues
that are expressible as content products
\begin{equation}
G_n (z, \mathcal{J} ) F_\lambda= \prod_{(ij) \in \lambda} G(z(j-i)) F_\lambda, \quad |\lambda| =n .
\label{content_product_center}
\end{equation}
We also consider the ``dual'' generating function:
\begin{equation}
\tilde{G}(z) := {1\over G(-z)} = \prod_{i=1}^\infty (1 - c_i z)^{-1}
\label{tilde_G_inf_prod}
\end{equation}
and associated central element
\begin{equation}
\tilde{G}_n(z, \mathcal{J} ) :=\prod_{b=1}^n \tilde{G}(z\mathcal{J}_a),
\end{equation}
which similarly satisfies
\begin{equation}
\tilde{G}_n (z, \mathcal{J} ) F_\lambda= \prod_{(ij) \in \lambda} \tilde{G}(z(j-i)) F_\lambda, \quad |\lambda| =n .
\label{dual_content_product_center}
\end{equation}
This suggests comparison with the ``convolution symmetry'' elements in the fermionic representation
of the group $\mathfrak{G}} \def\grg{\mathfrak{g} \grl(\mathcal{H})$ and an extension of the Bose-Fermi equivalence, using the
characteristic map, to a correspondence between the direct sum $\oplus_{n\in \mathbf{N}} \mathbf{Z}(\mathbf{C}[S_n])$
and the $N=0$ sector $\mathcal{F}_0\subset \mathcal{F}$ of the fermonic Fock space.
\subsection{Bose-Fermi equivalence and $\oplus_{n\in \mathbf{N}} \,\mathbf{Z}(\mathbf{C}[S_n])$}
\label{hurwitz_numbers}
Composing the characteristic map with the Bose-Fermi equivalence we obtain
an endomorphism $\mathcal{E}$ from the direct sum $\oplus_{n \in \mathbf{N}} \,\mathbf{Z}(\mathbf{C}[S_n])$ of the centers of
the group algebras to the zero charge sector $\mathcal{F}_0$ in the Fermionic Fock space
\begin{eqnarray}
\mathcal{E} : \oplus_{n\in \mathbf{N}}\, \mathbf{Z}(\mathbf{C}[S_n])\, &\& \rightarrow \, \mathcal{F}_0 \cr
\mathcal{E}: F_\lambda \ &\& \ {\mapsto} \ h_\lambda^{-1} | \lambda; 0\rangle
\label{EE_def}
\end{eqnarray}
This provides an intertwining map between the central elements in the completion of
the group algebra formed from products of functions of a single variable, acting
by multiplications, and the convolution symmetries discussed in \autoref{hypergeometric_convolution_sym}.
Choosing the parameters $T_j$ in (\ref{conv_symm}) as
\begin{equation}
T^{G(z)}_j = \sum_{k=1}^j \mathrm {ln} G(zk), \quad T^{G(z)}_0(z) = 0, \quad T^{G(z)}_{-j}(z) = -\sum_{k=0}^{j-1} \mathrm {ln} G(-zk) \quad\text{for $j>0$}.
\end{equation}
so that
\begin{equation}
\hat{g} =\hat{C}_G := e^{\sum_{i \in \mathbf{Z}} T^G_i (z): \psi_i \psi_i^\dag:} ,
\label{conv_symm_G}
\end{equation}
it follows that
\begin{equation}
r_j(g) := r_j^{G(z)}= G(jz)
\end{equation}
and
\begin{equation}
\hat{C}_G \ket{\lambda; N} = r_\lambda^{G(z)}(N) \ket{\lambda; N}
\label{CG_lambda}
\end{equation}
with eigenvalues
\begin{equation}
r_\lambda^{G(z)}(N) \coloneqq r^{G(z)}_0(N) \prod_{(i,j)\in \lambda} G(z(N+ j-i)),
\end{equation}
where
\begin{equation}
r_0^{G(z)}(N) = \prod_{j=1}^{N-1} G((N-j)z)^j, \quad r_0(0) = 1, \quad r_0^{G(z)}(-N) = \prod_{j=1}^{N} G((j-N)z)^{-j},
\quad N>1
\end{equation}
The map $\mathcal{E}$ defined in ({\ref{EE_def}) therefore intertwines the action
of $\oplus_{n\in \mathbf{N}}G_n(z, \mathcal{J})$ on $\oplus_{n\in \mathbf{N}} \,\mathbf{Z}(\mathbf{C}[S_n])$ with that of $\hat{C}_G $ on $\mathcal{F}_0$.
The same applies to the dual generating functions $\tilde{G}(z)$, for which we obtain
the corresponding content product formula expression
\begin{equation}
r_\lambda^{\tilde{G}(z)}(N) \coloneqq r^{\tilde{G}(z)}_0(N) \prod_{(i,j)\in \lambda} \tilde{G}(z(N+ j-i)).
\end{equation}
For the following, we only have need of the $N=0$ case, for which we simplify the notation
for the content product coefficients to
\begin{eqnarray}
r_\lambda^{G(z)} &\& \coloneqq r_\lambda^{G(z)}(0) = \prod_{(i,j)\in \lambda} G(z( j-i)), \\
r_\lambda^{\tilde{G}(z)} &\& \coloneqq r_\lambda^{\tilde{G}(z)}(0) = \prod_{(i,j)\in \lambda} \tilde{G}(z( j-i)),
\end{eqnarray}
\section{Hypergeometric $\tau$-functions as generating functions for weighted Hurwitz numbers }
\label{hypergeom_hurwitz}
We are now ready to state the main results, which show that the
KP and 2D Toda $\tau$-functions of hypergeometric type
\begin{eqnarray}
\tau^{G(z)}({\bf t}) &\& = \sum_{\lambda} r^{G(z)}_\lambda h_\lambda^{-1}s_\lambda ({\bf t}), \\
\tau^{G(z)} ({\bf t}, {\bf s})) &\& = \sum_{\lambda} \ r^{G(z)}_\lambda s_\lambda ({\bf t}) s_\lambda ({\bf s}),
\end{eqnarray}
when expanded in bases of (products of) the power sum symmetric functions $\{p_\mu\}$, are
interpretable as generating functions for suitably defined infinite parametric weighted Hurwitz numbers, both in the enumerative geometric and the combinatorial sense. The details and proofs may be found in \cite{GH1, GH2, H1, H2, HO3}.
\subsection{The Cauchy-Littlewood formula and dual bases for $\Lambda$}
We have already encountered the two bases consisting of Schur functions $\{s_\lambda\}$ and power sum symmetric
functions $\{p_\lambda\}$ for the ring $\Lambda$ of symmetric functions in an arbitrary number
of indeterminates \cite{Mac}. In addition to these, there are four other useful bases, consisting of the products of the elementary symmetric functions
\begin{equation}
e_\lambda ({\bf x}) := \prod_{i=1}^{\ell(\lambda} e_{\lambda_i}
\end{equation}
the complete symmetric functions
\begin{equation}
h_\lambda( {\bf x} ):= \prod_{i=1}^{\ell(\lambda} h_{\lambda_i},
\end{equation}
with generating functions
\begin{equation}
E(z) = \prod_{ij} (1+z x_i) = \sum_{i=0}^\infty e_i z^i, \quad H(z) = \prod_{ij} (1-z x_i)^{-1} = \sum_{i=0}^\infty h_i z^i,
\end{equation}
the monomial sum symmetric functions
\begin{equation}
m_\lambda ({\bf x}) :=
\frac{1}{\abs{\aut(\lambda)}} \sum_{\sigma\in S_k} \sum_{1 \le i_1 < \cdots < i_k}
x_{i_\sigma(1)}^{\lambda_1} \cdots x_{i_\sigma(k)}^{\lambda_k},
\end{equation}
and the ``forgotten'' symmetric functions
\begin{equation}
f_\lambda ({\bf x}) :=
\frac{(-1)^{\ell^*(\lambda)}}{\abs{\aut(\lambda)}} \sum_{\sigma\in S_k} \sum_{1 \le i_1 \le \cdots \le i_k}
x_{i_\sigma(1)}^{\lambda_1}, \cdots x_{i_\sigma(k)}^{\lambda_k},
\end{equation}
where
\begin{equation}
{\bf x} := (x_1, x_2, \dots )
\end{equation}
is an infinite sequence of indeterminates, and defining $m_i(\lambda)$ to be the number of parts of $\lambda$
equal to $i$,
\begin{equation}
\abs{\aut(\lambda)} := \prod_{i=1}^{\ell(\lambda)}(m(\lambda_i))!
\end{equation}
is the order of the automorphism group of the conjugacy class of type $\lambda$ under conjugation.
These bases have the following duality and orthogonality properties with respect to the standard
scalar product $(\, , \, )$ in which the Schur functions are orthonormal \cite{Mac}:
\begin{equation}
(s_\lambda, s_\mu) = \delta_{\mu \nu}, \quad (p_\lambda, p_\mu) = z_\mu \delta_{\mu \nu},
\quad (e_\lambda, m_\mu) = \delta_{\mu \nu}, \quad \quad (f_\lambda, h_\mu) = \delta_{\mu \nu}.
\end{equation}
It follows \cite{Mac} that the Cauchy-Littlewood formula is expressible bilinearly in terns of l these dually paired bases
\begin{eqnarray}
\prod_{i=1}^\infty\prod_{j=1}^\infty (1-x_i y_j) ^{-1}&\&= \sum_\lambda s_\lambda({\bf x}) s_\lambda({\bf y}) \\
&\&= \sum_\lambda z_\mu^{-1}p_\lambda({\bf x}) p_\lambda({\bf y}) \\
&\&= \sum_\lambda e_\lambda({\bf x}) m_\lambda({\bf y})= \sum_\lambda e_\lambda({\bf y}) m_\lambda({\bf x})\\
&\&= \sum_\lambda f_\lambda({\bf x}) h_\lambda({\bf y}) = \sum_\lambda f_\lambda({\bf y}) h_\lambda({\bf x}) .
\end{eqnarray}
The dual Cauchy-Littlewood generating function is similarly expressed in terms these in a dual way \cite{Mac}:
\begin{eqnarray}
\prod_{i=1}^\infty\prod_{j=1}^\infty (1+x_i y_j) &\&= \sum_\lambda s_\lambda({\bf x}) s_{\lambda'}({\bf y}) \\
&\&= \sum_\lambda (-1)^{\ell^*(\lambda)}z_\mu^{-1}p_\lambda({\bf x}) p_\lambda({\bf y}) \\
&\&= \sum_\lambda e_\lambda({\bf x}) f_\lambda({\bf y})= \sum_\lambda e_\lambda({\bf y}) f_\lambda({\bf x})\\
&\&= \sum_\lambda h_\lambda({\bf x}) m_\lambda({\bf y}) = \sum_\lambda h_\lambda({\bf y}) m_\lambda({\bf x}) ,
\end{eqnarray}
where $\lambda'$ is the partition whose Young diagram is the transpose of that for $\lambda$ and
\begin{equation}
\ell^*(\lambda) = |\lambda| -\ell(\lambda)
\end{equation}
is the {\em colength} of $\lambda$ (i.e., the complement of its length).
We make use of these formulae in the following way: for the indeterminates $(x_1, x_2, \dots )$
we substitute the parameters $(c_1, c_2, \dots )$ defining the weight generating function $G(z)$
as an infinite product, or its dual $\tilde{G}(z)$, while for the indeterminates $(y_1, y_2, \dots )$,
we substitute the $x$ times the Jucys-Murphy elements $(z\mathcal{J}_1, z\mathcal{J}_2, \dots )$ up to a finite number $n$ of
these, and $0$ for the rest, to obtain a finite sum in $j$.
This gives the following central elements, expressed as sums of products of these bases,
either evaluated on the contents $(c_1, c_2, \dots )$ or the commuting elements $(\mathcal{J}_1, \mathcal{J}_2, \dots)$:
\begin{eqnarray}
G_n(z,\mathcal{J}) &\&= \prod_{i=1}^n \prod_{a=1}^n(1 + zc_i \mathcal{J}_a)\\
&\&= \sum_{d=0}^\infty z^d\sum_{\lambda, \, |\lambda|=d} e_\lambda({\bf c}) m_\lambda(\mathcal{J})
\label{G_e_lambda_m_lambda}
\\
&\&= \sum_{d=0}^\infty z^d\sum_{\lambda, \, |\lambda|=d}m_\lambda({\bf c}) e_\lambda(\mathcal{J})
\label{G_m_lambda_e_lambda}
\end{eqnarray}
and
\begin{eqnarray}
\tilde{G}_n(z,\mathcal{J}) &\&= \prod_{i=1}^n \prod_{a=1}^n(1 - zc_i \mathcal{J}_a)^{-1} \\
&\& =\sum_{d=0}^\infty z^d \sum_{\lambda, \, |\lambda|=d} h_\lambda({\bf c}) m_\lambda(\mathcal{J})
\label{tilde_G_h_lambda_m_lambda}
\\
&\&= \sum_{d=0}^\infty z^d \sum_{\lambda, \, |\lambda|=d} f_\lambda({\bf c}) e_\lambda(\mathcal{J}) .
\label{tilde_G_f_lambda_e_lambda}
\end{eqnarray}
Recall that the elements $G_n(z,\mathcal{J}) , \tilde{G}_n(z,\mathcal{J}) \in \mathbf{Z}(\mathbf{C}[S_n])$ are diagonal
in the basis of orthogonal idempotents, with the contact product coefficients as eigenvalues
\begin{equation}
G_n(z,\mathcal{J}) F_\lambda = r_\lambda^{G(z)} F_\lambda, \quad
\tilde{G}_n(z,\mathcal{J}) F_\lambda = r_\lambda^{\tilde{G}}(z) F_\lambda
\end{equation}
where
\begin{eqnarray}
r_\lambda^{G(z)} &\&= \prod_{(ij) \in \lambda} \prod_{k=1}^\infty (1 + z c_k (j-i)) \\
r_\lambda^{\tilde{G}(z)} &\&= \prod_{(ij) \in \lambda} \prod_{k=1}^\infty (1 - z c_k (j-i))^{-1}.
\end{eqnarray}
\subsection{Multiplication by $m_\lambda(\mathcal{J}) $ and $e_\lambda(\mathcal{J}) $ in the $C_\mu$ basis}
In order to proceed further, we need to compute the effect of multiplication
by $G_n(z,\mathcal{J})$ and $\tilde{G}_n(z,\mathcal{J}) $ in the basis $\{C_\mu\}$ of $\mathbf{Z}(\mathbf{C}[S_n])$ consisting
of cycle sums. Combinatorially, this requires the notion of the {\em signature} of a path in
the Cayley graph of $S_n$ generated by the transpositions.
\begin{definition}
Given a $d$-step path in the Cayley graph of $S_n$ generated by transpositions $(ab)$,
$a, b \in \{1, \dots, n \}$, $a<b$, consisting of the sequence:
\begin{equation}
h, \rightarrow (a_1 b_1)h \rightarrow (a_2 b_2) (a_1 b_1) h\rightarrow \dots \rightarrow (a_d b_d) \cdots (a_1 b_1) h,
\end{equation}
its {\em signature} $\lambda$ is the partition of weight $|\lambda|=d$ whose length $\ell(\lambda)$ equals the number of distinct second elements appearing in the sequence, and whose parts $\{ \lambda_i\}_{i =1, \dots, d}$ consist of the number of times each given second element occurs.
\end{definition}
The effect of multiplication of $C_\mu$ by the central element $m_\lambda(\mathcal{J}) \in \mathbf{Z}(\mathbf{C}[S_n])$
is given by the following easily proved lemma \cite{GH2}:
\begin{lemma}
\label{m_lambda_JJ_C_mu}
\label{generating_weighted_paths}
Multiplication by $m_\lambda(\mathcal{J})$ defines an endomorphism of $\mathcal{Z}(\mathbf{C}[S_n])$ which, expressed in the $\{C_\mu\}$ basis, is given by
\begin{equation}
m_\lambda(\mathcal{J}) C_\mu = {1\over |\mu|!} \sum_{\nu, \, \abs{\nu}=\abs{\mu}} m^\lambda_{\mu \nu} z_\nu C_\nu,
\end{equation}
where $m^\lambda_{\mu \nu}$ is the number of monotonic $\abs{\lambda}$-step paths in the Cayley graph of $S_n$ generated by all transpositions, starting from an element $h$ in the conjugacy class $\cyc(\nu)$ to $\cyc(\mu)$ with signature $\lambda$. Equivalently,
\begin{equation}
m^\lambda_{\mu \nu} :=
\frac{\prod_{i=1}^{\ell(\lambda)} \lambda_i!}{\abs{\lambda}!} \, \tilde{m}^\lambda_{\mu \nu}
\label{m_lambda_C_mu}
\end{equation}
where $\tilde{m}^\lambda_{\mu \nu} $ is the number of $|\lambda|$ step paths of signature $\lambda$ in the Cayley graph of $S_n$ generated by transpositions, starting at the conjugacy class $\cyc{\mu}$ and ending in the class $\cyc(\nu)$
\end{lemma}
On the other hand, the effect of multiplication of $C_\mu$ by the central element
$e_\lambda(\mathcal{J}) \in \mathbf{Z}(\mathbf{C}[S_n])$ is given by the following \cite{GH2}:
\begin{lemma}
Multiplication by $e_\lambda(\mathcal{J})$ defines an endomorphism of $\mathcal{Z}(\mathbf{C}[S_n])$ which, expressed in the $\{C_\mu\}$ basis, is given by
\label{e_lambda_JJ_C_mu}
\label{generating_weighted_covers}
\begin{eqnarray}
e_\lambda(\mathcal{J}) C_\mu &\& = \sum_{\substack {\mu^{(1)}, \dots , \, \mu^{(k)} , \\ \{\ell^*(\mu^{(i}) = \lambda_i\}} }
\left(\prod_{i=1}^k C_{\mu^{(i)}} \right)C_\mu
\label{e_lambda_C_mu} \\
&\& = \sum_{\substack {\mu^{(1)}, \dots , \, \mu^{(k)} , \\ \{\ell^*(\mu^{(i}) = \lambda_i\}} }
H(\mu^{(1)}, \dots, \mu^{(k)}, \mu, \nu) z_\nu C_\nu,
\label{f_lambda_C_mu}
\end{eqnarray}
(where the identity (\ref{C_mu_prod}) has been used in the second line).
\end{lemma}
\subsection{Weighted double Hurwitz numbers: enumerative geometric and combinatorial }
We now proceed to the enumerative geometrical definition of weighted Hurwitz numbers.
For a fixed pair of branch points, say at $(0, \infty)$, with ramification profiles $(\mu, \nu)$
and an additional set of $k$ branch points $(q_1, \dots, q_k)$ with ramification profiles
$(\mu^{(1)}, \dots , \mu^{(k)})$, we define the
weights to be given by the evaluation of the monomial sum and ``forgotten'' symmetric functions
at the parameter values ${\bf c} =(c_1, c_2, \dots)$ for the two cases corresponding to the
dual weight generating functions $G(z)$ and $\tilde{G}(z)$.
\begin{eqnarray}
W_G(\mu^{(1)}, \dots, \mu^{(k)}) &\& := m_\lambda ({\bf c})
\label{W_G_def}
\\
W_{\tilde{G}}(\mu^{(1)}, \dots, \mu^{(k)}) &\& :=
f_\lambda ({\bf c}).
\label{W_G_tilde_def}
\end{eqnarray}
The weighted geometrical Hurwitz numbers $H^d_G(\mu, \nu)$, giving the weighted count
of such $n$-sheeted branched coverings of the Riemann sphere, having a pair of specified
branch points with ramification profiles $\mu$ and $\nu$ and any number $k$ of further
branch points, with arbitrary ramification profiles $(\mu^{(1)}, \dots, \mu^{(k)})$, but fixed genus,
are defined to be the weighted sum
\begin{eqnarray}
H^d_G(\mu, \nu) &\&\coloneqq \sum_{k=0}^\infty \sideset{}{'}\sum_{\substack{\mu^{(1)}, \dots \mu^{(k)} \\ \sum_{i=1}^k \ell^*(\mu^{(i)})= d}}
W_{G}(\mu^{(1)}, \dots, \mu^{(k)}) H(\mu^{(1)}, \dots, \mu^{(k)}, \mu, \nu)
\label{Hd_G}
\\
H^d_{\tilde{G}}(\mu, \nu) &\&\coloneqq \sum_{k=0}^\infty \sideset{}{'}\sum_{\substack{\mu^{(1)}, \dots \mu^{(k)} \\ \sum_{i=1}^d \ell^*(\mu^{(i)})= d}}
W_{\tilde{G}}(\mu^{(1)}, \dots, \mu^{(k)}) H(\mu^{(1)}, \dots, \mu^{(k)}, \mu, \nu),
\label{Hd_tildeG}
\end{eqnarray}
where $\sum'$ denotes the sum over all partitions other than the cycle type of the identity element.
The genus $g$ of the covering cover is given by the Riemann-Hurwitz formula \cite{LZ},
\begin{equation}
2 -2g = \ell(\mu) + \ell(\nu) - d.
\label{riemann_hurwitz}
\end{equation}
where
\begin{equation}
d:= \sum_{i=1}^k \ell^*(\mu^{(i)}).
\end{equation}
The weighted combinatorial Hurwitz numbers $F^d_G(\mu, \nu)$ give weighted enumerations of the paths
in the Cayley graph of $S_n$ generated by transpositions.
Expanding the weight generating functions $G(z)$ as a Taylor series
\begin{equation}
G(z) = 1 + \sum_{i=1}^\infty G_i z^i, \quad \tilde{G}(z) = 1 + \sum_{i=1}^\infty \tilde{G}_i z^i,
\end{equation}
the weight for a given path depends only upon the signature $\lambda$, and is chosen to be the product
of the coefficients of the Taylor series of the generating functions $G(z)$ (or $\tilde{G}(z))$ corresponding to the parts $\lambda_i$
\begin{eqnarray}
G_\lambda := &\&\prod_{i=1}^{\ell(\lambda)} G_{\lambda_i} = e_\lambda({\bf c}) \\
\tilde{G}_\lambda := &\&\prod_{i=1}^{\ell(\lambda)} \tilde{G}_{\lambda_i} =h_\lambda({\bf c}) .
\end{eqnarray}
The path weights for signature $\lambda$, are thus chosen to be either
the products $e_\lambda({\bf c})$ of the elementary symmetric functions evaluated
at the weighting parameters ${\bf c} =(c_1, c_2, \dots)$ entering in the infinite product representation of $G(z)$ or,
in the dual case $\tilde{G}(z)$, the products $h_\lambda({\bf c})$ of the complete symmetric functions.
The weighted combinatorial Hurwitz numbers $F^d_G(z) (\mu, \nu)$ and
$F^d_{\tilde{G}} (\mu, \nu)$ are defined to be the weighted number of $d$-step paths, starting in the
conjugacy class $\cyc(\mu)$ and ending in $\cyc(\nu)$
\begin{eqnarray}
F^d_G(\mu, \nu)&\& \coloneqq {1\over n!} \sum_{\lambda, \ \abs{\lambda}=d} e_\lambda({\bf c}) m^\lambda_{\mu \nu}
\label{Fd_G_def}
\\
F^d_{\tilde{G}}(\mu, \nu)&\& \coloneqq {1\over n!} \sum_{\lambda, \ \abs{\lambda}=d} f_\lambda({\bf c}) m^\lambda_{\mu \nu}
\label{Fd_tilde_G_def}.
\end{eqnarray}
\subsection{Hypergeometric 2D Toda $\tau$-functions as generating functions for weighted Hurwitz number}
Applying the central elements $G_n(z,\mathcal{J}) $, $\tilde{G}_n(z,\mathcal{J}) $ to the cycle sums $C_\mu$, $|\mu|=n$,
and using (\ref{G_e_lambda_m_lambda}), (\ref{G_m_lambda_e_lambda}), (\ref{tilde_G_h_lambda_m_lambda}),
( \ref{tilde_G_f_lambda_e_lambda}), (\ref{e_lambda_C_mu}) and (\ref{f_lambda_C_mu}), gives \cite{GH2}
\begin{proposition}
\label{G_n_C_mu}
\begin{eqnarray}
G_n(z,\mathcal{J}) C_\mu &\&=\sum_{d=1}^\infty z^d \sum_{\substack{\nu \\ |\nu|=|\mu|=n}} F^d_G(\mu, \nu) \, C_\nu
=\sum_{d=1}^\infty z^d \sum_{\substack{\nu \\ |\nu|=|\mu|=n}}H^d_G(\mu, \nu) \, C_\nu \\
\tilde{G}_n(z,\mathcal{J}) C_\mu &\&=\sum_{d=1}^\infty z^d \sum_{\substack{\nu \\ |\nu|=|\mu|=n}} F^d_{\tilde{G}}(\mu, \nu) \, C_\nu
=\sum_{d=1}^\infty z^d \sum_{\substack{\nu \\ |\nu|=|\mu|=n}}H^d_{\tilde{G}}(\mu, \nu) \, C_\nu
\end{eqnarray}
\end{proposition}
This implies, in particular, that the two definitions of weighted Hurwitz numbers coincide:
\begin{corollary}
\begin{equation}
F^d_G(\mu, \nu) = H^d_G(\mu, \nu).
\end{equation}
\end{corollary}
Since
\begin{eqnarray}
G_n (z, \mathcal{J} ) F_\lambda &\& = r_\lambda^{G(z)} F_\lambda, \quad |\lambda| =n , \\
\tilde{G}_n (z, \mathcal{J} ) F_\lambda &\& = r_\lambda^{\tilde{G}(z)} F_\lambda,
\label{content_product_center}
\end{eqnarray}
the change of basis formulae ({\ref{F_lambda_C_mu}), (\ref{C_mu_F_lambda}) together with
Proposition~\ref{G_n_C_mu} imply that
\begin{equation}
\sum_{d=0}^\infty z^d H^d_G(\mu, \nu) = \sum_{d=0}^\infty z^d F^d_G(\mu, \nu)
= \sum_{\substack{\lambda \\ |\lambda|=|\mu|=|\nu|}} r_\lambda^{G(z)}z_\mu^{-1} z_\nu^{-1} \chi_\lambda(\mu) \chi_\lambda(\nu)
\end{equation}
This leads to our main result \cite{GH2}
\begin{theorem}
\label{tau_H_G_generating function}
The 2D Toda $\tau$-functions $\tau^{G(z)}({\bf t}, {\bf s})$, $\tau^{\tilde{G}(z)}({\bf t}, {\bf s})$
can be expressed as
\begin{eqnarray}
\tau^{G(z)}({\bf t}, {\bf s}) &\&=\sum_{d=0}^\infty z^d \sum_{\substack{\mu, \nu \\ |\mu|=|\nu|}} H^d_G(\mu, \nu) p_\mu({\bf t}) p_\nu({\bf s})
= \sum_{d=0}^\infty z^d \sum_{\substack{\mu, \nu \\ |\mu|=|\nu|}} F^d_G(\mu, \nu) p_\mu({\bf t}) p_\nu({\bf s})
\label{tau_G_H_G}
\\
\tau^{\tilde{G}(z)}({\bf t}, {\bf s}) &\&=\sum_{d=0}^\infty z^d \sum_{\substack{\mu, \nu \\ |\mu|=|\nu|}} H^d_{\tilde{G}}(\mu, \nu) p_\mu({\bf t}) p_\nu({\bf s})
= \sum_{d=0}^\infty z^d \sum_{\substack{\mu, \nu \\ |\mu|=|\nu|}} F^d_{\tilde{G}}(\mu, \nu) p_\mu({\bf t}) p_\nu({\bf s}),
\label{tau_tilde_G_H_G}
\end{eqnarray}
and hence are generating functions for the weighted Hurwitz numbers $H^d_G(\mu, \nu) $, $F^d_G(\mu, \nu)$,
$H^d_{\tilde{G}}(\mu, \nu) $, $F^d_{\tilde{G}}(\mu, \nu)$.
\end{theorem}
\section{Examples of weighted double Hurwitz numbers}
\label{examples_classical}
We now consider several examples of different types of weighted Hurwitz numbers that are special cases
of this approach. All these have appeared in the recent literature on the subject \cite{Pa, Ok, GGN1, GGN2, GH1, GH2, HO3, AC1, AC2, H1, H2, Z, KZ, AMMN}, and new examples are easily constructed. Further details for all cases may be found in \cite{GH1, GH2, HO3, H1, H2}
\subsection{Double Hurwitz numbers for simple branchings; enumeration of $d$-step paths in the Cayley graph with equal weight \cite{Pa, Ok}}
\label{ex1}
This was the original case studied by Okounkov \cite{Ok}, extending an earlier result of Pandharipande \cite{Pa}.
The weight generating function in this case is just the exponential $G=\exp$
\begin{equation}
G(z)= \tilde{G}(z) = e^z.
\end{equation}
The central element $G(z, \mathcal{J}) \in \mathbf{Z}(\mathbf{C}[S_n])$ is therefore
\begin{equation}
exp_n(z, \mathcal{J}) = e^{z\sum_{b=1}^n \mathcal{J}_b}
\end{equation}
and the content product formula and fermionic exponent coefficients are given by
\begin{eqnarray}
r^{\exp(z)}_j &\&= e^{jz}, \quad r_\lambda^{\exp} (z)= e^{\frac{z}{2} \sum_{i=1}^{\ell(\lambda)}\lambda_i (\lambda_i - 2i +1)} \\
T^{\exp(z)}_j &\& = {1\over2 } j(j+1)z.
\end{eqnarray}
The generating hypergeometric 2D Toda $\tau$-function is thus
\begin{equation}
\tau^{\exp(z)}({\bf t})
= \sum_{\lambda} e^{\frac{z}{2} \sum_{i=1}^{\ell(\lambda)}\lambda_i (\lambda_i - 2i +1)} s_\lambda ({\bf t}) s_\lambda ({\bf s}),
\end{equation}
For this case, the infinite product form \eqref{G_inf_prod} of the generating function must be interpreted as a limit
\begin{equation}
e^z =\lim_{m \rightarrow \infty} \left(1+ {z \over m}\right)^m,
\end{equation}
and the expression (\ref{W_G_def}) for the geometrical weighting becomes:
\begin{equation}
W_{\exp} (\mu^{(1)}, \dots, \mu^{(k)}) = \prod_{i=1}^k\delta_{(\ell^*(\mu^{(i)}) , 1)}
\end{equation}
(since we require $\ell^*(\mu^{(i)}) \ge 1, \, \forall i$).
\autoref{tau_H_G_generating function} therefore gives the generating function
\begin{equation}
\tau^{\exp(z)}({\bf t}) = \sum_{d=0}^\infty z^d \sum_{\substack{\mu, \nu \\ |\mu|=|\nu|}} H^d_{\exp}(\mu, \nu) p_\mu({\bf t}) p_\nu({\bf s})
\end{equation}
function for the (weighted) numbers
\begin{equation}
H^d_{\exp}(\mu, \nu) := H(\underbrace{(2, (1)^{n-2}), \dots , (2, (1)^{n-2})}_{d \text{ times }}, \mu, \nu)
\label{H_exp_d_mu_nu}
\end{equation}
of $n$-sheeted branched coverings of the Riemann sphere having $d$ branch points with simple ramification
(i.e. , profile $(2, (1)^{n-2})$ and two more (say, at $0$ and $\infty$) with profiles $\mu$ and $\nu$
(weighted, as usual, by the inverse of the automorphism group).
The combinatorial definition of the weighted Hurwitz number (\ref{Fd_G_def}) gives
\begin{eqnarray}
F^d_{\exp}(\mu, \nu) &\&= {1\over n!}\sum_{\lambda, \ \abs{\lambda}=d}{1\over \prod_{i=1}^{\ell(\lambda)}\lambda_i!} m^\lambda_{\mu \nu}\cr
&\&= {1\over d! n!} \times (\#\ d\text{-step paths from an element } h\in\cyc(\mu) \text{ to } \cyc(\nu)).
\end{eqnarray}
\subsection{Coverings with three branch points (Belyi curves): strongly monotonic paths
\cite{Z, KZ, AC1, GH2, HO3}}
\label{ex2}
In this case the weight generating function is
\begin{equation}
G(z) = E(z)\coloneqq 1+ z,
\end{equation}
so
\begin{equation}
c_1 =1, \quad c_i = 0, \quad i>1.
\end{equation}
Therefore the central element $E_n(z, \mathcal{J})\in \mathbf{Z}(\mathbf{C}[S_n])$ is
\begin{eqnarray}
E_n(z, \mathcal{J}) &\&= \prod_{a=1}^n (1+z \mathcal{J}_a), \\
r^{E(z)}_j&\& = 1 + zj, \quad r^{E(z)}_\lambda = \prod_{(i,j)\in \lambda} (1 + z(j-i)) = z^{\abs{\lambda}} \, (1/z)_{\lambda}, \\
T^{E(z)}_j &\& = \sum_{i=1}^j \mathrm {ln}(1+iz), \quad T^{E(z)}_{-j} = -\sum_{i=1}^{j-1}\mathrm {ln}(1-iz), \quad j > 0,
\end{eqnarray}
where
\begin{equation}
(u)_\lambda := \prod_{i=1}^{\ell(\lambda)}(u-i+1)_{\lambda_i}
\end{equation}
is the multiple Pochhammer symbol corresponding to the partition $\lambda$.
The generating $\tau$-function is thus \cite{GH1, HO3}
\begin{eqnarray}
\tau^{E(z)} ({\bf t}, {\bf s}) &\& = \sum_{\lambda}
z^{\abs{\lambda}} (1/z)_\lambda s_\lambda({\bf t}) s_\mu({\bf s}) \cr
&\& = \sum_{d=0}^\infty z^d \sum_{\mu, \nu,\; \abs{\mu}=\abs{\nu}} H_E^d(\mu, \nu) p_\mu ({\bf t}) p_\nu({\bf s}),
\end{eqnarray}
where
\begin{equation}
H_E^d(\mu, \nu) = \sideset{}{'}\sum_{\mu^{(1)}, \ \ell^*(\mu_1) =d}
H(\mu^{(1)}, \mu, \nu)
\end{equation}
is the number of $n=\abs{\mu}=\abs{\nu}=\abs{\mu^{(1)}}$ sheeted branched covers with branch
points of ramification type $(\mu,\nu)$ at $(0,\infty)$,
and one further branch point, with colength $\ell^*(\mu^{(1)}) =d$;
These are the double Hurwitz numbers for Belyi curves \cite{Z, KZ, AC1, GH2, HO3}, which enumerate $n$-sheeted branched coverings of the Riemann sphere having three ramification points, with ramification profile type $\mu$ and $\nu$ at $0$ and $\infty$, and a single additional branch point, with ramification profile $\mu^{(1)}$ of colength
\begin{equation}
\ell^*(\mu^{(1)}) \coloneqq n - \ell(\mu^{(1)}) = d,
\end{equation}
i.e., with $n-d$ preimages. The genus is again given by the Riemann-Hurwitz formula \eqref{riemann_hurwitz}.
Combinatorially, we have the weight
\begin{equation}
e_\lambda({\bf c}) = \delta_{\lambda, (1)^{|\lambda|}}
\end{equation}
and therefore
\begin{equation}
\sum_{\lambda, \abs{\lambda}=d} e_\lambda({\bf c}) m_\lambda(\mathcal{J}) = \sum_{b_1 < \dots< b_d} \mathcal{J}_{b_1} \cdots \mathcal{J}_{b_d}.
\end{equation}
The coefficient $ F_{E}^d(\mu, \nu)$ is thus
\begin{equation}
F_{E}^d(\mu, \nu) = m^{(1)^{d}}_{\mu \nu},
\label{F_Ed}
\end{equation}
which enumerates all $d$-step paths in the Cayley graph of $S_n$ starting at an element in the conjugacy class of cycle type $\nu$ and ending in the class of type $\mu$, that are strictly monotonically increasing in their second elements \cite{GH2, HO3}.
\subsection{Fixed number of branch points and genus: multimonotonic paths \cite{HO3} }
\label{ex3}
In this case the weight generating function is :
\begin{equation}
G(z) =E^k(z)\coloneqq (1+ z)^k,
\end{equation}
and hence
\begin{equation}
c_i =1, \ 1\le i \le k, \quad c_i =0, \forall i>k.
\end{equation}
Therefore the central element is
\begin{equation}
E_n(z, \mathcal{J})^k = \prod_{a=1}^n (1+z \mathcal{J}_a)^k,
\end{equation}
and
\begin{eqnarray}
r^{E^k(z)}_j &\&= (1 + zj)^k, \quad r_\lambda^{E^k(z)}
= \prod_{(i,j)\in \lambda} (1 + z(j-i))^k = z^{k \abs{\lambda}} ((1/z)_{\lambda})^{k}, \\
T^{E^k(z)}_j &\& = k \sum_{i=1}^j \mathrm {ln}(1+iz), \quad T^{E^k(z)}_{-j} = - k\sum_{i=1}^{j -1}\mathrm {ln}(1-iz), \quad j > 0.
\end{eqnarray}
The generating $\tau$-function is
\begin{eqnarray}
\tau^{E^k (z)} ({\bf t}, {\bf s}) &\& = \sum_{\lambda}
z^{\abs{\lambda}} (1/z)_\lambda s_\lambda({\bf t}) s_\mu({\bf s}) \cr
&\& = \sum_{d=0}^\infty z^d \sum_{\mu, \nu, \ \abs{\mu}=\abs{\nu}} H_{E^k}^d(\mu, \nu) p_\mu ({\bf t}) p_\nu({\bf s}),
\end{eqnarray}
where
\begin{equation}
H_{E^k}^d(\mu, \nu) = \sideset{}{'}\sum_{\substack{\mu^{(1)}, \dots, \mu^{(k)} \\ \sum_{i=1}^k\ell^*(\mu_i) =d}}
H(\mu^{(1)}, \dots \mu^{(k)}, \mu, \nu)
\end{equation}
is the number of $n=\abs{\mu}=\abs{\nu}=\abs{\mu^{(i)}}$ sheeted branched covers with branch points of ramification type $(\mu,\nu)$ at $(0,\infty)$,
and (at most) $k$ further branch points, such that the sum of the colengths of their ramification profile type
(i.e., the ``defect" in the Riemann Hurwitz formula \eqref{riemann_hurwitz}) is equal to $d$:
\begin{equation}
\sum_{i=1}^k \ell^*(\mu^{(i)}) = kn - \sum_{i=1}^k \ell(\mu^{(i)}) = d.
\end{equation}
This amounts to counting covers with the genus fixed by \eqref{riemann_hurwitz}
and the number of additional branch points fixed at $k$, but no restriction on their simplicity.
The combinatorial weighting for paths of signature $\lambda$ is
\begin{equation}
e_\lambda({\bf c})= \prod_{i=1}^{\ell(\lambda)}
\binom{k}{\lambda_i}
\end{equation}
and hence,
\begin{equation}
\sum_{\lambda, \abs{\lambda}=d} \left(\prod_{i=1}^{\ell(\lambda)} \binom{k}{\lambda_i}\right) m_\lambda(\mathcal{J})
= [z^d] \prod_{a=1}^n(1+z J_a)^k
\end{equation}
where $[z^d]$ means the coefficient of $z^d$ in the polynomial.
The weighted combinatorial Hurwitz number
\begin{equation}
F_{E^k}^d(\mu, \nu) = \sum_{\lambda, \abs{\lambda}=k} \left(\prod_{i=1}^{\ell(\lambda)} \binom{k}{\lambda_i}\right) m^\lambda_{\mu \nu}
\label{F_Ed}
\end{equation}
is thus the number of $(d+1)$-term products $(a_1\, b_1) \cdots (a_d\, b_d) h$ such that $h\in \cyc(\mu)$, while $(a_1\, b_1) \cdots (a_d\, b_d) h \in \cyc(\nu)$, which consist of a product of $k$ consecutive subsequences, each of which is strictly monotonically increasing in their second elements \cite{GH1, HO3}.
\subsection{Signed Hurwitz numbers at fixed genus: weakly monotonic paths \cite{GGN1, GGN2, GH1}}
\label{ex4}
This case was studied from the combinatorial viewpoint, and related to the HCIZ internal
in \cite{GGN1, GGN2, GH1}. It is the dual $\tilde{E}$ of the weight generating function of \autoref{ex2}.
\begin{equation}
\tilde{E}(z) := H(z)\coloneqq \frac{1}{1- z}
\end{equation}
and hence we have
\begin{equation}
c_i =1, \ 1\le i \le k, \quad c_i =0, \ \forall i>k
\end{equation}
as before, but the relevant combinatorial weighting factor is
\begin{equation}
h_\lambda({\bf c}) =1 \quad \forall \lambda .
\end{equation}
The corresponding central element is
\begin{equation}
H_n(z, \mathcal{J}) = \prod_{a=1}^n (1-z \mathcal{J}_a)^{-1},
\end{equation}
and therefore
\begin{eqnarray}
r^{H(z)}_j&\& = (1 - zj)^{-1}, \quad r_\lambda^{H(z)} (z) =
\prod_{(i,j)\in \lambda} (1 - z(j-i))^{-1} = (-z)^{-\abs{\lambda}}((-1/z)_\lambda)^{-1}, \\
T^{H(z)}_j &\& = - \sum_{i=1}^j \mathrm {ln}(1-iz), \quad T^{H(z)}_{-j} = \sum_{i=1}^{j -1}\mathrm {ln}(1+iz), \quad j > 0.
\end{eqnarray}
The generating $\tau$-function for this case is \cite{GH1, HO3}
\begin{eqnarray}
\tau^{H(z)} ({\bf t}, {\bf s}) &\& = \sum_{\lambda}
(-z)^{-\abs{\lambda}} \left(-1/z\right)^{-1}_{\lambda}
s_\lambda({\bf t}) s_\mu({\bf s}) \cr
&\& = \sum_{d=0}^\infty z^d\sum_{\mu, \nu,\; \abs{\mu}=\abs{\nu}} H^d_{H}(\mu, \nu) p_\mu ({\bf t}) p_\nu({\bf s})
\label{tau_H_s_lambda_exp}
\end{eqnarray}
where
\begin{equation}
H^d_{H}(\mu, \nu) = (-1)^{n+d}\sum_{k=1}^\infty (-1)^k \sideset{}{'} \sum_{\substack{\mu^{(1)},\dots,\mu^{(k)} \\ \sum_{i=1}^k\ell^*(\mu_i) = d}}
H(\mu^{(1)}, \dots \mu^{(k)}, \mu, \nu)
\end{equation}
is the signed enumeration of $n =\abs{\mu}=\abs{\nu}$ sheeted branched covers with branch points of ramification type $(\mu,\nu)$ at $(0,\infty)$,
and any number further branch points, the sum of whose colengths is $d $, with sign determined by the parity of the number of branch points \cite{HO3}.
These are thus double Hurwitz numbers for $n$-sheeted branched coverings of the Riemann sphere with branch points having ramification profile type $(\mu, \nu)$ at $(0,\infty)$ and an arbitrary number of further branch points,
such that the sum of the colengths of their ramification profile lengths is again equal to $d$
\begin{equation}
\sum_{i=1}^k \ell^*(\mu^{(i)}) = kn - \sum_{i=1}^k \ell(\mu^{(i)}) = d.
\end{equation}
The latter are counted with a sign, which is $ (-1)^{n+d}$ times the parity of the number of branch points \cite{HO3}. The genus is again given by \eqref{riemann_hurwitz}.
The combinatorial Hurwitz number $F_{H}^d(\mu, \nu)$, derived from
\begin{equation}
\sum_{\lambda, \abs{\lambda}=d} h_\lambda({\bf c}) m_\lambda(\mathcal{J}) = \sum_{b_1 \le \dots \le b_d} \mathcal{J}_{b_1} \cdots \mathcal{J}_{b_d}.
\end{equation}
is therefore is given by
\begin{equation}
F_{H}^d(\mu, \nu) = \sum_{\lambda, \ \abs{\lambda}=k} m^{\lambda}_{\mu \nu},
\end{equation}
which is the number of products of the form $(a_1\, b_1) \cdots (a_d\, b_d) h$ for $g\in \cyc(\mu)$ that are weakly monotonically increasing, such that
$(a_1\, b_1) \cdots (a_d\, b_d) h \in \cyc(\nu)$.
These thus enumerate the $d$-step paths in the Cayley graph of $S_n$ from an element in the conjugacy class of cycle type $\mu$ to the class cycle type $\nu$, that are weakly monotonically increasing in their second elements \cite{GH1}.
Equivalently, they are double Hurwitz numbers for $n$-sheeted branched coverings of the Riemann sphere with branch points at $0$ and $\infty$ having ramification profile type $\mu$ and $\nu$,
and an arbitrary number of further branch points, such that the sum of the colengths of their ramification profile lengths is again equal to $d$
\begin{equation}
\sum_{i=1}^k \ell^*(\mu^{(i)}) = kn - \sum_{i=1}^k \ell(\mu^{(i)}) = d.
\end{equation}
The latter are counted with a sign, which is $ (-1)^{n+d}$ times the parity of the number of branch points \cite{HO3}. The genus is again given by \eqref{riemann_hurwitz}.
This case is known to have a matrix model representation \cite{GGN1, GGN2} when the flow parameters
${\bf t}$ and ${\bf s}$ are restricted to be trace invariants of a pair of $N\times N$ normal matrices $A$, $B$:
\begin{equation}
t_i = {1\over i} \mathrm {tr}(A^i), \quad s_i = {1\over i} \mathrm {tr}(B^i),
\label{t_i_A_B}
\end{equation}
Within a normalization, setting
\begin{equation}
z = -{1\over N}
\label{z_1_over_N}
\end{equation}
as the expansion parameter, we have equality with the HCIZ double matrix integral
\begin{equation}
\tau^{H(-{1\over N})} ({\bf t}, {\bf s}) = \mathcal{I}_N(A, B)
:= \int_{\mathrlap{U \in U(N)}} \, e^{ \mathrm {tr}(UAU^\dag B)} d\mu(U)
= \left(\prod_{k=0}^{N-1} k!\right) \frac{\mathrm {det}\big(e^{a_i b_j}\big)_{1 \leq i,j \leq N}}{\Delta({\bf a}) \Delta ({\bf b})},
\label{HCIZ}
\end{equation}
where $d\mu(U)$ is the Haar measure on $U(N)$, ${\bf a} = (a_1, \ldots, a_N)$, ${\bf b} = (b_1, \ldots, b_N)$ are the eigenvalues of $A$ and $B$ respectively, and $\Delta({\bf a})$, $\Delta({\bf b})$ are the Vandermonde determinants.
The identification (\ref{z_1_over_N}), however, gives rise to a cutoff in the expansion (\ref{tau_H_s_lambda_exp}),
giving a sum only over partitions $\lambda$ of length $\ell(\lambda) \le N$:
\begin{equation}
\tau^{H(-{1\over N})} ({\bf t}, {\bf s}) = \sum_{\lambda, \ \ell(\lambda)\le N}
{N^{|\lambda|} \over (N)_{\lambda} }s_\lambda({\bf t}) s_\mu({\bf s}) .
\end{equation}
\subsection{Quantum weighted branched coverings and paths \cite{GH2}}
\label{ex5}
In \cite{GH2} three variants of quantum Hurwitz numbers were studied, with
weight generating functions denoted $E(q,z)$, $H(q,z)$ and $E'(q,z)$. We only consider the
case $E'(q,z)$, which has the most interesting interpretation in relation to Bosonic gases.
The other two are developed in detail in \cite{GH2} and may also be obtained as special
cases of the MacDonald polynomial approach to quantum Hurwitz numbers developed in \cite{H2} which is summarized in \autoref{macdonald_quantum_hurwitz} below.
The weight generating function is
\begin{eqnarray}
E'(q,z) &\& \coloneqq \prod_{i=1}^\infty (1+ q^i z) =1 + \sum_{i=0}^\infty E'_i(q) z^i,\\
E'_i(q) &\& \coloneqq \frac{q^{\frac{1}{2}i(i+1)}}{\prod_{j=1}^i (1-q^j)}, \quad i \ge 1,
\end{eqnarray}
where $q$ is viewed as a quantum deformation parameter that may interpreted (see below) in terms
of the energy distribution of Bosonic gases with a linear energy spectrum. This is related to the quantum dilogarithm function by
\begin{equation}
(1+z) E'(q, z) = e^{-\Li_2(q, -z)}, \quad \Li_2(q, z) \coloneqq \sum_{k=1}^\infty \frac{z^k}{k (1- q^k)}.
\end{equation}
We thus have
\begin{equation}
c_i = q^i, \quad i \ge 1, \quad
e_\lambda({\bf c}) = :E'_\lambda(q) = \prod_{i=1}^{\ell(\lambda)}\frac{q^{\frac{1}{2}\lambda_i(\lambda_i +1)}}{\prod_{j=1}^{\lambda_i} (1-q^j)} .
\end{equation}
The central element $E'_n(q, x\mathcal{J}) \in \mathbf{Z}(\mathbf{C}[S_n])$ is given by
\begin{equation}
E'_n(q, z\mathcal{J}) = \prod_{a=1}^n \prod_{k=1}^\infty (1+q^k z\mathcal{J}_a),
\end{equation}
and hence that content product coefficient is
\begin{eqnarray}
r^{E'(q, z)}_j &\&= \prod_{k=1}^\infty (1+ q^k z j), \\
r^{E'(q, z)}_\lambda(z) &\&= \prod_{k=1}^\infty \prod_{(i,j)\in \lambda} (1+ q^k z (j-i))
= \prod_{k=1}^\infty (zq^k)^{\abs{\lambda}} (1/(zq^k))_\lambda.
\end{eqnarray}
The generating $\tau$-function is therefore \cite{GH2}
\begin{eqnarray}
\tau^{E'(q, z)} ({\bf t}, {\bf s}) &\& = \sum_{\lambda}
\left(\prod_{k=1}^\infty (zq^k)^{\abs{\lambda}} (1/(zq^k))_\lambda \right)
s_\lambda({\bf t}) s_\mu({\bf s}) \cr
&\& = \sum_{d=0}^\infty z^d\sum_{\mu, \nu,\; \abs{\mu}=\abs{\nu}} H^d_{E'(q)}(\mu, \nu) p_\mu ({\bf t}) p_\nu({\bf s})
\label{tau_e_prime_q_s_lambda_exp}
\end{eqnarray}
where
\begin{equation}
H^d_{E'(q)}(\mu, \nu) \coloneqq \sum_{k=0}^\infty \sideset{}{'}\sum_{\substack{\mu^{(1)}, \dots \mu^{(k)} \\ \sum_{i=1}^k \ell^*(\mu^{(i)})= d}}
W_{E(q)}(\mu^{(1)}, \dots, \mu^{(k)}) H(\mu^{(1)}, \dots, \mu^{(k)}, \mu, \nu)
\label{Hd_Eq}
\end{equation}
is the quantum weighted enumeration of $n =\abs{\mu}=\abs{\nu}$ sheeted branched coverings with genus $g$ given by \eqref{riemann_hurwitz}
and weight $W_{E'(q)}(\mu^{(1)}, \dots, \mu^{(k)})$ for branched coverings of type $ (\mu^{(1)}, \dots, \mu^{(k)}, \mu, \nu)$ given by
\begin{eqnarray}
W_{E'(q)} (\mu^{(1)}, \dots, \mu^{(k)}) &\& \coloneqq {1\over\abs{\aut(\lambda)}}
\sum_{\sigma\in S_k} \sum_{1 \le i_1 < \cdots < i_k}^\infty q^{i_1 \ell^*(\mu^{(\sigma(1))})} \cdots q^{i_k \ell^*(\mu^{(\sigma(k))})} \cr
&\&= {1\over \abs{\aut(\lambda)}}\sum_{\sigma\in S_k} \frac{q^{k \ell^*(\mu^{(\sigma(1))})} \cdots q^{\ell^*(\mu^{(\sigma(k))})}}{
(1- q^{\ell^*(\mu^{(\sigma(1))})}) \cdots (1- q^{\ell^*(\mu^{(\sigma(1))}} \cdots q^{\ell^*(\mu^{(\sigma(k))})})} \cr
&\& ={1\over\abs{\aut(\lambda)}} \sum_{\sigma\in S_k} \frac{1}{
(q^{-\ell^*(\mu^{(\sigma(1))})} -1) \cdots (q^{-\ell^*(\mu^{(\sigma(1))})} \cdots q^{-\ell^*(\mu^{(\sigma(k))})}-1)}, \cr
&\&
\label{W_Eprime_q}
\end{eqnarray}
where $\lambda$ is the partition with parts $\{\ell^*(\mu^{(i)})\}_{i=1. \dots, k}$
The combinatorial Hurwitz number $F_{E'(q)}^d(\mu, \nu)$ giving the weighted enumeration of paths is
\begin{equation}
F_{E'(q)}^d(\mu, \nu) = \sum_{\lambda, \ \abs{\lambda}=d} \frac{q^{\frac{1}{2}i(i+1)}}{\prod_{j=1}^i (1-q^j)} \, m^\lambda_{\mu \nu}.
\end{equation}
and we have the usual equality
\begin{equation}
H^d_{E'(q)}(\mu, \nu) = F^d_{E'(q)}(\mu, \nu) .
\end{equation}
\begin{remark}\small \rm
{\bf Relation to Bosonic gas distribution.}
If we identify
\begin{equation}
q \coloneqq e^{-\beta \hbar \omega_0}, \quad \beta = k_B T,
\end{equation}
where $\hbar \omega_0$ is the lowest energy state in a gas of identical Bosonic particles,
assume the energy spectrum to consist of integer multiples of $\hbar \omega_0$
\begin{equation}
\epsilon_k = k \hbar \omega_0,
\end{equation}
and assign the energy
\begin{equation}
\epsilon(\mu^{(1)}, \dots, \mu^{(k)}) = \sum_{i=1}^k \epsilon_{\ell^*(\mu^{(i)})}
\end{equation}
to a configuration with branching profiles $(\mu^{(1)}, \dots, \mu^{(k)}, \mu, \nu)$,
the distribution function for Bosonic gases gives the weight
\begin{equation}
W(\mu^{(1)}, \dots, \mu^{(k)})= \frac{1}{e^{\beta \epsilon(\mu^{(1)}, \dots, \mu^{(k)})}-1}.
\end{equation}
The weighting factor $W_{E'(q)} (\mu^{(1)}, \dots, \mu^{(k)})$ in eq.~\eqref{W_Eprime_q}
is thus the symmetrized product
\begin{equation}
W_{E'(q)} (\mu^{(1)}, \dots, \mu^{(k)}) = {1\over \abs{\aut(\lambda)}}\sum_{\sigma\in S_k}
W(\mu^{(\sigma(1)}) \cdots W(\mu^{\sigma(1)}, \dots, \mu^{\sigma(k)})
\label{W_bosonic_gas_weight}
\end{equation}
of that for each subconfiguration.
\end{remark}
In \cite{GH2}, a dual pair of similar weight generating functions $E(q,z)$, $H(q,z)$ were introduced, which correspond to two slightly different definitions of quantum Hurwitz numbers. These are the $q$-analogs of what, when extended to the Cauchy-Littlewood formula, become the generating functions of the elementary and the complete symmetric functions:
\begin{eqnarray}
E(q,z) &\&:= \prod_{k=0}^\infty (1+z q^k) \\
H(q,z) &\& := \prod_{k=0}^\infty (1-z q^k)^{-1}
\end{eqnarray}
The corresponding weights for branched covers with ramification profiles $(\mu^{(1)}, \dots, \mu^{(k)}) $ and
$ (\nu^{(1)}, \dots, \nu^{(\tilde{k})})$ at the branch points are:
\begin{eqnarray}
W_{E(q)} (\mu^{(1)}, \dots, \mu^{(k)}) &\&{1\over \abs{\aut(\lambda)}}
\sum_{\sigma\in S_k} \sum_{0 \le i_1 < \cdots < i_k}^\infty q^{i_1 \ell^*(\mu^{(\sigma(1))})} \cdots q^{i_k \ell^*(\mu^{(\sigma(k))})} \cr
&\&= {1\over \abs{\aut(\lambda)}}\sum_{\sigma\in S_k} \frac{q^{(k-1) \ell^*(\mu^{(\sigma(1))})} \cdots q^{\ell^*(\mu^{(\sigma(k-1))})}}{
(1- q^{\ell^*(\mu^{(\sigma(1))})}) \cdots (1- q^{\ell^*(\mu^{(\sigma(1))})} \cdots q^{\ell^*(\mu^{(\sigma(k))})})}, \cr
&\&
\label{W_E_q}
\end{eqnarray}
where $\lambda$ is the partition with parts $(\ell^*(\mu^{(1)}), \dots, \ell^*(\mu^{({\tilde{k}})}))$, and
\begin{eqnarray}
W_{H(q)} (\nu^{(1)}, \dots, \nu^{(\tilde{k})}) &\& \coloneqq
{(-1)^{\ell^*{(\lambda)}}\over\abs{\aut(\lambda)}}\sum_{\sigma\in S_{\tilde{k}}} \sum_{0 \le i_1 \le \cdots \le i_{\tilde{k}}}^\infty q^{i_1 \ell^*(\nu^{(\sigma(1))})} \cdots q^{i_k \ell^*(\nu^{(\sigma({\tilde{k}}))})} \cr
&\&= {(-1)^{\ell^*{(\lambda)}}\over \abs{\aut(\lambda)}}\sum_{\sigma\in S_{\tilde{k}}} \frac{1}{
(1- q^{\ell^*(\nu^{(\sigma(1))})}) \cdots (1- q^{\ell^*(\nu^{(\sigma(1))})} \cdots q^{\ell^*(\nu^{(\sigma({\tilde{k}}))})})}, \cr
&\&
\label{W_H_q}
\end{eqnarray}
where $\lambda$ is the partition with parts $(\ell^*(\nu^{(1)}), \dots, \ell^*(\nu^{({\tilde{k}})}))$.
The associated hypergeometric $\tau$-functions $\tau^{E(q,z))}({\bf t} , {\bf s})$, and $\tau^{H(q,z))}({\bf t} , {\bf s})$ are defined similarly to $\tau^{E'(q,z))}({\bf t} , {\bf s})$ and are generating functions
for the correspondingly modified Hurwitz numbers $F^d_{E(q)}(\mu, \nu) = H^d_{E(q)}(\mu, \nu)$
and $F^d_{H(q)}(\mu, \nu) = H^d_{H(q)}(\mu, \nu)$. (See \cite{GH2} for further details.)
For later use, we denote the product of these
\begin{equation}
W_{Q(q)} (\mu^{(1)}, \dots, \mu^{(k)}; \nu^{(1)}, \dots, \nu^{({\tilde{k}})})
:= W_{E(q)} (\mu^{(1)}, \dots, \mu^{(k)})W_{H(q)} (\nu^{(1)}, \dots, \nu^{({\tilde{k}})}).
\label{W_{Q(q)}_mu_nu}
\end{equation}
\section{Multispecies weighted Hurwitz numbers}
\label{multispecies}
\subsection{Hybrid signed Hurwitz numbers at fixed genus: hybrid monotonic paths \cite{GH1, GH2, HO3}}
\label{ex6.1}
This case is just a hybrid product of the cases of \autoref{ex2} and \autoref{ex4}.
We choose as generating function
\begin{equation}
Q(w,z):= {1+ w \over 1-z}
\end{equation}
taking power series in both parameters $(w,z)$.
The associated central element is
\begin{equation}
Q(w, z, \mathcal{J}) = E_n(w, \mathcal{J}) H_n(z, \mathcal{J}) = \prod_{a=1}^n{ 1+w \mathcal{J}_a \over 1- z\mathcal{J}_a},
\end{equation}
and therefore
\begin{eqnarray}
r^{Q(w,z)}_j&\& = {1 + jw \over 1-jz} , \\
r^{Q(w,z)}_\lambda &\&= \prod_{(i,j)\in \lambda} {1 + (j-i)w\over 1-(j-i)z}
= (-w/z)^{\abs{\lambda}} \, {(1/w)_{\lambda} \over (-1/ z)_\lambda}, \\
T^{Q(w,z)}_j &\&= \sum_{i=1}^j \mathrm {ln}{1+iw\over 1-iz}, \quad T^{Q(w,z)}_{-j} (w,z)= \sum_{b=1}^{j-1}\mathrm {ln}{1+iz\over 1- iw}, \quad j > 0,
\end{eqnarray}
The generating $\tau$-function is thus \cite{GH1, HO3}
\begin{eqnarray}
\tau^{Q(w, z)} ({\bf t}, {\bf s}) &\& = \sum_{\lambda}
(-w/z)^{\abs{\lambda}}{ (1/w)_\lambda \over (-1/z)_\lambda} s_\lambda({\bf t}) s_\mu({\bf s}) \cr
&\& = \sum_{c=0}^\infty \sum_{d=0}^\infty w^c z^d \sum_{\mu, \nu,\; \abs{\mu}=\abs{\nu}} H^c_d(\mu, \nu) p_\mu ({\bf t}) p_\nu({\bf s}),
\end{eqnarray}
where
\begin{equation}
H^c_d(\mu, \nu) = \sum_{k=0}^\infty\sum_{\substack{\mu^{(1)} \\ \ell^*(\mu_1) =c}} (-1)^{k+d}
\sideset{}{'}\sum_{\substack{\nu^{(1)}, \dots, \nu^{(k)}\\ \sum_{i=1}^k \ell^*(\nu^{(i)})=d}}
H(\mu^{(1)}, \nu^{(1)}, \dots , \nu^{(k)}, \mu, \nu)
\end{equation}
is the number of $n=\abs{\mu}=\abs{\nu}=\abs{\mu^{(1)}}= |\nu^{(1)}|= \cdots = |\nu^{(k)}|$ sheeted branched
covers with branch points of ramification type $(\mu,\nu)$ at $(0,\infty)$,
one further branch point, of ``first class'', with colength $\ell^*(\mu^{(1)}) =c$ and $k$ further branch points,
$(\nu^{1)}, \dots, \nu^{(k)})$ of ``second class'' with total colength equal to $d$,
\begin{equation}
\sum_{i=1}^k \ell^*(\mu^{(i)})=d
\label{k_ell_d_constraint}
\end{equation}
counted with sign $(-1)^{k+d}$ determined by the parity of $k$. Note that the
sum over $k$ is actually finite, because of the constraint (\ref{k_ell_d_constraint}).
As usual, the Riemann-Hurwitz formula
\begin{equation}
2-2g = \ell(\mu) +\ell(\nu) -d
\end{equation}
determines the genus $g$ of the covering surface.
The meaning of the combinatorial Hurwitz number $F^c_d(\mu, \nu) = H^c_d(\mu, \nu)$
in this case is clear from combining its meaning for the cases considered in \autoref{ex2} and \autoref{ex4};
it is the number of $c +d$ step paths in the Cayley graph of $S_n$ starting at an element $h\in \cyc(\mu)$
in the conjugacy class of type $\cyc(\mu)$ and ending in $\cyc(\nu)$ such that the first $c$ steps are
strictly monotonic and the next $d$ steps are weakly monotonic.
This case also has a matrix integral representation, analogous to the HCIZ integral when the flow parameters
${\bf t}$ and ${\bf s}$ are again restricted to equal the trace invariants of a pair $A$, $B$ of normal matrices
as in (\ref{t_i_A_B}), and the expansion parameters are equated to
\begin{equation}
w = {1\over N -\alpha}, \quad z = -{1\over N}
\end{equation}
for some parameter $\alpha$.
\begin{eqnarray}
\tau^{Q({1 / (N-\alpha}), {-1/ N})} ({\bf t}, {\bf s})
&\& = \int_{\mathrlap{U \in U(N)}} \, \mathrm {det}(\mathbf{I} - \zeta UAU^\dag B)^{\alpha-N} d\mu(U)\\
&\& = \left(\prod_{k=0}^{N-1} {k! \over (1 -\alpha)_k}\right) {\mathrm {det}\left(1- \zeta a_i b_j\right)^{\alpha-1}_{1 \leq i,j \leq N} \over \Delta({\bf a}) \Delta ({\bf b})}
\label{HCHO}
\end{eqnarray}
where
\begin{equation}
\zeta:= {N \over N-\alpha}.
\end{equation}
The identification (\ref{z_1_over_N}) again gives rise to a cutoff in the expansion (\ref{tau_H_s_lambda_exp}),
restricting the sum to partitions $\lambda$ of length $\ell(\lambda) \le N$:
\begin{equation}
\tau^{Q({1 / (N-\alpha}), {-1/ N})} ({\bf t}, {\bf s}) = \sum_{\lambda, \ \ell(\lambda)\le N}
\left({1 - {\alpha\over N}}\right)^{|\lambda|} {(N-\alpha)_\lambda\over (N)_{\lambda}} s_\lambda({\bf t}) s_\mu({\bf s}) .
\end{equation}
\subsection{Signed multispecies Hurwitz numbers: hybrid multimonotonic paths \cite{HO3} }
\label{ex6.2}
Now consider the multiparametric generalization of the previous example. We introduce $l +m$
expansion parameters
\begin{equation}
{\bf w} := (w_1, \dots, w_l), \quad {\bf z} =(z_1, \dots, z_m).
\end{equation}
The weight generating functions $G$ is chosen to be products of those for the previous case:
\begin{equation}
Q^{(l,m)}({\bf w}, {\bf z}) := \prod_{\alpha=1}^l E(w_\alpha) \prod_{\beta=1}^m H(z_\beta)
= {\prod_{\alpha=1}^l (1+ w_\alpha)\over \prod_{\beta=1}^m (1 - z_\beta)}
\end{equation}
The corresponding element of the center $\mathbf{Z}(\mathbf{C}[S_n])$ is
\begin{equation}
Q^{(l,m)}_n({\bf w}, {\bf z}, \mathcal{J}) = \prod_{a=1}^nQ^{(l,m)}({\bf w}\mathcal{J}_a, {\bf z}\mathcal{J}_a),
\end{equation}
and therefore the eigenvalues of $Q^{(l,m)}_n({\bf w}, {\bf z}, \mathcal{J})$ are
\begin{equation}
r^{Q^{(l,m)}({\bf w}, {\bf z})}_\lambda = \prod_{(i,j)\in \lambda} {\prod_{\alpha=1}^l(1 + (j-i)w_\alpha)\over \prod_{\beta=1}^m (1-(j-i)z_\beta)} ={\prod_{\alpha=1}^l(w_\alpha)^{|\lambda|}(1/w_\alpha)_{\lambda} \over \prod_{\beta=1}^m(-z_\beta)^{|\lambda| }(-1/ z_\beta)_\lambda},
\end{equation}
while the diagonal exponential fermionic coefficients are
\begin{equation}
T_j^{Q^{(l,m)}({\bf w}, {\bf z})} = \sum_{i=1}^j \mathrm {ln}{\prod_{\alpha=1}^l(1+iw_\alpha)\over \prod_{\beta=1}^m(1-iz_\beta)},
\quad T_{-j}^{Q^{(l,m)}({\bf w}, {\bf z})}
=- \sum_{i=0}^{j-1}\mathrm {ln}{\prod_{\alpha=1}^l(1-iw_\alpha)\over\prod_{\beta=1}^m( 1+ iz_\beta)}, \quad j > 0.
\end{equation}
The generating $\tau$-function is thus \cite{HO3}
\begin{eqnarray}
\tau^{Q^{(l,m)}({\bf w}, {\bf z})} ({\bf t}, {\bf s}) &\& = \sum_{\lambda}
{\prod_{\alpha=1}^l(w_\alpha)^{|\lambda|}(1/w_\alpha)_{\lambda} \over \prod_{\beta=1}^m(-z_\beta)^{|\lambda| }(-1/ z_\beta)_\lambda} s_\lambda({\bf t}) s_\lambda({\bf s}) \cr
&\& = \sum_{{\bf d} \in \mathbf{N}^l} \sum_{\tilde{\bf d} \in \mathbf{N}^m} {\bf w}^{\bf d} {\bf z}^{\tilde{{\bf d}}}\sum_{\mu, \nu,\; \abs{\mu}=\abs{\nu}} H_{Q^{(l,m)}}^{({\bf d}, \tilde{\bf d})}(\mu, \nu) p_\mu ({\bf t}) p_\nu({\bf s}),
\label{multi_colour_hybrid_signed_tau}
\end{eqnarray}
where multi-index notation has been used:
\begin{equation}
{\bf w}^{\bf d} := \prod_{\alpha=1}^l w_\alpha^{d_\alpha}, \quad {\bf z}^{\tilde{\bf d}}
:= \prod_{\beta=1}^m z_\beta^{\tilde{d}_\beta}
\end{equation}
with
\begin{equation}
{\bf d} := (d_1, \dots, d_l), \quad \tilde{\bf d} := (\tilde{d}_1, \dots , \tilde{d}_m), \quad d_\alpha, \tilde{d}_\beta \in \mathbf{N}.
\end{equation}
Here
\begin{eqnarray}
H_{Q^{(l,m)}}^{({\bf d}, \tilde{\bf d})}(\mu, \nu) &\&= (-1)^{D } \sum_{\{k_\beta\}_{\beta=1}^m} \sum_{\substack{\{\mu^{(\alpha)} \}\\ \ell^*(\mu^{(\alpha)})=d_\alpha}}
\sideset{}{'}\sum_{\substack{\{ \nu^{(\beta, i_\beta)} \}\\\quad \sum_{i_\beta =1}^{k_\beta}\ell^*(\nu^{(\beta, i_\beta)}) =\tilde{d}_\beta}}
{\hskip -20 pt}(-1)^C H(\{ \mu^{(\alpha)}\}, \{{\nu^{(\beta, i_{\beta})}\}_{i_\beta=1}^{k_\beta},
\mu, \nu) },\cr
&\&
\label{signed_coloured_multihurwitz}
\end{eqnarray}
is the signed total number of branched coverings, weighted by the inverses of their automorphism groups,
with branch points at $(0, \infty)$ having ramification profiles $(\mu, \nu)$,
and further branch points divided into two types: $l$ ``plain'' branch points $\{\mu^{(\alpha)}\}_{\alpha=1, \dots , l}$
with colengths
\begin{equation}
\ell^*(\mu^\alpha) = d_\alpha
\end{equation}
and
\begin{equation}
C= \sum_{\beta=1}^m k_\beta, \quad
\end{equation}
``coloured'' branch points with colours labeled by $\beta=1, \dots, m$ and
ramification profiles $\{ \nu^{(\beta,i_\beta)}\}_{\beta=1, \dots, m; \ i_{\beta}=1, \dots k_{\beta}}$,
of total ramification type colengths
\begin{equation}
\sum_{i_\beta=1}^{k_{\beta}} \ell^*(\nu^{(\beta, i_{\beta})} )=\tilde{d}_\beta
\end{equation}
in each colour group, and
\begin{equation}
D = \sum_{\beta=1}^m \tilde{d}_\beta
\end{equation}
is the sum of these colengths over all colours.
The genus $g$ of the covering surface is determined by the Riemann-Hurwitz formula:
\begin{equation}
2-2g = \ell(\mu) + \ell(\nu) -\sum_{\alpha=1}^l d_\alpha - D.
\label{riemann_hurwitz_multi_colour_signed}
\end{equation}
The combinatorial significance of the weighted Hurwitz number $F^{({\bf d}, \tilde{\bf d})}_{Q^{(l,m)}}(\mu, \nu)$ in this case is given (see \cite{HO3}) by:
\begin{theorem}
\label{combinatorial_interpretation}
The coefficients $ H^{({\bf d}, \tilde{\bf d})}_{Q^{(l,m)}}(\mu, \nu)=F^{({\bf d}, \tilde{\bf d})}_{Q^{(l,m)}}(\mu, \nu)$ in the expansion
(\ref{multi_colour_hybrid_signed_tau}) are equal to the number of paths in the Cayley graph of $S_n$ generated by transpositions $(a\, b)$, $a<b$, starting at an element in the conjugacy class with cycle type given by the partition $\mu$ and ending in the conjugacy class with cycle type given by partition $\nu$, such that the paths consist of a sequence of
\begin{equation}
k:= \sum_{\alpha=1}^l d_\alpha + \sum_{\beta=1}^m \tilde{d}_{\beta}
\end{equation}
transpositions $(a_1 b_1) \cdots (a_k b_k)$, divided into $l+m$ subsequences, the first $l$ of which
consist of $\{d_1, \dots, d_l\}$ transpositions that are strictly monotonically increasing
(i.e.\ $ b_i < b_{i+1}$ for each neighbouring pair of transpositions within the subsequence),
followed by $\{\tilde{d}_1, \dots, \tilde{d}_m\}$ subsequences within each of which the
transpositions are weakly monotonically increasing (i.e. $b_i \le b_{i+1}$ for each neighbouring pair)
\end{theorem}
\subsection{General weighted multispecies Hurwitz numbers \cite{H1, H2} }
\label{ex6.3}
We may extend the multispecies signed Hurwitz numbers considered in the preceding section to general multispecies weighting \cite{H1, H2} by replacing the factors $E(w_\alpha)$ and $H(z_\beta)$ in the above by arbitrary weight generating functions of type $G^\alpha(w_\alpha)$ and dual type $\tilde{G}^\beta(z_\beta)$.
The partitions are divided into two classes: those corresponding to the weight factors of type $G(w)$,
labelled $\{\mu^{(\alpha, u_\alpha)}\}$, and those corresponding to dual type $\tilde{G}(z)$,
labelled $\{\nu^{(\beta, v_\beta}\}$, These are further subdivided into $l$ ``colours'', or ``species'' for the first class,
denoted by the label $\alpha =1, \dots , l$ and $m$ in the second, denoted by $\beta=1, \dots , m$.
Any given configuration
$\{ \{\mu^{(\alpha, u_\alpha)}\}_{1\le u_\alpha \le k_\alpha}, \{\nu^{(\beta, v_\beta}\}_{1\le v_\beta \le \tilde{k}_\beta}\}$
has $k_\alpha$ elements of colour $\alpha$ in the first class and $\tilde{k}_\beta$ elements of colour $\beta$
in the second class, for a total of
\begin{equation}
k = \sum_{\alpha=1}^l k_\alpha + \sum_{\beta=1}^m \tilde{k}_\beta
\end{equation}
partitions.
Denoting the $l+m$ expansion parameters again as
\begin{equation}
{\bf w} = (w_1, \dots, w_l), \quad {\bf z} = (z_1, \dots, z_m),
\end{equation}
the multispecies weight generating function is formed from the product
\begin{equation}
G^{(l,m)}({\bf w}, {\bf z}) := \prod_{\alpha=1}^l G^{\alpha}(w_\alpha) \prod_{\beta=1}^m\tilde{G}^\beta(z_\beta),
\end{equation}
where each factor has an infinite product representation that is of one of the two types
\begin{eqnarray}
G^{\alpha}(w) &\&= \prod_{i=1}^\infty (1 + c_i^\alpha w), \ \alpha =1, \dots , l\\
\tilde{G}^{\beta}(w) &\&= \prod_{i=1}^\infty (1 - \tilde{c}_i^\beta w), \ \beta =1, \dots , m.
\end{eqnarray}
for $l+m$ infinite sequences of parameters
\begin{eqnarray}
{\bf c}^\alpha &\&= (c^\alpha_1, c^\alpha_2, \dots), \quad \alpha =1, \dots , l \\
\tilde{\bf c}^\alpha &\&= (\tilde{c}^\alpha_1, \tilde{c}^\alpha_2, \dots), \quad \beta =1, \dots m.
\end{eqnarray}
The corresponding central element, denoted
\begin{equation}
G^{(l,m)}_n({\bf w}, {\bf z}, \mathcal{J}) := \prod_{a=1}^n\left(\prod_{\alpha=1}^lG^\alpha(w_\alpha \mathcal{J}_a) \right)
\left(\prod_{\beta=1}^m \tilde{G}^\beta(z_\beta \mathcal{J}_a)\right)
\end{equation}
has eigenvalues
\begin{equation}
r_\lambda^{G^{(l,m)}({\bf w}, {\bf z}) }= \prod_{\alpha=1}^l r_\lambda^{G^\alpha}(w_\alpha)
\prod_{\beta=1}^mr_\lambda^{\tilde{G}^\beta}(z_\beta)
\end{equation}
in the $\{F_\lambda\}$ basis where, as before,
\begin{equation}
r_\lambda^{G^\alpha}(w_\alpha) := \prod_{(ij)\in \lambda} G(w_\alpha(j-i)), \quad r_\lambda^{\tilde{G}^\beta}(z_\beta)
:= \prod_{(ij)\in \lambda} \tilde{G}(z_\beta(j-i)).
\end{equation}
The diagonal exponential fermionic coefficients are
\begin{eqnarray}
T_j^{G^{(l,m)}({\bf w}, {\bf z})}&\&= \sum_{i=1}^j\left(\prod_{\alpha=1}^lG^\alpha(iw_\alpha)\prod_{\beta=1}^m \tilde{G}^\beta(iz_\beta)\right), \cr
T_{-j}^{G^{(l,m)}({\bf w}, {\bf z})}
&\&=- \sum_{i=0}^{j-1}\mathrm {ln} \left(\prod_{\alpha=1}^lG^\alpha(-iw_\alpha)\prod_{\beta=1}^m \tilde{G}^\beta(-iz_\beta)\right), \quad j > 0.
\end{eqnarray}
The generating hypergeometric $\tau$-function is \cite{H1}
\begin{eqnarray}
\tau^{G^{(l,m)}({\bf w}, {\bf z})}({\bf t}, {\bf s}) &\& = \sum_{\lambda}
\prod_{\alpha=1}^l r_\lambda^{G^\alpha(w_\alpha)}
\prod_{\beta=1}^mr_\lambda^{\tilde{G}^\beta((z_\beta)} \, s_\lambda({\bf t}) s_\lambda({\bf s}) \cr
&\& = \sum_{{\bf d} \in \mathbf{N}^l} \sum_{\tilde{\bf d} \in \mathbf{N}^m} {\bf w}^{\bf d} {\bf z}^{\tilde{{\bf d}}}\sum_{\mu, \nu,\; \abs{\mu}=\abs{\nu}} H_{G(l,m)}^{({\bf d}, \tilde{\bf d})}(\mu, \nu) p_\mu ({\bf t}) p_\nu({\bf s}),
\label{multi_colour_hybrid_weighted_tau}
\end{eqnarray}
where
\begin{eqnarray}
H_{G^{(l,m)}}^{({\bf d}, \tilde{\bf d})}(\mu, \nu)
&\&:= \sum_{k_1, \dots, k_l} \sum_{\tilde{k}_1, \dots , \tilde{k}_m} {\hskip -10 pt} \sum_{\substack{\{\mu^{(\alpha, u_\alpha)}\} \\ |\mu^{(\alpha, u_\alpha)}|=n \\
\sum_{u_\alpha=1}^{k_\alpha} \ell^*(\mu^{(\alpha, u_\alpha)}) = d_\alpha}}
{\hskip 10 pt}\sideset{}{'}\sum_{\substack{\{\nu^{(\beta, v_\beta)}\} \\ |\nu^{(\beta, u_\beta)}|=n \\
\sum_{v_\beta=1}^{\tilde{k}_\beta} \ell^*(\nu^{(\beta, v_\beta)}) = \tilde{d}_\beta}}\cr
&\&{\hskip 60 pt} \times W_{G^{(l,m)}} (\{\mu^{(\alpha, u_\alpha)}\}, \{\nu^{(\beta, v_\beta}\})
H(\{\mu^{(\alpha, u_\alpha)}\}, \{\nu^{(\beta, v_\beta}\}, \mu, \nu)
\cr
&\&
\end{eqnarray}
is the geometrical multispecies Hurwitz number giving the weighted enumeration
of $n$-sheeted branched coverings with $l+m$ branch points of type
$\{ \{\mu^{(\alpha, u_\alpha)}\}_{1\le u_\alpha \le k_\alpha}, \{\nu^{(\beta, v_\beta}\}_{1\le v_\beta \le \tilde{k}_\beta}\}$
and $(\mu, \nu)$ at $(0, \infty)$, with weighting factor equal to the product of those for single species
\begin{eqnarray}
W_{G(^{l,m)}} (\{\mu^{(\alpha, u_\alpha)}\}, \{\nu^{(\beta, v_\beta}\}) &\&=
\prod_{\alpha=1}^l m_{\lambda^{(\alpha)}} ({\bf c}^{(\alpha)})
\prod_{\beta=1}^m m_{\tilde{\lambda}^{(\beta)}} (\tilde{\bf c}^\beta),
\end{eqnarray}
Here the partitions $\{\lambda^{(\alpha)}\}_{\alpha=1, \dots, l}$,
and $\{\tilde{\lambda}^{(\beta)}_{\beta=1, \dots, m}\}$ have lengths
\begin{equation}
\ell(\lambda^{(\alpha)}) = k_\alpha, \quad \ell(\tilde{\lambda}^{(\beta)}) = \tilde{k}_\beta,
\end{equation}
weights
\begin{equation}
|\lambda^{(\alpha)})| = d_\alpha, \quad |\tilde{\lambda}^{(\beta)}| = \tilde{d}_\beta,
\end{equation}
and parts equal to the colengths $\ell^*(\mu^{(\alpha, u_\alpha)})$ and
$\ell^*(\mu^{(\beta, v_\beta)})$, for $\lambda^{(\alpha)}$ and $\tilde{\lambda}^{(\beta)}$
respectively.
The combinatorial multispecies Hurwitz number $F_{G^{(l,m)}}^{({\bf d}, \tilde{\bf d})}(\mu, \nu) $ is
determined as follows \cite{H1}. Let $D_n$ denote the number of partitions of weight $n$.
For each generating function $G^\alpha(w_\alpha)$ or $\tilde{G}^\beta(z_\beta)$, let ${\bf F}_{G^\alpha}^{d_\alpha}$
and ${\bf F}_{\tilde{G}^\beta}^{\tilde{d}_\beta}$ denote the $D_n \times D_n$ matrices whose elements
are $F^{d_\alpha}_{G^\alpha}(\mu, \nu)$ and $F^{\tilde{d}_\beta}_{\tilde{G}^\beta}(\mu, \nu)$,
respectively, as defined in (\ref{Fd_G_def}}), (\ref{Fd_tilde_G_def}). From the fact that the central elements $\{G^\alpha(w_\alpha, \mathcal{J}), \tilde{G}^\beta (z_\beta, \mathcal{J})\}$
all commute, it follows that so do the matrices $\{{\bf F}^{d_\alpha}_{G^\alpha},
{\bf F}^{\tilde{d}_\beta}_{\tilde{G}^\beta}\}$. Denoting the product of these in any order,
\begin{equation}
{\bf F}^{({\bf d}, \tilde{\bf d})}_{G^{(l,m)}}:= \prod_{\alpha=1}^l {\bf F}_{G^\alpha}^{d_\alpha} \prod_{\beta=1}^m {\bf F}_{\tilde{G}^\beta}^{\tilde{d}_\beta},
\end{equation}
the $(\mu, \nu)$ matrix element $F^{({\bf d}, \tilde{\bf d})}_{G^{(l, m)}}(\mu, \nu)$ is the combinatorial multispecies
weighted Hurwitz number, and is equal to the geometrically defined one.
\begin{equation}
F^{({\bf d}, \tilde{\bf d})}_{G^{(l, m)}}(\mu, \nu) = H^{({\bf d}, \tilde{\bf d})}_{G^{(l, m)}}(\mu, \nu)
\end{equation}
The combinatorial meaning of $F^{({\bf d}, \tilde{\bf d})}_{G^{(l, m)}}(\mu, \nu)$ is as follows,
Let
\begin{equation}
d := \sum_{\alpha=1}^l d_\alpha + \sum_{\beta=1}^m \tilde{d}_\beta.
\end{equation}
Then $F^{({\bf d}, \tilde{\bf d})}_{G^{(l,m)}} (\mu, \nu)$, may be interpreted as the weighted sum
over all sequences of $d$ step paths in the Cayley graph from an element $h \in \cyc(\mu)$ in
the conjugacy class of cycle type $\mu$ to one $(a_db_d) \cdots (a_1b_1)h$ of cycle type $\nu$,
in which the transpositions appearing are subdivided into subsets consisting of $(d_1, \dots, d_l, \tilde{d}_1, \dots , \tilde{d}_m)$ transpositions in all $ {d! \over (\prod_{\alpha=1}^l d_\alpha !) (\prod_{\beta=1}^l \tilde{d}_\beta !)}$ possible ways, and to each of these, if the signatures are $(\lambda^{(1)}, \dots, \lambda^{(l)}, \tilde{\lambda}^{(1)} , \dots , \tilde{\lambda}^{(m)})$, a weight is given that is equal to the product
\begin{equation}
\prod_{\alpha=1}^l e_{\lambda^{(\alpha)}} ({\bf c}^\alpha)
\prod_{\beta =1}^m h_{\tilde{\lambda}^{(\beta)}} (\tilde{{\bf c}}^\beta) .
\end{equation}
\begin{theorem}
\begin{equation}
F^{({\bf d}, \tilde{\bf d})}_{G^{(l, m)}} (\mu, \nu) = H^{({\bf d}, \tilde{\bf d})}_{G^{(l, m)}} (\mu, \nu) .
\end{equation}
\end{theorem}
For proofs of these results, see \cite{H1}.
\section{Quantum weighted Hurwitz numbers and Macdonald polynomials \cite{H2} }
\label{macdonald_quantum_hurwitz}
Only a summary of the results will be given here; for details see \cite{H2}.
\subsection{Generating functions for Macdonald polynomials}
Following \cite{Mac}, for two infinite sets of indeterminates ${\bf x} := (x_1, x_2, \dots )$, ${\bf y} := (y_1, y_2, \dots )$, we define the generating function
\begin{equation}
\Pi ({\bf x}, {\bf y}, q, t) := \prod_{a=1}^\infty \prod_{b=1}^\infty {(t x_a y_b; q)_\infty \over (x_a y_b; q)_\infty},
\end{equation}
where
\begin{equation}
(t; q)_\infty := \prod_{i=0}^\infty (1 - t q^i).
\end{equation}
is the infinite $q$-Pochhammer symbol.
$\Pi (q, t, {\bf x}, {\bf y})$ can be expanded in a number of ways in terms of products of dual bases
for the algebra $\Lambda$ of symmetric functions
\begin{eqnarray}
\Pi ({\bf x}, {\bf y}, q, t) &\&= \sum_\lambda P_\lambda( {\bf x}, q,t) P_\lambda({\bf y}, q,t) \\
&\&= \sum_\lambda z_\mu^{-1}(q,t)p_\lambda({\bf x}) p_\lambda ({\bf y}) \\
&\&= \sum_\lambda g_\lambda({\bf x}, q,t) m_\lambda ({\bf y})
\label{Pi_qt_mg}
\\
&\&= \sum_\lambda g_\lambda( {\bf y}, q,t) m_\lambda ({\bf x})
\label{Pi_qt_gm}
\end{eqnarray}
where
\begin{equation}
z_\mu(q,t) := z_\mu n_\mu(q,t), \quad n_\mu :=\prod_{i=1}^{\ell(\mu)} {1 - q^\mu_i \over 1 - t^\mu_i}.
\end{equation}
Here $\{P_\lambda(q,t, {\bf x})\} $ are the MacDonald symmetric functions, which are orthogonal
\begin{equation}
(P_\lambda(q,t), P_\mu(q,t))_{(q,t) }= 0, \quad \lambda \neq \mu
\end{equation}
with respect to the inner product $( \ , \ ) _{(q,t)}$ in which the power sum symmetric functions $\{p_\lambda\} $ satisfy
\begin{equation}
(p_\lambda, p_\mu)_{(q,t) } =z^{-1}_\mu (q,t)\delta_{\lambda \mu},
\end{equation}
$ \{m_\lambda\}$ are the basis of monomial sum symmetric functions and
\begin{equation}
g_\lambda({\bf x}, q,t) :=\prod_{i=1}^{\ell(\lambda)}g_{\lambda_i}({\bf x}, q,t) , \quad g_j( {\bf x}, q,t) := (P_j, P_j)^{-1} P_j(q,t, {\bf x}),
\end{equation}
is the $(q,t)$ analog of the interpolating function between the elementary $e_\lambda({\bf x})$
and complete $h_\lambda({\bf x})$ symmetric function bases.
\subsection{Quantum families of central elements and weight generating functions}
We now consider an extended infinite parametric family of generating functions $M(q, t, {\bf c}, z)$, depending on
the infinite set of ``classical'' parameters ${\bf c}$ appearing in the infinite product representations
as in (\ref{G_inf_prod}), (\ref{tilde_G_inf_prod}) as well as the further pair of
``quantum deformation'' parameters $(q,t)$ appearing in the MacDonald polynomials \cite{Mac}.
For a dual pair of ``classical'' generating function $G(z)$, $\tilde{G}(z)$, with
infinite product representations (\ref{G_inf_prod}), (\ref{tilde_G_inf_prod})
we introduce a $(q,t)$ deformed parametric family of weight generating functions $M(q,t, {\bf c}, z)$ as follows
\begin{equation}
M(q, t, {\bf c}, z) := \prod_{k=0}^\infty G(t q^k z) \tilde{G}( q^k z)
= \prod_{k=0}^\infty \prod_{i=1}^\infty {1 - t zq^k c_i \over 1 - zq^k c_i}.
\label{M_qtcz}
\end{equation}
The associated central element $M_n(q,t, {\bf c}, z,\mathcal{J})\in \mathbf{Z}(\mathbf{C}[S_n])$ is defined as
\begin{equation}
M_n(q,t, {\bf c}, z \mathcal{J}) := \prod_{a=1}^b M(q,t, {\bf c}, z\mathcal{J}_a) = \Pi({\bf c}, z\mathcal{J}, q, t).
\end{equation}
The eigenvectors are the orthogonal idempotents:
\begin{equation}
M_n(q,t, {\bf c}, z\mathcal{J}) F_\lambda = r_\lambda^{M(q,t,{\bf c}, z )} F_\lambda
\end{equation}
and the eigenvalues $r_\lambda^{M(q, t, {\bf c}, z)}$ have the usual content product form
\begin{equation}
r_\lambda^{M(q, t, {\bf c}, z)} =\prod_{(i, j)\in \lambda} M(q,t, {\bf c}, z(j-i))
=\prod_{i=1}^\infty {(zt c_i; q)_\lambda \over (zc_i; q)_\lambda}
\label{r_lambda_q_p}
\end{equation}
where, for a partition $\lambda = (\lambda_ \ge \cdots \ge \lambda_{\ell(\lambda)} >0)$,
the $q$-Pochhammer symbol $(t;q)_\lambda$ is defined as
\begin{equation}
(t;q)_\lambda := \prod_{i=1}^{\ell(\lambda)} (t q^{-i+1};q)_{\lambda_i},
\quad (t;q)_l := \prod_{k=0}^{l-1}(1- tq^k).
\end{equation}
\subsection{MacDonald family of quantum weighted Hurwitz numbers}
Using the content product formula (\ref{r_lambda_q_p}), we define the associated
2D Toda $\tau$-function for $N=0$ as
\begin{equation}
\tau^{M(q,t, {\bf c}, z)}({\bf t}, {\bf s}):= \sum_{\lambda} r_\lambda^{M(q,t, {\bf c}, z)} s_\lambda({\bf t}) s_\lambda({\bf s}).
\end{equation}
Substitution of the parameters ${\bf c}$ and the Jucys-Murphy elements $(\mathcal{J}_1, \dots , \mathcal{J}_n)$
for the indeterminates ${\bf x}$ and ${\bf y}$ in (\ref{Pi_qt_mg}), (\ref{Pi_qt_gm}) gives
\begin{eqnarray}
M_n(q,t, {\bf c}, z \mathcal{J}) &\&= \sum_\lambda g_\lambda(q,t, {\bf c}) m_\lambda ({\bf \mathcal{J}})
\label{Mqt_lambda_cJ} \\
&\&= \sum_\lambda g_\lambda(q,t, {\bf \mathcal{J}}) m_\lambda ({\bf c}).
\label{Mqt_lambda_Jc}
\end{eqnarray}
Applying $M_n(q,t, {\bf c}, z \mathcal{J})$ to $C_\mu$ and using (\ref{Mqt_lambda_cJ}) and \autoref{m_lambda_JJ_C_mu},
gives
\begin{equation}
M_n(q,t, {\bf c}, z\mathcal{J}) C_\mu = \sum_{d=0}^\infty z^d\sum_{\nu, |\nu|=|\mu|} F^d_{M(q,t,{\bf c})} (\mu, \nu) z_\nu C_\nu
\label{Pi_Cmu_F_Gd}
\end{equation}
where we define, as before, the combinatorial quantum weighted Hurwitz numbers $F^d_{M(q,t, {\bf c})}(\mu, \nu)$
associated to the weight generating function $M(q,t, {\bf c}, z)$ as
\begin{equation}
F^d_{M(q,t,{\bf c})}(\mu, \nu) \coloneqq {1 \over \abs{n}!} \sum_{\lambda, \ \abs{\lambda}=d} g_\lambda({\bf c},q,t) m^\lambda_{\mu \nu}.
\label{Fd_Mqt_def}
\end{equation}
It follows as before that when the $\tau$-function $\tau^{M(q,t, {\bf c}, z)}({\bf t}, {\bf s})$ is
expanded in the basis of products of power sum symmetric functions, the coefficients
are the quantum weighted Hurwitz combinatorial numbers $F^d_{M(q,t,{\bf c})}(\mu, \nu) $:
\begin{equation}
\tau^{M(q,t, {\bf c}, z)} ({\bf t}, {\bf s})
= \sum_{d=0}^\infty \sum_{\substack{\mu, \nu \\ \abs{\mu} =
\abs{\nu}}} z^d F^d_{M(q,t,{\bf c})}(\mu, \nu) p_\mu({\bf t}) p_\nu({\bf s}).
\label{tau_GM_F}
\end{equation}
The corresponding geometrically defined quantum weighted Hurwitz numbers are somewhat
more intricate. Let $\{\{\mu^{i, u_i}\}_{u_i =1, \dots , k_i}, \{\nu^{i, v_i}\}_{v_ i= 1, \dots, \tilde{k}_i}, \mu, \nu \}_{i=1, \dots, l}$
denote the branching profiles of an $n$-sheeted covering, of the Riemann sphere,
with two specified branch points of ramification profile types $(\mu, \nu)$, at $(0, \infty)$, and the rest divided into two classes I and II,
denoted $\{\mu^{(i,u_i)}\}_{u_i =1, \dots , k_i}$ and $\{\nu^{(i, v_ i)}\}_{v_ i= 1, \dots, \tilde{k}_i}$, respectively. These are further subdivided into $l $ species, or ``colours'', labelled by $i=1, \dots l$, the elements within each colour group distinguished by the labels $(u_i =1, \dots , k_i)$
and $(v_ i =1, \dots , \tilde{k}_ i)$. To such a grouping, we assign a partition $\lambda$ of length
\begin{equation}
\ell(\lambda) =: l
\end{equation}
and weight
\begin{equation}
d := |\lambda| = \sum_{i =1}^l \left (\sum_{u_i =1}^{k_i}\ell^*(\mu^{(i, u_i)})
+ \sum_{v_i =1}^{\tilde{k}_i}\ell^*(\nu^{(i, v_i)}) \right) = \sum_{i=1}^l d_i,
\end{equation}
whose parts $(\lambda_1\ge \cdots \ge \lambda_l > 0)$ are equal the total colengths
\begin{equation}
d_i := \sum_{u_i =1}^{k_i} \ell^*(\mu^{(i, u_i)}) + \sum_{v_i=1}^{\tilde{k}_i}\ell^*(\nu^{(i, v_i)})
\end{equation}
in weakly decreasing order.
By the Riemann-Hurwitz formula, the genus $g$ of the covering curve is given by
\begin{equation}
2-2g = \ell(\mu) +\ell(\nu) - d.
\end{equation}
We now assign a weight $W_{Q(q)} (\{\mu^{(i,u_i)}, \nu^{( i, v_ i)}\}, {\bf c} ) $
to each such covering as in (\ref{W_{Q(q)}_mu_nu}), consisting of the product of all the weights
$W_{E(q)}(\{\mu^{(i, u_i)}\}_{u_i = 1, \dots, k_i})$,
$W_{H(q)}(\{\nu^{( i, v_ i)}\}_{v_ i = 1, \dots, \, \tilde{k}_ i})$
for the subsets of different colour and class with the weight $m_\lambda({\bf c})$
given by the monomial symmetric functions evaluated at the parameters ${\bf c}$
\begin{equation}
W_{Q(q)} (\{\mu^{(i,u_i)}, \nu^{( i, v_ i)}\}, {\bf c} )
:= W_{Q(q)} (\{\mu^{(i,u_i)}, \nu^{( i, v_ i)}\} )m_\lambda({\bf c})
\end{equation}
where
\begin{equation}
W_{Q(q)} (\{\mu^{(i,u_i)}, \nu^{( i, v_ i)}\} ) :=
\prod_{i=1}^lW_{E(q)}(\{\mu^{(i,u_i)}\}_{u_i = 1, \dots, \, k_i})
W_{H(q)}(\{\nu^{( i, v_ i)}\}_{ i = 1, \dots, \, \tilde{k}_ i})
\end{equation}
Using these weights, for every pair $(d,e)$ of non-negative integers and $(\mu, \nu)$
of partitions of $n$, we define the geometrical quantum weighted Hurwitz numbers
$H^{(d,e)}_{({\bf c}, q)}(\mu, \nu) $ as the sum
\begin{equation}
H^{(d,e)}_{({\bf c}, q)}(\mu, \nu) := z_\nu \sum_{l=0}^{d} {\hskip -35 pt}
\sideset{}{'} \sum_{\substack{\{\mu^{(i, u_i)}, \nu^{(i, v_i)}\} , \ k_i\ge 1, \ \tilde{k}_i \ge 1\\
\sum_{i=1}^l \sum_{u_i =1}^{k_i}\ell^*(\mu^{(i,u_i)}) = e, \\
\sum_{i=1}^l\left( \sum_{u_i =1}^{k_i}\ell^*(\mu^{(i, u_i)} )
+ \sum_{v_ i =1}^{\tilde{k}_ i}\ell^*(\nu^{( i, v_ i)})\right) =d}} {\hskip - 20 pt}
{\hskip-50 pt}W_{Q(q)} (\{\mu^{(i, u_i)}, \nu^{( i, v_ i})\}, {\bf c}) \
H(\{\mu^{(i, u_i)}\}_{\substack{u_i = 1, \dots, k_i \\ i =1, \dots , l}} ,
\{\nu^{( i, v_ i)}\}_{\substack{v_ i = 1, \dots, \tilde{k}_ i \\ i =1, \dots , l}}, \mu, \nu).
\label{Hde_c_q_mu_nu}
\end{equation}
We then have the following theorem, which is proved in \cite{H2}:
\begin{theorem}
\label{geometric_hurwitz}
The combinatorial Hurwitz numbers $F^d_{M(q,t,{\bf c})} (\mu, \nu) $ are degree $d$ polynomials in $t$, whose
coefficients are equal to the geometrical quantum weighted Hurwitz numbers $H^{(d,e)}_{({\bf c}, q)}(\mu, \nu)$
\begin{equation}
F^d_{M(q,t,{\bf c})} (\mu, \nu) = \sum_{e=0}^d H^{(d,e)}_{({\bf c}, q)}(\mu, \nu) t^e.
\label{Fd_G_Hde_G}
\end{equation}
Hence $\tau^{M(q,t,{\bf c}, z)} ({\bf t}, {\bf s})$, when expanded in
the basis of products of power sum symmetric functions and power series in $z$ and $t$
is the generating function for the $H^{(d,e)}_{({\bf c}, q)}(\mu, \nu)$'s:
\begin{equation}
\tau^{M(q,t,{\bf c}, z)} ({\bf t}, {\bf s}) = \sum_{d=0}^\infty \sum_{e=0}^d z^d t^e H^{(d,e)}_{({\bf c}, q)}(\mu, \nu) p_\mu({\bf t}) p_\nu({\bf s}).
\label{tau_G_H}
\end{equation}
\end{theorem}
\subsection{Examples}
We now give several examples of special classes of weighted Hurwitz numbers
that arise through restrictions or limits involving the parameters $(q,t,z)$. The details for all these examples are provided in \cite{H2}; we limit ourselves to specifying the restrictions and limits involved, giving only the generating functions, $\tau$-functions and quantum weighted Hurwirz formulae for each case.
\subsubsection{Elementary quantum weighted Hurwitz numbers}
In this case, we take the limits $z\rightarrow 0$, $t\rightarrow \infty$, but keeping the value of $-tz$ fixed
at a finite value, that is renamed $z$. The resulting weight generating function is
\begin{equation}
E(q, {\bf c}, z) := \prod_{k=0}^\infty \prod_{i=1}^\infty(1 +zq^k c_i) = \prod_{i=1}^\infty (-zc_i; q)_{\infty}.
=:\sum_{j=0}^\infty e_j(q,{\bf c}) z^j
\end{equation}
Here $e_j(q, {\bf c})$ is the quantum deformation of the elementary symmetric function $e_j({\bf c})$.
The corresponding central element $E_n(q, {\bf c}, z, \mathcal{J}) \in \mathbf{Z}(\mathbf{C}[S_n])$ is:
\begin{equation}
E_n(q, {\bf c}, z \mathcal{J}) := \prod_{a=1}^n E(q, {\bf c}, z\mathcal{J}_a)
=\sum_{\lambda} z^{|\lambda|} e_\lambda(q, {\bf c}) m_\lambda(\mathcal{J})
= \sum_{\lambda} z^{|\lambda|} m_\lambda(\mathcal{J}) e_\lambda(q, {\bf c}) ,
\end{equation}
where
\begin{equation}
e_\lambda(q, {\bf c}) := \prod_{i=1}^{\ell(\lambda)} e_{\lambda_i} ({\bf c}).
\end{equation}
Applying $E_n(q, {\bf c}, z\mathcal{J})$ to the orthogonal idempotents
$\{F_\lambda\}$ to obtain the content product coefficients
and to the cycle sums $\{C_\mu\}$ to obtain the Hurwitz numbers,
the resulting hypergeometric $2D$ Toda $\tau$-function is
\begin{eqnarray}
\tau^{E(q, {\bf c}, z)}({\bf t}, {\bf s}) &\&= \sum_\lambda r_\lambda^{E(q,{\bf c}, z)} s_\lambda({\bf t}) s_\lambda({\bf s}) \\
&\&= \sum_{d=0}^\infty z^d\sum_\lambda F^d_{E(q, {\bf c})}(\mu, \nu) p_\mu({\bf t}) p_\nu({\bf s}),
\end{eqnarray}
where
\begin{equation}
r_\lambda^{E(q, {\bf c}, z)} := \prod_{(ij) \in \lambda} \prod_{k=0}^\infty (-zc_k; q)_\infty
\end{equation}
is the content product coefficient and
\begin{equation}
F^d_{E(q, {\bf c})}(\mu, \nu) := \sum_{|\lambda|=d}e_\lambda(q, {\bf c}) m^\lambda_{\mu \nu}
\label{F_dE_qc}
\end{equation}
is the weighted number of $d$-step paths in the Cayley graph of $S_n$ generated by transpositions,
starting at the conjugacy class $\cyc(\mu)$ and ending at $\cyc(\nu)$, with
weight $e_\lambda(q, {\bf c})$ for a path of signature $\lambda$.
Now consider again $n$-fold branched coverings of $\mathbf{C} \mathbf{P}^1$ with a fixed pair of branch points at $(0, \infty)$
with ramification profiles $(\mu, \nu)$ and a further $ \sum_{i=1}^l k_i $ branch points $\{\mu^{(i,u_i)}\}_{u_i = 1, \dots, \, k_i}$ of $l$
different species (or ``colours''), labelled by $i=1, \dots , l$, with non trivial ramifications profiles.
The weight $W_{E^l(q)} (\{\mu^{(i,u_i)}\}_{\substack{u_i = 1, \dots, k_i \\ i=1, \dots , l}}, {\bf c} )$
for such a covering is defined by
\begin{equation}
W_{E^l(q)} (\{\mu^{(i,u_i)}\}_{\substack{u_i = 1, \dots, k_i \\ i=1, \dots , l}}, {\bf c} )
:= W_{E^l(q)} (\{\mu^{(i,u_i)}\}_{\substack{u_i = 1, \dots, k_i \\ i=1, \dots , l}} )\, m_\lambda({\bf c})
\end{equation}
where
\begin{equation}
W_{E^l(q)}(\{\mu^{(i,u_i)}\}_{\substack{u_i = 1, \dots, k_i \\ i=1, \dots , l}} ) :=
\prod_{i=1}^lW_{E(q)}(\{\mu^{(i,u_i)}\}_{u_i = 1, \dots, \, k_i}).
\end{equation}
It follows from the general result that
\begin{equation}
F^d_{E(q,{\bf c})}(\mu, \nu) =H^d_{E(q, {\bf c})} (\mu, \nu),
\end{equation}
where
\begin{equation}
H^d_{E({\bf c}, q)}(\mu, \nu) := z_\nu \sum_{l=0}^{d} {\hskip -10pt}
\sideset{}{'} \sum_{\substack{\{\mu^{(i, u_i)}\} , \ k_i\ge 1, \ \\
\sum_{i=1}^l \sum_{u_i =1}^{k_i}\ell^*(\mu^{(i,u_i)}) = d
}}
{\hskip-20 pt}W_{E^l(q)} (\{\mu^{(i, u_i)}\}_{\substack{u_i = 1, \dots, k_i \\ i=1, \dots , l}}, {\bf c}) \
H(\{\mu^{(i, u_i)}\}_{\substack{u_i = 1, \dots, k_i \\ i =1, \dots , l}} ,
\mu, \nu)
\label{Eq_d_c}
\end{equation}
is the geometrically defined quantum weighted Hurwitz number for this case.
\subsubsection{Complete quantum weighted Hurwitz numbers}
This is the dual of the preceding case, obtained by setting $t=0$. The weight generating function
becomes
\begin{equation}
H(q, {\bf c}, z) := \prod_{k=0}^\infty \prod_{i=1}^\infty(1 - zq^k c_i)^{-1} = \prod_{i=1}^\infty (zc_i; q)^{-1}_{\infty}
=:\sum_{j=0}^\infty h_j(q,{\bf c}) z^j,
\end{equation}
where $h_j(q, {\bf c})$ is the quantum deformation of the complete symmetric function $h_j({\bf c})$.
The corresponding central element $H_n(q, {\bf c}, z, \mathcal{J}) \in \mathbf{Z}(\mathbf{C}[S_n])$ is:
\begin{equation}
H_n(q, {\bf c}, z \mathcal{J}) := \prod_{a=1}^n H(q, {\bf c}, \mathcal{J}_a)
=\sum_{\lambda}z^{|\lambda|} h_\lambda(q, {\bf c} )m_\lambda(\mathcal{J})
= \sum_{\lambda} z^{|\lambda|} m_\lambda({\bf c}) h_\lambda(q, \mathcal{J}) ,
\end{equation}
where
\begin{equation}
h_\lambda(q, {\bf c}) := \prod_{i=1}^{\ell(\lambda)} h_{\lambda_i} (q, {\bf c}).
\end{equation}
By specializing the general case by setting $t=0$, the resulting hypergeometric $2D$ Toda $\tau$-function is
\begin{eqnarray}
\tau^{H(q,{\bf c}, z)}({\bf t}, {\bf s}) &\&= \sum_\lambda r_\lambda^{H(q, {\bf c}, z)} s_\lambda({\bf t}) s_\lambda({\bf s}) \\
&\&= \sum_{d=0}^\infty z^d\sum_\lambda F^d_{H(q,{\bf c})}(\mu, \nu) p_\mu({\bf t}) p_\nu({\bf s}),
\end{eqnarray}
where
\begin{equation}
r_\lambda^{H(q,{\bf c}, z)} := \prod_{(ij) \in \lambda} \prod_{k=0}^\infty (z(j-i)c_k; q)^{-1}_\infty
\end{equation}
and
\begin{equation}
F^d_{H(q,{\bf c})}(\mu, \nu) := \sum_{|\lambda|=d}h_\lambda(q, {\bf c}) m^\lambda_{\mu \nu}
\label{F_dH_qc}
\end{equation}
is the weighted number of paths in the Cayley graph of $S_n$ generated by transpositions,
starting at the conjugacy class $\cyc(\mu)$ and ending at $\cyc(\nu)$, with
weight $h_\lambda(q, {\bf c})$ for a path of signature $\lambda$.
Consider again the $n$-fold branched coverings of $\mathbf{C} \mathbf{P}^1$, with a fixed pair of branch points at $(0, \infty)$
with ramification profiles $(\mu, \nu)$ and a further
$ \sum_{i=1}^l \tilde{k}_i $ branch points $\{\nu^{(i,v_i)}\}_{v_i = 1, \dots, \, \tilde{k}_i}$
of $l$ different species (or ``colours''), labelled by $i=1, \dots , l$, with nontrivial ramifications profiles.
The weight $W_{H^l(q)} (\{\nu^{(i,v_i)}\}_{\substack{v_i = 1, \dots, \tilde{k}_i \\ i=1, \dots , l}}, {\bf c} )$
for such a covering is defined by
\begin{eqnarray}
W_{H^l(q)} (\{\nu^{(i,v_i)}\}_{\substack{v_i = 1, \dots, \tilde{k}_i \\ i=1, \dots , l}}, {\bf c} )
&\&:= W_{H^l(q)} (\{\nu^{(i,v_i)}\}_{\substack{v_i = 1, \dots, \tilde{k}_i \\ i=1, \dots , l}} ) \, m_\lambda({\bf c})\\
W_{H^l(q)}(\{\nu^{(i,v_i)}\}_{\substack{v_i = 1, \dots, k_i \\ i=1, \dots , l}} ) &\&:=
\prod_{i=1}^lW_{H(q)}(\{\nu^{(i,v_i)}\}_{v_i = 1, \dots, \, k_i}).
\end{eqnarray}
We again have
\begin{equation}
F^d_{H(q,{\bf c})}(\mu, \nu) =H^d_{H(q, {\bf c})} (\mu, \nu),
\end{equation}
where
\begin{equation}
H^d_{H({\bf c}, q)}(\mu, \nu) := z_\nu \sum_{l=0}^{d} {\hskip -10pt}
\sideset{}{'} \sum_{\substack{\{\nu^{(i, v_i)}\} , \ \tilde{k}_i\ge 1, \ \\
\sum_{i=1}^l \sum_{v_i =1}^{k_i}\ell^*(\nu^{(i,v_i)}) = d
}}
{\hskip-20 pt}W_{H^l(q)} (\{\nu^{(i, v_i)}\}_{\substack{v_i = 1, \dots, \tilde{k}_i \\ i=1, \dots , l}}, {\bf c}) \
H(\{\nu^{(i, v_i)}\}_{\substack{v_i = 1, \dots, \tilde{k}_i \\ i =1, \dots , l}} ,
\mu, \nu).
\label{Eq_d_c}
\end{equation}
\subsubsection{Hall-Littlewood function weighted Hurwitz numbers}
The generating function for Hall-Littlewood polynomials $P_\lambda({\bf x}, t)$, is obtained
by setting $q=0$ in eq.~(\ref{M_qtcz}). The orthogonality relations \cite{Mac} become
\begin{equation}
(P_\lambda, P_\mu)_t= \delta_{\lambda \mu} (b_\lambda(t))^{-1}, \quad b_\lambda (t) := \prod_{i\ge 1} \prod_{k=1}^{m_i(\lambda)} (1 - t^k)
\end{equation}
with respect to the scalar product $(\ , \ )_t$ defined by
\begin{equation}
(p_\lambda, p_\mu)_t = \delta_{\lambda \mu} z_\lambda n_\lambda (t), \quad n_\lambda:= \prod_{i=1}^{\ell(\lambda)} {1\over 1 - t^{\lambda_i}}.
\end{equation}
Defining, as in \cite{Mac},
\begin{equation}
q_\lambda({\bf x}, t) := b_\lambda(t) \prod_{i=1}^{\ell(\lambda)} P_j({\bf x}, t)
\label{q_lambda}
\end{equation}
we obtain the following expansion
\begin{equation}
L(t, {\bf x}, {\bf y})= \prod_{i, j}^\infty {1 - t x_i y_j \over 1- x_i y_j}
= \sum_{\lambda}^\infty q_\lambda({\bf x}, t) m_\lambda({\bf y})
= \sum_{\lambda}^\infty q_\lambda({\bf y}, t) m_\lambda({\bf x}).
\label{Lt_yx_t}
\end{equation}
The corresponding central element of $\mathbf{C}[S_n]$ is then
\begin{equation}
L(t, {\bf c}, z\mathcal{J}):= \prod_{i=1}^\infty \prod_{a=1}^n {1 - t c_i z \mathcal{J}_a \over 1- c_i z \mathcal{J}_a}
= \sum_{\lambda}^\infty z^{|\lambda|}q_\lambda({\bf c}, t) m_\lambda({\bf \mathcal{J}}) \\
= \sum_{\lambda}^\infty z^{|\lambda|} q_\lambda({\bf \mathcal{J}}, t) m_\lambda({\bf c}).
\label{HL_generating_element}
\end{equation}
The hypergeometric $2D$ Toda $\tau$-function then reduces to
\begin{eqnarray}
\tau^{L(t,{\bf c}, z)}({\bf t}, {\bf s}) &\&:= \sum_\lambda r_\lambda^{L(t,{\bf c}, z)} s_\lambda({\bf t}) s_\lambda({\bf s}) \\
&\&= \sum_{d=0}^\infty z^d\sum_\lambda F^d_{L(t,{\bf c})}(\mu, \nu) p_\mu({\bf t}) p_\nu({\bf s})\\
\end{eqnarray}
where the content product coefficient $r_\lambda^{L(t,{\bf c}, z)}$
is
\begin{equation}
r_\lambda^{L(t,{\bf c}, z)} := \prod_{(ij) \in \lambda} \prod_{k=1}^\infty { 1 - t z (j-i) c_k \over 1 - z (j-i) c_k}
= \prod_{k=1}^\infty (-t)^{|\lambda|} {(- 1/(tz c_k))_\lambda \over (- 1/(z c_k))_\lambda },
\end{equation}
and
\begin{equation}
F^d_{L(t,{\bf c})}(\mu, \nu) := \sum_{|\lambda|=d}q_\lambda({\bf c}, t) m^\lambda_{\mu \nu}
\end{equation}
is the weighted number of paths in the Cayley graph with the
weight $q_\lambda({\bf c},t)$ for a path of signature $\lambda$.
As in the general case, we also have
\begin{equation}
F^d_{L(t,{\bf c})}(\mu, \nu) = \sum_{e=0}^d H^{(d,e)}_{{\bf c}} (\mu, \nu) t^e
\end{equation}
where, denoting the total number of branch points,
\begin{equation}
K:= \sum_{i=1}^l (k_i +\tilde{k}_i)
\end{equation}
the weighted generalization of the multispecies hybrid signed Hurwitz numbers studied in \cite{HO2} is
\begin{equation}
H^{(d,e)}_{L({\bf c})}(\mu, \nu) := z_\nu \sum_{l=0}^{d} {\hskip -35 pt}
\sideset{}{'} \sum_{\substack{\{\mu^{(i, u_i)}, \nu^{(i, v_i)}\} , \ k_i\ge 1, \ \tilde{k}_i \ge 1\\
\sum_{i=1}^l \sum_{u_i =1}^{k_i}\ell^*(\mu^{(i,u_i)}) = e, \\
\sum_{i=1}^l\left( \sum_{u_i =1}^{k_i}\ell^*(\mu^{(i, u_i)} )
+ \sum_{v_ i =1}^{\tilde{k}_ i}\ell^*(\nu^{( i, v_ i)})\right) =d}} {\hskip - 20 pt}
{\hskip-50 pt} (-1)^{K+d-e}
H(\{\mu^{(i, u_i)}\}_{\substack{u_i = 1, \dots, k_i \\ i =1, \dots , l}} ,
\{\nu^{( i, v_ i)}\}_{\substack{v_ i = 1, \dots, \tilde{k}_ i \\ i =1, \dots , l}}, \mu, \nu).
\label{H_de_c}
\end{equation}
Its interpretation in terms of weighted enumerations of multispecies Hurwitz numbers of two classes
is the same as in the general Macdonald case, with the general quantum weighting factor
reducing to a sign times the standard classical one $m_\lambda({\bf c})$.
\subsubsection{Jack function weighted Hurwitz numbers}
The Jack polynomials $P^{(\alpha)}_\lambda$ are obtained by setting $t=q^\alpha$ and taking the limit $q \rightarrow 1$.
These satisfy the orthogonality relations \cite{Mac}
\begin{equation}
\langle P^{\alpha}_\lambda, P^{\alpha}_\mu\rangle_\alpha= \delta_{\lambda \mu} z_\lambda b_\lambda(t), \quad b_\lambda (t) := \prod_{i=1}^{\ell(\lambda)} \prod_{j=1}^{\lambda_i} (1 - t^j)
\end{equation}
where the scalar product $\langle \ , \ \rangle_\alpha$ is defined by \cite{Mac}
\begin{equation}
\langle p_\lambda, p_\mu\rangle_\alpha = \delta_{\lambda \mu} z_\lambda \alpha^{\ell(\lambda)}.
\end{equation}
The corresponding family of weight generating functions becomes
\begin{equation}
J(\alpha, {\bf c}, z) :=\prod_{k=1}^\infty (1 - z c_i)^{-1/\alpha}
\end{equation}
and the central elements
\begin{equation}
J(\alpha, z \mathcal{J}):= \prod_{i=1}^\infty \prod_{a=1}^n(1 - z c_i \mathcal{J}_a)^{-1/\alpha}
= \sum_{\lambda} z^{|\lambda|} g^{\alpha}_\lambda(\mathcal{J}) m_\lambda({\bf c})
= \sum_{\lambda} z^{|\lambda|} g^{\alpha}_\lambda({\bf c}) m_\lambda({\mathcal{J}}) ,
\end{equation}
where the symmetric functions $g^{\alpha}_\lambda({\bf x})$ are the analogs of the $e_\lambda({\bf x})$ or $h_\lambda ({\bf x})$ bases formed from products of elementary or complete symmetric functions \cite{Mac}
\begin{equation}
g^{\alpha}_\lambda({\bf x}) = \alpha^{\ell(\lambda)}\prod_{i=1}^{\ell(\lambda} P^{(\alpha)}_{(\lambda_i)}({\bf x}).
\end{equation}
The associated hypergeometric $2D$ Toda $\tau$-function is
\begin{equation}
\tau^{J(\alpha, {\bf c}, z)}({\bf t}, {\bf s}) = \sum_{\lambda} r_\lambda^{J(\alpha, {\bf c}, z)} s_\lambda({\bf t}) s_\lambda({\bf s})
\end{equation}
where the content product coefficients are
\begin{equation}
r_\lambda^{J(\alpha, {\bf c}, z)} := \prod_{(ij)\in \lambda} \prod_{k=0}^\infty (1- z (j-i) c_k)^{-1/\alpha}
= \prod_{k=1}^\infty (1-zc_k)^{|\lambda|\over \alpha} (-1/z c_k)_\lambda^{-1/\alpha}.
\end{equation}
Expanding over products of power sum symmetric functions gives
\begin{equation}
\tau^{J(\alpha, {\bf c}, z)} ({\bf t}, {\bf s})
= \sum_{d=0}^\infty \sum_{\substack{\mu, \nu \\ \abs{\mu} =
\abs{\nu}=n}} z^d F^d_{J(\alpha, {\bf c})}(\mu, \nu) p_\mu({\bf t}) p_\nu({\bf s})
\label{tau_GJ_F}
\end{equation}
where
\begin{equation}
F^d_{J(\alpha, {\bf c})} (\mu, \nu) = \sum_{\lambda} g^{\alpha}_\lambda({\bf c})m^\lambda_{\mu \nu}
\end{equation}
is the combinatorial Hurwitz number giving the weighted number of $d$-step paths of signature $\lambda$ in the Cayley
graph of $S_n$, starting in the conjugacy class $\cyc(\mu)$ and ending in $\cyc(\nu)$, with weight $g^{\alpha)}_\lambda({\bf c})$.
We again have equality with the weighted geometrical Hurwitz number
\begin{equation}
F^d_{J(\alpha, {\bf c})} (\mu, \nu) = H^d_{J(\alpha, {\bf c})} (\mu, \nu) ,
\end{equation}
where
\begin{equation}
H^d_{J(\alpha, {\bf c})} (\mu, \nu) := \sum_{k=0}^\infty \left({-{1\over \alpha} \atop k}\right)
\sum_{\substack{\mu^{(1)}, \dots , \mu^{((k)} \\ |\mu^{(i)}|=n \\ \sum_{i=1}^k \ell^*(\mu^{(i)})=d }}
m_\lambda({\bf c}) H(\mu^{(1)}, \dots , \mu^{(k)}, \mu, \nu)
\end{equation}
with the sum is over partitions $\lambda$ of length $k$, and weight $d$ whose parts are $\{\ell^*(\mu^{(1)}), \dots, \ell^*(\mu^{(k)})\}$.
\bigskip
\noindent
\small{ {\it Acknowledgements.} The author would like to thank G. Borot, L.~Chekhov, M.~Guay-Paquet,
M.~Kazarian, S.~Lando and A.Yu.~Orlov for helpful discussions.}
\newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{arXiv:{#1}}}
|
{
"timestamp": "2015-09-22T02:22:26",
"yymm": "1504",
"arxiv_id": "1504.03408",
"language": "en",
"url": "https://arxiv.org/abs/1504.03408"
}
|
\section{Introduction}
\subsection{Overview}
Let $\eta$ be a Poisson random measure over some measurable space $(\mathbb{X}, \mathcal{X})$ such that $\mathcal{X}$ is countably generated, and assume that $\eta$ has a $\sigma$-finite intensity $\mu$. Let $F = F(\eta)$ be a real-valued functional of $\eta$ having finite expectation. In this paper, we are interested in proving several novel estimates for the upper and lower tails
$$
\P(F \geq \mathbb{E} F+ r) \quad \mbox{and} \quad \P(F \leq \mathbb{E} F -r), \quad r>0,
$$
that are well adapted for geometric applications, with particular emphasis on quantities appearing in the modern theory of random geometric graphs -- see e.g. \cite{Penrose_book}.
\medskip
Our techniques are based on many variations of the so-called {\it Herbst argument} (see e.g. \cite{blm_book, BLM_2003, ledoux_book, M_2000}), basically consisting in using a logarithmic Sobolev inequality (or, alternatively, an integration by parts formula) in order to deduce a differential inequality involving the moment generating function $u\mapsto K(u) := \mathbb{E}[e^{uF}]$; solving the inequality then yields an upper bound on $K(u)$, implying in turn a tail estimate for $F$ by means of Markov's inequality. The main insight developed in the present paper is that, by carefully combining the {\it Mecke formula} for Poisson point processes (see Section \ref{s:frame}) with logarithmic Sobolev inequalities such as the one in Theorem \ref{t:wu} below, one can deduce bounds on $K(u)$ involving quantities of a fundamental geometric nature. As discussed below, other approaches to concentration via the Herbst argument on the Poisson space (see e.g. \cite{BHP_2007, HP_2002, W_2000}) do not yield conditions that are amenable to geometric analysis.
\subsection{Logarithmic Sobolev inequalities and a motivating example}
Our starting point is the following powerful Theorem \ref{t:wu}, proved by Wu in \cite{W_2000} (see also \cite{chafai}), and extending previous breakthrough findings contained in \cite{AL_2000, BL_1998}. Such a result involves two objects: (i) the {\it entropy} of a random variable $Z > 0$ with $\mathbb{E} Z<\infty$, that is defined as
\[
\Ent(Z) := \mathbb{E}(Z \log(Z)) - \mathbb{E}(Z) \log(\mathbb{E} Z ),
\]
and (ii) the {\it difference} (or {\it add-one cost}) operator $DF$, that is defined for any $x \in\mathbb{X}$ as
\begin{align*}
D_xF(\eta) = F(\eta + \delta_x) - F(\eta),
\end{align*}
where $\delta_x$ denotes the Dirac mass at $x\in\mathbb{X}$.
\begin{thm}[(See Corollary 2.3 in \cite{W_2000})]\label{t:wu} For all $\lambda \in\mathbb{R}$ satisfying $\mathbb{E} (e^{\lambda F})<\infty$ we have
\begin{equation}\label{e:wu}
\Ent(e^{\lambda F}) \leq \mathbb{E} \left[ e^{\lambda F} \left(\int_{\mathbb{X}} \psi(\lambda D_x F(\eta)) \ d\mu(x)\right)\right],
\end{equation}
where $\psi(z) = ze^z - e^z + 1$.
\end{thm}
A typical way of applying \eqref{e:wu} to concentration estimates (via the Herbst argument) is demonstrated e.g. in \cite[Proposition 3.1]{W_2000}, where it is proved that, if $DF$ and $\int_\mathbb{X} (DF)^2 d\mu $ are almost surely bounded by positive constants $\beta$ and $c$, respectively, then the upper tail of $F$ is bounded by the function $$r\mapsto \exp\left[ - \frac{r}{2\beta}\log\left(1+\frac{\beta r}{c}\right) \right], \quad r>0.$$ Letting $\beta\to 0$, one deduces from this estimate that, if $DF\leq 0$ and $\int_\mathbb{X} (DF)^2 d\mu\leq c$, then,
$$
\P[F\geq \mathbb{E} F+ r] \leq \exp\left(- \frac{r^2}{2c}\right),\quad r>0,
$$
that is: such a concentration result only captures a Gaussian behaviour for the upper tail in the case of a non-increasing functional $F$ such that $\int_\mathbb{X} (DF)^2 d\mu$ is deterministically bounded. Apart from the monotonicity requirement on $F$, a crucial limitation of a result of this kind is that, in most examples where $F$ is a quantity arising in stochastic geometry (for instance, $F$ is an {\it edge-counting statistic} such as the ones considered in Section \ref{s:edge} below), the quantity $\int_\mathbb{X} (DF)^2 d\mu$ {\it does not admit any meaningful geometric interpretation} -- roughly because averaging $DF$ over the deterministic measure $\mu$ completely cancels the special role played by those points in $\mathbb{X}$ that belong to the support of $\eta$. One should contrast such a situation with the following statement, that will be proved later on as a special case of Corollary \ref{DevIneq1} (such a result also implies the already quoted Proposition 3.1 in \cite{W_2000}):
\begin{prop}\label{p:lh} Assume that there exists a finite constant $c>0$ such that, almost surely,
{
\begin{equation}
V^+ := \int_\mathbb{X} (D_xF)^2_- d\mu(x) + \int_{\mathbb{X}} (F(\eta) - F(\eta-\delta_x))^2_+ d\eta(x)\leq c,
\end{equation}
}
where $(u)_-$ and $(u)_+$ stand for the negative and positive part of $u\in \mathbb{R}$, respectively. Then,
$$
\P(F\leq \mathbb{E} F +r) \leq \exp\left( - \frac{r^2}{2c}\right).
$$
\end{prop}
Proposition \ref{p:lh} is particularly interesting when $F$ is {\it non-decreasing}, {that is, when $DF\geq 0$}. Indeed, in this case one has that {$V^+ = \int_{\mathbb{X}} (F(\eta) - F(\eta-\delta_x))^2_+ d\eta(x)$} and, since the role of $\mu$ is now immaterial, the relation $V_+\leq c$ can be in principle verified by means of arguments of a purely geometric or combinatorial nature. For instance, we implement this strategy in Proposition \ref{convDistProp} below, where we use Proposition \ref{p:lh} in order to deduce a novel intrinsic proof of the {Gaussian} upper tail behaviour of the convex distance for point processes -- as recently introduced by Reitzner in \cite{R_2013}.
As anticipated, the principal aim of this paper is to prove a large collection of statements with the same flavour as Proposition \ref{p:lh} (see Section 3), and then to apply them to random variables arising in the theory of random geometric graphs.
\subsection{Plan} Our work is organised as follows. After some preliminary facts discussed in Section 2, the subsequent Section 3 contains the statements of our main concentration estimates. Our results involve random variables having a form similar to the quantity $V^+$ introduced above, and largely generalise Proposition \ref{p:lh}. Proofs are detailed in Section 4.
\smallskip
Section 5 presents several applications of the results of Section 3 to Poisson U-statistics of arbitrary order -- as defined in the seminal reference \cite{RS_2013} (see also \cite{BP_2012, DFRV, ET, LRP1, LRP2, PT_alea, ST_spa2012, LR_2015}). As a by-product of our analysis, in Proposition \ref{p:p} we also establish a new characterisation of square-integrable Poisson U-statistics.
\smallskip
The results of Section 5 are specialised in Section 6 to the case of edge-counting statistics associated with general random geometric graphs. Several careful comparisons with the existing literature (in particular \cite{ERS_2015, RST_2013}) are presented.
\smallskip
Section 7 contains further estimates on U-statistics of order two, that are proved by adapting some techniques introduced in \cite{HRB, R_2003}. Geometric applications to edge-length functionals are discussed in detail.
\smallskip
As anticipated, in Section 8 we apply the estimates of Section 3, in order to deduce a novel intrinsic proof of the concentration estimates for the convex distance for random point measures established in \cite{R_2013}. Such a fundamental object generalises to the framework of random point processes the celebrated {\it convex distance} introduced by Talagrand in \cite{T_1995}; see \cite{RST_2013, LR_2015} for several applications. We stress that the problem of finding an intrinsic proof of the striking concentration results from \cite{R_2013} has been one of the main motivations for elaborating the theory developed in the present paper.
\subsection{Further remarks on the literature}
The inequalities obtained in this paper (as well as some techniques exploited in the proofs) are very close in spirit to those appearing in the seminal references \cite{blm_book, BLM_2003, M_2000}, where the so-called {\it entropy method} (roughly corresponding to a combination of the Herbst argument and of logarithmic Sobolev inequalities -- see e.g. \cite{ledoux_book}) is developed in the framework of functions of finite vectors of independent random elements. We recall that the results from \cite{blm_book, BLM_2003, M_2000} typically apply to random variables with the form $F = f(X_1,...,X_n)$, where $X = (X_1,...,X_n)$ is a vector of independent random elements and $f$ is some deterministic measurable function, and are based on a pervasive use of {\it random difference operators} of the type $$\Delta_if(X) = f(X) - f(X_1,...,X_{i-1}, X'_i, X_{i+1}, ...,X_n), \quad i=1,...,n,$$
where $X'$ is an independent copy of $X$. By inspection of the results presented below, it is not difficult to show that, in the case where the intensity $\mu$ of the Poisson measure $\eta$ is {\it finite} and {\it non-atomic}, some versions of the main results of the present paper could be obtained by implementing the following rough strategy:
\begin{enumerate}
\item Select a sequence of measurable partitions $\{B_1^{n},...,B_{k(n)}^{n} : n\geq 1\}$ of $\mathbb{X}$, in such a way that $k(n)\to\infty$ and $\max_{i=1,...,k(n)} \mu(B_i^{n})\to 0$, as $n\to \infty$.
\item Consider a random variable $F = F(\eta)$ and represent it in the form $F = f_n(X_{n,1},..,X_{n,k(n)})$, where $X_{n,i}$ is defined as the restriction of $\eta$ to the set $B_i^n$, and $f_n$ is some appropriate measurable mapping.
\item For a fixed $n$, prove a concentration estimate for $F(\eta)$ by applying the results from \cite{blm_book, BLM_2003, M_2000} to $f_n(X_{n,1},..,X_{n,k(n)})$, in particular by considering an independent copy of $X_{n,1},..,X_{n,k(n)}$ defined in terms of an independent Poisson measure $\eta'$ on $\mathbb{X}$ with intensity $\mu$.
\item Let $n\to \infty$, and recover a bound involving quantities related to the add-one cost operator $DF$ described above, by exploiting the fact that independent Poisson measures with non-atomic intensities have almost surely disjoint supports.
\end{enumerate}
Apart from the fact that this approach only works with finite intensity measures without atoms, some investigations in this direction have convincingly shown us that (to the best of our expertise), in order for the step described at Point (iv) to take place in a meaningful way, one should systematically add to our statements some additional technical assumptions, that are indeed {\it not required} if one implements the direct approach based on the Mecke formula that is systematically adopted in this paper. An analogous phenomenon can be observed for instance in \cite[Theorem 4]{hr_2009}, where a weaker version of the Poincar\'e inequality on the Poisson space is deduced by means of a discretisation procedure similar to the one outlined above, and of the use of the classical Efron-Stein inequality. In view of these remarks, we decided not to directly exploit the connection with the entropy method on product spaces in the proofs of our main results.
\medskip
Another collection of results that is relevant for our paper is contained in references \cite{BHP_2007, HP_2002}, where the authors obtain concentration estimates by applying integration by parts techniques, in particular by using the properties of the so-called {\it Ornstein-Uhlenbeck semigroup} associated with a given Poisson measure -- see also \cite{surg}. As in the already discussed examples from \cite{W_2000}, the estimates contained in these references have an equally problematic geometric interpretation, since they involve integrals of add-one cost operators with respect to the underlying intensity measure $\mu$. Moreover, in order to exploit some probabilistic representation of the Ornstein-Uhlenbeck semigroup, one has also to work on {\it extended} probability spaces. It is a natural question to ask whether the Mecke formula could be combined with some of the estimates from \cite{BHP_2007, HP_2002} in order to obtain concentration inequalities that are adapted to a geometric framework. We prefer to think of this issue as a separate problem, and leave it open for further research.
\subsection{Acknowledgments} The authors wish to thank M. Reitzner and Ch.$\!$ Th\"ale for useful discussions. S. Bachmann is partially supported by the German Research Foundation DFG-GRK 1916. G. Peccati is partially supported by the grant F1R-MTH-PUL-12PAMP (PAMPAS) at Luxembourg University.
\section{Framework}\label{s:frame}
For the rest of the paper, we shall denote by $(\mathbb{X}, \mathcal{X}, \mu)$ a $\sigma$-finite measure space, such that the $\sigma$-field $\mathcal{X}$ is countably generated and $\mu(\mathbb{X})>0$. We write $\eta$ to indicate a Poisson point process on $(\mathbb{X}, \mathcal{X})$. This means that $\eta = \{\eta(A) : A\in \mathcal{X}_0\}$ is a collection of random variables, defined on some probability space $(B, \mathcal{B},\mathbb{P})$ and indexed by the elements of $\mathcal{X}_0 = \{A\in \mathcal{X} : \mu(A)<\infty\}$, such that the following properties are satisfied: {\bf (i)} for every fixed $A\in \mathcal{X}_0$, $\eta(A)$ is a Poisson random variable with parameter $\mu(A)$, and {\bf (ii)} for every collection of pairwise disjoint $A_1,...,A_n\in \mathcal{X}_0$, one has that the random variables $\eta(A_1),...,\eta(A_n)$ are stochastically independent.
\medskip
As usual, we interpret $\eta$ as a random element in the space
${\bf N} = {\bf N}(\mathbb{X})$ of integer-valued $\sigma$-finite measures $\xi$
on $\mathbb{X}$ equipped with the smallest $\sigma$-field $\mathcal{N} $
making the mappings $\xi\mapsto\xi(B)$ measurable for all
$B\in\mathcal{X}$; see e.g. \cite{SW_2008} or \cite{LP_2011}. The standard notation $x\in \eta$ is shorthand in order to indicate that the point $x$ is an element of the support of $\eta$. We write $\hat{\eta}$ for the compensated random (signed) measure $\eta-\mu$. We shall write $F = F(\eta)$ to indicate that a given random variable $F$ can be written in the form $F=\mathfrak{f}(\eta)$, $\mathbb{P}$-a.s., for some measurable function
$\mathfrak{f}: {\bf N} \rightarrow\mathbb{R}$; such a function $\mathfrak{f}$ (which is uniquely determined by $F$ up to sets of $\mathbb{P}$-measure zero) is customarily called a {\it representative} of $F$, and $F$ is called a {\it Poisson functional}.
\begin{rem}[(Some conventions)]\label{r:talich} In what follows, we will indifferently use the notation $F$ and $F(\eta)$ if there is no ambiguity. Also, for any $\xi\in{\bf N}$, the notation $F(\xi)$ refers to $\mathfrak{f}(\xi)$, where $\mathfrak{f}$ is a fixed representative of $F$. Finally, we observe that, in the statements of some of our main results, we will often work under the assumption that the add-one cost operator $D_xF(\xi)$ verifies a given property $\mathcal{P}$ (for instance, $D_xF(\xi)\leq 0$) for every $x\in \mathbb{X}$ and every $\xi \in {\bf{N}}$: this requirement means of course that there exists a representative $\mathfrak{f}$ of $F$ such that the quantity $\mathfrak{f}(\xi + \delta_x) - \mathfrak{f}(\xi)$ verifies $\mathcal{P}$ for every $x\in \mathbb{X}$ and every $\xi \in {\bf{N}}$.
\end{rem}
\medskip
We will systematically use the following standard notation: for every $\xi \in {\bf N}$,
\begin{align}\label{xiNeq}
\xi_{\neq}^k := \{{\bf x} = (x_1,...,x_k) : x_i \in \xi, \, \forall i=1,...,k, \, \text{and}\, x_i\neq x_j, \, i\neq j\}.
\end{align}
A result that we shall use in several occasions (and that in some sense represents the backbone of our approach) is the following well-known {\it Slivnyak-Mecke formula}: for every $m\geq 1$ and every non-negative measurable function $H$ on ${\bf N}\times \mathbb{X}^m$, one has that
{
\begin{eqnarray}\label{e:smecke}
&&\mathbb{E}\left[\int_{\mathbb{X}^m} H(\eta, x_1,...,x_m) d\eta^{(m)}(x_1,\ldots,x_m) \right] \\ \notag &&= \int_{\mathbb{X}}\cdots \int_{\mathbb{X}} \mathbb{E}\left[H\left(\eta+\sum_{i=1}^m \delta_{x_i},x_1, \ldots , x_m\right)\right] d\mu(x_1)\cdots d\mu(x_m),
\end{eqnarray}
where $\delta_x$ stands for the Dirac mass at $x$ and $\eta^ {(m)}$ is the point process on $\mathbb{X}^m$ with support $\eta_{\neq}^m$ and $\eta^{(m)}(\{{\bf x}\}) = \eta(\{x_1\})\cdots\eta(\{x_m\})$ for any ${\bf x}=(x_1,\ldots,x_m)\in\eta_{\neq}^m$}. A standard proof of the fundamental relation \eqref{e:smecke} can be found e.g. in \cite[Theorem 3.2.5 and Corollary 3.2.3]{SW_2008}, in the case of a non-atomic intensity $\mu$. The result extends straightforwardly to the case of a general $\sigma$-finite measure $\mu$ -- see e.g. \cite{LP_2011}. When specialised to the case $m=1$, relation \eqref{e:smecke} is known as {\it Mecke formula}, and boils down to the following identity: for every non-negative measurable function $H$ on ${\bf N}\times \mathbb{X}$, one has that
{
\begin{equation}\label{e:mecke}
\mathbb{E}\left[\int_\mathbb{X} H(\eta, x) d\eta(x)\right] = \int_{\mathbb{X}} \mathbb{E}[H(\eta+\delta_x,x)] d\mu(x).
\end{equation}
}
\smallskip
For the rest of the paper, for every integer $k\geq 1$ and every real $p>0$, we will write $L^p(\mu^k) := L^p(\mathbb{X}^k , \mathcal{X}^{\otimes k}, \mu^k)$, and also use the shorthand notation $L^p(\mu^1) = L^p(\mu)$. In Section \ref{ss:zut}, the symbol $L^2(\mu^0)$ is used to denote the real line $\mathbb{R}$, endowed with the usual Euclidean inner product.
\section{Deviation Inequalities for Poisson Functionals} \label{DIPF}
In the following, we are going to develop new tools for proving deviation inequalities of Poisson functionals. Our approach is an adaptation of the entropy method for product space functionals that was particularly investigated in \cite{BLM_2003}.
\medskip
\medskip
The heart of the method we are about to present is the modified logarithmic Sobolev inequality stated below. {For the remainder of the section, we consider a Poisson functional $F$.}
As above, we will use the difference (or add-one cost) operator $DF$, that is defined for any $(x,\xi)\in\mathbb{X}\times{\bf N}$ by
\begin{align*}
D_xF(\xi) = F(\xi + \delta_x) - F(\xi),
\end{align*}
where $\delta_x$ denotes the Dirac mass at $x\in\mathbb{X}$. To shorten notations, we write
\begin{align*}
D_x^IF(\xi) = D_xF(\xi) \mathbbm{1}\{(x,\xi)\in I\}
\end{align*}
whenever $I\subseteq \mathbb{X} \times {\bf N}$ is measurable, $x\in\mathbb{X}$ and $\xi \in{\bf N}$. In the same spirit we will also use the notations
\begin{align*}
D_x^{\geq\beta}F(\xi) &= D_xF(\xi) \mathbbm{1}\{D_xF(\xi)\geq \beta\},\\
D_x^+F(\xi) &= D_xF(\xi) \mathbbm{1}\{D_xF(\xi)\geq 0\}.
\end{align*}
The quantities $D_x^{\leq\beta}F(\xi)$ and $D_x^-F(\xi)$ are defined analogously, and so are the operators $D_x^{>\beta}F$ and $D_x^{<\beta}F$ (with strict inequalities). The following observation is derived by combining Wu's modified logarithmic Sobolev inequality for Poisson point processes \eqref{e:wu} with the Mecke formula \eqref{e:mecke}.
\begin{prop} \label{EntIneq}
Let $I\subseteq \mathbb{X}\times{\bf N}$ be a measurable set. Then for all $\lambda \in\mathbb{R}$ satisfying $\mathbb{E} (e^{\lambda F})<\infty$ we have
\[
\Ent(e^{\lambda F}) \leq \mathbb{E} \left[ e^{\lambda F} \left(\int_{\mathbb{X}} \psi(\lambda D^I_x F(\eta)) \ d\mu(x) + \int_\mathbb{X} \phi(-\lambda D^{I^c}_x F(\eta-\delta_x)) \ d\eta(x)\right)\right],
\]
where $\phi(z) = e^z - z - 1$ and $\psi(z) = ze^z - e^z + 1$.
\end{prop}
For any $\beta\in\mathbb{R}$ we define the random variables $V_\beta^+ = V_\beta^+(F)$ and $V_\beta^- = V_\beta^-(F)$ by
\begin{align*}
V_\beta^+ &= \int_{\mathbb{X}} (D^{\leq \beta}_x F(\eta))^2 \ d\mu(x) + \int_\mathbb{X} (D^{ > \beta}_x F(\eta-\delta_x))^2 d\eta(x),\\
V_\beta^- &= \int_{\mathbb{X}} (D^{\geq \beta}_x F(\eta))^2 \ d\mu(x) + \int_\mathbb{X} (D^{ < \beta}_x F(\eta-\delta_x))^2 d\eta(x).
\end{align*}
Note that we will write $V^+ = V_0^+$ and $V^- = V_0^-$. The notation is in correspondence with \cite{BLM_2003} where the entropy method for product spaces was investigated. The upcoming result can be regarded as a generalized analogue of \cite[Theorem 2]{BLM_2003} for the Poisson space. The proof is similar to the product space version where Proposition \ref{EntIneq} takes now the role of the log Sobolev inequality. The generalization is achieved using arguments similar to those in the proof of \cite[Proposition 3.1]{W_2000}.
To get prepared for the presentation of the theorem, for $\beta\in\mathbb{R}$ and $z>0$, let
\begin{align*}
\Phi_\beta(z) = \begin{cases}
\psi(z\beta)/(z\beta^2)& \beta>0\\
z/2& \beta=0\\
\phi(-z\beta)/(z\beta^2) & \beta<0,
\end{cases}
\end{align*}
and
\begin{align*}
\Psi_\beta(z) = \begin{cases}
\phi(z\beta)/(z\beta^2)& \beta>0\\
z/2& \beta=0\\
\phi(-z\beta)/(z\beta^2) & \beta<0,
\end{cases}
\end{align*}
where $\phi$ and $\psi$ are as in Proposition \ref{EntIneq}. Note that we will frequently use the fact that these functions are non-decreasing.
\begin{thm} \label{EntIneqs}
{Assume that the Poisson functional $F$ is integrable}. Let $\lambda>0$ be such that $\mathbb{E} \exp(\lambda F)<\infty$. Then for any $\theta>0$ satisfying $\Phi_\beta(\lambda)\theta < 1$, we have
\begin{align}\label{EntIneq1}
\log \mathbb{E}[\exp(\lambda(F-\mathbb{E} F))] &\leq \frac{\Psi_\beta(\lambda) \theta}{1-\Phi_\beta(\lambda)\theta} \log \mathbb{E} \left[ \exp\left(\frac{\lambda V_\beta^+}{\theta}\right)\right].
\end{align}
Let $\lambda>0$ be such that $\mathbb{E} \exp(-\lambda F)<\infty$. Then for any $\theta>0$ with $\Phi_{-\beta}(\lambda)\theta < 1$, we have
\begin{align}\label{EntIneq2}
\log \mathbb{E}[\exp(-\lambda(F-\mathbb{E} F))] &\leq \frac{\Psi_{-\beta}(\lambda) \theta}{1-\Phi_{-\beta}(\lambda)\theta} \log \mathbb{E} \left[\exp\left(\frac{\lambda V_\beta^-}{\theta}\right)\right].
\end{align}
Assume that $F$ is not necessarily integrable and that one of the following conditions is satisfied:
\begin{enumerate}
\item $\beta>0$ and $D_x F(\xi) \leq \beta$ holds for all $(x,\xi)\in\mathbb{X}\times{\bf N}$,
\item $\beta<0$ and $D_x F(\xi) \geq \beta$ holds for all $(x,\xi)\in\mathbb{X}\times{\bf N}$,
\item $\beta = 0$.
\end{enumerate}
Then, the relation $\mathbb{E} \exp(\lambda V_\beta^+ / \theta)<\infty$ implies that $\mathbb{E} |F|<\infty$ and $\mathbb{E} \exp(\lambda F)<\infty$. Also, the relation $\mathbb{E} \exp(\lambda V_\beta^- / \theta)<\infty$ implies that $\mathbb{E} |F|<\infty$ and $\mathbb{E} \exp(-\lambda F)<\infty$.
\end{thm}
With the above results the methods for deriving deviation inequalities presented in \cite{BLM_2003} naturally carry over to the Poisson space. In the following we present some variations of these techniques that will be used for the applications later on.
In the case when $V_\beta^+$ and $V_\beta^-$ for $\beta\neq 0$ are almost surely bounded by a constant, the above entropy inequalities yield exponential tails for the random variable $F(\eta)$. If $V^+$ and $V^-$ are almost surely bounded (i.e. $\beta = 0$), we even obtain Gaussian tails. Note that Wu's deviation inequality \cite[Proposition 3.1]{W_2000} is implied by {the following} more general results.
\begin{cor} \label{DevIneq1}
Assume that $F$ satisfies $V_\beta^+ \leq c$ almost surely. Then $F$ is integrable and the following statements hold:
\begin{enumerate}
{
\item If either condition (i) or (ii) of Theorem \ref{EntIneqs} is satisfied, then for all $r\geq 0$,
\begin{align*}
\P(F \geq\mathbb{E} F + r) &\leq \exp\left(-\left(\frac{c}{\beta^2} + \frac{r}{|\beta|}\right)\log\left(1+\frac{|\beta| r}{c}\right) + \frac{r}{|\beta|}\right)\\
&\leq \exp \left(-\frac{r}{2|\beta|} \log\left(1 + \frac{|\beta| r}{c}\right)\right).
\end{align*}
}
\item If $\beta = 0$, that is if $V^+\leq c$ holds almost surely, then for all $r\geq 0$,
\begin{align*}
\P(F\geq\mathbb{E} F + r) \leq \exp\left(-\frac{r^2}{2c}\right).
\end{align*}
\end{enumerate}
\end{cor}
We continue with the corresponding version for the lower tail. This corollary is obtained in the same way as the above one where inequality (\ref{EntIneq2}) is used instead of (\ref{EntIneq1}). The proof is therefore omitted.
\begin{cor} \label{DevIneq12}
Assume that $F$ satisfies $V_\beta^- \leq c$ almost surely. Then $F$ is integrable and the following statements hold:
\begin{enumerate}
{
\item If either condition (i) or (ii) of Theorem \ref{EntIneqs} is satisfied, then for all $r\geq 0$,
\begin{align*}
\P(F \leq\mathbb{E} F - r) &\leq \exp\left(-\left(\frac{c}{\beta^2} + \frac{r}{|\beta|}\right)\log\left(1+\frac{|\beta| r}{c}\right) + \frac{r}{|\beta|}\right)\\
&\leq \exp \left(-\frac{r}{2|\beta|} \log\left(1 + \frac{|\beta| r}{c}\right)\right).
\end{align*}
}
\item If $\beta = 0$, that is if $V^-\leq c$ holds almost surely, then for all $r\geq 0$,
\begin{align*}
\P(F\leq\mathbb{E} F - r) \leq \exp\left(-\frac{r^2}{2c}\right).
\end{align*}
\end{enumerate}
\end{cor}
The following result is useful to obtain deviation inequalities under less restrictive boundedness conditions on $V^+$.
\begin{cor} \label{cor2}
Assume that $F\geq 0$ and that there is a {random variable $G\geq 0$} and an $\alpha\in[0,2)$ such that almost surely
\begin{align*}
V^+ \leq G F^\alpha.
\end{align*}
Let $\theta >0$ and $\lambda\in (0,2/\theta)$ be such that $\mathbb{E}\exp(\lambda G/\theta) < \infty$. Then $\mathbb{E} F^{1-\alpha/2} <\infty$ and
\begin{align*}
\log \mathbb{E}(\exp(\lambda(F^{1-\alpha/2}-\mathbb{E} F^{1-\alpha/2}))) &\leq \frac{\lambda \theta}{2-\lambda\theta} \log \mathbb{E} \left( \exp\left(\frac{\lambda G}{\theta}\right)\right).
\end{align*}
\end{cor}
In the case when the random variable $G$ in the above corollary is just a constant, we obtain the following deviation inequality for the upper tail.
\begin{cor} \label{DevIneq2}
{Assume that $F\geq 0$} and that for some $\alpha\in[0,2)$ and $c>0$ we have almost surely
\begin{align*}
V^+ \leq c F^\alpha.
\end{align*}
Then $F$ is integrable and for all $r\geq 0$,
\begin{align*}
\P(F\geq\mathbb{E} F + r) \leq \exp\left(-\frac{((r+\mathbb{E} F)^{1-\alpha/2} - (\mathbb{E} F)^{1-\alpha/2})^2}{2c}\right).
\end{align*}
\end{cor}
The next result is a variation of Corollary \ref{cor2} for Poisson functionals that are not necessarily non-negative. This is the Poisson space analogue of \cite[Theorem 5]{BLM_2003}.
\begin{thm}\label{arbSignThm}
Assume that the Poisson functional $F$ is integrable and that for some $a>0$ and $b \geq 0$ we have almost surely
\begin{align*}
V^+ \leq a F + b.
\end{align*}
Then for any $\lambda\in (0,2/a)$ we have $\mathbb{E}(e^{\lambda F})<\infty$ and
\begin{align*}
\log \mathbb{E} (\exp( \lambda(F-\mathbb{E} F))) \leq \frac{\lambda^2}{2-a\lambda} (a\mathbb{E} F+b).
\end{align*}
Moreover, for any $r\geq 0$,
\begin{align*}
\P(F\geq \mathbb{E} F + r) \leq \exp\left(-\frac{r^2}{2a\mathbb{E} F + 2b + ar/3}\right).
\end{align*}
\end{thm}
{We continue with a result that applies whenever $F$ is non-decreasing and $V^-$ is non-decreasing and integrable. In this case, the random variable $F$ has a Gaussian lower tail:}
\begin{thm} \label{LowerDevIneq}
Assume that the Poisson functional $F$ satisfies
\begin{align*}
{ D_x F(\xi), \ D_x V^-(\xi) \geq 0 \ \ \text{for all} \ \ (x,\xi)\in\mathbb{X}\times{\bf N}} \ \ \text{and} \ \ \mathbb{E} V^-(\eta) < \infty.
\end{align*}
Then $F = F(\eta)$ is integrable and for all $r\geq 0$ we have
\begin{align*}
\P(F\leq \mathbb{E} F -r) \leq \exp\left(-\frac{r^2}{2 \mathbb{E} V^-}\right).
\end{align*}
\end{thm}
\begin{rem} \label{incVRem}
{Assume that the Poisson functional $F$ is non-decreasing.} A sufficient condition for the assumption $DV^-\geq 0$ in the above theorem is that the second interation of the difference operator of $F$ is non-negative. Indeed, assume that
\begin{align*}
D_zD_x F(\xi) \geq 0 \ \ \text{for all} \ \ (z,x,\xi)\in\mathbb{X}\times\mathbb{X}\times{\bf N}.
\end{align*}
Then, $D_xF(\xi+\delta_z) \geq D_xF(\xi)\geq 0$ and hence $D_xF(\xi+\delta_z)^2 \geq D_xF(\xi)^2$ for all $(z,x,\xi)\in\mathbb{X}\times\mathbb{X}\times{\bf N}$. So we see that also $D(DF)^2$ is non-negative, thus yielding
\begin{align*}
D V^- = D \int_\mathbb{X} (DF)^2d\mu = \int_\mathbb{X} D(DF)^2d\mu\geq 0.
\end{align*}
\end{rem}
\bigskip
{We conclude this section with a result that deals with the situation when $F$ is non-decreasing and the difference operator $DF$ is bounded. In this case}, we obtain a deviation inequality for the lower tail by controlling the random variable $V^+$. This is a Poisson space analogue of \cite[Theorem 13]{M_2006}.
\begin{thm}\label{vPlusLowerThm}
{Assume that $F\geq 0$ and }that for some $a> 0$ we have
\begin{align*}
{ 0\leq D_x F(\xi)\leq 1 \ \ \text{for any} \ \ (x,\xi)\in\mathbb{X}\times{\bf N}} \ \ \text{and almost surely} \ \ V^+(\eta) \leq aF(\eta).
\end{align*}
Then $F$ is integrable and for any $r\geq 0$ we have
\begin{align*}
\P(F\leq \mathbb{E} F - r) \leq \exp\left(-\frac{r^2}{2 \max(a,1) \mathbb{E} F}\right).
\end{align*}
\end{thm}
\section{Proofs}
We begin with the proof of the crucial logarithmic Sobolev type inequality, namely Proposition \ref{EntIneq}, that is the foundation of our techniques.
\begin{proof}[Proof of Proposition \ref{EntIneq}]
By \eqref{e:wu}, the inequality
\begin{align*}
\Ent(e^{\lambda F}) &\leq \mathbb{E} \left[ e^{\lambda F}\int_{\mathbb{X}}\psi(\lambda D_x F) \ d\mu(x)\right]
\end{align*}
holds. Now, since $\psi(0) = 0$, $\phi(0)=0$ and $\psi(z) = e^z \phi(-z)$ for any $z\in\mathbb{R}$, we have
\[
\psi(\lambda D_xF(\eta)) = \psi(\lambda D^I_xF(\eta)) + e^{\lambda D_xF(\eta)}\phi(-\lambda D^{I^c}_xF(\eta)).
\]
Hence, we compute
\begin{align*}
\Ent(e^{\lambda F}) &\leq \mathbb{E} \ \int_{\mathbb{X}} e^{\lambda F}\psi(\lambda D^I_x F) \ d\mu(x) + \mathbb{E} \int_{\mathbb{X}} e^{\lambda D_x F + \lambda F} \phi(-\lambda D^{I^c}_x F) \ d\mu(x)\\
&= \mathbb{E} \ \int_{\mathbb{X}} e^{\lambda F}\psi(\lambda D^I_x F) \ d\mu(x) + \mathbb{E} \int_{\mathbb{X}} e^{\lambda F(\eta + \delta_x)} \phi(-\lambda D^{I^c}_x F) \ d\mu(x)\\
&= \mathbb{E} \ \int_{\mathbb{X}} e^{\lambda F}\psi(\lambda D^I_x F) \ d\mu(x) + \mathbb{E} \int_\mathbb{X} e^{\lambda F} \phi(-\lambda D^{I^c}_x F(\eta -\delta_x)) d\eta(x).
\end{align*}
At this, the last equality holds by the Mecke formula \eqref{e:mecke}.
\end{proof}
The following lemma will be used occasionally in the upcoming proofs.
\begin{lem} \label{ExpLem}
Assume that for some $\beta\in\mathbb{R}$ we have $\mathbb{E} V_\beta^+ < \infty$ or $\mathbb{E} V_\beta^- < \infty$. Then $F$ is integrable.
\end{lem}
\begin{proof}
The proof uses a truncation argument that is standard in this context, see e.g. the proof of \cite[Proposition 3.1]{W_2000}. The statement for $V_\beta^-$ is proved in the same way than for $V_\beta^+$. Consider for any $n\in\mathbb{N}$ the truncation
\[
F_n = \min(\max(F,-n),n).
\]
Then $\mathbb{E} (F_n)^2<\infty$, hence the Poincar\'e inequality for Poisson point processes (see e.g. \cite[Remark 1.4]{W_2000}) together with the Mecke formula \eqref{e:mecke} yield
\begin{align*}
\mathbb{V} F_n \leq \mathbb{E} \int (D_x F_n(\eta))^2 d \mu(x) \leq \mathbb{E} \int (D_x F(\eta))^2d \mu(x) = \mathbb{E} V_\beta^+.
\end{align*}
Therefore, $\sup_{n\in\mathbb{N}}\mathbb{V} F_n \leq \mathbb{E} V_\beta^+ < \infty$. Now, if there is a subsequence $\{F_{n_k}\}_{k\in\mathbb{N}}$ satisfying $\lim_{k\to\infty}\mathbb{E} F_{n_k} = \pm\infty$, then $F_{n_k}\to \pm\infty$ in probability. This would be a contradiction to $F_n \to F$ in probability. We see that the family $\{\mathbb{E} F_n\}_{n\in\mathbb{N}}$ is bounded. Together with $\sup_{n\in\mathbb{N}}\mathbb{V} F_n < \infty$ this implies also $ \sup_{n\in\mathbb{N}}\mathbb{E} (F^2_n) < \infty$. Thus, the family $\{F_n\}_{n\in\mathbb{N}}$ is uniformly integrable. In particular, as desired, we have $\mathbb{E} |F| < \infty$.
\end{proof}
We continue with the proof of Theorem \ref{EntIneqs}. As in the proof of the product space version \cite[Theorem 2]{BLM_2003}, we also need \cite[Lemma 11]{M_2000}. This result states that for any $\lambda>0$ and any two random variables $X$ and $Y$ satisfying $\mathbb{E} (e^{\lambda X}), \mathbb{E} (e^{\lambda Y}) <\infty$, we have
\begin{align} \label{decoupling}
\frac{\lambda \mathbb{E}(X e^ {\lambda Y})}{\mathbb{E}(e^ {\lambda Y})} \leq \frac{\lambda \mathbb{E}(Y e^ {\lambda Y})}{\mathbb{E}(e^ {\lambda Y})} + \log \mathbb{E}(e^{\lambda X}) - \log \mathbb{E}(e^{\lambda Y}).
\end{align}
\begin{proof}[Proof of Theorem \ref{EntIneqs}]
We prove (\ref{EntIneq1}). We only deal with the case $\beta\neq 0$, whereas the case $\beta = 0$ can be obtained with by similar arguments. To prove the desired inequality we adapt the proof of \cite[Theorem 2]{BLM_2003} and combine this with arguments from the proof of \cite[Proposition 3.1]{W_2000}. Let $\phi$ and $\psi$ be as in Proposition \ref{EntIneq}. Then $\psi(z)/z^2$ and $\phi(z)/z^2$ are non-decreasing. Hence, for any $u\in(0,\lambda]$ we have
\begin{align*}
\psi(u z) & \leq \rlap{$(\psi(u\beta)/\beta^2)z^2$}\phantom{(\phi(-u\beta)/\beta^2)z^2} \leq u \Phi_\beta(u) z^2 \ \ \text{for} \ z \leq \beta,\\
\phi(-u z) &\leq (\phi(-u\beta)/\beta^2)z^2 \leq u \Phi_\beta(u) z^2 \ \ \text{for} \ z \geq \beta.
\end{align*}
Together with $\psi(0)=\phi(0)=0$ this gives
\begin{align*}
\psi(u D^{\leq\beta}_x F(\eta)) &\leq u \Phi_\beta(u) (D^{\leq\beta}_x F(\eta))^2,\\
\phi(-u D^{ >\beta}_x F(\eta-\delta_x)) &\leq u \Phi_\beta(u) (D^{ >\beta}_x F(\eta-\delta_x))^2.
\end{align*}
Hence, taking $I=\{(x,\xi) \in \mathbb{X}\times{\bf N} : D_xF(\xi) \leq \beta\}$, it follows from Proposition \ref{EntIneq} that
\begin{align*}
\Ent(e^{u F}) &\leq \mathbb{E} \left[e^{u F} \left(\int_{\mathbb{X}} \psi(u D^{\leq\beta}_x F(\eta)) \ d\mu(x) + \int_\mathbb{X} \phi(-u D^{>\beta}_x F(\eta-\delta_x)) d\eta(x)\right)\right]\\
&\leq u \Phi_\beta(u) \ \mathbb{E} \left[ e^{u F} \left(\int_{\mathbb{X}} (D^{\leq \beta}_x F(\eta))^2 \ d\mu(x) + \int_\mathbb{X} (D^{ >\beta}_x F(\eta-\delta_x))^2 d\eta(x)\right)\right]\\
&= u\Phi_\beta(u) \ \mathbb{E}(V_\beta^+e^{u F}).
\end{align*}
Moreover, taking $X =V_\beta^+/\theta$ and $Y = F$, it follows from (\ref{decoupling}) that
\begin{align*}
\frac{\mathbb{E}(V_\beta^+ e^ {u F})}{\mathbb{E}(e^ {u F})} \leq \frac{\theta \mathbb{E}(F e^ {u F})}{\mathbb{E}(e^ {u F})} + \frac{\theta}{u}\log \mathbb{E}(e^{u V_\beta^+ / \theta}) - \frac{\theta}{u}\log \mathbb{E}(e^{u F}).
\end{align*}
{Invoking the definition of the entropy, it follows from the last two displays that}
\begin{align*}
\frac{\mathbb{E}(F e^{u F})}{u \mathbb{E}(e^{u F})} - \frac{\log \mathbb{E} (e^{u F})}{u^2}
\leq
\Phi_\beta(u)\left(\frac{\theta \mathbb{E}(F e^ {u F})}{u\mathbb{E}(e^ {u F})} + \frac{\theta}{u^2}\log \mathbb{E}(e^{u V_\beta^+ / \theta}) - \frac{\theta}{u^2}\log \mathbb{E}(e^{u F})\right).
\end{align*}
Since by assumption $\Phi_\beta(u)\theta\leq \Phi_\beta(\lambda)\theta < 1$, the latter inequality is equivalent to
\begin{align*}
\frac{\mathbb{E}(F e^{u F})}{u \mathbb{E}(e^{u F})} - \frac{\log \mathbb{E} (e^{u F})}{u^2}
\leq
\frac{\Phi_\beta(u)\theta \log \mathbb{E}(e^{u V_\beta^+ / \theta})}{u^2 (1-\Phi_\beta(u) \theta)}.
\end{align*}
Defining $h(u) = \frac{1}{u} \log \mathbb{E}(e^{u F})$ and $g(u) = \log\mathbb{E}(e^{u V_\beta^+})$, the above estimate can be restated as follows:
\begin{align*}
h'(u) \leq \frac{\Phi_\beta(u)\theta g(u/\theta)}{u^2(1-\Phi_\beta(u)\theta)}
\end{align*}
for any $u\in(0,\lambda]$. Since $\lim_{u\to 0+} h(u) = \mathbb{E} F$, integration from $0$ to $\lambda$ gives
\begin{align}\label{diffineq}
h(\lambda) \leq \mathbb{E} F + \int_0^\lambda \frac{\Phi_\beta(u)\theta g(u/\theta)}{u^2(1-\Phi_\beta(u)\theta)} \ du.
\end{align}
It's a well known fact that the logarithm of a moment generating function is convex, hence $g$ is convex on the interval $[0,\lambda/\theta]$. In particular, we have for any $u\in (0,\lambda]$ that
\begin{align*}
\frac{g(u/\theta)}{u(1-\Phi_\beta(u)\theta)} \leq \frac{\tfrac{g(\lambda/\theta)}{\lambda} \ u}{u(1-\Phi_\beta(u)\theta)} \leq \frac{g(\lambda/\theta)}{\lambda(1-\Phi_\beta(\lambda) \theta)}.
\end{align*}
Hence,
\begin{align*}
\int_0^\lambda \frac{\Phi_\beta(u)\theta g(u/\theta)}{u^2(1-\Phi_\beta(u)\theta)} \ du \leq \frac{\theta g(\lambda/\theta)}{\lambda(1-\Phi_\beta(\lambda) \theta)} \int_0^\lambda \frac{\Phi_\beta(u)}{u} \ du.
\end{align*}
In the case $\beta < 0$, we bound the integral on the right hand side by $\Phi_\beta(\lambda) = \Psi_\beta(\lambda)$. This works since $\Phi_\beta(u)/u$ is non-decreasing. In the case $\beta > 0$, the integral can be explicitly computed {and one obtains $\int_0^\lambda (\Phi_\beta(u)/u) du = \phi(\lambda\beta)/(\lambda\beta^ 2) = \Psi_\beta(\lambda)$}. Combining this with (\ref{diffineq}) gives
\begin{align*}
\log \mathbb{E}(e^{\lambda F}) \leq \lambda\mathbb{E} F + \frac{\Psi_\beta(\lambda) \theta }{1-\Phi_\beta(\lambda) \theta} \ g(\lambda/\theta).
\end{align*}
This proves inequality (\ref{EntIneq1}). Repeating the above reasoning for $-F$ instead of $F$ where the set $I$ is replaced by its complement proves inequality (\ref{EntIneq2}).
To prove the second part of the theorem, assume that one of the conditions (i) to (iii) is satisfied. For $n\in\mathbb{N}$ consider the truncated random variables
\[
F_n = \min(\max(F,-n),n).
\]
We will now conclude that if $\mathbb{E}\exp(\lambda V_\beta^ + / \theta) < \infty$, then $F$ is integrable and the family of random variables
\begin{align}\label{famRV}
\{\exp(\lambda(F_n-\mathbb{E} F_n))\}_{n\in\mathbb{N}}
\end{align}
converges in probability to $\exp(\lambda(F-\mathbb{E} F))$ and is uniformly integrable. Thus, it follows that $\mathbb{E} \exp(\lambda(F-\mathbb{E} F)) < \infty$ and hence also $\mathbb{E}\exp(\lambda(F))<\infty$.
Integrability of $F$ follows from Lemma \ref{ExpLem} since the assumption $\mathbb{E}(\exp(\lambda V_\beta^+/\theta)) < \infty$ implies that $\mathbb{E} V_\beta^+ < \infty$.
By {dominated} convergence, integrability of $F$ now implies the convergence in probability of the sequence in (\ref{famRV}).
To prove the uniform integrability, first observe, that if (i), (ii) or (iii) holds, then
\begin{align*}
V_\beta^+(F_n)\leq V_\beta^+(F) = V_\beta^+.
\end{align*}
Also note that we can choose $\nu>1$ such that $\Phi_\beta(\nu\lambda)\nu\theta < 1$. Then $\mathbb{E}(e^{\lambda\nu F_n})<\infty$ for all $n\in\mathbb{N}$, so it follows from (\ref{EntIneq1}) that
\begin{align*}
\log \mathbb{E}(\exp(\lambda\nu(F_n-\mathbb{E} F_n))) &\leq \frac{\Psi_\beta(\nu\lambda)\nu \theta}{1-\Phi_\beta(\nu\lambda)\nu \theta} \log \mathbb{E} \left( \exp\left(\frac{\lambda V_\beta^+(F_n)}{\theta}\right)\right)\\
&\leq \frac{\Psi_\beta(\nu\lambda)\nu \theta}{1-\Phi_\beta(\nu\lambda)\nu \theta} \log \mathbb{E} \left( \exp\left(\frac{\lambda V_\beta^+}{\theta}\right)\right) < \infty.
\end{align*}
Denoting the map $x\mapsto x^\nu$ by $\Lambda$, the above inequality yields
\begin{align*}
\sup_{n\in\mathbb{N}}\mathbb{E}[\Lambda(\exp(\lambda(F_n-\mathbb{E} F_n)))] < \infty.
\end{align*}
By the Theorem of de la Vall\'ee-Poussin this implies uniform integrability of the family in (\ref{famRV}).
Repeating the above reasoning for $-F$ instead of $F$ where inequality (\ref{EntIneq2}) is used instead of (\ref{EntIneq1}) proves the corresponding statement for $V_\beta^-$.
\end{proof}
{
\begin{proof}[Proof of Corollary \ref{DevIneq1}]
It follows from Lemma \ref{ExpLem} that $F$ is integrable. Theorem \ref{EntIneqs} yields
\[
\log \mathbb{E} (e^{\lambda (F - \mathbb{E} F)}) \leq \inf_{\theta\in (0,1/\Phi_\beta(\lambda))}\frac{\Psi_\beta(\lambda) \lambda c}{1-\Phi_\beta(\lambda) \theta} = \Psi_\beta(\lambda) \lambda c.
\]
Markov's inequality now gives for any $\lambda>0$,
\[
\P(F \geq \mathbb{E} F + r) = \P(e^{\lambda (F-\mathbb{E} F)} \geq e^{\lambda r}) \leq \frac{\mathbb{E}(e^{\lambda (F-\mathbb{E} F)})}{e^{\lambda r}} \leq \exp\left(\Psi_\beta(\lambda) \lambda c -\lambda r\right).
\]
Optimizing in $\lambda$ yields the desired deviation bounds.
\end{proof}
}
\begin{proof}[Proof of Corollary \ref{cor2}]
Here we adapt and combine the proofs of \cite[Theorem 8 and Theorem 9]{BLM_2003}. {For $\alpha=0$, the statement follows directly from Theorem \ref{EntIneqs}, so let $\alpha\in(0,2)$.} Let $\gamma = 1-\alpha/2$. Then, {on the event $\{F\neq 0\}$, we have}
\begin{align*}
&\int_\mathbb{X} (D^+_x F^\gamma(\eta-\delta_x))^2 d\eta(x)\\ &= \int_\mathbb{X} \mathbbm{1}\{F(\eta)^\gamma \geq F(\eta-\delta_x)^\gamma >0 \} (F(\eta)^ \gamma - F(\eta - \delta_x)^ \gamma)^2 d\eta(x) \\
& \quad\quad + \int_\mathbb{X} \mathbbm{1}\{F(\eta)^\gamma \geq F(\eta-\delta_x)^\gamma { =0} \} F(\eta)^ {2\gamma} d\eta(x)
\\ &= \int_\mathbb{X} \mathbbm{1}\{F(\eta) \geq F(\eta-\delta_x) >0\}\left(\frac{F(\eta)}{F(\eta)^{1-\gamma}} - \frac{F(\eta - \delta_x)}{F(\eta - \delta_x)^ {1-\gamma}}\right)^2 d\eta(x)\\
& \quad\quad+ \int_\mathbb{X} \mathbbm{1}\{F(\eta) \geq F(\eta-\delta_x) { =0} \} F(\eta)^ {2\gamma} d\eta(x).
\end{align*}
Since $1-\gamma >0$, we have that $F(\eta) \geq F(\eta - \delta_x)$ implies $F(\eta)^{1-\gamma} \geq F(\eta - \delta_x)^{1-\gamma}$. Hence, the above expression does not exceed
\begin{align*}
&\int_\mathbb{X} \mathbbm{1}\{F(\eta) \geq F(\eta-\delta_x) >0\}\left(\frac{F(\eta)}{F(\eta)^{1-\gamma}} - \frac{F(\eta - \delta_x)}{F(\eta)^ {1-\gamma}}\right)^2 d\eta(x)\\
& \quad\quad+ \int_\mathbb{X} \mathbbm{1}\{F(\eta) \geq F(\eta-\delta_x) { =0} \} F(\eta)^ {2\gamma} d\eta(x)\\
= \ &\frac{1}{F(\eta)^{\alpha}}\int_\mathbb{X} \left(D^+_x F(\eta - \delta_x)\right)^2 d\eta(x).
\end{align*}
Quite similarly one obtains {that on the event $\{F\neq 0\}$,}
\begin{align*}
\int_{\mathbb{X}} (D^-_x F^\gamma(\eta))^2 \ d\mu(x) \leq \frac{1}{F(\eta)^\alpha} \int_{\mathbb{X}} (D^-_x F(\eta))^2 \ d\mu(x).
\end{align*}
Hence, it follows that {on the event $\{F\neq 0, V^+ \leq G F^\alpha\}$,}
\begin{align*}
V^+(F^\gamma) = \int_{\mathbb{X}} (D^-_x F^\gamma(\eta))^2 \ d\mu(x) + \int_\mathbb{X} (D^+_x F^\gamma(\eta-\delta_x))^2 d\eta(x) \leq \frac{V^+}{F(\eta)^\alpha} \leq G.
\end{align*}
{Moreover, it is easy to check that on the event $\{F=0, V^+\leq GF^\alpha\}$, one has that $V^+(F^\gamma) = 0 = V^+$. Therefore, by virtue of the assumption that almost surely $V^+\leq GF^\alpha$, it follows that almost surely $V^+(F^\gamma)\leq G$.} Applying Theorem \ref{EntIneqs} to the random variable $F^\gamma$ yields the result.
\end{proof}
\begin{proof}[Proof of Corollary \ref{DevIneq2}]
{For $\alpha=0$, the statement follows directly from Corollary \ref{DevIneq1} (ii), so let $\alpha\in(0,2)$.} Let $\gamma = 1 - \alpha/2$. Continuing in the same way as in the proof of Corollary \ref{cor2} yields {that almost surely}
\begin{align*}
V^+(F^\gamma) = \int_{\mathbb{X}} (D^-_x F^\gamma(\eta))^2 \ d\mu(x) + \int_\mathbb{X} (D^+_x F^\gamma(\eta-\delta_x))^2 d\eta(x) \leq c.
\end{align*}
We conclude that Corollary \ref{DevIneq1} (ii) applies to $F^\gamma$. So $F^\gamma$ is non-negative and has an exponentially decaying upper tail. Thus, by virtue of \cite[Lemma 3.4]{K_2002}, all moments of $F^\gamma$ exist. In particular, $F$ is integrable. As it was pointed out in \cite[p. 1588]{BLM_2003}, we can now write
\begin{align*}
\P(F\geq\mathbb{E} F + r) &= \P(F^ \gamma \geq (r + \mathbb{E} F)^ \gamma) \leq \P(F^ \gamma - \mathbb{E} (F^ \gamma) \geq (r + \mathbb{E} F)^ \gamma - (\mathbb{E} F)^ \gamma)\\
&\leq \exp\left(-\frac{((r+\mathbb{E} F)^{\gamma} - (\mathbb{E} F)^{\gamma})^2}{2c}\right).
\end{align*}
\end{proof}
We continue with the proof of Theorem \ref{arbSignThm}. To get prepared for this, we first establish the following lemma.
\begin{lem}\label{truncLem}
Let $n\in\mathbb{N}$ and consider $F_n = \min(\max(F,-n),n)$ and $V^+_{(n)}=V^+(F_n)$. Then for any real number $b\geq 0$, almost surely
\begin{align*}
F (V^+_{(n)} - b) \leq F_n (V^+ - b) \ \ \text{if} \ \ F,F_n\geq 0,\\
F (V^+_{(n)} - b) \geq F_n (V^+ - b) \ \ \text{if} \ \ F,F_n\leq 0.\\
\end{align*}
\end{lem}
\begin{proof}
It is easy to see that $V^+_{(n)} \leq V^+$. Hence, the desired statement holds on the event $\{F=F_n\}$. If $F\neq F_n$, then either $F_n = n < F$ or $F_n = -n > F$. The latter case implies $V^+_{(n)} = 0$ and $F,F_n \leq 0$, hence the desired statement holds. So consider the case $F_n = n < F$ and let $A=F/n$. Then the desired inequality is equivalent to
\begin{align*}
A V^+_{(n)} \leq V^+ +(A-1) b.
\end{align*}
Since $b\geq 0$ and $A>1$, the above inequality is implied by $AV^+_{(n)}\leq V^+$, i.e. by
\begin{align*}
A\left(\int_\mathbb{X} (n-F_n(\eta - \delta_x))_+^2 d\eta(x) + \int_\mathbb{X} (F_n(\eta + \delta_x ) - n)_-^2 d\mu(x)\right)\\
\leq \int_\mathbb{X} (An - F(\eta - \delta_x))_+^2 d\eta(x) + \int_\mathbb{X} (F(\eta + \delta_x) - An)_-^2 d\mu(x).
\end{align*}
To prove this, it suffices to conclude
\begin{align}
\label{ineq1}A(n-F_n(\eta-\delta_x))_+^2 \leq (An - F(\eta - \delta_x))_+^2,\\
\label{ineq2}A(F_n(\eta+\delta_x) - n)_-^2 \leq (F(\eta + \delta_x) - An)_-^2.
\end{align}
We prove (\ref{ineq1}). If $F(\eta - \delta_x)>n$, then
\begin{align*}
A (n-F_n(\eta - \delta_x))_+^2 = 0 \leq (An-F(\eta - \delta_x))_+^2.
\end{align*}
If $F(\eta - \delta_x)\leq n$, then
\begin{align*}
F(\eta - \delta_x) \leq F_n(\eta - \delta_x) =: m.
\end{align*}
Now, since $|m| \leq n$, we have $A n^2 - m^2 \geq 0$. This gives $(A^2- A)n^2 + (1-A)m^2\geq 0$, thus
\begin{align*}
(An-m)^2 = A^2 n^2 - 2 Anm + m^2 \geq An^2 - 2Anm + Am^2 = A(n-m)^2.
\end{align*}
Hence,
\begin{align*}
A(n-F_n(\eta-\delta_x))_+^2 \leq
(An-F_n(\eta-\delta_x))_+^2 \leq
(An-F(\eta-\delta_x))_+^2,
\end{align*}
where the last inequality follows from $F(\eta - \delta_x) \leq F_n(\eta - \delta_x)<An$. This proves (\ref{ineq1}) and analogously one obtains (\ref{ineq2}). The result follows.
\end{proof}
\begin{proof}[Proof of Theorem \ref{arbSignThm}]
For the case when $F$ is bounded, we adapt the proof of \cite[Theorem 5]{BLM_2003}. Here we can argue in the same way as in the beginning of the proof of Theorem \ref{EntIneqs} to obtain for any $u\in(0,\lambda]$,
\begin{align*}
\Ent(e^{u F}) \leq \frac{1}{2} u^2 \mathbb{E}(V^+ e^{u F}).
\end{align*}
Invoking the assumption on $V^+$ yields
\begin{align*}
u \mathbb{E}(F e^{u F}) - \mathbb{E}(e^{u F})\log \mathbb{E}(e^{u F}) \leq \frac{1}{2} u^2 (a\mathbb{E}(F e^{u F}) + b\mathbb{E}(e^{u F})).
\end{align*}
With $h(u) = \tfrac{1}{u} \log \mathbb{E}(e^{u F})$ this can be rearranged as
\begin{align*}
h'(u) \leq \frac{1}{2}(a \log(\mathbb{E} (e^{u F}))' + b).
\end{align*}
Integrating this from $0$ to $\lambda$ gives
\begin{align*}
h(\lambda) - \mathbb{E} F \leq \frac{1}{2}\left( a \log(\mathbb{E} (e^{\lambda F}) +\lambda b\right).
\end{align*}
Noting that $a\lambda<2$ and rearranging the above inequality, we obtain the result for the bounded case.
For the unbounded case, consider for any $n\in\mathbb{N}$ the truncated random variables
\begin{align*}
F_n = \min(\max(F, -n), n).
\end{align*}
It follows from the assumptions and Lemma \ref{truncLem} that almost surely
\begin{align*}
F(V^+_{(n)} - b) \leq aF_nF \ \ \text{if} \ \ F\geq 0,\\
F(V^+_{(n)} - b) \geq aF_nF \ \ \text{if} \ \ F\leq 0.
\end{align*}
Note also that for $F = 0 = F_n$ we have $V^+_{(n)} \leq V^+ \leq b$. Therefore, almost surely
\begin{align*}
V^+_{(n)} \leq a F_n + b,
\end{align*}
so the result holds for all $F_n$. By dominated convergence, the sequence $\mathbb{E} F_n$ is convergent, hence bounded above by some constant $C$. Moreover, we can choose a $\nu> 1$ such that $\nu\lambda < 2/a$. Thus, since we already proved that the result applies to all the $F_n$, we conclude
\begin{align*}
\sup_{n\in\mathbb{N}}\mathbb{E}[\exp(\lambda(F_n-\mathbb{E} F_n))^\nu] \leq \exp \left(\frac{\nu^2\lambda^2}{2 - a\nu\lambda} (a C + b) \right) < \infty.
\end{align*}
By the Theorem of de la Vall\'ee-Poussin this implies that the family of random variables
\begin{align*}
\{\exp(\lambda(F_n-\mathbb{E} F_n))\}_{n\in\mathbb{N}}
\end{align*}
is uniformly integrable. Continuing as in the proof of Theorem \ref{EntIneqs} gives
\begin{align*}
\lim_{n\to\infty}\mathbb{E} \exp(\lambda(F_n-\mathbb{E} F_n)) = \mathbb{E} \exp(\lambda(F-\mathbb{E} F))<\infty.
\end{align*}
We note again that the result is already proved for the $F_n$ and that $\mathbb{E} F_n \to \mathbb{E} F$ as $n\to\infty$. This concludes the proof of the first inequality.
The deviation inequality now follows using the inequality we just proved together with Markov's inequality and \cite[Lemma 11]{BLM_2003}.
\end{proof}
For the proof of Theorem \ref{LowerDevIneq} we use the following FKG inequality for Poisson point processes, taken from \cite[Lemma 2.1]{J_1984}, see also \cite[Theorem 1.4]{LP_2011}.
\begin{lem} \label{FKG}
Let $F$ and $G$ be bounded Poisson functionals and assume that
\begin{align*}
D_xF(\xi), D_xG(\xi)\geq 0 \ \ \text{for all} \ \ (x,\xi)\in\mathbb{X}\times{\bf N}.
\end{align*}
Then
\[
\mathbb{E}(FG)\geq (\mathbb{E} F)(\mathbb{E} G).
\]
\end{lem}
It was also remarked in \cite{J_1984} that under conditions like $F,G\geq 0$ or $\mathbb{E} F^2, \mathbb{E} G^2 < \infty$, the above result easily extends to unbounded functionals by monotone convergence. For our purpose we need the following extension.
\begin{cor} \label{FKGCor}
Let $F, G \geq 0$ be Poisson functionals. Assume that $F$ is bounded and $G$ is integrable. Moreover, assume that
\begin{align*}
D_xF(\xi)\leq 0 \ \ \text{and} \ \ D_xG(\xi)\geq 0 \ \ \text{for all} \ \ (x,\xi)\in\mathbb{X}\times{\bf N}.
\end{align*}
Then
\[
\mathbb{E}(FG)\leq (\mathbb{E} F)(\mathbb{E} G).
\]
\end{cor}
\begin{proof}
Since $F$ is bounded, it follows from $\mathbb{E} G<\infty$ that also $\mathbb{E}(FG) < \infty$. Now consider for any $n\in\mathbb{N}$ the truncations $G_n = \min(G, n)$. Then we have almost surely $G_n\to G$ and $F G_n \to FG$ as $n\to \infty$. By monotone convergence, $\mathbb{E} G_n\to \mathbb{E} G$ and $\mathbb{E}(F G_n)\to\mathbb{E}(FG)$ as $n\to\infty$. It follows from Lemma \ref{FKG} that for any $n\in\mathbb{N}$,
\begin{align*}
\mathbb{E}(F G_n) \leq (\mathbb{E} F)(\mathbb{E} G_n).
\end{align*}
The result follows.
\end{proof}
\pagebreak
The following proof is inspired by ideas from the proof of \cite[Theorem 6]{BLM_2003}.
\begin{proof}[Proof of Theorem \ref{LowerDevIneq}]
For any $n\in\mathbb{N}$ consider the truncations
\begin{align*}
F_n = \min(\max(F, -n), n).
\end{align*}
Then the $F_n$ are again non-decreasing. Let $\lambda < 0$. It follows from Proposition \ref{EntIneq} with $I = \mathbb{X}\times{\bf N}$ that for any $u\in[\lambda,0)$ we have
\begin{align*}
\Ent(e^{u F_n}) \leq \mathbb{E} \left[e^{u F_n} \int_{\mathbb{X}} \psi(u D_x F_n) \ d\mu(x)\right].
\end{align*}
Since $\psi(-z) \leq (1/2)z^2$ for $z\geq 0$, the right hand side of the above expression does not exceed
\begin{align*}
\frac{1}{2}\mathbb{E} \left[ e^{u F_n} \int_{\mathbb{X}} (u D_x F_n)^2 \ d\mu(x)\right] = \frac{1}{2} u^ 2 \ \mathbb{E} (e^{u F_n} V^-(F_n)).
\end{align*}
We have $V^-(F_n)\leq V^-$ almost surely, hence $\mathbb{E}(e^{uF_n}V^-(F_n))$ in the above display can be upper bounded by $\mathbb{E}(e^{uF_n}V^-)$. Now, since $F_n$ is non-decreasing and $u<0$, the functional $e^{u F_n}$ is non-increasing and bounded. Moreover, by assumption the functional $V^-$ is non-decreasing and $\mathbb{E} V^- <\infty$. Hence, by Corollary \ref{FKGCor} we have
\begin{align*}
\mathbb{E} (e^{u F_n} V^-) \leq \mathbb{E} (e^{u F_n}) \ \mathbb{E} V^-.
\end{align*}
It follows that
\[
h'(u)\leq \frac{1}{2}\mathbb{E} V^- \ \ \text{where} \ \ h(u) = \frac{1}{u} \log\mathbb{E}(e^{u F_n}).
\]
Integrating from $\lambda$ to $0$ yields
\begin{align*}
\log \mathbb{E}[\exp(\lambda(F_n-\mathbb{E} F_n))] \leq \frac{1}{2}\lambda^2 \mathbb{E} V^-.
\end{align*}
Since $\mathbb{E} V^- < \infty$, by Lemma \ref{ExpLem} we have $\mathbb{E} |F|<\infty$. Thus, applying the Theorem of de la Vall\'ee-Poussin similarly as in the proof of Theorem \ref{EntIneqs}, we conclude that the inequality in the last display also holds for the random variable $F$. Using Markov's inequality and optimizing in $\lambda$ yields the result.
\end{proof}
To prove the statement of Theorem \ref{vPlusLowerThm} for bounded $F$, we adapt the proof of the product space version \cite[Theorem 13]{M_2006}. To extend the result to unbounded $F$, Lemma \ref{truncLem} is used similarly as it was done in the proof of Theorem \ref{arbSignThm}.
\begin{proof}[Proof of Theorem \ref{vPlusLowerThm}]
First consider the case when $F$ is bounded. Let $\lambda < 0 $ and $u\in[\lambda,0)$. Then by Proposition \ref{EntIneq} {with $I=\emptyset$} we have
\begin{align*}
\Ent(e^{u F}) &\leq \mathbb{E} \left[ e^{u F}\int_\mathbb{X} \phi(-u D_x F(\eta-\delta_x)) d\eta(x)\right].
\end{align*}
Moreover, since $0\leq D F \leq 1$, we have $-u D_x F(\eta-\delta_x) \leq -u$ and since the map $z\mapsto \phi(z)/z^2$ is increasing, this implies
\begin{align*}
\int_\mathbb{X} \phi(-u D_x F(\eta-\delta_x)) d\eta(x) &=u^2 \int_\mathbb{X} \frac{\phi(-u D_x F(\eta-\delta_x))}{u^2 D_x F(\eta-\delta_x)^2} D_x F(\eta-\delta_x)^2 d\eta(x)\\
&\leq u^2 \int_\mathbb{X} \frac{\phi(-u) }{u^2} D_x F(\eta-\delta_x)^2 d\eta(x).
\end{align*}
Now, since $V^+ \leq aF$, we obtain
\begin{align*}
\Ent(e^{u F}) \leq \phi(-u) \mathbb{E}(e^{u F} V^+) \leq \phi(-u) a \mathbb{E}(Fe^{u F}).
\end{align*}
Dividing by $u^2\mathbb{E}(e^{u F})$ and integrating from $\lambda$ to $0$ yields
\begin{align*}
\mathbb{E} F - \frac{1}{\lambda} \log\mathbb{E}(e^{\lambda F}) \leq -\frac{\phi(-\lambda)}{\lambda^ 2} a \log\mathbb{E}(e^ {\lambda F}).
\end{align*}
{Since $1-a\phi(-\lambda)/\lambda>1$}, this can be rearranged as
\begin{align*}
\log\mathbb{E}[\exp(\lambda(F-\mathbb{E} F))] \leq \lambda^2 \frac{a\phi(-\lambda)/\lambda^2}{1- a\phi(-\lambda)/\lambda} \mathbb{E} F.
\end{align*}
Similarly as in the proof of Theorem \ref{arbSignThm}, the above inequality can be extended to the case when $F$ is unbounded. Here one should notice that according to Corollary \ref{DevIneq2}, the condition $V^+\leq aF$ guarantees $\mathbb{E}|F|<\infty$. It was pointed out in \cite{M_2006} that for any $\lambda<0$,
\begin{align*}
\frac{a\phi(-\lambda)/\lambda^2}{1- a\phi(-\lambda)/\lambda} \leq \frac{\max(a,1)}{2}.
\end{align*}
Markov's inequality now gives
\begin{align*}
\P(F\leq \mathbb{E} F - r) \leq \mathbb{E}(e^{\lambda(F-\mathbb{E} F)})e^{\lambda r} \leq \exp\left(\lambda^2 \frac{\max(a,1)}{2} \mathbb{E} F + \lambda r\right).
\end{align*}
Optimizing in $\lambda$ concludes the proof.
\end{proof}
\section{Applications to U-Statistics} \label{AppUstat}
{
\subsection{General remarks} The aim of the present section is to investigate the concentration properties of Poisson U-statistics. For this purpose, we need to specialize the very general framework that was in order so far. Throughout this section, the intensity measure $\mu$ on the space $\mathbb{X}$ is assumed to be non-atomic, that is, $\{x\}\in \mathcal{X}$ and $\mu(\{x\}) = 0$, for every $x\in \mathbb{X}$. This assumption is equivalent to the fact that the Poisson process $\eta$ on $\mathbb{X}$ is \emph{simple}, meaning that almost surely $\eta(\{x\})\leq 1$ for all $x\in\mathbb{X}$. It is common practice in this setting to identify the simple point process $\eta$ with its support, which now corresponds to a random set in $\mathbb{X}$. Plainly, the integral of a map $f:\mathbb{X}\to\mathbb{R}$ with respect to $\eta$ is now exactly given by the (possibly infinite) sum
\begin{align*}
\int_\mathbb{X} f \ d\eta = \sum_{x\in\eta} f(x).
\end{align*}
\begin{rem} \label{representativeRem}
Consider a Poisson functional $F$ together with some representative $\mathfrak{f}:{\bf N}\to\mathbb{R}$. For any $\xi\in{\bf N}$, we denote by $[\xi]$ the integer-valued measure uniquely determined by its value on singletons via the relation $[\xi](\{x\}) = \mathbbm{1}\{\xi(\{x\})>0\}$, for all $x\in\mathbb{X}$. Then, since $\eta$ is simple, we have that almost surely $F = \mathfrak{f}([\eta])$. It follows that another representative of $F$ is given by $\mathfrak{f}':{\bf N}\to\mathbb{R}$, where $\mathfrak{f}'(\xi) = \mathfrak{f}([\xi])$. Therefore, without loss of generality, we can assume that $F(\xi)=F([\xi])$ for all $\xi\in{\bf N}$, that is, given an arbitrary functional $\xi \mapsto F(\xi)$, in this section we will systematically select a representative of $F$ that only depends on $\xi$ via the mapping $\xi\mapsto [\xi]$. With this convention, one has that $D_xF(\xi) = D_x F([\xi])$, and also that $D_x F(\xi) = 0$ whenever $\xi(\{x\})>0$. Finally we observe that, again by virtue of the above convention and in accordance with the content of Remark \ref{r:talich}, the fact that the quantity $D_xF(\xi)$ verifies some property $\mathcal{P}$ for every $x\in \mathbb{X}$ and every $\xi \in {\bf N}$ is equivalent to the fact that $\mathcal{P}$ is verified for all $(x,\xi)\in \mathbb{X}\times{\bf N}$ such that $\xi$ charges each singleton with a mass at most equal to 1.
\end{rem}
We now recall some relevant definitions. Let $f:\mathbb{X}^k \to \mathbb{R}_{\geq 0}$ be a symmetric measurable map and define the functional $S_f:{\bf N}\to[0,\infty]$ by
\begin{align}\label{e:1}
S_f(\xi) = \sum_{\textbf{x}\in\xi_{\neq}^k} f(\textbf{x}).
\end{align}
A \emph{(Poisson) U-statistic} $F$ of order $k$ with kernel $f$ is a random variable such that almost surely $F = S_f(\eta)$. According to the Slyvniak-Mecke formula \eqref{e:smecke}, the expectation of a U-statistic $F$ is given by
$$
E[F] = \int_\mathbb{X}\cdots \int_\mathbb{X} f(x_1,...,x_m) d\mu(x_1) \cdots d\mu(x_m);
$$
see e.g. \cite[Section 3]{RS_2013} for more details as well as for an introduction to U-statistics with kernels that may have arbitrary sign.
\subsection{Choice of a representative} \label{s:representative}
In order to apply results from Section \ref{DIPF} to a Poisson U-statistic $F$ with kernel $f\geq 0$, we first need to choose a suitable representative of $F$ as defined in Section \ref{s:frame}. Whenever the considered U-statistic $F$ is almost surely finite, we can choose as a representative of $F$ the map $\mathfrak{f}:{\bf N}\to\mathbb{R}$ defined by $\mathfrak{f}(\xi)=S_f(\xi)$ if $S_f(\xi)<\infty$ and $\mathfrak{f}(\xi)=0$ if $S_f(\xi)=\infty$.
In order to avoid technical problems arising from the choice of this representative, we will often assume that a given U-statistic with kernel $f$ is \emph{well-behaved}. By this we mean that there exists a measurable set $B\subseteq {\bf N}$ with $\P(\eta\in B)=1$, such that
\begin{enumerate}
\item $S_f(\xi)<\infty$ for all $\xi\in B$,
\item $\xi+\delta_x\in B$ whenever $\xi\in B$ and $x\in\mathbb{X}$,
\item $\xi-\delta_x\in B$ whenever $\xi\in B$ and $x\in\xi$.
\end{enumerate}
If $F$ is well-behaved, then we will choose as a representative of $F$ the map $\mathfrak{f}:{\bf N}\to \mathbb{R}$ defined by
$\mathfrak{f}(\xi) = S_f(\xi)$ if $\xi\in B$ and $\mathfrak{f}(\xi)=0$ if $\xi\in B^c$. Then, for any $(x,\xi)\in\mathbb{X}\times{\bf N}$ one has $D_xF(\xi) = S_f(\xi+\delta_x) - S_f(\xi) < \infty$ if $\xi\in B$ and $D_xF(\xi) = 0$ if $\xi\in B^c$.
Note that by virtue of (\ref{xiNeq}) and (\ref{e:1}), the above choices of a representative imply $F(\xi) = F([\xi])$ for all $\xi\in{\bf N}$ which is consistent with Remark \ref{representativeRem}. Finally, note that U-statistics that arise in typical applications (in particular, all $U$-statistics considered in this paper) are usually well-behaved in the sense described above.
\subsection{General results}
We will use an explicit expression for the difference operator of a U-statistic that was established in \cite{RS_2013}.
The following result gathers together several results from \cite[Lemma 3.3 and Theorem 3.6]{RS_2013}, in a form that is adapted to our setting.
\begin{prop} \label{DUstatProp} Let the above assumptions and notation prevail, let $F$ be a U-statistic with non-negative kernel $f$ and let $S_f$ be as in (\ref{e:1}). Then, for any $\xi\in{\bf N}$ and $x\in\xi$, one has
\[
S_f(\xi)-S_f(\xi-\delta_x) = k F(x,\xi) \ \ \text{whenever} \ \ S_f(\xi)<\infty,
\]
where, for any $\xi \in {\bf N}$ and every $x\in\xi$ such that $\xi(\{x\})=1$, the \emph{local version} of $F$ is defined as
\begin{align}\label{e:genug}
F(x,\xi) := \sum_{\textbf{y}\in(\xi\setminus x)^{k-1}_{\neq}} f(x,\textbf{y}),
\end{align}
where $\xi\setminus x$ is shorthand for the set obtained by deleting $x$ from the support of $\xi$, and $F(x,\xi)=0$ whenever $\xi(\{x\})>1$. Moreover, if $\mathbb{E} F^2<\infty$, then $f\in L^1(\mu^k)\cap L^2(\mu^k)$.
\end{prop}
{
As a direct consequence of the above result together with our canonical choices of a representative, described in Section \ref{s:representative}, we obtain:
\begin{cor}\label{DCor}
Let $F$ be a U-statistic with non-negative kernel $f$. Then the following statements hold:
\begin{enumerate}
\item If $F$ is almost surely finite, then there exists a measurable set $B\subseteq {\bf N}$ that satisfies $\P(\eta\in B)=1$ such that for any $\xi\in B$ and $x\in\xi$, the local version $F(x,\xi)$ is finite and
\begin{align*}
D_xF(\xi - \delta_x) = k F(x,\xi).
\end{align*}
\item If $F$ is well-behaved, then there exists a measurable set $B\subseteq {\bf N}$ that satisfies $\P(\eta\in B)=1$ such that the following holds:
\medskip
\begin{enumerate}
\item For any $\xi\in B$, $x\in\xi$ and $z\in\mathbb{X}$, the local versions $F(x,\xi)$ and $F(z,\xi+\delta_z)$ are finite, and moreover
\begin{align*}
D_xF(\xi-\delta_x) = k F(x,\xi) \ \ \text{and} \ \ D_zF(\xi) = kF(z,\xi+\delta_z).
\end{align*}
\item For any $\xi\in B^c$, $x\in\xi$ and $z\in\mathbb{X}$, one has
\begin{align*}
D_xF(\xi-\delta_x) = 0 = D_zF(\xi).
\end{align*}
\end{enumerate}
\end{enumerate}
\end{cor}
}
The previous Corollary \ref{DCor} implies that, if $F$ is an almost surely finite U-statistic with kernel $f\geq 0$, then almost surely
\begin{align*}
V^+ &= k^2 \sum_{x\in \eta} F(x,\eta)^2.
\end{align*}
If $F$ is in addition well-behaved, then almost surely
\begin{align*}
V^- &= k^2 \int_\mathbb{X} F(x,\eta+\delta_x)^2 d \mu(x).
\end{align*}
We have therefore the following consequences of Corollary \ref{DevIneq2} and Theorem \ref{LowerDevIneq}.
\begin{cor}
Consider an almost surely finite U-statistic $F$ of order $k$ with non-negative kernel $f$. Assume that for some $\alpha\in[0,2)$ and $c>0$ we have almost surely
\begin{align*}
\sum_{x\in \eta} F(x,\eta)^2 \leq c F^\alpha.
\end{align*}
Then $F$ is integrable and for all $r> 0$,
\begin{align*}
\P(F\geq\mathbb{E} F + r) \leq \exp\left(-\frac{((r+\mathbb{E} F)^{1-\alpha/2} - (\mathbb{E} F)^{1-\alpha/2})^2}{2ck^2}\right).
\end{align*}
\end{cor}
\begin{cor} \label{LowerDevIneqUStat}
Consider a well-behaved U-statistic $F$ of order $k$ with non-negative kernel $f$. Assume that
\begin{align}\label{e:2}
V := \mathbb{E} \int_\mathbb{X} F(x,\eta+\delta_x)^2 d \mu(x)<\infty.
\end{align}
Then, for all $r> 0$ we have
\begin{align*}
\P(F\leq\mathbb{E} F -r) \leq \exp\left(-\frac{r^2}{2k^2V}\right).
\end{align*}
\end{cor}
\begin{proof}
We have $\mathbb{E} V^-(F) = k^2 \mathbb{E} V < \infty$ and since $F$ is well-behaved, it follows from {Corollary \ref{DCor} (ii)} that $D_xF(\xi)\geq 0$ for any $(x,\xi)\in\mathbb{X}\times{\bf N}$. So the result follows from Theorem \ref{LowerDevIneq} together with Remark $\ref{incVRem}$ once we proved that $DD F \geq 0$. According to \cite{RS_2013}, and since $F$ is well-behaved, for any $(z,x,\xi)\in\mathbb{X}\times\mathbb{X}\times{\bf N}$, the second iteration of the difference operator either satisfies $D_zD_xF(\xi) = 0$ or it can be written as
\begin{align*}
D_zD_x F(\xi) = k(k-1) \sum_{\textbf{y}\in\xi^{k-2}_{\neq}} f(z,x,\textbf{y}).
\end{align*}
The right-hand side of the above display is non-negative since $f\geq 0$.
\end{proof}
}
\subsection{Computing $V$ in formula \eqref{e:2}}\label{ss:zut}
We will now provide a direct proof that condition \eqref{e:2} is equivalent to the fact that $F$ is a square-integrable U-statistic and that one can obtain a rather explicit expression for $V$ in terms of some set of auxiliary kernels built from $f$.
\begin{defi}\label{d:fi} Let $f$ be a symmetric element of $L^1(\mu^k)$, for some $k\geq 1$. For $i=1,...,k$, we define the kernels $f_i$ as follows:
\begin{equation}\label{e:kw}
f_i(y_1,...,y_i) := \binom{k}{i} \int_{\mathbb{X}^{k-i}} f(y_1,...,y_i, z_1,...,z_{k-i}) d\mu^{k-i}(z_1,...,z_{k-i}),
\end{equation}
if the integral on the right-hand side is well defined, and $f_i(y_1,...,y_i)= 0$ otherwise. Observe that, since $f$ is in $L^1(\mu^k)$, then the class of those $(y_1,...,y_i)$ such that the integral on the right-hand side of \eqref{e:kw} is not defined has measure $\mu^i$ equal to zero, for every $i=1,...,k$. Plainly, each $f_i$ is a symmetric mapping from $\mathbb{X}^i$ into $\mathbb{R}$ and $f_i \in L^1(\mu^i)$, for every $i=1,...,k$, and $f_k = f$ by definition. \end{defi}
The upcoming result provides new necessary and sufficient conditions for the square-integrability of U-statistics. {Although the investigations in the present paper (and hence also in the result below) are restricted to U-statistics with non-negative kernels, we stress that this assumption is not needed in the forthcoming proof, and thus, after appropriately adapting the notion of a well-behaved U-statistic for kernels with arbitrary sign, the presented characterization for square-integrable U-statistics also applies when the kernels are not necessarily assumed to be non-negative.}
\begin{prop}[(Characterization of square-integrable $U$-statistics)]\label{p:p} Consider a {well-behaved} U-statistic $F$ of order $k\geq 1$, with {non-negative} kernel $f\in L^1(\mu^k)$. Then, the following assertions are equivalent:
\begin{enumerate}
\item $F$ is square-integrable;
\item for every $i=1,...,k$, $f_i\in L^2(\mu^i)\cap L^1(\mu^i) $, where the kernels $f_i$ have been introduced in Definition \ref{d:fi};
\item $V<\infty$, where $V$ is defined in \eqref{e:2}.
\end{enumerate}
If either one of conditions {\rm (i)}, {\rm (ii)} or {\rm (iii)} is verified, then
\begin{equation}\label{e:vv}
k^2\, V = \sum_{i=1}^k ii! \| f_i\|^2_{L^2(\mu^i)} \quad\mbox{and}\quad \mathbb{V} F = \sum_{i=1}^k i! \| f_i\|^2_{L^2(\mu^i)},
\end{equation}
so that, in particular, $V\leq k^{-1} \times \mathbb{V} F$.
\end{prop}
\begin{proof}
{}[Step 1: (i) $\to$ (ii), (iii) ] According to \cite[Theorem 3.6]{RS_2013}, if $F$ is a U-statistic as in the statement and $F$ is square-integrable, then necessarily $f_i\in L^1(\mu^{i})\cap L^2(\mu^{i})$ for every $i=1,...,k$, and moreover $F$ admits the following representation:
$$
F=\mathbb{E} F + \sum_{i=1}^k I_i(f_i),
$$
where $I_i$ denotes a multiple Wiener-It\^o integral of order $i$, with respect to the compensated Poisson measure $\hat{\eta} = \eta-\mu$ (see e.g. \cite[Chapter 5]{PT_2010} for definitions). Note that, exploiting the standard orthonormality properties of multiple integrals, one has also that
$$
\mathbb{V} F = \sum_{i=1}^k i! \| f_i\| ^2_{L^2(\mu^i)},
$$
which corresponds to the first relation in \eqref{e:vv}. Combining \cite[Theorem 3.3]{LP_2011} with the previous discussion, one also infers that, if $F$ is square-integrable, then a version of the add-one cost operator $DF$ is given by
\begin{equation}\label{e:gum}
D_xF = \sum_{i=1}^k i I_{i-1} (f_i(x, \cdot) ),
\end{equation}
where $I_{i-1}(f_i(x, \cdot))$ indicates a multiple Wiener-It\^o integral of order $i-1$, with respect to $\hat\eta = \eta-\mu$, of the kernel $f_i(x,\cdot) : \mathbb{X}^{i-1} \to \mathbb{R}$, obtained from $f_i$ (see Definition \ref{d:fi}) by setting one of the variables in its argument equal to $x$; observe that, as usual, the right-hand side of \eqref{e:gum} is implicitly set equal to zero on the exceptional set of those $x\in \mathbb{X}$ such that $f_i(x, \cdot)\notin L^2(\mu^{i-1})$ for at least one $i\in \{1,...,k\}$. Exploiting the standard orthonormality properties of multiple integrals, one has therefore that
\begin{equation}\label{e:zo}
\mathbb{E}[(D_xF)^2] = \sum_{i=1}^k i^2 (i-1)! \| f_i(x,\cdot)\|^2_{L^2(\mu^{i-1})},
\end{equation}
so that the conclusion (as well as the explicit expression of $k^2V =\mathbb{E} \int_\mathbb{X} (D_xF)^2 d\mu(x)$ appearing in \eqref{e:vv}) follows from an application of the Fubini Theorem.
\smallskip
[Step 2: (ii) $\to$ (i)] Assume that, for every $i=1,...,k$, $f_i\in L^2(\mu^i)\cap L^1(\mu^i)$. Then, according to \cite[Theorem 4.1]{surg} the multiple integral $I_i(f_i)$ is a well-defined square-integrable random variable, and moreover
$$
I_i(f_i) = \sum_{j=0}^i(-1)^{i-j} \binom{i}{j} \sum_{(x_1,...,x_j)\in \eta^j_{\neq}} f_i^{(j)}(x_1,...,x_j),
$$
where $f_i^{(j)} :=\binom{k}{i} \times \binom{k}{j} ^{-1}\times f_j$, and each (possibly infinite) sum in the previous expression converges in $L^1(\P)$. Now write
$$
u(j) := \sum_{(x_1,...,x_j)\in \eta^j_{\neq}} f_{j}(x_1,...,x_j), \quad j=0,...,k.
$$
The previous discussion yields that
\begin{eqnarray*}
\sum_{i=0}^k I_i(f_i) &=& \sum_{i=0}^k \binom{k}{i} \sum_{j=0}^i(-1)^{i-j} \binom{i}{j} u(j)\\
& =& \sum_{j=0}^k u(j)\left\{ \sum_{i=j}^k \binom{k}{i}\binom{i}{j} (-1)^{i-j}\right\} = u(k)=F,
\end{eqnarray*}
where we have used the fact that the sum $\sum_{i=j}^k \binom{k}{i}\binom{i}{j} (-1)^{i-j}$ equals one if $j=k$, and vanishes otherwise. It follows that $F$ is square-integrable, since it is equal to a finite sum of square-integrable random variables.
\smallskip
[Step 3: (iii) $\to$ (ii)] If $V<\infty$, then there exists a measurable set $B\subset \mathbb{X}$ such that $\mu(B^c)=0$, and $\mathbb{E}(D_xF)^2<\infty$, for every $x\in B$. {Using \cite[Lemma 3.5, Theorem 3.6]{RS_2013} together with the {fact that, since $F$ is well-behaved,} $D_xF$ is the {(well-behaved)} U-statistic of order $k-1$ defined in Proposition \ref{DUstatProp}}, we immediately deduce that, for $x\in B$, one has that (adopting the same notation as in Step 1) $f_i(x, \cdot) \in L^2(\mu^{i-1})$, and also
$$
D_xF = \sum_{i=1}^k i I_{i-1} (f_i(x, \cdot) ).
$$
The conclusion follows by using once again \eqref{e:zo} and the Fubini Theorem.
\end{proof}
\begin{rem} According e.g. to \cite[Lemma 3.1]{PT_alea}, the condition $\mathbb{E} \int_\mathbb{X} (D_xF)^2 d\mu(x)<\infty$ is equivalent to the fact that $F$ belongs to the domain of the {\it Malliavin derivative} associated with $\eta$. This fact is consistent with the fact that {square-integrable} $U$-statistics have a finite Wiener-It\^o chaotic expansion, and therefore belong automatically to the domain of the Malliavin derivative.
\end{rem}
An application of Proposition \ref{p:p} to the estimation of lower tails for edge-counting in random geometric graphs (involving in particular U-statistics of order $k=2$) is presented in Section \ref{ss:eclower}.
\section{Applications to edge counting}\label{s:edge}
In this section, we let $\eta$ denote a Poisson point process on $(\mathbb{R}^d, \mathcal{B}(\mathbb{R}^d))$, with intensity given by a Borel measure $\mu$ (in particular, $\mu(K)<\infty$ for every compact set $K$). {We also assume again} that $\mu$ has no atoms, that is, $\mu(\{x\}) = 0$ {for every $x\in \mathbb{R}^d$}. For a fixed $\rho>0$, we shall consider the graph $\mathfrak{G}$ (often called the {\it Gilbert graph}, or the {\it disk Graph} with radius $\rho$ associated with $\eta$) obtained as follows: the vertex set of $\mathfrak{G}$ is given by the points in the support of $\eta$, and two vertices $x, y$ are linked by an edge {(in symbols, $x\leftrightarrow y$)} whenever $0< \lVert x-y \rVert \leq \rho$ (in particular, $\mathfrak{G}$ has no loops). For technical reasons clarified below, we will assume for the rest of the section that the following condition on $\mu$ is verified: denoting $B(x,\rho)$ the {closed} ball of radius $\rho$ centered in $x$,
\begin{equation}\label{e:r}
\int_{\mathbb{R}^d} \, \mu(B(x,\rho)) \, d\mu(x)<\infty.
\end{equation}
Relation \eqref{e:r} is verified whenever $\mu(\mathbb{R}^d)<\infty$, but such a finiteness condition is not necessary for \eqref{e:r} to hold \footnote{ Consider for instance the measure $\mu$ on $\mathbb{R}^2$ having density $p(x)=(\lVert x\rVert + 1)^{-2}$ together with an arbitrary radius $\rho>0$}. Note that, if $\mu$ is Borel and \eqref{e:r} is in order, then the mapping $x\mapsto \mu(B(x,\rho))$ is necessarily bounded. To see this, choose $\gamma>0$ such that the ball $B(0,\rho)$ can be written as a union of $\lfloor 1/\gamma\rfloor$ many sets with diameter less than $\rho$. Then the pigeonhole principle yields that for any $y\in\mathbb{R}^d$ we can choose a set $C_y\subset B(y,\rho)$ satisfying: (i) $C_y\subseteq B(x,\rho)$ for all $x\in C_y$, and (ii) $\mu(C_y) \geq \gamma \mu(B(y,\rho))$. Now,
\begin{align*}
\int_{\mathbb{R}^d} \mu(B(x,\rho)) d\mu(x) &\geq \sup_{y\in\mathbb{R}^d} \int_{C_y} \mu(B(x,\rho))d\mu(x) \geq \gamma \sup_{y\in\mathbb{R}^d} \int_{C_y} \mu(B(y,\rho)) d\mu(x)\\
& = \gamma \sup_{y\in\mathbb{R}^d} \mu(C_y) \mu(B(y,\rho)) \geq \gamma^2 \sup_{y\in\mathbb{R}^d} \mu(B(y,\rho))^2.
\end{align*}
\medskip
Originally introduced in 1959 by Gilbert in the seminal work \cite{G_1959}, the disk graph $\mathfrak{G}$ is the archetypical example of a {\it random geometric graph}. Since then, the study of such an object has been at the center of a formidable collective effort, both at a theoretical and applied level. We refer the reader to the fundamental monograph \cite{Penrose_book} for a detailed overview of the literature on Gilbert graphs up to the year 2003. Recent developments that are relevant for our work are discussed e.g. in \cite{BP_2012, DFRV, LRP1, LRP2, RS_2013, RST_2013, ST_spa2012}.
\medskip
In this section, we will provide new concentration estimates for the random variable $$N =N(\eta):= \# \big\{\{x,y\}\subseteq \eta : x\leftrightarrow y\big\}, $$ corresponding to the number of edges of $\mathfrak{G}$. It is immediately seen that $N$ is a Poisson U-statistic of order $2$ with positive kernel $f(x,y) = \tfrac{1}{2} \mathbbm{1}\{\lVert x-y \rVert \leq \rho\}$. In particular, the Slivniak-Mecke formula \eqref{e:smecke} together with a standard use of the Fubini Theorem yields that the assumption \eqref{e:r} is actually equivalent to integrability of $N$ and that
$$
\mathbb{E} N =\frac12 \int_{\mathbb{R}^d} \mu(B(x,\rho)) \, d\mu(x).
$$
We also see that assumption \eqref{e:r} implies that $N<\infty$ almost surely, yielding in turn that $N$ is well-behaved.
\subsection{Preparation: optimal rates} \label{optimalExp}
Let the above notation and assumptions prevail. In the forthcoming Section \ref{ss:rggut}, we will provide estimates for the upper tail of $N$ having the form
\begin{equation}\label{e:ir}
\P(N\geq \mathbb{E} N + r)\leq \exp(-I(r)), \quad r>0,
\end{equation}
where $r\mapsto I(r)$ is a positive mapping verifying
\begin{equation}\label{e:rc}
\lim_{r\to\infty}I(r) = \infty.
\end{equation}
The next statement contains a universal necessary condition on the asymptotic behaviour of $I(r)$.
\begin{prop}\label{p:ny}
Let $I(r)$ verify \eqref{e:ir} and \eqref{e:rc}. Then,
$$
\limsup_{r\to \infty} \frac{I(r)}{r^{1/2} \log r}\leq \frac{1}{\sqrt{2}}.
$$
\end{prop}
\begin{proof} Let $x\in\mathbb{R}^d$ such that $m = \mu(B(x,\rho/2))>0$. Then $\hat{N} = \eta(B(x,\rho/2))$ is Poisson distributed with expectation $m$. Moreover, the distance between any $y,z\in B(x,\rho / 2)$ is at most $\rho$, thus any two vertices in $B(x,\rho / 2)$ are connected by an edge. This implies that almost surely $\hat{N}(\hat{N}-1)/2 \leq N$. Hence, for any $r\geq 0$ we have
\begin{align*}
\P(N\geq \mathbb{E} N + r) \geq \P(\hat{N}^2 - \hat{N} \geq 2\mathbb{E} N + 2r) \geq \P(\hat{N}\geq \gamma(r)),
\end{align*}
where $\gamma(r) = \sqrt{2\mathbb{E} N + \tfrac{1}{4} + 2 r} + \tfrac{1}{2}$. It is well known that for a Poisson random variable $X$ the upper tail satisfies $\P(X\geq r) \sim \exp(-r\log(r/\mathbb{E} X) - \mathbb{E} X)$ as $r\to \infty$, see for example \cite{G_1987}. Hence,
\begin{align*}
\liminf_{r\to\infty} \frac{\P(N\geq \mathbb{E} N + r)}{\exp(-\gamma(r)\log(\gamma(r)/m) - m)} \geq 1.
\end{align*}
The above considerations yield that {there exists a constant $C\geq 0$ such that}, for $r$ large enough along any subsequence diverging to infinity,
\begin{align*}
\gamma(r)\log(\gamma(r)/m)+m \geq I(r) - C.
\end{align*}
Dividing this inequality by $I(r)$ and letting $r$ diverge to infinity gives
\begin{align}\label{e:g}
\liminf_{r\to\infty} \frac{\gamma(r)\log(\gamma(r)/m)}{I(r)} \geq 1.
\end{align}
The conclusion is obtained by observing that, as $r\to\infty$,
\begin{align*}
\gamma(r)\log(\gamma(r)/m) \sim (r/2 )^{1/2} \log r.
\end{align*}
\end{proof}
The following statement is an elementary consequence of Proposition \ref{p:ny}.
\begin{cor}\label{c:o} Let $r\mapsto I(r)$ be a positive mapping verifying \eqref{e:ir}, and assume that there exist constants $a, b>0$ such that, as $r\to\infty$, $I(r)\sim b\, r^a $. Then, necessarily, $a \leq \frac12$.
\end{cor}
\subsection{Deviation inequalities for the upper tail}\label{ss:rggut}
We will now deal with bounds on the upper tail of $N$. We start by observing that, for every $x\in \eta$, the local version $N(x, \eta)$, as defined in \eqref{e:genug}, is exactly given by {the quantity $\deg(x)/2$, where $\deg(x)=\#\{y\in\eta:x\leftrightarrow y\}$ is the \emph{degree} of the vertex $x$}. Our aim in what follows is to show that, for some constant $c>0$, one has that almost surely
\begin{align}\label{edgeIneq}
\sum_{x\in\eta} \deg(x)^ 2 \leq c N^ {3/2}.
\end{align}
Hence, Theorem \ref{DevIneq2} yields the following deviation inequality for the upper tail:
\begin{equation}\label{e:copa}
\P(N\geq\mathbb{E} N + r) \leq \exp\left(-\frac{((r+\mathbb{E} N)^{1/4} - (\mathbb{E} N)^{1/4})^2}{ 2 c}\right).
\end{equation}
Observe that the right-hand side of \eqref{e:copa} has the form $\exp(-I(r))$, where $I(r)\sim r^{1/2}/2c$, as $r\to\infty$. According to Corollary \ref{c:o}, the power $1/2$ for $r$ is optimal in this situation. We will see in Section \ref{ss:lpf} that, by adopting an alternative approach, the rate of decay of $I(r)$ can indeed be improved by the square root of a logarithmic factor.
Also notice that by virtue of relation \eqref{edgeIneq} together with Theorem \ref{DevIneq2}, almost sure finiteness of $N$ is equivalent to integrability of $N$. Hence, relation \eqref{e:r} actually holds if and only if $N$ is almost surely finite.
\medskip
We start by proving a geometric lemma, focussing on deterministic point configurations. In what follows, we shall write $\mathfrak{p} = \mathfrak{p}(d)$ to indicate the smallest integer $\mathfrak{p}$ such that the half ball $B=\{x\in\mathbb{R}^d: \lVert x\rVert \leq \rho, x_1>0\}$ can be written as a union $B = B_1\cup\ldots \cup B_{\mathfrak{p}}$ of disjoint sets such that $\diam(B_i)\leq \rho$ for all $i=1,...,\mathfrak{p}$. Note that the value of $\mathfrak{p}$ depends on the dimension $d$ of the surrounding Euclidean space. In the plane $\mathbb{R}^2$ one has for example that $\mathfrak{p} = 3$. The picture below illustrates the situation described in the proof of the upcoming lemma.
\medskip
\begin{figure}[ht]
\includegraphics[scale=0.3]{vertex_half}
\caption{Partitioning of the half ball}
\end{figure}
\begin{lem} \label{edgeLem}
Let $\xi\subset\mathbb{R}^d$ be a countable set. Denote the disk graph with radius $\rho>0$ associated with $\xi$ by $\mathfrak{G}_\xi$. For all $x\in\xi$, define the \emph{right-degree} and the \emph{left-degree} of $x$ as
\[
\deg_r(x) = \#((x+B)\cap \xi) \ \ \text{ and } \ \ \deg_l(x) = \#((x-B)\cap \xi).
\]
Let $T_\xi$ and $N_\xi$ denote the number of triangles and edges in $\mathfrak{G}_\xi$, respectively. Then
\[
2\mathfrak{p} T_\xi + \mathfrak{p} N_\xi \geq \sum_{x\in\xi} \deg_r(x)^2.
\]
This inequality also holds for the left-degree instead of the right-degree.
\end{lem}
\begin{proof}
For the rest of the proof, we write $\mathfrak{G}, \, T$ and $N$, without the subscript $\xi$, to simplify the notation. Without loss of generality, we can assume that $\xi$ only contains a finite number of non-isolated points in the topology of the graph; otherwise $N=\infty$ and the estimate in the statement is trivially satisfied. Let $x\in\xi$ and denote by $T(x)$ the number of those triangles $\{x,y,z\}$ in $\mathfrak{G}$ incident to $x$, and such that $x_1<\min(y_1,z_1)$, where $a_1$ indicates the first coordinate of a given vector $a\in\mathbb{R}^ d$. Moreover, for $i=1,\ldots,\mathfrak{p}$ let $n_i$ denote the number of elements of $\xi$ contained in $B_i+x$. Then, since any two vertices contained in the same $B_i+x$ yield an edge and thus a triangle incident to $x$ in the sense described above, we have
\[
T(x)\geq \sum_{i=1}^{\mathfrak{p}} \binom{n_i}{2}.
\]
In view of the relation $\sum_i n_i = \deg_r(x)$, the right-hand side of the previous expression can be further bounded from below, thus yielding the relation:
\[
T(x) \geq \frac{1}{2}\mathfrak{p} \cdot \frac{\deg_r(x)}{\mathfrak{p}}\left(\frac{\deg_r(x)}{\mathfrak{p}}-1\right).
\]
Hence,
\[
2\mathfrak{p} T(x) \geq \deg_r(x)^2-\mathfrak{p}\deg_r(x).
\]
Summing up over all $x\in\xi$ yields
\[
2\mathfrak{p} \sum_{x\in\xi} T(x) + \mathfrak{p}\sum_{x\in\xi}\deg_r(x) \geq \sum_{x\in\xi}\deg_r(x)^2.
\]
Observe that in the sum $\sum_{x\in\xi} T(x)$ each triangle is counted at most once, thus $\sum_{x\in\xi} T(x)\leq T$. Also, in the sum $\sum_{x\in\xi}\deg_r(x)$ each edge is counted at most once, so this sum is less than $N$.
\end{proof}
We now deduce a bound on $\sum_{x\in\xi}\deg(x)^2$ for any countable point configuration $\xi$.
\begin{cor} \label{edgeCor}
Let $\xi\subset\mathbb{R}^d$ be countable. Let $\mathfrak{G}_\xi$ be the disk graph (with some arbitrary radius $\rho>0$) associated with $\xi$ and denote the number of edges of $\mathfrak{G}_\xi$ by $N_\xi$. Then,
\[
\sum_{x\in\xi}\deg(x)^2 \leq \frac{8 \sqrt{2}}{3} \mathfrak{p} N_\xi^{3/2} +4\mathfrak{p}N_\xi
\leq \left(\frac{8 \sqrt{2}}{3} + 4\right) \mathfrak{p} N_\xi^{3/2}.
\]
\end{cor}
\begin{proof}
First we observe that without loss of generality it can be assumed that the first coordinates of all elements in $\xi$ are distinct. Obviously, the combinatorial structure of $\mathfrak{G}_\xi$ is invariant under rotation of the set $\xi$. Also, the assumption in question can be achieved to hold by rotating the set $\xi$ with respect to a direction $a\in\mathbb{R}^d, \lVert a \rVert = 1$ that satisfies $(x-y) / \lVert x-y \rVert \neq \pm a$ for all distinct $x,y\in \xi$. Such a direction exists since the set of directions
\begin{align*}
\{(x-y)/ \lVert x-y \rVert : (x,y)\in\xi_{\neq}^2\}
\end{align*}
is countable and hence a strict subset of all directions $\{a\in\mathbb{R}^d:\lVert a\rVert = 1\}$. So, it can be assumed that the elements of $\xi$ have distinct first coordinates. In particular, for any $x\in\xi$ we have $\deg(x) = \deg_r(x) + \deg_l(x)$. Thus
\[
\sum_{x\in\xi}\deg(x)^2 = \sum_{x\in\xi}(\deg_r(x)+\deg_l(x))^2 \leq
2 \sum_{x\in\xi} (\deg_r(x)^2 + \deg_l(x)^2).
\]
By Lemma \ref{edgeLem}, the latter expression does not exceed
\[
8 \mathfrak{p} T_\xi + 4\mathfrak{p} N_\xi,
\]
where $T_\xi$ stands for the number of triangles in $\mathfrak{G}_\xi$. The result follows by using an estimate taken from \cite{R_2002}, where it was proven that
\[
3T_\xi \leq \sqrt{2}N_\xi^{3/2}.
\]
\end{proof}
The next statement is one of the main achievements in the present section.
\begin{thm}
\begin{enumerate}
\item Let $\xi\subset\mathbb{R}^d$ be countable. Let $\mathfrak{G}_\xi$ be a disk graph with arbitrary radius $\rho$ associated with $\xi$, and denote the number of edges of $\mathfrak{G}_\xi$ by $N_\xi$. Then,
\[
\sum_{x\in\xi}\deg(x)^2 \leq \left(\frac{8 \sqrt{2}}{3} + \frac{4}{D}\right) \mathfrak{p} N_\xi^{3/2},
\]
where
\[
D=\frac{4\sqrt{2}}{3} \mathfrak{p} + \sqrt{\frac{32}{9}\mathfrak{p}^2 + 4\mathfrak{p} - 1}.
\]
\item Now let $\eta$ be the Poisson measure on $\mathbb{R}^d$ with non-atomic intensity $\mu$ considered in this section, and denote by $\mathfrak{G}_\eta = \mathfrak{G}$ the random disk graph (with arbitrary radius $\rho$) associated with $\eta$. Let $N_\eta = N$ be the number of edges of {the random disk graph $\mathfrak{G}$. Then relation \eqref{edgeIneq} holds almost surely, for $c=\left(\frac{8 \sqrt{2}}{3} + \frac{4}{D}\right) \mathfrak{p}(d)$.} In particular, the tail estimate \eqref{e:copa} is verified.
\end{enumerate}
\end{thm}
\begin{proof}
\noindent{[Proof of (i)}] By Corollary \ref{edgeCor} we have
\[
\sum_{x\in\xi}\deg(x)^2 \leq \left(\frac{8 \sqrt{2}}{3} +\frac{4}{\sqrt{N_\xi}}\right) \mathfrak{p}N_\xi^{3/2}.
\]
Observe that among all graphs with $N_\xi$ edges, the star, i.e. the graph with $\deg(x) = N$ for one vertex $x$ and $\deg(y)=1$ for all other vertices $y$, maximises the sum of the squared degrees. Thus, we also have
\[
\sum_{x\in\xi} \deg(x)^2 \leq N_\xi^2 + N_\xi = \left(\sqrt{N_\xi} + \frac{1}{\sqrt{N_\xi}}\right) N_\xi^{3/2}.
\]
Now, since
\[
\left(\frac{8 \sqrt{2}}{3} +\frac{4}{\sqrt{N_\xi}}\right) \mathfrak{p}
\]
is monotonically decreasing in $N_\xi$ and
\[
\sqrt{N_\xi} + \frac{1}{\sqrt{N_\xi}}
\]
is monotonically increasing in $N_\xi$,
the minimum of both functions is always less or equal to the value at the intersection of them. Computing this value yields the result.
\smallskip
\noindent{[Proof of (ii)}] This follows directly from part (i) of the statement.
\end{proof}
\subsection{Deviation inequalities for the lower tail}\label{ss:eclower} We now focus on lower tails. In order to do that, we introduce the notation
$$
K := \sup_{x\in \mathbb{R}^d} \mu(B(x, \rho))<\infty,
$$
and we define the parameter ${\mathfrak v}$ as
$$
{\mathfrak v}:= 2(K+1)\, \mathbb{E} N.
$$
Our main estimate is the following:
\begin{thm}
For every $r>0$ one has the estimate
\begin{align}\label{e:gglt}
\P(N\leq\mathbb{E} N -r) \leq \exp\left(-\frac{r^2}{2{\mathfrak v}}\right).
\end{align}
\end{thm}
\begin{proof} We will freely use the notation and definitions introduced in Section \ref{ss:zut}. Since $N$ is a U-statistic of order $2$ with kernel $f(x,y) = \tfrac{1}{2} \mathbbm{1}\{\lVert x-y \rVert \leq \rho\}$, by virtue of Proposition \ref{p:p} it is sufficient to show that, in this case, $4 V\leq \mathfrak{v}$. In order to do that, observe first that $f_1(x) = \mu(B(x,\rho))$ and $f_2=f$. As a consequence,
\begin{align*}
&\|f_1\|^2_{L^2(\mu^1)} = \int_{\mathbb{R}^d} \mu(B(x,\rho))^2 \, d\mu(x) \leq 2 K \mathbb{E} N;\\
&\|f_2 \|^2_{L^2(\mu^2)} = \tfrac 12\mathbb{E} N.
\end{align*}
Plugging the above upper bounds into the definition of $V$ yields the inequality $4V \leq \mathfrak{v}$, and therefore the desired conclusion.
\end{proof}
\subsection{Comparison with the literature}
We shall now briefly compare the concentration inequalities for edge counting presented above with the results already existing in the literature. To the best of our knowledge, the only concentration inequalities known so far that apply to the setting of edge counting in random disk graphs over Poisson point configurations, are established in \cite{RST_2013} and \cite{ERS_2015}. Both papers deal with the case where the {intensity measure $\mu$ is finite.}
\smallskip
\noindent{\it Comparison with {\rm \cite{ERS_2015}}}. We shall use some computations from \cite{RST_2013}, where it is explained how the general results about stabilizing functionals from \cite{ERS_2015} apply to random graph statistics. In the special case of edge counting, using the notation and assumptions of the present section, one deduces indeed from \cite[Proposition 5.1]{RST_2013} an estimate of the type
$$
\P( |N - \mathbb{E} N| \geq r) \leq \exp(-I_0(r)), \quad \mbox{where } I_0(r) \sim a r^{1/3},
$$
for some positive constant $a$, as $r\to\infty$. As far as the asymptotic behaviour of $r\mapsto I_0(r)$ is concerned, this result is worse than the estimates that one can obtain from \eqref{e:copa} (where the argument of the exponential bound on the upper tail is asymptotic to $-r^{1/2}/2c$, and therefore optimal in the sense of Corollary \ref{c:o}), and than those given by \eqref{e:gglt} (where we have proved a Gaussian upper bound on the lower tail).
\smallskip
\noindent{\it Comparison with {\rm \cite{RST_2013}}.} Writing $m$ for a median of the law of $N$, in \cite[Theorem 5.2]{RST_2013} one can find an estimate of the type
$$
\P( |N - m | \geq r) \leq \exp(-I_1(r)), \quad\mbox{where }\, I_1(r)\sim b r^{1/2},
$$
as $r\to \infty$, for some positive constant $b$. Note that, as $r\to\infty$, the upper tail bound determined by $I_1$ has the same order as our estimate \eqref{e:copa}, whereas the lower tail estimate is worse than our Gaussian upper bound \eqref{e:gglt}. We also stress that \cite[Theorem 5.2]{RST_2013} has a different nature than our results, since it gives a concentration inequality around the median, and not the expectation. This might be a drawback for applications since the median of the edge count is harder to deal with than the expectation -- which can be easily expressed using the Slivniak-Mecke formula. Finally, a major advantage of our results over those presented in \cite[Theorem 5.2]{RST_2013} is that the latter only applies to disk graphs built on finite intensity measure Poisson processes, whereas our tail estimates merely require that {the number of edges is almost surely finite.}
\subsection{Consistency with CLT}
In the following, we compare the deviation inequality for the upper tail with a CLT that was proven in \cite{RS_2013}. Let $\eta_1$ be a Poisson point process in $\mathbb{R}^d$ with intensity measure $\mu_1$ and fix some radius $\rho > 0$. For any $n\in\mathbb{N}$, let $\eta_n$ be a Poisson point process with intensity measure $\mu_n = n\mu_1$ and denote the number of edges in the corresponding random geometric graph by $N_n$. Assume that $\mathbb{E} N_1 < \infty$ (and hence $\mathbb{E} N_n < \infty$ for all $n$). Then by \cite[Theorem 5.2]{RS_2013}, the sequence of random variables $N_n$ satisfies a central limit theorem, i.e. $(N_n - \mathbb{E} N_n) / \sqrt{\mathbb{V} N_n}$ converges to a standard Gaussian distribution, where $\mathbb{V} N_n$ stands for the variance of $N_n$. Therefore, as $n\to\infty$, the sequence of probabilities $\P(N_n\geq \mathbb{E} N_n + \sqrt{\mathbb{V} N_n} r)$, $n\geq 1$, converges to the quantity
$$
\frac{1}{\sqrt{2\pi}}\int_r^\infty e^{-y^2/2} dy\leq \frac{e^{-r^2/2}}{r\sqrt{2\pi}}.
$$
According to the next result, the asymptotic behavior of the upper tail deviation inequality is consistent with this Gaussian tail.
\begin{thm}
{Let $c>0$ be a constant satisfying \eqref{edgeIneq}.} Then there exists a constant $C>0$ and a sequence $(x_n)_{n\in\mathbb{N}}$ with $x_n \to \infty$ as $n\to\infty$ such that for any $n\in\mathbb{N}$,
\begin{align*}
\exp\left(-\frac{(((\mathbb{V} N_n)^{1/2}r+\mathbb{E} N_n)^{1/4} - (\mathbb{E} N_n)^{1/4})^2}{2c}\right) \leq \exp(-C r^2) \ \ \text{for all} \ \ r\in[0,x_n].
\end{align*}
\end{thm}
\begin{proof}
It was pointed out in \cite{RS_2013} that there are constants $\alpha,\beta > 0$ such that
\begin{align*}
\mathbb{V} N_n \sim \alpha n^3,\\
\mathbb{E} N_n \sim \beta n^2.
\end{align*}
{Let $A = \sqrt{2Cc}$}. Then the desired inequality is equivalent to
\begin{align*}
(\mathbb{V} N_n)^{1/2} r &\geq \left(A r + (\mathbb{E} N_n)^{1/4}\right)^4 - \mathbb{E} N_n\\
&= A^4 r^4 + 4 A^3 r^3 (\mathbb{E} N_n)^{1/4} + 6 A^2 r^2(\mathbb{E} N_n)^{2/4}
+ 4 A r (\mathbb{E} N_n)^{3/4}.
\end{align*}
This holds if and only if
\begin{align*}
(\mathbb{V} N_n)^{1/2}(\mathbb{E} N_n)^{-3/4} - 4 A \geq A^4r^3 (\mathbb{E} N_n)^{-3/4} + 4A^3 r^2 (\mathbb{E} N_n)^{-2/4} + 6 A^2 r (\mathbb{E} N_n)^{-1/4}.
\end{align*}
Now, choose $C>0$ such that {$4A<\alpha^{1/2}\beta^{-3/4}$}. Consider the equality corresponding to the inequality in the above display. For any $n\in\mathbb{N}$ let $x_n$ be the (unique) positive solution of this equality in case such a solution exists, and let $x_n = 0$ otherwise. Then, since the right-hand side of the last display is increasing in $r$, the desired inequality holds for all $r\in[0,x_n]$. Moreover, the left hand side converges to {$\alpha^{1/2}\beta^{-3/4} - 4 A>0$} while $(\mathbb{E} N_n)^{-3/4}, (\mathbb{E} N_n)^{-2/4}, (\mathbb{E} N_n)^{-1/4} \to 0$, as $n\to\infty$. From this it follows that $x_n\to\infty$ as $n\to\infty$.
\end{proof}
\section{Another look at U-Statistics of order two}
In this section, we develop a different {approach } for obtaining deviation inequalities for the upper tail of U-statistics of order 2, that is partially inspired by the results from \cite{HRB, R_2003}. {Throughout this section, we let the assumptions of Section \ref{AppUstat} prevail; in particular, the intensity $\mu$ of $\eta$ is {a non-atomic positive measure on $(\mathbb{X}, \mathcal{X})$}. We begin by generalizing \cite[Theorem 3]{R_2003} to Poisson processes with possibly non-finite intensity measure:
\begin{thm} \label{supThm}
Consider a countable family $\{f_j\}_{j\in J}$ of functions $\mathbb{X}\to[0,1]$ and let
\begin{align*}
G = \sup_{j\in J} \sum_{x\in\eta} f_j(x).
\end{align*}
Assume that $\mathbb{E} G < \infty$. Then for any $\lambda>0$ we have
\begin{align*}
\log \mathbb{E}[\exp(\lambda(G-\mathbb{E} G))] \leq \phi(\lambda)\mathbb{E} G,
\end{align*}
where $\phi(\lambda) = e^\lambda - \lambda - 1$.
\end{thm}
\begin{proof}
First note that by monotone convergence, we can assume without loss of generality that $|J|<\infty$. For each $n\in\mathbb{N}$ let $G_n = \min(G,n)$. Then $\mathbb{E} (e^{\lambda G_n}) < \infty$, hence Proposition \ref{EntIneq} with $I = \emptyset$ gives
\begin{align}\label{supEntIneq}
\Ent(e^{\lambda G_n}) \leq \mathbb{E} \left(e^{\lambda G_n} \sum_{x\in\eta} \phi(-\lambda D_x G_n(\eta-\delta_x))\right).
\end{align}
Consider some realization of $\eta$. Since we assumed $|J|<\infty$, it follows that for some $j^* \in J$ we have
\begin{align*}
G(\eta) = \sum_{x\in\eta} f_{j^*}(x).
\end{align*}
Now, for any $x\in\eta$ we have
\begin{align*}
0\leq D_x G(\eta - \delta_x) \leq f_{j^*}(x) \leq 1.
\end{align*}
Moreover, if $G(\eta-\delta_x) \geq n$, then $D_xG_n(\eta-\delta_x) = 0$ and if $G(\eta - \delta_x) < n$, then
\begin{align*}
D_xG_n(\eta-\delta_x) = D_xG(\eta-\delta_x) - \max(0,G(\eta)-n).
\end{align*}
From this we obtain
\begin{align*}
\sum_{x\in\eta} D_xG_n(\eta-\delta_x) &\leq \left(\sum_{x\in\eta} f_{j^*}(x)\right) - \max(0,G(\eta) - n)\\ &= G(\eta) - \max(0, G(\eta)-n) = G_n(\eta).
\end{align*}
Since $\phi(-\lambda z) \leq \phi(-\lambda) z$ for $\lambda>0$ and $0\leq z \leq 1$, it follows from the above considerations that
\begin{align*}
\Ent(e^{\lambda G_n}) \leq \phi(-\lambda) \mathbb{E} \left(e^{\lambda G_n} \sum_{x\in\eta} D_x G_n(\eta-\delta_x)\right) \leq \phi(-\lambda) \mathbb{E}(e^{\lambda G_n} G_n).
\end{align*}
Continuing in the same way as in the proof of \cite[Theorem 10]{M_2000} gives
\begin{align}\label{truncSupIneq}
\log \mathbb{E} [\exp(\lambda(G_n - \mathbb{E} G_n))] \leq \phi(\lambda)\mathbb{E} G_n \leq \phi(\lambda)\mathbb{E} G.
\end{align}
{Now, since $\lambda>0$ and $\mathbb{E} G < \infty$, by monotone convergence we have}
\begin{align*}
\lim_{n\to\infty}\mathbb{E}[\exp(\lambda(G_n - \mathbb{E} G_n))] = \mathbb{E}[\exp(\lambda(G - \mathbb{E} G))].
\end{align*}
Invoking (\ref{truncSupIneq}) yields the result.
\end{proof}
We continue with an analytic lemma that is used in the proof of the upcoming theorem, but which might be of independent interest in similar situations. The proof is inspired by the proof of \cite[Corollary 2.12]{L_1999}.
\begin{lem}\label{analyticLem}
For any $z>0$,
\begin{align*}
\sup_{\lambda>0}\left[\lambda z - e^{\lambda^2} + 1\right] \geq \frac{\sqrt{\log(z+1)} z^ {3/2}}{4\sqrt{z} + 8}.
\end{align*}
\end{lem}
\begin{proof}
We begin with a preparing observation. Let $a>1$ and consider the map $\lambda \mapsto 1 - e^{\lambda^2} + a \lambda^2$. Then this map takes $0$ to $0$ and it has a unique local extremum (a maximum) at $\sqrt{\log(a)}$ on the positive reals. Hence, whenever $1- e^{x^2}+ax^2 \geq 0$ for some $x>0$, we have $1-e^{\lambda^2} \geq -a \lambda^2$ for any $\lambda\in(0,x]$.
Now assume that $0<z\leq 4$. Then we take $\lambda = z/4$. Since $\lambda\leq 1$ and $1-e + 2\geq 0$, the above observation implies
\begin{align*}
1-e^{\lambda^2} \geq -2\lambda^2.
\end{align*}
Hence, we have
\begin{align*}
\lambda z - e^{\lambda^2} + 1 \geq
\lambda z -2\lambda^2 = \frac{z^2}{8} \geq \frac{\sqrt{\log(z+1)} z^ {3/2}}{4\sqrt{z} + 8}.
\end{align*}
For $4<z\leq 12$ we take $\lambda = z / 8$. Then $\lambda\leq 3/2$ and since $1-e^{9/4} + 9\geq 0$, the initial observation gives
\begin{align*}
1-e^{\lambda^2}\geq -4 \lambda^2.
\end{align*}
Thus,
\begin{align*}
\lambda z - e^{\lambda^2} + 1 \geq
\lambda z -4\lambda^2 = \frac{z^2}{16} \geq \frac{\sqrt{\log(z+1)} z^ {3/2}}{4\sqrt{z} + 8}.
\end{align*}
It remains to prove that the result holds for $z>12$. Here we can take $\lambda = \sqrt{\log(z)}$ and see that the supremum is lower bounded by
\begin{align*}
z(\sqrt{\log(z)} - 1) + 1.
\end{align*}
Now, since $\sqrt{\log(z)} - \sqrt{\log(z+1)}$ is increasing for $z\geq 12$, we have
\begin{align*}
\sqrt{\log(z)} \geq \sqrt{\log(z+1)} + A,
\end{align*}
where $A=\sqrt{\log(12)} - \sqrt{\log(13)} < 0$. Hence,
\begin{align*}
z(\sqrt{\log(z)} - 1) + 1 \geq z(\sqrt{\log(z+1)} + A - 1) + 1.
\end{align*}
We claim that
\begin{align*}
z(\sqrt{\log(z+1)} + A - 1) + 1 \geq \frac{1}{4}z(\sqrt{\log(z+1)}) \ \ \text{for} \ \ z>12.
\end{align*}
This will imply the result since
\begin{align*}
\frac{1}{4}z(\sqrt{\log(z+1)}) \geq \frac{1}{4}\frac{z(\sqrt{\log(z+1)}) \sqrt{z}}{\sqrt{z} + 2} = \frac{\sqrt{\log(z+1)}z^{3/2}}{4\sqrt{z} + 8}.
\end{align*}
To prove the claim, define
\begin{align*}
h(z) &= z(\sqrt{\log(z+1)} + A - 1) + 1 -\frac{1}{4}z(\sqrt{\log(z+1)})\\
&= \frac{3}{4}z \sqrt{\log(z+1)} + z(A - 1) + 1.
\end{align*}
The derivative of $h$ satisfies
\begin{align*}
h'(z) = \frac{3}{4} \left(\frac{z+ 2 \log(z+1) (z+1)}{2 \sqrt{\log(z+1)} (z+1)}\right) + A - 1 \geq \frac{3}{4} \sqrt{\log(z+1)} + A - 1.
\end{align*}
Since the right hand side is increasing in $z$ and positive for $z = 12$, we have that $h'(z)\geq 0$ for any $z>12$. So $h$ is increasing for $z>12$ and since also $h(12)\geq 0$, it follows that $h(z)\geq 0$ for all $z>12$. This proves the claim and concludes the proof of the result.
\end{proof}
\begin{thm}\label{UStatDevIneq}
{Let $F$ be a $U$-statistic} of order $2$ with kernel $f\geq 0$ such that $\mathbb{E} F<\infty$. Assume that there is a countable family $\{g_j\}_{j\in J}$ of functions $\mathbb{X}\to[0,1]$ and a constant $c>0$ such that
\begin{align*}
G = \sup_{j\in J} \sum_{x\in\eta} g_j(x)
\end{align*}
satisfies almost surely
\begin{align*}
\sup_{y\in\eta} \sum_{x\in\eta\setminus y} f(y,x) \leq c G
\end{align*}
and $\mathbb{E} G < \infty$. Then for any $r>0$ we have
\begin{align*}
\P(F \geq \mathbb{E} F+r) &\leq \exp\left( -\mathbb{E} (G)\ \chi\left(\frac{\sqrt{\mathbb{E} F + r} - \sqrt{\mathbb{E} F}}{\sqrt{4c}\ \mathbb{E} G}\right)\right),
\end{align*}
where
\begin{align*}
\chi(z)= \frac{\sqrt{\log(z+1)} z^ {3/2}}{4\sqrt{z} + 8}.
\end{align*}
\end{thm}
\begin{proof}
The assumptions imply {that almost surely}
\begin{align*}
V^+ = 4 \sum_{y\in\eta}\left(\sum_{x\in\eta\setminus y} f(y,x)\right)^2 \leq 4cG F.
\end{align*}
By Corollary \ref{cor2} and Theorem \ref{supThm} this gives for any $\lambda> 0$,
\begin{align*}
\log \mathbb{E} [\exp(\lambda(\sqrt{F} - \mathbb{E} \sqrt{F}))] &\leq \inf_{\theta\in(0,2/\lambda)} \frac{\lambda\theta}{2-\lambda\theta} \log \mathbb{E} \left[\exp\left(\frac{4c\lambda}{\theta} G\right)\right]\\
&\leq \inf_{\theta\in(0,2/\lambda)} \frac{\lambda\theta}{2-\lambda\theta} \mathbb{E} G (\exp(4c\lambda/\theta) - 1)\\
&\leq \mathbb{E} G (\exp(4c\lambda^2) - 1).
\end{align*}
Let $r>0$. Then, using the above computation and Markov's inequality, we obtain for any $\lambda>0$,
\begin{align*}
\P(F \geq \mathbb{E} F + r) &\leq \P(e^{\lambda(\sqrt{F} - \mathbb{E}\sqrt{F})} \geq e^{\lambda( \sqrt{\mathbb{E} F + r} - \sqrt{\mathbb{E} F})})\\
&\leq \exp\left(\mathbb{E} G (\exp(4c\lambda^2) - 1) - \lambda( \sqrt{\mathbb{E} F + r} - \sqrt{\mathbb{E} F})\right).
\end{align*}
Hence, writing $z = (\sqrt{\mathbb{E} F + r} - \sqrt{\mathbb{E} F})/(\sqrt{4c}\mathbb{E} G)$ and substituting $\lambda$ by $\lambda / \sqrt{4c}$, we obtain
\begin{align*}
\P(F\geq \mathbb{E} F + r) \leq \exp\left(-\mathbb{E} G \sup_{\lambda>0} \left[\lambda z -\exp(\lambda^2) + 1 \right]\right).
\end{align*}
The result now follows from Lemma \ref{analyticLem}.
\end{proof}
\subsection{Length power functionals}\label{ss:lpf} As an application, we now focus on length power functionals in random geometric graphs. These estimates contain as special case the edge counting statistics that we studied in Section \ref{s:edge}. We will see in particular that one can take advantage of the upper tail estimate stated in Theorem \ref{UStatDevIneq} in order to provide an alternate bound to that appearing in \eqref{e:copa}, which actually displays a {\it strictly faster} rate of decay in $r$.
As we did in Section \ref{s:edge}, we consider a Poisson measure on $\mathbb{R}^d$ with $\sigma$-finite {and non-atomic Borel intensity measure $\mu$}. We let $\rho>0$ be some radius and consider again the disk graph $\mathfrak{G}(\eta)$ associated with $\eta$. For any $\alpha\in[0,1]$ the {\it length power functional} $L^{(\alpha)}$ is the $U$-statistic of order $2$ with kernel
\begin{align*}
f_\alpha(x,y) = \tfrac 12 \mathbbm{1}\{\lVert x - y \rVert\leq \rho\} \lVert x - y \rVert^\alpha.
\end{align*}
Note that $L^{(0)} = N$, the number of edges in $\mathfrak{G}(\eta)$, and $L^{(1)}$ is just the {(total edge)} length of the graph. One easily sees that
\begin{align*}
\sup_{y\in\eta} \sum_{x\in\eta\setminus y} f_\alpha(x,y) \leq 2^{d-1} \rho^\alpha \sup_{j\in \mathbb{N}^d} \sum_{x\in\eta} \mathbbm{1}\{x \in[0,2\rho]^d + 2\rho j\}.
\end{align*}
It is also straightforward to check that, if $\mathbb{E} N<\infty$, then the expectation of the right-hand side of the above inequality is finite. We stress also that, if $\mathbb{E} N<\infty$, then $L^{(\alpha)}$ is trivially well-behaved for every $\alpha\in [0,1]$; see Section \ref{s:representative}. The following consequence of Theorem \ref{UStatDevIneq} therefore holds.
\begin{cor}\label{c:we} Let the above notation prevail, fix $\alpha\in [0,1]$, assume that $\mathbb{E} N<\infty$, and let
\begin{align*}
G = \sup_{j\in \mathbb{N}^d} \eta([0,2\rho]^d + 2\rho j).
\end{align*}
Then, $\mathbb{E} G<\infty$ and, for all $r>0$,
\begin{align}\label{e:cd}
\P(L^{(\alpha)} \geq \mathbb{E} L^{(\alpha)} + r) \leq \exp\left( -\mathbb{E} (G)\ \chi\left(\frac{\sqrt{\mathbb{E} L^{(\alpha)} + r} - \sqrt{\mathbb{E} L^{(\alpha)}}}{\sqrt{2^{d+1} \rho^\alpha} \ \mathbb{E} G}\right)\right),
\end{align}
where $\chi(z)$ is as in Theorem \ref{UStatDevIneq}.
\end{cor}
\begin{rem} The right-hand side of \eqref{e:cd} has the form $\exp(-I(r))$, where $I(r) \sim b \sqrt{r\log r}$, for some $b>0$, as $r\to \infty$. Such a rate of decay is better than the one we can deduce from \cite[Proposition 5.1]{RST_2013} (which is indeed a translation of the results from \cite{ERS_2015}), that applies to the case where $\mu$ is a multiple of the restriction of the Lebesgue measure to a convex body, and implies an upper bounds of the form $\exp(-I_0(r))$, with $I_0(r) \sim b_0 r^{1/3}$. Our result provides also a rate of decay that is faster than the one appearing in \cite[Theorem 5.5]{RST_2013}, where the bound has the form $\exp(-I_1(r))$, with $I_1(r) \sim b_1 r^{1/2}$. It is remarkable that, in the case $\alpha =0$ and as far as the rate of decay (as $r\to\infty$) is concerned, the estimate \eqref{e:cd} is also strictly better than \eqref{e:copa}, and that this comes at the cost of somewhat more complicated constants. Finally, we observe that the asymptotic relation $I(r) \sim b \sqrt{r\log r}$ is consistent with Proposition \ref{p:ny}.
\end{rem}
\subsection{Length in more general graph models}
In the following, we consider a slightly more general model of random geometric graphs. For this, let $\eta$ be a Poisson point process on $\mathbb{R}^d$ {with $\sigma$-finite and non-atomic Borel intensity measure $\mu$}, and let $\rho:\mathbb{R}^d\to\mathbb{R}_+$ be given by
\begin{equation}\label{e:rho}
\rho(x) =\left( \frac{1}{ \lVert x \rVert + 1}\right)^{\gamma},
\end{equation}
for some $\gamma>0$. We define $\mathfrak{H}(\eta)$ as the graph with vertex set $\eta$ and an edge between vertices $x,y\in\eta$ whenever $0<\lVert x-y \rVert \leq \rho(x)+\rho(y)$. Note that the graph $\mathfrak{H}= \mathfrak{H}(\eta)$ is obtained by implementing the following two-step procedure: (a) for every $x\in \eta$, draw the closed ball {$B(x,\rho(x))$}, centered at $x$ and with radius $\rho(x)$, and (b) connect two distinct points $x,y\in \eta$ with an edge, if and only if $$B(y,\rho(y))\cap B(x,\rho(x)) \neq \emptyset.$$ In other words, $\mathfrak{H}$ is the {\it intersection graph} of the balls centered at the points of $\eta$ with (decaying) radii given by $\rho(x), x\in\eta$. We will see that this model allows situations where $\mathfrak{H}$ has almost surely infinitely many edges but still a finite length. Interestingly, even if there are infinitely many edges, the length can have an exponentially decaying upper tail. Before we analyse concentration properties of the length, we present an illustration of how the considered graph might look like in the plane.
\begin{figure}[h]
\includegraphics[scale=0.26]{graphLarge}
\caption{A realisation of the intersection graph $\mathfrak{H}(\eta)$}
\end{figure}
Let $L$ be the length of $\mathfrak{H}$. Then $L$ is a U-statistic of order $2$ with kernel
\begin{align*}
f_L(x,y) = \tfrac 12 \lVert x-y \rVert \mathbbm{1}\{\lVert x-y \rVert \leq \rho(x) + \rho(y)\}.
\end{align*}
{Moreover, if $\mu$ guarantees that almost surely $L<\infty$, then $L$ is even well-behaved. As a consequence of Theorem \ref{UStatDevIneq} we obtain the following result.
\begin{cor} \label{lengthConc}
Let $c = 3^\gamma + 1$ and define
\begin{align}\label{defG}
G = \sup_{x\in\mathbb{Q}^d} \rho(x) \eta(B(x, c\rho(x))).
\end{align}
Assume that $\mathbb{E} G<\infty$ and $\mathbb{E} L < \infty$. Then for any $r\geq 0$,
\begin{align*}
\P(L\geq \mathbb{E} L + r) \leq \exp\left( -\mathbb{E} (G)\ \chi\left(\frac{\sqrt{\mathbb{E} L + r} - \sqrt{\mathbb{E} L}}{\sqrt{2c}\ \mathbb{E} G}\right)\right),
\end{align*}
where $\chi$ is defined as in Theorem \ref{UStatDevIneq}.
\end{cor}
\begin{proof}
Let $x,y\in\mathbb{R}^d$ be such that $\lVert x-y \rVert \leq \rho(x) + \rho(y)$. Then we have $\lvert\lVert x \rVert - \lVert y \rVert \rvert \leq 2$ since $\rho$ is upper bounded by $1$. Hence,
\begin{align}\label{rhoBound}
\frac{\rho(y)}{\rho(x)} = \left(\frac{\lVert x\rVert + 1}{\lVert y\rVert + 1}\right)^\gamma \leq \left(\frac{\lVert y\rVert + 3}{\lVert y\rVert + 1}\right)^\gamma \leq 3^\gamma.
\end{align}
Therefore, $\lVert x-y \rVert \leq (3^\gamma + 1)\rho(x)$. It follows that the local version of $L$ satisfies
\begin{align}\label{lengthObs}
\nonumber L(x,\eta) &= \tfrac 12 \sum_{y\in\eta\setminus x} \lVert x-y\rVert \mathbbm{1}\{\lVert x-y \rVert \leq \rho(x) + \rho(y)\}\\
&\leq \tfrac {3^\gamma + 1}{2} \sum_{y\in\eta\setminus x} \rho(x) \mathbbm{1}\{\lVert x-y \rVert \leq (3^\gamma + 1)\rho(x)\}.
\end{align}
For $x,y\in\mathbb{R}^d$ we define $g_x(y) = \rho(x) \mathbbm{1}\{\lVert x-y \rVert \leq (3^\gamma + 1)\rho(x)\}$. Then the above reasoning gives {that, almost surely,}
\begin{align*}
\sup_{x\in\eta} \sum_{y\in\eta\setminus x} f_L(x,y) \leq \tfrac {3^\gamma + 1}{2} \sup_{x\in\mathbb{Q}^d} \sum_{y\in\eta} g_x(y).
\end{align*}
Let
\begin{align*}
G = \sup_{x\in\mathbb{Q}^d} \sum_{y\in\eta} g_x(y).
\end{align*}
We see that Theorem \ref{UStatDevIneq} applies to $L$ whenever $\mathbb{E} L, \mathbb{E} G<\infty$ and this concludes the proof.
\end{proof}
Next we will prove a sufficient condition for the finiteness of the expectations appearing in Corollary \ref{lengthConc} for the case when the Poisson process $\eta$ is homogeneous, that is, when the intensity measure of $\eta$ has the form $t\times \lambda$, where $\lambda$ is the Lebesgue measure.
\begin{prop}\label{suffCondLG}
Assume that $\eta$ is a homogeneous Poisson point process on $\mathbb{R}^d$ with intensity $t\lambda$, $t>0$. Let $L$ be the length of $\mathfrak{H}$ and define the random variable $G$ as in (\ref{defG}). Then $G$ and $L$ are integrable, provided that
\begin{align}\label{finiteIntCond}
\int_{\mathbb{R}^d}\rho(x)^{d+1} dx < \infty.
\end{align}
\end{prop}
\begin{proof}
First observe, quite similarly as it was done in (\ref{rhoBound}), that for any $x\in\mathbb{Q}^d$ and $\hat{x} \in B(x,(3^\gamma + 1) \rho(x))$,
\begin{align*}
\rho(x) \leq (3^\gamma + 2)^\gamma\rho(\hat{x}).
\end{align*}
Hence, writing $c = 3^\gamma + 1$ and $c' = (3^\gamma + 2)^\gamma$, we have
\begin{align*}
\sum_{y\in\eta} g_x(y) &= \rho(x) \sum_{y\in\eta} \mathbbm{1}\{ \lVert x -y \rVert \leq c\rho(x)\}\\
&\leq c'\rho(\hat{x}) \sum_{y\in\eta} \mathbbm{1}\{ \lVert \hat{x}-y \rVert \leq 2cc'\rho(\hat{x})\}.
\end{align*}
It follows that $\mathbb{E} G<\infty$ if the expectation of the following is finite:
\begin{align} \label{lengthObs2}
\nonumber &\sup_{x\in \eta} \sum_{y\in\eta\setminus x} \rho(x) \mathbbm{1}\{\lVert x-y\rVert \leq 2cc' \rho(x)\}\\
&\leq \sum_{(x,y)\in\eta_{\neq}^2} \rho(x)\mathbbm{1}\{\lVert x-y\rVert \leq 2cc' \rho(x)\}.
\end{align}
Using the Slivniak-Mecke formula \eqref{e:smecke}, we obtain that the expectation of the latter expression equals
\begin{align*}
t^2 \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \rho(x)\mathbbm{1}\{\lVert x-y\rVert \leq 2cc' \rho(x)\}\ dx\ dy
= t^2 (2cc')^d \kappa_d \int_{\mathbb{R}^d}\rho(x)^{d+1} dx,
\end{align*}
where $\kappa_d$ denotes the Lebesgue measure of the unit ball in $\mathbb{R}^d$. So we have $\mathbb{E} G<\infty$ provided that (\ref{finiteIntCond}) holds. This condition also guarantees $\mathbb{E} L<\infty$ (to see this, just perform a computation similar to the one above where the estimate (\ref{lengthObs}) is used instead of (\ref{lengthObs2})).
\end{proof}
We initially claimed that $\mathfrak{H}$ can have a.s. infinitely many edges but still $\mathbb{E} L < \infty$. The following result, when {combined} with Proposition \ref{suffCondLG}, substantiates this statement.
\begin{prop} \label{infiniteEdgesProp}
Assume that $\eta$ is a homogeneous Poisson point process in $\mathbb{R}^d$ with intensity $t>0$. Denote by $N$ the number of edges in $\mathfrak{H}$. Then almost surely $N = \infty$, provided that
\begin{align}\label{infEdgesCond}
\int_{\mathbb{R}^d} \rho(x)^d dx = \infty.
\end{align}
\end{prop}
\begin{rem}
The phenomenon described above is remarkable since it allows for situations where the U-statistic
\begin{align*}
L = \sum_{(x,y)\in\eta_{\neq}^2} f_L(x,y)
\end{align*}
is indeed (as opposed to the edge counting statistics) almost surely an infinite series, i.e. we have almost surely $f_L(x,y)>0$ for infinitely many $(x,y) \in\eta_{\neq}^2$. Intuitively, one might expect that in this situation strong concentration properties for $L$ are more difficult to establish. However, as we have seen above, our method works without problems and yields exponential tail bounds for the graph length regardless of whether finitely or infinitely many edges are present.
\end{rem}
\begin{proof}[Proof of Proposition \ref{infiniteEdgesProp}]
Note that
\begin{align*}
N = \tfrac 12 \sum_{(x,y)\in\eta^2_{\neq}} \mathbbm{1}\{\lVert x-y \rVert \leq \rho(x) + \rho(y)\}.
\end{align*}
For all $x\in\mathbb{N}^d$ define the cube
\begin{align*}
Q_x = [0,\tfrac{1}{\sqrt{d}}\rho(x + \textbf{1})]^d \subset\mathbb{R}^d,
\end{align*}
where $\textbf{1} = (1,\ldots,1)\in\mathbb{N}^d$. Observe that for any $x\in\mathbb{N}^d$, we can place
\begin{align*}
\left\lfloor \frac{\sqrt{d}}{\rho(x + \textbf{1})}\right\rfloor^d =: r(x)
\end{align*}
many disjoint translated copies of $Q_x$ into the cube $x+[0,1]^d$. Denote these copies by $Q_x^1,\ldots Q_x^{r(x)}$. Observe also that for the diameter of each $Q_x^i$ {one has $\diam(Q_x^i) = \rho(x + \textbf{1})$.} Hence, any two distinct vertices $x,y\in \eta$ within the same cube are connected by an edge. Therefore,
\begin{align*}
N\geq \sum_{x\in\mathbb{N}^d} \sum_{i=1}^{r(x)}\binom{\eta(Q_x^i)}{2}.
\end{align*}
Now, the $\eta(Q_x^i)$ are independent Poisson random variables. Thus, by the second Borel-Cantelli lemma, the right hand side in the above display is almost surely non-finite if
\begin{align*}
\sum_{x\in\mathbb{N}^d} \sum_{i=1}^{r(x)} \P(\eta(Q_x^i)\geq 2) = \infty.
\end{align*}
The expectation of $\eta(Q_x^i)$ is given by
\begin{align*}
\lambda_x = \frac{t}{d^{d/2}} \rho(x + \textbf{1})^d.
\end{align*}
Using this we obtain
\begin{align*}
\sum_{x\in\mathbb{N}^d}\sum_{i=1}^{r(x)} \P(\eta(Q_x^i)\geq 2) &= \sum_{x\in\mathbb{N}^d} r(x)(1 - e^{-\lambda_x} - \lambda_x e^{-\lambda_x}).
\end{align*}
Since $1-e^{-z}-ze^{-z} \geq (1/4) z^2$ for $z\in[0,1]$ and since $\lambda_x\to 0 $ as $\lVert x\rVert\to\infty$, the above series is non-finite if
\begin{align*}
\sum_{x\in\mathbb{N}^d} r(x) \lambda_x^2 = \infty \ \ \ \text{or equivalently} \ \ \ \sum_{x\in\mathbb{N}^d} \rho(x + \textbf{1})^d = \infty.
\end{align*}
It is easy to see that the above is implied by condition (\ref{infEdgesCond}).
\end{proof}
\section{Concentration for the convex distance in Poisson-based models}\label{ss:convex}
The convex distance for product spaces that was introduced by M. Talagrand in \cite{T_1995} has proved to be a very useful tool in the context of concentration inequalities -- see e.g. \cite[Chapter 11]{DP}, \cite[Chapter 6]{steele}, \cite[Chapter 2]{Tao} and the references therein. In the recent paper \cite{R_2013} by M. Reitzner, this notion has been adapted for models based on Poisson point processes with finite intensity measure. For both the product space and the Poisson space version, the method of using the convex distance to establish concentration properties is based on an isoperimetric inequality. {First applications of this method for Poisson-based models are worked out in \cite{RST_2013, LR_2015} where concentration inequalities for Poisson U-statistics are presented.}
The proof of the convex distance isoperimetric inequality in \cite{R_2013} uses an approximation of the Poisson process by binomial processes. The goal in this section is to give an alternative proof for this inequality. Apart from slightly worse constants, we entirely recover Reitzner's result \cite[Theorem 1.1]{R_2013} with the tools developed in the present work. In particular, we only use methods from Poisson process theory, thus answering the question proposed in \cite{R_2013} of whether such a direct proof is possible. Moreover, the assumptions on the space $\mathbb{X}$ for our results are less restrictive than in \cite{R_2013} where only locally compact second countable Hausdorff spaces are considered.
The upcoming presentation is based on \cite{BLM_2009} and \cite{BLM_2003} where the convex distance for product spaces is recovered using the entropy method.
\subsection{Convex distance for Poisson processes}
To introduce the convex distance for Poisson point processes, let ${\bf N}_{\rm fin} \subset {\bf N}$ denote the space of finite integer-valued measures on $\mathbb{X}$ which is equipped with the $\sigma$-algebra $\mathcal{N}_{\rm fin}$ obtained by restricting $\mathcal{N}$ to ${\bf N}_{\rm fin}$. We will write $\xi(x)=\xi(\{x\})$ whenever $\xi\in{\bf N}_{\rm fin}$ and $x\in\mathbb{X}$ in order to simplify notations. For any two measures $\xi,\nu\in{\bf N}_{\rm fin}$, we define the measure $\xi\setminus\nu$ by
\begin{align*}
\xi\setminus\nu = \sum_{x\in\xi}(\xi(x) - \nu(x))_+ \delta_x,
\end{align*}
where $x\in\xi$ indicates that $x$ belongs to the support of $\xi$. The \emph{convex distance} $d_T(\xi, A)$ is now defined for any measurable set $A\in\mathcal{N}_{\rm fin}$ and $\xi\in{\bf N}_{\rm fin}$ by
\[
d_T(\xi,A) = \sup_{\lVert u \rVert_\xi \leq 1}\inf_{\nu\in A}\int_{\mathbb{X}} u\ d(\xi\setminus \nu),
\]
where the supremum ranges over all measurable maps $u:\mathbb{X} \to \mathbb{R}$ such that $\lVert u \rVert_\xi\leq 1$ and $\lVert \cdot \rVert_\xi$ denotes the $2$-norm with respect to the measure $\xi$. It is immediate from the above definition that
\begin{align}\label{e:tuba}
d_T(\xi,A) = \sup_{\lVert u \rVert_\xi \leq 1}\inf_{\nu\in A}\sum_{x\in\xi} u(x)(\xi(x)-\nu(x))_+.
\end{align}
The following result gives an alternative characterization for the convex distance which will be crucial for our proof of the isoperimetric inequality later on.
\begin{prop} \label{ConvDistProp}
Let $A\in\mathcal{N}_{\rm fin}$ and denote by $\mathcal{M}(A)$ the set of probability measures on $A$. Then, for any $\xi\in {\bf N}_{\rm fin}$ we have
\begin{align*}
d_T(\xi,A) &= \max_{\lVert u \rVert_\xi \leq 1}\min_{\zeta \in \mathcal{M}(A)}\sum_{x\in\xi} u(x) \mathbb{E}_{\zeta(\nu)}[(\xi(x) - \nu(x))_+]\\
&= \min_{\zeta \in \mathcal{M}(A)}\max_{\lVert u \rVert_\xi \leq 1}\sum_{x\in\xi} u(x) \mathbb{E}_{\zeta(\nu)}[(\xi(x) - \nu(x))_+],
\end{align*}
where, here and for the rest of the section, we use the shorthand notation
$$
\mathbb{E}_{\zeta(\nu)}[ h(\nu) ] = \int_A h(\nu ) d\zeta(\nu),
$$
for every positive measurable mapping $h : A\to \mathbb{R}_+$.
\end{prop}
\begin{proof}
Here we adapt arguments from the proof of \cite[Proposition 13]{BLM_2003}. We begin by proving that
\begin{align}\label{convDistEquation}
d_T(\xi,A) &= \sup_{\lVert u \rVert_\xi \leq 1}\inf_{\zeta \in \mathcal{M}(A)}\sum_{x\in\xi} u(x) \mathbb{E}_{\zeta(\nu)}[(\xi(x) - \nu(x))_+].
\end{align}
For any $\nu\in A$ consider the probability measure $\zeta_\nu\in \mathcal{M}(A)$ that is concentrated on $\nu$. Then
\[
\sum_{x\in\xi} u(x) \mathbb{E}_{\zeta_\nu(\nu')} [(\xi(x) - \nu'(x))_+] = \sum_{x\in\xi} u(x)(\xi(x) - \nu(x))_+.
\]
Hence, for any $u$ with $\lVert u \rVert_\xi \leq 1$, we have
\[
\inf_{\zeta \in \mathcal{M}(A)}\sum_{x\in\xi} u(x) \mathbb{E}_{\zeta(\nu)} [(\xi(x) - \nu(x))_+] \leq \inf_{\nu\in A}\sum_{x\in\xi} u(x)(\xi(x) - \nu(x))_+.
\]
On the other hand, for all $\zeta\in\mathcal{M}(A)$ we have
\begin{align*}
\inf_{\nu\in A} \sum_{x\in\xi} u(x)(\xi(x) - \nu(x))_+ &\leq \mathbb{E}_{\zeta(\nu)}\sum_{x\in\xi} u(x)(\xi(x) - \nu(x))_+\\
&= \sum_{x\in\xi} u(x) \mathbb{E}_{\zeta(\nu)}[(\xi(x) - \nu(x))_+].
\end{align*}
Thus,
\[
\inf_{\nu\in A} \sum_{x\in\xi} u(x)(\xi(x) - \nu(x))_+ \leq \inf_{\zeta\in\mathcal{M}(A)}\sum_{x\in\xi} u(x) \mathbb{E}_{\zeta(\nu)}[(\xi(x) - \nu(x))_+].
\]
This establishes equation (\ref{convDistEquation}).
We aim at applying Sion's minimax theorem \cite[Corollary 3.3]{S_1958}. To get prepared for this, first note that the supremum in \eqref{convDistEquation} can obviously by performed with respect to those functions $u:\mathbb{X}\to\mathbb{R}$ satisfying $u(x) = 0$ whenever $x\notin\xi$. Note also that these functions form a finite dimensional real vector space (whose dimension is given by $\#\{x\in\xi\}$) which will be denoted by $U$. So the supremum is actually taken over
\begin{align*}
U_{\leq 1} = \{u \in U : \lVert u \rVert_\xi \leq 1\}
\end{align*}
which is a convex and compact subset of $U$. Denote by $Q$ the finite set of maps $q:\mathbb{X}\to\mathbb{N}_0$ satisfying $q(x)\leq \xi(x)$ for all $x\in\xi$ and $q(x)=0$ whenever $x\notin\xi$. Moreover, define the map $I$ by
\begin{align*}
I:A \to Q, \nu \mapsto (x\mapsto (\xi(x) - \nu(x))_+).
\end{align*}
Then, for any $\zeta\in\mathcal{M}(A)$ and $x\in\xi$ we have
\[
\mathbb{E}_{\zeta(\nu)} [(\xi(x) - \nu(x))_+] = \mathbb{E}_{I\zeta(q)}[q(x)],
\]
where $I\zeta$ denotes the pushforward measure of $\zeta$ with respect to $I$. Now, instead of taking the infimum in $\mathcal{M}(A)$ we can also minimize in the set of pushforward measures $I\mathcal{M}(A)$. The set $I\mathcal{M}(A)$ coincides with the set of probability measures on $I(A)$, denoted by $\mathcal{M}I(A)$. Observe that $\mathcal{M} I(A)$ is a convex and compact subset in the finite dimensional real vector space of all signed measures on $I(A)$ which we denote by $\mathcal{S}I(A)$. Obviously, the map
\[
U\times \mathcal{S}I(A)\to\mathbb{R}, \ (u,\zeta)\mapsto \sum_{x\in\xi} u(x) \mathbb{E}_{\zeta(q)} [q(x)],
\]
is both linear in $u$ and $\zeta$. Hence, it is also upper semicontinuous and quasi-concave in $u$ and lower semicontinuous and quasi-convex in $\zeta$. According to the above considerations, the assumptions of Sion's theorem are satisfied and we obtain
\begin{align*}
&\sup_{\lVert u \rVert_\xi \leq 1}\inf_{\zeta \in \mathcal{M}(A)}\sum_{x\in\xi} u(x) \mathbb{E}_{\zeta(\nu)} [(\xi(x) - \nu(x))_+] = \sup_{u\in U_{\leq 1}}\inf_{\zeta \in \mathcal{M}I(A)} \sum_{x\in\xi} u(x) \mathbb{E}_{\zeta(q)} [q(x)]\\ &= \inf_{\zeta \in \mathcal{M}I(A)} \sup_{u\in U_{\leq 1}} \sum_{x\in\xi} u(x) \mathbb{E}_{\zeta(q)} [q(x)] = \inf_{\zeta \in \mathcal{M}(A)} \sup_{\lVert u \rVert_\xi \leq 1} \sum_{x\in\xi} u(x) \mathbb{E}_{\zeta(\nu)} [(\xi(x) - \nu(x))_+].
\end{align*}
Since both $U_{\leq 1}$ and $\mathcal{M} I(A)$ are compact, the suprema and infima are actually maxima and minima.
\end{proof}
\subsection{Convex distance inequality}
In what follows, we will give the announced new proof of the convex distance inequality for Poisson point processes. The result we aim to prove is the following:
{
\begin{thm}\label{convDistIneq}
Let $\eta$ be a Poisson point process in $\mathbb{X}$ with finite intensity measure $\mu$. Let $A\in\mathcal{N}_{\rm fin}$ be arbitrary. Then
\begin{align*}
\P(\eta\in A) \mathbb{E}( e^{d_T(\eta, A)^2/ 10}) \leq 1.
\end{align*}
In particular, for any $r\geq 0$,
\begin{align}\label{e:r/9}
\P(\eta\in A)\P(d_T(\eta,A)\geq r) \leq e^{-r^2/10}.
\end{align}
\end{thm}
}
Note that in \cite[Theorem 1.1]{R_2013} (under more restrictive assumptions on the space $(\mathbb{X}, \mathcal{X}, \mu)$), an inequality stronger than \eqref{e:r/9} is proved, where the {constant 1/10} is {replaced by 1/4}. To get prepared for the proof of Theorem \ref{convDistIneq}, we first establish the following result. This is interesting in its own right since it particularly states that the variance of the convex distance is bounded by $1$.
\begin{prop}\label{convDistProp}
Let $\eta$ be a Poisson point process on $\mathbb{X}$ with finite intensity measure $\mu$. Then for any $A\in\mathcal{N}_{\rm fin}$ almost surely
\begin{align*}
V^+(d_T(\eta, A)) = \int_\mathbb{X}(D_x d_T(\eta - \delta_x, A))^2 d\eta(x) \leq 1.
\end{align*}
In particular, $\mathbb{V} d_T(\eta, A)\leq 1$.
\end{prop}
\begin{proof}
For this proof we adapt arguments from the proof of \cite[Proposition 13]{BLM_2003}. According to Proposition \ref{ConvDistProp} we can choose a map $\hat{u}:\mathbb{X}\to\mathbb{R}$ with $\lVert \hat{u}\rVert_\xi\leq 1$ and a probability measure $\hat{\zeta}$ on $A$ satisfying
\[
d_T(\xi,A) = \sum_{x\in\xi} \hat{u}(x) \mathbb{E}_{\hat{\zeta}(\nu)} [(\xi(x) - \nu(x))_+].
\]
Then, for any $z\in\xi$, we have
\[
d_T(\xi - \delta_z, A)\geq \min_{\zeta\in\mathcal{M}(A)} \sum_{x\in\xi-\delta_z} \hat{u}(x) \mathbb{E}_{\zeta (\nu)} [((\xi - \delta_z)(x) - \nu(x))_+].
\]
Choose some $\tilde{\zeta}\in\mathcal{M}(A)$ that achieves the minimum in the above right hand side. Then
\[
d_T(\xi,A)\leq \sum_{x\in\xi} \hat{u}(x) \mathbb{E}_{\tilde{\zeta} (\nu)} [(\xi(x) - \nu(x))_+].
\]
It follows that
\begin{align*}
&d_T(\xi,A) - d_T(\xi- \delta_z,A)\\ &\leq \sum_{x\in\xi} \hat{u}(x) \mathbb{E}_{\tilde{\zeta} (\nu)} [(\xi(x) - \nu(x))_+] - \sum_{x\in\xi-\delta_z} \hat{u}(x) \mathbb{E}_{\tilde{\zeta} (\nu)} [((\xi - \delta_z)(x) - \nu(x))_+]\\
&= \hat{u}(z) \mathbb{E}_{\tilde{\zeta} (\nu)} [(\xi(z) - \nu(z))_+ - (\xi(z) - \nu(z) - 1)_+]\\
&= \hat{u}(z) \mathbb{E}_{\tilde{\zeta} (\nu)} [\mathbbm{1}\{\xi(z) > \nu(z)\}] \leq \hat{u}(z).
\end{align*}
This yields
\[
\int_\mathbb{X} (d_T(\xi,A) - d_T(\xi- \delta_z,A))^2 d\xi(z) \leq \int_\mathbb{X} \hat{u}(z)^2 \xi(z) = \lVert \hat{u}\rVert_\xi^2 \leq 1.
\]
Hence, almost surely
\[
\int_\mathbb{X} (D_x d_T(\eta - \delta_x, A))^2 d\eta(x)\leq 1.
\]
Applying the Poincare Inequality for Poisson processes (see e.g. \cite[Remark 1.4]{W_2000}) yields $\mathbb{V} d_T(\eta, A) \leq 1$.
\end{proof}
As a final ingredient for the upcoming proof of the convex distance inequality, we derive the following consequence of the Cauchy-Schwarz Inequality.
\begin{lem} \label{cauchySchwarzApp}
Let $\xi\in{\bf N}_{\rm fin}$ and consider the measure space $(\mathbb{X}, \mathcal{X}, \xi)$. Then for any measurable map $h:\mathbb{X}\to\mathbb{R}$,
\begin{align} \label{cauchySchwarzEq}
\sup_{\lVert u \rVert_\xi \leq 1} \int_\mathbb{X} u(x)h(x) d\xi(x) = \lVert h \rVert_\xi.
\end{align}
\end{lem}
\begin{proof}
Note that $h$ is of course square-integrable with respect to $\xi$. Hence, by the Cauchy-Schwarz Inequality, for any $u$ such that $\lVert u\rVert_\xi\leq 1$,
\begin{align*}
\int_\mathbb{X} u(x)h(x) d\xi(x) \leq \lVert u\rVert_\xi \lVert h\rVert_\xi \leq \lVert h\rVert_\xi.
\end{align*}
We see that the LHS in (\ref{cauchySchwarzEq}) is less or equal to the RHS. Moreover, we can take $u = h / \lVert h \rVert_\xi$ to conclude that the RHS is less or equal to the LHS.
\end{proof}
\begin{proof}[Proof of Theorem \ref{convDistIneq}]
Here we adapt arguments from the proofs of \cite[Lemma 1 and Corollary 1]{BLM_2009}. We will prove below that
\begin{align}
\label{cond1}&0\leq D_x (d_T(\xi,A)^2) \leq 2 \ \ \text{for any} \ \ (x,\xi)\in\mathbb{X}\times{\bf N}_{\rm fin}\\
\label{cond2}&\text{and almost surely} \ \ V^+(d_T(\eta,A)^2) \leq 4 d_T(\eta,A)^2.
\end{align}
Hence, if follows from Theorem \ref{arbSignThm} and Theorem \ref{vPlusLowerThm} {(where the latter result is applied to the Poisson functional $\tfrac 12 d_T(\eta,A)^2$)} that
\begin{enumerate}
\item For any $\lambda\in(0,1/2)$,
\begin{align*}
\log \mathbb{E}(\exp(\lambda(d_T(\eta,A)^2 - \mathbb{E} d_T(\eta,A)^2))) \leq \frac{2\lambda^2 \mathbb{E} d_T(\eta,A)^2}{1 - 2\lambda},
\end{align*}
\item For any $r\geq 0$,
\begin{align*}
\P(d_T(\eta,A)^2\leq \mathbb{E} d_T(\eta,A)^2 - r) \leq \exp\left(-\frac{r^2}{8\mathbb{E} d_T(\eta,A)^2}\right).
\end{align*}
\end{enumerate}
{Taking $\lambda = 1/10$} we obtain from (i) that
\begin{align*}
\mathbb{E} e^{d_T(\eta,A)^2/{10}} \leq \exp\left(\frac{\mathbb{E} d_T(\eta,A)^2}{8}\right).
\end{align*}
Moreover, since $\eta\in A$ implies $d_T(\eta,A) = 0$, it follows from (ii) with $r = \mathbb{E} d_T(\eta,A)^2$ that
\begin{align*}
\P(\eta\in A) \leq \P\left(d_T(\eta, A)^2 \leq \mathbb{E} d_T(\eta,A)^2 - \mathbb{E} d_T(\eta,A)^2\right) \leq \exp\left(-\frac{\mathbb{E} d_T(\eta,A)^2}{8}\right).
\end{align*}
So, the result follows once we have proven (\ref{cond1}) and (\ref{cond2}). To prove (\ref{cond2}), first observe that $d_T(\cdot,A)$ is a non-decreasing functional. Using this and Proposition \ref{convDistProp} we compute
\begin{align*}
V^+(d_T(\eta,A)^2) &= \int_\mathbb{X} (d_T(\eta,A)^2 - d_T(\eta-\delta_x,A)^2)^2 \ d\eta(x)\\
&= \int_\mathbb{X} (d_T(\eta,A) - d_T(\eta-\delta_x,A))^2 (d_T(\eta,A) + d_T(\eta-\delta_x,A))^2 \ d\eta(x)\\
&\leq \int_\mathbb{X} (d_T(\eta,A) - d_T(\eta-\delta_x,A))^2 4d_T(\eta,A)^2 \ d\eta(x)\\
&= 4d_T(\eta,A)^2 \int_\mathbb{X} (D_x d_T(\eta-\delta_x,A))^2 \ d\eta(x) \leq 4d_T(\eta,A)^2.
\end{align*}
It remains to prove (\ref{cond1}). For this, let $(z,\xi)\in\mathbb{X}\times {\bf N}_{\rm fin}$. Then, according to Proposition \ref{ConvDistProp}, we can write
\begin{align*}
d_T(\xi,A) &= \max_{\lVert u \rVert_{\xi} \leq 1} \sum_{x\in\xi} u(x) \mathbb{E}_{\hat{\zeta}(\nu)} [(\xi(x)-\nu(x))_+]\\
&=\max_{\lVert u \rVert_{\xi} \leq 1} \int_\mathbb{X} u(x) \mathbb{E}_{\hat{\zeta}(\nu)} \left[\left(1-\frac{\nu(x)}{\xi(x)}\right)_+\right] \ d\xi(x)
\end{align*}
for some probability measure $\hat{\zeta}$ on $A$. By virtue of Lemma \ref{cauchySchwarzApp}, the latter expression equals
\begin{align*}
\sqrt{\int_\mathbb{X} \left(\mathbb{E}_{\hat{\zeta}(\nu)} \left[\left(1-\frac{\nu(x)}{\xi(x)}\right)_+\right]\right)^2 d\xi(x)}.
\end{align*}
Invoking Proposition \ref{ConvDistProp} and again Lemma \ref{cauchySchwarzApp}, we also obtain
\begin{align*}
d_T({\xi + \delta_z},A) &\leq \max_{\lVert u \rVert_{\xi + \delta_z} \leq 1} \sum_{x\in\xi+\delta_z} u(x) \mathbb{E}_{\hat{\zeta}(\nu)} [((\xi + \delta_z)(x)-\nu(x))_+]\\
&= \sqrt{\int_\mathbb{X} \left(\mathbb{E}_{\hat{\zeta}(\nu)} \left[\left(1-\frac{\nu(x)}{(\xi + \delta_z)(x)}\right)_+\right]\right)^2 d(\xi + \delta_z)(x)}.
\end{align*}
From this it follows that
\begin{align*}
D_zd_T^2 \leq \left(\mathbb{E}_{\hat{\zeta}(\nu)} \left[\left(1-\frac{\nu(z)}{\xi(z)+1}\right)_+\right]\right)^2 (\xi(z)+1) - \left(\mathbb{E}_{\hat{\zeta}(\nu)} \left[\left(1-\frac{\nu(z)}{\xi(z)}\right)_+\right]\right)^2 \xi(z)
\end{align*}
where the subtrahend vanishes whenever $\xi(z)=0$. {Clearly, if $\xi(z)=0$, then the RHS in the above display is less or equal to $1$. So assume that $\xi(z)>0$.} Then, using the abbreviations $G(\nu,z) = (\xi(z)-\nu(z)+1)_+$ and $G'(\nu,z) = (\xi(z)-\nu(z))_+$, one observes that the RHS in the last display can be upper bounded by
\begin{align*}
\frac{(\mathbb{E}_{\hat{\zeta}(\nu)} G(\nu,z))^2 - (\mathbb{E}_{\hat{\zeta}(\nu)} G'(\nu,z))^2}{\xi(z)+1} &= \frac{\mathbb{E}_{\hat{\zeta}(\nu)} [G(\nu,z) - G'(\nu,z)]\ \mathbb{E}_{\hat{\zeta}(\nu)}[G(\nu,z) + G'(\nu,z)]}{\xi(z)+1}\\
&\leq \frac{\mathbb{E}_{\hat{\zeta}(\nu)}[G(\nu,z) + G'(\nu,z)]}{\xi(z)+1}\\
&=\mathbb{E}_{\hat{\zeta}(\nu)}\left[ \frac{(\xi(z) - \nu(z) + 1)_+ + (\xi(z) - \nu(z))_+}{\xi(z)+1}\right]\\
&\leq 2.
\end{align*}
It follows that $D_z(d_T(\xi,A)^2) \leq 2$.
The functional $d_T(\cdot, A)$ is non-decreasing and non-negative, thus $D_z(d_T(\xi,A)^2)\geq 0$. This concludes the proof.
\end{proof}
\renewcommand{\bibname}{References}
\defbibheading{bibliography}[\refname]{\section*{#1}}
\printbibliography
\end{document}
|
{
"timestamp": "2015-04-14T02:13:42",
"yymm": "1504",
"arxiv_id": "1504.03138",
"language": "en",
"url": "https://arxiv.org/abs/1504.03138"
}
|
\section{Introduction}\label{sec:introduction}
The cosmological principle, which is one of the foundations in modern cosmology, says that the Universe is homogeneous and isotropic at large enough scales \citep{Dodelson:2003ft}. It is well consistent with the present observational data, such as the cosmic microwave background (CMB) radiation from Wilkinson Microwave Anisotropy Probe (WMAP) \citep{Bennett:2012zja,Hinshaw:2012aka} and Planck satellite \citep{Ade:2015xua,Ade:2015lrj}. Until now, the cosmological observations are still in accordance with the cosmological constant plus cold dark matter ($\Lambda$CDM) model which is based on the cosmological principle. Thus, the $\Lambda$CDM model becomes the leading model in modern cosmology.
Despite the great successes it achieved, the $\Lambda$CDM model still faces certain challenges \citep{Perivolaropoulos:2008ud,Perivolaropoulos:2011hp,Mariano:2012ia}. As the improvements of accuracy, it is found from a large amount of observations that the Universe might deviate from statistical isotropy. These include the alignment of low multipoles in the angular power spectrum of CMB temperature fluctuations \citep{Tegmark:2003ve,Bielewicz:2004en,Copi:2010na,Frommert:2009qw}, the hemispherical power asymmetry of CMB temperature anisotropy \citep{Bennett:2012zja,Ade:2013nlj,Eriksen:2003db,Hansen:2004vq,Akrami:2014eta,Quartin:2014yaa}, the spatial variation of the electromagnetic fine-structure constant \citep{Webb:2010hc,King:2012id,Molaro:2013saa}, the large-scale alignment of the quasar polarization vectors \citep{Hutsemekers:2000fv,Hutsemekers:2005iz}, the large-scale bulk flow beyond the prediction of $\Lambda$CDM model \citep{Kashlinsky:2008ut,Kashlinsky:2009dw,Watkins:2008hf}, and so on. All of these phenomena arouse us to rethink the validity of the cosmological principle. If the cosmological principle is proven to be failed, the modern cosmology should be rewritten.
Due to their consistent absolute magnitudes, type-Ia supernovae (SNe Ia) are regarded as the ideal distance indicators to trace the accelerated expansion of the Universe. In fact, they have been widely used to search for the anisotropic signals in the Universe. Especially, a statistic based on the extreme value theory shows that the gold data set is consistent with the isotropy \citep{Gupta:2007pb}. The study \citep{Blomqvist:2010ky} on the angular covariance function of supernova magnitude fluctuations is consistent with zero dark energy fluctuations by using the Union2 compilation \citep{Amanullah:2010vv}. A ``\,residual'' statistic shows that the isotropic $\Lambda$CDM model is not consistent with the Union2 data with $z<0.05$ at $2-3\sigma$ \citep{Colin:2010ds}. There are no significant evidence for deviations from the isotropy in the anisotropic Bianchi-I cosmology \citep{Campanelli:2010zx,Schucker:2014wca}, Bianchi-III and Kantowski-Sachs metrics \citep{Koivisto:2010dr}, and Randers-Finsler cosmology \citep{Chang:2014wpa,Chang:2013xwa,Chang:2013zwa}. The hemisphere comparison is used to study the Union2 data and shows certain preferred directions \citep{Schwarz:2007wf,Antoniou:2010gw,Cai:2011xs,Kalus:2012zu,Yang:2013gea,Chang:2014nca}. By dividing the Union2 supernovae into 12 subsets according to their galactic coordinates, a dipole of the deceleration parameter is preferred at more than $2\sigma$ level \citep{Zhao:2013yaa}. By combining the data of Union2 and gamma-ray bursts, the isotropic $\Lambda$CDM model is well permitted \citep{Cai:2013lja} while the anisotropic Finsler cosmology is preferred at around $2\sigma$ \citep{Chang:2014jza}. By using the data of Union2.1 \citep{Suzuki:2011hu} and gamma-ray bursts, a model-independent way shows a dipolar anisotropy at more than $2\sigma$ \citep{Wang:2014vqa}. It has been found that there may be certain correlation between the fine structure dipole and the dark energy dipole \citep{Mariano:2012wx,Li:2015uda}. A fully-Bayesian method was developed to remove the systematics in the Union datasets and the anisotropic cosmology does not seem to be reflected \citep{Heneka:2013hka}.
Recently, a new sample of SNe Ia was released by the SDSS collaboration, which is called the ``\,joint light-curve analysis'' (JLA) compilation \citep{Betoule:2014frx}. Compared to previous compilations such as Union2 \citep{Amanullah:2010vv} and Union2.1 \citep{Suzuki:2011hu}, the number of SNe Ia in the JLA compilation is highly enlarged and the systematic uncertainties are significantly reduced. Recently, the JLA SNe Ia have been used to probe the anisotropic Hubble diagram in Bianchi type I cosmology \citep{Schucker:2014wca} and test the cosmological principle \citep{Bengaly:2015dza}. However, the work \citep{Bengaly:2015dza} did not consider the full covariance matrix between SNe Ia. In this paper, we use the JLA compilation to restudy the anisotropic Hubble diagram of the Universe. Unlike certain previous works which have neglected the correlations between any two SNe Ia, we make use of the full covariance matrix to construct the likelihood (or chi-square). In addition, we use the method of MCMC sampling in our analysis. It has been shown that the statistical significance of the previously claimed evidence for a preferred direction could be highly lowered if the full covariance matrix of SNe Ia is considered in the Union2 compilation \citep{Jimenez:2014jma}. We want to see whether the anisotropic signals in the accelerated expansion of the Universe still exist in the newly released JLA compilation.
The rest of the paper is arranged as follows. In section \ref{sec:modelsandmethodology}, we briefly introduce the JLA dataset, and present the anisotropic cosmological models and the numerical method used in our analysis. In section \ref{sec:results}, we give constraints on the anisotropic amplitudes and directions for the anisotropic expansion of the Universe. Finally, our conclusions are given in section \ref{sec:conclusion}.
\section{Data and Methodology}\label{sec:modelsandmethodology}
The anisotropic expansion of the Universe can be induced by assuming that the dark energy has anisotropic repulsive force \citep{ArmendarizPicon:2004pm,Koivisto:2008xf,Salehi:2015ira}, or the background spacetime has a certain preferred direction \citep{Chang:2013xwa,Chang:2013zwa,Li:2013vea,Li:2015uda,Schucker:2014wca}, and so on. In this paper, we assume a dipole modulation to describe the Universe deviating from the isotropic background. Phenomenologically, the direction-dependent distance modulus can be given as
\begin{equation}
\label{muth}
\mu_{\rm th}=\bar{\mu}_{\rm th}\left(1+A_D (\hat{\textbf{n}}\cdot\hat{\textbf{p}})\right),
\end{equation}
where $A_D$ denotes the amplitude of the dipole modulation, $\hat{\textbf{n}}$ is the dipole direction, $\hat{\textbf{p}}$ is the unit 3-vector pointing towards the supernova, and $\bar{\mu}_{\rm th}$ denotes the theoretical distance modulus predicted by the isotropic $\Lambda$CDM, $w$CDM or CPL models. Here the anisotropic amplitude $A_D$ is assumed to be a constant over the whole redshift range. In the galactic coordinates, the dipole direction $\hat{\textbf{n}}$ can be parameterized as $(l,b)$, where $l$ and $b$ are the longitude and latitude, respectively. In such a parametrization, we have $\hat{\textbf{n}}=\cos(b)\cos(l)\hat{\textbf{i}}+\cos(b)\sin(l)\hat{\textbf{j}}+\sin(b)\hat{\textbf{k}}$, where $\hat{\textbf{i}}$, $\hat{\textbf{j}}$, $\hat{\textbf{k}}$ are the unit vectors along the axes of a Cartesian coordinates system. The position of the $i$th supernova with galactic coordinates $(l_i,b_i)$ can be written as
$\hat{\textbf{p}}_i=\cos(b_i)\cos(l_i)\hat{\textbf{i}}+\cos(b_i)\sin(l_i)\hat{\textbf{j}}+\sin(b_i)\hat{\textbf{k}}$.
In the spatially-flat isotropic background spacetime, we can express the luminosity distance $d_L(z)$ of a supernova in terms of the redshift $z$,
\begin{equation}\label{luminositydistance}
d_L(z)=\frac{1+z}{H_0}\int_0^{z} \frac{dz^\prime}{E(z^\prime)}\ ,
\end{equation}
where $H_0=100h~\rm{km}~\rm{s}^{-1}~\rm{Mpc}^{-1}$ is the Hubble constant, and $E(z)$ is a function of redshift. The isotropic distance modulus $\bar{\mu}_{\rm th}$ can be given by
\begin{equation}
\bar{\mu}_{\rm th}=5\log_{10}\frac{d_L}{10~\rm{pc}}.
\end{equation}
In equation (\ref{luminositydistance}), the quantity $E(z)$ is a function of redshift $z$. The expression of $E(z)$ depends on a specific cosmological model. In the $\Lambda$CDM model, it can be expressed as
\begin{equation}
E^2(z)=\Omega_{m}(1+z)^3+(1-\Omega_{m})\ ,
\end{equation}
where $\Omega_{m}$ is the energy density of matter today. In the $w$CDM model, it can be expressed as
\begin{equation}
E^2(z)=\Omega_{m}(1+z)^3+(1-\Omega_{m})(1+z)^{3(1+w)}\ ,
\end{equation}
where $w\equiv p/\rho$ denotes the equation of state of dark energy. In the Chevallier-Polarski-Linder (CPL) parametrization \citep{Chevallier:2000qy,Linder:2002et}, the equation of state of dark energy is redshift-dependent, and it is parameterized by $w=w_0+w_1z/(1+z)$. In this case, $E(z)$ can be expressed as
\begin{equation}
E^2(z)=\Omega_{m}(1+z)^3+(1-\Omega_{m})(1+z)^{3(1+w_0+w_1)}\exp{\left(-3w_1\frac{z}{1+z}\right)}\ .
\end{equation}
In this paper, we use the most recently published JLA compilation \citep{Betoule:2014frx} of SNe Ia to constrain the anisotropy of the Universe. The JLA compilation consists of 740 well-calibrated SNe Ia in the redshift range of $z\in[0.01,1.30]$. It is a collection of several low-redshift samples, all three seasons from the SDSS-II, three years from SNLS, and a few high-redshift samples from the Hubble Space Telescope (HST). All of the SNe Ia have high-quality light curves, so their distance moduli can be abstracted with high precision. The positions of SNe Ia in the sky of equatorial coordinates system can be found at the website of IAU Central Bureau for Astronomical Telegrams\footnote{http://www.cbat.eps.harvard.edu/lists/Supernovae.html}.
To compare with others' work, we transform the positions of SNe Ia into the galactic coordinates.
From the observational point of view, the distance modulus of a SN Ia can be abstracted from its light curve through the empirical linear relation \citep{Betoule:2014frx}
\begin{equation}\label{muobs}
\hat{\mu}=m_B^{*}-(M_B-\alpha\times X_1+\beta \times \mathcal{C}),
\end{equation}
where $m_B^*$ is the observed peak magnitude in rest frame $B$ band, $M_B$ is the absolute magnitude depending on the host galaxy properties complexly, $X_1$ is the time stretching of the light curve, and $\mathcal{C}$ is the supernova color at maximum brightness. The three light-curve parameters $m_B^*$, $X_1$ and $\mathcal{C}$ are different from one supernova to other one and can be derived directly from the light curves. The two nuisance parameters $\alpha$ and $\beta$ are assumed to be constants for all the supernovae.
For the JLA samples, the isotropic luminosity distance of a supernova can be given as
\begin{equation}
d_L(z_{\rm hel},z_{\rm cmb})=\frac{1+z_{\rm hel}}{H_0}\int_0^{z_{\rm cmb}} \frac{dz'}{E(z')},
\end{equation}
where $z_{\rm cmb}$ and $z_{\rm hel}$ denote the CMB frame redshift and heliocentric redshift, respectively. Then we can obtain the anisotropic distance modulus $\mu_{\rm th}$ in equation (\ref{muth}). Using the observed distance modulus $\hat{\mu}$ in equation (\ref{muobs}), the anisotropic cosmological models can be fitted to the JLA dataset by using the chi-square as
\begin{equation}\label{chijla}
\chi^2_{\rm JLA}=\left(\hat{\mu}-\mu_{\rm th}\right)^{\dagger}C^{-1}\left(\hat{\mu}-\mu_{\rm th}\right),
\end{equation}
where $C$ is the covariance matrix of $\hat{\mu}$, and it is presented in \citet{Betoule:2014frx}.
In order to directly compare with previous works, we also apply our method to the Union2 \citep{Amanullah:2010vv} dataset. The Union2 data set consists of 557 SNe Ia with well-observed redshift in the range of $z \in [0.015, 1.4]$. The distance moduli and their uncertainties are extracted from the SALT2 light-curve fitter. The directions of SNe Ia are well localized in the sky of the equatorial coordinates \citep{Blomqvist:2010ky}. The data in Union2 are usually assumed to be uncorrelated. Thus, the chi-square of Union2 is simplified to
\begin{equation}\label{union2}
\chi^2_{\rm Union2}=\sum\left(\frac{\hat{\mu}-\mu_{\rm th}}{\sigma_{\mu}}\right)^2,
\end{equation}
where the summation runs over all the SNe Ia data.
In this paper, we employ the Markov Chain Monte Carlo (MCMC) method to estimate the model parameters. The joint likelihood is given by $\mathcal{L}\propto\exp(-\chi^2/2)$. The nuisance parameters such as $\alpha$ and $\beta$ are marginalized. We modify the publicly available Cosmological Monte Carlo sampler (CosmoMC) \citep{Lewis:2002ah} to estimate the background parameters and anisotropic parameters. For $\Lambda$CDM model, the isotropic parameter can be well constrained by supernovae data. For $w$CDM and CPL models, however, the isotropic parameters can't be well constrained by using the supernovae data only. Following the method in \citet{Betoule:2014frx}, therefore, we combine the supernovae data with the Planck~2013 results of CMB temperature anisotropy (Planck2013), the WMAP9 observations of CMB polarizations (WP), and the SDSS-III BOSS DR11 Baryon acoustic oscillations data (BAO) to constrain the isotropic parameters. Once the isotropic parameters are given, we can fix them and fit the anisotropic parameters with JLA SNe Ia only.
\section{Results}\label{sec:results}
As was mentioned above, we study the anisotropic signals of the dipole-modulated $\Lambda$CDM, $w$CDM and CPL models by using the JLA sample. The nuisance parameters such as $\alpha$ and $\beta$ can be marginalized, since they are not model parameters with significant meanings. We just focus on studying the anisotropic signals, thus neglect the topic of model comparison. Our final results are listed in Table \ref{tab:parameters1}, where the models and the anisotropic amplitudes and preferred directions are given. The likelihood distributions of the anisotropic parameters $A_D$, $l$ and $b$ in three cosmological models are plotted in Figure~\ref{fig:figure1}. In the last panel of Figure~\ref{fig:figure1}, we also plot the distribution of $\chi^2_{\rm JLA}$.
\begin{table}
\centering
\begin{tabular}{cccc}
\hline\hline
parameters & $\Lambda$CDM & $w$CDM & CPL \\
\hline
$A_D$ & $<1.98\times10^{-3}$ & $<2.09\times10^{-3}$ & $<2.05\times10^{-3}$ \\
$l[^{\circ}]$ & $316_{-110}^{+107}$ & $320_{-104}^{+107}$ & $318_{-183}^{+177}$ \\
$b[^{\circ}]$ & $-5_{-60}^{+41}$ & $-4_{-61}^{+45}$ & $-8_{-54}^{+36}$ \\
\hline
\end{tabular}
\caption{The 95\% upper bound of dipole amplitude $A_D$, and the preferred direction $(l,b)$ with $1\sigma$ uncertainty in three cosmological models.} \label{tab:parameters1}
\end{table}
\begin{figure}
\centering
\subfigure[]{
\label{fig:subfig:a1}
\includegraphics[width=1.6in]{d_1d.eps}}
\subfigure[]{
\label{fig:subfig:b1}
\includegraphics[width=1.56in]{l_1d.eps}}
\subfigure[]{
\label{fig:subfig:c1}
\includegraphics[width=1.6in]{b_1d.eps}}
\subfigure[]{
\label{fig:subfig:c1}
\includegraphics[width=1.5in]{chi2_1d.eps}}
\caption{Likelihood distributions for the amplitude $A_D$ and direction $(l,b)$ of dipole modulation in three cosmological models. The distribution of $\chi^2_{\rm JLA}$ is also showed in the last panel.}
\label{fig:figure1}
\end{figure}
In the modulated $\Lambda$CDM model, the isotropic parameter can be well constrained by using the JLA dataset only. The result is $\Omega_M=0.295\pm 0.034$ \citep{Betoule:2014frx}, which is computed for a fixed fiducial value of $H_0 = 70~{\rm km}~{\rm s}^{-1}~{\rm Mpc}^{-1}$. The change of $H_0$ does not affect the best-fit value of $\Omega_M$. By using the MCMC approach, and fixing $\Omega_M$ at 0.295, the anisotropic amplitude is constrained as $A_D<1.98\times10^{-3}$ at $95\%$ C.L., which is consistent with the isotropy within $1\sigma$ uncertainty. This implies that the JLA compilation shows no significant evidence for the deviations from isotropy. In the galactic coordinates, the dipole direction points towards F $(l,b)=(316^{\circ}\,_{-110^{\circ}}^{+107^{\circ}}, -5^{\circ}\,_{-60^{\circ}}^{+41^{\circ}})$ at $68\%$ C.L. By contrast, the Union2 sample gives constraints on the dark energy dipole as $A_D=(1.3\pm0.6)\times10^{-3}$ and $(l,b)=(309.4^\circ\pm 18.0^\circ,-15.1^\circ\pm11.5^\circ)$ at $68\%$ C.L., which implies that the anisotropic expansion of the Universe is permitted at more than $2\sigma$ \citep{Mariano:2012wx}. However, the statistical significance can be highly lowered if the full covariance matrix of SNe Ia is considered in the Union2 compilation \citep{Jimenez:2014jma}. Our constraints on the anisotropic parameters are listed in the second column in Table~\ref{tab:parameters1}. Their likelihood distributions are illustrated by the solid curves in Figure~\ref{fig:figure1}.
In the modulated $w$CDM model, there is an extra background parameter, i.e., the equation of state of dark energy $w$. This parameter can't be well constrained by using the JLA data only. The combined constraint from Planck2013+WP+BAO+JLA gives $\Omega_M=0.303\pm 0.012$, $w=-1.027\pm 0.055$ and $H_0=68.50\pm 1.27~{\rm km}~{\rm s}^{-1}~{\rm Mpc}^{-1}$ \citep{Betoule:2014frx}. By fixing these parameters at their best-fitting central values, we use the MCMC approach to constrain the anisotropic parameters. Similarly to the above discussions, we obtain the $95\%$ upper bound on anisotropic amplitude $A_D<2.09\times10^{-3}$, and preferred direction $(l,b)=(320^{\circ}\,_{-104^{\circ}}^{+107^{\circ}}, -4^{\circ}\,_{-61^{\circ}}^{+45^{\circ}})$ at $68\%$ C.L., which are listed in the third column in Table~\ref{tab:parameters1}. Their likelihood distributions are illustrated by the dashed curves in Figure~\ref{fig:figure1}.
In the modulated CPL parametrization, there are two extra background parameters, i.e., the equation of state of dark energy parameters $w_0$ and $w_1$. These two parameters also can't be well constrained by using the JLA data only. The combined constraint from Planck2013+WP+BAO+JLA gives $\Omega_M=0.304\pm 0.012$, $w_0=-0.957\pm 0.124$, $w_1=-0.336\pm 0.552$ and $H_0=68.59\pm 1.27~{\rm km}~{\rm s}^{-1}~{\rm Mpc}^{-1}$ \citep{Betoule:2014frx}. By fixing these parameters at their best-fitting central values, the anisotropic amplitude is constrained to be $A_D<2.05\times10^{-3}$ at $95\%$ C.L., and the preferred direction $(l,b)=(318^{\circ}\,_{-183^{\circ}}^{+177^{\circ}}, -8^{\circ}\,_{-54^{\circ}}^{+36^{\circ}})$ at $68\%$ C.L. These results are listed in the last column in Table~\ref{tab:parameters1}. Their likelihood distributions are illustrated by the dash-dotted curves in Figure~\ref{fig:figure1}.
From Table~\ref{tab:parameters1}, we can see that all the three cosmological models give consistent results. At first glimpse, the dipole directions we obtained seem to be consistent with that of \citet{Mariano:2012wx}. However, this is in fact not the case. In our calculation, we constrain dipole amplitude to be non-negative, i.e., $A_D\geq 0$, and let the dipole direction runs over the whole sky. \citet{Mariano:2012wx} parameterized the dipole modulation as $\mu_{\rm th}=\bar{\mu}_{\rm th}(1-A_D (\hat{\textbf{n}}\cdot\hat{\textbf{p}}))$, which has a sign difference from our parametrization of equation (\ref{muth}). Therefore, our dipole direction is actually opposite to that obtained by \citet{Mariano:2012wx}.
To test if the discrepancy between our results and previous works is due to the different dataset or different method. We apply our method to Union2 dataset, such that direct comparison with previous results can be made. The constraint on isotropic $\Lambda$CDM model from Union2 dateset gives $\Omega_M=0.274\pm 0.040$, with fiducial parameter $H_0 = 70~{\rm km}~{\rm s}^{-1}~{\rm Mpc}^{-1}$. Then we use the MCMC approach to constrain the anisotropic parameters. We obtain $A_D=(0.54_{-0.54}^{+0.13})\times 10^{-3}$ and $(l,b)=(142^{\circ}\,_{-72^{\circ}}^{+41^{\circ}}, 11^{\circ}\,_{-27^{\circ}}^{+42^{\circ}})$ at $68\%$ C.L. Regardless of the much larger uncertainty, the dipole direction we obtained is consistent with that of \citet{Mariano:2012wx}. However, our results show that the Union2 dataset is consistent with isotropy at $1\sigma$ C.L., and the $95\%$ upper bound on anisotropy is $A_D<1.40\times 10^{-3}$. On the other hand, if we apply the least-square method, we obtain $A_D=1.10\times 10^{-3}$, and $(l,b)=(126^{\circ},18^{\circ})$. Both the anisotropic amplitude and preferred direction are well consistent with the results of \citet{Mariano:2012wx}.
\section{Conclusions}\label{sec:conclusion}
In this paper, we probed the possibly anisotropic expansion of the Universe by using the recently released JLA compilation of SNe Ia. We considered the dipole-modulated deviation from the isotropy in three different dark energy models. We obtained similar constraints on the anisotropic amplitude and direction in three cases. This indicates that the preferred direction anisotropy is insensitive to the isotropic background models. Especially, our MCMC studies show that the anisotropic amplitude has an upper bound $D<1.98\times 10^{-3}$ at $95\%$ C.L., and the dipole direction points towards $(l,b)=(316^{\circ}\,_{-110^{\circ}}^{+107^{\circ}},-5^{\circ}\,_{-60^{\circ}}^{+41^{\circ}})$ for the dipole-modulated $\Lambda$CDM model. These results imply that the there is no significant evidence for anisotropy in the JLA dataset. For comparison, we also applied MCMC method to the Union2 dataset, and we got $A_D<1.40\times 10^{-3}$ at $95\%$ C.L. The dipole direction of the Union2 points towards $(l,b)=(142^{\circ}\,_{-72^{\circ}}^{+41^{\circ}}, 11^{\circ}\,_{-27^{\circ}}^{+42^{\circ}})$ at $68\%$ C.L., which is consistent with previous results. We surprisingly found that the dipole direction derived from the JLA is approximately opposite to that from the Union2.
\section*{Acknowledgements}
We are grateful to Prof. Perivolaropoulos L. for useful comments and suggestions. This work has been funded by the National Natural Science Fund of China under grants Nos. 11375203, 11305181, 11322545, 11335012 and 11575271.
|
{
"timestamp": "2015-11-30T02:15:37",
"yymm": "1504",
"arxiv_id": "1504.03428",
"language": "en",
"url": "https://arxiv.org/abs/1504.03428"
}
|
\section{Acknowledgment}
The authors wish to thank the anonymous reviewers for their helpful feedback. The research presented in this paper is supported in part by the National Natural Science Foundation (61221063, U1301254), 863 High Tech Development Plan (2012AA011003) and 111 International Colaboration Program of China.
\bibliographystyle{abbrv}
\section{Introduction}
By decoupling the control plane from the data plane, Software-Defined Network (SDN) makes programmability a built-in feature for networks, thereby introducing automaticity and flexibility to the networking management. SDN has therefore been foreseen as the key technology that enables the next generation of networking paradigm. Despite its promise, one of the most significant barriers towards SDN's wide practical deployment resides in overwhelming security concerns. Therefore, proactively detecting, quantifying, and mitigating its security vulnerabilities becomes of fundamental importance.
In spite of its novelty, SDN indeed reuses various design and implementation elements ranging from architectures and protocols to systems from traditional network. It is not surprising that SDN inheres the vulnerabilities intrinsic to these elements. For example, similar to any networked service, secure channels between controllers and switches might be disrupted by DDoS attacks; like firewall rules, the flow entries may also conflict with each other, leaking unwanted traffic; malicious arp spoofing generated by attackers may poison the controller MAC table, disturbing the normal topology information gathering and packet forwarding; untrusted applications may instrument SDN controller to perform malicious behaviors without proper access control, which is one of the design objectives for modern operating systems. In response, existing research in the context of SDN security mainly focuses on detecting and mitigating these vulnerabilities. For example, ~\cite{OpenFlow Vulnerability Assessment} evaluates man-in-the-middle attacks that target at SDN/OpenFlow secure channels; FortNOX~\cite{FortNOX} brings security enforcement module into NOX~\cite{NOX} and enables real-time flow entry conflict check; VeriFlow~\cite{VeriFlow} detects network-wide invariant violations by acting as a transparent layer between control plane and data plane.
In this paper, we introduce a novel SDN vulnerability. The novelty of this vulnerability stems from the feedback-loop nature of SDN, a fundamental difference compared with traditional networks.
Specifically, most commercial SDN/OpenFlow switches have limited flow table capacities, ranging from hundreds to thousands~\cite{PAST}. Such capacity is usually insufficient to handle millions of flows that are typical for enterprise and data center networks~\cite{Network traffic characteristics of data centers in the wild}. Nevertheless, the flow table capacity was just considered as a potential bottleneck of resource consuming attacks in the past, motivating researches on flow caching systems like \cite{Flow caching for high entropy packet fields}, \cite{CAB} and \cite{CacheFlow}. But according to our analysis, the flow table capacity can lead to inference attack and privacy leakage under certain circumstances.
As a consequence of flow table overflow, the SDN controller needs to dynamically maintain the flow table by inserting and deleting flow entries. The maintaining process typically include packet information transferring, routing rule calculation and flow entry deployment, which leads to measurable network performance decrease.
Particularly, once the flow table is full, extra interactions between controller and switch are needed to remove certain existing flow entries to make room for newly generated flow entries, resulting in further network performance decrease. An attacker can therefore leverage the perceived performance change to deduce the internal state of the SDN. To be more specific, we consider the scenario that an attacker resides in a network that is managed by a SDN. The attacker can then actively generate network traffic, triggering the interactions between the controller and switch with respect to flow entry insertion and deletion. The attacker can then measure the change of the network performance to estimate the internal state of the SDN including the flow table capacity and flow table usage. We have designed innovative algorithms to exploit this vulnerability and quantify their effectiveness on exploiting this vulnerability based on extensive evaluation.
To summarize, in this paper we made the following contributions:
\begin{itemize}
\item We have identified a novel vulnerability introduced by the limited flow table capacities of SDN/OpenFlow switches and formalized that threat.
\item We have designed effective algorithms that can successfully exploit this vulnerability to accurately infer the internal states of the SDN network including flow table capacity and flow table usage.
\item We have performed extensive evaluation to quantify the effectiveness of proposed algorithms. The experimental results have demonstrated that the discovered vulnerability indeed leads to significant security concerns: our algorithm can infer the network parameters with an accuracy of 80\% or higher across various network settings.
\end{itemize}
The rest of this paper is organized as follows. Section \ref{chap:background} gives an overview of some background information. Section \ref{chap:problem-statement} gives an overall statement of the inference attack problem. Section \ref{chap:fifo-inference-algorithm} and \ref{chap:lru-inference-algorithm} give detailed inference algorithms targeting at FIFO and LRU replacement algorithms respectively. Section \ref{chap:evaluation} gives a detailed evaluation of the simulation results. Section \ref{chap:discussion} is a brief discussion about our findings and future research. Finally, section \ref{chap:conclusion} concludes this paper.
\section{Background}\label{chap:background}
\subsection{Software-Defined Network}
Software Defined Network (SDN) is a competitive solution for next-generation network. SDN offers network programmability by separating the control plane from the data plane. Network functions like routing calculation and link discovery are extracted from switches (data plane) and implemented by centralized controllers (control plane). OpenFlow~\cite{OpenFlow Introduction} is the most prominent SDN implementation.
In a SDN network, the controller gathers network topology information and makes high-level routing decisions while the switches only perform the functionality of packet forwarding according to routing rules assigned by the controller. The dedicated link connecting controller and switch is called secure channel. Controller and switch communicates via secure channel using the OpenFlow protocol. Controller also exposes network control APIs or north-bound interfaces so network administrators can write their own network management applications to more effectively run their networks.
\subsection{SDN Datacenter Network}
The decoupled nature of SDN introduces programmability, automaticity and flexibility to the networking management, making SDN a popular solution for large-scale datacenter networks.
Figure \ref{fig:sdn-datacenter} is a network structure comparison showing the difference between traditional datacenter network and SDN-based datacenter network.
\begin{figure}[H]
\centering
\includegraphics[scale=0.3]{figures/sdn-datacenter.png}
\caption{SDN/OpenFlow Structure}
\label{fig:sdn-datacenter}
\end{figure}
In traditional datacenter network, the administrator needs to configure switches separately. Switches from different vendors with different managing tools become a major obstacle in network management. What's worse, each switch only handles a fragment of the whole network, the lack of global network topology makes it impossible to optimize network traffic dynamically and globally.
Compared with traditional network, SDN reveals great potential in datacenter networks because of its attractive advantages. The SDN controller stores global network topology so that the network can be efficiently optimized. The unified network APIs provided by controller make it easy to perform network management. There have been several successful commercial deployments of SDN in datacenter networks. B4~\cite{B4} network implemented by Google utilizes SDN to schedule network traffic between Google's global datacenters and achieves the maximum link usage of almost $100$ percent; Microsoft deploys SWAN~\cite{SWAN} to dynamically re-configure routing paths of network traffic to optimize inter-datacenter network utilization. Evaluation shows that SWAN can carry 60\% more traffic than traditional network.
\subsection{Information Leakage in Datacenter Network}
In modern datacenter networks like Microsoft Azure and Amazon EC2, customer VMs are usually multiplexed across shared physical infrastructures. Besides achieving a high utilization of hardware and software resources, this approach also introduces new vulnerabilities. Previously published researches have shown that transparently shared physical infrastructures can lead to potential cross-VM information leakage.
At first glance information leakage might seem innocuous, but in fact it is quite useful for clever attackers and will bring security issues in many aspects. The leakage of cache miss information is used in extraction and inference of RSA~\cite{RSA} and AES~\cite{AES} secret keys. The leakage of inter-keystroke time information can be used to perform recovery of the password in keystroke timing attack. The leakage of network topology might provide weapons to attackers because some attacks are only possible when attacker's VM is executed on the same physical server with victim's VM~\cite{Information Leakage}. The leakage of network performance information might make it possible for commercial spies to estimate the number of visitors to a co-resident server belonging to competitors and further infer the operation situation of their company. Thus information leakage in datacenter networks is drawing more and more attentions in network security and privacy researches.
\subsection{Flow Table Capacity}
Flow table is a hardware structure in OpenFlow switch, it stores hundreds to thousands of routing rules called flow entries. These flow entries are generated and assigned by the controller. Every time a network packet arrives in the switch, the switch will look up its flow table to find corresponding flow entries. If there exists a corresponding flow entry, the switch will forward the network packet according to the actions associated with that flow entry. If there is no flow entry matching this network packet, the switch will send the packet to the controller through the secure channel, then controller will calculate and generate a new flow entry and assign it to the switch.
Previous works typically assume that the flow table of each switch can hold an infinite number of flow entries, which makes the controller easy to design. In practice, however, this assumption does not hold, and the switch flow table capacity can become a significant bottleneck to scaling SDN networks. SDN/OpenFlow switch flow tables often cannot scale beyond a few hundred entries, because they typically include wildcards, and therefore are implemented using either complex and slow data structures, or expensive and extremely power-hungry ternary content-addressable memories (TCAM). Typical SDN/OpenFlow switches have rather limited flow table capacities from 750 to 3000 flow entries while handling about 100,000 concurrent flows in data centers. The flow table capacity bottleneck leads to potential flow table overflow, which is unacceptable.
Combine the flow table capacity issue of SDN switches and the resource sharing phenomenon in SDN-based datacenter networks, we discover the possibility of performing inference attacks targeting at SDN vulnerabilities. A formalized problem statement will be given in next section.
\section{Problem Statement}\label{chap:problem-statement}
Considering the wide use of SDN/OpenFlow in data center networks, we assume the inference attack scene to be in a SDN-based multi-tenant datacenter network like Amazon EC2 or Microsoft Azure. In a SDN-based datacenter network, different tenants connected to the same switch will share the flow table space. The flow table capacity's importance as key network parameter and the possibility of inference attack hidden behind the flow table sharing phenomenon make it natural for us to take flow table capacity as our primary inference target. Besides inferring intrinsic and static property of the switch (flow table capacity), further inference should be performed on the flow table usage condition of other tenants in the same datacenter, which reflects the real-time dynamic resource consuming situation in datacenter networks. So we choose flow table usage as our secondary inference target. But inferring flow table capacity and flow table usage is not that easy.
As cloud computing infrastructures, data center networks are typically well-managed and equipped with advanced firewalls and intrusion detection systems. In order to avoid triggering the IDS, we must behave like an ordinary tenant, which means we cannot gather sensitive information or directly hack into the controller. What's worse, with the constraints of passiveness and concealment, the available parameters for our inference attack are further limited.
After analyzing current structure and implementations of SDN/OpenFlow, its decoupled nature gives us inspiration: the interactions between control plane and data plane will lead to network performance decrease, which can be measured through performance parameters like round trip time (RTT).
\begin{figure}[H]
\centering
\includegraphics[scale=0.58]{figures/flowchart.png}
\caption{OpenFlow Packet Processing Flowchart}
\label{fig:packet-processing-flowchart-of-openflow}
\end{figure}
Fig \ref{fig:packet-processing-flowchart-of-openflow} gives an overall flowchart of packet processing in an OpenFlow switch. The three rectangular regions surrounded by dotted line stand for three possible packet processing branches respectively. When the switch encounters an incoming packet, it will parse it and send the parsed packet into the subsequent processing pipeline.
Then as the first step of the pipeline, the switch will lookup its flow table to search flow entries matching the packet. When there is a match, the switch will directly forward the packet according to actions associated with the corresponding flow entry. This branch is illustrated in the innermost rectangle of fig \ref{fig:packet-processing-flowchart-of-openflow}.
When there is no corresponding flow entry in the flow table, extra steps will be introduced into the procedure. Additional interactions between the switch and the controller will happen to acquire corresponding routing rules, including packet information transferring, routing rule calculation and flow entry deployment. The middle rectangle of fig \ref{fig:packet-processing-flowchart-of-openflow} illustrates this process.
Before the switch inserts the newly generated flow entry, it has to check the flow table status to make sure that there is enough space in the flow table. When the flow table is full, the controller has to perform flow table replacement operations to make room for the upcoming flow entry. These operations include deciding which old flow entry to delete according to certain flow table replacement algorithm and flow entry deletion. The outermost rectangle in fig \ref{fig:packet-processing-flowchart-of-openflow} stands for this branch.
That is exactly where the vulnerability lies. In traditional networks, the switches and routers are autonomous, which means they can maintain their routing tables locally without interacting with an external device. But due to the decoupled nature of SDN/OpenFlow, maintaining switch flow tables needs frequent interactions between switches and controllers, making it possible for an attacker to leverage the perceived performance change to deduce the internal state of the SDN network.
As shown in fig \ref{fig:packet-processing-flowchart-of-openflow}, the rectangular regions surrounded by dotted line correspond to different possible packet processing branches. The larger a rectangle is, the longer the processing time of that branch will be because of the extra steps that rectangle contains. When there is a match in the flow table, the processing time will be the shortest; when there is no match in the flow table and the flow table is not full, the processing time will be longer because of addition routing calculation and flow entry deployment; when there is no match in the flow table and the flow table is full, the processing time will be the longest because a flow table replacement operation has to be performed. So as a network parameter directly influenced by the processing time, the RTT of a packet can serve as an indicator of flow table state and flow entry state.
The process of deciding RTT thresholds for flow table state detection is shown in figure \ref{fig:rtt-measurement-of-different-flow-table-state}.
\begin{figure}[H]
\centering
\includegraphics[scale=0.45]{figures/RTT-measurement.png}
\caption{RTT Measurement of Different Flow Table State}
\label{fig:rtt-measurement-of-different-flow-table-state}
\end{figure}
The two sub-figures in figure \ref{fig:rtt-measurement-of-different-flow-table-state} represent two cooperating threads, the x-axis represents the packet sequence and the y-axis represents the recorded RTT of every packet. Firstly, in the upper thread, we generate a packet with a specific \{src\_ip, dst\_ip, src\_mac, dst\_mac\} combination, calling it $Pkt_1$. Send $Pkt_1$ to the target OpenFlow switch and record the corresponding RTT as $T_2$. Currently there is no corresponding flow entry in the OpenFlow switch because $Pkt_1$ is a new packet. After a time span $TS_1$, send $Pkt_1$ to the target OpenFlow switch again and record the corresponding RTT as $T_1$. If $TS_1$ is chosen properly, the newly installed flow entry matching $Pkt_1$ should still exist in the OpenFlow switch. Next, in the lower thread, we continuously generate packets $Pkt_2$, $Pkt_3$, $\cdots$ $Pkt_N$, each with a different combination of \{src\_ip, dst\_ip, src\_mac, dst\_mac\}, and send these packets to the target OpenFlow switch with the time span of $TS_2$. Because there are no flow entries matching there packets in the OpenFlow switch, the recorded RTTs will be approximately the same as $T_2$. Keep generating and sending packets until we observe a sudden increase of the RTT, which indicates that the flow table is full. Then in the upper thread we send $Pkt_1$ again immediately and record the RTT as $T_3$. To achieve higher precision, we can repeat the process and use average values of $T_1$, $T_2$ and $T_3$ as final results.
From the process above we can see $T_1$, $T_2$, $T_3$ will serve as thresholds for flow table state detection: when the measured RTT is around $T_1$, we can infer that there is corresponding flow entry in the flow table; when the measured RTT is around $T_2$, we can infer that there is no corresponding flow entry in the flow table and the flow table is not full; when the measured RTT is around $T_3$, we can infer that there is no corresponding flow entry in the flow table and the flow table is full. The measured RTT thresholds corresponding to different flow table states are shown in table \ref{tab:rtt-comparison}.
\begin{table}[!ht]
\centering
\caption{RTT Comparison}
\label{tab:rtt-comparison}
\begin{tabular}{c c c}
\hline
Flow Table State & Flow Entry State & RTT\\
\hline
NotFull & Exist & $0.2-0.3ms$\\
NotFull & NotExist & $3-5ms$\\
\hline
Full & Exist & $0.2-0.3ms$\\
Full & NotExist & $8-10ms$\\
\hline
\end{tabular}
\end{table}
Having got a method to detect flow table state and flow entry state using the bootstrap process above, our inference model will follow a "probe--observe--infer" pattern. We model the SDN/OpenFlow network as a black box and observe its response (RTT) to different input (network packets), then we use the response to estimate the flow table state and flow entry state and perform further inference. The whole process comes in three steps.
Firstly, we send probing packets into the network to trigger the interaction. As there is still no mature routing aggregation algorithm or hierarchical routing rule solution, current SDN/OpenFlow switches typically use exact-match rules. That means if we send $n$ packets with different faked meta information like src\_ip and dst\_ip, there will be $n$ newly generated flow entries inserted into the flow table. If we send excessive probing packets in a short period of time, the flow table will overflow and then the interaction process will be triggered. Secondly, we measure RTTs of the responded packets and infer the flow table state and flow entry state. Thirdly, we use observed flow table states and flow table states as controlling signals in our inference algorithm and perform flow table capacity inference.
But there is still one problem: how will the flow table deal with the flow entries when the flow table is full. In other words, we need to know the flow table replacement algorithm. The algorithm decides the internal state transition of a SDN network thus it's an essential part of our inference model. Though flow table replacement algorithms of commercial SDN/OpenFlow switches are stored in their firmwares, making it impossible for researchers to read, we can still get illuminations of what the algorithms will be like by analyzing the functionalities of flow tables.
Flow table in a SDN/OpenFlow switch stores the local router tables assigned by the controller. Having to achieve a hit rate as high as possible in a rather limited space, flow table serves like a "cache" in operating systems and web proxy servers. So we have reason to believe that the flow table replacement algorithm will be some of the most popular cache replacement algorithms or their variations. In this paper we choose FIFO\cite{FIFO} and LRU\cite{LRU} because they are common and popular.
So far we have finished our inference model targeting at flow table capacity and there is still another target: flow table usage inference. Though flow table usage inference seems impossible because network traffic is isolated among different tenants, the information leakage attack mentioned in section \ref{chap:background} and our previous analysis provide us with possible breakthrough points: we can infer the flow table capacity and we can record generated flow entries during a time period, the difference of these two values will be the flow table usage during that time period.
Last but not least, we have to discuss the feasibility of our inference attack as well as some key parameters. In order to facilitate the description, we will first introduce the flow entry deletion mechanism of OpenFlow.
Flow entry deletion mechanism of OpenFlow includes two main parts: active flow deletion invoked by controller and passive flow deletion invoked by timeout. According to OpenFlow specification, each flow entry has two timeout values --- hard\_timeout and idle\_timeout. Hard\_timeout decides how long a flow entry will live after it has been inserted while idle\_timeout means the longest time of no packet matching before a flow entry is deleted. The feasibility of our inference attack will be associated with timeout values and the feasibility analysis consists of two parts: RTT bootstrap feasibility and probing feasibility.
\subsection{RTT Bootstrap Feasibility}
In RTT bootstrap process illustrated in figure \ref{fig:rtt-measurement-of-different-flow-table-state}, there are three key parameters: $TS_1$, $TS_2$ and $TS_3$. Both $TS_1$ and $TS_3$ should not exceed the minimum of hard\_timeout and idle\_timeout because we must prevent the flow entry matching $Pkt_1$ from being deleted in the whole process, which depends on $TS_1$ and $TS_3$ respectively. In order to shorten $TS_3$, $TS_2$ should not be too long because its the time span between every packet and $TS_3$ is made up of $(n-1)$ numbers of $TS_2$. In conclusion, the feasibility constraints for RTT bootstrap are as follows:
\begin{equation}
(n-1)TS_2 = TS_3
\end{equation}
\begin{equation}
TS_1 \leq min\{hard\_timeout, idle\_timeout\}
\end{equation}
\begin{equation}
TS_3 \leq min\{hard\_timeout, idle\_timeout\}
\end{equation}
\subsection{Probing Feasibility}
The key part of our inference model is triggering interactions between switches and controllers by sending probing packets in a short period of time, so the feasibility of our inference attack depends on the feasibility of triggering flow table overflow to a large extent. If we use $V_{gen}$ to represent our packet generating speed and use $V_{del}$ to represent the packet deletion speed and $C$, $T$ to represent the flow table capacity and the probing time, the feasibility formulation will be:
\begin{equation}
V_{gen} \times T - V_{del} \times T \geq C
\end{equation}
Or in another form:
\begin{equation}
V_{gen} \geq V_{del} + C / T
\end{equation}
As can be seen from the formulation, the minimum packet generating speed required are associated with the flow entry deletion speed and flow table capacity. If we set $T$ to be shorter than the minimum of hard\_timeout and idle\_timeout, which means our inference can be completed in a timeout circle, the flow entries deleted during our inference due to timeout will be ignorable. The feasibility constraint will be:
\begin{equation}
V_{gen} \geq C / min\{hard\_timeout, idle\_timeout\}
\end{equation}
From the feasibility constraints above, we can conclude that the feasibility of our inference attack depends on the timeout measurement. It is essential for us to measure hard\_timeout and idle\_timeout not only for inference time limit but also for packet generating speed adjustment.
Idle\_timeout can be measured by sending a train of packets with gradually increasing packet intervals. The process is shown in Figure \ref{fig:idle-timeout-measurement}.
\begin{figure}[H]
\centering
\includegraphics[scale=0.45]{figures/measurable_parameter/idle_timeout.png}
\caption{Idle\_timeout Measurement}
\label{fig:idle-timeout-measurement}
\end{figure}
We choose an initial value of packet interval $\Delta T_1$, for example $0.1$s. Then we gradually increase the packet interval to $\Delta T_2$, $\Delta T_3$ and so on using "binary search" or other algorithms. Keep sending these packets with increasing intervals until we observe a significantly high RTT value. The corresponding packet interval at that time will be the idle\_timeout.
Hard\_timeout can be measured by sending a train of packets with constant packet intervals far smaller that the previously measured idle\_timeout. Keep sending packets with a packet interval of $\Delta T$, for example $0.1$s. When observing a significantly high RTT value, the corresponding packet interval will be the hard\_timeout. The process is shown in Figure \ref{fig:hard-timeout-measurement}.
\begin{figure}[H]
\centering
\includegraphics[scale=0.45]{figures/measurable_parameter/hard_timeout.png}
\caption{Hard\_timeout Measurement}
\label{fig:hard-timeout-measurement}
\end{figure}
In this section we set our inference targets to be flow table capacity and flow table usage, we also discussed the inference model and its feasibility. In section \ref{chap:fifo-inference-algorithm} and \ref{chap:lru-inference-algorithm} we'll give detailed descriptions of inference algorithms for FIFO and LRU respectively.
\section{FIFO Inference Algorithm}\label{chap:fifo-inference-algorithm}
As mentioned in section \ref{chap:problem-statement}, the inference process of FIFO algorithm will be as follows: we generate and send a huge amount of probing packets each with a different combination of {src\_ip, dst\_ip, src\_mac, dst\_mac}, the newly inserted flow entries matching the generated packets will "push" the other users' flow entries out of the flow table. We can detect if the flow table is full and the existence of our flow entries. Combined with the number of inserted flow entries we recorded, we can infer the flow table capacity and flow table usage. The process of flow table state transformation is shown in figure \ref{fig:fifo-step}.
\begin{figure}[H]
\centering
\includegraphics[scale=0.3]{figures/principle/FIFO-step.png}
\caption{FIFO Inference Principle}
\label{fig:fifo-step}
\end{figure}
We use $F_{our}$ to represent the number of our inserted flow entries and use $F_{other}$ to represent the number of flow entries from other users in the flow table. Both $F_{our}$ and $F_{other}$ are functions of time. We use $T_A$, $T_B$, $T_C$ and $T_D$ to represent four time points corresponding to four sub figures respectively and use $C$ to respresent the flow table capacity.
Figure \ref{fig:fifo-step} (A) shows the flow table and the flow entries it contains just before the experiment starts. The rectangle items represent the flow entries from other users sharing the OpenFlow switch. The current number of other users' flow entries can be expressed as $F_{other}(T_A)$.
Figure \ref{fig:fifo-step} (B) illustrates the time when we start to send generated packets, inserting new flow entries into the flow table. The grey rectangles represent the flow entries inserted by us. As we can see, our flow entries keep pushing other users' flow entries to the front of the FIFO queue. During the experiment, we should keep a record of the generated packets, including their attributes and serial numbers.
Figure \ref{fig:fifo-step} (C) shows the time when we detect the flow table is full. At this point of time, flow entries from us and other users add up to fill the whole flow table precisely. We have:
\begin{equation}
F_{our}(T_C) + F_{other}(T_C) = C
\end{equation}
Figure \ref{fig:fifo-step} (D) shows the time when we detect that one of our inserted flow entries has been deleted. That means the flow table is now full of our flow entries, without any flow entries from other users. We have:
\begin{equation}
F_{our}(T_D) = C
\end{equation}
Combine the two equations above, we have:
\begin{equation}
\begin{aligned}
&F_{other}(T_A) = F_{other}(T_C) \\
&= C - F_{our}(T_C) = F_{our}(T_D) - F_{our}(T_C)
\end{aligned}
\end{equation}
According to the analysis above, we describe the inference process for FIFO algorithm as shown below.
\begin{algorithm}[H]
\caption{FIFO Inference Algorithm}
\begin{algorithmic}[1]
\Require\\
Packet-Sending Function: $SendPacket\left(\right)$;\\
List of IP: $IP$;
\Ensure\\
The flow table capacity: $F_{capacity}$;\\
The number of other users' flow entries: $F_{other}$;
\Statex
\State $F_{capacity} \gets 0$
\State $F_{other} \gets 0$
\State $N \gets 0$
\State $N_1 \gets 0$
\State $N_2 \gets 0$
\While {$N<length(IP)$}
\State $ip \gets IP[N]$
\State $\Call{SendPacket}{ip}$
\State $N \gets N+1$
\If {Flow table is full}
\State $N_1 \gets N$
\State continue
\EndIf
\If {One of our flow entries is deleted}
\State $N_2 \gets N$
\State break
\EndIf
\EndWhile
\State $F_{capacity} \gets N_2$
\State $F_{other} \gets N_2-N_1$
\State \Return {$F_{capacity}$, $F_{other}$}
\end{algorithmic}
\end{algorithm}
The main error of the inference comes from the flow entries inserted by other users when our insertion is in progress. We assume that our flow entry insertion speed is fast enough so that during the period of experiment, the newly inserted flow entries are all from us. But that is not always the truth. Ignoring the possible flow entries inserted by other users will make our inference result smaller than the actual value.
Considering the flow entries inserted by other users, the actual equations are listed below.
When we detect the flow table is full, if we use $E(A, B)$ to represent the number of just inserted flow entries from other users from time point $A$ to time point $B$, the equation becomes:
\begin{equation}
F_{our}(T_C) + F_{other}(T_C) + E(T_A, T_C) = C
\end{equation}
And when we detect one of our inserted flow entries is deleted, the equation becomes:
\begin{equation}
F_{our}(T_D) + E(T_A, T_C) + E(T_C, T_D) = C
\end{equation}
Combine the two equations above, we have:
\begin{equation}
F_{other}(T_C) = F_{our}(T_D) - F_{our}(T_C) + E(T_C, T_D)
\end{equation}
So the actual equation considering flow entry insertions during inference should be:
\begin{equation}
\begin{aligned}
C &= F_{our}(T_D) + E(T_A, T_C) + E(T_C, T_D) \\
F_{other}(T_C) &= F_{our}(T_D) - F_{our}(T_C) + E(T_C, T_D)
\end{aligned}
\end{equation}
Compared with our former equation ignoring flow entry insertions:
\begin{equation}
\begin{aligned}
C &= F_{our}(T_D) \\
F_{other}(T_C) &= F_{our}(T_D) - F_{our}(T_C)
\end{aligned}
\end{equation}
We can see that the inferred flow table usage $F_{other}$ and the inferred flow table capacity $F_{capacity}$ will both be smaller than the actual value.
\section{LRU Inference Algorithm}\label{chap:lru-inference-algorithm}
The experiment principle of LRU algorithm has something in common with that of FIFO algorithm, because under these two circumstances we can both keep our flow entries stay in the back of the cache queue using certain operations.However, there are still differences lies in the flow entry maintaining process.
The nature of FIFO algorithm ensures that the position of the flow entries only depends on the time they are inserted. The earlier inserted flow entries are sure to be nearer to the front of the cache queue compared with the later inserted flow entries. But in LRU algorithm, the positions of the flow entries depend not only on the time they are inserted, but also on the last time they are accessed. In order to keep our flow entries stay in the back of the cache queue, we need to continuously access the previously inserted flow entries.
During the maintain process, every time we insert a new flow entry, we need to access all previously inserted flow entries for one time to "lift" them to the back of the cache queue. The access history may be like $\{P_1\},\ \{P_1,\ P_2\},\ \{P_1,\ P_2,\ P_3\},\ \{P_1,\ P_2,\ P_3,\ P_4\}, \cdots$, we call it a "rolling" maintaining process. The maintaining algorithm is shown in algorithm 2.
\begin{algorithm}[H]
\caption{Rolling Maintaining Algorithm}
\begin{algorithmic}[1]
\Require\\
Packet-Sending Function: $SendPacket()$;\\
List of Inserted IP: $IP_{inserted}$;
\Statex
\Function {RollingPacketSender}{$IP_{inserted}$}
\State $i \gets 1$
\While {$i < length(IP_{inserted})$}
\For {$j \gets 0;j < i;j++$}
\State $ip \gets IP_{inserted}[j]$
\State \Call{SendPacket}{$ip$}
\EndFor
\State $i \gets i+1$
\EndWhile
\EndFunction
\end{algorithmic}
\end{algorithm}
According to the analysis above, we describe the inference process for LRU algorithm as shown below.
\begin{algorithm}[H]
\caption{LRU Inference Algorithm}
\begin{algorithmic}[1]
\Require\\
Packet-Sending Function: $SendPacket\left(\right)$;\\
List of IP: $IP$;
\Ensure\\
The flow table capacity: $F_{capacity}$;\\
The number of other users' flow entries: $F_{other}$;
\Statex
\State $F_{capacity} \gets 0$
\State $F_{other} \gets 0$
\State $N \gets 0$
\State $N_1 \gets 0$
\State $N_2 \gets 0$
\State $IP_{inserted} \gets [\ ]$
\While {$N<length(IP)$}
\State $ip \gets IP[N]$
\State $IP_{inserted} \gets IP_{inserted} + ip$
\State \Call{RollingPacketSender}{$IP_{inserted}$}
\State $N \gets N+1$
\If {Flow table is full}
\State $N_1 \gets N$
\State continue
\EndIf
\If {One of our flow entries is deleted}
\State $N_2 \gets N$
\State break
\EndIf
\EndWhile
\State $F_{capacity} \gets N_2$
\State $F_{other} \gets N_2-N_1$
\State \Return {$F_{capacity}$, $F_{other}$}
\end{algorithmic}
\end{algorithm}
The feasibility and error analysis of LRU algorithm is similar with that of FIFO algorithm. The inferred flow table usage $F_{other}$ and the inferred flow table capacity $F_{capacity}$ will both be smaller than the actual value because of ignoring the flow entries inserted by other users during the experiment.
\section{Evaluation}\label{chap:evaluation}
\subsection{Implementation}
The emulation environment of our experiment consists of three parts: a network prototyping system used to emulate host and switch, a network controller, and our inference attack toolkit.
We choose Mininet~\cite{Mininet} as the network prototyping system because it encapsulates host and switch emulation and thus easy to use. Our emulated network prototype for evaluation uses a star topology, consisting of $20$ hosts connected to a single OpenFlow switch. We build FIFO and LRU controller applications using Python on the basis of POX~\cite{POX} OpenFlow controller. As for the inference attack toolkit, we use libnet~\cite{libnet} to generate probing packets, and libpcap~\cite{libpcap} to capture replied packets. Experiments are conducted on a computer with Intel i5-2400 3.1 GHz (4 cores) processor and 8GB RAM.
\subsection{RTT Measurement}
As we have mentioned in section \ref{chap:problem-statement}, the difference between traditional network and SDN/OpenFlow network in handling previously unseen packets gives us a possible indicator of the flow table state and the flow entry living state -- RTT. When there isn't corresponding flow entry existing in the flow table, the RTT of a packet will significantly increase due to the interactions between controller and switch in order to acquire new flow entries. That is the case when there is still space in the flow table. Once the flow table is full, the RTT of a packet will further increase as a result of extra flow table replacement operations. To prove the effectiveness of using RTT as the flow table state and flow entry state indicator, we measured packet RTTs corresponding to different flow table state and flow entry state.
\begin{figure}[H]
\centering
\includegraphics[scale=0.3]{figures/rtt/rtt.png}
\caption{RTT Measurement}
\label{fig:rtt-measurement}
\end{figure}
Figure \ref{fig:rtt-measurement} gives the RTT measurement result showing the difference. The points with different colors represent the total $300$ times of RTT measurements we have conducted, $100$ times of measurement for each combination of flow table state and flow entry state. The green points stand for RTTs when flow entry exists in flow table. The blue points and red points both stand for RTTs when flow entry doesn't exist in flow table, the only difference is the flow table is not full when measuring the blue points.
As can be seen from the figure, when flow entry exists in flow table, the packet RTTs are highly concentrated in the range of $0.2\sim0.3$ ms; when flow entry doesn't exist in flow table and flow table is not full, the packet RTTs will increase to about $3\sim5$ ms; when flow entry doesn't exist in flow table and flow table is full, the packet RTTs will be the highest, ranging from $6$ms to $8$ms. These three groups of RTTs all distribute intensively in a small range without overlapping other groups, showing the excellent discrimination of using RTT as a flow table state and flow entry state indicator.
To better illustrate the distribution of measured RTTs, we plot their CDF curves in figure \ref{fig:rtt-measurement-distribution}. Apparently RTT can be used to deduce the internal state of the SDN network effectively.
\begin{figure}[H]
\centering
\includegraphics[scale=0.3]{figures/rtt/rtt_cdf.png}
\caption{RTT Measurement Distribution}
\label{fig:rtt-measurement-distribution}
\end{figure}
\subsection{Timeout}
\subsubsection{Default Timeout Values}
According to our previous analysis, the feasibility of our inference attack depends on whether we can generate enough flow entries to fulfill the flow table within a single timeout cycle. That means we must have the ability to generate as many flow entries as the flow entry can hold during a timeout period. So we analyze several popular open-source controllers and search for their default timeout values in the built-in applications. The result is presented in table \ref{tab:default-timeout-values}. The zero values in the table mean the corresponding timeout will not take effect, or in other words the timeout value is "permanent". As can be seen from the table, most available controllers have timeout values in the range of $5$s to $30$s.
\begin{table}[!ht]
\centering
\caption{Default Timeout Values}
\label{tab:default-timeout-values}
\begin{tabular}{l c c}
\hline
Controller & Hard\_timeout & Idle\_timeout\\
\hline
Ryu & $0$ & $0$\\
Beacon & $0$ & $5s$\\
Floodlight & $0$ & $5s$\\
NOX & $0$ & $5s$\\
POX & $30s$ & $10s$\\
Trema & $0$ & $60s$\\
Maestro & $180s$ & $30s$\\
\hline
\end{tabular}
\end{table}
If we take the flow table capacity of $2000$ flow entries as an example, the minimum packet generating speed required will be $2000/5=400$ packets per second, while libnet can generate tens of thousand packets per second. So the default timeout values ensure the feasibility of our inference attack.
\subsubsection{Timeout Measurement}
Though default timeout values of mainstream OpenFlow controllers can be read from their source codes, it's still possible for SDN network administrators to manually change the default timeout values. In order to handle non-default timeout values and provide basis for adjusting packet generating speed, it's essential to examine the accuracy of passive timeout measurement.
Figure \ref{fig:timeout-relative-error} illustrates relative errors of hard\_timeout and idle\_timeout measurement respectively. We manually modify hard\_timeout and idle\_timeout values of POX OpenFlow controller to $5$s, $10$s, $15$s, $20$s, $25$s and $30$s, then we use timeout measurement algorithm mentioned in section \ref{chap:problem-statement} to measure these timeout values and calculate relative errors.
\begin{figure}[H]
\centering
\includegraphics[scale=0.23]{figures/timeout/timeout_relative_error.png}
\caption{Timeout Relative Error}
\label{fig:timeout-relative-error}
\end{figure}
Every line in the two sub-figures corresponds to $10$ times of repeated measurements conducted under a certain timeout setting from $5$s to $30$s. The margin stays in the range of plus-or-minus $10$ percent, showing the effectiveness and high accuracy of our timeout measurement algorithm.
\subsection{Flow Table Capacity}
Flow capacity is the primary target of our inference attack. It reflects the hardware specification of an OpenFlow switch. Figure \ref{fig:fifo-capacity} illustrates the flow table capacity measurement result when controller adopts FIFO replacement algorithm. We manually limited the switch flow table capacity to $10$ different values from $100$ flow entries to $1000$ flow entries and used our framework to perform the inference.
\begin{figure}[H]
\centering
\includegraphics[scale=0.25]{figures/capacity_usage/flow_table_capacity_fifo.png}
\caption{FIFO Flow Table Capacity}
\label{fig:fifo-capacity}
\end{figure}
The pink bars represent the manually set flow table capacities or \textit{real} capacities. The blue bars represent the measured flow table capacities. For every manually set flow table capacity, we conduct $10$ times of repeated measurements and take their mean value as the final result. From the figure we can see that the measured capacities is quite close to the real capacities, indicating the high accuracy of our inference framework. For example, when the real capacity is $400$ flow entries, our measured capacity is $408$ flow entries with an error of only $8$ flow entries. As the real capacity grows, the packet generating speed required becomes faster, placing higher requirements on packet sending -- receiving synchronization and accurate timing. But our inference algorithm shows unbelievable stability and accuracy: when the real capacity is $1000$ flow entries, our measured capacity is $973$ flow entries with an error of just $27$ flow entries.
Like figure \ref{fig:fifo-capacity}, figure \ref{fig:lru-capacity} also illustrates the flow table capacity measurement results, with the only difference of being performed under LRU replacement algorithm instead of FIFO.
\begin{figure}[H]
\centering
\includegraphics[scale=0.25]{figures/capacity_usage/flow_table_capacity_lru.png}
\caption{LRU Flow Table Capacity}
\label{fig:lru-capacity}
\end{figure}
According to our previous analysis, the inference principle of LRU replacement algorithm is more complex because of the unavoidable mixed nature of flow entries in the flow table and the rolling maintaining process. But our inference framework still shows high accuracy and reliability. Even when the real flow table capacities are set to be rather large values like $900$ and $1000$, the errors of our measure capacities are just around $20$ flow entries.
Only illustrating the mean value of measured flow table capacities may not be enough: the mean value may be the result of error compensations and hide the detailed measurement errors of every separate experiment. So in figure \ref{fig:flow-table-capacity-relative-error} we illustrates the relative error of every single flow table capacity measurement.
\begin{figure}[H]
\centering
\includegraphics[scale=0.23]{figures/error_analysis/capacity_relative_error.png}
\caption{Flow Table Capacity Relative Error}
\label{fig:flow-table-capacity-relative-error}
\end{figure}
We choose $5$ groups of different flow table capacities from $200$ flow entries to $1000$ flow entries and perform $10$ times of measurements under every single flow table capacity value. The left sub-figure stands for relative error of flow table capacity measurements conducted under FIFO replacement algorithm, showing that the margin is no larger than plus-or-minus $10$ percent. The right sub-figure stands for relative error of flow table capacity measurements conducted under LRU replacement algorithm. Due to the more complex inference principle and the rolling maintaining process, the margin becomes larger but still hasn't exceeded $15$ percent even in the worst case.
\subsection{Flow Table Usage}
In this section we evaluated our framework's efficiency of inferring the number of flow entries from other users sharing the same flow table, or the flow table usage. Flow table usage is our secondary inference target, it reflects the network resource consuming condition of other tenants in the same SDN network. Figure \ref{fig:fifo-usage} and figure \ref{fig:lru-usage} illustrate the flow table usage measurement results conducted under FIFO and LRU replacement algorithm respectively.
\begin{figure}[H]
\centering
\includegraphics[scale=0.25]{figures/capacity_usage/flow_table_usage_fifo.png}
\caption{FIFO Flow Table Usage}
\label{fig:fifo-usage}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[scale=0.25]{figures/capacity_usage/flow_table_usage_lru.png}
\caption{LRU Flow Table Usage}
\label{fig:lru-usage}
\end{figure}
Again we manually set $10$ different flow table usage values from $100$ to $1000$ flow entries by manually generating and inserting corresponding number of flow entries into the flow table beforehand. Then we use our inference algorithm to infer the flow table usage and take mean values of every $10$ times of measurements as the final results. The errors of all these measurements show the high accuracy, stability and reliability of our inference algorithm.
The relative errors are shown in figure \ref{fig:flow-table-usage-relative-error}. We emulate $5$ groups of different flow table usage values and conducted $10$ times of flow table usage inference for every single value. For both FIFO and LRU replacement algorithm, the relative errors of flow table usage inference stay in a quite small range. The results prove that our algorithm can infer other tenants' flow table usage condition in high accuracy.
\begin{figure}[H]
\centering
\includegraphics[scale=0.23]{figures/error_analysis/usage_relative_error.png}
\caption{Flow Table Usage Relative Error}
\label{fig:flow-table-usage-relative-error}
\end{figure}
\section{Discussion}\label{chap:discussion}
SDN/OpenFlow has become a competitive solution for next generation network and is being more and more widely used in modern datacenters. But considering its key role as the datacenter fundamental infrastructure, we have to admit that the security issues of SDN/OpenFlow haven't been explored to a large extent. Especially, the flow table capacity of SDN/OpenFlow switch is only considered as a vulnerable part for DDoS and flooding attacks in published researches. But according to our analysis in previous sections, the flow table capacity can lead to potential inference attack if combined with reasonable assumptions and RTT measurements.
Firstly, we found in section \ref{chap:problem-statement} that exact match flow entries as well as the lack of route aggregation would consume a lot of flow table space, making it impossible to process millions of flows per seconding using SDN/OpenFlow. Secondly, we found in section \ref{chap:evaluation} that assigning the decision making job of flow table replacement to the controller would lead to significant network performance decrease, which had to be changed in time. Thirdly, there is currently no mature attack detection mechanism for SDN/OpenFlow network, so it's quite easy for criminals to exploit system vulnerabilities or invoke DDoS attacks.
All these security issues call for improvements to current OpenFlow switch and flow table design. The improvements should at least contain three aspects: (1)new flow table maintaining mechanism, like transferring the flow entry deleting workload from controller to switch. Switch itself can decide which flow entry to delete and then sync with controller. In the newest OpenFlow switch specification 1.4~\cite{OpenFlow Switch Specification 1.4}, this mechanism has been added as an optional feature, but without any mature implementation so far; (2) routing aggregation. Routing aggregation can match a group of flows using one flow entry, which will reduce the flow table consuming significantly compared with exact match; (3) inference attack detection. Administrators can develop inference attack detecting applications and then perform defences like port speed limiting or network address validation.
\section{Related Work}\label{chap:related-work}
The inference attack proposed in this paper is motivated by the limited flow table capacity of SDN/OpenFlow switches. The flow table capacity issue has been presented in many previous works like~\cite{Sezer},~\cite{Kreutz} and~\cite{Scott}. They all point out the limitation of switch flow table memory and potential scalability and security issue. However, these work don't give further analysis on the inference attack and information leakage caused by the limited flow table capacity.
Kloti et al.~\cite{Kloti} presents potentially problematic issues in SDN/OpenFlow including information disclosure through timing analysis. However, this information disclosure requires disclosing existing flows with side channel attack, which is hard to perform in real world. Compared with their approach, our inference attack is self-contained and requires no prior knowledge.
Gong et al.~\cite{Gong} presents a kind of inference attack using RTT measurement to infer which website the victim is browsing. They recover victims' network traffic patterns based on the queuing side channel happened at the Internet router. However, the scenario of their work is in the public Internet, while our approach focus on SDN/OpenFlow infrastructures in datacenter network. Compared with public Internet and website inference, the inference attack and information leakage in modern data centers is more sensitive and valuable.
Shin et al.~\cite{Shin} demonstrate a novel attack targeting at SDN networks. This attack includes fingerprinting SDN networks and further flooding the data plane flow table by sending specifically crafted fake flow requests in high speed. In the fingerprinting phase, header field change scanning is used to collect the different response time (RTT) for new flow and existing flow. The fingerprinting result is then analyzed to estimate if the target network used SDN technology. The RTT measurement and analysis they used in fingerprinting is similar with our approach. But they just perform DoS attacks to the SDN network, without performing any further information leakage or network parameter inference.
\section{Conclusion}\label{chap:conclusion}
In this paper, we have explored the structure of SDN/\\OpenFlow network and some of the possible security issues it brings. After out detailed analysis of the SDN/\\OpenFlow network, we proposed a novel inference attack model targeting at the SDN/OpenFlow network, which is the first proposed inference attack model of this kind in the SDN/OpenFlow area. This inference attack is introduced by the OpenFlow switch, especially by its limited flow table capacity. The inference attack can be done in a completely passive way, making it hard to detect and defence. We also implemented the inference attack framework and examined the efficiency and accuracy of it using network traffic data from different sources. The simulation results show that the inference attack framework can infer the network parameter(flow table capacity and flow table usage) with an accuracy of up to $80\%$ or higher.
|
{
"timestamp": "2015-04-14T02:12:16",
"yymm": "1504",
"arxiv_id": "1504.03095",
"language": "en",
"url": "https://arxiv.org/abs/1504.03095"
}
|
\section{Introduction}
It is well known that the solution of the Yamabe problem on a
compact Riemannian manifold is unique in the case of negative or
vanishing scalar curvature. The proof of these results, which rely
on the maximum principle, extend readily to sub-Riemannian
settings such as the CR and quaternionic contact (abbr. qc) Yamabe
problems due to the sub-ellipticity of the involved operators. The
positive (scalar curvature) case is of continued interest since it
presents considerable difficulties due to the possible
non-uniqueness. The most important positive case in each of these
geometries is given by the corresponding round sphere due to its
role in the general existence theorem and also because of its
connection with the corresponding $L^2$ Sobolev type embedding
inequality. Through the corresponding Cayley transforms, the
sphere cases are equivalent to the problems of finding all
solutions to the respective Yamabe equation on the flat models
given by Euclidean space or Heisenberg groups. The Riemannian and
CR sphere cases were settled in \cite{Ob} and \cite{JL3}. It
should be noted that the Euclidean case can be handled
alternatively by a reduction to a radially symmetric solution
\cite{GNN} and \cite{Ta}. Furthermore, \cite{Ob} established a
uniqueness result in every conformal class of an Einstein metric.
In this paper we solve the qc Yamabe problem on the $4n+3$
dimensional round sphere and quaternionic Heisenberg group and
establish a uniqueness result in every qc-conformal class
containing a 3-Sasakain metric.
We continue by giving a brief background and the statements of our results.
It is well known that the sphere at infinity of a any non-compact
symmetric space $M$ of rank one carries a natural
Carnot-Carath\'eodory structure, see \cite{M,P}. A quaternionic
contact (qc) structure, \cite{Biq1,Biq2}, appears naturally as the
conformal boundary at infinity of the quaternionic hyperbolic
space. {Following Biquard, a quaternionic contact structure
(\emph{qc structure}) on a real (4n+3)-dimensional manifold $M$ is
a codimension three distribution $H$ (\emph{the horizontal distribution}) locally given as the kernel of a $%
\mathbb{R}^3$-valued one-form $\eta=(\eta_1,\eta_2,\eta_3)$, such
that, the three two-forms $d\eta_i|_H$ are the fundamental forms
of a quaternionic Hermitian structure on $H$.
The 1-form $\eta$
is determined up to a conformal factor and the action of $SO(3)$ on $\mathbb{R}^3$, and therefore $%
H$ is equipped with a conformal class $[g]$ of quaternionic
Hermitian metrics. To every metric in the fixed conformal class
one can associate a linear connection with torsion preserving the
qc structure, see \cite{Biq1}, which is called the Biquard
connection. } For a fixed metric in the conformal class of metrics
on the horizontal space one associates the horizontal Ricci-type
tensor of the Biquard connection, which is called the qc Ricci
tensor. This is a symmetric tensor \cite{Biq1} whose trace-free
part is determined by the torsion endomorphism of the Biquard
connection \cite{IMV} while the trace part is determined by the
scalar curvature of the qc-Ricci tensor, called the qc-scalar
curvature. It was shown in \cite{IMV} that the torsion
endomorphism of the Biquard connection is completely determined by
the trace-free part of the horizontal Ricci tensor whose vanishing
defines the class of qc-Einstein manifolds. A basic example of a
qc manifold
is a 3-Sasakian space which can be defined as a $(4n+3)$%
-dimensional Riemannian manifold whose Riemannian cone is a
hyperK\"ahler manifold and the qc structure is induced from that
hyperK\"ahler structure. It was shown in \cite{IMV,IMV4} that the qc-Einstein manifolds
of positive qc-scalar curvature are exactly the locally 3-Sasakian manifolds, up to a multiplication with a constant factor and a $SO(3)$%
-matrix. In particular, every 3-Sasakian manifold has vanishing
torsion endomorphism and is a qc-Einstein manifold.
The quaternionic contact Yamabe problem on a compact qc manifold
$M$ is the problem of finding a metric $\bar g\in [g]$
on $H$ for which the qc-scalar curvature is constant. A natural
question is to determine the possible uniqueness or non-uniqueness
of such qc-Yamabe metrics.
The question reduces to the solvability of the quaternionic
contact (qc) Yamabe equation \eqref{e:conf change
scalar curv}. Taking the conformal factor in the form $\bar\eta=u^{4/(Q-2)}%
\eta$, $Q=4n+6$, turns \eqref{e:conf change scalar curv} into the
equation
\begin{equation*}
\mathcal{L} u\ \equiv\ 4\frac {Q+2}{Q-2}\ \triangle u -\ u\, Scal
\ =\ - \ u^{2^*-1}\ \overline{Scal},
\end{equation*}
where $\triangle $ is the horizontal sub-Laplacian, $\triangle h\
=\ tr^g(\nabla^2h)$, $Scal$ and $\overline{Scal}$ are the
qc-scalar curvatures
correspondingly of $(M,\, \eta)$ and $(M, \, \bar\eta)$, and $2^* = \frac {2Q%
}{Q-2},$ with $Q=4n+6$--the homogeneous dimension.
Another motivation for studying the qc Yamabe equation comes from
its connection with the determination of the norm and extremals
in the $L^2$ Folland-Stein \cite{FS}
Sobolev-type embedding on the quaternionic Heisenberg group $\boldsymbol{%
G\,(\mathbb{H})}$, \cite{GV}, \cite{Va1}, \cite{Va2} and
completed in \cite{IMV2}. The qc Yamabe equation is essentially
the Euler-Lagrange equation of the extremals for the $L^2$ case of the
Folland-Stein inequality \cite{FS} on the quaternionic Heisenberg group $\boldsymbol{%
G\,(\mathbb{H})}$.
\begin{comment}
It follows from the work
of Folland and Stein that on any Carnot group $\boldsymbol{G}$ of
homogeneous dimension $Q$ {\ and Haar measure $dH$} there exists a
positive constant $S_p$ such that the following Sobolev type
inequality holds for all functions $u$ for which both sides of the
inequality are finite
\begin{equation} \label{FS}
\left(\int_{\boldsymbol{G}}\ |u|^{p^*}\ dH(g)\right)^{1/p^*} \leq\
S_p\ \left(\int_{\boldsymbol{G}}|Xu|^p\ dH(g)\right)^{1/p},
\end{equation}
where $|Xu|=\sum_{j=1}^m |X_ju|^2$ with $X_1,\dots, X_m$ denoting
a basis of the first layer of $\boldsymbol{G}$.
The qc Yamabe equation is essentially the
Euler-Lagrange equation of the extremals for the $L^2$ case
($p=2$) of
\eqref{FS} on the quaternionic Heisenberg group $\boldsymbol{%
G\,(\mathbb{H})}$.
\end{comment}
{On a compact quaternionic contact manifold $M$ with a fixed
conformal class $[\eta]$ the qc Yamabe equation characterizes the
non-negative extremals of the qc Yamabe functional defined by
\begin{equation*}
\Upsilon (u)\ =\ \int_M\Bigl(4\frac {Q+2}{Q-2}\ \lvert \nabla u
\rvert^2\ +\ {Scal}\, u^2\Bigr) dv_g,\qquad \int_M u^{2^*}\,
dv_g \ =\ 1, \ u>0.
\end{equation*}
Here $dv_g$ denotes the Riemannian volume form of the
Riemannian metric
on $M$ extending in a natural way the horizontal metric associated to $\eta$. Considering $M$ equipped with a fixed qc structure, hence, a
conformal class $[\eta]$,} the Yamabe constant is defined as the
infimum
\begin{equation*}
\lambda(M)\ \equiv \ \lambda(M, [\eta])\ =\ \inf \{ \Upsilon (u)
:\ \int_M u^{2^*}\, dv_g \ =\ 1, \ u>0\}.
\end{equation*}
The main result of \cite{Wei} is that the qc Yamabe equation has a
solution on a compact qc manifold provided
$\lambda(M)<\lambda(S^{4n+3})$, where $S^{4n+3}$ is the standard
unit sphere in the quaternionic space $\mathbb H^n$.
In this paper we consider the qc Yamabe problem on the {\ unit $(4n+3)$%
-dimensional sphere in $\mathbb H^n$. The standard 3-Sasaki
structure on the sphere $\tilde\eta$ has a constant qc-scalar
curvature $\widetilde{\text{Scal}}=16n(n+2)$ and vanishing
trace-free part of its qc-Ricci tensor, i.e., it is a qc-Einstein
space. The images under conformal quaternionic contact
automorphisms are again qc-Einstein structures and, in
particular, have constant qc-scalar curvature. In \cite{IMV} we
conjectured that these are the only solutions to the Yamabe
problem on the quaternionic sphere and proved it in dimension
seven in \cite{IMV1}. One of the main goals of this paper is to
prove this conjecture in full generality.
\begin{thrm}
\label{main2} Let $\tilde\eta=\frac{1}{2h}\eta$ be a qc conformal
transformation of the standard qc-structure $\tilde\eta$ on a
3-Sasakian sphere of dimension $4n+3$. If $\eta$ has constant
qc-scalar curvature, then up to a multiplicative constant $\eta$
is obtained from $\tilde\eta$ by a conformal quaternionic contact
automorphism.
\end{thrm}
We note that Theorem~\ref{main2} together with the results of
\cite{IMV} allows the determination of \emph{all} solutions of the
qc Yamabe problem on the sphere and on the quaternionic Heisenberg
group $\boldsymbol {G\,(\mathbb{H})}$. In fact, as a consequence of Theorem~\ref{main2}, we
obtain here that all solutions to the qc Yamabe equation are given
by the functions which realize the equality case of the $L^2$
Folland-Stein inequality found in \cite{IMV2} with the help
of the center of mass technique
developed for the CR case in \cite{FL} and \cite{BFM}.
Recall that the quaternionic Heisenberg group $\boldsymbol {G\,(\mathbb{H})}$ of homogeneous
dimension $Q=4n+6$ is given by
$\boldsymbol {G\,(\mathbb{H})}=\mathbb{H}^n\times\text{Im}\mathbb{H}$, $\quad
(q=(t^a,x^a,y^a,z^a)\in \mathbb H^n,\omega=(x,y,z)\in
\text{Im}\mathbb{H})$ with the group low
$
(q_o, \omega_o)\circ(q, \omega)\ =\ (q_o\ +\ q, \omega\ +\
\omega_o\ + \ 2\ \text {Im}\
q_o\, \bar q). $$
The "\emph{standard}" qc contact form in quaternion variables is $
\tilde\Theta= (\tilde\Theta_1,\ \tilde\Theta_2, \ \tilde\Theta_3)=
\frac 12\ (d\omega - q \cdot d\bar q + dq\, \cdot\bar q) $.
\begin{comment}
The left-invariant horizontal vector fields are
\begin{equation}\label{qHh}
\begin{aligned}
T_{\alpha} = \dta {} +2x^{\alpha}\dx {}+2y^{\alpha}\dy
{}+2z^{\alpha}\dz {} , \
X_{\alpha} = \dxa {}-2t^{\alpha}\dx {}-2z^{\alpha}\dy
{}+2y^{\alpha}\dz {} ,\\
Y_{\alpha} = \dya {} +2z^{\alpha}\dx {}-2t^{\alpha}\dy
{}-2x^{\alpha}\dz {}, \ Z_{\alpha} = \dza {} -2y^{\alpha}\dx
{}+2x^{\alpha}\dy {}-2t^{\alpha}\dz {}\,,
\end{aligned}
\end{equation}
\end{comment}
The corresponding sub-Laplacian
\triangle_{\tilde\Theta} u=\sum_{a=1}^n \left (T_{\alpha}^2u+
X_{\alpha}^2u+Y_{\alpha}^2u+Z_{\alpha}^2u \right ), $ where
$T_a,X_a,Y_a,Z_a$ denote the left-invariant horizontal vector
fields on $\boldsymbol {G\,(\mathbb{H})}$.
Theorem \ref{main2} shows, in particular, the following
\begin{cor}
If $\Phi$ satisfies the qc Yamabe equation on the quaternionic
Heisenberg group $\boldsymbol {G\,(\mathbb{H})}$,
\begin{equation*
\frac {4(Q+2)}{Q-2}\triangle_{\tilde\Theta} \Phi = -S_{\Theta}\,
\Phi^{2^*-1},
\end{equation*}
for some constant $S_{\Theta}$, then up to a left translation the
function $\Phi=(2h)^{-(Q-2)/4}$ and $h$ is given by
\begin{equation}\label{e:Liouville conf factor}
h(q,\omega) \ =\ c_0\ \Big [ \big ( \sigma\ +\
|q+q_0|^2 \big )^2\ +\ |\omega\ +\ \omega_o\ +
\ 2\ \text {Im}\ q_o\, \bar q|^2 \Big ],
\end{equation}
for some fixed $(q_o,\omega_o)\in \boldsymbol {G\,(\mathbb{H})}$ and constants $c_0>0$ and
$\sigma>0$. Furthermore, the qc-scalar curvature of $\Theta$ is
$
S_{\Theta}=128n(n+2)c_0\sigma. $
\end{cor}
This confirms the Conjecture made after \cite[Theorem 1.1]{GV}. In
\cite[Theorem 1.6]{GV} the above result is proved on all groups of
Iwasawa type, but with the assumption of partial-symmetry of the
solution. Here with a completely different method from \cite{GV}
we show that the symmetry assumption is superfluous. The
corresponding solutions on the 3-Sasakain sphere are obtained via
the Cayley transform, see for example \cite{IMV,IMV1,IMV2},
\cite[Sections 2.3 \& 5.2.1]{IV3} for an account and history.
Finally, it should be observed that the functions
\eqref{e:Liouville conf factor} with $c_0\in \mathbb{R}$ give all
conformal factors for which $\Theta$ is also qc-Einstein.
We derive Theorem \ref{main2} from a more general result in which we solve the qc Yamabe problem on a locally 3-Sasakian
compact manifolds.
By the results of \cite{IMV} and \cite{IMV4} a qc-Einstein
manifold is of constant qc-scalar curvature, hence as far as the
qc Yamabe equation is concerned only the uniqueness of solutions
needs to be addressed. As mentioned earlier, the interesting case
is when the qc-scalar curvature is a positive constant, hence we
focus exclusively on the locally 3-Sasakian case.
\begin{thrm}\label{mainth}
Let $(M, \bar\eta)$ be a compact locally 3-Sasakian qc manifold of qc-scalar curvature $16n(n+2)$. If $\eta=2h\bar\eta$ is qc-conformal to $\bar\eta$ structure which is also of constant
qc-scalar curvature, then up to a homothety $(M,\eta)$ is locally 3-Sasakian
manifold. Furthermore, the function $h$ is constant unless
$(M,\bar\eta)$ is the unit 3-Sasakian sphere.
\end{thrm}
The proof of Theorem \ref{mainth} consists of two steps. The first step is a divergence formula
Theorem~\ref{t:div formulas} which shows that if $\bar\eta$ is of constant qc-curvature and is qc-conformal to a locally
3-Sasakian manifold, then $\bar\eta$ is also a locally 3-Sasakian manifold.
The general idea to search for such
a divergence formula goes back to Obata \cite{Ob} where the
corresponding result on a Riemannian manifold was proved for a
conformal transformation of an Einstein space. However, our
result is motivated by the (sub-Riemannian) CR case where a
formula of this type was introduced in the ground-breaking paper
of Jerison and Lee \cite{JL3}.
As far as the qc case is concerned in \cite[Theorem 1.2]{IMV} a
weaker results was shown, namely Theorem~\ref{mainth} holds
provided the vertical space of $\eta$ is integrable. In dimension
seven, the $n=1$ case, this assumption was removed in
\cite[Theorem 1.2]{IMV1} where the result was established with the
help of a suitable divergence formula. The general case $n>1$
treated here presents new difficulties due to the extra non-zero
torsion terms that appear in the higher dimensions, which
complicate considerably the search of a suitable divergence
formula. In the seven dimensional case the [3]-component of the
traceless qc-Ricci tensor vanishes which decreases the number of
torsion components.
The proof of the second part of Theorem~\ref{mainth} builds on
ideas of Obata in the Riemannian case, who used that the gradient
of the (suitably taken) conformal factor is a conformal vector
field and the characterization of the unit sphere through its
first eigenvalue of the Laplacian among all Einstein manifolds. We
show a similar, although a more complicated relation between the
conformal factor and the existence of an infinitesimal qc
automorphism (qc vector field). Our divergence formula found in
Theorem~\ref{t:div formulas} involves a smooth function $f$, c.f.
\eqref{e:f}, expressed in terms of the conformal factor and its
horizontal gradient. Remarkably, we found that the horizontal
gradient of $f$ is precisely the horizontal part of the qc vector
field mentioned above and the sub-Laplacian of $f$ is an
eigenfuction of the sub-Laplacian with the smallest possible
eigenvalue $-4n$ thus showing a geometric nature of $f$ (cf
Remark~\ref{mys}). Then we use the characterization of the
3-Sasakian sphere by its first eigenvalue of the sub-Laplacian
among all locally 3-Sasakian manfolds established in
\cite[Theorem~1.2]{IPV1} for $(n>1)$ and in
\cite[Corollary~1.2]{IPV2} for $n=1$.
\begin{rmrk}
Remarkably, a similar arguments also work in the CR case
describing the geometric nature of the mysterious function in the
Jerison-Lee's divergence formula in \cite{JL3}. Indeed, the
CR-Laplacian of the real part of the function $f$ defined in
\cite[Proposition~3.1]{JL3} turns out to be an eigenfuction of
the CR-Laplacian with the smallest possible eigenvalue $-2n$ thus
showing a geometric nature of the real part of $f$.
\end{rmrk}
\begin{comment}
\textbf{Organization of the paper.} The paper uses some results from \cite%
{IMV}. In order to make the present paper self-contained, in Section \ref%
{s:review} we give a review of the notion of a quaternionic contact
structure and collect formulas and results from \cite{IMV} that will be used
in the subsequent sections.
Section \ref{s:conf transf} and \ref{s:div formulas} are of
technical nature. In the former we find some transformations
formulas for relevant tensors, while in the latter we prove
certain divergence formulas. The key result is Theorem \ref{t:div
formulas}, with the help of which in the last Section we prove the
main Theorems.
\end{comment}
\begin{conv} \label{conven}
We use the following
\begin{itemize}
\item[1.] $\{e_1,\dots,e_{4n}\}$ denotes an orthonormal basis of
the horizontal space $H$; \item[2.] The capital letters X,Y,Z...
denote horizontal vectors, $X,Y,Z...\in H$.
\item[3.] The summation convention over repeated vectors from the basis $%
\{e_1,\dots,e_{4n}\}$ will be used. For example, for a
(0,4)-tensor $P$, $k=P(e_b,e_a,e_a,e_b)$ means
$k=\sum_{a,b=1}^{4n}P(e_b,e_a,e_a,e_b).$
\item[4.] The triple $(i,j,k)$ denotes any cyclic permutation of
$(1,2,3)$.
\end{itemize}
\end{conv}
\textbf{Acknowledgements} S.Ivanov is visiting University of
Pennsylvania, Philadelphia. S.I. thanks UPenn for providing the
support and an excellent research environment during the whole
stages of the paper. S.I. and I.M. are partially supported by
Contract DFNI I02/4/12.12.2014 and Contract 168/2014 with the
Sofia University "St.Kl.Ohridski". I.M. is supported by a SoMoPro II Fellowship which is co-funded by
the European Commission\footnote{This article reflects only the author's views and the EU is not liable for any use that may be made of the information contained therein.} from \lq\lq{}People\rq\rq{} specific programme (Marie Curie Actions) within the EU
Seventh Framework Programme on the basis of the grant agreement REA No. 291782. It is further co-financed by the South-Moravian Region. D.V. was partially supported by Simons Foundation grant \#279381.
\section{Quaternionic contact manifolds}
\label{s:review} In this section we will briefly review the basic notions of
quaternionic contact geometry and recall some results from \cite{Biq1} and
\cite{IMV}, see \cite{IV3} for a more leisurely exposition.
A quaternionic contact (qc) manifold $(M, \eta,g, \mathbb{Q})$ is a $4n+3$%
-dimensional manifold $M$ with a codimension three distribution
$H$ locally given as the kernel of a 1-form
$\eta=(\eta_1,\eta_2,\eta_3)$ with values in $\mathbb{R}^3$. In
addition $H$ has an $Sp(n)Sp(1)$ structure, that is, it is
equipped with a Riemannian metric $g$ and a rank-three bundle
$\mathbb{Q}$ consisting of endomorphisms of $H$ locally generated
by three almost complex structures $I_1,I_2,I_3$ on $H$ satisfying
the identities of the imaginary unit quaternions,
$I_1I_2=-I_2I_1=I_3, \quad I_1I_2I_3=-id_{|_H}$ which are
hermitian compatible with the metric $g(I_s.,I_s.)=g(.,.)$ and the
following contact condition holds $$\qquad 2g(I_sX,Y)\ =\
d\eta_s(X,Y).$$
A special phenomena, noted in \cite{Biq1}, is that the contact
form $\eta$ determines the quaternionic structure and the metric
on the horizontal distribution in a unique way.
The transformations preserving a given quaternionic contact
structure $\eta$, i.e., $\bar\eta=\mu\Psi\eta$ for a positive
smooth function $\mu$ and an $SO(3)$ matrix $\Psi$ with smooth
functions as entries are called \emph{quaternionic contact
conformal (qc-conformal) transformations}. If the function $\mu$
is constant $\bar\eta$ is called qc-homothetic to $\eta$. The qc
conformal curvature tensor $W^{qc}$, introduced in \cite{IV}, is
the obstruction for a qc structure to be locally qc conformal to
the standard 3-Sasakian structure on the $(4n+3)$-dimensional
sphere \cite{IMV,IV}.
\begin{dfn}
\label{d:3-ctct auto} A diffeomorphism $\phi$ of a QC manifold $(M,[g],%
\mathbb{Q})$ is called a \emph{conformal quaternionic contact
automorphism (conformal qc-automorphism)} if $\phi$ preserves the
QC structure, i.e.
\begin{equation*}
\phi^*\eta=\mu\Phi\cdot\eta,
\end{equation*}
for some positive smooth function $\mu$ and some matrix $\Phi\in
SO(3)$ with smooth functions as entries and
$\eta=(\eta_1,\eta_2,\eta_3)^t$ is a local 1-form considered as a
column vector of three one forms as entries.
\end{dfn}
On a qc manifold with a fixed metric $g$ on $H$ there exists a
canonical
connection defined first by O. Biquard in \cite{Biq1} when the dimension $(4n+3)>7$, and in \cite%
{D} for the 7-dimensional case. Biquard showed that there is a
unique connection $\nabla$ with torsion $T$ and a unique
supplementary subspace $V$ to $H$ in $TM$, such that:
\begin{enumerate}[(i)]
\item $\nabla$ preserves the decomposition $H\oplus V$ and the $
Sp(n)Sp(1)$ structure on $H$, i.e. $\nabla g=0, \nabla\sigma
\in\Gamma( \mathbb{Q})$ for a section
$\sigma\in\Gamma(\mathbb{Q})$, and its torsion on $H$ is given by
$T(X,Y)=-[X,Y]_{|V}$; \item for $\xi\in V$, the endomorphism
$T(\xi,.)_{|H}$ of $H$ lies in $ (sp(n)\oplus sp(1))^{\bot}\subset
gl(4n)$; \item the connection on $V$ is induced by the natural
identification $ \varphi$ of $V$ with the subspace $sp(1)$ of the
endomorphisms of $H$, i.e. $ \nabla\varphi=0$.
\end{enumerate}
This canonical connection is also known as \emph{the Biquard
connection}. When the dimension of $M$ is
at least eleven \cite{Biq1} also described the supplementary distribution $V$%
, which is (locally) generated by the so called Reeb vector fields $%
\{\xi_1,\xi_2,\xi_3\}$ determined by
\begin{equation} \label{bi1}
\eta_s(\xi_k)=\delta_{sk}, \qquad (\xi_s\lrcorner
d\eta_s)_{|H}=0,\qquad (\xi_s\lrcorner
d\eta_k)_{|H}=-(\xi_k\lrcorner d\eta_s)_{|H},
\end{equation}
where $\lrcorner$ denotes the interior multiplication.
If the dimension of $%
M $ is seven Duchemin shows in \cite{D} that if we assume, in
addition, the existence of Reeb vector fields as in \eqref{bi1},
then the Biquard result holds. Henceforth, by a qc structure in
dimension $7$ we shall mean a qc structure satisfying \eqref{bi1}.
The fundamental 2-forms $\omega_s$ of the quaternionic contact
structure $Q$ are defined by
\begin{equation*}
2\omega_{s|H}\ =\ \, d\eta_{s|H},\qquad
\xi\lrcorner\omega_s=0,\quad \xi\in V.
\end{equation*}
Notice that equations \eqref{bi1} are invariant under the natural
$SO(3)$
action. Using the triple of Reeb vector fields we extend the metric $g$ on $%
H $ to a metric $h$ on $TM$ by requiring
$span\{\xi_1,\xi_2,\xi_3\}=V\perp H \text{ and }
h(\xi_s,\xi_k)=\delta_{sk}. $ The Riemannian metric $h$ as well
as the Biquard connection do not depend on the action of $SO(3)$
on $V$, but both change if $\eta$ is multiplied by a conformal
factor \cite{IMV}. Clearly, the Biquard connection preserves the
Riemannian metric on $TM, \nabla h=0$.
The properties of the Biquard connection are encoded in the
torsion endomorphism $T_{\xi}\in(sp(n)+sp(1))^{\perp}$. We recall
the $Sp(n)Sp(1)$ invariant decomposition. An endomorphism $\Psi$
of $H$ can be decomposed with respect to the quaternionic
structure $(\mathbb{Q},g)$ uniquely into four $Sp(n)$-invariant
parts
$\Psi=\Psi^{+++}+\Psi^{+--}+\Psi^{-+-}+\Psi^{--+},$
where the superscript $+++$
means commuting with all three $I_{i}$, $+--$ indicates commuting with $%
I_{1} $ and anti-commuting with the other two and etc. The two $Sp(n)Sp(1)$%
-invariant components $\Psi_{[3]}=\Psi^{+++}, \quad
\Psi_{[-1]}=\Psi^{+--}+\Psi^{-+-}+\Psi^{--+} $ are determined by
\begin{equation*}
\begin{aligned} \Psi=\Psi_{[3]} \quad \Longleftrightarrow 3\Psi+I_1\Psi
I_1+I_2\Psi I_2+I_3\Psi I_3=0,\\ \Psi=\Psi_{[-1]}\quad
\Longleftrightarrow \Psi-I_1\Psi I_1-I_2\Psi I_2-I_3\Psi I_3=0.
\end{aligned}
\end{equation*}
With a short calculation one sees that the $Sp(n)Sp(1)$-invariant
components are the projections on the eigenspaces of the Casimir
operator $\Upsilon =\ I_1\otimes I_1\ +\ I_2\otimes I_2\ +\
I_3\otimes I_3$ corresponding, respectively, to the eigenvalues
$3$ and $-1$, see \cite{CSal}. If $n=1$ then the space of
symmetric endomorphisms commuting with all $I_s$ is 1-dimensional,
i.e. the [3]-component of any symmetric endomorphism $\Psi$
on $H$ is proportional to the identity, $\Psi_{[3]}=-\frac{tr\Psi}{4}Id_{|H}$%
. Note here that each of the three 2-forms $\omega_s$ belongs to
its [-1]-component, $\omega_s=\omega_{s[-1]}$ and constitute a
basis of the Lie algebra $sp(1)$.
\subsection{The torsion tensor}
Decomposing the endomorphism $T_{\xi }\in (sp(n)+sp(1))^{\perp }$
into its symmetric part $T_{\xi }^{0}$ and skew-symmetric part
$b_{\xi },T_{\xi }=T_{\xi }^{0}+b_{\xi }$, O. Biquard shows in
\cite{Biq1} that the torsion $T_{\xi }$ is completely trace-free,
$tr\,T_{\xi }=tr\,T_{\xi }\circ I_{s}=0$, its symmetric part has
the properties $T_{\xi _{i}}^{0}I_{i}=-I_{i}T_{\xi _{i}}^{0}\quad
I_{2}(T_{\xi _{2}}^{0})^{+--}=I_{1}(T_{\xi _{1}}^{0})^{-+-},\quad
I_{3}(T_{\xi _{3}}^{0})^{-+-}=I_{2}(T_{\xi _{2}}^{0})^{--+},\quad
I_{1}(T_{\xi _{1}}^{0})^{--+}=I_{3}(T_{\xi _{3}}^{0})^{+--}$. The
skew-symmetric part can be represented as $b_{\xi _{i}}=I_{i}u$,
where $u$ is a traceless symmetric (1,1)-tensor on $H$ which
commutes with $I_{1},I_{2},I_{3}$. Therefore we have $T_{\xi
_{i}}=T_{\xi _{i}}^{0}+I_{i}u$. If $n=1$ then the tensor $u$
vanishes identically, $u=0$, and the torsion is a symmetric
tensor, $T_{\xi }=T_{\xi }^{0}$.
\noindent The two $Sp(n)Sp(1)$-invariant trace-free symmetric
2-tensors $T^0(X,Y)= g((T_{\xi_1}^{0}I_1+T_{\xi_2}^{0}I_2+T_{
\xi_3}^{0}I_3)X,Y)$, $U(X,Y) =g(uX,Y)$ on $H$, introduced in
\cite{IMV}, have the properties:
\begin{equation} \label{propt}
\begin{aligned} T^0(X,Y)+T^0(I_1X,I_1Y)+T^0(I_2X,I_2Y)+T^0(I_3X,I_3Y)=0, \\
U(X,Y)=U(I_1X,I_1Y)=U(I_2X,I_2Y)=U(I_3X,I_3Y). \end{aligned}
\end{equation}
In dimension seven $(n=1)$, the tensor $U$ vanishes identically,
$U=0$.
\noindent These tensors determine completely the torsion endomorphism of the Biquard connection due to
the following identity \cite[Proposition~2.3]{IV} $%
4T^0(\xi_s,I_sX,Y)=T^0(X,Y)-T^0(I_sX,I_sY)$ which implies
\begin{equation*}
4T(\xi_s,I_sX,Y)=4T^0(\xi_s,I_sX,Y)+4g(I_suI_sX,Y)=
T^0(X,Y)-T^0(I_sX,I_sY)-4U(X,Y).
\end{equation*}
\subsection{The qc-Einstein condition and Bianchi identities}
We explain briefly the consequences of the Bianchi identities and
the notion of qc-Einstein manifold introduced in \cite{IMV} since
it plays a crucial role in solving the Yamabe equation in the
quaternionic sphere (see \cite{IMV1} for dimension seven). For
more details see \cite{IMV}.
Let $R=[\nabla,\nabla]-\nabla_{[\ ,\ ]}$ be the curvature of the
Biquard connection $\nabla$. The Ricci tensor and the scalar
curvature, called \emph{qc-Ricci tensor} and \emph{qc-scalar
curvature}, respectively, are defined by
\begin{equation*} \label{e:horizontal ricci}
Ric(X,Y)={g(R(e_a,X)Y,e_a)}, \qquad
Scal=Ric(e_a,e_a)=g(R(e_b,e_a)e_a,e_b).
\end{equation*}
According to \cite{Biq1} the Ricci tensor restricted to $H$ is a
symmetric tensor. If the trace-free part of the qc-Ricci tensor is
zero we call the quaternionic structure \emph{a qc-Einstein
manifold} \cite{IMV}. It is shown in \cite{IMV} that the qc-Ricci
tensor is completely determined by the components of the torsion.
\noindent{\ Theorem~1.3, Theorem~3.12 and Corollary~3.14 in \cite{IMV} imply%
}
that on a qc manifold $(M^{4n+3},g,\mathbb{Q})$ the qc-Ricci
tensor and the qc-scalar curvature satisfy
\begin{equation*} \label{sixtyfour}
\begin{aligned} Ric(X,Y) \ & =\ (2n+2)T^0(X,Y)
+(4n+10)U(X,Y)+\frac{Scal}{4n}g(X,Y)\\ Scal\ & =\
-8n(n+2)g(T(\xi_1,\xi_2),\xi_3) \end{aligned}
\end{equation*}
Hence, the qc-Einstein condition is equivalent to the vanishing of
the torsion endomorphism of the Biquard connection and in this
case the qc scalar curvature is constant \cite{IMV,IMV4}. If $Scal
>0$ {\ the latter} holds exactly when the qc-structure is locally
3-Sasakian up to a multiplication by a constant and an
$SO(3)$-matrix with smooth entries.
We remind that a (4n+3)-dimensional
Riemannian manifold $(M,g)$ is called 3-Sasakian if the cone metric $%
g_N=t^2g+dt^2$ on $N=M\times \mathbb{R}^+$ is a hyperk\"ahler
metric, namely, it has holonomy contained in $Sp(n+1)$. The
3-Sasakian manifolds are Einstein with positive Riemannian scalar
curvature.}
The following vectors will be important for our considerations,
\begin{equation} \label{d:A_s}
A_i\ =\ I_i[\xi_j, \xi_k],\qquad A\ = \ A_1\ +\ A_2\ +\ A_3.
\end{equation}
\noindent We denote with the same letter the corresponding
horizontal 1-form and recall the action of $I_s$ on it
\begin{equation*}
{A} (X)\ = \ g(I_1[\xi_2,\xi_3]+I_2[\xi_3,\xi_1]+I_3[\xi_1,\xi_2],
X),\quad I_sA(X)=-A(I_sX).
\end{equation*}
The horizontal divergence $\nabla^*P$ of a (0,2)-tensor field $P$
on $M$ with respect to Biquard connection is defined to be the
(0,1)-tensor field
\nabla^*P(.)=(\nabla_{e_a}P)(e_a,.). $
We have from \cite[Theorem 4.8]{IMV} that on a
$(4n+3)$-dimensional QC manifold with constant qc-scalar curvature
the next identities hold
\begin{equation} \label{div:To}
\nabla^*T^0=(n+2){A}, \qquad \nabla^*U=\frac{1-n}{2}{A}.
\end{equation}
For any smooth function $h$ on a qc manifold with constant qc
scalar curvature the following formulas are valid \cite[Lemma~4.1
]{IMV1}
\begin{equation}\label{l:div of I_sA}
\begin{aligned} \nabla^*\, \Bigl (\sum_{s=1}^3 dh(\xi_s) I_sA_s\Bigr )\
=\ \sum_{s=1}^3 \ \nabla dh\,(I_s e_a, \xi_s)A_s(e_a);\\ \nabla^*\, \Bigl (\sum_{s=1}^3 dh(\xi_s) I_sA \Bigr )\
=\ \sum_{s=1}^3 \ \nabla dh\,(I_s e_a, \xi_s)A(e_a). \end{aligned}
\end{equation}
\subsection{Qc conformal transformations}
\label{s:conf transf}
Let $h$ be a positive smooth function on a qc manifold $(M, \eta)$. Let $%
\bar\eta=\frac{1}{2h}\eta$ be a conformal deformation of the qc structure $%
\eta$. We will denote the objects related to $\bar\eta$ by over-lining the
same object corresponding to $\eta$. Thus, $d\bar\eta=-\frac{1}{2h^2}%
\,dh\wedge\eta\ +\ \frac{1}{2h\,}d\eta$ and $\bar
g=\frac{1}{2h}g$. The new triple
$\{\bar\xi_1,\bar\xi_2,\bar\xi_3\}$ is determined by the
conditions defining the Reeb vector fields as follows $\bar\xi_s\
=\ 2h\,\xi_s\ +\ I_s\nabla h$, where $\nabla h$ is the horizontal
gradient defined by $g(\nabla h,X)=dh(X)$. The components of the
torsion tensor transform according to the following formulas from
\cite[Section 5]{IMV}
\begin{equation} \label{e:U conf change}\begin{split}
\overline T^0(X,Y) \ =\ T^0(X,Y)\ +\ h^{-1}\,[\nabla dh]_{[sym][-1]}(X,Y), \\
\bar U(X,Y) \ =\ U(X,Y)\ +\ (2h)^{-1}[\ {\nabla dh}-2h^{-1}dh\otimes
dh]_{[3][0]}(X,Y),
\end{split}
\end{equation}
where the symmetric part is given by \begin{equation*}
\label{symdh} [\ {\nabla dh}]_{[sym]}(X,Y)\ =\ \ {\nabla dh}(X,Y)\
+ \ \sum_{s=1}^3 dh(\xi_s)\,\omega_s(X,Y) \end{equation*} {and
$_{[3][0]}$ indicates the trace free part of the [3]-component of
the corresponding tensor. }
In addition, the qc-scalar curvature changes according to the
formula \cite{Biq1}
\begin{equation} \label{e:conf change scalar curv}
\overline {\text{Scal}}\ =\ 2h\,(\text{Scal})\ -\ 8(n+2)^2\,h^{-1}|\nabla
h|^2\ +\ 8(n+2)\,\triangle h.
\end{equation}
\section{Qc conformal transformations on qc Einstein manifolds}
Throughout this section $h$ is a positive smooth function on a qc
manifold $(M, g, \mathbb{Q})$ {\ with constant qc-scalar curvature
$Scal=16n(n+2)$} and $\bar\eta\ =\ \frac{1}{2h}\, \eta$ is a qc
Einstein structure which is a conformal deformation of the qc
structure $\eta$. We recall some formulas from \cite{IMV1} which
we need here.
First we write the expressions of the 1-forms $A_s, A$ in terms of
$h$ (see \cite[Lemma~ ]{IMV1})
\begin{multline} \label{e:A_s}
A_i(X)\ =\ -\frac12 h^{-2}dh(X)\ -\ \frac 12h^{-3}\lvert \nabla h
\rvert^2dh(X) -\ \frac 12 h^{-1}\Bigl (\ {\nabla dh} (I_jX,
\xi_j)\ +\ \ {\nabla dh} (I_kX, \xi_k) \Bigr )\\ +\ \frac 12
h^{-2}\Bigl (dh(\xi_j)\,dh (I_jX)\ +\ dh(\xi_k)\,dh (I_kX) \Bigr )
+\ \frac 14 h^{-2}\Bigl ( \ {\nabla dh} (I_jX, I_j \nabla h)\ +\ \ {\nabla dh%
} (I_kX, I_k \nabla h) \Bigr ).
\end{multline}
Thus, we have also
\begin{multline} \label{e:A}
A(X)\ =\ -\frac32 h^{-2}dh(X)\ -\ \frac 32h^{-3}\lvert \nabla h \rvert^2dh(X)
\\
-\ h^{-1}\sum_{s=1}^3\ {\nabla dh} (I_sX, \xi_s)\ +\
h^{-2}\sum_{s=1}^3dh(\xi_s)\,dh (I_sX)\ +\ \frac 12 h^{-2}\sum_{s=1}^3\ {%
\nabla dh} (I_sX, I_s \nabla h)\
\end{multline}
Second we consider the following one-forms
\begin{equation} \label{d:D_s}
\begin{aligned} D_s(X
=-\frac1{2h}\Big[T^0(X,\nabla h)+T^0(I_sX,I_s\nabla h)\Big]
\end{aligned}
\end{equation}
For simplicity, using the musical isomorphism, we will denote with
$D_1, \, D_2, \, D_3$ the corresponding (horizontal) vector
fields, for example \hspace{3mm} $g(D_1, X)=D_1(X)$. Using
\eqref{propt}, we set
\begin{equation} \label{d:def of D}
D\ =\ D_1\ +\ D_2\ +\ D_3\ =\ - h^{-1}\,T^{0}(X,\nabla h).
\end{equation}
Setting $\bar T^0=0$ in \eqref{e:U conf change}, we obtain from
equations \eqref{d:D_s} the expressions (cf. \cite{IMV1} or
\cite{IV})
\begin{equation} \label{Ds}
\begin{aligned} D_i(X)\ =\ h^{-2}\, dh(\xi_i)\,dh(I_iX)+ \ \frac14h^{-2}\, \bigl [\ \nabla dh\, (X,\nabla h)+\ \nabla dh\, (I_iX,I_i\nabla h)
\\ - \ \nabla dh\, (I_jX,I_j\nabla h)\ -\ \nabla dh\,
(I_kX,I_k\nabla h)\bigr ] .
\end{aligned}
\end{equation}
The equalities \eqref{d:def of D} together with \eqref{Ds} yield
\cite[Lemma~4.2]{IMV1}
\begin{equation}\label{n21
D(X)\ =\ \frac 14 h^{-2}\Bigl (3\ {\nabla dh} (X, \nabla h)\ -\
\sum_{s=1}^3\ {\nabla dh} (I_sX, I_s\nabla h) \Bigr )\ +\
h^{-2}\sum_{s=1}^3dh(\xi_s)\,dh (I_sX).
\end{equation}
Third, we consider the following one-forms (and corresponding
vectors)
\begin{equation*} \label{d:F_s}
F_s(X)\ =\ - h^{-1}\, {T^0}(X,I_s\nabla h).
\end{equation*}
\noindent From the definition of $F_i$ and \eqref{d:D_s} we find
\begin{equation}\label{e:F_s by D_s}
F_i(X)\ =\ - h^{-1}{T^0}(X,I_i\nabla h)
=\ -D_i(I_iX)\ +\ D_j(I_i X)\ +\ D_k(I_iX).
\end{equation}
We recall the next divergence
formulas established in \cite[Lemma~4.2, Lemma~4.3]{IMV1} with the
help of the contracted second Bianchi identity \eqref{div:To}.
\begin{equation}\label{divD}
\nabla^*\, D\ =\ \lvert T^0 \rvert^2\ -h^{-1}g(dh,D)\ -\ h^{-1}
(n+2)\,g(dh,A).
\end{equation}
\begin{multline}\label{p:div of F_s}
\nabla^*\, \Bigl (\sum_{s=1}^3 dh(\xi_s) F_s\Bigr )\ =\ \sum_{s=1}^3 \Bigl [%
\ \nabla dh\, (I_se_a,\xi_s)F_s(I_se_a)\Bigr] \\
+ \ h^{-1}\sum_{s=1}^3 \Bigl[dh(\xi_s)dh (I_se_a)D(e_a)\ +(n+2)\,dh(\xi_s)dh
(I_s e_a)\, A(e_a)\Bigr ].
\end{multline}
\section{The divergence formula}
{\ Following is our main technical result.}{\ As mentioned in the
introduction, we were motivated to seek a divergence formula of
this type based on the Riemannian, CR and seven dimensional qc
cases of the considered problem. The main difficulty was to find a
suitable vector field with non-negative divergence containing the
norm of the torsion. The fulfilment of this task was facilitated
by the results of \cite{IMV}. In particular,
similarly to the CR case, but unlike the Riemannian case, we were
not able to achieve a proof based purely on the Bianchi
identities, see \cite[Theorem
4.8]{IMV}.}
Using $\overline{Scal}=Scal=16n(n+2)$ in the Yamabe equation
\eqref{e:conf change scalar curv} we have
\begin{equation} \label{n1}
\triangle h=2n-4nh +h^{-1}(n+2)|\nabla h|^2.
\end{equation}
The equation \eqref{e:U conf change} in the case $\bar T^0=\bar
U=0$ and \eqref{n1} motivate the definition of the following
symmetric (0,2) tensors
\begin{multline} \label{n2}
\mathbf{D}(X,Y)=-T^0(X,Y)=\frac{h^{-1}}{4}\Big[3\nabla^2h(X,Y)-%
\sum_{s=1}^3\nabla^2h(I_sX,I_sY)
+4\sum_{s=1}^3dh(\xi_s)\omega_s(X,Y)\Big]
\end{multline}
\begin{multline} \label{n3}
\mathbf{E}(X,Y)=-2U(X,Y)=\frac{h^{-1}}{4}\Big[\nabla^2h(X,Y)+%
\sum_{s=1}^3\nabla^2h(I_sX,I_sY)\Big] \\
-\frac{2h^{-2}}{4}\Big[dh(X)dh(Y)+\sum_{s=1}^3dh(I_sX)dh(I_sY)\Big] -\frac{h^{-1}}{4}%
\Big(2-4h+h^{-1}|\nabla h|^2\Big)g(X,Y).
\end{multline}
The one form $D$ defined in \eqref{d:def of D} and expressed
in terms of $h$ in \eqref{n21} satisfies $D(X)=h^{-1}%
\mathbf{D}(X,\nabla h).$
Consider the 1-form $E(X)=h^{-1}\mathbf{E}(X,\nabla h)$. We obtain
from \eqref{n2} and \eqref{n3} the expression
\begin{equation} \label{n31}
E(X)=\frac{h^{-2}}{4}\Big[\nabla^2h(X,\nabla h)+\sum_{s=1}^3\nabla^2h(I_sX,I_s\nabla h)+%
\Big(-2+4h-3h^{-1}|\nabla h|^2\Big)dh(X)\Big].
\end{equation}
We also define the (0,3)-tensors $\mathbb{D}$ and $\mathbb{E}$ by
\begin{multline}\label{ddd3}
\mathbb{D}(X,Y,Z)=-\frac{h^{-1}}{8}\Big[dh(X)T^0(Y,Z)+dh(Y)T^0(X,Z)\\+\sum_{s=1}^3dh(I_sX)T^0(I_sY,Z)+\sum_{s=1}^3dh(I_sY)T^0(I_sX,Z)\Big]
\end{multline}
\begin{multline}\label{eee3}
\mathbb{E}(X,Y,Z)=\frac{h^{-1}}{8}\Big\{dh(X)\mathbf{E}(Y,Z)+
dh(Y)\mathbf{E}(X,Z)\\+\sum_{s=1}^3dh(I_sX)\mathbf{E}(I_sY,Z)
+\sum_{s=1}^3dh(I_sY)\mathbf{E}(I_sX,Z)\Big\}.
\end{multline}
After this preparations we are ready to state the main result.
\begin{thrm}
\label{t:div formulas} {\ Suppose $(M^{4n+3},\eta)$ is a
quaternionic contact
structure conformal to a 3-Sasakian structure $(M^{4n+3},\bar\eta)$}, $%
\tilde\eta\ =\ \frac{1}{2h}\, \eta.$ If $Scal_{\eta}=Scal_{\tilde%
\eta}=16n(n+2)$, then with $f$ given by
\begin{equation} \label{e:f}
f\ = \ \frac 12\ +\ h\ +\ \frac 14 h^{-1}\lvert \nabla h \rvert^2,
\end{equation}
\noindent the following identity holds
\begin{multline} \label{divgen}
\nabla^*\Bigl(f(D+E)\ +\ \sum_{s=1}^3dh(\xi_s)I_sE \ +\ \sum_{s=1}^3 dh(\xi_s)\, F_s \ +\
4\sum_{s=1}^3 dh(\xi_s)I_sA_s \ -\ \frac {10}{3}\sum_{s=1}^3
dh(\xi_s)\,I_s A \Bigr)\\ =\ \Big(\frac12 +h\Big)\Big(\lvert T^0
\rvert^2+\lvert\textbf{E}\rvert^2\Big)+2h|\mathbb{D}+\mathbb{E}|^2
+\ h\,\langle Q V,\, V\rangle .
\end{multline}
where $Q$ is equal to
\begin{equation*}
Q := \left[ {%
\begin{array}{ccccccc}
{\displaystyle \frac {5}{2} } & -{\displaystyle \frac {1}{2} } &-{\displaystyle \frac {1}{2} } &-{\displaystyle \frac {1}{2} } & -2 & -2 &-2 \\[2ex]
-{\displaystyle \frac {1}{2} } &{\displaystyle \frac {5}{2} }\, & -{\displaystyle \frac {1}{2} } & -{\displaystyle \frac {1}{2} } & {\displaystyle \frac {10}{3}} \, & - {\displaystyle \frac {2}{3%
}} \, & - {\displaystyle \frac {2}{3} } \, \\[2ex]
-{\displaystyle \frac {1}{2} } &-{\displaystyle \frac {1}{2} } &{\displaystyle \frac {5}{2} }\, & -{\displaystyle \frac {1}{2} } & - {\displaystyle \frac {2}{3}} \, & {\displaystyle \frac {10}{3%
}} \, & - {\displaystyle \frac {2}{3 }} \, \\[2ex]
-{\displaystyle \frac {1}{2} } & -{\displaystyle \frac {1}{2} } & -{\displaystyle \frac {1}{2} } & {\displaystyle \frac {5}{2} }\, & - {\displaystyle \frac {2}{3}} \, & - {\displaystyle \frac {2}{%
3}} \, & {\displaystyle \frac {10}{3}} \, \\[2ex]
-2 &{\displaystyle \frac {10}{3}} \, & - {\displaystyle \frac {2}{3 }} \, & - {%
\displaystyle \frac {2}{3}} \, & {\displaystyle \frac {22}{3}} \, & - {%
\displaystyle \frac {2}{3}} \, & - {\displaystyle \frac {2}{3}} \, \\[2ex]
-2 &- {\displaystyle \frac {2}{3}} \, & {\displaystyle \frac {10}{3 }} \, & - {%
\displaystyle \frac {2}{3}} \, & - {\displaystyle \frac {2}{3}} \, & {%
\displaystyle \frac {22}{3}} \, & - {\displaystyle \frac {2}{3}} \, \\[2ex]
-2 &- {\displaystyle \frac {2}{3}} \, & - {\displaystyle \frac {2 }{3}} \, & {%
\displaystyle \frac {10}{3}} \, & - {\displaystyle \frac {2}{3}} \, & - {%
\displaystyle \frac {2}{3} } \, & {\displaystyle \frac {22}{3}} \,%
\end{array}%
} \right]
\end{equation*}
Here, $Q$ is a positive definite matrix with eigenvalues $1$,
$\frac92\pm\frac{\sqrt {73}}{2}$ and $\frac{11}{2}\pm\frac{\sqrt
{89}}{2}$ and
$V=(E, D_1, D_2, D_3,A_1,
A_2, A_3)$ with $E$, $D_s$, $A_s$ defined, correspondingly, in \eqref{n31} \eqref{d:D_s} and %
\eqref{d:A_s}.
\end{thrm}
\begin{proof} For the sake of making some formulas more compact, in the proof we will use sometimes the notation $XY=g(X,Y)$ for the product of two horizontal vector fields $X$ and $Y$ and the similar abbreviation for horizontal 1-forms.
We begin by recalling \eqref{n21}, \eqref{n31} and \eqref{e:A}, which imply
\begin{multline} \label{ead1}
A(X)=\frac{3E(X)-D(X)}2-h^{-1}\sum_{s=1}^3\nabla^2h(I_sX,\xi_s) \\
+\frac32h^{-2}\sum_{s=1}^3dh(\xi_s)dh(I_sX)-\frac32h^{-2}\Big(\frac12+h+\frac14h^{-1}|%
\nabla h|^2\Big)dh(X).
\end{multline}
Using the function $f$ defined in \eqref{e:f}, we write \eqref{ead1} in the
form
\begin{equation} \label{ead2}
2\sum_{s=1}^3\nabla^2h(I_sX,\xi_s)=h(3E(X)-D(X)-2A(X))+3h^{-1}\sum_{s=1}^3dh(%
\xi_s)dh(I_sX)-3h^{-1}fdh(X).
\end{equation}
The sum of \eqref{n21} and \eqref{n31} yields
\begin{equation} \label{ed1}
(E+D)(X)=h^{-2}\nabla^2h(X,\nabla h)+h^{-2}\sum_{s=1}^3dh(\xi_s)dh(I_sX)+\frac{h^{-2}}4%
\Big(-2+4h-3h^{-1}|\nabla h|^2\Big)dh(X)\Big].
\end{equation}
Using \eqref{e:f} and \eqref{ed1}, we obtain
\begin{equation} \label{df}
2\nabla_X
= h(E+D)(X) -h^{-1}\sum_{s=1}^3dh(\xi_s)dh(I_sX)+h^{-1}fdh(X).
\end{equation}
We calculate the divergences of $E$ using \eqref{div:To} as
follows
\begin{multline} \label{divE}
\nabla^*E=2h^{-2}dh(e_a)U(e_a,\nabla h)- 2h^{-1}(\nabla_{e_a}U)(e_a,\nabla
h) - 2h^{-1}U(e_a,e_b)\nabla^2h(e_a,e_b) \\
=-h^{-1}(1-n)A(\nabla h)+U(e_a,e_b)(-2h^{-1})\Big[%
\nabla^2h(e_a,e_b)-2dh(e_a)dh(e_b)\Big]+h^{-1}E(\nabla h) \\
=|\mathbf{E}|^2+h^{-1}dh(e_a)E(e_a)-h^{-1}(1-n)dh(e_a)A(e_a).
\end{multline}
Similarly, we have
\begin{multline} \label{divIE}
-\nabla^*I_sE=2h^{-2}dh(e_a)U(I_se_a,\nabla h)+
2h^{-1}(\nabla_{e_a}U)(e_a,I_s\nabla h)
- 2h^{-1}U(I_se_a,e_b)\nabla^2h(e_a,e_b) \\%\quad use \quad \eqref{div:To} \\
=h^{-1}(1-n)A(I_s\nabla h)+U(I_se_a,e_b)(-2h^{-1})\Big[%
\nabla^2h(e_a,e_b)-2dh(e_a)dh(e_b)\Big]+h^{-1}E(I_s\nabla h) \\
=U(I_se_a,e_b)U(e_a,e_b)-h^{-1}(1-n)dh(I_se_a)A(e_a)
=-h^{-1}(1-n)dh(I_se_a)A(e_a),
\end{multline}
since $U(I_se_a,e_b)U(e_a,e_b)=E(I_s\nabla h)=0$ due to
\eqref{propt}.
Now we are prepared to calculate the divergence of the first four terms. Using %
\eqref{divD}, \eqref{p:div of F_s}, \eqref{divE}, \eqref{df},
\eqref{divIE} and \eqref{ead2}, we have
\begin{multline} \label{newdiv1}
\nabla_{e_a}\Big[f(D+E)(e_a)-\sum_{s=1}^3dh(\xi_s)E(I_se_a)+\sum_{s=1}^3dh(\xi_s)F_s(e_a)\Big] \\
= \Big(\frac{h}2(E+D)(e_a) -\frac{h^{-1}}2\sum_{s=1}^3dh(\xi_s)dh(I_se_a)+\frac{h^{-1}}%
2fdh(e_a)\Big)(D+E)(e_a) \\
+f\Big[-h^{-1}D(\nabla h)-h^{-1}(n+2)A(\nabla h)+|T^0|^2+
|E|^2+h^{-1}dh(e_a)E(e_a)-h^{-1}(1-n)dh(e_a)A(e_a)\Big] \\
+h^{-1}(1-n)\sum_{s=1}^3dh(\xi_s)dh(I_se_a)A(e_a)+\sum_{s=1}^3\nabla^2h(I_se_a,\xi_s)E(e_a)
\\
+ \sum_{s=1}^3\nabla^2h\, (I_se_a,\xi_s)F_s(I_se_a) + \
h^{-1}\sum_{s=1}^3dh(\xi_s)dh
(I_se_a)D(e_a)\ +(n+2)\,\sum_{s=1}^3dh(\xi_s)dh (I_s e_a)\, A(e_a) \\
= f(|T^0|^2+|\mathbf{E}|^2)+\frac{h}2|D+E|^2 +%
\frac{h}2(3E-D-2A)(e_a)E(e_a) \\
+h^{-1}\Big[\sum_{s=1}^3dh(\xi_s)dh(I_se_a)-fdh(e_a)\Big]\Big(\frac12D(e_a)+3A(e_a)\Big)
+ \sum_{s=1}^3\nabla^2 h\, (I_se_a,\xi_s)F_s(I_se_a).
\end{multline}
\begin{comment}
Now we add the term $\sum_{s=1}^3dh(\xi_s)F_s$ and use
\eqref{p:div of F_s}. We have using \eqref{newdiv}
\begin{multline} \label{newdiv1}
\nabla_{e_a}\Big[f(D+E)(e_a)-\sum_{s=1}^3dh(\xi_s)E(I_se_a)+\sum_{s=1}^3dh(\xi_s)F_s(e_a)\Big] \\
= f(|T^0|^2+|\mathbf{E}|^2)+\frac{h}%
2|D+E|^2-3h^{-1}fdh(e_a)A(e_a)+h^{-1}(1-n)\sum_{s=1}^3dh(\xi_s)dh(I_se_a)A(e_a) \\
-\Big[\frac{h^{-1}}2\sum_{s=1}^3dh(\xi_s)dh(I_se_a)+\frac{h^{-1}}2fdh(e_a)\Big]D(e_a) +%
\frac{h}2(3E-D-2A)(e_a)E(e_a) \\
+ \sum_{s=1}^3\nabla^2h\, (I_se_a,\xi_s)F_s(I_se_a) + \
h^{-1}\sum_{s=1}^3dh(\xi_s)dh
(I_se_a)D(e_a)\ +(n+2)\,\sum_{s=1}^3dh(\xi_s)dh (I_s e_a)\, A(e_a) \\
= f(|T^0|^2+|\mathbf{E}|^2)+\frac{h}2|D+E|^2-3h^{-1}\Big[fdh(e_a)-\sum_{s=1}^3dh(%
\xi_s)dh(I_se_a)\Big]A(e_a) \\
+\Big[\frac{h^{-1}}2\sum_{s=1}^3dh(\xi_s)dh(I_se_a)-\frac{h^{-1}}2fdh(e_a)\Big]D(e_a) +%
\frac{h}2(3E-D-2A)(e_a)E(e_a) \\
+ \sum_{s=1}^3\nabla^2 h\, (I_se_a,\xi_s)F_s(I_se_a).
\end{multline}
\end{comment}
Applying \eqref{l:div of I_sA} and \eqref{ead2} we obtain
\begin{multline} \label{newdiv2}
\nabla_{e_a}\Big[f(D+E)(e_a)-\sum_{s=1}^3dh(\xi_s)E(I_se_a)+\sum_{s=1}^3dh(\xi_s)F_s(e_a)-2\sum_{s=1}^3dh(%
\xi_s)I_sA(e_a)\Big] \\
=
f(|T^0|^2+|\mathbf{E}|^2)+\frac{h}2|D+E|^2+\frac{h}2(3E-D-2A)E-h(3E-D-2A)A
\\
+\frac{h^{-1}}2\Big[\sum_{s=1}^3dh(\xi_s)dh(I_se_a)-fdh(e_a)\Big]D(e_a)
+ \sum_{s=1}^3\nabla^2 h\, (I_se_a,\xi_s)F_s(I_se_a)
\end{multline}
According to \eqref{e:F_s by D_s}, the last term in \eqref{newdiv2} reads
\begin{multline} \label{newDs}
\sum_{s=1}^3\nabla^2 h\, (I_se_a,\xi_s)F_s(I_se_a)=D_1(e_a)\Big[\nabla^2h(I_1e_a,\xi_1)-%
\nabla^2h(I_2e_a,\xi_2)-\nabla^2h(I_3e_a,\xi_3) \Big] \\
+D_2(e_a)\Big[-\nabla^2h(I_1e_a,\xi_1)+\nabla^2h(I_2e_a,\xi_2)-%
\nabla^2h(I_3e_a,\xi_3) \Big] \\
+D_3(e_a)\Big[-\nabla^2h(I_1e_a,\xi_1)-\nabla^2h(I_2e_a,\xi_2)+%
\nabla^2h(I_3e_a,\xi_3) \Big].
\end{multline}
Using \eqref{newDs} we rewrite the last line in \eqref{newdiv2} as follows
\begin{multline} \label{newDs1}
\Big[\frac{h^{-1}}2\sum_{s=1}^3dh(\xi_s)dh(I_se_a)-\frac{h^{-1}}2fdh(e_a)\Big]D(e_a)
+
\sum_{s=1}^3\nabla^2 h\, (I_se_a,\xi_s)F_s(I_se_a) \\
= D_1(e_a)\Big[\nabla^2h(I_1e_a,\xi_1)-\nabla^2h(I_2e_a,\xi_2)-%
\nabla^2h(I_3e_a,\xi_3)+\frac{h^{-1}}2\sum_{s=1}^3dh(\xi_s)dh(I_se_a)-\frac{h^{-1}}%
2fdh(e_a) \Big] \\
+D_2(e_a)\Big[-\nabla^2h(I_1e_a,\xi_1)+\nabla^2h(I_2e_a,\xi_2)-%
\nabla^2h(I_3e_a,\xi_3)+\frac{h^{-1}}2\sum_{s=1}^3dh(\xi_s)dh(I_se_a)-\frac{h^{-1}}%
2fdh(e_a) \Big] \\
+D_3(e_a)\Big[-\nabla^2h(I_1e_a,\xi_1)-\nabla^2h(I_2e_a,\xi_2)+%
\nabla^2h(I_3e_a,\xi_3)+\frac{h^{-1}}2\sum_{s=1}^3dh(\xi_s)dh(I_se_a)-\frac{h^{-1}}%
2fdh(e_a) \Big].
\end{multline}
The equalities \eqref{n31}, \eqref{Ds} and \eqref{e:A_s} imply
\begin{multline} \label{newAsdsE}
\nabla^2h(I_2X,\xi_2)+\nabla^2h(I_3X,\xi_3)\\=h(E-D_1-2A_1)(X)
+h^{-1}\sum_{s=1}^3dh(\xi_s)dh(I_sX)-h^{-1}fdh(X),
\end{multline}
Subtracting two times \eqref{newAsdsE} from \eqref{ead2} we obtain
\begin{multline} \label{fin1}
\nabla^2h(I_1e_a,\xi_1)-\nabla^2h(I_2e_a,\xi_2)-\nabla^2h(I_3e_a,\xi_3)+%
\frac{h^{-1}}2\sum_{s=1}^3dh(\xi_s)dh(I_se_a)-\frac{h^{-1}}2fdh(e_a) \\
=\frac{h}2\Big[-E-D+4D_1-2A+8A_1\big](e_a)
\end{multline}
The left-hand side of the above identity is the second line in
\eqref{newDs1}. The other two lines are evaluated similarly and
the formulas are obtained from the above by a cyclic rotation of
$\{1,2,3\}$.
\begin{comment}
\begin{equation} \label{fin}
\begin{aligned} Third =
\frac{h}2\Big[-E-D+4D_2-2A+8A_2\big]; \quad Fourth =
\frac{h}2\Big[-E-D+4D_3-2A+8A_3\big]. \end{aligned}
\end{equation}
A substitution of \eqref{fin}, \eqref{fin1} in
\end{comment}
A substitution of the resulting new form of \eqref{newDs1}
in \eqref{newdiv2} give
\begin{multline} \label{finn}
\nabla_{e_a}\Big[f(D+E)(e_a)-\sum_{s=1}^3dh(\xi_s)E(I_se_a)+\sum_{s=1}^3dh(\xi_s)F_s(e_a)-2\sum_{s=1}^3dh(%
\xi_s)I_sA(e_a)\Big]
\\=
f\Big(|T^0|^2+|\mathbf{E}|^2\Big)+\frac{4h}2\Big[%
E^2+A^2+D_1^2+D_2^2+D_3^2-2AE+2A_1D_1+2A_2D_2+2A_3D_3\Big].
\end{multline}
In view of \eqref{l:div of I_sA} for any non-zero
constant $c$ we calculate the following divergences as follows
\begin{multline} \label{divnew}
\nabla_{e_a}\Big(c\sum_{s=1}^3dh(\xi_s)I_sA_s(e_a) -\frac{c}3\sum_{s=1}^3dh(\xi_s)I_sA(e_a)\Big) \\
=\frac{c}3\Big[2\nabla^2h(I_1e_a,\xi_1)-\nabla^2h(I_2e_a,\xi_2)-%
\nabla^2h(I_3e_a,\xi_3) \Big]A_1(e_a) \\
+\frac{c}3\Big[2\nabla^2h(I_2e_a,\xi_2)-\nabla^2h(I_1e_a,\xi_1)-%
\nabla^2h(I_3e_a,\xi_3) \Big]A_2(e_a) \\
+\frac{c}3\Big[2\nabla^2h(I_3e_a,\xi_3)-\nabla^2h(I_2e_a,\xi_2)-%
\nabla^2h(I_1e_a,\xi_1) \Big]A_3(e_a)
\end{multline}
subtracting
\eqref{newAsdsE} from twice \eqref{ead2} yields
\begin{multline} \label{divnew1}
2\nabla^2h(I_1e_a,\xi_1)-\nabla^2h(I_2e_a,\xi_2)-\nabla^2h(I_3e_a,\xi_3)
\\=h\Big[2D_1-D_2-D_3+4A_1-2A_2-2A_3\Big](e_a)
\end{multline}
Now, taking into account \eqref{divnew1}, \eqref{divnew} and
\eqref{finn} we obtain
\begin{multline} \label{finnn}
\nabla^*\Big[f(D+E)(X)-\sum_{s=1}^3dh(\xi_s)E(I_sX)+\sum_{s=1}^3dh(\xi_s)F_s(X)-2\sum_{s=1}^3dh(\xi_s)I_sA(X)\Big]\\+\nabla^*\Big[c\quad
\sum_{s=1}^3dh(\xi_s)I_sA_s(X) -\frac{c}3\sum_{s=1}^3dh(\xi_s)I_sA(X)\Big] \\
= f\Big(|T^0|^2+|\mathbf{E}|^2\Big)+\frac{4h}2\Big[%
E^2+A^2+D_1^2+D_2^2+D_3^2-2AE+2A_1D_1+2A_2D_2+2A_3D_3\Big] \\
+h\frac{c}3\Big[(2D_1-D_2-D_3+4A_1-2A_2-2A_3)A_1\Big]
+h\frac{c}3\Big[(2D_2-D_1-D_3+4A_2-2A_1-2A_3)A_2\Big] \\
+h\frac{c}3\Big[(2D_3-D_2-D_1+4A_3-2A_2-2A_1)A_3\Big]
\end{multline}
In the next Lemma, as in the proof of Theorem \ref{t:div formulas}, we shall use again the notation $XY=g(X,Y)$ for the product of two horizontal vector fields $X$ and $Y$ and the similar abbreviation for horizontal 1-forms.
\begin{lemma}\label{lemDE}
For the (0,3)-tensors $\mathbb{D}$ and $\mathbb{E}$ defined by
\eqref{ddd3} and \eqref{eee3} we have
\begin{comment}
\begin{equation*}
\mathbb{D}(X,Y,Z)=-\frac{h^{-1}}{8}\Big\{dh(X)T^0(Y,Z)+dh(Y)T^0(X,Z)+dh(I_sX)T^0(I_sY,Z)+dh(I_sY)T^0(I_sX,Z)\Big\}
\end{equation*}
and
\begin{equation*}
\mathbb{E}(X,Y,Z)=\frac{h^{-1}}{8}\Big\{dh(X)\mathbf{E}(Y,Z)+ dh(Y)\mathbf{E}(X,Z)+dh(I_sX)\mathbf{E}(I_sY,Z)+dh(I_sY)\mathbf{E}(I_sX,Z)\Big\},
\end{equation*}
then we have
\end{comment}
\begin{equation}\label{calDE}
\begin{aligned}
|\mathbb{D}|^2\ =\ \frac{1}{8}h^{-2}|\nabla h|^2|T^0|^2-\frac14\sum_{s=1}^3|D_s|^2+\frac12(D_1D_2+D_1D_3+D_2D_3),\\
|\mathbb{E}|^2\ =\ \frac{1}{8}h^{-2}|\nabla
h|^2|\mathbf{E}|^2-\frac14|E|^2,\qquad \mathbb{D}\mathbb{E}\ =\
\frac14\sum_{s=1}^3 ED_s.\qquad
\end{aligned}
\end{equation}
Consequently,
\begin{multline}\label{ednew}
\frac14h^{-2}|\nabla h|^2(|T^0|^2+|\mathbf{E}|^2)\ =\
2|\mathbb{D}+\mathbb{E}|^2-\sum_{s=1}^3 ED_s\\\ +\
\frac12|E|^2+\frac12\sum_{s=1}^3|D_s|^2-(D_1D_2+D_1D_3+D_2D_3)
\end{multline}
\end{lemma}
\begin{proof} We shall repeatedly apply \eqref{propt}, the defining equations \eqref{ddd3}, \eqref{eee3}, \eqref{d:A_s} and \eqref{d:def of D}. We have
\begin{multline}\label{d2}
|\mathbb{D}|^2\
=\frac{h^{-2}}{8}|\nabla
h|^2|T^0|^2+\frac{h^{-2}}{8^2}\Big(2T^0(\nabla h,e_c)T^0(\nabla
h,e_c)\\-4\sum_{s=1}^3T^0(I_s\nabla h,e_c)T^0(I_s\nabla h,e_c)
+2\sum_{s,t=1}^3T^0(I_sI_t\nabla h,e_c)T^0(I_tI_s\nabla
h,e_c)\Big)\\%=\frac{h^{-2}}{8}|\nabla
=\frac{h^{-2}}{8}|\nabla
h|^2|T^0|^2+\frac{1}{4}\Big(-\sum_{s=1}^3D_s^2+2(D_1D_2+D_1D_3+D_2D_3)\Big)
\end{multline}
which is the first line of \eqref{calDE}. For example, the third
term in \eqref{d2} is calculated as follows
\begin{multline*}
\sum_{s,t=1}^3T^0(I_sI_t\nabla h,e_c)T^0(I_tI_s\nabla h,e_c)=
\sum_{s=1}^3\Big[T^0(\nabla h,e_c)T^0(\nabla h,e_c)-2T^0(I_s\nabla h,e_c)T^0(I_s\nabla h,e_c)\Big]\\
=6|D|^2-12\sum_{s=1}^3D_s^2+8(D_1D_2+D_1D_3+D_2D_3)
=-6\sum_{s=1}^3D_s^2+20(D_1D_2+D_1D_3+D_2D_3).
\end{multline*}
Similarly, we obtain the second line of \eqref{calDE}. The
equality \eqref{ednew} follows from \eqref{calDE} which completes
the proof of Lemma~\ref{lemDE}.
\end{proof}
Finally, the proof of Theorem~\ref{t:div formulas} follows by
letting $c=4$ in \eqref{finnn}and using \eqref{ednew} and
\eqref{d:A_s}.
\end{proof}
\section{Proof of Theorem\protect[\ref{mainth}] and Theorem\protect[\ref{main2}]}
We begin with the proof of Theorem \ref{mainth}. The first step of
the proof relies on Theorem~\ref{t:div formulas}. By a homothety
we can suppose that both qc-scalar curvatures are equal to
$16n(n+2)$. Integrating the divergence formula of
Theorem~\ref{t:div formulas}
and then using the divergence theorem established in \cite[Proposition 8.1%
]{IMV} shows that the integral of the left-hand side is zero. Thus, the
right-hand side vanishes as well, which shows that the
quaternionic contact structure $\bar\eta$ has vanishing torsion,
i.e., it is also qc-Einstein according to
\cite[Proposition~4.2]{IMV}. This proves the first part of
Theorem~\ref{mainth}.
To prove the second part, we develop a sub-Riemannian extension of
the result of \cite{Ob}, see also \cite{BoEz87} and the review
\cite[Theorem 2.6]{IV14}, on the relation between the Yamabe
equation and the Lichnerowicz-Obata first eigenvalue estimate. We
begin by recalling some results from \cite[Section 7.2]{IMV}. A
vector field $Q$ on a qc manifold $(M, \eta)$ is a \emph{qc vector
field}\index{qc vector field} if its flow preserves the horizontal
distribution $H=\ker \eta$. Since the conformal class of the qc
structure on $\text{span}\{\eta_1,\eta_2,\eta_3\}$ is uniquely
determined by $H$ (cf. \cite{Biq1}
, we have that
\begin{equation*}
\mathcal{L}_Q\, \eta=(\nu I+O)\cdot\eta,
\end{equation*}
where $\nu$ is a smooth function and $O\in so(3)$ is a matrix
valued function with smooth entries. Since the exterior derivative
$d$ commutes with the Lie derivative $\mathcal{L}_Q\, $, any qc vector field
$Q$ satisfies
\begin{gather*}
\mathcal{L}_Q\, g =\nu g,\qquad \mathcal{L}_Q\, I=O\cdot I,
\qquad I=(I_1,I_2,I_3)^t,
\end{gather*}
which is equivalent to saying that the flow of $Q$ preserves the
conformal class $[g]$ of the horizontal metric and the
quaternionic structure $\mathbb Q$ on $H$. The function $\nu$ can
be easily expressed in terms of the divergence (with respect to
$g$) of the horizontal part $Q_H$ of the vector field $Q$. Indeed,
from \cite[Lemma 7.12]{IMV} we have
\begin{gather*}
g(\nabla_XQ_H,Y)\ +\ g(\nabla_YQ_H,X)\ + \ 2\eta_s(Q)g(T^0_{\xi_s} X,Y)=\nu\, g(X,Y),
\end{gather*}
hence
\begin{equation*}
\nu=\frac {1}{2n}\nabla^* Q_H.
\end{equation*}
This gives a geometric interpretation for the quantity $(\nabla^*
Q_H)$, namely, the flow of a qc vector field $Q$ preserves a fixed
metric $g\in[g]$ if and only if $\nabla^* Q_H=0$.
As an infinitesimal version of the qc Yamabe equation we obtain the
following general fact concerning the divergence of a QC vector
field.
\begin{lemma}\label{l:lapdiv} Let $(M,\eta)$ be a qc manifold.
For any qc vector field $Q$ on $M$ we have
\begin{equation*}
\Delta(\nabla^*Q_H)\ =\ -\ \frac{n}{2(n+2)}Q(\text{Scal})\ -\ \frac{\text{Scal}}{4(n+2)}\nabla^*Q_H,
\end{equation*}
where Scal, $\nabla^*$, $\Delta$ and the projection $Q_H$ correspond to the contact form $\eta$.
\end{lemma}
\begin{proof}
Suppose $Q$ is a qc vector field and
let $\phi_t$ be the corresponding (local) 1-parameter group of diffeomorphisms generated by its flow. Then $$\phi_t^*(\eta)\ =\ {\displaystyle \frac{1}{ 2h_t}}\,\eta\quad\text{and}\quad \phi_t^*(g)\ =\ {\displaystyle \frac{1}{ 2h_t}}\,g$$ for some positive function $h_t$, depending smoothly on the parameter $t$. The qc scalar curvature $\text{Scal}_t$ of the pull back contact form $\phi_t^*(\eta)$ is given by $\text{Scal}_t=\text{Scal}\circ\phi_t$. Then, formula \eqref{e:conf change scalar curv} yields
\begin{equation}\label{e:conf change scalar curv-t}
\text{Scal}\circ\phi_t\ =\ 2h_t\,(\text{Scal})\ -\ 8(n+2)^2\,h_t^{-1}|\nabla
h_t|^2\ +\ 8(n+2)\,\triangle h_t.
\end{equation}
Clearly, we have $h_0=\frac12$, and from
\begin{gather*}
\frac{1}{2n}(\nabla^*Q_H)\,g\ =\ \mathcal L_Q\,g\ =\ \frac{d}{dt}\vert_{ t=0}\left(\frac{1}{2h_t}g\right)\ =\ -\ \frac{h'_0}{2h_0}\,g\ =\ -\ 2h'_0\, g
\end{gather*}
we obtain that $h'_0=-\frac{1}{4n}\nabla^*Q_H$, where $h'_0$ denotes the derivative of $h_t$ at $t=0$.
A differentiation at $t=0$ in \eqref{e:conf change scalar curv-t} gives the lemma.
\end{proof}
\begin{lemma}\label{l:qc conf Einst} Let $(M,\eta)$ and $(M,\bar\eta)$ be qc-Einsten manifolds with equal qc-scalar curvatures
$16n(n+2)$. If $\eta$ and $\bar\eta$ are qc conformal to each other, $\overline \eta=\frac{1}{2h}\eta$ for some smooth positive function $h$, then \begin{equation}\label{e:qc field}
Q=\frac12\nabla f+\sum_{s=1}^3dh(\xi_s)\xi_s
\end{equation}
is a qc vector field on $M$, where the function $f$ is defined in
\eqref{df}
\end{lemma}
\begin{proof} The assumption of the lemma implies that
$E=D=D_s=A_s=0$. Using \eqref{newAsdsE},
\eqref{fin1} and \eqref{df} we obtain
$\nabla^2h(I_sX,\xi_s)=-df(X)$ and thus $\nabla^2h(X,\xi_s)=df(I_sX).$ It follows that
\begin{equation*
\sum_{s=1}^3\nabla_X(dh(\xi_s)\xi_s)=\sum_{s=1}^3df(I_sX)\xi_s.
\end{equation*}
To show that the flow of the vector field $Q$, defined by
\eqref{e:qc field}, preserves the horizontal distribution $H$, for
any $X\in H$, we have
\begin{multline*}
\mathcal L_Q(X
= \frac12 \,[\nabla f,X]+\sum_{s=1}^3[dh(\xi_s)\xi_s,X] =
\frac12\nabla_{\nabla f} X\ -\ \frac12\nabla_X(\nabla f)\ -\
\sum_{s=1}^3\omega_s(\nabla f,X)\xi_s\ \\+\ \sum_{s=1}^3 \left
[dh(\xi_s)\nabla_{\xi_s}X- \nabla_X(dh(\xi_s)\xi_s)-
dh(\xi_s)T_{\xi_s}(X)\right ] =
\frac12\nabla_{\nabla f} X\ -\
\frac12\nabla_X(\nabla f)+\sum_{s=1}^3 dh(\xi_s)\nabla_{\xi_s}X \ \in H.
\end{multline*}
\end{proof}
\begin{comment}
\textbf{REMOVE THIS PARAGRAPH - THE NEGATIVE AND ZERO CASES WHICH ARE DIFFERENT IN VIEW OF THE OPENING OF THE PAPER AND THEN WE STATE EVERYTHING IN THE POSITIVE CASE.
????At this point we are ready to complete the proof of Theorem
\ref{mainth}. We give the details for the more interesting case
when the qc-scalar curvature is positive since the uniqueness in
remaining cases follows from the maximum principle as in the
Riemannian case due to the sub-ellipticity of the sub-Laplacian.????}
\end{comment}
At this point we are ready to complete the proof of Theorem
\ref{mainth}. Consider the qc vector field $Q$ defined in Lemma
\ref{l:qc conf Einst}. By Lemma \ref{l:lapdiv}, {the function
$\phi=\frac12\triangle f$ is either an eigenfunction of the
sub-Laplacian with eigenvalue $-4n$, $\triangle\phi=-4n\phi$}, or
it vanishes identically. In the first case, using the quaternionic
contact version of the Lichnerowicz-Obata eigenfunction sphere
theorem \cite[Theorem
1.2]{IPV2} and \cite[Corollary~1.2]{IPV1} (see also
\cite{BauKim14}), we conclude
that $(M, \eta)$ is the 3-Sasakain sphere.
{In the other case, we have that $\Delta f = 0$, hence the
function $f=\frac 12 + h+\frac 14 h^{-1}\lvert \nabla h
\rvert^2=const$ since $M$ is compact. It follows that $h=1/2$ by
considering the points where $h$ achieves its minimum and maximum
and taking into account the qc Yamabe equation \eqref{n1}. }The
proof of Theorem~\ref{mainth} is complete.
\begin{rmrk}\label{mys} {Lemma~\ref{l:qc conf Einst} provides also a certain geometric
insight for the mysterious function $f$ in \eqref{e:f}. In fact,
up to an additive constant, $f$ is the unique function on $M$ for
which $Q_H=\frac 12\nabla f$ is the horizontal part of a qc vector
field $Q$ with vertical part $Q_V=dh(\xi_s)\xi_s$, $Q=Q_H+Q_V$.
This assertion is an easy consequence of the computation given in
the proof of Lemma~\ref{l:qc conf Einst}. Moreover, it implies
that on the 3-Sasakain sphere $\phi=\triangle f$ is an
eigenfuction of the sub-Laplacian realizing the smallest possible
eigenvalue $-4n$ on a compact locally 3-Sasakian manifold.}
\end{rmrk}
Theorem~\ref{main2} is a direct corollary from Theorem
\ref{mainth}. Alternatively, as in the proof of
Theorem~\ref{mainth}, we can use in the first step Theorem
\ref{t:div formulas} which shows that the "new" structure is also
qc-Einstein. The second step of the proof of Theorem \ref{main2}
follows then also by taking into account \cite[Theorem 1.2]{IMV}
where all locally 3-Sasakian structures of positive constant
qc-scalar curvature which are qc-conformal to the standard
3-Sasakian structure on the sphere were classified (we note that
this classification extends easily to the case when no sign
condition of the "new" qc-structure is assumed, see \cite{IV14}).
|
{
"timestamp": "2015-04-14T02:13:44",
"yymm": "1504",
"arxiv_id": "1504.03142",
"language": "en",
"url": "https://arxiv.org/abs/1504.03142"
}
|
\section{Introduction}
\label{sec:introduction}
Despite decades of effort, we still lack a thorough understanding of how galaxies assemble and evolve over cosmic time. This is true not only for distant galaxies but also for our own Milky Way. In the current paradigm, galaxies such as the Milky Way form from smaller pieces \citep[e.g.,][]{sea78}, driven by the hierarchical growth of dark matter structures \citep[e.g.,][]{pee71, pre74}. Much of the most exciting phases of star formation and galaxy assembly appear to have taken place at early times, perhaps before $z\sim2$. If true, this puts much of the most interesting phases of galaxy formation beyond direct detailed study. For this reason much effort has focused on reconstructing the past based on present-day observations of stars, in particular in the Galaxy. For example, studies of the Galactic stellar halo provides clues to the assembly history of dwarf galaxies \citep[e.g.][]{egg62, sea78}. The properties of stars in the thin and thick disks provide clues to the formation history of these Galactic components. The abundance patterns of the most metal poor stars probe star formation and supernovae conditions during the first generation of stars. And the evolutionary histories of star clusters, both intact, dissolving, and long destroyed, offer clues not only into the star formation process (by reconstructing the CMF), but also the dynamical history of the Galaxy \citep[e.g.,][]{kol07,all12,web13}.
However, reconstructing disrupted star clusters is difficult because most star clusters dissolve quickly upon their formation due to dynamical interactions, such as intracluster $N$-body interaction and external tidal stripping from ram pressure. In fact, most clusters are not expected to survive for more than 10 Myrs \citep{lad03}. For this reason, the study of young embedded clusters \citep[e.g.,][]{bic03,por03,kop08,bor11} is typically restricted to the study of star formation conditions at the present time. Although most star clusters are quickly disrupted, they retain their identity in kinematic phase space for a longer period of time. Several examples of clusters identified in phase space are known, such as HR1614, the Argus association and the Wolf 360 group \citep[e.g.,][]{des07a,des13,bub10}, with an age of 2-3 Gyrs. This implies that at least some clusters can maintain their phase space identity for a few disk dynamical times. Within a few dynamical times these groups will phase mix with the background stars, which implies that the timescale over which groups can be identified in phase space is still a small fraction of the age of the Galaxy.
While dynamical information is mostly short-lived, elemental abundances are expected to leave a more permanent fossil record of star clusters. The idea of ``chemical tagging'', first proposed by \citet{fre02} \citep[also see][]{bla14}, is to use elemental abundances to identify stars that are now widely separated in phase space to a common birth site. If such an association could be made, even for a small fraction of stars, it would provide an extraordinary new view into both the early star formation process and the subsequent dynamical history of the Galaxy.
Observations have shown that satellite galaxies exhibit different chemical evolution histories compared to stars either in the disk, bulge, or halo of the Galaxy \citep[e.g.,][]{ven04, pom08, ven08, tol09, let10}. As a consequence, stars accreted into the Galaxy from different satellite systems should show distinct chemistry from e.g., disk stars. It has been proposed that these variations could be used in chemical tagging to find the remnants of disrupted satellite galaxies \citep{fre02}. The possibility of reconstructing disrupted satellite galaxies via chemical tagging could for example provide important clues to the missing satellite problem \citep{moo99}.
Previous studies of high-resolution stellar spectroscopy were limited to a few hundred stars \citep[e.g.,][]{bar05, red06, ben14}. The small samples restricted the possibility of chemical tagging for reasons that will become clear in later sections. But this situation is rapidly changing. Recent and on-going large-scale surveys, such as GALAH \citep{des15}, Gaia-ESO \citep{ran13} and APOGEE \citep{zas13} aim to observe $10^5-10^6$ stars with resolution $R > 20,000$ in order to measure $\sim 15 - 30$ elements for each star. These surveys were motivated, at least in part, by the idea of chemical tagging and the prospects for uncovering the distribution of stars in their $N-$dimensional chemical space, spanned by the elemental abundances.
There are several conditions that must be met for chemical tagging to work \citep[see][for details]{fre02, bla04, bla10a, bla10b, des15}. First, clusters must be internally chemically homogeneous. Open clusters have been found to be chemically homogeneous at the level of $\sigma_{[X/{\rm Fe}]} < 0.05$ dex \citep[e.g.,][]{des07b, des09, tin12b, fri14, one14}. Theoretical arguments from \citet{bla10b} showed that the chemical signature within a protocloud should have sufficient time to homogenize before the first supernova goes off, for clusters with mass $10^5 - 10^7 \, M_\odot$. Simulations by \citet{fen14} showed that turbulent mixing, even for a loosely bound cluster, could homogenize the elemental abundances of a protocloud. Their simulations showed that turbulent mixing creates an intracluster chemical dispersion at least five times more homogenized than the protocloud. Both observations and theory agree that clusters less massive than $\sim10^7 \, M_\odot$ should be chemical homogeneous, except perhaps for the confounding internal abundance trends observed in the light elements of all known globular clusters \citep[e.g.,][]{car09,mar11}, though many globular clusters show a high degree of chemical uniformity \citep[e.g.,][]{roe15} in all heavy elements.
In addition to cluster homogeneity, the existence of substantial cloud-to-cloud variation in elemental abundances is another requirement. For example, if all star clusters shared the same elemental abundances, it would not be possible to separate them in chemical space. We know that this condition is broadly satisfied given the sizable spread in abundance ratios in existing spectroscopic samples \citep[e.g.,][]{edv93,ben14}. Quantitatively, an important parameter is the volume of abundance space that is available for a particular survey. This volume depends both on Galactic chemical evolution and on the particular survey design. The latter is important both in determining the target sample and in the number of elements that can be spectroscopically measured. Combining the available chemical volume with the measurement uncertainty on individual abundances allows us to define the concept of the total number of distinct cells in chemical space. As we will see below, this is a key concept in chemical tagging \citep[see also][]{fre02}.
\citet{tin12a} presented an empirical estimate of cloud-to-cloud variation in elemental abundances. They performed principal component analysis and estimated that there are $7-9$ independent dimensions among the $\sim 25$ elements that will be measured by surveys such as GALAH and Gaia-ESO, and $4-5$ independent dimensions for an APOGEE-like survey. From this one can estimate the number of distinguishable cloud-to-cloud variations in the chemical space, denoted $N_{\rm cells}$. As discussed in detail in \S\ref{subsec:chem-model} below, the result is that modern surveys should be able to reach $N_{\rm cells} \sim 10^{3-4}$, at least, implying that there is a decent cloud-to-cloud variation.
The goal of this paper is to explore the prospects for identifying long disrupted star clusters based on their clustering in chemical space. We follow \citet{fre02}, \citet{bla10a}, and \citet{des15} in identifying the global survey parameters and the shape of the CMF as key parameters. Our emphasis on the information contained in the distribution (i.e., clumpiness) of stars in chemical space echoes the results found in \citet{bla10a}. In the present work we consider a wide array of parameters in order to identify optimal regions of parameter space for chemical tagging. In addition, for the first time we analyze the local properties of cells in chemical space that appear as high sigma fluctuations and find that in many cases these high overdensities in chemical space are not the result of a single star cluster but instead are comprised of stars from many distinct birth sites.
The rest of this paper is organized as follows. In \S\ref{sec:overview} we review several basic arguments relevant for chemical tagging and in \S\ref{sec:models} we describe the model used in the present work. In \S\ref{sec:results} we present the results and discuss how these assumptions and survey strategies affect the chemical tagging detections. In \S\ref{sec:discussion} we discuss various caveats, limitations and future directions. We conclude in \S\ref{sec:conclusions}. It is difficult to present the full set of results from a multidimensional parameter space and so we urge readers to explore the online interactive applet \footnote{\href{www.cfa.harvard.edu/~yuan-sen.ting/chemical_tagging.html}{www.cfa.harvard.edu/$\sim$yuan-sen.ting/chemical$\_$tagging.html}} created in the course of this project (see Appendix~\ref{sec:interactive} for details).
\begin{figure*}
\center
\includegraphics[width=\textwidth,natwidth=1700,natheight=900]{Fig1.pdf}
\caption{Flow chart demonstrating the main components of the model. Sections defining or describing certain components of the model are indicated in the chart.}
\label{fig:flow-chart}
\end{figure*}
\section{Basic Arguments}
\label{sec:overview}
As we will show quantitatively below, the prospects for chemical tagging largely depends on the number of stars sampled per cluster. This number in turn primarily depends on the number of stars in the survey divided by the integrated star formation rate (SFR), over cosmic history, in the volume sampled by the survey. We will denote the former number as $N_\star$, the latter number as $M_{\rm annulus}$. Ongoing and upcoming surveys are targeting primarily FGK stars, which have on average $\langle M \rangle \approx 1 M_\odot$. This implies that $N_\star$ stars in a survey corresponds to $N_\star$ in solar masses and therefore numerically $M_{\rm annulus}\approx N_{\rm annulus}$. The ratio of $N_\star$ and $N_{\rm annulus}$ defines the sampling rate. In this section, we motivate why the sampling rate largely defines the number of stars sampled per cluster \citep[see also][]{des15}.
First, let's consider a simple case where there is no radial migration and stellar excursion, i.e., stars stay in the annulus in which they were born. The integrated SFR in the Solar annulus, with a survey width $\Delta R_{\rm survey} = \pm 3 \, {\rm kpc}$, is $\sim 2 \times 10^{10} \, M_\odot$ (see model detail in \S\ref{sec:models}).\footnote{The survey width $\Delta R_{\rm survey}$ defines the Solar annulus by $|R-R_0| < |\Delta R_{\rm survey}|$. The survey width should not be confused with the line-of-sight depth from the Sun, which is $|{\bf R} - {\bf R_0}| < 3 \, {\rm kpc}$.} For a survey of $10^6$ stars with $\langle M \rangle = 1 \, M_\odot$, the sampling rate can thus be calculated to be $(10^6 \, M_\odot)/(2 \times 10^{10} \, M_\odot) = 1/(2 \times 10^{4})$. In other words, assuming all stellar mass (including stellar mass loss) is now fully mixed in the annulus, we would have only sampled, on average, $1/(2 \times 10^{4})$ of the original zero age mass from each cluster. Thus, we would expect to observe, on average, only one star from a $2 \times 10^4 \, M_\odot$ cluster. If we define the ``detection'' of a cluster to include the identification of at least 10 stars, then for a survey of $10^6$ random stars in the solar annulus we would be able to probe clusters more massive than $2 \times 10^5 \, M_\odot$.
In practice, the sample is affected by the process of radial migration \citep[e.g.,][]{bla10b}. Some stars are migrated away from their birth annulus while others that were born outside the Solar annulus will now reside within the Solar annulus. In other words, the number of stars that could end up in the Solar annulus increases with radial migration (another way of thinking of this effect is that the effective volume of the Solar annulus increases as the strength of radial migration increases). Given that the number of stars in the survey stays the same, the sampling rate decreases with radial migration. For a fixed survey strategy, the minimum cluster mass that one can probe increases in the presence of radial migration.
We must also consider the fact that we have limited resolution in separating groups in terms of their elemental abundance variations due to measurement uncertainties on the abundances. Multiple clusters might share the same cell in chemical space \citep[e.g.,][]{bla10a}. If we assume a CMF over the range $50 \, M_\odot$ to $10^6 \, M_\odot$ and a CMF slope of $-2$ (see details in \S\ref{sec:models}), the mean cluster mass is $\sim 5 \times 10^2 \, M_\odot$. Since the integrated SFR is $\sim 2 \times 10^{10} \, M_\odot$, we deduce that there are $\sim 4 \times 10^7$ clusters in the Solar annulus. Fully resolving clusters in chemical space would require roughly as many distinct chemical cells \citep{fre02}, but it was argued in the Introduction that the actual number of chemical cells spanned by the data may be $2-3$ orders of magnitude lower. This suggests that most cells in chemical space will be occupied by many clusters, each with a small number of stars sampled per cluster. One of the key goals of this paper is to understand the distribution of clusters in chemical space under different scenarios.
The simple calculations in this section already demonstrate that key parameters include the number of stars in a survey, $N_\star$, the geometry of the survey (via $M_{\rm annulus}$), the strength of radial migration, the shape of the CMF (which sets the typical cluster size), and the number of cells in chemical space ($N_{\rm cells}$).
\begin{table*}
\begin{center}
\caption{List of constraints in this study.\label{table:constraints}}
\begin{tabular}{lll}
\tableline \tableline
\\[-0.2cm]
Property & Value & References \\[0.1cm]
\tableline
\\[-0.2cm]
Galactocentric radius of the Sun, $R_0$ & $8 \, {\rm kpc}$ & \citet{ghe08,gil09,rei14} \\[0.1cm]
Stellar surface density, $\Sigma_\star (R_0,z=0)$ & $38 \, M_\odot {\rm pc}^{-2}$ & \citet{fly06,bov13,zha13} \\[0.1cm]
Gas surface density, $\Sigma_{\rm gas} (R_0,z=0)$ & $13 \, M_\odot {\rm pc}^{-2}$ & \citet{fly06} \\[0.1cm]
Total stellar mass in the disk, $M_\star (z=0)$ & $4.5 \times 10^{10} \, M_\odot$ & \citet{fly06,bin08,bov13} \\[0.1cm]
Halo virial mass, $M_{\rm halo} (z=0)$ & $10^{12} \, M_\odot$ & \citet{wil99,kly02,xue08,kaf12} \\[0.1cm]
Global SFR ($z=0$) & $0.5 - 2 \, M_\odot {\rm yr}^{-1}$ & \citet{rob10,cho11,ven13} \\[0.1cm]
Solar neighborhood SFR, $\Sigma_{\rm SFR} (R_0,t)$ & $3-6 \, M_\odot {\rm Gyr}^{-1} {\rm pc}^{-2}$ & \citet{her00,ber01} \\[0.1cm]
Stellar disk scale length, $R_\star (z=0)$ & $2.2 \, {\rm kpc}$ & \citet{bov13} \\[0.1cm]
SFR scale length, $R_{\rm SFR} (z=0)$ & $2.6 \, {\rm kpc}$ & \citet{sch11} on NGC 6946 \\[0.1cm]
Gas scale length, $R_{\rm gas} (z=0)$ & $4.2 \, {\rm kpc}$ & \citet{sch11} on NGC 6946 \\[0.1cm]
Radial size growth & $R_\star \propto M_\star^{0.27}$ & \citet{van13} \\[0.1cm]
\tableline
\end{tabular}
\end{center}
\end{table*}
\begin{table}
\begin{center}
\caption{List of parameters in the model.\label{table:parameters}}
\begin{tabular}{llll}
\tableline \tableline
\\[-0.2cm]
Parameter & Fiducial & Range \\[0.1cm]
\tableline
\\[-0.2cm]
In-situ fraction, $f_{\rm in-situ} (\Delta R_{\rm survey} = \pm 1 \, {\rm kpc})$ & $50\%$ & $15\%-100\%$ \\[0.1cm]
Survey width, $\Delta R_{\rm survey}$ & $\pm 3 \, {\rm kpc}$ & $\pm 0.6 - 5 \, {\rm kpc}$ \\[0.1cm]
CMF slope, $\alpha$ & $-2.0$ & $-1.5$ to $-2.5$ \\[0.1cm]
CMF low mass cutoff, $M_{\rm cluster}^{\rm min}$ & $50 \, M_\odot$ & $10-100 \, M_\odot$ \\[0.1cm]
CMF high mass cutoff, $M_{\rm cluster}^{\rm max}$ & see Figure \ref{fig:model-properties} & see Figure \ref{fig:model-properties} \\[0.1cm]
Number of chemical cells, $N_{\rm cells}$ & $10^4$ & $10^3-10^5$ \\[0.1cm]
Number of stars in the survey, $N_\star$ & $10^6$ & $10^4-10^6$ \\[0.1cm]
\tableline
\end{tabular}
\end{center}
\end{table}
\begin{table*}
\begin{center}
\caption{Meaning of other important symbols in this paper that are not listed in Table~\ref{table:constraints} and \ref{table:parameters}.\label{table:other-symbols}}
\begin{tabular}{llll}
\tableline \tableline
\\[-0.2cm]
Symbols & Meanings \\[0.1cm]
\tableline
\\[-0.2cm]
$k_{\rm ch}$ & Churning strength in the radial migration prescription \\[0.1cm]
$\eta$ & Gas fraction; the ratio of gas mass over total dynamical mass \\[0.1cm]
$\sigma_{\rm [X/Fe]}$ & Elemental measurement uncertainty in [X/Fe] \\[0.1cm]
$\sigma$ & Elemental measurement uncertainty along the chemical space principal components \\[0.1cm]
$N_{\rm dim}$ & Number of independent/informative dimensions in chemical space \\[0.1cm]
$M_{\rm gas}$ & Total gas mass in the Milky Way \\[0.1cm]
$M_{\rm cluster}$ & Zero age stellar mass of a star cluster \\[0.1cm]
$M_{\rm annulus}$ & Integrated SFR, over cosmic history, in the volume sampled by the survey \\[0.1cm]
$N_{\rm annulus}$ & Total number of stars (including stellar mass loss) in the volume sampled by the survey \\[0.1cm]
$N_i$ & Total number of stars sampled in a chemical cell \\[0.1cm]
$N_{\rm mean}$ & Average number of stars sampled per chemical cell \\[0.1cm]
$N_{\rm cluster}$ & Number of stars sampled from a cluster \\[0.1cm]
$N_{\rm dominant}$ & Number of stars sampled from the most dominant cluster in a chemical cell \\[0.1cm]
local S/N & Number of stars sampled from the most dominant cluster over the total number of other stars in a chemical cell \\[0.1cm]
$f_{\rm sub}$ & Sampling rate of a certain stellar subpopulation \\[0.1cm]
\tableline
\end{tabular}
\end{center}
\end{table*}
\section{Model Description}
\label{sec:models}
In this section we describe the ingredients of our model for the Milky Way in some detail. The model is spatially two dimensional (though we assume that stars are uniformly distributed in the azimuthal angle), time-dependent, and statistical in nature. For the present study we are only interested in the disk; the bulge and halo are not included in the model below. We do not follow dynamics nor do we include a treatment of chemical evolution (these will be subjects of future work). The present aim is to build a model that is computationally very fast to allow the exploration of a large multi-dimensional parameter space.
The model specifies the star formation history (SFH) and evolution in time of the size of the Milky Way disk and the gas mass distribution. We define the SFH to be the total SFR in the Milky Way as a function of cosmic time. These quantities are used to model the effects of radial migration and an evolution in the cutoff of the CMF. The model is illustrated in a flow chart in Figure \ref{fig:flow-chart}. Table \ref{table:constraints} lists observational constraints that we employ to constrain the model. Free parameters in the model and their adopted fiducial values are listed in Table \ref{table:parameters}. We now proceed to explain the details of the model.
\subsection{Star formation history and radial size growth of the disk}
\label{subsec:sfh}
The SFH in the Solar neighborhood, $\Sigma_{\rm SFR} (R_0,t)$, has been estimated by analyzing the color-magnitude diagram from the Hipparcos catalog. Results from, for e.g., \citet{her00} and \citet{ber01} showed a rather flat SFH near $R_0$, ranging from $3 - 6 \, M_\odot {\rm Gyr}^{-1} {\rm pc}^{-2}$ through $0-8$ Gyr in lookback time. The current total SFR in the Milky Way has been estimated to be $0.5 - 2 \, M_\odot {\rm yr}^{-1}$ from the study of young stellar objects \citep[e.g.,][]{rob10,cho11,ven13}.
In comparison to the Solar neighborhood, the Galactic global SFH is less well understood. We therefore adopt cosmological semi-empirical modeling from \citet{beh13}, assuming a Milky Way halo virial mass of $M_{\rm halo} \equiv M_{200} = 10^{12} \, M_\odot$ \citep[e.g.,][]{wil99, kly02, xue08, kaf12}. \citet{beh13} investigated the best-fitting global SFH as a function halo mass that is consistent with the observed galaxy stellar mass function, specific SFR, and cosmic SFR. We fit their result for Milky Way-like halos with a Schechter function,
\noindent
\begin{equation}
{\rm SFR} [M_\odot {\rm yr}^{-1}] = A \, (t[{\rm Gyr}]/C)^B \, \exp( - t[{\rm Gyr}]/C).
\end{equation}
Given a global SFH, the stellar mass evolution is calculated assuming the stellar population synthesis code from \citet{con09}, with a Kroupa IMF \citep{kro02} from $0.08 - 125 \, M_\odot$. The synthesis code is used to take into account secular stellar mass loss, etc. The normalization of the global SFH is further adjusted such that the present-day stellar mass (long-lived stars + remnant stars) agrees with observations, $M_\star (z=0) = 4.5 \times 10^{10} \, M_\odot$ \citep[e.g.,][]{bin08, bov13}. In this study, we only trace long-lived stars with $0.5 - 1.5 \, M_\odot$ because almost all FGK stars in chemical tagging surveys are within this mass range.
We consider two SFH models in this study, with parameters from equation (1) as follows: (1) $A=1.4$, $B=4.4$, $C=1.3$, which is the best fitting SFH model from Behroozi et al.; (2) $A=15.5$, $B=2$, $C=2.7$, which produces better agreement with the observed $\Sigma_{\rm SFR} (R_0,t)$. Both models are within the uncertainty quoted by Behroozi et al. We adopt the latter as the fiducial model and the former to be the optimistic model (see Figure~\ref{fig:model-properties} and Table~\ref{table:models}). The former coins the term ``optimistic model'' as its more highly peaked SFR entails a higher total gas mass (see \S\ref{subsec:gas-mass}). The higher total gas mass in turn predicts a larger cluster high mass cutoff (see \S\ref{subsec:cmf}) than the ``fiducial model.'' We emphasize that while the optimistic and fiducial models assume different SFHs, the integrated SFRs of these models over cosmic time are the same. Since the total integrated SFRs are the same, they both produce the same $M_\star(z=0)$ and $\Sigma_\star (R_0,z=0)$. Therefore, the sampling rate is the same for both cases. The global SFR and $\Sigma_{\rm SFR} (R_0,t)$ in these two models are compared in the upper panels in Figure \ref{fig:model-properties}. The main differences of these models are summarized in Table~\ref{table:models} (the ``quiescent model'' will be defined in \S\ref{subsec:cmf}).
With the stellar mass evolution in hand, we then derive the radial size growth of the Milky Way using the empirical relation from \citet{van13}. By studying the evolution of galaxies at a fixed comoving number density at different redshifts, \citet{van13} found that the effective radius $R_\star$ of Milky Way-like galaxies grow with the total stellar mass according to the relation $R_\star \propto M_\star^{0.27}$.
Finally, to fully specify the star formation at different radii, we also require the star formation scale length, $R_{\rm SFR}$, and its evolution. Unfortunately, determining $R_{\rm SFR}$ for the Milky Way is observationally challenging. Therefore, we resort to $R_{\rm SFR}$ from extragalactic studies where the external vantage point provides an easier measurement of scale lengths. NGC 6946 has long been thought to be a Milky Way counterpart \citep[e.g.,][]{ken12}. We find the SFR and the (atomic and molecular) gas mass of NGC 6946 from \citet{sch11} can be fitted with an exponential model. We find scale lengths $R_{\rm SFR}(z=0) = 2.6 \, {\rm kpc}$ and $R_{\rm gas}(z=0) = 4.2 \, {\rm kpc}$, which we adopt in our model of the Milky Way. To compute the evolution $R_{\rm SFR}(z)$ and $R_{\rm gas}(z)$ through cosmic time, we assume all scale lengths trace the stellar effective radius. We find that this adopted $R_{\rm SFR}(z)$ leads to a stellar disk scale length of $R_\star (z=0) = 2.2 \, {\rm kpc}$ and $\Sigma_\star (R_0,z=0) = 38 \, M_\odot {\rm pc}^{-2}$. These values agree with existing observations \citep{fly06, bov13, zha13}. Furthermore, the model implies $R_{\rm gas}(z=0) \simeq 2 R_\star(z=0)$, agreeing with \citet{bov13}.
\begin{table}
\begin{center}
\caption{Summary of the three model variants in this study.\label{table:models}}
\begin{tabular}{llll}
\tableline \tableline
\\[-0.2cm]
Property & Optimistic & Fiducial & Quiescent \\[0.1cm]
\tableline
\\[-0.2cm]
CMF cutoff & $\sim 10^7 \, M_\odot$ & $\sim 10^6 \, M_\odot$ & $10^5 \, M_\odot$ \\[0.1cm]
Global SFR & Peaks in the past & More flat & More flat \\[0.1cm]
$\Sigma_{\rm SFR} (R_0,t)$ & Too high in the past & Agrees with obs. & Agrees with obs. \\[0.1cm]
Integrated SFR & The same & The same & The same \\[0.1cm]
\tableline
\end{tabular}
\end{center}
\end{table}
\begin{figure*}
\center
\includegraphics[width=\textwidth,natwidth=1400,natheight=1000]{Fig2.pdf}
\caption{{\em Bottom right panel}: Evolution of CMF high mass cutoff. The CMF evolves according to \citet{esc08}. The CMF cutoff is the main property that defines the quiescent, fiducial and optimistic models that we will discuss throughout this study. For example, the optimistic CMF allows the formation of larger clusters ($M_{\rm cluster} \sim 10^7 \, M_\odot$). We adopt an upper limit of $M_{\rm cluster}^{\rm max} = 10^7 \, M_\odot$, above which clusters are not expected to be homogeneous. {\em Bottom left panel}: Stellar and gas mass evolutions. The gas mass at $z=0$ is calculated from $\Sigma_{\rm gas} (R_0,z=0) = 13 \, M_\odot {\rm pc}^{-2}$. The gas mass evolution is calculated from the global SFH, following a Kennicutt-Schmidt law with $\alpha_{\rm KS} = 1.5$. {\em Top left panel}: Global SFH models in this study, assuming $M_{\rm halo}=10^{12} \, M_\odot$ adjusted to produce $M_\star (z=0) = 4.5 \times 10^{10} \, M_\odot$. The two SFHs have the same integrated SFR. The SFHs mainly come into play in determining the gas mass evolution and subsequently the CMF cutoff evolution. Since the quiescent CMF cutoff is constant through cosmic time without evolution, employing the optimistic SFH or fiducial SFH for the quiescent model does not change its results as they have the same integrated SFR. We choose to follow the fiducial SFH for the quiescent model as it fits the $\Sigma_{\rm SFR} (R_0,t)$ better. {\em Top right panel}: $\Sigma_{\rm SFR} (R_0,t)$ calculated from the global SFHs.}
\label{fig:model-properties}
\end{figure*}
\subsection{Gas mass distribution \& evolution}
\label{subsec:gas-mass}
The mass of gas in the disk comes into play in two aspects of the model, namely the radial migration prescription and the CMF evolution. We assume $\Sigma_{\rm gas} (R_0,z=0) = 13 \, M_\odot {\rm pc}^{-2}$ \citep{fly06}, which, when combined with $R_{\rm gas} (z=0) = 4.2 \, {\rm kpc}$, yields a total gas mass of $M_{\rm gas} (z=0) = 9.7 \times 10^9 \, M_\odot$. We the estimate the redshift evolution of the gas mass $M_{\rm gas} (z)$ by inverting the Kennicutt-Schmidt relation with $\alpha_{\rm KS} = 1.5$ and the SFR evolution described in the previous section. The distribution of gas is fully specified by $M_{\rm gas} (z)$ and $R_{\rm gas}(z)$. The total stellar mass and the total gas mass evolution are shown in the bottom left panel in Figure \ref{fig:model-properties}. For this work we do not need to specify the disk scale height because all quantities of interest are related to surface mass densities.
\begin{figure}
\includegraphics[width=0.45\textwidth]{Fig3.pdf}
\caption{Probability of position of a star after evolving over 13 Gyr, assuming $f_{\rm in-situ} = 50\%$. The solid lines show the final positions, whereas the dashed lines show the corresponding initial positions.}
\label{fig:radial-migration}
\end{figure}
\subsection{Radial migration}
\label{subsec:migration}
Radial migration describes the phenomenon of stars in the disk moving, either inward or outward, in radius from their birth radius. Studies of processes giving rise to radial migration have a long history. In the past decade, radial migration has gained increasing attention as playing a key role in driving the chemodynamical evolution of the Milky Way \citep[e.g.,][]{sel02, hay08, sch09, min10, bla10b, dim13}. Due to its role in changing stellar orbiting radii, radial migration provides tentative explanations to some observational puzzles. For example, the upturn in the stellar population age at the outer part of some galaxies \citep[e.g.,][]{bak08, zhe15}, the wide range of stellar metallicity in the Solar neighborhood \citep[e.g.,][]{hay08,sch09}; and perhaps even the formation of the thick disk \citep[e.g.,][]{loe11} can be explained by appealing to the process of radial migration.
An important physical process giving rise to radial migration is known as ``churning'' \citep{sel02}. In the process of churning, stars that co-rotate with transient non-axisymmetric features can increase their angular momentum while maintaining the ellipticity of the orbit, effectively bumping stars from an orbiting radius to the other. \citet{sch09} proposed a simple analytic formula for churning that we will adopt in this study. In this prescription, the probability of moving from the $i$-th to the $j$-th annulus, $P_{ij}$, where $j = i \pm 1$, is given by
\noindent
\begin{equation}
P_{ij} = k_{\rm ch} \frac{M_j}{M_{\rm max}},
\end{equation}
\noindent
where $M_j$ denotes the total (stellar + gas) mass of the $j$-th annulus and $k_{\rm ch}$ is a free parameter governing the strength of the churning.
In the present work we discretize the model galaxy into annuli with width of $0.2 \, {\rm kpc}$ and apply the churning exchange every 0.5 Gyr. We define in-situ fraction, $f_{\rm in-situ}$, as the fraction of stars that were born in-situ in a Solar annulus with $\Delta R_{\rm survey} = \pm 1 \, {\rm kpc}$. Clearly, $f_{\rm in-situ}$ depends on the choice of $\Delta R_{\rm survey}$. We choose $\Delta R_{\rm survey} = \pm 1 \, {\rm kpc}$ to calculate the in-situ fraction, instead of our fiducial value $\pm 3 \, {\rm kpc}$ in the model for ease of comparing to hydrodynamics simulations \citep[e.g.,][]{ros08}. We note that the free parameter $k_{\rm ch}$ maps directly into the variable $f_{\rm in-situ}$, and we choose to express the effect of radial migration in terms of the latter value. We consider a range of $k_{\rm ch}$ corresponding to $f_{\rm in-situ} = 15\%- 100\%$ and we choose $f_{\rm in-situ} = 50 \%$ to be the fiducial value, as suggested by simulations \citep[e.g.,][]{ros08, hal15}. To illustrate the radial migration prescription adopted in this study, solid lines in Figure \ref{fig:radial-migration} show the PDF of the final position of a star after 13 Gyr of evolution starting from various initial positions.
In addition to churning, scattering, e.g., from interactions with molecular clouds, can also diffuse stars from their birth radii. This scattering is known as ``blurring'' \citep{sel02}. For simplicity, we do not include blurring in the model. However, we note for our purposes only the fraction $f_{\rm in-situ}$ is important; the details of migration, either through churning or blurring are largely irrelevant in this study.
\subsection{Cluster mass function evolution}
\label{subsec:cmf}
We have discussed in \S\ref{sec:overview} that the number of stars sampled {\it per cluster} is governed primarily by the sampling rate and the in-situ fraction. However, knowing the detections per cluster is insufficient. To determine the number of detectable groups, we also need to understand the relative number of massive clusters compared to their smaller counterparts. Therefore, the CMF is another key factor \citep[see also][]{bla10a}. In this study, we assume a CMF that is characterized by a power law slope $\alpha$, high mass cutoff $M_{\rm cluster}^{\rm max}$ and low mass cutoff $M_{\rm cluster}^{\rm min}$, where
\begin{equation}
\frac{{\rm d} N}{{\rm d} M} \propto M^{-\alpha}.
\end{equation}
\noindent
Note that cluster masses refer to zero age masses; clusters will lose at least a factor of two mass after a Hubble time due to stellar evolution effects and the evaporation of stars.
\citet{lad03} analyzed young embedded clusters within $2.5 \, {\rm kpc}$ from the Sun and found a CMF slope $\alpha \approx -2.0$. We take this as the fiducial value in the model. The fact that $\alpha \approx -2$ is important in chemical tagging. In this case, the total mass in a survey sample coming from clusters within a mass bin $\delta M$, can be calculated to be
\begin{equation}
M \, {\rm d}N/{\rm d}M \, \delta M = M^2 \, {\rm d}N/{\rm d}M \, \delta \log M \propto \delta \log M.
\end{equation}
\noindent
Quantitatively, this means that the chance of sampling a star from the logarithmic bin $[10 \, M_\odot,100 \, M_\odot]$ is the same as the probability of sampling from the logarithmic bin $[100 \, M_\odot, 1000 \, M_\odot]$, and so forth. Since we adopt a maximum cluster mass $M_{\rm cluster}^{\rm max} = 10^5 - 10^7 \, M_\odot$ in this model, we have $4-6$ orders of dynamical range in the cluster mass. This large range of cluster mass implies that clusters with $[10 \, M_\odot,100 \, M_\odot]$ contribute only $\sim 10\% - 25\%$ of the total stellar mass. \citet{lad03} determined that the CMF low mass cutoff occurs around $M_{\rm cluster} = 50 \, M_\odot$, which we will adopt as the fiducial value.
Although not shown in this paper, we find that changing the low mass cutoff to $10 \, M_\odot$ or $100 \, M_\odot$ has a negligible effect on the results. First, as we have discussed, the small clusters only contribute $\sim 10\%-25\%$ of the population. Furthermore, changing the low mass cutoff will alter the number of small clusters and hence the background in each cell, however since the signal is concentrated in $\sim 0.1\%-1\%$ of the chemical cells, as we will show in \S\ref{subsec:local-results}, only $< 1\%$ of this background change is affecting the signal.
The high mass cutoff $M_{\rm cluster}^{\rm max}$ has a dramatic effect on the results because massive clusters dominate the signal, as shown in later sections. We therefore consider several different scenarios for the high mass cutoff and its evolution with redshift (see the lower right panel of Figure \ref{fig:model-properties}). The largest open clusters observed in the Milky Way appear to be Westerlund 1 \citep[e.g.,][]{bra08}, Berkeley 39 \citep[e.g.,][]{bra12} and Arches \citep[e.g.,][]{esp09}, with a mass few times of $10^4 \, M_\odot$. Noting the fact that the cluster could have gone through a period of rapid mass loss in its formation phase \citep[e.g.,][]{lad03}, we adopt $M_{\rm cluster}^{\rm max} \simeq 10^5 \, M_\odot$ at $z=0$ as the nominal mass cutoff at $z=0$ in the Milky Way disk.
A number of arguments suggest that the CMF high mass cutoff could have been higher in the past. For instance, the existence of massive globular clusters with surviving mass of $10^{4.5} - 10^{6.5} \, M_\odot$ \citep[e.g.,][]{har94} suggests that early conditions in the Galaxy favored the formation of more massive clusters. Observations of high-redshift disk galaxies also suggests a high frequency, relative to $z=0$, of very massive gas clumps of $10^7-10^9 M_\odot$ \citep[e.g.,][]{gen06,for09,jon10,liv12}.
\citet{esc08} provided a simple model for the maximum cluster mass by studying gravitational instability in disks, similar to Toomre's classic analysis \citep{too64}. They calculate the maximum unstable mass to be $M_{\rm cluster}^{\rm max} = \Sigma_{\rm gas} (\lambda_{\rm rot}/2)^2$, where $\lambda_{\rm rot} = \pi^2 G \Sigma_{\rm gas}/\Omega^2$. From this formula, they further found that the maximum cluster mass can be determined by the gas fraction $\eta$ (i.e., gas mass to the total gravitational mass) and the total gas mass $M_{\rm gas}$ alone, where
\begin{equation}
M_{\rm cluster}^{\rm max} \propto M_{\rm gas} \eta^2.
\end{equation}
\noindent
The normalization of this formula depends on a variety of unknown parameters and so we choose instead to fix the normalization by hand at $z=0$. The dynamics of the Milky Way disk can be explained without appealing to dark matter, at least within the Solar radius. We therefore ignore the influences of dark matter when computing the upper mass cutoff, i.e., we define $\eta = M_{\rm gas}/(M_{\rm gas} + M_\star)$. The evolution of $M_{\rm gas}$ and $M_\star$ follow the discussion in \S\ref{subsec:sfh} and \ref{subsec:gas-mass}.
We consider three scenarios for the evolution of the upper mass cutoff, which we will denote as the quiescent, fiducial and optimistic models (see Figure \ref{fig:model-properties}). In the quiescent model, we consider the fiducial SFH and fix $M_{\rm cluster}^{\rm max} (z) = 10^5 \, M_\odot$ through cosmic time. In the fiducial and optimistic cases, we consider the SFHs labeled as fiducial and optimistic in Figure \ref{fig:model-properties} and allow $M_{\rm cluster}^{\rm max} (z)$ to evolve. We set $M_{\rm cluster}^{\rm max} (z=0) = 10^5 \, M_\odot$ for the fiducial case, and $M_{\rm cluster}^{\rm max} (z=0) = 3 \times 10^5 \, M_\odot$ for the optimistic case. We use the term ``optimistic'' because this model allows the formation of very massive clusters, which is favorable for chemical tagging. Finally, we impose a maximum upper limit of $10^7 \, M_\odot$. Clusters with mass larger than this cutoff are unlikely to be homogeneous \citep{bla10b} in their elemental abundances due to self-enrichment. The evolution of $M_{\rm cluster}^{\rm max} (z)$ in these three cases are plotted in the bottom right panel in Figure \ref{fig:model-properties}. The main differences of these three CMF models are summarized in Table \ref{table:models}. The range of CMFs we consider is similar to the range explored by \citet{bla10a}, although the authors do not consider a time-dependent CMF as we do here (for the optimistic and fiducial models).
\subsection{Chemical space}
\label{subsec:chem-model}
The last model ingredient is multi-dimensional space of elemental abundances, often referred to as the ``chemical space''. The chemical space is spanned by the elemental abundances [Fe/H], [$X_1$/Fe], $\ldots$, [$X_n$/Fe], where $X_1$ to $X_n$ are $n$ different elements measured. Since stars that were born together are expected to share the same abundances, they should reside at the same location in chemical space.
As we will show below, the number of chemical cells in chemical space $N_{\rm cells}$ is a key variable in chemical tagging. To understand its importance, let's consider the case where we have an infinite number of chemical cells, in other words we have infinite resolution in the chemical space. In this case, all clusters from various birth sites can be easily identified. However, as the number of cells decreases, the probability that two clusters occupy the same chemical cell increases. In this case, the smaller clusters (in terms of the number of stars sampled per cluster) become contaminants in the detection. They dilute the number of genuine members of the massive clusters.
$N_{\rm cells}$ depends on two ingredients: (a) The chemical space spanned by the sample. This volume is governed by Galactic chemical evolution and survey design, including the number of elements of each star the survey can extract. Note that the volume does not scale in a simple way with the number of elements measured because of the strong correlation between various subgroups of elements. (b) The abundance measurement uncertainty $\sigma_{[X/{\rm Fe}]}$, which sets the volume of each cell. Regarding (b), in this study, we assume that the width of chemical cell is $1.5 \, \sigma$, i.e., two different distinct groups in chemical space can be recovered if their separation is larger than $1.5 \, \sigma$, where $\sigma$ represents the uncertainties along the principal components/independent dimensions.\footnote{As these component vectors are comprised of various elements, the uncertainties along these directions require the full covariance matrix of $\sigma_{[X/{\rm Fe}]}$.} Note that, given a chemical space of $N_{\rm dim}$ (independent) dimensions, the volume of each cell is proportional to $\sigma^{N_{\rm dim}}$. As a consequence, the number of cells is extremely sensitive to the abundance measurement uncertainties. We therefore stress that not only are small uncertainties favorable, but also accurate measurement of the uncertainties and their covariances are equally important.
The chemical space spanned by the sample, in principal, can be modeled through chemodynamical simulations. However, we note that chemical evolution models are still rather uncertain for many elements and are often limited to a relatively small number of elements \citep[e.g.,][]{kob06, min13}. \citet{kob11} include more elements, but they do not include neutron capture elements. Therefore, we are not aware of an existing chemical evolution model that encompasses all $\sim 25$ elements measured by the GALAH and Gaia-ESO surveys. For these reasons, and for simplicity, we choose here to adopt empirical results in estimating the volume and defer a chemical modeling approach to future work.
We make use of the estimated chemical space volume of Milky Way disk stars from \citet{tin12a} \citep[also see][for a similar study on bulge stars]{and12}. Using principal components analysis, \citet{tin12a} searched for directions in the chemical space that are orthogonal to each other and contain most variances of the data. These principal components define a n-dimensional cube spanned by the data. By definition, the number of cells is the volume of the cube divided by the volume spanned by each cell. As for the latter, given the assumption that the width of chemical cell is $1.5 \, \sigma$, the volume of the chemical cell is $(1.5 \, \sigma)^{N_{\rm dim}}$. The volume of the n-dimensional cube can be estimated from the width of edges in each dimension, which can be calculated from the principal components axial ratios. Here we use the axial ratios of the principal components to estimate the volume that will be spanned by the GALAH data, as an example. The axial ratios of the first 6 dimensions are 1, 0.4, 0.25, 0.25, 0.1, 0.1. Apart from the obvious additional dimension from [Fe/H], \citet{tin12a} speculated that there should be another dimension associated with neutron capture elements. This last dimension was not available in the data analyzed by Ting et al. but will be probed by both GALAH and Gaia-ESO.
We can safely assume that the first principal component spans at least 1.5 dex as it is the diagonal direction of the 17 dimension in study. Let's further assume that [Fe/H] and both of the additional dimensions span $1 \, {\rm dex}$, and the uncertainties along the independent dimensions are $\sigma = 0.1\,$dex. A simple calculation using the axial ratios yields: $N_{\rm cells} = (1.5 \, {\rm dex})^6 \times (1 \cdot 0.4 \cdot 0.25 \cdot 0.25 \cdot 0.1 \cdot 0.1) \times (1 \, {\rm dex})^2/(1.5 \, \sigma)^8= 10^4$ for GALAH. The Gaia-ESO survey spans a comparable list of elements and should therefore contain a similar number of $N_{\rm cells}$. An APOGEE-like survey should have $2-3$ fewer independent dimensions than GALAH \citep{tin12a}. All other parameters being the same, APOGEE should have $N_{\rm cells} \sim 10^3$.
The above calculations are simple estimates for the number of chemical cells that could easily be off by an order of magnitude. Hence, in the analysis below we consider a wide range in this important parameter, ranging from $10^3-10^5$.
\begin{figure*}
\center
\includegraphics[width=\textwidth,natwidth=1400,natheight=1050]{Fig4.pdf}
\caption{Number of stars sampled per cluster as a function of cluster mass, assuming $\Delta R_{\rm survey} = \pm 3 \, {\rm kpc}$. The left panels assume $N_\star = 10^5$, whereas the right panels assume $N_\star = 10^6$. The top panels show the cases where there is no radial migration ($f_{\rm in-situ} = 100\%$), while the bottom panels illustrate the cases with radial migration and an in-situ fraction $f_{\rm in-situ} = 50\%$. The solid lines show the median and the shaded regions in color show the $1\sigma$ range of the results from simulations. In the limit of no radial migration, the number of stars sampled per cluster can be predicted analytically from equation (6). The predictions from the analytic formula are shown in dashed lines and gray shaded regions. The $1\sigma$ range from simulations follows very well the Poisson expectations. However, the analytic formula does not work in the case with radial migration because ex-situ clusters tend to have fewer stars sampled and bring down the number (see text and Figure \ref{fig:average-num-detail} for details).}
\label{fig:average-num}
\end{figure*}
\section{Results}
\label{sec:results}
With the model for the Milky Way disk stars now in hand, we turn to using that model to explore what ongoing and future massive spectroscopic surveys of stars may expect to reveal in the context of chemical tagging. In \S\ref{subsec:average-num} we investigate how many stars we expect to sample from the same cluster for different number of stars surveyed and both with and without the effect of radial migration. The main results are presented in \S\ref{subsec:local-results}, where we simulate the number of detectable groups in different scenarios. We study how observations of the distribution of stars in chemical space may encode information on the shape of the CMF. We also investigate whether each detectable group in chemical space is dominated by a single cluster or is comprised of a wide range of clusters.
\subsection{Number of stars sampled per cluster}
\label{subsec:average-num}
In this section we study the number of stars sampled per cluster for several idealized surveys. In particular, we are interested in how many stars will be sampled per cluster after the cluster is dispersed and mixed with the background sea of other clusters, and how the process of radial migration influences the sampling. Note that since we consider quantities as a function of cluster mass in this section, for a fixed $\Sigma_\star (R_0,z=0)$ the results will be independent of the CMF. However, the results do depend on $\Delta R_{\rm survey}$ and $f_{\rm in-situ}$ as these parameters change the sampling rate and the radial migration prescription. Here we assume $\Delta R_{\rm survey} = \pm 3 \, {\rm kpc}$ and $f_{\rm in-situ} = 50 \%$.
In Figure \ref{fig:average-num}, we plot the number of stars sampled per cluster as a function of cluster mass. The solid lines show the median of the results in each cluster mass bin and the shaded color regions show the $1\sigma$ range. In the top panels, we consider the case without radial migration, i.e., stars stay in the orbiting radii that they formed, while the bottom panels show the case with radial migration. The left and right panels show results for $N_\star = 10^5$ and $N_\star = 10^6$. A horizontal line at $N=10$ stars is meant to serve as a reference point.
While the results in Figure \ref{fig:average-num} clearly show that the {\it typical} sampling rate (within $\pm 1\sigma$ range) per cluster is quite low, except in the case of large $N_\star$ and high cluster mass, we emphasize that the {\it distribution} of the number of stars sampled per cluster has a long tail toward high values. We return to this point below.
\begin{figure}
\center
\includegraphics[width=0.45\textwidth]{Fig5.pdf}
\caption{Distribution of the number of stars sampled per cluster for $M_{\rm cluster} = (0.7 - 1.3) \times 10^6 \, M_\odot$. The top panel shows the result for $N_\star = 10^5$ and the bottom panel shows $N_\star = 10^6$. We assume $\Delta R_{\rm survey} = \pm 3 \, {\rm kpc}$ and $f_{\rm in-situ} = 50 \%$. We separate the cluster population into two - the in-situ and ex-situ populations. The ex-situ clusters have much smaller number of stars sampled per cluster compared to the in-situ population, indicating that ex-situ stars are mostly contaminants in chemical tagging. The red vertical line shows the 75 percentile of the combined results from in-situ and ex-situ clusters.}
\label{fig:average-num-detail}
\end{figure}
In the limit where there is no radial migration, the average number of stars (with $\langle M \rangle =1 \, M_\odot$) sampled per cluster can be analytically derived \citep[see also][]{des15}. The number of stars sampled per cluster is simply
\noindent
\begin{equation}
N_{\rm cluster} = M_{\rm cluster} \frac{N_\star}{M_{\rm annulus}}.
\end{equation}
\noindent
Recall that $M_{\rm annulus}$ is the total integrated SFR in the Solar annulus and $N_\star/M_{\rm annulus}$ is proportional to the sampling rate. This analytic model is shown in the top panels of Figure \ref{fig:average-num} and clearly predicts very well the results of the simulations. The grey shaded region demarks the $1\sigma$ from this analytic model.
\begin{figure*}
\center
\includegraphics[width=\textwidth,natwidth=1400,natheight=500]{Fig6.pdf}
\caption{Standardized number of stars in each cell compared to a Poisson distribution, where the mean of Poisson distribution is $N_{\rm mean} = N_\star/N_{\rm cells}$ and the standard deviation follows $\sigma = \sqrt{N_{\rm mean}}$. Cells in which the number of stars sampled exceeds $5\sigma$ are considered as detectable groups. The y-axis shows the probability of a detected group having a certain deviation from the Poisson distribution, quantified by the standardized number of stars. The integral under each curve is one. Unless stated of otherwise, we assume fiducial values for all the model parameters, as listed in Table~\ref{table:parameters}. Different CMFs show different degrees of deviation from Poisson statistics. The clumpiness of the chemical space may therefore be a useful tool to probe the underlying CMF. }
\label{fig:local-poisson}
\end{figure*}
Although illustrative, this analytic formula is unfortunately not applicable when radial migration is included. First, radial migration increases the number of stars that could end up in the Solar annulus, which has the effect of increasing the effective volume of the survey. We can define an effective radius of the observed annulus to be the mean distance,
\begin{equation}
R_{\rm effective} = \frac{1}{n} \sum_{i=1}^n |R_{i,{\rm birth}} - R_0|,
\end{equation}
\noindent
where we sum over all the stars in the Solar annulus at the present-day. This equation takes into account the fact that, with radial migration, the actual sampled volume is larger than the observed volume because $|R_{i,{\rm birth}} - R_0| \geq |\Delta R_{\rm survey}|$. The effective integrated SFR $M_{\rm annulus}'$ within this effective volume is strictly larger than the one without radial migration due to the migration of ex-situ population, and therefore the number of stars per cluster will generally be lower than in the case without radial migration.
Moreover, clusters that were born ex-situ are unlikely to have a significant number of stars migrated into the Solar annulus. As shown in Figure \ref{fig:radial-migration}, while stars born $5 \, {\rm kpc}$ from the Galactic center can move into the Solar annulus at $R_0 = 8 \, {\rm kpc}$, only a small fraction of this population is in the Solar annulus. Figure \ref{fig:radial-migration} suggests that most of the ex-situ stars, even from massive clusters, will tend to enter as ``contaminants'' in the sense that they will have only $\mathcal{O}(1)$ stars sampled per cluster. In addition, some stars that were born in-situ will migrate outside the Solar annulus, further diluting the number of members of in-situ clusters. All of these effects work in the same direction of reducing the number of stars per cluster compared to a model without radial migration.
In Figure \ref{fig:average-num-detail} we show the distribution of the number of stars sampled per cluster for two choices of $N_\star$. This figure shows the distribution for a vertical slice in Figure \ref{fig:average-num} at a cluster mass of $\sim10^6\,M_\odot$. By separating the in-situ and ex-situ populations, Figure \ref{fig:average-num-detail} shows that the ex-situ population has on average a much smaller number of stars sampled per cluster, in agreement with the arguments described above. Although not shown, we checked that the in-situ population is only marginally influenced by radial migration --- only a small fraction of in-situ stars leave the Solar annulus. The mild effect on in-situ clusters is likely due to the fact that we consider a fairly large Solar annulus width of $\Delta R_{\rm survey} = \pm 3 \, {\rm kpc}$. In the radial migration prescription in this study, a typical radial migration length is $\sim 2 \, {\rm kpc}$, which is smaller than $|\Delta R_{\rm survey}|$. Although the typical radial migration length is still largely unconstrained from observations, some studies have suggested that since $R_0$ is beyond the outer Limblad resonance of the Galactic bar \citep{deh00}, a typical radial migration length is $< 2 \, {\rm kpc}$ \citep{hal15}.
Another feature evident in Figure \ref{fig:average-num-detail} is the tail of clusters with a large number of stars sampled per cluster. This highlights that median statistics are not sufficient to capture the full variety of expected behavior. These rare clusters may end up being the most valuable from the standpoint of chemical tagging as they should stand out as strong concentrations of stars in chemical space. The following section explores this effect in detail.
\subsection{Finding and counting groups in chemical space}
\label{subsec:local-results}
Observational uncertainties on elemental abundances impose a finite resolution in chemical space that can have important consequences for chemical tagging \citep{bla10a}. In this section, we simulate observational results by studying detections on a chemical cell-by-cell basis. In the following, for each generated sample, we distribute sampled clusters uniformly (on average) into $N_{\rm cells}$ cells. We perform Monte Carlo simulations and take the mean from 100 realizations. By jack-knife estimation, we find that the uncertainties on the mean is $\ll 10\%$ for $N_{\star} = 10^5 - 10^6$.
We define several terms that will be important in this section. A cell that contains a high density of stars compared to the mean defines a ``group''. We distinguish between ``group'' and ``cluster'' because the former can be comprised of multiple clusters. The cluster with the most stars sampled in each cell is referred to as the dominant cluster. Stars from the dominant cluster define the ``local signal''. The rest of the stars in the cell are referred to as ``local noise''.
\subsubsection{Identifying groups in chemical space}
\label{subsec:search-poisson}
If we were to randomly distribute $N_\star$ stars into $N_{\rm cells}$ chemical cells, the number of stars per cell should follow a Poisson distribution with a mean $N_{\rm mean} = N_\star/N_{\rm cells}$ and a $1\sigma$ range of $\sqrt{N_{\rm mean}}$. Since stars are born in clusters, there will be clumping in chemical space that is larger than Poisson expectations. The degree of clumpiness depends on several factors, chief among them is the form of the CMF \citep{bla10a}.
Operationally we define a cell as containing a ``detected'' group of stars if that cell deviates from Poisson expectations by at least $5\sigma$ and the total number of stars in that cell $>1$. Figure \ref{fig:local-poisson} shows the deviations from Poisson statistics for different CMFs and numbers of stars in the survey. In the right panel, we assume $N_\star = 10^6$. In this case, both the fiducial and optimistic CMFs show substantial numbers of cells exceeding $5\sigma$ from the average. By contrast, when $N_\star = 10^5$ (left panel), only the optimistic CMF shows substantial deviation from Poisson expectations.
Figure \ref{fig:local-poisson} demonstrates that the deviation from Poisson is minimal for a quiescent CMF. This lack of deviation is not unexpected because clusters with $M_{\rm cluster} < 10^5 \, M_\odot$ have $\mathcal{O}(1)$ stars detected per cluster even for $N_\star = 10^6$ (see Figure \ref{fig:average-num}). Hence, randomly distributing clusters in $N_{\rm cells}$ cells for a quiescent CMF is close to randomly distributing $N_\star$ in $N_{\rm cells}$ cells.
Figure~\ref{fig:local-poisson} also shows that the distribution of deviations can be a sensitive probe of the CMF. CMFs with a higher mass cutoff produce more clumpiness in chemical space. Although not shown, a flatter CMF also entails a larger number of massive clusters and hence a clumpier chemical space, echoing the results of \citet{bla10a,bla14}. The effect of the CMF on the distribution of deviations could potentially be exploited to reconstruct the CMF (and the physical processes that the CMF depends on, such as the SFH) from observational samples. This will be the subject of future work.
\subsubsection{What are groups in chemical space comprised of?}
\label{subsec:not-individual}
In this section we investigate the properties of the``detected'' groups in chemical space (consisting of $>5 \sigma$ fluctuations). Figure \ref{fig:local-poisson-2} shows the distribution of the local ``S/N'' for those cells exceeding $5\sigma$ from Poisson statistics. Recall that the local S/N is defined as the ratio of stars coming from the most massive cluster in the cell to the remaining stars in that cell. A cell dominated by a single massive cluster will have high local S/N. In the left panel, we assume $N_{\rm cells} = 10^4$ and consider three different CMFs. Clearly most of the detected groups have local S/N $< 1$, especially for the quiescent and fiducial CMFs.
This result is not surprising in light of the mean number of stars per cell ($100$ for $N_\star=10^6$ and $N_{\rm cells} = 10^4$). In this regime, in order for the S/N to be $\gg1$, we would require that a single dominant cluster contribute $\gg100$ stars in a particular cell. However, as shown in Figure \ref{fig:average-num}, the average number of stars sampled per cluster for the most massive clusters is $\sim100$ for $N_\star=10^6$. The relatively low sampling rate, combined with the high average number of stars per cell, essentially guarantees that the local S/N will never be much larger than one. As we discuss in \S\ref{sec:key-params}, the prospects for finding higher local S/N cells can be improved by searching in regions of chemical space in which the mean number of stars per cell is low.
The result in the left panel of \ref{fig:local-poisson-2} is fairly insensitive to $N_\star$. Increasing $N_\star$ increases both the number of stars sampled per cluster and the ``background'' comprised of stars from small clusters and hence the local S/N is left largely unchanged. In fact, the local S/N slightly decreases as we increase $N_\star$. This is not unexpected. As $N_\star$ decreases, it becomes more difficult to exceed the Poisson threshold. Therefore for smaller $N_\star$, the clumping of detected groups are mostly comprised of more massive clusters (e.g., $\sim 10^7 \, \rm M_\odot$), which implies a better local S/N. By contrast, for a larger $N_\star$, the clumping could either be due to a massive cluster or a few moderately massive clusters (e.g., $\sim 10^4 - 10^6 M_\odot$). While the S/N is somewhat negatively impacted by increasing $N_\star$, the total number of detected groups greatly increases with increasing $N_\star$, as shown in \S\ref{subsec:parameter-effect}.
The right panel of Figure \ref{fig:local-poisson-2} shows the median local S/N as a function of the number of chemical cells. Increasing $N_{\rm cells}$ results in a dramatic (almost linear) improvement in the local S/N. An increase in $N_{\rm cells}$ results in a decrease in the local background while keeping the signal unchanged. This panel also shows the effect of changing the definition of a ``detected'' group from $2\sigma$ to $10\sigma$. Increasing the threshold has a modest effect on the local S/N but of course has a dramatic effect on the total number of resulting detected clusters. Although not shown, we have explored the effect of varying the slope of the CMF from $\alpha=-2.0$ to $-1.5$. This has only a modest effect on the trends shown in Figure \ref{fig:local-poisson-2}.
Note that the ($5\sigma$) deviation with respect to Poisson statistics is measurable in reality as it only requires the expected average number of stars in each cell. On the other hand, the local S/N is not measurable.\footnote{For readers who want to understand the number of groups that consist mainly a dominant cluster (e.g., having local S/N $\geq 1$), we urge readers to explore the interactive online applet (see Appendix~\ref{sec:interactive} for details). In the applet, we allow users to impose a local S/N criteria.} In this paper we only define ``detected groups'' according to a measurable parameter, and we emphasize again that we use the term ``group'' rather than ``cluster'' when describing clumps in chemical space because of the effect discussed in this section. The ambiguity that can arise, even when a cell deviates by more than $5\sigma$ argues strongly that interpretation of the data from ongoing and upcoming surveys will require models such as the one presented in this work.
\begin{figure*}
\center
\includegraphics[width=\textwidth,natwidth=1400,natheight=500]{Fig7.pdf}
\caption{{\em Left panel}: Local S/N ratio in chemical cells with $5\sigma$ more stars than the average. The number of stars sampled from the dominant cluster is considered signal in each cell, whereas the rest are considered noise. The y-axis shows the probability of a detected group having a certain local S/N. The integral under each curve is one. We assume $N_{\rm cells} = 10^4$. In this case, most detectable groups have local S/N $< 1$, showing that at least half of the stars in the detectable groups are not from dominant clusters. The difference between $N_\star = 10^5$ and $10^6$ is small, illustrating that sampling more stars increases the number of stars per cell, but it does not change the S/N. {\em Right panel:} Median of local S/N for different $N_{\rm cells}$. We assume a fiducial CMF in this panel. Unlike $N_\star$, increasing $N_{\rm cells}$ boosts the local S/N, and hence increases the chance of recovering individual clusters through chemical tagging.}
\label{fig:local-poisson-2}
\end{figure*}
\subsubsection{Number of detectable groups as a function of model parameters}
\label{subsec:parameter-effect}
In this section we present the total number of detected groups in chemical space as a function of a variety of model parameters, including the in-situ fraction, $f_{\rm in-situ}$, CMF slope, $\alpha$, survey width $\Delta R_{\rm survey}$ number of chemical cells $N_{\rm cells}$, and number of stars in the survey, $N_\star$. We vary one of these model parameters at a time while adopting the fiducial values for the other model parameters (see Table~\ref{table:parameters}); modifying more than one parameters at once is allowed in the online applet. The results are presented in Figures \ref{fig:local-results-2} and \ref{fig:local-results-3}.
\noindent
\paragraph{Number of chemical cells}
As the number of chemical cells increases, more moderately massive (e.g., $\sim 10^4 - 10^6 M_\odot$) clusters start to occupy different cells instead of sharing the same cell. The total number of detectable groups thus increases, approximately linearly for the fiducial and optimistic CMFs. However, the gain is more drastic for CMFs with a smaller high mass cutoff. This trend is due to the fact that, given the same $N_\star$, moderately massive clusters are more abundant for CMFs with a smaller high mass cutoff. These clusters might not be detected with a smaller $N_{\rm cells}$. Including more cells benefits these moderate clusters the most.
Since both the number of detectable groups and the local S/N (see \S\ref{subsec:not-individual}) are sensitive to $N_{\rm cells}$, it is clear that $N_{\rm cells}$ is one of the most important parameters in the context of chemical tagging. Recall that the number of cells scales as $\sigma^{-N_{\rm dim}}$, where $N_{\rm dim} \sim 8$ is the number of independent dimensions in the chemical space we can expect for upcoming optical surveys (GALAH and Gaia-ESO). Therefore, if we improve the abundance measurement uncertainties by a factor two, the number of chemical cells is improved by a factor of $2^8 \sim 250$. On the other hand, this also means that the number of chemical cells decreases by a factor $\sim2$ for every $10\%$ increase in the measurement uncertainties. Substantial effort should therefore go into decreasing (and characterizing!) the uncertainties in abundance measurements in upcoming spectroscopic surveys.
\begin{figure*}
\center
\includegraphics[width=\textwidth,natwidth=1400,natheight=1000]{Fig8.pdf}
\caption{Total number of cells that exceed $5\sigma$ from Poisson statistics as a function of a variety of model parameters. We vary each of these model parameters while fixing the rest to the fiducial values as listed in Table~\ref{table:parameters}. The three different solid lines show results from three CMF evolutions as illustrated in Figure \ref{fig:model-properties}. The dashed lines show linear relations for reference. The solid symbols show the results assuming fiducial values for all model parameters. See text for discussion.}
\label{fig:local-results-2}
\end{figure*}
\noindent
\paragraph{Survey width}
As $\Delta R_{\rm survey}$ increases the number of detectable groups decreases. To understand this trend, it suffices to note that as we increase $\Delta R_{\rm survey}$ there are more stars in the annulus. As a result, the chance that we sample from the same cluster decreases (i.e., the sampling rate decreases). Since each cluster is sampled with fewer stars, the chance to observe signal spikes in chemical space also decreases. Therefore, the total number of detectable groups decreases as the survey width widens. In fact, since the volume of the Solar annulus is proportional to $\Delta R_{\rm survey}$, the number of stars in the annulus is also roughly proportional to $\Delta R_{\rm survey}$. Therefore, the sampling rate is, to first order, inversely proportional to $\Delta R_{\rm survey}$.
Interestingly, the survey width has less effect on CMFs with a larger higher mass cutoff. This trend is due to the fact that as we increase the survey width, we also increase the number of clusters, roughly in proportion to $\Delta R_{\rm survey}$. The most massive clusters are the least susceptible to change in sampling rate because a large number of stars from such clusters are already sampled in the fiducial case. For CMFs with a larger high mass cutoff, the decrease in sampling rate caused by an increase in $\Delta R_{\rm survey}$ is partly compensated by the increase in the number of massive clusters, resulting in a weak dependence of the number of detected groups on $\Delta R_{\rm survey}$.
\noindent
\paragraph{In-situ fraction}
As the in-situ fraction decreases, the number of cells exceeding $5\sigma$ decreases because there are more contaminants from ex-situ clusters (see Figure \ref{fig:average-num-detail}). However, the effect of in-situ fraction is rather marginal for CMFs with a larger high mass cutoff. This effect is best understood from Figure \ref{fig:local-poisson}. Most of the detectable groups for a quiescent CMF or a fiducial CMF are at the edge of the detection level of $5 \sigma$. Hence adding in additional background noise in the form of ex-situ stars can have a much larger effect for model with a quiescent CMF compared to an optimistic CMF, in which many of the cells far exceed the $5\sigma$ detection threshold.
\noindent
\paragraph{CMF slope}
As we vary the CMF slope, we are essentially redistributing mass between smaller clusters and massive clusters. This has two effects that act in tandem: a shallower CMF results in more massive clusters, which will have more stars sampled per cluster. In addition, a shallower CMF results in fewer low mass clusters that contribute primarily to the ``noise'' in a cell. The chemical space becomes much clumpier as $\alpha$ increases
\citep[also see][]{bla10a}, and as a result there are many more detected groups.
\noindent
\paragraph{Number of stars in the survey}
Since the number of stars sampled for massive clusters is roughly proportional to $N_\star$ while the Poisson threshold only grows as $\sqrt{N_{\rm mean}} \propto \sqrt{N_\star}$, increasing $N_\star$ improves the number of detectable groups, as shown in Figure \ref{fig:local-results-3}. In the left panel, the gain is approximately linear in $N_\star$ for the optimistic and fiducial CMFs. The right panel shows the gain in the number of detected groups as a function of $N_\star$ and $N_{\rm cells}$. The stochasticity at $N_\star \sim 10^4$ is likely due to the uncertainties in our Monte Carlo procedures.
\subsubsection{Selecting subpopulations}
\label{subsec:subpops}
As we argued in \S\ref{sec:overview}, the sampling rate, which is proportional to the number of stars in the survey divided by the number of stars in the survey volume, is a key parameter determining the number of stars sampled per cluster. In the limit where the sampling rate is 100\%, the main limiting factor for chemical tagging is the resolution in chemical space. One way to increase the sampling rate is to increase $N_\star$; this was discussed in the previous section. A second way is to decrease the number of stars in the survey volume. The latter will be effective only if one is able to identify a subpopulation of stars that corresponds to a subpopulation of clusters. For example, selecting on stellar age satisfies this criterion, while selecting a random subsample does not.
Figure \ref{fig:local-results-4} considers the case where only stars above certain stellar ages are targeted in a survey. Since the number of older stars is smaller, there are not as many survey candidates compared to the case where we sample all disk stars uniformly. As a consequence, given the same $N_\star$, the chance that we sample from the same cluster improves. In addition to improving the total number of detectable groups, as we consider a more selective stellar subpopulation the number of clusters is reduced. The dominant cluster therefore contributes a greater fraction of the total stars in each detectable group because there are not as many clusters sharing the same cell. As shown in the right panel of Figure \ref{fig:local-results-4}, if the survey sample is collected randomly from all populations (the red solid line), most of the detectable groups have a local S/N of $0.3$. This local S/N value implies that only $0.3/(0.3+1) \simeq 25\%$ of the members of detectable groups are from the dominant cluster. However, if we only target old stars with stellar age $> 12 \, {\rm Gyr}$, the local S/N is $\sim 2$, indicating that $2/(2+1) \simeq 70\%$ members of each of the detectable groups are from the dominant cluster.
As a caveat, we caution that the interpretation of Figure \ref{fig:local-results-4} is complicated by the fact that the selection of older clusters also preferentially selects a population of stars forming from a CMF with a higher mass cutoff (at least for the fiducial model used in the figure). So not only is the sampling rate increasing but so also is the characteristic cluster mass. Future work is required to disentangle these effects.
\begin{figure*}
\center
\includegraphics[width=\textwidth,natwidth=1400,natheight=500]{Fig9.pdf}
\caption{Total number of cells exceeding $5\sigma$ from Poisson statistics as a function of the number of stars in the survey. We assume a survey width of $\Delta R_{\rm survey} = \pm 3 \, {\rm kpc}$ and $f_{\rm in-situ} = 50\%$. The red solid lines in both panels represent the reference results assuming a fiducial CMF and $N_{\rm cells} = 10^4$. The dashed lines show linear relations for reference. Different solid lines in the left panel show the results assuming different CMFs, whereas the right panel shows the results for different $N_{\rm cells}$. See text for discussion.}
\label{fig:local-results-3}
\end{figure*}
\section{Discussion}
\label{sec:discussion}
\subsection{Summary of the key parameters affecting chemical tagging}
\label{sec:key-params}
The key parameters governing both the ability to detect groups in chemical space and the ``purity'' of those recovered groups (i.e., the local S/N) are the number of stars in the survey, $N_\star$, the number of chemical cells, $N_{\rm cells}$, the CMF, and the sampling rate. Table \ref{table:effects} presents a summary of the key variables and their effect on various quantities of interest.
Several of these parameters are either outside of the control of the observer, including the form and evolution of the CMF, or are trivially in control of the observer, such as $N_\star$. Others require further consideration. For example, the number of chemical cells depends on both the volume of chemical space and the size of each cell. The former depends on chemical evolution of the stellar population(s) under consideration, and can be influenced by the survey strategy. The latter is proportional to $\sigma^{-N_{\rm dim}}$ where $\sigma$ is the observational uncertainty on abundance measurements and $N_{\rm dim}$ is the number of effective dimensions in the chemical volume.
Perhaps the most conceptually complex parameter is the sampling rate. For a fixed $N_\star$ the sampling rate is inversely proportional to the total number of stars available within the survey design. The phrase ``survey design'' was chosen to highlight not only the survey volume but also the subpopulation under consideration. Moreover, with regards to the survey volume, this must be considered in an orbit-averaged sense. For example, a survey targeting stars within 1 kpc of the Sun has a survey volume in this definition that encompasses the entire annulus of the Galactic disk with a width of $\pm1$ kpc. Likewise, a pencil beam survey of bulge stars has a survey volume of the entire bulge. As we showed in \S\ref{subsec:subpops}, selecting subpopulations of stars can be very effective provided that the selection picks out a subset of clusters. Selecting on stellar age can achieve this, and so will effectively boost the average number of stars sampled per cluster. On top of that, selecting subsample reduces the number of clusters in each cell, and thus improves the local S/N in each detectable group. In contrast, a random subsample of stars will simply result in a smaller number of stars per cluster.
These parameters affect different aspects of chemical tagging. As shown in Table~\ref{table:effects}, increasing the number of stars or reducing the survey volume increases the number of detected groups and improves the reconstruction of the CMF because it increases the sampling rate, but it has little effect on the local S/N ratio. Even though the sampling rate increases in these cases, both the local signal and noise increase in similar proportions. In contrast, decreasing $\sigma_{[X/{\rm Fe}]}$ and/or selecting subpopulation reduces the average number of stars per cell, while maintaining the same signal. Therefore the local S/N improves as well.
In this work we focused on idealized surveys of stars in the Milky Way disk. In such situations the ratio of the number of stars in the annulus, $N_{\rm annulus}$ to $N_{\rm cells}$ is $\gg1$. However, there are regimes in which this ratio can be closer to or less than unity. \citet{bla10a} considered the regime of metal poor stars in dwarf galaxies. Such subpopulations could easily have a total number less than $N_{\rm cells}$. In this case the mean number of stars per cell will be $\ll1$ and so significant overdensities in chemical space will much more likely reflect a single cluster, rather than a superposition of multiple clusters \citep[see example in][]{kar12}. As argued by \citet{bla10a}, in this regime one can in principle find clusters in chemical space with a relatively modest number of stars surveyed, provided that the CMF is not too steep. Similarly, for a survey targeting disk stars, one might imagine the first chemical-tagging detections to come from the less populated regime in chemical space with a smaller contaminated background $N_{\rm mean}$ (i.e., outliers), as discussed in \citet{bla15}.
\begin{figure*}
\center
\includegraphics[width=\textwidth,natwidth=1400,natheight=500]{Fig10.pdf}
\caption{{\em Left panel:} Total number of cells exceeding $5\sigma$ from Poisson statistics as a function of the number of stars in the survey. We assume a fiducial CMF, with $N_{\rm cells} = 10^4$, $\Delta R_{\rm survey} = \pm 3 \, {\rm kpc}$ and $f_{\rm in-situ} = 50 \%$. Different lines in this panel show the results assuming a variety of subpopulation selections. The subpopulations are selected through the stellar age criteria of $> 0 \, {\rm Gyr}$ (the lowest line), $> 3 \, {\rm Gyr}$, $> 6 \, {\rm Gyr}$, $> 9 \, {\rm Gyr}$ and $> 12 \, {\rm Gyr}$ (the highest line), respectively. The corresponding sampling rates, $f_{\rm sub}$, for $N_\star = 10^6$ are stated in each line. {\em Right panel:} Local S/N in each of the detected cells for different subpopulations, assuming $N_\star = 10^6$ and a fiducial CMF. The number of stars sampled from the dominant cluster is considered signal in each cell, whereas the rest are considered noise. See text for discussion.}
\label{fig:local-results-4}
\end{figure*}
\begin{table}
\begin{center}
\caption{The effects of various survey strategies on chemical tagging detections.\label{table:effects}}
\begin{tabular}{lccc}
\tableline \tableline
\\[-0.2cm]
& Improve the & Improve chance & Improve \\
& number of & of recovering & reconstruction \\
& detectable groups & single cluster & of CMF \\[0.1cm]
\tableline
\\[-0.2cm]
Increase $N_\star$ & \checkmark & & \checkmark \\[0.1cm]
Decrease $\sigma_{[X/{\rm Fe}]}$ & \checkmark & \checkmark & \checkmark \\[0.1cm]
Reduce $\Delta R_{\rm survey}$ & \checkmark & & \checkmark \\[0.1cm]
Subpopulations & \checkmark & \checkmark & \checkmark \\[0.1cm]
\tableline
\end{tabular}
\end{center}
\end{table}
\subsection{Strategies for optimizing the potential for chemical tagging}
\label{sec:design-survey}
The influence of key parameters on various observables allows us to consider ways in which one could optimize a spectroscopic survey of stars for the purposes of chemical tagging.
A survey that could reach $N_\star \sim 10^6$ and $N_{\rm cells} \gtrsim 4 \times 10^4$ could potentially achieve three major goals: (a) producing a sizable number ($\sim 10^3$) of detectable groups; (b) the detected groups would consist primarily of a single dominant cluster; and (c) reconstructing the CMF for $M_{\rm cluster}^{\rm max} \simeq 10^5 \, M_\odot$. These goals could be realized if the CMF is somewhere in the range between our ``fiducial'' and ``optimistic'' scenarios.
The GALAH survey \citep{des15} aims to observe $N_\star=10^6$; a key question will be whether or not the number of chemical cells is closer to $10^4$ or $10^5$ (see Section~\ref{sec:key-params} for the key dependencies).
Even if not all three goals are realized in the context of a massive spectroscopic survey, one could imagine a tiered approach. A survey of $10^6$ could be used to identify overdensities in the chemical space. One could then follow up those overdensities with higher quality spectroscopy to obtain more precise abundance constraints, or one could appeal to differential techniques to increase the relative abundance precision. One could also use other information to separate multiple clusters within a single cell, e.g., kinematics or color-magnitude diagrams.
Given that both $N_\star$ and $N_{\rm cells}$ affect the number of detected groups in chemical space in similar ways, is there an advantage to spending more time collecting greater numbers of stars, or more time obtaining higher quality spectra could lead to smaller $\sigma_{[X/{\rm Fe}]}$, more elements, and hence larger $N_{\rm cells}$? In the simplest scenario (assuming for example that one has not already exhausted the input catalog at a particular apparent magnitude), $N_\star$ is roughly proportional to the integration time. On the other hand, since $N_{\rm cells} \propto \sigma^{-N_{\rm dim}}$, there is an enormous gain in $N_{\rm cells}$ for even a modest improvement in the abundance uncertainties. For $N_{\rm dim} \sim 8$ independent dimensions (likely appropriate for e.g., GALAH), one could improve $N_{\rm cells}$ by a factor of two for a 10\% reduction in the abundance uncertainties (\S\ref{subsec:parameter-effect}). Therefore, if the goal is to find as many local peaks in chemical space (i.e., detectable groups) as possible and/or to increase the odds of those peaks being dominated by a single massive cluster, it might be more advantageous to seek strategies that reduce the abundance uncertainties rather than simply acquiring more stars.
An effective way to improve chemical tagging detections is by targeting a stellar subpopulation exclusively. As we have shown in \S\ref{subsec:subpops} and discussed in \S\ref{sec:key-params}, targeting a subpopulation not only improves the sampling rate but also reduces the number of clusters per chemical cell. It improves chances of the reconstructing the CMF because there are more stars sampled per cluster and more significant deviations from Poisson statistics. It also improves the local S/N and hence the chance of recovering individual clusters within detected groups in chemical space.
A variety of properties could be used to select special subpopulations from a larger parent sample, including age, metallicity, and kinematics. One could envision pilot surveys at modest spectral resolution designed to select stars in a narrow range in [Fe/H]. Kinematics from Gaia could be used to separate hot and cold components, for example thin and thick disk stars \citep[e.g.,][]{red06}. Stars could also be selected according to their age once age measurements are available for large samples of stars, e.g., from isochrone fitting and/or asteroseismic constraints. Finally, in an optically selected survey such as GALAH, which is biased to higher Galactic latitudes, it preferentially observes thick disk stars \citep{des15}. Since the total number of thick disk stars is smaller than thin disk stars, this preference argues that the sampling rate in these surveys could be larger than the one we assume in this study as we adopt an uniform sampling strategy \citep[see also][]{bla15}.
\subsection{Caveats, limitations, \& future directions}
A variety of assumptions and simplifications were made in this study. Here we highlight the most important limitations and comment on future directions.
When populating the chemical space we assumed that clusters are (statistically) homogeneously distributed in all $N_{\rm cells}$ chemical cells available. From both observations and chemical evolution models we know that this assumption is not true in detail. Of course, there are many more high metallicity stars than low metallicity stars, but also we expect the size of the chemical space to vary systematically with metallicity (for example, due to certain nucleosynthetic pathways, e.g., in AGB stars, that only become important some time after the initial burst of star formation). Because of these complexities, the space cannot be completely described by the parameter $N_{\rm cells}$. A more accurate approach would be to include a model for chemical evolution and then to define overdensities in chemical space with respect to a local background, either using neighboring cells or a more sophisticated group finding algorithm \citep[e.g.,][]{sha09,mit13}.
This study focused on idealized surveys targeting Milky Way disk stars. We did not consider the bulge, stellar halo, disrupted satellite galaxies, nor nearby dwarf galaxies. Each of these populations offers a unique set of challenges and opportunities. These components will be included in future versions of the model.
We did not follow the actual orbits of stars in a live Galactic potential, and the treatment of radial migration is quite simplistic. One could imagine an extension to the current model that follows the dynamical disruption of star clusters and the sequent orbital histories of the individual stars. This would be very valuable for exploring the potential gains of folding in kinematic information, such as will soon be available from Gaia and/or from the spectroscopic surveys themselves. \citet{mit14} found that kinematics information does not improve the detectability, but it is likely due to the limitation of their small sample with $< 10^3$ stars. As we have demonstrated in this study, detected groups in small sample are not likely to be co-natal, agreeing with their assessment.
The adopted model for the gas mass is fairly simplistic. However, we emphasize that the gas mass distribution only influences the radial migration prescription and the evolution of the CMF. The former is parameterized via the in-situ fraction, $f_{\rm in-situ}$. In both cases we consider a range of possible scenarios, which in some sense is equivalent to exploring the effects of varying the underlying gas mass model directly.
We assume that the spatial frequency of star formation follows an exponential disk characterized by the scale length $R_{\rm SFR}$. We are aware that this assumption might not be true in detail. At a given time, stars might form in some large scale molecular rings \citep[e.g.,][]{blo06,gor06} or spiral arms \citep[e.g.,][]{rix93,bik03}. However, we are only interested in the integrated star formation rate over the cosmic history. Since these transient complexes, at least for the molecular rings, are expected to be short lived and rapidly dissipate \citep[$< 100$ Myr; e.g.,][]{bas05,gor06}, the smooth star forming assumption is likely to do fine.
\section{Conclusions}
\label{sec:conclusions}
In this study we explored the prospects for chemically tagging stars in idealized spectroscopic surveys of the Solar vicinity. We constructed a simple two dimensional time-dependent model of the Milky Way disk including the effects of radial migration and evolution in the CMF. We explored a number of important parameters affecting the detectability of groups of stars in chemical space and we studied the composition of the detected groups. We now summarize our principle conclusions.
\begin{itemize}
\item The key parameters affecting the number of detected groups in chemical space, and whether or not those groups are dominated by a single massive cluster, are: the shape and evolution of the CMF; the number of chemical cells; and the survey sampling rate. The sampling rate is proportional to the number of stars in the survey divided by the total number of stars belonging to a particular (sub)population. The latter two parameters are strongly influenced by observational survey design choices.
\item The clumpiness in chemical space is strongly influenced by the CMF and by the survey sampling rate. This implies that one can probe the CMF of long disrupted clusters by statistically analyzing the clumpiness in chemical space.
\item Confidently identifying {\it individual} clusters through chemical tagging will be challenging even for $N_\star = 10^6$, if disk stars are uniformly sampled. Fundamentally this is because the sampling rate is inherently small in such cases ($\sim 10^{-4}$) implying that one expects to collect on average 10 stars per cluster for clusters with $M_{\rm cluster} \gtrsim 10^5 \, M_\odot$. This is born out by our modeling, where we find that even very large overdensities in chemical space are typically not comprised of stars from a single dominant cluster. In the fiducial case with $N_{\rm cells}=10^4$, the dominant cluster contributes only 25\% of the stars in the detected group. Additional follow-up of the stars within large overdensities in chemical space may provide additional discriminating power, either by decreasing the measurement uncertainties on the abundances, or by folding in color magnitude diagram or kinematic information.
\end{itemize}
\acknowledgments
The authors thank Joss Bland-Hawthorn, Ken Freeman, Sanjib Sharma, Charlie Lada and Eve Ostriker for helpful discussions, and Andrea Schruba and Peter Behroozi for sharing their data in electronic format. The computations in this paper were run on the Odyssey cluster supported by the FAS Division of Science, Research Computing Group at Harvard University.
|
{
"timestamp": "2015-06-01T02:01:00",
"yymm": "1504",
"arxiv_id": "1504.03327",
"language": "en",
"url": "https://arxiv.org/abs/1504.03327"
}
|
\section{Introduction}
\paragraph{\bf Summary.} \ We consider a semantic class, {\em weakly-chase-sticky} (WChS), and a syntactic subclass, {\em jointly-weakly-sticky} (JWS), of {\em Datalog}$^\pm$~ programs. Both extend that of weakly-sticky (WS) programs, which appear in
our applications to data quality. For WChS programs we propose a practical, polynomial-time query answering algorithm (QAA). We establish that the two classes are closed under magic-sets rewritings. As a consequence, QAA can be applied to the optimized programs. QAA takes as inputs the program (including the query) and semantic information about the ``finiteness" of predicate positions. For the syntactic subclasses JWS and WS of WChS, this additional information is computable.
\vspace{-2mm}
\paragraph{\bf {\em Datalog}$^\pm$~\!\!.} \ {\em Datalog}, a rule-based language for query and view-definition in relational databases \cite{ceri}, is not expressive enough to logically represent interesting and useful ontologies, at least of the kind needed to specify conceptual data models. {\em Datalog}$^\pm$~ extends {\em Datalog}~ by allowing
existentially quantified variables in rule heads ($\exists$-variables), equality atoms in rule heads, and program constraints \cite{AC09}.
Hence the ``$+$" in {\em Datalog}$^\pm$~, while the ``$-$" reflects syntactic restrictions on programs, for better computational properties.
A typical {\em Datalog}$^\pm$~ program, $\mc{P}$, is a finite set of rules, $\Sigma \cup E \cup N$, and an extensional database (finite set of {\em facts}), $D$.
The rules in $\Sigma$ are {\em tuple-generating-dependencies} ({\em tgds}) of the form $\exists \bar{x}\!P(\bar{x},\bar{x}') \leftarrow P_1(\bar{x}_1), \ldots, P_n(\bar{x}_n)$, where $\bar{x}' \subseteq
\bigcup \bar{x}_i$, and $\bar{x}$ can be empty. $E$ is a set of {\em equality-generating-dependencies} ({\em egds}) of the form $x = x' \leftarrow P_1(\bar{x}_1), \ldots, P_n(\bar{x}_n)$, with $\{x,x'\} \subseteq \bigcup \bar{x}_i$. Finally, $N$ contains {\em negative constraints} of the form $\bot \leftarrow P_1(\bar{x}_1), \ldots, P_n(\bar{x}_n)$, where $\bot$ is false.
\begin{example} \label{ex:program} The following {\em Datalog}$^\pm$~ program shows a tgd, an egd, and a negative constraint, in this order:
$\exists x\;{\it Assist}(y,x) \leftarrow {\it Doctor}(y)$; \ \ $x=x' \leftarrow {\it Assist}(y,x), \ {\it Assist}(y,x')$; \ \
$\bot \leftarrow {\it Specialist}(y,x,z), \ {\it Nurse}(y,z)$.
\hfill $\Box$\vspace*{0.2cm}
\end{example}
\vspace{-3mm}Below, when we refer to a class of {\em Datalog}$^\pm$~ programs, we consider only $\Sigma$, the tgds.
Due to different syntactic restrictions, {\em Datalog}$^\pm$~ can be seen as a class of sublanguages
of {\em Datalog}$^\exists$~\!\!, which is the extension of {\em Datalog}~ with tgds with $\exists$-variables \cite{LE11}.
The rules of a {\em Datalog}$^\pm$~ program can be seen as an ontology $\mc{O}$ on top of $D$, which can be {\em incomplete}. $\mc{O}$ plays the role of: (a) a ``query layer" for $D$, providing ontology-based data access (OBDA) \cite{lenzerini12}, and (b) the specification of
a completion of $D$, usually carried out through the {\em chase} mechanism that, starting from $D$, iteratively enforces the rules in $\Sigma$, generating new tuples. This leads to a possibly infinite
instance extending $D$, denoted with $\nit{chase}(\Sigma, D)$.
The answers to a conjunctive query $\mc{Q}(\bar{x})$ from $D$ wrt. $\Sigma$ is a sequence of constants $\bar{a}$, such that
$\Sigma \cup D \models \mc{Q}(\bar{a})$ (or $\nit{yes}$ or $\nit{no}$ in case $\mc{Q}$ is boolean). The answers can be obtained by querying as usual the {\em universal} instance $\nit{chase}(\Sigma, D)$.
The chase may be infinite, which leads, in some cases, to undecidability of query answering \cite{JO84}. However, in some cases where the chase is infinite, query answering (QA) is still
computable (decidable), and even tractable in the size of $D$. Syntactic classes of {\em Datalog}$^\pm$~ programs with tractable QA have been identified and investigated, among them: {\em sticky}~\cite{AC12,tods14}, and {\em weakly-sticky} \cite{AC12} {\em Datalog}$^\pm$~ programs.
\vspace{-2.5mm}
\paragraph{\bf Our Need for QA Optimization.} \ In our work, we concentrate on the {\em stickiness} and {\em weak-stickiness} properties, because these programs appear in our applications to quality data specification and extraction
\cite{desweb}, with the latter task accomplished through QA, which becomes crucial.
Sticky programs~\cite{AC12} satisfy a syntactic restriction on the multiple occurrences of variables (joins) in the body of a {\it tgd}.
Weakly-sticky (WS) programs form a class that extends that of sticky programs \cite{AC12}. WS-{\em Datalog}$^\pm$~ is more expressive than sticky {\em Datalog}$^\pm$~\!\!, and results from applying the notion of
{\em weak-acyclicity} (WA) as found in data exchange \cite{FG03}, to relax acyclicity conditions on stickiness. More precisely, in comparison with sticky programs, WS programs require a milder condition on join variables, which is based on a program's {\em dependency graph} and the positions in it with finite rank~\cite{FG03}.\footnote{A position refers to a predicate attribute, e.g. ${\it Nurse}[2]$.}
For QA, sticky programs enjoy {\em first-order rewritability} \cite{tods14}, i.e. a conjunctive query $\mc{Q}$ posed to $\Sigma \cup D$ can be rewritten into a new first-order (FO) query
$\mc{Q}'$, and correctly answered by posing $\mc{Q}'$ to $D$, and answering
as usual. For WS programs, QA is $\nit{PTIME}$-complete in data, but the polynomial-time algorithm provided for the proof in ~\cite{AC12} is not a practical one.
\vspace{-2.5mm}
\paragraph{\bf Stickiness of the Chase.}
\ In addition to (syntactic) stickiness, there is a ``semantic" property of programs, which is relative to the chase (and the data, $D$), and is called ``chase-stickiness" (ChS).
Stickiness implies semantic stickiness (but not necessarily the other way around) \cite{AC12}. \ For chase-sticky programs, QA is tractable \cite{AC12}.
Intuitively, a program has the chase-stickiness property if, due to the application of a tgd $\sigma$: When a value replaces a repeated variable in the body of a rule, then that value also appears in all the
head atoms obtained through the iterative enforcement of applicable rules that starts with $\sigma$. So, that value is propagated all the way down through all the possible subsequent steps.
\vspace{-7mm}
\begin{figure}[h]
\begin{center}
\includegraphics[width=3.25cm]{nchs.eps}
\includegraphics[width=3.25cm]{chs.eps}
\end{center}
\vspace{-0.7cm}
\caption{The chase for a non-ChS program and the chase for a ChS program, resp.}
\label{fig:chase}
\vspace{-0.8cm}
\end{figure}
\begin{example}\label{exp:chs} Consider $D=\{{\nit Assist}(a,b),{\nit Assist}(b,c)\}$, and the following set, $\Sigma_1$, of tgds: \
${\it Nurse}(y,z)\leftarrow {\it Assist}(x,y),{\it Assist}(y,z)$; \ \
$\exists z\;{\it Specialist}(x,y,z)\leftarrow {\it Nurse}(x,y)$; \ \
${\nit Doctor}(y)\leftarrow {\nit Specialist}(x,y,z)$. \
$\Sigma_1$ is not ChS, as the chase on the LHS of Figure\ref{fig:chase} shows: value $b$ is not propagated all the way down to $\nit{Doctor}(c)$.
However, program $\Sigma_2$, which is $\Sigma_1$ without its third rule, is ChS, as shown on the RHS of Figure\ref{fig:chase}.
\hfill $\Box$\vspace*{0.2cm}\end{example}
\vspace{-7mm}
\paragraph{\bf Weak-Stickiness of the Chase.} Weak-stickiness also has a semantic version, called ``weak-chase-stickiness" (WChS); which is implied by the former. So as for chase-stickiness, weak-chase-sticky
programs have a tractable QA problem, even with a possibly infinite chase. This class is one of the two we introduce and investigate. They appear in double-edged boxes in Figure~\ref{fig:gener},
with dashed edges indicating a semantic class.
By definition, weak-chase-stickiness is obtained by relaxing the condition for ChS: it applies only to values for repeated variables in the body of $\sigma$
that appear in so-called {\em infinite positions}, which are semantically defined. A position is infinite if there is an instance $D$ for which an unlimited number of different values appear in $\nit{Chase}(\Sigma,D)$.
Given a program, deciding if a position is infinite is unsolvable, so as deciding in general if the chase terminates. Consequently, it is also undecidable if a program is WChS\ignore{since WChS is defined based on the notion of (in)finite positions}. However, there are syntactic conditions on programs~\cite{FG03,RUD11} that determine some (but not necessarily all) the finite positions. For example, the notion of position {\em rank}, based on
the program's {\em dependency graph}, are used in~\cite{FG03,AC12} to identify a (sound) set of finite positions, those with {\em finite rank}. Furthermore,
finite-rank positions are used in~\cite{AC12} to define weakly-sticky (WS) programs as a syntactic subclass of WChS.
\vspace{-2mm}
\paragraph{\bf Finite Positions and Program Classes.} In principle, any set-valued function $S$ that, given a program, returns a subset of the program's finite positions can be used to define a subclass WChS($S$) of WChS.
This is done by applying the definition of WChS above with ``infinite positions" replaced by ``non-$S$-finite positions". Every class WChS($S$) has a tractable QA problem.
$S$ could be computable on the basis of the program syntax or not. In the former case, it would be a ``syntactic class". \ignore{${\it WChS}(.)$ is a mapping that returns a class of programs by replacing the (in)finite positions in the definition of WChS programs with the positions specified by $S$.}
Class ${\it WChS}(S)$ grows monotonically with $S$ in the sense that if $S_1 \subseteq S_2$ (i.e. $S_1$ always returns a subset of the positions returned by $S_2$), then ${\it WChS}(S_1) \subseteq {\it WChS}(S_2)$.
In general, the more finite positions are (correctly) identified (and the consequently, the less finite positions are treated as infinite), the more general the subclass of WChS that is identified or characterized.
For example, the function $S^\bot$ that always returns an empty set of finite positions, ${\it WChS}(S^\bot)$ is the class of sticky programs, because stickiness must hold no matter what the (in)finite positions are. At the other extreme, for function $S^\top$ that returns all the (semantically) finite positions, ${\it WChS}(S^\top)$ becomes the class WChS. (As mentioned above, $S^\top$ is in general uncomputable.) Now, if $S^{\it rank}$ returns the set of finite-rank positions (for a program $\mc{P}$, usually denoted by $\Pi_F(\mc{P})$ \cite{FG03}), ${\it WChS}(S^{\it rank})$ is the class of WS programs.
\vspace{-2mm}
\paragraph{\bf Joint-Weakly-Stickiness.} The {\em joint-weakly-sticky} (JWS) programs we introduce form a syntactic class strictly between WS and WChS. Its definition appeals to the notions of {\em joint-acyclicity} and {\em existential dependency graphs} introduced in~\cite{RUD11}. Figure~\ref{fig:gener} shows this syntactic class, and the inclusion relationships between classes of {\em Datalog}$^\pm$~ programs.\footnote{Rectangles with dotted-edges show semantic classes, and double-edged rectangles show the classes introduced in this work. Notice that programs in semantic classes include the instance $D$, but syntactic classes are data-independent (for any instance as long as the syntactic conditions apply).}
If $S^{\it ext}$ denotes the function that specifies finite positions on the basis of the {\em existential dependency graphs} (EDG), implicitly defined in~\cite{RUD11}, the JWS class is, by definition, the class ${\it WChS}(S^{\it ext})$. EDGs provide a finer mechanism for capturing (in)finite positions in comparison with positions ranks (defined through dependency graphs): \ $S^{\it rank} \subseteq S^{\it ext}$. Consequently, the class of JWS programs, i.e. ${\it WChS}(S^{\it ext})$, is a strict superclass of WS programs, i.e. ${\it WChS}(S^{\it rank})$.\footnote{The JWS class is different from (and incomparable with) the class of {\em weakly-sticky-join} programs (WSJ) introduced in~\cite{ACL10}, which extends the one of WS programs with consideration that are different from those used for JWS programs. WSJ generalizes WS on the basis of the weakly-sticky-join property of the chase~\cite{ACL10,AC12} and is related to repeated variables in single atoms.}
\vspace{-2mm}
\paragraph{\bf QAA for WChS.} Our query answering algorithm for WChS programs is parameterized by a (sound) finite-position function $S$ as above. It is denoted with ${\it AL}^S$, and takes as input $\Sigma, D$, query $\mc{Q}$, and $S(\Sigma)$, which is a subset of the program's finite positions (the other are treated as infinite by default).
The customized algorithm ${\it AL}^S$ is guaranteed to be sound and complete only when applied to programs in ${\it WChS}(S)$: \ ${\it AL}^S(\Sigma,D,\mc{Q})$ returns all and only the query answers. (Actually, ${\it AL}^S$ is still sound for any program in WChS.) ${\it AL}^S$ runs in polynomial-time in data; and
can be applied to both the WS and the JWS syntactic classes. For them the finite-position functions are computable.
${\it AL}^S$ is based on the concepts of {\em parsimonious chase} ({\em pChase}) and {\em freezing nulls}, as used for QA with {\em shy Datalog}, a fragment of {\em Datalog}$^\exists$~ \cite{LE11}. At a {\em pChase} step, a new atom is added only if a homomorphic atom is not already in the chase. Freezing a null is promoting it to a constant (and keeping it as such in subsequent chase steps). So, it cannot take (other) values under homomorphisms, which may create new {\em pChase} steps. Resumption of the {\em pChase}
means freezing {\em all} nulls, and continuing {\em pChase} until no more {\em pChase} steps are applicable.
Query answering with shy programs has a first phase where the {\em pChase} runs until termination (which it does). In a second phase, the {\em pChase} iteratively resumes for a number of times that depends on the number of distinct $\exists$-variables in the query. This second phase is required to properly deal with joins in the query.
Our QAA for WChS programs (${\it AL}$) is similar, it has the same two phases, but a {\em pChase} step is modified: after every application of a {\em pChase} step that generates nulls, the latter that appear in $S$-finite positions are immediately frozen.
\vspace{-2mm}
\paragraph{\bf Magic-Sets Rewriting.} \ It turns out that JWS, as opposed to WS, is closed under the quite general magic-set rewriting method \cite{ceri}
introduced in \cite{AL12}. As a consequence, ${\it AL}$ can be applied to both the original JWS program and its magic rewriting. (Actually, this also holds for the superclass WChS.)
\newpage
\begin{figure}[t]
\begin{center}
\includegraphics[width=7.5cm]{generalizations.eps}
\end{center}
\vspace{-0.7cm}
\caption{Generalization relationships among program classes.}
\label{fig:gener}\vspace{-5mm}
\end{figure}
It can be proved that (our modification of) the magic-sets rewriting method in \cite{AL12} does not change the character of the original finite or infinite positions.
The specification of (in)finiteness character of positions in magic predicates is not required by ${\it AL}$, because no new nulls appear in them during the ${\it AL}$ execution. As consequence, the MS method rewriting can be perfectly integrated with our QAA, introducing additional efficiency.
\ignore{
\paragraph{\bf An Alternative QAA.} Specially for WS-{\em Datalog}$^\pm$~, we also studied a different approach for query answering based on grounding variables. We propose a partial grounding algorithm that given a WS-{\em Datalog}$^\pm$~ program replaces some of the variables in the body of the {\it tgd} rules with constants and null values. That is to say, the partial grounding algorithm removes problematic repeated variables that violate syntactic restriction of sticky programs. Consequently, the result of the partial grounding algorithm on a WS-{\em Datalog}$^\pm$~ program is a simpler sticky {\em Datalog}$^\pm$~ program and therefore fo-rewritability of sticky programs allows us to employ query rewriting for answering conjunctive queries. We prove that this partial grounding algorithm generates polynomially many new {\it tgd} rules with respect to the size of {\it edb}.
}
\vspace{1mm}
\noindent \small {\bf Acknowledgments:} \ We are very grateful to Mario Alviano and the DLV team for providing us with information and support in relation to existential Datalog. We also appreciate
useful conversations with Andrea Cali and Andreas Pieris on Datalog$\pm$, and important comments from Andrea Cali on an earlier version of this paper.
\vspace{-4mm}
|
{
"timestamp": "2015-04-15T02:02:35",
"yymm": "1504",
"arxiv_id": "1504.03386",
"language": "en",
"url": "https://arxiv.org/abs/1504.03386"
}
|
\section{ Introduction} The study of codes over finite rings was initiated by Blake \cite{4, 5} in the year 1970. During the last decades of the twentieth century a great deal of attention has been given to codes over finite rings because of their new role in algebraic coding theory and their successful applications. In a landmark paper \cite{13}, it has been shown that certain good non-linear binary codes can be constructed from cyclic codes over $Z_4$ via the Gray map. Since then, codes over finite rings have been studied by many authors \cite{9, 10, 14, 17}. A lot of work has been done in this direction, but codes over $\mathbb{Z}_4$ remain a special topic of interest in the field of algebraic coding theory because of their relation to lattices, designs, cryptography and their many applications. There is a vast literature on codes over $\mathbb{Z}_4$ and their applications for detail see the references \cite{1, 7, 8, 12, 20, 21}.\\
Recently, Yildiz and Karadeniz considered linear and cyclic codes over the non-ring $F_2+uF_2+vF_2+uvF_2$ of size $16$ with $u^2=v^2=0$ and $uv=vu$ in \cite{22} and \cite{23}, where some good binary codes have been obtained as the images under two Gray maps. Motivated by this work the same authors \cite{24}, studied linear codes over the ring $\mathbb{Z}_4+u\mathbb{Z}_4,~u^2=0,$ which is also a non-chain ring of size $16.$ Further, Yildiz and Aydin \cite{25}, determined algebraic structure of cyclic codes over this ring. They obtained many new linear codes over $\mathbb{Z}_4$ as the Gray images of cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4.$ Bandi and Bhaintwal \cite{6} gave the structural properties of cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$ of odd length in different aspect. They provided the general form of the generators of a cyclic code over $\mathbb{Z}_4+u\mathbb{Z}_4.$\\
Constacyclic codes constitute a remarkable generalization of cyclic codes, hence form an important class of linear codes in the coding theory. Constacyclic codes can be efficiently encoded using shift registers, which explains their preferred role in engineering. Qian et al. \cite{19}, introduced $(1+u)$-constacyclic codes of odd length over the ring $F_2+uF_2,$ where $u^2=0.$ with the help of cyclic codes over this ring. Later on Abualrub and Siap \cite{3}, studied $(1+u)$-constacyclic codes of arbitrary length over this ring and they proved that the Gray image of a $(1+u)$-constacyclic code is a binary cyclic code of length $2n.$ $(1+v)$-constacyclic codes of odd length over the ring $F_2+uF_2+vF_2+uvF_2$ were studied by Karadeniz and Yildiz in \cite{15}. A lot of work has been done on constacyclic codes over different structure of rings by several authors we refer to \cite{2, 11, 16, 18, 26} and the references therein. This is the motivation to study $(1+2u)$-constacyclic codes of odd length over $\mathbb{Z}_4+u\mathbb{Z}_4$ with $u^2=0.$\\
\vspace{.4cm}
\parindent=7mm
\section{Preliminaries} Let $\mathbb{Z}_4$ be the ring of integers modulo $4$. Consider the ring $R=\mathbb{Z}_4+u\mathbb{Z}_4=\{a+ub|~a,~b\in \mathbb{Z}_4\}$ with $u^2=0$. $R$ is a commutative ring with characteristic $4$ and it can be viewed as the quotient ring $\mathbb{Z}_4[u]/\langle u^2\rangle$. The units and non-units in $R$ can be characterized by $``a+bu$ is a unit in $R$ if and only if $a$ is a unit in $\mathbb{Z}_4$". Therefore $\{1, 3, 1+u, 1+2u, 1+3u, 3+u, 3+2u, 3+3u\}$ is a set of units of $R$ while $\{0, 2, u, 2u, 2+u, 2+2u, 3u, 2+3u\}$ is a set of non-units of $R$.\\
$R$ has a total of six ideals given by $\langle0\rangle=\{0\},~\langle 2u\rangle=\{0, 2u\},~\langle u\rangle=\{0, u, 2u, 3u\},~\langle 2\rangle=\{0, 2, 2u, 2+2u\},~\langle 2+u\rangle=\{0, 2u, 2+u, 2+3u\}$ and $\langle 2, u\rangle=\{0, 2, u, 2u, 3u, 2+u, 2+2u, 2+3u\}$. It is clear that $R$ is a local Frobenius ring with $\langle 2, u\rangle$ as its maximal ideal. The residue field is given by $R/\langle 2, u\rangle,$ which is isomorphic to binary field $F_2$.\\
A commutative ring $R$ is called a chain ring if its ideals form a chain under the relation of inclusion. From the ideals of $R$, we can see that they do not form a chain; for instance, the ideals $\langle u\rangle$ and $\langle 2\rangle$ are not comparable. Therefore, $R$ is a non-chain extension of $\mathbb{Z}_4$. Also $R$ is not a principle ideal ring; for example, the ideal $\langle 2, u\rangle$ is not generated by any single element of $R$.\\
Let $R^n$ be the set of all $n$-tuples over $R$, then a nonempty subset $C$ of $R^n$ is called a code of length $n$ over $R$. $C$ is called linear code of length $n$ over $R$ if it is an $R$-submodule of $R^n$. Each codeword $\overline c$ in such a code $C$ is just an $n$-tuple of the form $\overline{c}=(c_0, c_1, \cdots, c_{n-1})\in R^n $ and can be represented by a polynomial in $R[x]$ as follows:$$\overline{c}=(c_0, c_1, \cdots, c_{n-1})~\mbox{if and only if}~c(x)=\sum\limits_{i=0}^{n-1}c_ix^i\in R[x].$$
\vspace{.4cm}
\parindent=7mm
\section{Gray map over $R$} Here, we give the definition of the Gray map on $R^n$. Observe that any element $c\in R$ can be expressed as $c=a+ub$, where $a,~b\in \mathbb{Z}_4$. The Gray map $\phi:R\longrightarrow \mathbb{Z}_{4}^{2}$ is given by $\phi(c)=\phi(a+ub)=(b, 2a+b)$. This map can be extended to $R^n$ in a natural way as follows: $$\phi:R^n\longrightarrow \mathbb{Z}_{4}^{2n}$$
$$(c_0, c_1, ..., c_{n-1})\longmapsto(b_0, b_1, ..., b_{n-1}, 2a_0+b_0, 2a_1+b_1, ..., 2a_{n-1}+b_{n-1}),$$ where $c_i=a_i+ub_i,~0\leq i\leq n-1$.\\
\noindent Also the Lee weight on $R$ is defined as $$w_L(a+ub)=w_L(b, 2a+b),$$ where $w_L(b, 2a+b)$ denotes the usual Lee weight on $\mathbb{Z}_{4}^{2}.$ This weight can be extended to $R^n$ componentwise. The Lee weight of a codeword $\overline{c}=(c_0, c_1, ..., c_{n-1})\in R^n$ is the rational sum of the Lee weights of its components, that is, $w_L=\sum\limits_{i=0}^{n-1}{w_L(c_i)}$. For any $\overline{c}_1,~\overline{c}_2\in R^n$, the Lee distance $d_L$ is given by $d_L(\overline{c}_1, \overline{c}_2)=w_L(\overline{c}_1-\overline{c}_2)$. The minimum Lee distance of $C$ is the smallest nonzero Lie distance between all pairs of distinct codewords of $C$. The minimum Lee weight of $C$ is the smallest nonzero Lee weight among all codewords of $C$.\\
\noindent Now with the definitions of Gray map and Lee weight, we have the following obvious theorem:
\begin{thm} The map $\phi:R^n\longrightarrow \mathbb{Z}_{4}^{2n}$ is a distance preserving linear isometry. Thus, if $C$ is a linear code over $R$ of length $n$, then $\phi(C)$ is a linear code over $\mathbb{Z}_4$ of length $2n$ and the two codes have the same Lee weight enumerators.
\end{thm}
\vspace{.4cm}
\parindent=7mm
\section{$(1+2u)$-constacyclic codes of odd length over $R$} A cyclic shift on $R^n$ is a permutation $\sigma$ such that $$\sigma(c_0, c_1, \cdots, c_{n-1})=(c_{n-1}, c_0, c_1, \cdots, c_{n-2}).$$ A linear code $C$ over $R$ of length $n$ is called cyclic code if it is invariant under the cyclic shift $\sigma$, that is, $\sigma(C)=C.$\\
\noindent A $(1+2u)$-constacyclic shift $\tau$ on $R^n$ acts as $$\tau(c_0, c_1, \cdots, c_{n-1})=((1+2u)c_{n-1}, c_0, c_1, \cdots, c_{n-2}).$$ A linear code $C$ over $R$ of length $n$ is called $(1+2u)$-constacyclic code if it is invariant under the $(1+2u)$-constacyclic shift $\tau$, that is, $\tau(C)=C.$\\
Using the polynomial representation of codewords of $R^n$ in $R[x]$, we see that for a codeword $\overline c\in R^n,~\sigma(\overline{c})$ corresponds to $xc(x)$ in $R[x]/\langle x^n-1\rangle$ while $\tau(\overline{c})$ corresponds to $xc(x)$ in $R[x]/\langle x^n-(1+2u)\rangle$. The following propositions are the analogy of a well-known result for cyclic codes over finite fields. The proofs are also similar, therefore, we are omitting the proofs.
\begin{prop} A subset $C$ of $R^n$ is a linear cyclic code of length $n$ over $R$ if and only if its polynomial representation is an ideal of $R[x]/\langle x^n-1\rangle.$
\end{prop}
\begin{prop} A subset $C$ of $R^n$ is a linear $(1+2u)$-constacyclic code of length $n$ over $R$ if and only if its polynomial representation is an ideal of $R[x]/\langle x^n-(1+2u)\rangle.$
\end{prop}
It is noted that $(1+2u)^n=1+2u$ if $n$ is odd and $(1+2u)^n=1$ if $n$ is even. Therefore, we only study the properties of $(1+2u)$-constacyclic codes of odd length over $R$. Cyclic codes over $R$ of odd length are classified in \cite{25}. Using this classification we study $(1+2u)$-constacyclic codes over $R$ of odd length by introducing the following isomorphism from $R[x]/\langle x^n-1\rangle$ to $R[x]/\langle x^n-(1+2u)\rangle$:
\begin{thm} Let $\mu$ be the map of $R[x]/\langle x^n-1\rangle$ into $R[x]/\langle x^n-(1+2u)\rangle$ defined by $\mu(c(x))=c((1+2u)x)$. If $n$ is odd, then $\mu$ is a ring isomorphism.
\end{thm}
\noindent{\bf{Proof.}} Since $(1+2u)$ is a unit in $R$, $(1+2u)^n=1$. Also we know that if $n$ is odd, then $(1+2u)^n=1+2u.$ Now suppose $a(x)\equiv b(x)(\mbox{mod}x^n-1)$, that is, $a(x)-b(x)=(x^n-1)q(x)$ for some $q(x)\in R[x]$. Then
$$\begin{array}{lll}
a((1+2u)x)-b((1+2u)x)& = & ((1+2u)^nx^n-1)q((1+2u)x)\\
& = & ((1+2u)x^n-(1+2u)^2)q((1+2u)x)\\
& = & (1+2u)(x^n-(1+2u))q((1+2u)x),
\end{array}$$ which means if $a(x)\equiv b(x)(\mbox{mod}x^n-1),$ then $a((1+2u)x)\equiv b((1+2u)x)(\mbox{mod}x^n-(1+2u)).$ But the converse can easily be shown as well which means $$a(x)\equiv b(x)(\mbox{mod}x^n-1)\Leftrightarrow a((1+2u)x)\equiv b((1+2u)x)(\mbox{mod}x^n-(1+2u)).$$ Note that one side of the implication tells us that $\mu$ is well defined and the other side tells us that it is injective, but since the rings are finite this proves that $\mu$ is an isomorphism.\\
\noindent The following corollary is an immediate consequence of the above theorem:
\begin{cor} $I$ is an ideal of $R[x]/\langle x^n-1\rangle$ if and only if $\mu(I)$ is an ideal of\linebreak $R[x]/\langle x^n-(1+2u)\rangle$, where $n$ is odd.
\end{cor}
\noindent Before stating our next result, we need the following known lemma:
\begin{lem}{\cite[Theorem 4]{24}}, Let $n$ be odd and $C$ be a cyclic code of length $n$ over the ring $R$. Then $C$ is an ideal in $R[x]/\langle x^n-1\rangle$ generated by $$C=\langle a_1(x)(b_1(x)+2), ua_2(x)(b_2(x)+2)\rangle,$$ for some $a_i(x), b_i(x)\in \mathbb{Z}_4[x]$ such that $x^n-1=a_i(x)b_i(x)c_i(x)$ and $a_i(x), b_i(x), c_i(x)$ are monic coprime polynomials.
\end{lem}
\noindent Using the isomorphism $\mu$ and the above lemma, we characterize $(1+2u)$-constacyclic codes over $R$ of odd length as follows:
\begin{thm} Let $n$ be odd and $C$ be a $(1+2u)$-constacyclic code of length $n$ over the ring $R$. Then $C$ is an ideal in $R[x]/\langle x^n-(1+2u)\rangle$ generated by $$C=\langle a_1(\tilde{x})(b_1(\tilde{x})+2), ua_2(\tilde{x})(b_2(\tilde{x})+2)\rangle,$$ where $\tilde{x}=(1+2u)x$, $a_i(x), b_i(x)$ are the polynomials in $\mathbb{Z}_4[x]$ such that $x^n-1=a_i(x)b_i(x)c_i(x)$ and $a_i(x), b_i(x), c_i(x)$ are monic coprime polynomials.
\end{thm}
Now, we define a map ${\overline\mu}:R^n\longrightarrow R^n$ such that $${\overline\mu}(c_0, c_1, \cdots, c_{n-1})=(c_0, (1+2u)c_1, (1+2u)^2c_2, \cdots, (1+2u)^{n-1}c_{n-1}).$$ It is worth mentioning that ${\overline\mu}$ acts as the vector equivalent of $\mu$ on $R^n$. Therefore, we can restate Corollary 4.4 in terms of vectors as well.
\begin{cor} $C$ is a linear cyclic code over $R$ of odd length $n$ if and only if ${\overline\mu}(C)$ is a linear $(1+2u)$-constacyclic code of length $n$ over $R.$
\end{cor}
Note that if $c=a+ub\in R$, then $(1+2u)c=a+u(2a+b).$ Thus $$\begin{array}{lll}
w_L(c)&=&w_L(b, 2a+b)\\
&=&w_L((1+2u)c).
\end{array}$$
\noindent In view of the Corollary 4.7, we have the following result:
\begin{cor} $C$ is a cyclic code over $R$ of length $n$ with Lee distance $d_L$ if and only if ${\overline\mu}(C)$ is a $(1+2u)$-constacyclic code over $R$ of length $n$ with same Lee distance, where $n$ is odd.
\end{cor}
\section{Gray images of $(1+2u)$-constacyclic codes over $R$} Even length cyclic codes over $\mathbb{Z}_4$ were characterized by Dougherty and Ling in \cite{12}. Here, we study even length cyclic codes over $\mathbb{Z}_4$ as the Gray images of $(1+2u)$-constacyclic codes over $R.$
\begin{prop} Let $\tau$ be the $(1+2u)$-constacyclic shift of $R^n$ and $\sigma$ be the cyclic shift of $\mathbb{Z}_4^{2n}$. If $\phi$ is the Gray map from $R^n$ to $\mathbb{Z}_4^{2n},$ then $\phi\tau=\sigma\phi$.
\end{prop}
\noindent{\bf{Proof.}} Let $\overline{c}=(c_0, c_1, \cdots, c_{n-1})\in R^n$, where $c_i=a_i+ub_i$ with $a_i,~b_i\in \mathbb{Z}_4$ for $0\leq i\leq n-1$. Taking $(1+2u)$-constacyclic shift of $\overline{c}$, we have
$$\begin{array}{lll}
\tau(\overline{c}) & = & ((1+2u) c_{n-1}, c_0, \cdots, c_{n-2}) \\
& = & ((1+2u)(a_{n-1}+ub_{n-1}), a_0+ub_0, \cdots, a_{n-2}+ub_{n-2}) \\
& = & (a_{n-1}+u(2a_{n-1}+b_{n-1}), a_0+ub_0, \cdots, a_{n-2}+ub_{n-2}).
\end{array}$$ Now, using the definition of Gray map $\phi$, we can deduce that $$\phi(\tau(\overline{c}))=(2a_{n-1}+b_{n-1}, b_0, b_1, \cdots, b_{n-2}, b_{n-1}, 2a_0+b_0, 2a_1+b_1, \cdots, 2a_{n-2}+b_{n-2}).$$
On the other hand, $$\phi(\overline{c})=(b_0, b_1, \cdots, b_{n-1}, 2a_0+b_0, 2a_1+b_1, \cdots, 2a_{n-1}+b_{n-1}).$$
Hence, $$\sigma(\phi(\overline{c}))=(2a_{n-1}+b_{n-1}, b_0, b_1, \cdots, b_{n-2}, b_{n-1}, 2a_0+b_0, 2a_1+b_1, \cdots, 2a_{n-2}+b_{n-2}).$$
Therefore, $$\phi\tau=\sigma\phi.$$
\noindent As a consequence of Proposition 5.1, we get the following main result:
\begin{thm} The Gray image of a linear $(1+2u)$-constacyclic code over $R$ of length $n$ is a distance invariant linear cyclic code over $\mathbb{Z}_4$ of length $2n.$
\end{thm}
\noindent{\bf{Proof.}} Let $C$ be a linear $(1+2u)$-constacyclic code over $R$ of length $n.$ Then $\tau(C)=C,$ and therefore $\phi(\tau(C))=(\phi\tau)(C)=\phi(C).$ It follows from Proposition 5.1 that $(\sigma\phi)(C)=\sigma(\phi(C))=\phi(C),$ which means that $\phi(C)$ is a linear cyclic code over $\mathbb{Z}_4$ of length $2n.$\\
Yildiz and Aydin in \cite{25} obtained table for cyclic codes over $R$ of length $7.$ By modifying the generators with isomorphism $\mu$, we get the following table for $(1+2u)$-constacyclic codes of length $7$ over $R$. The generators of $(1+2u)$-constacyclic codes given by the Theorem 4.6 are taken to be $C=\langle g_1(\tilde{x}), ug_2(\tilde{x})\rangle$, where $\tilde{x}=(1+2u)x.$
\newpage
\noindent Some $(1+2u)$-constacyclic codes of length $7$ and their $\mathbb{Z}_4$ images.
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
$g_1(\tilde{x})$ & $g_2(\tilde{x})$ & $\mathbb{Z}_4$ Parameters\\
\hline
$0$ & $3x^4+2x^3+x^2+3(1+2u)x+3$ & $[14, 4^32^0, 12]$ \\
$0$ & $x^4+(1+2u)x^3+3x^2+3$ & $[14, 4^32^3, 8]$ \\
$3x^4+2x^3+x^2+(3+2u)x+3$ & $x^4+(1+2u)x^3+3x^2+3$ & $[14, 4^62^3, 6]$ \\
$(3+2u)x^3+x^2+2x+1$ & $(1+2u)x^3+2x^2+(1+2u)x+1$ & $[14, 4^82^3, 4]$ \\
$(3+2u)x^3+x^2+2x+1$ & $(3+2u)x+1$ & $[14, 4^{10}2^0, 4]$ \\
$(3+2u)x^3+x^2+2x+1$ & $(1+2u)x+1$ & $[14, 4^{10}2^1, 4]$\\
$(1+2u)x^3+2x^2+(1+2u)x+1$ & $(1+2u)x+1$ & $[14, 4^{10}2^{4}, 2]$\\
$(3+2u)x+1$ & $(3+2u)x+1$ & $[14, 4^{12}2^0, 2]$\\
$(3+2u)x+1$ & $(1+2u)x+1$ & $[14, 4^{12}2^1, 2]$\\
$(1+2u)x+1$ & $(1+2u)x+1$ & $[14, 4^{12}2^2, 2]$\\
$(3+2u)x+1$ & $3$ & $[14, 4^{13}2^0, 2]$\\
\hline
\end{tabular}
\end{center}
\vspace{.6cm}
\noindent Now, we give some examples of $(1+2u)$-constacyclic codes over $R$ of odd lengths other than length $7$. Also, we find the $\mathbb{Z}_4$ images of these constacyclic codes.\\
\noindent\textbf{Example 5.3} Let $n=9$ and $g_1(x)=0,~g_2(x)=x^8+(1+2u)x^7+x^6+(1+2u)x^5+x^4+(1+2u)x^3+3x^2+(3+2u)x+3.$ Then $C=\langle g_1(x), ug_2(x)\rangle$ is a $(1+2u)$-constacyclic code of length $9$ over $R$ with minimum Lee distance $8.$ In view of Theorem 5.2, the Gray image $\phi(C)$ of $C$ is a cyclic code over $\mathbb{Z}_4$ with parameters $[18, 4^12^6, 8]$.\\
\noindent\textbf{Example 5.4} Let $n=15$ and $g_1(x)=2x^{10}+2x^{8}+2x^{5}+2x^{4}+2x^2+2x,~g_2(x)=(3+2u)x^{13}+x^{12}+3x^{10}+(1+2u)x^9+(3+2u)x^7+x^6+3x^4+(1+2u)x^3+(3+2u)x+1.$ Then $C=\langle g_1(x), ug_2(x)\rangle$ is a $(1+2u)$-constacyclic code of length $15$ over $R$ with minimum Lee distance $8.$ In view of Theorem 5.2, the Gray image $\phi(C)$ of $C$ is a cyclic code over $\mathbb{Z}_4$ with parameters $[30, 4^22^{10}, 8]$.\\
\noindent\textbf{Example 5.5} Let $n=23$ and $g_1(x)=0,~g_2(x)=x^{22}+(1+2u)x^{21}+x^{20}+(1+2u)x^{19}+x^{18}+(1+2u)x^{17}+x^{16}+(1+2u)x^{15}+x^{14}+(1+2u)x^{13}+x^{12}+(3+2u)x^{11}
+3x^{10}+(1+2u)x^{9}+x^8+(1+2u)x^7+3x^6+(3+2u)x^5+3x^4+(1+2u)x^3+3x^2+(1+2u)x+3.$ Then $C=\langle g_1(x), ug_2(x)\rangle$ is a $(1+2u)$-constacyclic code of length $23$ over $R$. In view of Theorem 5.2, the Gray image $\phi(C)$ of $C$ is a cyclic code over $\mathbb{Z}_4$ with parameters $[46, 4^12^{11}, 28]$.\\
One-generator cyclic codes over $R$ of length $n$ have been studied by Yildiz and Aydin in \cite{25}. Motivated by this study we consider a one generator $(1+2u)$-constacyclic code over $R$ of length $n$, that is, we take an ideal in $R[x]/\langle x^n-(1+2u)\rangle$ generated by some polynomial $a(x)+ub(x)\in R[x]/\langle x^n-(1+2u)\rangle,$ where $a(x),~b(x)\in \mathbb{Z}_4[x]$ and deg$(a(x))<n,$ deg$(b(x))<n.$ In light of this, we give the following result about the Gray image of such a code in $\mathbb{Z}_4$:\\
\noindent\textbf{Theorem 5.6} Let $C=\langle a(x)+ub(x)\rangle$ be a $(1+2u)$-constacyclic code over $R$ of length $n.$ Then $\phi(C)$ is a cyclic code over $\mathbb{Z}_4$ of length $2n$ generated by the polynomial $b(x)+x^n(2a(x)+b(x))$ and $a(x)+x^na(x).$\\
\noindent{\bf{Proof.}} First we define a Gray map for polynomials as follows: $$\phi:R[x]/\langle x^n-(1+2u)\rangle\longrightarrow\mathbb{Z}_{4}[x]/\langle x^n-1\rangle\times\mathbb{Z}_{4}[x]/\langle x^n-1\rangle$$
$$\phi(a(x)+ub(x))=(b(x), 2a(x)+b(x)).$$ One can easily proved that $\phi$ is well defined. It may be noted that $(b(x), 2a(x)+b(x))$ gives us the same vector as $b(x)+x^n(2a(x)+b(x))$ in $\mathbb{Z}_{4}[x]/\langle x^{2n}-1\rangle.$ Since $C$ is an ideal in $\mathbb{Z}_{4}[x]/\langle x^{n}-(1+2u)\rangle,$ $ur(x)(a(x)+ub(x))\in C$ for all $r(x)\in \mathbb{Z}_4[x],$ and therefore we get $ur(x)a(x)\in C.$ Also $\phi(ur(x)a(x))=r(x)(a(x), a(x)),$ which gives us the same vector as $r(x)(a(x)+x^nr(x))$ in $\mathbb{Z}_{4}[x]/\langle x^{2n}-1\rangle.$ Again it may be noted that $$(r(x)+uq(x))(a(x)+ub(x))=r(x)(a(x)+ub(x))+uq(x)a(x).$$ Thus, we have
$$\phi[(r(x)+uq(x))(a(x)+ub(x))]=r(x)[b(x)+x^n(2a(x)+b(x))]+q(x)[a(x)+x^na(x)].$$ Since $r(x)$ and $q(x)$ are arbitrary polynomials in $\mathbb{Z}_4[x]$ of degree $<n,$ this proves the theorem.\\
\noindent We close our discussion with the following example of even length $(1+2u)$-constacyclic code over $R$:\\
\noindent\textbf{Example 5.7} Let $n=6$ and the one-generator $(1+2u)$-constacyclic code $C=\langle (x+x^2+x^4+x^5)+u(2x+2x^3+2x^5)\rangle.$ Then by Theorem 5.6, $\phi(C)$ is a cyclic code over $\mathbb{Z}_4$ of length $12$ with Lee distance $8$ generated by the polynomials $2x^{11}+2x^{10}+2x^9+2x^8+2x^5+2x^3+2x$ and $x^{11}+x^{10}+x^8+x^7+x^5+x^4+x^2+x.$\\
\section{Conclusion} In this paper, we have studied $(1+2u)$-constacyclic codes of odd length over $R=\mathbb{Z}_4+u\mathbb{Z}_4,$ where $u^2=0.$ We have defined a new Gray map from $R^n$ to $\mathbb{Z}_4^{2n}$ to study linear codes over $\mathbb{Z}_4$ as the Gray images of $(1+2u)$-constacyclic codes over $R.$ Further, we have proved that the Gray image of $(1+2u)$-constacyclic codes of length $n$ over $R$ are cyclic codes of length $2n$ over $\mathbb{Z}_4$.\\
We have used the fact that $(1+2u)^n=1$ when $n$ is even and $(1+2u)^n=(1+2u)$ when $n$ is odd, these two conditions are also true for the unit $(3+2u)$ in $R.$ Therefore, the study that we have done for $(1+2u)$-constacyclic codes of odd length over $R$ can be obtained analogously for $(3+2u).$ Also, one may generalized this study of constacyclic codes over the large ring $\mathbb{Z}_q+u\mathbb{Z}_q,$ where $q=p^m.$
\vspace{.3cm}
\begin{center}
|
{
"timestamp": "2015-04-15T02:06:10",
"yymm": "1504",
"arxiv_id": "1504.03445",
"language": "en",
"url": "https://arxiv.org/abs/1504.03445"
}
|
\section{Introduction} \label{Sec1}
Most organic or anorganic surfaces of mesoscopic objects (macromolecules
or colloids) become charged when immersed in polar solvents such as water.
These solvents provide favorable environments for free charges
(``counter-ions'' to the charged surfaces) which intermediate
an effective interaction among the mesoscopic objects.
At low temperatures, and in particular at $T=0$ when the system is in its
ground state, counter-ions between two charged plates crystallize into
bilayer Wigner structures which are important in understanding anomalous
phenomena such as like-charge attraction or overcharging
\cite{Lau,Grosberg,Levin,Naji,Samaj}.
Bilayer Wigner crystals describe several real physical systems in
condensed and soft matter, such as semiconductors \cite{Fil}, quantum dots
\cite{Imamura} and dusty plasmas \cite{Teng}.
Confined systems of charged colloidal particles were reviewed recently
\cite{review}, both from experimental and theoretical point of view.
From the particle models studied in this paper, we start with
the neutral Coulomb system of say elementary pointlike charges $-e$ with
$1/r$ interaction between two parallel plates of the same homogeneous surface
charge density $\sigma e$ at distance $d$, the phase diagram at $T=0$
depends on a single dimensionless parameter $\eta = d\sqrt{\sigma}$.
According to the Earnshaw theorem \cite{Earnshaw}, particles will stick
symmetrically on the surface of the plates.
Five distinct phases were detected to be stable, i.e. providing global
minimum of the energy, as $\eta$ is changing from 0 to $\infty$
\cite{Falko,Esfarjani,Goldoni,Schweigert,Weis}.
The lattice structures are the same on both plates and they are shifted
laterally with respect to one another.
Structures I, III and V are rigid (Fig. \ref{fig:Structures}), i.e. they
have fixed ($\eta$-independent) primary cells.
Structures II and IV are soft (Fig. \ref{fig:Structures24}),
the shape of their primary cells is varying with $\eta$.
\begin{figure}[thb]
\begin{center}
\includegraphics[clip,width=0.35\textwidth]{fig1.eps}
\caption{Rigid structures I, III and V of particles on two parallel plates;
open and filled symbols correspond to particle positions on the opposite
layers.}
\label{fig:Structures}
\end{center}
\end{figure}
\bigskip
\bigskip
\begin{figure}[thb]
\begin{center}
\includegraphics[clip,width=0.40\textwidth]{fig2.eps}
\caption{Soft structures II and IV.}
\label{fig:Structures24}
\end{center}
\end{figure}
Structures I, II and III correspond to the staggered rectangular lattice,
see Fig. \ref{fig:Structures24} left.
The primitive translation vectors of Bravais lattice are
\begin{equation} \label{structII}
{\bm a}_1 = a(1,0) , \quad {\bm a}_2 = a(0,\Delta) , \quad
a = \frac{1}{\sqrt{\sigma \Delta}} .
\end{equation}
The lattice spacing $a$ within one layer is determined by
the electroneutrality condition.
Two rectangular structures, one within each layer, are shifted
with respect to each other by the vector
\begin{equation} \label{vectorc}
{\bm c} = \alpha ({\bm a_1}+{\bm a_2})
\end{equation}
with $\alpha=1/2$.
The rigid structure I has the aspect ratio $\Delta=\sqrt{3}$ and it arises
for $\eta = 0$ because the two layers merge into a Wigner monolayer which
is known to be hexagonal (or, equivalently, equilateral triangular)
\cite{Bonsall77}.
Structure III consists of a square lattice with $\Delta=1$.
Phase II with $\sqrt{3}>\Delta>1$ interpolates continuously between
structures I and III.
Phase IV consists of two staggered rhombus lattices
(Fig. \ref{fig:Structures24} right).
One rhombus structure has the angle $\phi$ between the primitive translation
vectors
\begin{equation}
{\bm a}_1 = a(1,0) , \quad {\bm a}_2 = a(\cos\phi,\sin\phi) , \quad
a = \frac{1}{\sqrt{\sigma \sin\phi}} .
\end{equation}
In general, this phase has two variants according to the lateral shift
(\ref{vectorc}) between the opposite sublattices.
The version IVA has $\alpha=1/2$ whereas for IVB the shift parameter
$1/3<\alpha<1/2$.
For Coulomb bilayers, only phase IVA takes place.
Phase V corresponds to two shifted hexagonal lattices.
In a single layer, the elementary cell is the rhombus with the
primitive translation vectors
\begin{equation}
{\bm a}_1 = a(1,0) , \quad {\bm a}_2 = \frac{a}{2} (1,\sqrt{3}) , \quad
a = \frac{\sqrt{2}}{3^{1/4}\sqrt{\sigma}} .
\end{equation}
The lateral shift between the opposite lattices ${\bm c}$ is given by
(\ref{vectorc}) with $\alpha=1/3$.
The transitions between phases ${\rm II}\to {\rm III}$ and
${\rm III}\to {\rm IV}$ are continuous (of second order),
while the transition ${\rm IV}\to {\rm V}$ is discontinuous (of first order).
In order to describe these phase transitions, a new analytic approach
to Coulomb bilayers was proposed in Ref. \cite{Samaj1}.
Using a series of transformations with Jacobi theta functions, the
energy of the five phases was expressed as series of generalized Misra
functions which converge very quickly.
Near critical points, the generalized Misra functions can be expanded
easily in powers of the order parameter and the corresponding energies
posses the Ginsburg-Landau form.
This allows one to specify the critical points with an arbitrary
prescribed accuracy and to derive the mean-field critical behavior of
the order parameter.
Also the existence of phase I at $\eta=0$ only was confirmed.
This result was in contradiction with numerical approaches like
Ewald technique \cite{Goldoni} and Monte Carlo simulations \cite{Weis}
which predicted an extremely small, but finite, stability interval
of $\eta$'s for phase I.
Colloidal particles or particles in highly charged dusty plasmas usually
interact via Yukawa potential \cite{Nunomura} due to the Coulomb potential
screening by additional microions in the system.
The Yukawa pair potential of particles at distance $r$ is defined by
\begin{equation} \label{yukawa}
V(r)=V_0\ \frac{{\rm e}^{-\kappa r}}{\kappa r} ,
\end{equation}
where $\kappa$ is the inverse screening length and the amplitude
$V_0=Z^2\kappa\exp{(\kappa R)}/\epsilon(1+\kappa R)^2$, with $Z$ being
the charge of one particle and $\epsilon\approx \epsilon_0$ is
the permittivity for dusty plasma.
When $\kappa$ is large, $R$ becomes the radius of a hard sphere
as $V(r)\propto\exp{[\kappa (R-r)]}$ is exponentially large
for $r < R$ and negligible otherwise.
The relation $V_0\propto \kappa$ keeps the limit $\kappa\to 0$ of
Eq. (\ref{yukawa}) finite, yielding the proper Coulomb formula.
Thus the limiting cases $\kappa\to 0$ and $\kappa\to \infty$ correspond to
the unscreened Coulomb and hard-spheres interaction potentials, respectively.
We shall work in units of $V_0=1$.
For two parallel plates at distance $d$, the phase diagram depends on
two dimensionless parameters
\begin{equation}
\eta = \sqrt{\sigma} d , \qquad \lambda=\kappa d .
\end{equation}
A system of hard-sphere particles between two parallel hard plates was studied
by computer simulations in the past \cite{Murray,Schmidt,Neser,Fortini};
numerical methods were reviewed recently in Ref. \cite{Mazars11}.
For small values of $\eta$, the ground-state crystal structures involve
Wigner bilayers I-V, including phase IVB with two varying parameters
$\phi$ and $\alpha$.
For large values of $\eta$, phase-V bilayer transforms itself to crystalline
multilayers, with particles entering the region between the plates,
such as multiple square and hexagonal layers \cite{Murray},
rhombic \cite{Schmidt} and prism superlattices \cite{Neser}.
A similar phase diagram was obtained for the general Yukawa potential.
For small values of $\eta$, although Earnshaw theorem \cite{Earnshaw}
does not apply to Yukawa particles, the particles stick symmetrically
to plates and with increasing $\eta$ they constitute successively Wigner I-V
bilayers \cite{Messina}.
In the region of large values of $\eta$, in close analogy with confined
hard spheres, some of the particles will move in the interior of the domain
between the plates and create multilayers \cite{Oguz}.
In this paper, we shall concentrate on Wigner bilayers of pointlike
particles interacting via Yukawa potential.
The original numerical work of Messina and L\"owen \cite{Messina} determined
the phase diagram of the Yukawa system which exhibits single and double
reentrant transition.
We shall apply a straightforward extension of the recent analytic method
\cite{Samaj1} which provides us with high precision calculations
to shed more light on important tiny details of the phase diagram.
For any $\lambda$, the transition from phase I to II is shown to occur
directly at $\eta=0$, which solves a longtime controversy.
We recall that this scenario was anticipated only in the hard-spheres limit
$\lambda\to\infty$ \cite{Messina}.
We find a tricritical point where Messina and L\"owen suggested
a coexistence domain of several phases which should divide one staggered
rhombic phase into two separate regions.
Our calculations reveal one continuous region for this rhombic phase
with a narrow connecting channel.
Closed-form formulas for critical lines between various phases,
expressed in terms of generalized Misra functions, permit us to
determine the asymptotic Coulomb $\lambda\to 0$ and hard spheres
$\lambda\to\infty$ shapes of these lines.
The expansions of the structure energies around second-order transition
points and the determination of the order parameter can be done analytically,
which enables us to derive the critical behavior of the Ginzburg-Landau type.
In and close to the hard-spheres limit, the $\eta$-dependence of the internal
parameters of the phases is determined exactly.
The paper is organized as follows.
In Sec. \ref{Sec2}, we derive the expression for the energy per particle
of phase II (phases I and III being its special cases) in terms of
the generalized Misra functions.
The fact that going from phase I to II occurs at $\eta=0$ is shown
in Sec. \ref{Sec3}.
The second-order transition between phase II and III and the corresponding
mean-field critical behavior are described in detail in Sec. \ref{Sec4}.
The expression for the energy of phase IVB (with phases IVA and V as
its special cases) is derived in Sec. \ref{Sec5}.
The second-order transition between phases III and IVA is described
in Sec. \ref{Sec6}.
The first-order transitions between phases IVA-V, IVA-IVB and IVB-V
are discussed in Sec. \ref{Sec7}.
The dependence of the energy on the dimensionless distance $\eta$,
for fixed values of $\lambda$, is the subject of Sec. \ref{Sec8}.
The $\eta$-dependence of the internal structure parameters of phases
present in and near the hard-spheres limit is derived in Sec. \ref{Sec9}.
Sec. \ref{Sec10} is the Conclusion.
Auxiliary formulas for the generalized Misra functions and for the
critical lines are given in Appendices A-D.
\section{Energy of structures I, II and III} \label{Sec2}
We aim at deriving the interaction energy per particle $E_{\rm II}$ for
the structure II with the aspect ratio $\Delta$, phases I and III being
its special cases with $\Delta = \sqrt{3}$ and $\Delta=1$, respectively.
The energy consists of two parts: the intralayer energy $E_{\rm intra}$
sums the contributions from all particles in the same layer as
the reference one while the interlayer energy $E_{\rm inter}$ involves
all particles from the opposite layer.
To express the energy per particle as a quickly convergent series,
we shall apply a three-step method from Ref. \cite{Samaj1}.
The occupied lattice sites within one layer are numbered as
${\bm r} = j{\bm a}_1 + k {\bm a}_2$ where the primitive vectors
${\bm a}_1$ and ${\bm a}_2$ are defined in (\ref{structII}) and
$j, k$ run over all integers, except for the reference site $(0,0)$.
The intralayer interaction of a reference particle is thus given by
\begin{equation}
E_{\rm intra} = \frac{1}{2} \sum_{(j,k)\ne (0,0)}
\frac{\exp\left( -\kappa a\sqrt{j^2+k^2\Delta^2}\right)}{
\kappa a \sqrt{j^2+k^2\Delta^2}} .
\end{equation}
To evaluate lattice sums of Yukawa potentials, we shall often use
the integral representation (see e.g. \cite{Mazars11})
\begin{equation} \label{yukawatrans}
\frac{{\rm e}^{-\kappa r}}{\kappa r} = \frac{1}{\kappa\sqrt{\pi}}
\int_0^{\infty} \frac{{\rm d}t}{\sqrt{t}}
\exp\left( -\frac{\kappa^2}{4t}-r^2 t \right) .
\end{equation}
The intralayer energy per particle is then expressible as
\begin{eqnarray}
E_{\rm intra} = \frac{1}{2a\kappa\sqrt{\pi}}\int_0^\infty
\frac{{\rm d}t}{\sqrt{t}} {\rm e}^{-\frac{\kappa^2 a^2}{4t}}
\left( \sum_{j,k}{\rm e}^{-j^2t}{\rm e}^{-k^2\Delta^2t}-1 \right) \nonumber \\
= \frac{\eta}{2\sqrt{\pi}\lambda}\int_0^\infty \frac{{\rm d}t}{\sqrt{t}}
{\rm e}^{-\frac{\lambda^2}{4\eta^2 t}} \left[ \theta_3\left({\rm e}^{-t\Delta}\right)
\theta_3\left({\rm e}^{-\frac{t}{\Delta}}\right)-1\right] , \nonumber \\
\label{e2intra}
\end{eqnarray}
where we substituted $t\Delta\to t$ and introduced the Jacobi theta
function $\theta_3(q,0) \equiv \theta_3(q) = \sum_{j=-\infty}^{\infty} q^{j^2}$
\cite{Gradshteyn}.
The Wigner lattice on the opposite layer at distance $d$ is shifted by
the vector $({\bm a_1}+{\bm a_2})/2$.
The square of the distance between the reference particle and the particles
on the opposite layer becomes
$r_{jk}^2=(j-1/2)^2 a^2+(k-1/2)^2 a^2\Delta^2+d^2$.
Proceeding analogously as in the previous case,
we get for the interlayer energy
\begin{equation} \label{e2inter}
E_{\rm inter} =\frac{\eta}{2\sqrt{\pi}\lambda}
\int_0^\infty \frac{{\rm d}t}{\sqrt{t}}
{\rm e}^{-\frac{\lambda^2}{4\eta^2 t}-\eta^2 t}
\theta_2\left({\rm e}^{-t\Delta}\right)\theta_2
\left({\rm e}^{-t/\Delta} \right) ,
\end{equation}
where another Jacobi theta function $\theta_2(q)=\sum_j q^{(j-1/2)^2}$
was introduced.
The total energy per particle $E_{\rm II}$ is a sum $E_{\rm intra}+E_{\rm inter}$.
Using the Poisson summation formula
\begin{equation} \label{poisson}
\sum_{j=-\infty}^\infty {\rm e}^{-(j+\psi)^2 t}
= \sqrt{\frac{\pi}{t}}\sum_{j=-\infty}^{\infty}
{\rm e}^{2\pi {\rm i} j\psi}{\rm e}^{-(\pi j)^2 /t} ,
\end{equation}
it can be easily shown that in the limit $t\to 0$ the product of
theta functions $\theta_m({\rm e}^{-t})\theta_m({\rm e}^{-t})\approx \pi/t$
for both $m=2,3$.
In the unscreened Coulomb limit $\lambda\to 0$, this would lead to
the divergence of the corresponding integrals due to the lack of
the neutralizing background charge.
We ``artificially'' subtract the singular $\pi/t$ terms from the products
of theta functions and simultaneously add the same singular terms
and integrate them explicitly, with the result
\begin{eqnarray}
E_{\rm II} & = & \frac{\eta}{2\sqrt{\pi}\lambda}
\int_0^\infty \frac{{\rm d}t}{\sqrt{t}} {\rm e}^{-\frac{\lambda^2}{4\eta^2 t}}
\bigg\{ \Big[ \theta_3\left({\rm e}^{-t\Delta}\right)
\theta_3\left({\rm e}^{-t/\Delta}\right) \nonumber \\
& & - 1 -\frac{\pi}{t} \Big] +{\rm e}^{-\eta^2 t}
\left[\theta_2\left({\rm e}^{-t\Delta}\right)
\theta_2\left({\rm e}^{-t/\Delta} \right)-\frac{\pi}{t} \right] \bigg\}
\nonumber\\ & & + \pi \frac{\eta^2}{\lambda^2}\left(1+{\rm e}^{-\lambda}\right).
\label{e2tot}
\end{eqnarray}
This corresponds to adding and subtracting the background interaction energy
\cite{Samaj1}
\begin{equation} \label{eb}
E^{\rm B} = -\pi \frac{\eta^2}{\lambda^2} \left(1+{\rm e}^{-\lambda}\right) .
\end{equation}
The procedure is inevitable in the Coulomb $\lambda\to 0$ limit.
For a positive $\lambda>0$, the procedure is not necessary but it enhances
substantially the convergence properties of the obtained series.
The integration region $[0,\infty]$ in (\ref{e2intra}) can be split into
intervals $[0,\pi]$ and $[\pi,\infty]$.
Using the Poisson summation formula (\ref{poisson}),
the integral over $[\pi,\infty]$ can be rewritten as
\begin{eqnarray}
\int_\pi^\infty \frac{{\rm d}t}{\sqrt{t}} {\rm e}^{-\frac{\lambda^2}{4\eta^2 t}}
\left[ \theta_3\left({\rm e}^{-t\Delta}\right)
\theta_3\left({\rm e}^{-t/\Delta}\right)-1-\frac{\pi}{t}\right] \nonumber\\
= \int_\pi^\infty \frac{{\rm d}t}{\sqrt{t}} {\rm e}^{-\frac{\lambda^2}{4\eta^2 t}}
\left( \frac{\pi}{t}\sum_j {\rm e}^{-\frac{(\pi j)^2}{t\Delta}}
\sum_k {\rm e}^{-(\pi k)^2\frac{\Delta}{t}} -1-\frac{\pi}{t}\right) \nonumber\\
= \int_0^\pi \frac{\pi\ {\rm d}t'}{t'^{3/2}}
{\rm e}^{-\frac{\lambda^2 t'}{4\eta^2 \pi^2}}
\left( \frac{\pi}{t'}\sum_j {\rm e}^{-\frac{j^2 t'}{\Delta}}
\sum_k {\rm e}^{- k^2 t'\Delta}-1-\frac{t'}{\pi} \right) \nonumber\\
= \int_0^\pi \frac{{\rm d}t}{\sqrt{t}} {\rm e}^{-\frac{\lambda^2 t}{4\eta^2 \pi^2}}
\left[\theta_3\left({\rm e}^{-t\Delta}\right)
\theta_3\left({\rm e}^{-t/\Delta}\right)-1-\frac{\pi}{t} \right]. \label{eq11}
\end{eqnarray}
Similarly,
\begin{eqnarray}
& & \int_\pi^\infty \frac{{\rm d}t}{\sqrt{t}}
{\rm e}^{-\frac{\lambda^2}{4\eta^2 t}}{\rm e}^{-\eta^2 t}
\left[\theta_2\left({\rm e}^{-t\Delta}\right)
\theta_2\left({\rm e}^{-\frac{t}{\Delta}}\right)-\frac{\pi}{t}\right] \nonumber\\
& = & \int_0^\pi\frac{{\rm d}t}{\sqrt{t}}
{\rm e}^{-\frac{\lambda^2 t}{4\eta^2 \pi^2}-\frac{\eta^2 \pi^2}{t}}
\left[\theta_4\left({\rm e}^{-t\Delta}\right)
\theta_4\left({\rm e}^{-\frac{t}{\Delta}}\right)-1\right],
\phantom{aaa} \label{eq12}
\end{eqnarray}
where we introduced the Jacobi theta function
$\theta_4(q)=\sum_j (-1)^j q^{j^2}$.
Finally, in close analogy with Ref. \cite{Samaj1} we apply once more
the Poisson summation formula (\ref{poisson}) for each term
in the integration from $[0,\pi]$.
The final formula for the energy reads
\begin{widetext}
\begin{eqnarray}
E_{\rm II} & = & \frac{\eta}{2\sqrt{\pi}\lambda}\bigg\{2\sum_{j=1}^\infty\left[
z_{3/2}\left(0,\lambda^2/(4 \pi^2 \eta^2)+j^2\Delta\right)
+z_{3/2}\left(0,\lambda^2/(4 \pi^2 \eta^2)+j^2/\Delta\right)\right]
-\pi z_{1/2}\left(0,\frac{\lambda^2}{4\eta^2\pi^2}\right) \nonumber\\
& & + 2\sum_{j=1}^\infty(-1)^j\left[
z_{3/2}\left(\pi^2\eta^2,\lambda^2/(4 \pi^2 \eta^2)+j^2\Delta\right)
+z_{3/2}\left(\pi^2\eta^2,\lambda^2/(4 \pi^2 \eta^2)+j^2/\Delta\right)\right]
-\pi z_{1/2}\left(\lambda^2/(4\eta^2),0\right) \nonumber\\
& & + 4\sum_{j,k=1}^\infty (-1)^j(-1)^k
z_{3/2}\left(\pi^2\eta^2,\lambda^2/(4 \pi^2 \eta^2)+j^2/\Delta+k^2\Delta\right)
+4\sum_{j,k=1}^\infty
z_{3/2}\left(0,\lambda^2/(4 \pi^2 \eta^2)+j^2/\Delta+k^2\Delta\right) \nonumber\\
& & + 2\sum_{j=1}^\infty(-1)^j\left[
z_{3/2}\left(\pi^2\eta^2,\lambda^2/(4 \pi^2 \eta^2)+j^2\Delta\right)
+z_{3/2}\left(\pi^2\eta^2,\lambda^2/(4 \pi^2 \eta^2)+j^2/\Delta\right)\right]
-\pi z_{1/2}\left(\lambda^2/(4\eta^2),\eta^2\right) \nonumber\\
& & + 2\sum_{j=1}^\infty\left[
z_{3/2}\left(\lambda^2/(4\eta^2),j^2\Delta\right)+
z_{3/2}\left(\lambda^2/(4\eta^2),j^2/\Delta\right)\right]
+4\sum_{j,k=1}^\infty z_{3/2}\left(\lambda^2/(4\eta^2),j^2/\Delta+k^2\Delta\right)
\nonumber\\
& & + 4\sum_{j,k=1}^\infty z_{3/2}\left[\lambda^2/(4\eta^2),\eta^2+(j-1/2)^2/\Delta
+(k-1/2)^2\Delta\right]\bigg\}
+\pi \frac{\eta^2}{\lambda^2}\left(1+{\rm e}^{-\lambda}\right). \label{e2}
\end{eqnarray}
\end{widetext}
Here, we introduced the function
\begin{equation} \label{znu}
z_{\nu}(x,y)=\int_0^{1/\pi} \frac{{\rm d}t}{t^{\nu}}{\rm e}^{-xt}{\rm e}^{-y/t} .
\end{equation}
It is a generalization of the well-known Misra function \cite{Misra},
corresponding to $x=0$, commonly used in lattice summations.
The functions $z_{\nu}(x,y)$ with half-integer values of $\nu$ can be
expressed in terms of the complementary error function, see Appendix A.
This permits us to use very effectively the MATHEMATICA software and
to derive in Appendix A their asymptotic forms for ($x$ finite, $y\to\infty$)
and ($y$ finite, $x\to\infty$).
The series in the generalized Misra function (\ref{e2}) is quickly converging;
for the known $\lambda=0$ Coulomb cases \cite{Samaj1}, the truncation of
the series over $j,k$ at $M=1,2,3,4$ reproduces the exact value of the energy
up to $2,5,10,17$ decimal digits, respectively.
This accuracy even improves itself for $\lambda>0$, so in our numerical
calculations we keep the truncation of the series at $M=5$.
The formula (\ref{e2}) is symmetric with respect to the transformation
$\Delta\to 1/\Delta$.
This symmetry corresponds to an obvious invariance of the energy with respect
to the lattice rotation around one point by $90$ degrees.
\section{Going from phase I to II} \label{Sec3}
As was mentioned in Introduction, numerical approaches \cite{Goldoni,Weis}
predicted that phase I has a region of stability $[0,\tilde{\eta}]$ with
a very small $\tilde{\eta}>0$ and there is a second-order transition between
phases I and II.
This small region was expected to vanish ($\tilde{\eta}=0$) in
the hard-spheres limit $\lambda\to\infty$ \cite{Messina}.
But in the paper \cite{Samaj1} it was shown both analytically and
numerically that $\tilde{\eta}=0$ in the unscreened Coulomb limit
$\lambda\to 0$, i.e. phase I exists only for $\eta=0$.
There is no singularity in the ground-state energy, so going from
phase I to phase II is not a phase transition in the usual sense.
In what follows, we derive the same results for any positive $\lambda$.
We know that $\Delta=\sqrt{3}$ for phase I at $\eta=0$.
Let us assume that for $\eta>0$ we have $\Delta=\sqrt{3}-\epsilon$
with a small $\epsilon$ and, in close analogy with Ref. \cite{Samaj1},
expand the energy (\ref{e2}) in Taylor series:
\begin{eqnarray}
E_{\rm II}(\sqrt{3}-\epsilon,\eta,\lambda) & = &
E_{\rm II}(\sqrt{3},\eta,\lambda) + f_1(\eta,\lambda)\epsilon \nonumber \\
& & + f_2(\eta,\lambda)\epsilon^2 + {\cal O}(\epsilon^3) , \label{e21}
\end{eqnarray}
where the expansion functions $f_1(\eta,\lambda)$ and $f_2(\eta,\lambda)$
are written explicitly in terms of the generalized Misra functions
in Appendix B.
For given $\eta$ and $\lambda$, the extremum of the energy (\ref{e21}) occurs
at $\epsilon^*$ given by
\begin{equation} \label{eps21eq}
\frac{\partial}{\partial \epsilon} E_{\rm II}(\sqrt{3}-\epsilon,\eta,\lambda)
\Big\vert_{\epsilon=\epsilon^*} \approx f_1(\eta,\lambda) +
2 f_2(\eta,\lambda)\epsilon^* = 0,
\end{equation}
implying
\begin{equation} \label{eps21}
\epsilon^*(\eta,\lambda) \equiv \sqrt{3} - \Delta^*(\eta,\lambda) =
-\frac{f_1(\eta,\lambda)}{2 f_2(\eta,\lambda)} .
\end{equation}
For the unscreened Coulomb case $\lambda=0$ it has been shown in
\cite{Samaj1} that
\begin{equation} \label{Coulombas}
\sqrt{3} - \Delta^*(\eta,0) = - \frac{f_1(\eta,0)}{2 f_2(\eta,0)}
= 7.14064\ldots \eta^2 + {\cal O}(\eta^4) .
\end{equation}
This extremum is the minimum of $E_{\rm II}(\epsilon)$.
In the case of $\lambda>0$, it is shown in Appendix B that for
$\eta\ll\lambda$ the coefficient functions can be approximated by
\begin{eqnarray}
f_1(\eta,\lambda) & \approx & -\frac{\eta\lambda}{3^{1/4}\ 4}
{\rm e}^{-\frac{\lambda}{3^{1/4}\eta}} +
{\cal O}\left(\eta^2 {\rm e}^{-\frac{\lambda}{3^{1/4}\eta}}\right), \nonumber\\
f_2(\eta,\lambda) & \approx & \frac{\lambda}{3^{1/4}\ 16\eta}
{\rm e}^{-\frac{\lambda}{3^{1/4}\eta}} +
{\cal O}\left(\eta {\rm e}^{-\frac{\lambda}{3^{1/4}\eta}}\right). \label{f1f2}
\end{eqnarray}
The extremum
\begin{equation} \label{extr}
\sqrt{3} - \Delta^*(\eta,\lambda)
= - \frac{f_1(\eta,\lambda)}{2 f_2(\eta,\lambda)}
= 2 \eta^2 + {\cal O}(\eta^4)
\end{equation}
interestingly does not depend in the leading order on $\lambda$.
It corresponds to the minimum of energy $E_{\rm II}(\epsilon)$ as
${\partial^2_{\epsilon}} E_{\rm II}(\sqrt{3}-\epsilon,\eta,\lambda)
\vert_{\epsilon=\epsilon^*} = 2 f_2(\eta,\lambda) > 0$.
For $\lambda=1$ and $\eta=0.01$, we checked the result (\ref{extr})
numerically in Fig. \ref{fig:trans12}.
One can see that $E_{\rm II}(\epsilon)$, calculated using the complete formula
(\ref{e2}) truncated at $M=5$ has a minimum rather close to the value
$\epsilon^*=0.0002$ predicted by our asymptotic formula (\ref{eps21}).
\begin{figure}[thb]
\begin{center}
\includegraphics[clip,width=0.42\textwidth]{fig3.eps}
\caption{$E_{\rm II}(\Delta,\eta)-E_{\rm I}(\eta)$ as a function of
$\sqrt{3}-\Delta$ for the fixed values of $\lambda=1$ and $\eta=0.01$.
The value of $\epsilon^*=0.0002$, which provides the energy minimum
according to the asymptotic formula (\ref{extr}) is depicted by the
vertical dashed line for comparison.
Note that the energy differences are extremely small.}
\label{fig:trans12}
\end{center}
\end{figure}
We conclude that phase I is stable only at $\eta=0$ and for
an arbitrarily small positive $\eta$ we enter the region of phase II.
It is interesting that the asymptotic $\eta\to 0$ predictions for
the unscreened Coulomb $\lambda=0$ case (\ref{Coulombas}) and for
$\lambda>0$ (\ref{extr}) exhibit the same $\eta^2$ dependence, but
there is a skip in the prefactors from $7.14064\ldots$ at $\lambda=0$
to $2$ for $\lambda>0$.
The fact that in the previous works \cite{Goldoni,Weis,Messina}
phase I was detected also for small positive values of $\eta$ is
probably related to extremely small deviation of
$\sqrt{3}-\Delta^*\propto \eta^2$ which are ``invisible'' by
standard numerical methods.
\section{Second-order transition between phases II and III} \label{Sec4}
Let us parametrize $\Delta=\exp({\epsilon})$.
The symmetry $\Delta\to 1/\Delta$ of the energy (\ref{e2}) is then
equivalent to the transformation $\epsilon\to -\epsilon$ and the energy
is an even function of $\epsilon$.
The Ginsburg-Landau form of its expansion around $\epsilon=0$ reads as
\begin{equation} \label{e2gl}
E_{\rm II}({\rm e}^\epsilon,\eta,\lambda) = E_{\rm III}(\eta,\lambda)
+ g_2(\eta,\lambda)\epsilon^2 + g_4(\eta,\lambda)\epsilon^4 + \ldots
\end{equation}
The explicit expression for $g_2$ is given in Appendix C and
a rather cumbersome expression for $g_4$ is also at our disposal.
The critical point is given by the vanishing of the prefactor
\begin{equation} \label{critline}
g_2(\eta^c,\lambda^c) = 0 .
\end{equation}
We used this equation to get the (dashed) critical line between phases II
and III in Fig. \ref{fig:phd}.
Our definition of $\eta$ differs from that of the dimensionless distance
in the paper of Messina and L\"owen \cite{Messina}, namely
$\eta^2 = \eta_{\rm ML}$.
To maintain the full comparability, we shall present the phase diagram
using the variable $\eta^2$.
\begin{figure}[thb]
\begin{center}
\includegraphics[clip,width=0.45\textwidth]{fig4.eps}
\caption{Phase diagram of the Yukawa bilayer. Dashed lines denote
the second-order phase transitions, solid lines correspond
to the first-order phase transitions.
The important data for the hard-sphere limit $\lambda\to\infty$
are added on the top.}
\label{fig:phd}
\end{center}
\end{figure}
\subsection{Critical behavior}
To obtain the critical behavior, we note that the functions $g_2$ and
$g_4$ in Eq. (\ref{e2gl}) behave in the vicinity of the critical point
$(\eta^c,\lambda^c)$ as follows
\begin{eqnarray}
g_2(\eta,\lambda) & = & g_{21}(\lambda^c)(\eta^c-\eta)
+ {\cal O}[(\eta^c-\eta)^2], \nonumber \\
g_4(\eta,\lambda) & = & g_{40}(\lambda^c)
+ {\cal O}(\eta^c-\eta), \label{g2g4}
\end{eqnarray}
where $g_{21}(\lambda^c)<0$ and $g_{40}(\lambda^c)>0$ for all $\lambda^c$.
The minimum energy is reached at $\epsilon^*\approx\Delta^*-1$ given by
\begin{equation} \label{epsceq}
\frac{\partial}{\partial \epsilon} E_{\rm II}({\rm e}^\epsilon,\eta,\lambda)
\big\vert_{\epsilon=\epsilon^*}
\approx 2 g_2(\eta,\lambda)\epsilon^* +4 g_4(\eta,\lambda)(\epsilon^*)^3 = 0 .
\end{equation}
For $\eta>\eta^c$, there is only one solution $\epsilon^*=0$ which
corresponds to the square lattice of phase III.
For $\eta<\eta^c$, we get one trivial (unphysical) solution $\epsilon^* = 0$
and two non-trivial conjugate solutions $\pm \epsilon^*$ with
\begin{equation} \label{epsc}
\epsilon^* = \left(-\frac{g_2(\eta,\lambda)}{2 g_4(\eta,\lambda)}\right)^{1/2}
\approx \left(-\frac{g_{21}(\lambda^c)}{2 g_{40}(\lambda^c)}\right)^{1/2}
\sqrt{\eta^c-\eta} ,
\end{equation}
$\eta\to(\eta^c)^-$.
The order parameter $\epsilon^*\propto\sqrt{\eta^c-\eta}$ is thus associated
with the mean-field critical index $\beta_{\rm MF}=1/2$ for every
$\lambda\ge 0$.
The dependence of $\Delta-1$ on $\eta^c-\eta$ is shown in
Fig. \ref{fig:trans23} for three values of $\lambda=1,10,100$.
Near the critical point ($\eta^c-\eta$ small), the asymptotic
relation (\ref{epsc}) (dashed lines) fits perfectly the numerical data
from minimization of the energy $E_{\rm II}$ (\ref{e2tot}) (full lines).
In the logarithmic plot, for all values of $\lambda$ the slope of $\Delta-1$
vs. $\eta^c-\eta$ is very close to 0.5 in the region of small and
intermediate values of $\eta^c-\eta$, confirming the value $1/2$ of
the mean-field critical index $\beta_{\rm MF}$ for all values of $\lambda$.
\begin{figure}[thb]
\begin{center}
\includegraphics[clip,width=0.44\textwidth]{fig5.eps}
\caption{Order parameter close to the critical point of the transition II-III
for three values of $\lambda=1,10,100$.
Full lines follow from numerical minimization of the energy (\ref{e2}).
The slope of lines is close to $\beta_{\rm MF} = 1/2$.
Dashed lines represent the asymptotic $\eta\to (\eta^c)^-$
relation (\ref{epsc}).}
\label{fig:trans23}
\end{center}
\end{figure}
From Eq. (\ref{e2gl}), the energy difference of phases
II and III close to the critical point is given by
\begin{equation}
E_{\rm II}({\rm e}^\epsilon,\eta,\lambda) - E_{\rm III}(\eta,\lambda)
\sim - \frac{g_{21}^2(\lambda^c)}{4 g_{40}(\lambda^c)} (\eta^c - \eta)^2 .
\end{equation}
The critical singularity should be of type $(\eta^c - \eta)^{2-\alpha}$
implying the mean-field critical index $\alpha_{\rm MF}=0$ for any $\lambda$.
To obtain another two critical indices, we add to the energy (\ref{e2gl})
the symmetry-breaking term $-h \epsilon$, where a small positive external
field $h\to 0^+$ is linearly coupled to the order parameter.
The optimization condition for the energy with respect to $\epsilon$
now takes the form
\begin{equation} \label{hcrit}
2 g_2(\eta,\lambda) \epsilon^* + 4 g_4(\eta,\lambda) (\epsilon^*)^3 - h = 0 .
\end{equation}
At the critical point, since $g_2(\eta^c,\lambda^c)=0$ and
$g_4(\eta^c,\lambda^c)= g_{40}(\lambda^c)$, we find from (\ref{hcrit}) that
\begin{equation}
\epsilon^* = \left[ \frac{h}{4 g_{40}(\lambda^c)} \right]^{1/3} .
\end{equation}
This critical singularity should be of type $h^{1/\delta}$,
which leads to the mean-field critical index $\delta_{\rm MF}=3$
for any $\lambda$.
Performing the derivative of Eq. (\ref{hcrit}) with respect to $h$,
we find for the field succeptibility close to the critical point:
\begin{equation}
\frac{\partial \epsilon^*}{\partial h} \Bigg\vert_{h=0} =
\frac{1}{- 4 g_{21}(\lambda^c)} \frac{1}{\eta^c-\eta} , \qquad
\eta\to(\eta^c)^- .
\end{equation}
The corresponding critical singularity $(\eta - \eta^c)^{-\gamma}$ leads to
the mean-field critical index $\gamma_{\rm MF}=1$ for arbitrary $\lambda$.
It is easy to verify that our mean-field critical indices
\begin{equation} \label{MFind}
\alpha_{\rm MF} = 0, \quad \beta_{\rm MF} = \frac{1}{2}, \quad
\gamma_{\rm MF} = 1, \quad \delta_{\rm MF} = 3
\end{equation}
fulfill two standard scaling relations \cite{Ma}
\begin{equation}
2-\alpha = 2\beta + \gamma = \beta (\delta+1) .
\end{equation}
Since there are no fluctuations in our system at zero temperature,
the critical indices $\eta$ and $\nu$, related to the particle
correlation function, are not defined.
\subsection{Coulomb $\lambda\to 0$ limit of the critical line}
We reproduce $\eta^c(0)=0.2627602682$ \cite{Samaj1}
in the Coulomb $\lambda\to 0$ limit.
It is shown in Appendix C that the asymptotic $\lambda\to 0$ shape of
the critical line between phases II and III is parabolic:
\begin{equation} \label{le23}
\lambda^2 \approx c_{23}[\eta^c-\eta^c(0)] , \qquad c_{23}\approx 24.173744.
\end{equation}
This formula is compared to the critical line evaluated numerically
by using the relation (\ref{critline}) in Fig. \ref{fig:trans23ll}.
\begin{figure}[thb]
\begin{center}
\includegraphics[clip,width=0.44\textwidth]{fig6.eps}
\caption{The critical line between phases II and III near the unscreened
Coulomb $\lambda\to 0$ limit.
Full line follows from the numerical evaluation by using
the relation $g_2(\eta,\lambda)=0$.
Dashed line corresponds to the asymptotic formula (\ref{le23}).}
\label{fig:trans23ll}
\end{center}
\end{figure}
\subsection{Hard-spheres $\lambda\to \infty$ limit of the critical line}
In the hard-spheres limit $\lambda\to\infty$, the critical point for
the ${\rm II}\to {\rm III}$ transition is $(\eta^c)^2\to 1/2$ \cite{Messina}.
The convergence to this value is extraordinarily slow.
Let us analyze this limit in the critical equation $g_2(\eta,\lambda)=0$.
Applying the asymptotic formulas for the generalized Misra functions
(\ref{znuasym}) and (\ref{znuasymp}) to $g_2(\eta,\lambda)$ given by
the series (\ref{g2}), most summands become exponentially small
compared to the few leading terms proportional to $\exp(-\lambda/\eta)$.
In particular, we can neglect completely the first four sums in Eq. (\ref{g2})
since all terms behave as $\exp(-c\lambda^2)$ and the sixth sum because
we get at least $\exp(-\sqrt{2}\lambda/\eta)$ for the $j=k=1$ term.
Those leading terms appear in the fifth sum with $j=1$ and the seventh sum
with $j=k=1$.
The ones with e. g. $j=2$, $k=1$ etc. are exponentially small again compared
to the leading ones.
In the last sum the $z_{7/2}(.,.)$ term has zero prefactor for $j=k$.
We are left with the three-terms expression
\begin{eqnarray}
g_2(\eta,\lambda) & \approx & z_{7/2}\left(\frac{\lambda^2}{4\eta^2},1\right)
-z_{5/2}\left(\frac{\lambda^2}{4\eta^2},1\right) \nonumber \\ & &
-\frac{1}{2}z_{5/2}\left(\frac{\lambda^2}{4\eta^2},\eta^2+\frac{1}{2}\right),
\qquad \lambda \gg 1 . \label{g2as0}
\end{eqnarray}
Applying the asymptotic formula (\ref{znuasymp}), we rewrite the rhs of
this expression as
\begin{equation} \label{g2as2}
\frac{\sqrt{\pi}\lambda {\rm e}^{-\frac{\lambda}{\eta}}}{4\eta}
\left[\frac{\lambda}{\eta}+1+\frac{\eta}{\lambda} -
\frac{1+\frac{\eta}{\lambda\sqrt{\eta^2+\frac{1}{2}}}}{\eta^2+\frac{1}{2}}
{\rm e}^{\frac{\lambda}{\eta}\left(1-\sqrt{\eta^2+\frac{1}{2}}\right)}\right].
\end{equation}
The critical condition $g_2(\eta,\lambda) = 0$ implies a transcendental
formula for $\eta(\lambda)$:
\begin{equation} \label{g2as2root}
\frac{\left(\frac{\lambda}{\eta}+1+\frac{\eta}{\lambda}\right)
\left(\eta^2+\frac{1}{2}\right)}{1+
\frac{\eta}{\lambda\sqrt{\eta^2+\frac{1}{2}}}} =
{\rm e}^{\frac{\lambda}{\eta}\left(1-\sqrt{\eta^2+\frac{1}{2}}\right)} .
\end{equation}
The exponential term can equal to the rational one only if
$1-\sqrt{\eta^2+1/2}$ is close to zero, i. e. $\eta^2\to 1/2$
in the $\lambda\to\infty$ limit as expected.
The next terms of the large-$\lambda$ expansion of $\eta(\lambda)$
can be derived straightforwardly, with the result
\begin{equation} \label{e23as}
\eta\approx \frac{1}{\sqrt{2}}-\frac{\ln{\lambda}}{\lambda}
-\frac{\ln{2}}{2\lambda}+{\cal O}\left(\frac{\ln^2{\lambda}}{\lambda^2}\right).
\end{equation}
In general, the series contains the terms of the form
$(\ln \lambda)^m/\lambda^n$ where $m$, $n$ are integers such that
$0\le m\le n$.
The first correction of type $(\ln\lambda)/\lambda$ explains a slow
convergence of the results as $\lambda\to\infty$.
\begin{figure}[thb]
\begin{center}
\includegraphics[clip,width=0.41\textwidth,height=0.43\textwidth]{fig7.eps}
\caption{An excerption of the phase diagram for Yukawa particles
for the second-order phase transitions ${\rm II}\to {\rm III}$
and ${\rm III}\to {\rm IVA}$.
The solid lines denote the critical lines obtained numerically by using
Eq. (\ref{critline}) and (\ref{crith}), respectively.
The dash-dotted lines correspond to the asymptotic large-$\lambda$
formulas (\ref{e23as}) and (\ref{e34aas}).}
\label{fig:phd2}
\end{center}
\end{figure}
The asymptotic formula (\ref{e23as}), taken for $\eta^2$, is plotted
in Fig. \ref{fig:phd2} by the dash-dotted line.
We see that it reproduces adequately the numerical results for
the critical line (solid line) in a large region of the phase diagram.
It can be shown that the next term of the series (\ref{e23as}) reads as
$3 \ln^2\lambda/(2^{3/2}\lambda^2)$; plotting the asymptotic formula
(\ref{e23as}) with this term included makes the difference with the numerical
solid line invisible by eye.
\section{Energy for structures IVA, IVB and V} \label{Sec5}
It was already mentioned that structures IVA, V and even III are
special cases of the most general phase IVB.
Hence we will sketch the derivation of the energy per particle for the latter.
The elementary cell is a rhombus with the angle $\phi$ between the vectors
${\bm a_1}$ and ${\bm a_2}$ of the same magnitude $a$,
see Fig. \ref{fig:Structures24}.
The density of particles on one plate is $\sigma=1/(a^2\sin\phi)$.
We will prefer the parametrization of the angle by $\delta=\tan (\phi/2)$.
Another free parameter is $\alpha\in [1/3,1/2]$ measuring the diagonal shift
${\bm c}$ of the lattice on the opposite layer, see formula(\ref{vectorc}).
The square of the lattice vector can be written as
$\vert{\bm r}_{jk}\vert^2=a^2[(j+k)^2\cos^2(\phi/2)+(j-k)^2\sin^2(\phi/2)]$.
Next we distinguish the cases when $j+k$ is an even or odd integer and
go to the summation over new indices $m$ and $n$; details of this
technicality and of the next steps can be found in Sec. III of
paper \cite{Samaj1}.
The main difference is that we get $(n+\alpha)^2$ and $(n-1/2+\alpha)^2$
instead of $n^2$ and $(n-1/2)^2$ for the interlayer contribution.
Applying the Poisson formula (\ref{poisson}) creates additional factors
$\exp(2\pi i n\alpha)$ and $\exp[2\pi i n(\alpha-1/2)]$.
Reducing the summation over $\{-\infty,\infty\}$ to $\{1,\infty\}$ turns
these factors to $2\cos(2\pi n\alpha)$ and $2\cos[2\pi n(\alpha-1/2)]$,
respectively.
The final formula for the energy per particle of phase IVB reads
\begin{widetext}
\begin{eqnarray}
E_{\rm IVB} & = & \frac{\eta}{2\sqrt{2\pi}\lambda}\Bigg(
2\sum_{j=1}^\infty\left[
z_{3/2}\left(0,\frac{\lambda^2}{2 \pi^2 \eta^2}+j^2\delta\right)
+z_{3/2}\left(0,\frac{\lambda^2}{2 \pi^2 \eta^2}+j^2/\delta\right)\right]
\bigg[1+(-1)^j\bigg] \nonumber\\
& & + 4\sum_{j,k=1}^\infty \left[ 1+(-1)^{j+k} \right]
z_{3/2}\left(0,\frac{\lambda^2}{2 \pi^2 \eta^2}+j^2/\delta+k^2\delta\right)
-\pi z_{1/2}\left(0,\frac{\lambda^2}{2\eta^2\pi^2}\right) \nonumber\\
& & + 2\sum_{j=1}^\infty\left[\cos(2\pi j\alpha)
z_{3/2}\left(\pi^2\eta^2/2,\frac{\lambda^2}{2 \pi^2 \eta^2}+j^2\delta\right)
+z_{3/2}\left(\pi^2\eta^2/2,\frac{\lambda^2}{2 \pi^2 \eta^2}+j^2/\delta\right)
\right] \nonumber\\ & &
+ 2\sum_{j=1}^\infty\left\{\cos\left[ 2\pi j\left(\alpha-\frac{1}{2}\right)\right]
z_{3/2}\left(\pi^2\eta^2/2,\frac{\lambda^2}{2 \pi^2 \eta^2}+j^2\delta\right)
+(-1)^j z_{3/2}\left(\pi^2\eta^2/2,\frac{\lambda^2}{2 \pi^2 \eta^2}+j^2/\delta
\right)\right\} \nonumber\\ & &
+ 4\sum_{j,k=1}^\infty\left\{\cos(2\pi j\alpha)
+\cos\left[2\pi j\left(\alpha-\frac{1}{2}\right)\right] (-1)^k\right\}
z_{3/2}\left(\pi^2 \eta^2/2,\frac{\lambda^2}{2 \pi^2 \eta^2}+j^2 \delta
+k^2/\delta\right) \nonumber\\ & &
+ 2\sum_{j=1}^\infty\left[ z_{3/2}\left(\frac{\lambda^2}{2\eta^2},j^2\delta\right)+
z_{3/2}\left(\frac{\lambda^2}{2\eta^2},j^2/\delta\right)\right]
+4\sum_{j,k=1}^\infty z_{3/2}\left(\frac{\lambda^2}{2\eta^2},j^2/\delta
+k^2\delta\right) -2\pi z_{1/2}\left(\frac{\lambda^2}{2\eta^2},0\right)
\nonumber\\ & & + \sum_{j,k=-\infty}^\infty
\left\{z_{3/2}\left[\frac{\lambda^2}{2\eta^2},\eta^2/2+\frac{1}{\delta}
(j+\alpha)^2+k^2\delta\right] + z_{3/2}\left[\frac{\lambda^2}{2\eta^2},
\eta^2/2+\frac{1}{\delta}(j+\alpha-1/2)^2+(k-1/2)^2\delta\right]\right\}
\nonumber\\ & & - 2\pi z_{1/2}\left(\frac{\lambda^2}{2\eta^2},\eta^2/2\right)
+ 4 \sum_{j,k=1}^\infty z_{3/2}\left[\frac{\lambda^2}{2\eta^2},
\frac{1}{\delta}\left(j-\frac{1}{2}\right)^2
+\left(k-\frac{1}{2}\right)^2\delta\right]\Bigg)
+\pi \frac{\eta^2}{\lambda^2}\left(1+{\rm e}^{-\lambda}\right). \label{e4b}
\end{eqnarray}
\end{widetext}
\section{Transition between phases III and IVA} \label{Sec6}
One can verify that for the structure IVA with $\alpha=1/2$
the energy (\ref{e4b}) possesses the symmetry $\delta\to 1/\delta$.
The case $\delta=1$ or $\phi=\pi/2$ is the fixed point of the transformation
$\delta\to 1/\delta$ and corresponds to the critical point between
phases III and IVA.
In full analogy with the transition between phases II and III,
we parametrize $\delta=\exp(-\epsilon)$ so that the energy of phase IVA
becomes an even function of $\epsilon$.
The expansion of the energy (\ref{e4b}) around the critical point $\delta=1$
in powers of small $\epsilon$ takes the form
\begin{eqnarray}
E_{\rm IVA}({\rm e}^{-\epsilon},\eta,\lambda) & = & E_{\rm III}(1,\eta,\lambda)
+ h_2(\eta,\lambda) \epsilon^2 \nonumber \\
& & + h_4(\eta,\lambda)\epsilon^4 + \ldots . \label{e4agl}
\end{eqnarray}
The explicit formula for $h_2$ in terms of the generalized Misra functions
is presented in Appendix D and $h_4$ is also at our disposal.
The critical line between phases III and IVA is once again given by
vanishing of the prefactor
\begin{equation} \label{crith}
h_2(\eta^c,\lambda^c) = 0 ,
\end{equation}
see Figs. \ref{fig:phd} and \ref{fig:phd2}.
\subsection{Critical behavior}
The expansion of the coefficients $h_2$ and $h_4$ around the critical point
$(\eta^c,\lambda^c)$ is analogous to the previous case of the
second-order transition between phases II and III.
The leading terms are $h_2(\eta,\lambda)\approx h_{21}(\lambda^c)(\eta-\eta^c)$
and $h_4(\eta,\lambda)\approx h_{40}(\lambda^c)$, where
$h_{21}(\lambda^c)<0$ and $h_{40}(\lambda^c)>0$ for all $\lambda_c$.
Optimizing the energy $E_{\rm IVA}$ with respect to $\epsilon$,
the stationary solution $\epsilon^* = 1-\delta^*$ behaves as
\begin{equation} \label{epsc34}
\epsilon^*=\left(-\frac{h_2(\eta,\lambda)}{2 h_4(\eta,\lambda)}\right)^{1/2}
\approx \left(-\frac{h_{21}(\lambda^c)}{2 h_{40}(\lambda^c)}\right)^{1/2}
\sqrt{\eta - \eta^c}
\end{equation}
with $\eta\to (\eta^c)^+$.
The order parameter $\epsilon^*$ has again the singular behavior
of mean-field type with critical index $\beta_{\rm MF} = 1/2$.
We tested this results numerically in a plot analogous to
Fig. \ref{fig:trans23} and got the slope $\beta\approx 0.499$.
Without going into details, also other critical indices attain their
mean-field values (\ref{MFind}).
\subsection{Coulomb $\lambda\to 0$ limit of the critical line}
The week screening (small $\lambda$) case of the phase transitions III-IVA
and IVA-V was studied by Monte Carlo methods in Ref. \cite{Mazars08}.
We reproduce $\eta^c(0) = 0.6214809246$ \cite{Samaj1} in the Coulomb
$\lambda\to 0$ limit.
The asymptotic shape of the critical line for small $\lambda$ is again
parabolic, see Appendix D:
\begin{equation} \label{le34a}
\lambda^2\approx c_{34}[\eta^c-\eta^c(0)],\qquad c_{34}\approx 149.7837254 .
\end{equation}
This asymptotic result is compared with the numerical calculation of
the critical line directly from the relation (\ref{crith})
in Fig. \ref{fig:trans34all}.
\begin{figure}[thb]
\begin{center}
\includegraphics[clip,width=0.44\textwidth]{fig8.eps}
\caption{Transition III-IVA near the Coulomb $\lambda\to 0$ limit.
The full line follows from the numerical treatment of
the relation $h_2(\eta,\lambda)=0$.
The dashed line corresponds to the asymptotic formula (\ref{le34a}).}
\label{fig:trans34all}
\end{center}
\end{figure}
\subsection{Hard-spheres $\lambda\to \infty$ limit of the critical line}
In the hard-spheres limit $\lambda\to\infty$, the critical point
for the ${\rm III}\to {\rm IVA}$ transition is $(\eta^c)^2\to 1/2$
\cite{Messina}, the same as in the previous case of
the ${\rm II}\to {\rm III}$ transition.
Let us analyze the large-$\lambda$ limit of the critical relation
$h_2(\eta,\lambda)=0$.
In the same way as for $g_2$, we get three leading terms from the seventh
and ninth (last) sums of Eq. (\ref{h2}):
\begin{eqnarray}
\frac{1}{16} z_{7/2}\left(\frac{\lambda^2}{2\eta^2},\frac{\eta^2}{2}
+\frac{1}{4}\right)
-\frac{1}{4}z_{5/2}\left(\frac{\lambda^2}{2\eta^2},\frac{\eta^2}{2}
+\frac{1}{4}\right) \nonumber\\
-\frac{1}{2}z_{5/2}\left(\frac{\lambda^2}{2\eta^2},\frac{1}{2}\right) = 0,
\qquad \lambda \gg 1 . \label{h2as0}
\end{eqnarray}
The application of the asymptotic relations (\ref{znuasymp})
to this equation implies
\begin{eqnarray}
\sqrt{\frac{\pi}{2}}\frac{\lambda}{\eta}{\rm e}^{-\frac{\lambda}{\eta}}
\Bigg\{\Bigg[\frac{\lambda}{8\eta \left(\eta^2+\frac{1}{2}\right)^{3/2}}
\Bigg(1+\frac{3\eta}{\lambda\sqrt{\eta^2+\frac{1}{2}}} \nonumber\\
+\frac{3\eta^2}{\lambda^2(\eta^2+\frac{1}{2})}\Bigg)
-\frac{1}{2(\eta^2+\frac{1}{2})}
\left(1+\frac{\eta}{\lambda\sqrt{\eta^2+\frac{1}{2}}}\right) \Bigg] \nonumber\\
\times {\rm e}^{\frac{\lambda}{\eta}\left(1-\sqrt{\eta^2+1/2}\right)}
-\left(1+\frac{\eta}{\lambda}\right) \Bigg\} = 0 . \label{h2as2}
\end{eqnarray}
The root of the expression in the largest parentheses yields
\begin{eqnarray}
\eta & \approx & \frac{1}{\sqrt{2}} + \frac{\ln \lambda}{\lambda} +
- \frac{5\ln 2}{2\lambda} \nonumber \\ & & +
\frac{3}{4} \sqrt{2}\left(\frac{\ln{\lambda}}{\lambda}\right)^2+
{\cal O}\left(\frac{\ln{\lambda}}{\lambda^2}\right) . \label{e34aas}
\end{eqnarray}
This asymptotic formula, taken for $\eta^2$, is plotted
in Fig. \ref{fig:phd2} by the dash-dotted line.
The comparison with the numerical results for the critical line (solid line)
is very good.
\section{Phase transitions IVA - V, IVA - IVB and IVB - V} \label{Sec7}
All phase transitions IVA - V, IVA - IVB and IVB - V are of first order
due to a discontinuous change of both structure parameters
$\delta$ and $\alpha$.
For phase IVB one has to minimize numerically the energy (\ref{e4b})
with respect to two parameters $\delta$ and $\alpha$, which is tedious
but feasible.
We found that for $\lambda < 27.4436$ phase IVA goes over directly to
phase V without entering the intermediate phase IVB.
On the transition line, the parameter $\alpha$ jumps from 1/2 to 1/3.
In the Coulomb limit $\lambda\to 0$ we get $\eta^t(0)=0.732416$,
$\delta^t=0.69334$ for phase IVA \cite{Samaj1} whereas for the rigid phase V
$\delta^t=\tan(\pi/6)=1/\sqrt{3}\approx 0.57735$.
The shape of the transition line is again parabolic, we can approximate
it empirically by
\begin{equation}
\lambda^2\approx c_{4A5} [\eta^t(0)-\eta^t] , \qquad c_{4A5}\approx 805.3 ,
\end{equation}
but now the parabola is reversed giving rise to the multiple reentrant
behavior, see Figs. \ref{fig:phd} and \ref{fig:phd3}.
\begin{figure}[thb]
\begin{center}
\includegraphics[clip,width=0.41\textwidth]{fig9.eps}
\caption{A detailed view of the phase diagram around the tricritical point.
The sector of phase IVA is connected by a narrow channel.}
\label{fig:phd3}
\end{center}
\end{figure}
The non-trivial $\delta^t$ for phase IVA increases to approximately 0.926
at $\lambda = 14$ and then slightly decreases to 0.853081 at $\lambda=27.4436$.
In Ref. \cite{Messina} it was anticipated that there exist two disjunct
regions of phase IVA in the phase diagram.
Our more precise calculations indicate that there exists a narrow connecting
channel merging these two regions into one, see Fig. \ref{fig:phd3}.
The maximum value of $\delta^t$ for phase IVA is achieved when the channel
is the most narrow so that it does not decrease too much from
the value $\delta=1$ for phase III.
Looking at Figs. \ref{fig:phd} and \ref{fig:phd3} we can confirm
the double reentrant scenario IVA-V-IVA-III-IVA-IVB \cite{Messina},
restricted to a more precise interval $0.5275<\eta^2<0.53643$.
For $\lambda>27.4436$, the phase IVB takes place and we have first-order
transitions IVA-IVB and IVB-V, see Figs. \ref{fig:phd} and \ref{fig:phd3}.
As concerns the transition line IVA - IVB, for $\lambda\to\infty$
it should asymptotically approach the value $(\eta^t)^2\to 1/2$ so that
phase IVA is absent in the hard-spheres limit \cite{Messina}.
The value of $\delta^t$ in phase IVA increases from 0.85308
at $\lambda=27.4436$ towards 1 for very large $\lambda$.
Concerning phase IVB, $\delta^t$ increases from 0.763284 at $\lambda=27.4436$
to 1 for very large $\lambda$ and the other parameter $\alpha^t$ from
0.41358 to 0.5 along the same transition line.
Thus, in the hard-spheres limit, the values $\delta^t\to 1$ and
$\alpha^t\to 1/2$ of phase III (see the top of Fig. \ref{fig:phd})
will be attained as expected.
Still in the hard-spheres limit $\lambda\to\infty$, the transition line
IVB-V should reach the point $\eta^t \approx 0.877\ldots$ \cite{Messina}.
Numerically, we got mere $\eta^t = 0.864133\ldots$ even for $\lambda = 500$.
The convergence is rather slow again, we have $\delta^t=0.58102$
and $\alpha^t=0.334428$ for phase IVB at the same $\lambda=500$ value,
gradually approaching the values 0.57735 and 1/3 of phase V, respectively,
with ${\cal O}(1/\lambda)$ corrections of both structure parameters.
Now we want to derive the above hard-spheres result from our formalism.
We recall that the energy $E_{\rm IVB}$ is given by Eq. (\ref{e4b}) and
$E_{\rm V}$ is its special case for $\delta=1/\sqrt{3}$ and $\alpha=1/3$.
We apply the asymptotic formulas (\ref{znuasym}) and (\ref{znuasymp})
to the $\lambda\to\infty$ limit of Eq. (\ref{e4b}) and neglect exponentially
small terms.
Five summands remain dominant; one from the sixth sum with $j=1$, three from
the eighths (last but one) sum, namely both terms with $j=k=0$ and the second
one with $j=0$, $k=1$ plus the $j=k=0$ term from the ninth (last) sum:
\begin{eqnarray}
E_{\rm IVB} & \approx & \frac{\eta}{2\sqrt{2\pi}\lambda}
\biggl\{2z_{3/2}\left(\frac{\lambda^2}{2\eta^2},\delta\right) \nonumber\\
& & +z_{3/2}\left(\frac{\lambda^2}{2\eta^2},
\frac{\eta^2}{2}+\frac{\alpha^2}{\delta}\right) \nonumber\\
& & + 2\ z_{3/2}\left[\frac{\lambda^2}{2\eta^2},
\frac{\eta^2}{2}+\frac{(\alpha-1/2)^2}{\delta}+\frac{\delta}{4}\right]
\nonumber\\
& & +4\ z_{3/2}\left(\frac{\lambda^2}{2\eta^2},
\frac{1}{4\delta}+\frac{\delta}{4}\right) \biggr\} . \label{eivbas}
\end{eqnarray}
Notice that two identical terms merged to the one on the third line.
All these summands should be of the same order for very large $\lambda$.
Since the asymptotic relations (\ref{znuasymp}) imply that
$z_{\nu}(x,y)\propto \exp(-2\sqrt{x y})$ for $x\to\infty$, and the first
argument $x=\lambda^2/(2\eta^2)$ is common for the summands, the second
arguments must coincide as well.
Thus we have
\begin{equation} \label{eta2hs}
\delta = \frac{\eta^2}{2}+\frac{\alpha^2}{\delta} = \frac{\eta^2}{2}
+\frac{(\alpha-\frac{1}{2})^2}{\delta}+\frac{\delta}{4}
= \frac{1}{4\delta}+\frac{\delta}{4} .
\end{equation}
This equalities yield the expected asymptotic values of the structure
parameters $\delta = 1/\sqrt{3}$ and $\alpha = 1/3$.
Simultaneously,
\begin{equation} \label{etat-hs}
\eta^t = \frac{2}{3^{3/4}}\approx 0.877383\ldots ,\qquad \lambda\to\infty.
\end{equation}
This value can be rederived from purely geometric considerations, too.
We have already mentioned that the particle density at one plate in phase V
is $\sigma=1/[a^2\sin(\pi/3)]=2/(a^2 \sqrt{3})$.
For dense packed hard spheres of radius $a$, the perpendicular distance of
two layers of triangular lattices is $d=\sqrt{2/3}\ a$,
see e.g. \cite{Schmidt}.
Inserting these values into $\eta = d \sqrt{\sigma}$ yields immediately
(\ref{etat-hs}).
We confirm that in the hard-spheres limit the transition IVB-V will undergo
no stepwise changes of structure parameters and it will be of the second order,
as expected.
In Fig. \ref{fig:deltaeta}, we present the transition values of the structure
parameters $\delta^t$ and $\alpha^t$ for phases IVA and IVB at first-order
transitions IVA - V (left, $\lambda\in [0,27.4436]$)
and IVB - V (right, $\lambda\in [27.4436,\infty]$).
The left and right line fragments are separated by a gap, illustrating
the step-wise change of structure parameters when going from phase IVA to IVB.
We recall that the parameters of phase V are always fixed to
$\delta^t=1/\sqrt{3}$ and $\alpha^t=1/3$.
We found a tricritical point at $\eta^c = 0.772814$ and $\lambda^c=27.4436$
where the three phases IVA, IVB and V coexist.
\begin{figure}[thb]
\begin{center}
\includegraphics[clip,width=0.45\textwidth]{fig10.eps}
\caption{The transition parameters $\delta^t$ and $\eta^t$ along
the phase transition lines IVA-V (left) and IVB-V (right).
Along two line fragments, $\lambda$ increases from 0 to $\infty$.
For the left fragment, the parameter $\alpha=1/2$ for phase IVA and
$\alpha=1/3$ for phase V.
There is a discontinuity in the parameters $\delta$ and $\alpha$
between phases IVA and IVB at the tricritical point with $\lambda=27.4436$.
The values of $\alpha$ on the right fragment correspond to phase IVB.
In the hard-spheres limit $\lambda\to\infty$, the line ends up at
the critical point $(\eta^c=0.877383,\delta^c=1/\sqrt{3})$.}
\label{fig:deltaeta}
\end{center}
\end{figure}
\section{The energy plot} \label{Sec8}
We want to compare the values of the optimized energy per particle
for various values of $\lambda$ and $\eta$.
We plot $E(\eta)$ for several fixed values of $\lambda$
in Fig. \ref{fig:energy-eta}.
These energies vary by orders of magnitude, thus we have chosen
semilogarithmic scale.
First we consider two limiting cases.
For $\eta\ll 1$ and $\lambda>0$, according to (\ref{e21}) and (\ref{f1f2})
the energy of the corresponding phase II
$\ln(E_{\rm II})\approx -3^{1/4}\lambda/\eta$
and so $\ln {E}$ diverges if $\eta\to 0$.
More interesting is the optimal energy of phase V, $E_{\rm V}$, for $\eta\gg 1$.
We were used to get the Coulomb limit as $\lambda\to 0$, but we can obtain
this limit also for medium $\lambda$ and very large $\eta$, as the ratio
$\lambda/\eta\to 0$ again.
For $\eta\gg \lambda$, using the asymptotic formulas for the generalized
Misra functions (Appendix A) we obtain from (\ref{e4b}) that
\begin{equation} \label{e5as}
E_{\rm V} = \pi \frac{\eta^2}{\lambda^2}\left(1+{\rm e}^{-\lambda}\right)
+ c_M\frac{\eta}{\lambda} +{\cal O}(1) ,
\end{equation}
where $c_M=-1.9605158...$ is the Madelung constant of the Coulomb potential
for the hexagonal lattice; for an explicit representation of the Madelung
constant in terms of $z_{\nu}(0,y)$ functions, see Eq. (24) with
$\Delta=\sqrt{3}$ and $\eta=0$ of Ref. \cite{Samaj1}.
The leading term is the (minus) background energy (\ref{eb}).
\begin{figure}[thb]
\begin{center}
\includegraphics[clip,width=0.45\textwidth]{fig11.eps}
\caption{The dependence of the energy per Yukawa particle $E$ on the
dimensionless distance $\eta$ for four values of $\lambda=1,10,20,35$,
in semilogarithmic scale.}
\label{fig:energy-eta}
\end{center}
\end{figure}
\begin{figure}[thb]
\begin{center}
\includegraphics[clip,width=0.45\textwidth]{fig12.eps}
\caption{The derivative $\partial E/\partial \eta$ for $\lambda=20$.
Full circle corresponds to the second-order transition III-IVA,
dashed line marks the discontinuity at the first-order transition IVA-V.}
\label{fig:de-eta}
\end{center}
\end{figure}
We see in Fig. \ref{fig:energy-eta} for few fixed values of $\lambda$ that
the energy is a monotonously increasing function of the dimensionless
distance $\eta$.
This means that the force between the plates is always attractive.
The non-analyticities at transition points are not clearly manifested
in this scale.
Therefore, for $\lambda=20$, we performed the derivative
$\partial E/\partial \eta$, directly for rigid structures and
numerically using $E_{\rm IVA}$ minimized with respect to $\delta$
for phase IVA.
The obtained results are plotted in Fig. \ref{fig:de-eta}.
We see the expected continuous but non-analytic behavior at
the second-order transition point III-IVA as well as
a jump discontinuity at the first order transition IVA-V.
\section{Internal parameters of the phases near hard-spheres limit} \label{Sec9}
In and close to the limit of hard spheres $\lambda\to\infty$, the expressions
for the energies of the structures in terms of the generalized Misra
functions admit an asymptotic analysis.
This fact permits us to determine the $\eta$-dependence of the structure
parameters of the present soft phases II and IVB in the $\lambda\to\infty$
limit and eventually to derive their leading correction for large but
finite $\lambda$.
\subsection{Aspect ratio $\Delta$ of phase II at and near hard spheres}
The dependence of the aspect ratio $\Delta_{\rm HS}$ on $\eta$ for phase II
is well known in the hard-spheres limit $\lambda\to\infty$ \cite{Messina}:
\begin{equation} \label{DeltaHS}
\Delta_{\rm HS}(\eta) = \sqrt{4\eta^4+3} - 2\eta^2 .
\end{equation}
In the following, we derive this result and the first $1/\lambda$
correction to it by using our method.
For $\lambda\gg 1$, most of terms in the energy of phase II (\ref{e2})
become exponentially small (we exclude from the discussion trivial terms
which do not depend on $\Delta$); only the term $j=1$ in the sixth sum
and the term $j=k=1$ in the eighth (last) sum contribute.
As soon as $\Delta>1$, using (\ref{znuasymp}) we get
\begin{eqnarray}
E_{\rm II} \approx \frac{\eta}{\sqrt{\pi}\lambda}
\left[z_{\frac{3}{2}}\left(\frac{\lambda^2}{4 \eta^2},\frac{1}{\Delta}\right)
+2 z_{\frac{3}{2}}\left(\frac{\lambda^2}{4 \eta^2},\eta^2
+\frac{\Delta}{4}+\frac{1}{4\Delta}\right)\right] \nonumber \\
\approx \frac{\eta}{\lambda}
\left(\sqrt{\Delta} {\rm e}^{-\frac{\lambda}{\eta\sqrt{\Delta}}}
+\frac{2}{\sqrt{\eta^2+\frac{\Delta}{4}+\frac{1}{4\Delta}}}
{\rm e}^{-\frac{\lambda}{\eta}\sqrt{\eta^2+\frac{\Delta}{4}+\frac{1}{4\Delta}}}
\right) . \nonumber \\ \label{e2hs}
\end{eqnarray}
The minimum of the energy is given by $\partial E_{\rm II} /\partial \Delta =0$,
which implies
\begin{eqnarray}
\left( \frac{1}{2\sqrt{\Delta}}+\frac{\lambda}{2\Delta\eta}\right)
{\rm e}^{-\frac{\lambda}{\eta\sqrt{\Delta}}} =
\frac{1-\frac{1}{\Delta^2}}{4 \eta^2+\Delta+\frac{1}{\Delta}}
\nonumber \\ \times \left(
\frac{1}{\sqrt{\eta^2+\frac{\Delta}{4}+\frac{1}{4\Delta}}}+\frac{\lambda}{\eta}
\right)
{\rm e}^{-\frac{\lambda}{\eta}\sqrt{\eta^2+\frac{\Delta}{4}+\frac{1}{4\Delta}}}. \label{del2}
\end{eqnarray}
If we want to reproduce just the hard-spheres limit $\lambda\to\infty$,
we can say that exponentials are by far more significant than rationals
and their arguments must become the same, i.e.
$\sqrt{\eta^2+\Delta/4+1/(4\Delta)}=1/\sqrt{\Delta}$ which
leads to the known result (\ref{DeltaHS}).
Numerics suggests that the correction is of the type $1/\lambda$, i. e.
\begin{equation} \label{delta}
\Delta \approx \Delta_{\rm HS} + \frac{a(\eta)}{\lambda}
=\sqrt{4\eta^4+3}-2\eta^2 + \frac{a(\eta)}{\lambda} .
\end{equation}
We put the exponentials on one side, insert (\ref{delta}) and expand
$\sqrt{\eta^2+\Delta/4+1/(4\Delta)}-1/\sqrt{\Delta}$ up to the order
$1/\lambda$.
The absolute term vanishes and we have
\begin{eqnarray}
\exp{\left[{-\frac{a(\eta)}{4\eta}\ \frac{3+4\eta^4-2\eta^2\sqrt{4\eta^4+3}}
{(\sqrt{4\eta^4+3}-2\eta^2)^{3/2}}}\right]} = \nonumber \\
\frac{\eta^2+\frac{\Delta}{4}+\frac{1}{4\Delta}}{2\Delta
\left(\frac{1}{4}-\frac{1}{4\Delta^2}\right)}
\approx \frac{1}{1+4\eta^4-2\eta^2\sqrt{4\eta^4+3}}, \label{del3}
\end{eqnarray}
where we considered $\Delta\approx\Delta_{\rm HS}$ on the second line.
From this relation we readily get
\begin{eqnarray}
a(\eta) & = & \frac{\left(\sqrt{4\eta^4+3}-2\eta^2\right)^{3/2}4\eta}{3
+4\eta^4-2\eta^2\sqrt{4\eta^4+3}} \nonumber \\ & &
\times\ln{\left[1+4\eta^4-2\eta^2\sqrt{4\eta^4+3}\right]}. \label{aeta}
\end{eqnarray}
The value of $a(\eta)$ is negative in the whole interval $0<\eta<1/\sqrt{2}$
of phase II.
We find $a(\eta)\approx-8\times 3^{1/4}\eta^3$ for $\eta\ll 1$,
confirming once more that phase II is entered directly from phase I
for any small positive $\eta$.
We tested the asymptotic result (\ref{delta}), (\ref{aeta})
numerically, see Fig. \ref{fig:del-lam}.
\begin{figure}[thb]
\begin{center}
\includegraphics[clip,width=0.45\textwidth]{fig13.eps}
\caption{The aspect ratio $\Delta$ of phase II vs. $\lambda$ for
four values of $\eta=0.1,0.3,0.4,0.5$.
The solid lines correspond to numerical calculations.
The asymptotic $\lambda\to\infty$ result (\ref{delta}), (\ref{aeta})
is represented by dashed lines.}
\label{fig:del-lam}
\end{center}
\end{figure}
\subsection{Parameters $\delta$ and $\alpha$ of phase IVB in
the hard-spheres limit}
As was shown above, in the hard-spheres limit $\lambda\to\infty$ phase IVB
takes place in the interval $\eta\in [1/\sqrt{2},2/3^{3/4}]$.
Let us study the $\lambda\to\infty$ limit of the energy $E_{\rm IVB}$
(\ref{e4b}); terms which do not depend on $\delta$ or $\alpha$ are
automatically excluded from the discussion.
For very large $\lambda$ and general $\eta$ only the last two sums contribute,
the remaining sums are exponentially small.
The eighth sum has three important terms - one from the first $z_{3/2}$ summand
with $j=k=0$ and two identical ones from the second $z_{3/2}$ summand
with $j=k=0$ and $j=0,k=1$.
From the ninth (last) sum we take the $j=k=0$ term.
The result is
\begin{eqnarray}
E_{\rm IVB} & \approx & \frac{\eta}{2\sqrt{2\pi}\lambda}\Bigg[
z_{3/2}\left(\frac{\lambda^2}{2\eta^2},\eta^2/2+\frac{\alpha^2}{\delta}\right)
\nonumber\\
&+&2z_{3/2}\left(\frac{\lambda^2}{2\eta^2},\eta^2/2+\frac{(\alpha-1/2)^2}{\delta}
+\frac{\delta}{4}\right)
\nonumber\\
&+&4z_{3/2}\left(\frac{\lambda^2}{2\eta^2},\frac{1}{4\delta}
+\frac{\delta}{4}\right) \Bigg]. \label{e4bapp}
\end{eqnarray}
We notice that two more terms can become important in special limits.
The first is the $j=-1,\ k=0$ term from the eighth sum,
the first $z_{3/2}$ summand, which contributes only in the limit $\delta\to 1$
(i. e. $\eta^2\to 1/2$).
The other possibly important term can be found in Eq. (\ref{eivbas})
as the first one in the bracket, but it plays role only if
$\eta^2\to 4/(3\sqrt{3})$ and it can be omitted for general $\eta$ as well.
Now we apply the asymptotic relations (\ref{znuasymp}) to
the energy (\ref{e4bapp}).
The optimization of the energy with respect to parameters $\delta$ and
$\alpha$ leads to the equations $\partial E/\partial \delta = 0$ and
$\partial E/\partial \alpha = 0$.
What we get are certain products of rational functions and exponentials.
To have a non-trivial solution in the $\lambda\to\infty$ limit,
the dominant exponentials must have the same arguments which yields
\begin{equation} \label{exparg}
\eta^2/2+\frac{\alpha^2}{\delta} =\eta^2/2+\frac{(\alpha-1/2)^2}{\delta}
+\frac{\delta}{4} =\frac{1}{4\delta}+\frac{\delta}{4} .
\end{equation}
This set set of equations can be readily rewritten as
\begin{equation} \label{deltaalpha}
\alpha = \frac{1}{4}\left(1+\delta^2\right), \qquad
\alpha^2 -\alpha+\frac{\eta^2}{2}\delta = 0.
\end{equation}
The quartic equation for $\delta$ follows
\begin{equation} \label{delta4}
\delta^4-2\delta^2+8\eta^2\delta-3 = 0 .
\end{equation}
The discriminant of this equation $-2^{12}(3-14\eta^4+27\eta^8)$ is negative
for any $\eta$.
Consequently, we get two complex roots and two real ones.
It turns out that one of the real roots is negative and the only
physical - real positive - root is given by Cardano formulas as follows
\begin{equation} \label{delta2}
\delta = -S(\eta)+\frac{1}{2}\sqrt{-4S^2(\eta)+4+\frac{8\eta^2}{S(\eta)}},
\end{equation}
where
\begin{equation} \label{S}
S(\eta) = \frac{1}{2}\sqrt{\frac{4}{3}+\frac{1}{3}
\left[Q(\eta)-\frac{32}{Q(\eta)}\right]}
\end{equation}
with
\begin{equation} \label{Q}
Q(\eta)=2^{5/3}\left(27\eta^4-7+3\sqrt{9-42\eta^4+81\eta^8}\right)^{1/3}.
\end{equation}
The value of $\alpha$ follows straightforwardly from the first of
Eqs. (\ref{deltaalpha}).
It is easy to check that the above formulas give the correct lattice
parameters at the endpoints of the phase IVB region, namely we have
$(\alpha=1/2,\delta=1)$ at $\eta=1/\sqrt{2}$ (phase III) and
$(\alpha=1/3,\delta=1/\sqrt{3})$ at $\eta=2/3^{3/4}$ (phase V).
We did not find in the literature the above specification of
the structural parameters of phase IVB in the hard-spheres limit.
The numerical test of the results for phase IV parameters
is depicted in Fig. \ref{fig:del-alf-lam}.
For a given $\eta$, the dependence of the parameters $\delta$ (top set)
and $\alpha$ (bottom set) on $1/\lambda$, obtained by numerics, is represented
by open symbols (connected by solid line), the asymptotic $\lambda\to\infty$
result given by our Eqs. (\ref{deltaalpha}) is depicted by full symbol.
It is seen that numerical data converge quickly to their asymptotic values.
\begin{figure}[thb]
\begin{center}
\includegraphics[clip,width=0.45\textwidth]{fig14.eps}
\caption{The structure parameters $\delta$ and $\alpha$ of phase IVB
vs. $\lambda$ for three values of $\eta=0.75,0.8,0.83$.
Numerical data are represented by open symbols (connected by solid lines),
the asymptotic $\lambda\to\infty$ result given by Eqs. (\ref{deltaalpha})
is depicted by full symbol.}
\label{fig:del-alf-lam}
\end{center}
\end{figure}
\section{Conclusion} \label{Sec10}
In this paper, we have studied the zero-temperature phase diagram of
bilayer Wigner crystals of Yukawa particles.
To calculate the energy per particle of the phases, we used the recent method
of lattice summations \cite{Samaj1} extended to Yukawa potentials.
The weak point of the method is that one has to know ahead the possible
phases from numerical simulations.
The strong point is that the truncation of the series of the generalized
Misra functions provides extremely precise estimates of the energy,
e.g. the truncation at the 5th term provides the accuracy within
17 decimal digits.
Another strong point of Misra functions is that they can be readily
expanded around the critical point, providing in this way closed-form
expressions for the critical lines between phases II-III (\ref{critline})
and III-IVA (\ref{crith}).
Only few Misra functions contribute in the equations for the critical
lines in the asymptotic Coulomb $\lambda\to 0$ and hard-spheres
$\lambda\to\infty$ limits.
The characteristic feature of the Coulomb limit is the parabolic shape of
the critical lines, see Eq. (\ref{le23}) with the corresponding plot
in Fig. \ref{fig:trans23ll} for the II-III phase transition and
Eq. (\ref{le34a}) with the corresponding plot in Fig. \ref{fig:trans34all}
for the III-IVA phase transition.
In the hard-spheres limit, the asymptotic formulas for the II-III
phase transition (\ref{e23as}) and the III-IVA phase transition (\ref{e34aas})
are pictures by dash-dotted lines in Fig. \ref{fig:phd2}.
It turns out that the second-order phase transitions II-III and
III-IVA exhibit the mean-field critical exponents (\ref{MFind}).
The most important features of the Yukawa phase diagram obtained
by Messina and L\"owen \cite{Messina} were confirmed.
On the contrary to previous suggestions, phase I goes directly to phase II
at $\eta=0$, i.e. there does not exist a finite interval of positive
$\eta$-values where phase I dominates.
Another important novelty is that instead of the suggested region of
phase coexistence, we found a narrow channel within one continuous region
of phase IVA.
This fact also lead to the tricritical point where
the phases IVA, IVB and V meet.
Another application of our formalism is the determination of the structure
parameters of soft phases II and IVB in and close to the hard-spheres limit.
For $\lambda\to\infty$, the $\eta$-dependence of the aspect ratio $\Delta$
of phase II has already been known \cite{Messina}, see Eq. (\ref{DeltaHS}).
We were able to derive the first $1/\lambda$ correction to this
asymptotic relation, see Eqs. (\ref{delta}) and (\ref{aeta}), which is
in perfect agreement with the numerical results (Fig. \ref{fig:del-lam}).
The derivation of the $\eta$-dependence of two structure parameters
$\delta$ and $\alpha$ of phase IVB in the limit $\lambda\to\infty$,
see the relations (\ref{exparg}) and Fig. \ref{fig:del-alf-lam},
is likely new as well.
As concerns future perspectives to apply our method to other systems,
the system of particles with $1/r^{\sigma}$ interactions \cite{Mazars11}
seems to be a good candidate.
\begin{acknowledgments}
The support received from the grant VEGA No. 2/0015/2015 is acknowledged.
\end{acknowledgments}
|
{
"timestamp": "2015-04-15T02:07:58",
"yymm": "1504",
"arxiv_id": "1504.03501",
"language": "en",
"url": "https://arxiv.org/abs/1504.03501"
}
|
\section{Introduction}
\subsection{Backgrounds and main results}
We study in this paper a problem that is in sprit of the works of the stability conjecture but with an ``intrinsic" nature. Let $M$ be a compact manifold without boundary, and $\diff^1(M)$ be the space of $C^1$-diffeomorphisms of $M$. Recall the stability conjecture formulated by Palis and Smale claims that if a diffeomorphism $f$ is structurally stable then it is hyperbolic. Here a diffeomorphism $f$ is called {\it hyperbolic} if the chain recurrent set $R(f)$ of $f$ (see Definition~\ref{chain recurrence}) is hyperbolic. A stronger version of the conjecture is to claim that if $f$ is $\Omega$-stable then it is hyperbolic. These two remarkable conjectures are solved by Ma\~{n}\'{e}~\cite{m3} and Palis~\cite{pa1}, respectively.
During the long way of study of the stability conjectures, the attention was more and more concentrated on periodic orbits of the (unperturbed) diffeomorphism $f$ as well as its perturbations $g$. Liao~\cite{l1} and Ma\~{n}\'{e}~\cite{m2} raised independently a conjecture (more precisely, a problem without a tentative answer), known as the star conjecture, stating that if $f$ has no, robustly, non-hyperbolic periodic orbits then it is hyperbolic. Being an assumption, the star condition is clearly weaker than the $\Omega$-stability. Hence the star conjecture is regarded another (strong) version of the stability conjecture. It is solved by Aoki and Hayashi~\cite{ao,h2}. To compare more precisely with our Main Theorem below we state their results in a generic version. Recall
that if $p$ is a periodic point with period $\tau$ of a diffeomorphism $f$, and if $\lambda_1,\lambda_2,\cdots,\lambda_d$ are the eigenvalues of $Df^{\tau}$ (counted by multiplicity), then the $d$ numbers $\chi_i=\frac{1}{\tau}\log|\lambda_i|$, $i=1,\cdots,d$ are called the \emph{Lyapunov exponents} of $orb(p)$.
\begin{theo}\emph{(Aoki and Hayashi)}\label{ah}
For a $C^1$-generic $f\in\diff^1(M)$, if $f$ is not hyperbolic, then there is a periodic orbit of $f$ that has a Lyapunov exponent arbitrarily close to 0.
\end{theo}
Now we state our main result. Recall that two hyperbolic periodic points are {\it homoclinically related} if $W^s(orb(p))$ has non-empty transverse intersection with $W^u(orb(q))$ and $W^u(orb(p))$ has non-empty transverse intersection with $W^s(orb(q))$. To be homoclinically related is an equivalent relation, and the \emph{homoclinic class} $H(p)$ of a hyperbolic periodic point $p$ is the closure of the union of periodic orbits that are homoclinically related to $p$. Two different homoclinic classes may intersect. Nevertheless by the result of~\cite{bc}, for $C^1$-generic diffeomorphisms, every homoclinic class is a maximal invariant compact set that is {\it chain transitive} (see Definition~\ref{pseudo-orbit}), hence they are pairwise disjoint. Homoclinic classes are generally infinite in number, even for generic diffeomorphisms.
\begin{maintheo}
For a $C^1$-generic $f\in\diff^1(M)$, if a homoclinic class $H(p)$ of $f$ is not hyperbolic,
then there is a periodic orbit of $f$ that is homoclinically related to $orb(p)$ and has a Lyapunov exponent arbitrarily close to 0.
\end{maintheo}
Note that here the ``weak" periodic orbit (the one with a Lyapunov exponent arbitrarily close to 0) is homoclinically related to $orb(p)$, that is, is ``inside" the homoclinic class $H(p)$. This is the main point of this paper. In fact, under the assumptions of the Main Theorem, it is straightforward to prove (following the classical proof of the stability conjecture) that, there must be a weak periodic orbit arbitrarily near $H(p)$. In contrast, here the Main Theorem claims there must be a weak periodic orbit not only near, but actually inside $H(p)$. Of course, if the homoclinic class $H(p)$ is assumed to be isolated, then being ``near" will be equivalent to being ``inside". The point is that here $H(p)$ is not known to be isolated hence, at each step, the periodic orbits created by perturbations have to be guaranteed to lie strictly inside the homoclinic class. It is in this sense we say the problem is of an ``intrinsic" nature, and the classical proof of the stability conjecture does not pass through.
There are other conjectures aimed to give a dichotomy of global dynamics. Recall that a \emph{homoclinic tangency} of a hyperbolic periodic point $p$ is a non-transverse intersection between $W^u(p)$ and $W^s(p)$. A diffeomorphism is with a \emph{heterodimensional cycle} if there are two hyperbolic periodic points $p$ and $q$ with different stable dimensions such that $W^s(p)\cap W^u(q)\neq \emptyset$ and $W^s(q)\cap W^u(p)\neq \emptyset$. It is obvious that any diffeomorphism with either a tangency or a heterodimensional cycle is not hyperbolic. Palis conjectured that these two phenomenons are the only obstacles for hyperbolicity. More precisely, the union of hyperbolic diffeomorphisms and diffeomorphisms with tangencies or heterodimensional cycles are dense in the space of diffeomorphisms, see~\cite{pa}. Based on the results afterwards, Bonatti and D\'iaz conjectured that the union of diffeomorphisms that are hyperbolic and those with heterodimensional cycles are dense in the space of diffeomorphisms, see~\cite{b,bd}. There are many works related to this subject, like~\cite{ps,c5,csy,cp}.~\cite{ps} solved this conjecture for dimension 2, and for higher dimension,~\cite{c5,csy,cp} got progress that far from homoclinic bifurcations, the systems has some weak hyperbolicity (partially hyperbolic or essentially hyperbolic).
By the Franks' lemma~\cite{f,g2}, we can perturb weak periodic orbits to get periodic orbits with different stable dimensions. But it is not clear whether these periodic orbits are still contained in the non-hyperbolic homoclinic class after perturbation. Thus we have the following conjecture, which is an intrinsic version of Palis conjecture for homoclinic classes.
\begin{conj}\emph{(\cite{b,bcdg})}\label{conj b}
There is a residual subset $\mathcal{R}\subset \diff^1(M)$, such that for all $f\in\mathcal{R}$, if a homoclinic class $H(p)$ is not hyperbolic, then there is a periodic point $q\in H(p)$, whose stable dimension is different from that of $p$.
\end{conj}
By~\cite{conley}, one can decompose the dynamics into pieces, and each piece is called a {\it chain recurrence class} (see Definition~\ref{chain class}). By~\cite{bc}, for $C^1$-generic diffeomorphism, a chain recurrence class is either a homoclinic class or contains no periodic point. We call a chain recurrence class without periodic point an \emph{aperiodic class}. Recall that a \emph{dominated splitting} $E\oplus F$ on an invariant compact $\Lambda$ set is an invariant splitting of $T_{\Lambda}M$ and the norm of $Df$ along $E$ is controlled by that along $F$, and $\Lambda$ is \emph{partially hyperbolic} if $T_{\Lambda}M$ splits into three bundles which is a dominated splitting such that the extremal bundles are hyperbolic and the center bundle is neutral (see Definition~\ref{dominated splitting}). By~\cite{c5,csy}, for $C^1$-generic diffeomorphisms far from homoclinic bifurcations (or just homoclinic tangencies), an aperiodic class is partially hyperbolic with center bundle of dimension 1. In~\cite{bd-aperiodic class}, they proved that if $dim(M)\geq 3$, then there are an open set $\mathcal{U}$ of $\diff^1(M)$, and a residual subset $\mathcal{V}$ of $\mathcal{U}$, such that any $g\in\mathcal{V}$ has infinitely many aperiodic classes, and each of them has no non-trivial dominated splitting. We state here a conjecture by S. Crovisier for aperiodic classes, which implies the non-existence of aperiodic classes for $C^1$-generic diffeomorphisms far from homoclinic bifurcations.
\begin{conj}\emph{(\cite{c3})}
Let $\Lambda$ be an aperiodic class for a $C^1$-generic diffeomorphism $f$, and $E^s \oplus E^c\oplus E^u$ the dominated splitting such that $E^s$ (resp. $E^u$) is the maximal uniformly contracted (resp. expanded) sub-bundle. Then $E^c$ has dimension larger than or equal to 2 and does not admit a finer dominated splitting.
\end{conj}
\subsection{Main theorem restated}
We give here a more general result rather than the main theorem. For a hyperbolic periodic point $p$, denote by $ind(p)$ its stable dimension.
\begin{theoa}
For $C^1$-generic $f\in \diff^1(M)$, assume that $p$ is a hyperbolic periodic point of $f$. If the homoclinic class $H(p)$ has a dominated splitting $T_{H(p)}M=E\oplus F$, with $dim E\leq ind(p)$, such that the bundle $E$ is not contracted, then there are periodic orbits in $H(p)$ with index $dim(E)$ that have the maximal Lyapunov exponents along $E$ arbitrarily close to 0.
\end{theoa}
\begin{rem}
If in the assumption of the Theorem A, $dim E=ind(p)$, then the periodic orbits $\mathcal{O}_k=orb(q_k)$ have the same index as $p$. Thus by the genericity assumption, they are homoclinically related with $orb(p)$.
\end{rem}
We give an explanation how Theorem A implies the main theorem. We assume that the second item of the main theorem does not happen, that is to say, all Lyapunov exponents of periodic orbits that are homoclinically related to $orb(p)$ are uniformly away from 0. Then by the genericity assumption, $H(p)$ has a dominated splitting $T_{H(p)}M=E\oplus F$, with $dim E= ind(p)$, (see~\cite{gy} and Proposition 4.8 of~\cite{bcdg}). By the conclusion of Theorem A and the assumption of no existence of weak periodic orbits homoclinically related to $orb(p)$, we get that the bundle $E$ is contracted. With the same argument for $f^{-1}$, we get that the bundle $F$ is expanded for $f$. Hence $T_{H(p)}M=E\oplus F$ is a hyperbolic splitting and we get the conclusion of the main theorem.
In~\cite{m3}, Ma\~n\'e introduced a very useful lemma (Theorem \textbf{II.1}) to get weak periodic orbits under certain hypothesis. The statement is very technical and the original proof of Ma\~{n}\'{e} is difficult, thus we will not state it here. Based on a modification of the proof of Ma\~{n}\'{e}, Bonatti, Gan and Yang have a result for homoclinic classes, see~\cite{bgy}.
Here we point out that, different from Theorem \textbf{II.1} of~\cite{m3} and the result of~\cite{bgy}, there is a genericity assumption in the main theorem and Theorem A. That is to say, the conclusion of the main theorem is a perturbation result and may not be valid for all diffeomorphisms. Thus one asks the following question naturally, whether the genericity assumption is essential in the main theorem.
\begin{ques}\label{question 1}
Is there a homoclinic class $H(p)$ for a diffeomorphism $f$ satisfying that all the Lyapunov exponents of all periodic orbits homoclinically related to $orb(p)$ are uniformly away from 0 but $H(p)$ is not hyperbolic?
\end{ques}
In some cases, we can give a positive answer to this question. In~\cite{r}, Rios proved that there is a diffeomorphism on the boundary of the set of hyperbolic diffeomorphisms on surface, with a homoclinic class containing a tangency inside. Hence it is not hyperbolic (it does not have a dominated splitting because of the existence of tangency). In~\cite{clr}, they proved that for this homoclinic class, all the Lyapunov exponents of all periodic orbits contained in the class are uniformly away form 0. In fact, they proved more that all the Lyapunov exponents of all ergodic measures are uniformly away from 0. Examples of non-hyperbolic homoclinic classes with a dominated splitting can be found in like~\cite{c4,dhrs,ps2}, but the homoclinic classes in these examples contain weak periodic orbits. For $C^2$ diffeomorphisms on surfaces, by the conclusions of~\cite{ps2}, one can not give a non-hyperbolic homoclinic class with domination and without weak periodic orbits, which is unknown in the $C^1$ dynamics. Hence we have the following question which is a stronger version of Question~\ref{question 1}.
\begin{ques}\label{question 2}
Is there a non-hyperbolic homoclinic class $H(p)$ with a non-trivial dominated splitting for a diffeomorphism $f$ satisfying that all the Lyapunov exponents of all periodic orbits homoclinically related to $orb(p)$ are uniformly away from 0?
\end{ques}
\subsection{Some applications of the main theorem}
In this subsection, we give some applications whose proof will be given after.
\subsubsection{Structural stability and hyperbolicity}
Recall that a diffeomorphism $f\in\diff^1(M)$ is \emph{structurally stable}, if there is a $C^1$ neighborhood $\mathcal{U}$ of $f$, such that, for any $g\in\mathcal{U}$, there is a homeomorphism $\phi:M\rightarrow M$, satisfying $\phi\circ f=g\circ\phi$. The orbital structure of a structurally stable diffeomorphism remains unchanged under perturbations. Ma\~{n}\'{e} proved that the chain recurrent set of a structurally stable diffeomorphism is hyperbolic, see~\cite{m3}. Here we give a local version about this result.
It is known that a hyperbolic periodic point has a continuation. More precisely, for a hyperbolic periodic point $p$ of a diffeomorphism $f$ with period $\tau$, there is a neighborhood $U$ of $orb(p)$ and a $C^1$ neighborhood $\mathcal{U}$ of $f$, such that, for any $g\in\mathcal{U}$, the maximal invariant compact set of $g$ in $U$ is a unique periodic orbit with period $\tau$ and with the same index as $p$. We denote this \emph{continuation} of $p$ by $p_g$ for such a diffeomorphism $g$, and denote the homoclinic class (and chain recurrence class resp.) of $p_g$ by $H(p_g)$ (and $C(p_g)$ resp.). Thus we say that a homoclinic class $H(p)$ of a diffeomorphism $f$ is \emph{structurally stable}, if there is a $C^1$ neighborhood $\mathcal{U}$ of $f$, such that, for any $g\in\mathcal{U}$, there is a homeomorphism $\phi:H(p)\rightarrow H(p_g)$, satisfying $\phi \circ f|_{H(p)}=g\circ\phi|_{H(p)}$, where $p_g$ is the continuation of $p$. Similarly we can define the structurally stability for a chain recurrence class $C(p)$ of a hyperbolic point. One asks naturally the following question, which can be seen as a ``local" version of the stability conjecture.
\begin{ques}\label{stability}
Assume $p$ is a hyperbolic point for a diffeomorphism, if $H(p)$ (or $C(p)$) is structurally stable, then is it hyperbolic?
\end{ques}
There are many works related to this question, see for example~\cite{gy,sv,ww,wenx}. In~\cite{ww} and~\cite{wenx}, they prove that structural stability implies hyperbolicity for the chain recurrence class and the homoclinic class respectively of a hyperbolic periodic point, under the hypothesis that the diffeomorphism is far away from tangency, or that the stable or the unstable dimension of this periodic point is 1. With the conclusions of the main theorem, we can give a complete answer to Question~\ref{stability}.
\begin{cor}\label{stuctually stable}
Assume $f$ is a diffeomorphism in $\diff^1(M)$ and $p$ is a hyperbolic periodic point of $f$. If the homoclinic class $H(p)$ is structurally stable, then $H(p)$ is hyperbolic. The conclusion is also valid for $C(p)$.
\end{cor}
\subsubsection{Partial hyperbolicity}
Next result is that for a homoclinic class with a dominated splitting of a $C^1$-generic diffeomorphism, if the dimensions of the two bundles in the splitting satisfy certain hypothesis, then the splitting is a partially hyperbolic splitting (at least one bundle is hyperbolic).
\begin{cor}\label{application 1}
For $C^1$-generic $f\in \diff^1(M)$, if a homoclinic class $H(p)$ has a dominated splitting $T_{H(p)}M=E\oplus F$, such that $dim(E)$ is smaller than the smallest index of periodic orbits contained in $H(p)$, then the bundle $E$ is contracted. Symmetrically, if $dim(E)$ is larger than the largest index of periodic orbits contained in $H(p)$, then the bundle $F$ is expanded.
\end{cor}
As another consequence of the main theorem, we can give a proof of Theorem 1.1 (2) in~\cite{csy} with a different argument. More precisely, we can prove that for a $C^1$-generic diffeomorphism far from tangency, a homoclinic class has a partially hyperbolic splitting whose center bundle splits into 1-dimensional subbundles, and the Lyapunov exponents of the periodic orbits along each the center subbundle can be arbitrarily close to 0. Denote $\mathcal{HT}$ the set of diffeomorphisms of $\diff^1(M)$ that exhibit a tangency.
\begin{cor}\emph{(\cite{csy})}\label{application 2}
For $C^1$-generic $f\in \diff^1(M)\setminus \overline{\mathcal{HT}}$, a homoclinic class $H(p)$ has a partially hyperbolic splitting $T_{H(p)}M=E^s\oplus E^c_1\oplus\cdots\oplus E^c_k\oplus E^u$ such that each of the center subbundles $E^c_i$ is neither contracted nor expanded and $dim(E^c_i)=1$, for all $i=1,\cdots,k$. Moreover, the minimal index of periodic points contained in $H(p)$ is $dim(E^s)$ or $dim(E^s)+1$, and symmetrically, the maximal index of periodic points contained in $H(p)$ is $d-dim(E^u)$ or $d-dim(E^u)-1$. For each $i=1,\cdots,k$, there exist periodic orbits contained in $H(p)$ with arbitrarily long periods with a Lyapunov exponent along $E^c_i$ arbitrarily close to $0$.
\end{cor}
\subsubsection{Lyapunov stable homoclinic classes}
Recall that an invariant compact set $\Lambda\subset M$ is \emph{Lyapunov stable for $f$}, if for any neighborhood $U$ of $\Lambda$, there is another neighborhood $V$ of $\Lambda$, such that $f^n(V)\subset U$ for all $n\geq 0$. We say that $\Lambda$ is \emph{bi-Lyapunov stable}, if $\Lambda$ is both Lyapunov stable for $f$ and for $f^{-1}$.
The following results are about $C^1$-generic Lyapunov stable homoclinic classes. First, for $C^1$-generic Lyapunov stable homoclinic classes, we can get a similar conclusion of Corollary~\ref{application 1} under a weaker hypothesis.
\begin{cor}\label{application 3}
For $C^1$-generic $f\in \diff^1(M)$, if a homoclinic class $H(p)$ is Lyapunov stable and has a dominated splitting $T_{H(p)}M=E\oplus F$ such that $dim(E)$ is larger than or equal to the largest index of periodic orbits contained in $H(p)$, then the bundle $F$ is expanded.
\end{cor}
With the conclusion of Corollary~\ref{application 3}, we can give a positive answer to Conjecture~\ref{conj b} for bi-Lyapunov stable homoclinic classes.
\begin{cor}\label{application 4}
For $C^1$-generic $f\in \diff^1(M)$, where $M$ is connected, if a homoclinic class $H(p)$ is bi-Lyapunov stable, then we have:
\begin{itemize}
\item either $H(p)$ is hyperbolic, hence $H(p)=M$ and $f$ is Anosov,
\item or $f$ can be $C^1$ approximated by diffeomorphisms that have a heterodimensional cycle.
\end{itemize}
\end{cor}
From~\cite{csy} (or Corollary~\ref{application 2}), we know that for $C^1$-generic diffeomorphisms far away from tangencies, a homoclinic class has a partially hyperbolic splitting with all central bundles dimension 1. We have the following result about the index of periodic orbits for Lyapunov stable homoclinic classes. It is a direct corollary of Corollary~\ref{application 3} and we omit the proof.
\begin{cor}
For $C^1$-generic $f\in \diff^1(M)\setminus \overline{\mathcal{HT}}$, if a homoclinic class $H(p)$ is Lyapunov stable and assume $T_{H(p)}M=E^s\oplus E^c_1\oplus\cdots\oplus E^c_k\oplus E^u$ is the partially hyperbolic splitting, then the largest index of periodic points contained in $H(p)$ equals $d-dim(E^s)$.
\end{cor}
\subsection{Propositions for the proof of Theorem A}
To prove Theorem A, we have to use the following three propositions. Proposition~\ref{time control} tells that for any hyperbolic periodic orbit $orb(p)$ and any invariant compact set $K$ of a diffeomorphism $f$ linked by heteroclinic orbits, we can get a periodic orbit that spends a given proportion of time close to $orb(p)$ and $K$ by arbitrarily $C^1$ small perturbation.
\begin{pro}\label{time control}
Assume $p$ is a hyperbolic periodic point of a diffeomorphism $f\in\diff^1(M)$ with period $\tau$ and $K$ is an invariant compact set of $f$. Assume there are two points $x,y\in M$ satisfying that:
\begin{itemize}
\item all periodic orbits in $K$ are hyperbolic and $p\not\in K$,
\item $x \in W^{u}(p)$ with $\omega(x)\cap K \neq \emptyset$, and $y \in W^{s}(p)$ with $\alpha(y)=K$.
\end{itemize}
Then for any neighborhood $\mathcal{U}$ of $f$ in $\diff^{1}(M)$, any neighborhood $U_{p}$ of $orb(p)$, and any neighborhoods $U_{K}$ of $K$, there are two integer $l$ and $n_0$, such that, for any integers $T_K$,
\begin{enumerate}
\item there is $h\in \mathcal{U}$ such that:
\begin{itemize}
\item $h$ coincides with $f$ on $orb(p)\cup orb^-(x)\cup orb^+(y)$ and outside $U_K$;
\item the point $y$ is on the positive orbit of $x$ under $h$, with $\sharp (orb(x,h)\cap U_K)\geq T_K$ and $\sharp ((orb(x,h)\setminus (U_K\cup U_p))\leq n_0$.
\end{itemize}
\item for any $m\in\mathbb{N}$, there is $h_m\in \mathcal{U}$ such that,
\begin{itemize}
\item $h_m$ coincides with $h$ on $orb(p)$ and outside $U_p$,
\item $h_m$ has a periodic orbit $O$, satisfying $O\setminus U_p=(orb(x,h))\setminus U_p$, and $\sharp (O\cap U_p)\in \{l+m\tau,l+m\tau+1,\cdots,l+(m+1)\tau-1\}$.
\end{itemize}
\end{enumerate}
\end{pro}
\begin{rem}
It is obvious that, in the settings of the proposition, if we change ``$\omega(x)\cap K \neq \emptyset$ and $\alpha (y) = K$'' to ``$\alpha (y)\cap K \neq \emptyset$ and $\omega(x) = K$'', the conclusion still holds.
\end{rem}
Proposition~\ref{asymptotic connecting 1} and~\ref{asymptotic connecting} are in some sense doing an asymptotic connecting process from a point to an invariant compact set. Proposition~\ref{asymptotic connecting 1} tells that if a point on the unstable manifold of a periodic orbit satisfies that its positive limit set intersects an invariant compact set, then we can make its positive limit set contained in this invariant compact set by a small perturbation. Moreover, the perturbation will not change certain pieces of orbit. In fact, we can get the first property directly by the conclusions of~\cite{c1}, but the second property is not a direct consequence.
\begin{pro}(\emph{A modified case of Proposition 10 in~\cite{c1}})\label{asymptotic connecting 1}
Assume $f$ is a diffeomorphism in $\diff^1(M)$. For any hyperbolic periodic point $p$ of $f$, for any invariant compact set $K$, and for any point $x\in M$, such that:
\begin{itemize}
\item all periodic orbits in $K$ are hyperbolic and $p\not\in K$,
\item $\overline{W^u(p)}\cap K\neq \emptyset$, and $\alpha(x)\subset K$,
\end{itemize}
then, for any neighborhood $\mathcal{U}$ of $f$ in $\diff^{1}(M)$, there are a diffeomorphism $g\in \mathcal{U}$, a point $y\in W^u(p,f)$, and an open set $V$ containing $orb^-(x)$, such that $g$ coincides with $f$ on the set $orb(p)\cup K\cup V\cup orb^-(y)$ and $\omega(y,g)\subset K$.
\end{pro}
In the assumptions of the above two propositions, the point and invariant compact sets are linked by true orbits. However, Proposition~\ref{asymptotic connecting} deals with the case that they are linked by pseudo-orbits which is more complicated. We use the technics of~\cite{bc,c1}.
\begin{pro}\label{asymptotic connecting}
Assume $f_0$ is a diffeomorphism in $\diff^1(M)$. For any neighborhood $\mathcal{U}$ of $f_0$ in $\diff^{1}(M)$, there are a smaller neighborhood $\mathcal{U}'$ of $f_0$ with $\overline{\mathcal{U}'}\subset\mathcal{U}$ and an integer $T$, with the following properties.\\
For any diffeomorphism $f\in\mathcal{U}'$, considering an invariant compact set $K$, a positively invariant compact set $X$ and a point $z\in X$, if the following conditions are satisfied:
\begin{itemize}
\item all periodic orbits contained in $K$ are hyperbolic,
\item all periodic orbits contained in $X$ with period less than $T$ are hyperbolic,
\item for any $\vep>0$, there is a $\vep$-pseudo-orbit contained in $X$ connecting $z$ to $K$,
\end{itemize}
then for any neighborhood $U$ of $X\setminus K$, there is a diffeomorphism $g\in \mathcal{U}$, such that: $g=f|_{M\setminus U}$ and $\omega(z,g)\subset K$. Moreover, the $C^0$ distance between $g$ and $f$ can be arbitrarily small.
\end{pro}
\begin{rem}
$(1)$ Proposition~\ref{asymptotic connecting 1} is not a direct corollary of Proposition~\ref{asymptotic connecting}, because we wish to keep the negative orbit of a point that accumulates to the invariant compact set unchanged after perturbation in Proposition~\ref{asymptotic connecting 1}.
$(2)$ In Proposition~\ref{asymptotic connecting}, we can see that $X\cap K\neq\emptyset$. Thus $X\setminus K$ is not a compact set and we have that $\overline{U}\cap K\neq\emptyset$, where $U$ is the neighborhood of $X\setminus K$.
\end{rem}
\subsection{Organization of the paper.}
In Section~\ref{preliminary}, we give some basic definitions and well known results that we will use in the proof. In Section~\ref{theorem b}, we give a slightly different version (Theorem B) of Theorem A, and we prove Theorem A using Theorem B. After, we give the proof of Theorem B from Propositions 1, 2 and 3 in Section~\ref{proof of theorem b}. The proofs of Proposition 1, 2 and 3 will be given in Section~\ref{proposition 1},~\ref{proposition 2} and~\ref{proposition 3} respectively. At last, we give the proofs of the applications of the main theorem in Section~\ref{applications}.
\section{Preliminary}\label{preliminary}
In this section, we give some definitions and some well known results. Denote by $\diff^1(M)$ the space of $C^1$-diffeomorphisms of $M$.
\subsection{Hyperbolicity and dominated splitting}
\begin{defi}
Assume that $f$ is a diffeomorphism in $\diff^1(M)$, $\Lambda$ is an invariant compact set of $f$ and $E$ is a $Df$-invariant subbundle of $T_{\Lambda}M$. We say that the bundle $E$ is \emph{$(C,\lambda)$-contracted} if there are constants $C>0$ and $\lambda\in (0,1)$, such that
\begin{displaymath}
\|Df^n|_{E(x)}\|<C\lambda^n,
\end{displaymath}
for all $x\in \Lambda$ and all $n\geq 1$. And we say that $E$ is \emph{$(C,\lambda)$-expanded} if it is \emph{$(C,\lambda)$-contracted} with respect to $f^{-1}$. If the tangent bundle of $\Lambda$ has an invariant splitting $T_{\Lambda}M=E^s\oplus E^u$, such that, $E^s$ is $(C,\lambda)$-contracted and $E^u$ is $(C,\lambda)$-expanded for some constants $C>0$ and $\lambda\in (0,1)$, then we call $\Lambda$ a \emph{hyperbolic set} and $dim(E^s)$ the \emph{index} of the hyperbolic splitting. Moreover, if a periodic orbit $orb(p)$ is a hyperbolic set, then we call $p$ a hyperbolic periodic point, and the dimension of the contracted bundle $E^s$ in the hyperbolic splitting is called the \emph{index} of $p$, denoted by $ind(p)$.
\end{defi}
\begin{defi}
For any point $x\in M$, any number $\delta>0$, we define the \emph{local stable set} and \emph{local unstable set} of $x$ of size $\delta$ respectively as follows:\\
$W^s_{\delta}(x)=\{y: \forall n\geq 0, d(f^n(x),f^n(y))\leq\delta; \text{ and } \lim_{n\rightarrow +\infty} d(f^n(x),f^n(y))=0\}$;\\
$W^u_{\delta}(x)=\{y: \forall n\geq 0, d(f^{-n}(x),f^{-n}(y))\leq\delta; \text{ and } \lim_{n\rightarrow +\infty} d(f^{-n}(x),f^{-n}(y))=0\}$.\\
We define the \emph{stable set} and \emph{unstable set} of $x$ respectively as follows:\\
$W^s(x)=\{y:\lim_{n\rightarrow +\infty} d(f^n(x),f^n(y))=0\}$;\\
$W^u(x)=\{y:\lim_{n\rightarrow +\infty} d(f^{-n}(x),f^{-n}(y))=0\}$.
\end{defi}
\begin{rem}
$(1)$ It is obvious that, for any $\delta>0$, we have
\begin{center}
$W^s(x)=\cup_{n\geq 0}f^{-n}(W^s_{\delta}(f^n(x)))$
\end{center}
and
\begin{center}
$W^u(x)=\cup_{n\geq 0}f^{n}(W^u_{\delta}(f^{-n}(x)))$.
\end{center}
$(2)$ To belong to a same stable set is an equivalent relation, thus two stable sets either coincide or are disjoint with each other. Similarly with the unstable set.
\end{rem}
For hyperbolic sets, the (local) stable (resp. unstable) set has the following properties, see for example~\cite{hps}.
\begin{lem}
If $\Lambda$ is a hyperbolic set and $T_{\Lambda}M=E^s\oplus E^u$ is the hyperbolic splitting, then there is a number $\delta>0$, such that, for any $x\in\Lambda$, the local stable (resp. unstable) set $W^s_{\delta}(x)$ (resp. $W^u_{\delta}(x))$ is an embedding disk with dimension $dim(E^s)$ (resp. $dim(E^u)$) and is tangent to $E^s$ (resp. $E^u$) at $x$. Moreover, the stable (resp. unstable) set $W^s(x)$ (resp. $W^u(x)$) of $x$ is an immersed submanifold of $M$.
\end{lem}
\begin{defi}
Assume that $f$ is a diffeomorphism in $\diff^1(M)$ and $p,q\in M$ are two hyperbolic periodic point of $f$, we say $p$ and $q$ are \emph{homoclinically related} and denote the relation by $p\sim q$, if $W^u(orb(p))$ has non-empty transverse intersections with $W^s(orb(q))$, and $W^s(orb(p))$ has non-empty transverse intersections with $W^u(orb(q))$, denoted by $W^u(orb(p))\pitchfork W^s(orb(q))\neq \emptyset$ and $W^s(orb(p))\pitchfork W^u(orb(q))\neq\emptyset$. We call the closure of the set of periodic orbits homoclinically related to $orb(p)$ the \emph{homoclinic class} of $p$ and denote it by $H(p,f)$ or $H(p)$ for simplicity.
\end{defi}
\begin{defi}\label{dominated splitting}
Assume $f\in \diff^1(M)$. An invariant compact set $\Lambda$ of $M$ is said to have an \emph{$(m,\lambda)$-dominated splitting}, if the tangent bundle has an Df-invariant splitting $T_{\Lambda}M=E\oplus F$ and there are an integer $m$ and a constant $\lambda\in (0,1)$ such that
\begin{center}
$\|Df^m|_{E(x)}\|\cdot \|Df^{-m}|_{F(f^mx)}\|<\lambda$.
\end{center}
We call $dim(E)$ the \emph{index} of the dominated splitting. Moreover, we say $\Lambda$ has a \emph{partially hyperbolic splitting}, if the tangent bundle has an invariant splitting $T_{\Lambda}M=E^s\oplus E^c\oplus E^u$, such that the two splittings $(E^s\oplus E^c)\oplus E^u$ and $E^s\oplus (E^c\oplus E^u)$ are both dominated splittings and, moreover, the bundle $E^s$ is contracted, the bundle $E^u$ expanded and the central bundle $E^c$ is neither contracted nor expanded.
\end{defi}
\begin{rem}\label{bundle of ds}
We point out here that if an invariant compact set $\Lambda$ has two dominated splittings $T_{\Lambda}M=E_1\oplus F_1=E_2\oplus F_2$ such that $dim(E_1)\geq dim(E_2)$, then we have $E_1\subset E_2$. Hence two dominated splittings on an invariant compact set with the same index would coincide.
\end{rem}
By~\cite{g1}, there is always an \emph{adopted metric} for a dominated splitting, that is to say, an $(m,\lambda)$-dominated splitting is a $(1,\lambda)$-dominated splitting by considering a metric equivalent to the original one. Also, it is obvious that an $(m,\lambda)$-dominated splitting is always an $(mN,\lambda)$-dominated splitting for any positive integer $N$.
\subsection{Recurrence}
We give some definitions of recurrence.
\begin{defi}\label{pseudo-orbit}
For a diffeomorphism $f\in\diff^1(M)$ and a number $\vep>0$, we call a sequence of points $\{x_i\}_{i=a}^b$ of $M$ an \emph{$\vep$-pseudo orbit of $f$}, if $d(f(x_i),x_{i+1})<\vep$ for any $i=a,a+1,\cdots,b-1$, where $-\infty\leq a<b\leq \infty$. An invariant compact set $K$ is called a \emph{chain transitive set}, if for any $\vep>0$, there is a periodic $\vep$-pseudo-orbit contain in $K$ and $\vep$-dense in $K$.
\end{defi}
\begin{defi}\label{chain recurrence}
Assume $f\in\diff^1(M)$. We say a point $y$ is \emph{chain attainable} from $x$, if for any number $\vep>0$, there is an $\vep$-pseudo orbit of $f$ $(x_0,x_1,\cdots,x_n)$ such that $x_0=x$ and $x_n=y$, and we denote it by $x\dashv y$. The \emph{chain recurrent set} of a diffeomorphism $f\in\diff^1(M)$, denoted by $R(f)$, is the union of the point $x$ such that $x$ is chain attainable from itself.
\end{defi}
It is well known that the chain recurrent set $R(f)$ of $f$ can be decomposed into a disjoint union of invariant compact "undecomposable" sets. More precisely, we give the definition as the following.
\begin{defi}\label{chain class}
Assume $f\in\diff^1(M)$. For any two points $x,y\in M$, denote $x\sim y$ if $x$ is chain attainable from $y$ and $y$ is chain attainable from $x$. Obviously $\sim$ is an equivalent relation on $R(f)$, and an equivalent class of $\sim$ is called a \emph{chain recurrence class}.
\end{defi}
\begin{defi}
Assume $f\in\diff^1(M)$ and $\Lambda$ is an invariant compact set of $f$. We say that $\Lambda$ is \emph{shadowable}, if for any $\vep>0$, there is $\delta>0$, such that for any $\delta$-pseudo orbit $\{x_i\}_{i=a}^b\subset \Lambda$ of $f$, where $-\infty\leq a<b\leq\infty$, there is a point $y\in M$, such that $d(f^i(y),x_i)<\vep$ for all $a\leq i\leq b$.
\end{defi}
Now we give another definition of a relation, which is denoted by $\prec$.
\begin{defi}
Assume $f$ is a diffeomorphism in $\diff^1(M)$ and $W$ is an open set of $M$. For any two points $x,y\in M$, we denote $x\prec y$ if for any neighborhood $U$ of $x$ and any neighborhood $V$ of $y$, there are a point $z\in M$ and an integer $n\geq 1$, such that $z\in U$ and $f^n(z)\in V$. We denote $x\prec_W y$ if for any neighborhood $U$ of $x$ and any neighborhood $V$ of $y$, there is a piece of orbit $(z,f(z),\cdots,f^n(z))$ contained in $W$ such that $z\in U$ and $f^n(z)\in V$. Moreover, let $K$ be a compact set of $M$, then we denote $x\prec K$ (resp. $x\prec_{W} K$) if there is a point $y\in K$, such that $x\prec y$ (resp. $x\prec_{W} y$).
\end{defi}
For the relation $\prec$, we have the following result, whose proof is similar to the proof of Lemma 6 in~\cite{c1}.
\begin{lem}\label{prec}
Assume that $K$ is an invariant compact set. Then for any two neighborhoods $U_2\subset U_1$ of $K$ and any point $y\in U_1$ satisfying $y\prec_{U_1} K$, there is a point $y'\in U_2$, such that $y\prec_{U_1} y'\prec_{U_2} K$ and the positive orbit of $y'$ is contained in $U_2$.
\end{lem}
It is obvious that $x\prec y$ implies $x\dashv y$, but the two relations are not equivalent. In~\cite{bc}, they have proved that for generic diffeomorphisms, the two relations are equivalent.
\begin{lem}\emph{\cite{bc}}
For generic diffeomorphism $f\in\diff^1(M)$, if $x\dashv y$, then $x\prec y$.
\end{lem}
\subsection{Pliss points and weak sets}
\begin{defi}\label{pliss point}
Assume that $\Lambda$ is an invariant compact set of a diffeomorphism $f$ in $\diff^1(M)$ and $E$ is an invariant sub-bundle of $T_{\Lambda}M$. For a constant $\lambda\in(0,1)$, we call $x\in \Lambda$ an \emph{$(m,\lambda)$-$E$-Pliss point}, if for any integer $n>0$, we have
\begin{displaymath}
\prod_{i=0}^{n-1} \|Df^{im}|_{E(f^{im}(x))}\|\leq {\lambda}^n.
\end{displaymath}
Particularly, if $m=1$, we call $x$ a \emph{$\lambda$-$E$-Pliss point} for short.
\end{defi}
\begin{defi}\label{weak set}
Consider a diffeomorphism $f\in \diff^1(M)$, an invariant compact set $K$ of $f$, an invariant sub-bundle $E$ of $T_K M$, an integer $m$ and a constant $\lambda\in (0,1)$. We say that $K$ is an \emph{$(m,\lambda)$-$E$-weak set}, if for any point $x\in K$, there is an integer $n_x$, such that
\begin{displaymath}
\prod_{i=0}^{n_x-1} \|Df^m|_{E(f^{im}(x))}\|> {\lambda}^{n_x}.
\end{displaymath}
We denote $N_x$ the smallest integer that satisfies the above inequality. Particularly, if $m=1$, we call $K$ a \emph{$\lambda$-$E$-weak set} for short.
\end{defi}
\begin{rem}\label{rem of weak set}
If $K$ is an $(m,\lambda)$-$E$-weak set, by the compactness of $K$, we can see that $N_x$ is bounded by an integer $N_K$ for all $x\in K$. Also from the definition, we can see that an invariant compact set $K$ is an $(m,\lambda)$-$E$-weak set if and only if $K$ does not contain any $(m,\lambda)$-$E$-Pliss point.
\end{rem}
One can obtain Pliss points by the following lemma given by V. Pliss, see~\cite{p,ps}.
\begin{lem}\emph{(Pliss lemma)}\label{pliss lemma}
Assume that $\Lambda$ is an invariant compact set of a diffeomorphism $f$ in $\diff^1(M)$ and $E$ is an invariant sub-bundle of $T_{\Lambda}M$. For any two numbers $0<\lambda_1<\lambda_2<1$, we have:
\begin{enumerate}
\item There are a positive integer $N=N(\lambda_1,\lambda_2,f)$ and a number $c=c(\lambda_1,\lambda_2,f)$ such that for any $x\in \Lambda$ and any number $n\geq N$, if
\begin{displaymath}
\prod_{i=0}^{n-1}\|Df|_{E(f^ix)}\|\leq {\lambda_1}^n,
\end{displaymath}
then there are $0\leq n_1<n_2<\cdots<n_l\leq n$ such that $l\geq cn$, and, for any $j=1,\cdots,l$ and any $k=n_j+1,\cdots,n$,
\begin{displaymath}
\prod_{i=n_j}^{k-1}\|Df|_{E(f^ix)}\|\leq {\lambda_2}^{k-n_j}.
\end{displaymath}
\item For any point $x\in \Lambda$, and any integer $m$, if for all $n\geq m$,
\begin{displaymath}
\prod_{i=0}^{n-1}\|Df|_{E(f^ix)}\|\leq {\lambda_1}^n,
\end{displaymath}
then there is an infinite sequence $0\leq n_1<n_2<\cdots$, such that
\begin{displaymath}
\prod_{i=n_j}^{k-1}\|Df|_{E(f^ix)}\|\leq {\lambda_2}^{k-n_j},
\end{displaymath}
for all $k>n_j$ and all $j=1,2,\cdots$.
\end{enumerate}
\end{lem}
\begin{cor}\label{cor of pliss}
For a diffeomorphism $f\in \diff^1(M)$ and an $f$-invariant continuous bundle $E\subset T_{\Lambda}M$ of an invariant compact set $\Lambda$, we have that, for any $x\in \Lambda$:
\begin{enumerate}
\item If $x$ is an $(m,\lambda)$-$E$-Pliss point, then there is a point $y\in \omega(x)$, such that $y$ is also a $(m,\lambda)$-$E$-Pliss point.
\item If for any $y\in\omega(x)$, there is an integer $n_y\in\mathbb{N}$, such that
\begin{displaymath}
\prod_{i=0}^{n_y-1}\|Df^m|_{E(f^{im}(y))}\|\leq {\lambda}^{n_y},
\end{displaymath}
then for any $\lambda'\in (\lambda,1)$, there are infinitely many $(m,\lambda')$-$E$-Pliss points on $orb^+(x)$.
\end{enumerate}
\end{cor}
\begin{proof}
By considering the diffeomorphism $f^m$ instead of $f$, we can assume that $m=1$. The proof of the general case is similar.
$(1)$ By item 2 of Pliss lemma, for any $\lambda'\in (\lambda,1)$, there are infinitely many $\lambda'$-$E$-Pliss points on $orb^+(x)$. Take a limit point of these $\lambda'$-$E$-Pliss points, denote it by $y_{\lambda'}$, then $y_{\lambda'}\in \omega(x)$ is a $\lambda'$-$E$-Pliss point. We take a sequence of numbers $(\lambda_n)_{n\geq 1}$ such that $\lambda_n\in (\lambda,1)$ and $\lambda_n\rightarrow \lambda$ when $n$ goes to infinity. Then for any $n\geq 1$, there is a $\lambda_n$-$E$-Pliss point $y_{\lambda_n}\in\omega(x)$. Taking a subsequence if necessary, we assume $(y_{\lambda_n})_{n\geq 1}$ converges to a point $y\in \omega(x)$. Then $y$ is a $\lambda_n$-$E$-Pliss point for any $n\geq 1$. Since $\lambda_n\rightarrow \lambda$, the point $y$ is a $\lambda$-$E$-Pliss point.
$(2)$ By the compactness of $\omega(x)$, there is an integer $N$, such that $n_y\leq N$ for any $y\in\omega(x)$. There is a constant $C>0$, such that, for any $y\in \omega(x)$, we have
\begin{displaymath}
\prod_{i=0}^{n-1}\|Df|_{E(f^i(y))}\|<C\lambda^n.
\end{displaymath}
Take a constant $\lambda'\in (\lambda,1)$. Take three constants $\lambda_1<\lambda_2<\lambda_3$ contained in $(\lambda,\lambda')$. There is $N\in\mathbb{N}$, such that $C\lambda^n<\lambda_1^n$ for any $n\geq N$. There is $\vep>0$, such that, for any two points $x_1,x_2\in \Lambda$, if $d(f(x_1),f(x_2))<\vep$, then $\frac{\|Df|_{E(f^i(x_1))}\|}{\|Df|_{E(f^i(x_2))}\|}<\frac{\lambda_2}{\lambda_1}$, for all $i=0,1,\cdots,N$. By considering an iterate of $x$ instead of $x$, we can assume that $d_H(\overline{orb^+(x)},\omega(x))<\vep$, where $d_H(\cdot,\cdot)$ is the Hausdorff distance. Then for any $n\geq 1$, we have
\begin{displaymath}
\prod_{i=0}^{nN}\|Df|_{E(f^i(x))}\|<(C\lambda^N)^n\left(\frac{\lambda_2}{\lambda_1}\right)^{nN}<\lambda_2^{nN}.
\end{displaymath}
There is $T>0$, such that, for any $k\geq T$, we have $\lambda_2^{kN}\|Df\|^j<\lambda_3^{kN+j}$ for all $j=0,1,\cdots,N-1$. Then for any $n>TN$, assume $n=kN+j$, where $0\leq j<N$, we have
\begin{displaymath}
\prod_{i=0}^{n}\|Df|_{E(f^i(x))}\|\leq\left(\prod_{i=0}^{kN}\|Df|_{E(f^i(x))}\|\right)\|Df\|^j<\lambda_2^{kN}\|Df\|^j<\lambda_3^n.
\end{displaymath}
Then by item $2$ of Pliss lemma, there are infinitely many $(m,\lambda')$-$E$-Pliss points on $orb^+(x)$.
\end{proof}
\begin{defi}\label{consecutive pliss}
Assume that $\Lambda$ is an invariant compact set of a diffeomorphism $f$ in $\diff^1(M)$ and $E$ is an invariant sub-bundle of $T_{\Lambda}M$. We call two $(m,\lambda)$-$E$-Pliss points $(f^{n}(x),f^{l}(x))$ on a single orbit \emph{consecutive} $(m,\lambda)$-$E$-Pliss points, if $n<l$ and for all $n<k<l$, $f^k(x)$ is not a $(m,\lambda)$-$E$-Pliss point. And if there is a dominated splitting $T_{\Lambda}M=E\oplus F$ on $\Lambda$, we call $x\in \Lambda$ an \emph{$(m,\lambda)$-bi-Pliss point}, if it is an $(m,\lambda)$-$E$-Pliss point for $f$ and an $(m,\lambda)$-$F$-Pliss point for $f^{-1}$.
\end{defi}
For Pliss-points, we have the following lemma. The technics of the proof can be found in many papers, for example~\cite{ps}.
\begin{lem}\label{property of pliss point}
Assume $\Lambda$ is an invariant compact set of a diffeomorphism $f\in \diff^1(M)$ with an $(m,\lambda^2)$-dominated splitting $T_{\Lambda}M=E\oplus F$. We have that, for any $\lambda'\in (\lambda,1)$:
\begin{enumerate}
\item If a sequence of consecutive $(m,\lambda')$-$E$-Pliss points $(f^{n_i}(x_i),f^{l_i}(x_i))_{i\geq 0}$ satisfies that $l_i-n_i\rightarrow +\infty$, then, take any limit point $y$ of the sequence $(f^{l_i}(x_i))$, we have that $y$ is a $(m,\lambda')$-bi-Pliss point.
\item If there are both $(m,\lambda')$-$E$-Pliss points for $f$ on $orb^+(x)$ and $(m,\lambda')$-$F$-Pliss points for $f^{-1}$ on $orb^-(x)$, then there is at least one $(m,\lambda')$-bi-Pliss point on $orb(x)$.
\item If $x\in \Lambda$ is an $(m,\lambda')$-$E$-Pliss point and there are no other $(m,\lambda')$-$E$-Pliss points on $orb^-(x)$, then $x$ is also an $(m,\lambda)$-$F$-Pliss point for $f^{-1}$. Thus $x$ is an $(m,\lambda')$-bi-Pliss point.
\end{enumerate}
\end{lem}
We have the following selecting lemma of Liao to get weak periodic orbits (see~\cite{l2},~\cite{w-selecting}).
\begin{lem}\emph{(Liao's selecting lemma).}\label{selecting}
Assume $f\in \diff^1(M)$. Consider an invariant compact set $\Lambda$ with a non-trivial $(m,\lambda)$-dominated splitting $T_\Lambda M=E\oplus F$, and $\lambda_0\in (\lambda,1)$, if the following two conditions are satisfied:
\begin{itemize}
\item There is a point $b\in \Lambda$, such that, for all $n\geq 1$, we have:
\begin{displaymath}
\prod_{i=0}^{n-1} \|Df^m|_{E(f^{im}(b))}\|\geq 1.
\end{displaymath}
\item For any invariant compact subset $K\subsetneqq \Lambda$, there is an $(m,\lambda_0)$-$E$-Pliss point $x\in K$.
\end{itemize}
Then for any neighborhood $U$ of $\Lambda$, for any $\lambda_1<\lambda_2$ contained in $(\lambda_0,1)$, there is a periodic orbit $orb(q)\subset U$ with period $\tau(q)$ a multiple of $m$, such that, for all $n=1,\cdots,\tau(q)/m$, the following two inequalities are satisfied:
\begin{displaymath}
\prod_{i=0}^{n-1} \|Df^m|_{E(f^{im}(q))}\|\leq {\lambda_2}^n,
\end{displaymath}
and
\begin{displaymath}
\prod_{i=n-1}^{\tau(q)/m-1} \|Df^m|_{E(f^{im}(q))}\|\geq {\lambda_1}^{\tau(q)/m-n+1}.
\end{displaymath}
Particularly, one can find a sequence of periodic points that are homoclinic related with each other and converges to a point in $\Lambda$. Similar assertions for $F$ hold with respect to $f^{-1}$.
\end{lem}
\subsection{Perturbation technics}
We give some tools for $C^1$-perturbation. First is the famous Hayashi's connecting lemma, see~\cite{h,wx}. The general connecting lemma deals with a single diffeomorphism and a given neighborhood. Here we give a uniform version that is valid to a neighborhood of a diffeomorphism, see~\cite{w2}.
\begin{theo}\emph{(A uniform connecting lemma, Theorem A of~\cite{w2})}\label{uniform connecting}
Assume that $f$ is a diffeomorphism in $\diff^1(M)$. For any $C^1$ neighborhood $\mathcal{U}$ of $f$ in $\diff^1(M)$, there are three numbers $\rho>1$, $\delta_0>0$ and $N\in\mathbb{N}$, together with a $C^1$ neighborhood $\mathcal{U}_1\subset\mathcal{U}$ of $f$ in $\diff^1(M)$, that satisfy the following property:\\
For any $f_1\in\mathcal{U}_1$, any point $z\in M$ and any number $0<\delta<\delta_0$, as long as the $N$ balls $(f_1^i(B(z,\delta)))_{0\leq i\leq N-1}$ are pairwise disjoint and each is of size smaller than $\delta_0$ (that is to say, $f_1^i(B(z,\delta))\subset B(f_1^i(z),\delta_0)$), then for any two points $x$ and $y$ that are outside the set $\Delta=\bigcup_{0\leq i\leq N-1}f_1^i(B(z,\delta))$, if there are two positive integers $n_x$ and $n_y$ such that $f_1^{n_x}(x)\in B(z,\delta/\rho)$ and $f_1^{-n_y}(y)\in B(z,\delta/\rho)$, then there are a diffeomorphism $g\in\mathcal{U}$ and a positive integer $m$ such that $g^m(x)=y$ and $g=f_1$ off $\Delta$. Moreover, the piece of orbit $\{x,g(x),\cdots,g^m(x)=y\}$ is contained in the set $\{x,f_1(x),\cdots,f_1^{n_x}(x)\}\cup \Delta\cup \{y,f_1^{-1}(y),\cdots,f_1^{-n_y}(y)\}$ and the number $m$ is no more than $n_x+n_y$.
\end{theo}
To control the perturbing neighborhood when connecting two points that are close, we have the following lemma, see~\cite{a}.
\begin{lem}\label{basic perturbation}\emph{(Basic perturbation lemma).} For any neighborhood $\mathcal{U}$ of a diffeomorphism $f\in \diff^1(M)$, there are two numbers $\theta>1$ and $r_0>0$ satisfying: for any two points $x,y\in M$ contained in a ball $B(z,r)$, where $r\leq r_0$, there is a diffeomorphism $g\in \mathcal{U}$, such that $g(x)=f(y)$, and $g$ coincides with $f$ outside the ball $B(z,\theta\cdot r)$.
\end{lem}
\begin{defi}
For a chart $\varphi:V\rightarrow \mathbb{R}^d$ of $M$, we call a set $C$ a \emph{cube} of $\varphi$ if $\varphi(C)$ is the image of $[-a,a]^d$ by a translation of $\mathbb{R}^d$, where $a$ is the radius of the cube. If a cube with radius $(1+\vep)a$ and the same center of $\varphi(C)$ is still contained in $\varphi(V)$, we denote by $(1+\vep)C$ its pre-image of $\varphi$.
\end{defi}
\begin{defi}
Consider a chart $\varphi:V\rightarrow \mathbb{R}^d$. A \emph{tiled domain} according to the chart of $\varphi$ is an open set $U\subset V$ and a family $\mathcal{C}$ of cubes of $\varphi$ (called \emph{tiles} of domain), such that:
\begin{enumerate}
\item the interior of the tiles are pairwise disjoint;
\item the union of all tiles of $\mathcal{C}$ equals to $U$;
\item the geometry of the tiling is bounded, i.e.
\end{enumerate}
\begin{itemize}
\item the number of tiles around each point is uniformly bounded (by $2^d$), that is to say, there is a neighborhood for each point that meets at most $2^d$ tiles,
\item for any two pairs $(C,C')$ of intersecting tiles, the rate of their diameters is uniformly bounded (by $2$).
\end{itemize}
\end{defi}
By a standard construction, any open set $U\subset V$ can be tiled according to the coordinates of $\varphi$ (e.g.~\cite{bc,c2}).
\begin{defi}\label{perturbation domain}
Assume $f\in\diff^1(M)$. Consider a neighborhood $\mathcal{U}\subset \diff^1(M)$ and a number $N$. A tiled domain $(U,\mathcal{C})$ is called a \emph{perturbation domain} of order $N$ of $(f,\mathcal{U})$, if the following properties are satisfied.
\begin{enumerate}
\item $U$ is disjoint from its $N$ first iterates of $f$.
\item For any finitely many sequence of pairs of points $\{(x_i,y_i)\}_{1\leq i\leq l}$ in $U$, such that for any $i=1,2,\cdots,l$, the points $x_i$ and $y_i$ are contained in the same tile of $\mathcal{C}$, then there exist:
\begin{itemize}
\item a diffeomorphism $g\in\mathcal{U}$, that coincides with $f$ outside $\bigcup_{0\leq i\leq N-1}f^i(U)$,
\item a strictly increasing sequence $1=n_0<n_1<\cdots<n_k\leq l$, such that $g^N(x_{n_i})=f^N(y_{n_{i+1}-1})$ for any $i\neq k$, and $g^N(x_{n_k})=f^N(y_l)$.
\end{itemize}
\end{enumerate}
The union $\bigcup_{0\leq i\leq N-1}f^i(U)$ is called the \emph{support} of the perturbation domain $(U,\mathcal{C})$ and denoted by $supp(U)$.
\end{defi}
\begin{defi}\label{jumps}
A pseudo-orbit $(x_0,x_1,\cdots,x_l)$ is said to \emph{keep the tiles} of a perturbation domain $(U,\mathcal{C})$ of order $N$ of $(f,\mathcal{U})$, if the intersection of the pseudo-orbit and $supp(U)$ is a union of segments $x_{n_i},x_{n_i+1},\cdots,x_{n_i+N-1}$ of the form that $x_{n_i}\in U$ and for any $j=1,2,\cdots,N-1$, $x_{n_i+j}=f^j(y_{n_i})$, where $y_{n_i}$ is a point contained in the same tile of $\mathcal{C}$ as $x_{n_i}$. A pseudo-orbit $(x_0,x_1,\cdots,x_k)$ is said to \emph{have jumps only in tiles} of a perturbation domain $(U,\mathcal{C})$ of order $N$ of $(f,\mathcal{U})$, if it keeps the tiles and for any $x_i\notin supp(U)$, we have $x_{i+1}=f(x_i)$. For a family of perturbation domains $(U_k,\mathcal{C}_k)_{k\geq 0}$ of order $N_k$ of $(f,\mathcal{U}_k)$ with disjoint support, we say that a pseudo-orbit $(x_0,x_1,\cdots,x_l)$ has \emph{jumps only in tiles} of the perturbation domains $(U_k,\mathcal{C}_k)_{k\geq 0}$, if it keeps the tiles of the perturbation domains and for any $x_i\notin\bigcup_k supp(U_k)$, we have $x_{i+1}=f(x_i)$.
\end{defi}
By the proof of connecting lemma in~\cite{ar}, the perturbation domain always exists (see also Th\'{e}or\`{e}me 2.1 of~\cite{bc} and Th\'{e}or\`{e}me 3.3 of~\cite{c2}).
\begin{theo}\emph{(Another statement of the connecting lemma)}\label{existence of perturbation domain}
For any neighborhood $\mathcal{U}$ of $f$, there is an integer $N\geq 1$, and for all point $p\in M$, there is a chart $\varphi:V\rightarrow \mathbb{R}^d$ such that any tiled domain $(U,\mathcal{C})$ according to $\varphi$ disjoint from its $N$ first iterates is a perturbation domain of order $N$ for $(f,\mathcal{U})$.
\end{theo}
From the definitions above, we can get the following lemma easily.
\begin{lem}\label{union of perturbation domains}\emph{(Lemme 2.3 of~\cite{bc})}
For a family of disjoint perturbation domains $(U_k,\mathcal{C}_k)$ of order $N_k$ of $(f,\mathcal{U}_k)$ with disjoint support, if there is a pseudo-orbit $(p=p_0,p_1,\cdots,p_m=q)$ that has only jumps in the tiles of $(U_k,\mathcal{C}_k)_{k\geq 0}$ and $p_0,p_m\not \in U_k\cup\cdots\cup f^{N_k-1}(U_k)$ for all $k\geq 0$, then for any $i$, there is $g_i\in \mathcal{U}_i$ and a new pseudo-orbit $(p=p_0',\cdots,p_{m'}'=q)$ of $g_i$ that has only jumps in the tiles of domains $(U_k,\mathcal{C}_k)_{k\geq 0,k\neq i}$. Moreover, $g_i=f|_{M\setminus (U_i\cup\cdots\cup f^{N_i-1}(U_i))}$ and $\{p_0',\cdots,p_{m'}'\}\setminus (U_i\cup\cdots\cup f^{N_i-1}(U_i))\subset \{p_0,p_1,\cdots,p_m\}$, and $m'\leq m$.
\end{lem}
\subsection{Topological towers}
In this subsection, we introduce two lemmas of~\cite{bc} that are useful to get a true orbit by perturbing a pseudo-orbit. These two lemmas are the key tools in the proof of Proposition~\ref{asymptotic connecting}. First we give the following lemma that is useful to choose perturbation neighborhoods. In fact, it is a general result of Lemma 3.7 in~\cite{bc}, but one can get the conclusion directly from the proof in~\cite{bc}.
\begin{lem}\label{choose neighborhoods}
There is a constant $\kappa_d>0$ (which only depends on the dimension d of $M$) satisfying the following property: assume $N>0$ is an integer and $W'$ and $V'$ are two compact sub-manifolds with boundary of $M$ of dimension $d$, if $V'$ is disjoint from its $\kappa_d N$ first iterates, then, for any neighborhood $U_1$ of $W'$ and any neighborhood $U_2$ of $V'$, there is an open set $S$, such that:
\begin{enumerate}
\item $V'\subset \bigcup_{i=0}^{\kappa_d N}f^{-i}(S)$.
\item $S=W\cup V$, where $W$ and $V$ satisfy the following:
\begin{itemize}
\item $W'\subset W\subset U_1$;
\item $V$ is contained in $U_2\cup f(U_2)\cup\cdots\cup f^{\kappa_dN}(U_2)$ and disjoint from its $N$ first iterates.
\item $\overline{W}\cap (\bigcup_{i=-N}^{N} f^{i}(\overline{V}))=\emptyset$.
\end{itemize}
\end{enumerate}
\end{lem}
Then, we give a lemma of~\cite{bc} for the construction of what they called \textit{topological tower} (see Th\'{e}or\`{e}me 3.1 and Corollaire 3.1 in~\cite{bc}). Denote by $Per_{N_0}(f)$ the set of periodic orbits contained with period less than $N_0$,
\begin{lem}\emph{(Topological Tower})\label{ttower}
There is a constant $\kappa_d>0$ (which only depends on the dimension d of $M$), such that, for any $N_0\in \mathbb{N}$, any constant $\delta>0$, any compact set $K$ of $f\in \diff^1(M)$ that does not contain any non-hyperbolic periodic orbits with periods less than $\kappa_d N_0$ and any neighborhood $U_0$ of $K$, there exist an open set $V$ and a compact set $D\subset V$, satisfying the following properties:
\begin{enumerate}
\item For any point $x\in K$ with $x\not \in \bigcup_{p\in Per_{N_0}(f)}W^s_{\delta}(p)$, there is $n>0$, such that $f^n(x)\in int(D)$.
\item The sets $\overline{V},f(\overline{V}),\cdots,f^{N_0}(\overline{V})$ are pairwise disjoint.
\item The set $\overline{V}$ is contained in $U_0\cup f(U_0)\cup \cdots \cup f^{\kappa_d N_0}(U_0)$.
\end{enumerate}
Moreover, the diameter of all connected components of $V$ can be arbitrarily small.
\end{lem}
\begin{rem}
$(1)$ In~\cite{bc}, the lemma is stated for an invariant compact set $K$, and the third property is not stated. But from the proof of the existence of topological tower, we can see that it is also true for non-invariant compact sets and also the third property is true.
$(2)$ We explain a sketch of the proof of Lemma~\ref{ttower}. Take $\kappa_d$ to be the constant in Lemma~\ref{choose neighborhoods}. First, one can take a compact sub-manifold $U_0$ of $M$ with boundary that is disjoint from its first $N_0$ iterates, such that, any point in a small neighborhood $O$ of $Per_{N_0}(f)$ that is not on the local stable manifold of $Per_{N_0}(f)$ has a positive iterate in $U_0$. Then one can take a finite cover of the compact set $K\setminus O$ by open sets that are disjoint from their first $\kappa_d N_0$ iterates (generally, they are not disjoint from each other). Finally, by Lemma~\ref{choose neighborhoods}, one can construct an open set that is disjoint from its first $N_0$ iterates by a finite induction, such that, any point that is not on the local stable manifold of $Per_{N_0}(f)$ has a positive iterate in it.
\end{rem}
\subsection{Generic properties}
A set $R$ of a topological Baire space $X$ is called a \emph{residual} set, if $R$ contains a dense $G_{\delta}$ set of $X$. We say a property is a \emph{generic} property of $X$, if there is a residual set $R\subset X$, such that each element contained in $R$ satisfies the property. We give some well known $C^1$-generic properties of diffeomorphisms in the following lemma. These results can be found in many papers like~\cite{bdv,c1,po}.
\begin{lem}\label{generic properties}
There is a residual set $\mathcal{R}$ in $\diff^1(M)$ of diffeomorphisms, such that any $f\in\mathcal{R}$ satisfies the following properties:
\begin{enumerate}
\item The diffeomorphism $f$ is Kupka-Smale: all periodic points of $f$ are hyperbolic and the stable and unstable manifolds of periodic orbits intersect transversely.
\item The periodic points are dense in the chain recurrent set and any chain recurrence class is either a homoclinic class or contains no periodic point.
\item For a periodic point $p$ of $f$, there exists a $C^1$-neighborhood $\mathcal{U}_1$ of $f$, such that every $g\in\mathcal{U}_1\cap\mathcal{R}$ is a continuity point for the map $g\mapsto H(p_g,g)$ where $p_g$ is the continuation of $p$ for $g$, where the continuity is with respect to the Hausdorff distance between compact subsets of $M$.
\item If $H(p)$ is a homoclinic class of $f$, then there exists an interval $[\alpha,\beta]$ of natural numbers and a $C^1$-neighborhood $\mathcal{U}_2$ of $f$, such that for every $g\in\mathcal{U}_2$, the set of indices of hyperbolic periodic points contained in $H(p_g,g)$ is $[\alpha,\beta]$. Also, all periodic points of the same index contained in $H(p)$ are homoclinically related.
\item If a homoclinic class $H(p)$ contains periodic orbits with different indices, then $f$ can be $C^1$ approximated by diffeomorphisms having a heterodimensional cycle.
\item If a homoclinic class $H(p)$ is Lyapunov stable, then there is a $C^1$ neighborhood $\mathcal{U}_3$ of $f$, such that for any $g\in\mathcal{U}_3\cap \mathcal{R}$, the homoclinic class $H(p_g,g)$ is also Lyapunov stable.
\end{enumerate}
\end{lem}
\section{Norm of products and product of norms: reduction of the proof of Theorem A}\label{theorem b}
Theorem A essentially follows from the theorem below.
\begin{theob}
For $C^1$-generic $f\in \diff^1(M)$, assume that $p$ is a hyperbolic periodic point of $f$ and that the homoclinic class $H(p)$ has a dominated splitting $T_{H(p)}M=E\oplus F$, with $dim E\leq ind(p)$, such that the bundle $E$ is not contracted. Then there are a constant $\lambda_0\in (0,1)$, an integer $m_0\in \mathbb{N}$, satisfying: for any $m\in \mathbb{N}$ with $m\geq m_0$, any constants $\lambda_1,\lambda_2\in (\lambda_0,1)$ with $\lambda_1<\lambda_2$, there is a sequence of different periodic orbits $\mathcal{O}_k=orb(q_k)$ with period $\tau(q_k)$ contained in $H(P)$, such that
\begin{displaymath}
{\lambda_1}^{\tau(q_k)}< \prod_{0\leq i<\tau(q_k)/m} \|Df^m|_{E(f^{im}(q_k))}\|< {\lambda_2}^{\tau(q_k)}.
\end{displaymath}
\end{theob}
From Theorem B, we can get periodic orbits that have certain controls of the product of norms along the bundle $E$. To control Lyapunov exponents of the periodic orbits, we have to control the norm of products along the bundle $E$. We have to use the following two lemmas. The first is a perturbation lemma for matrixes to control exponents, see~\cite{c2,p} (also see~\cite{l2,m2}).
\begin{lem}\label{matrix}
For any integer $d\geq 1$, $K\geq 1$, any constant $\vep>0$ and $\lambda>0$, there are two integers $N$ and $\tau_0$, such that for any $A_1,\cdots,A_{\tau}$ in $GL(d,\mathbb{R})$ with $\tau\geq \tau_0$, and $max_{1\leq i\leq \tau}\{\|A_i\|,\|A_i^{-1}\|\}\leq K$, if
\begin{displaymath}
\prod_{0\leq i< \tau/N}\|A_{(i+1)N}\cdots A_{iN+2}A_{iN+1}\|\geq \lambda^{\tau},
\end{displaymath}
then, there are $B_1,\cdots,B_{\tau}$ in $GL(d,\mathbb{R})$, with $\|B_i-A_i\|<\vep$ and $\|B_i^{-1}-A_i^{-1}\|<\vep$, for all $i=1,\cdots,\tau$, such that the maximal norm of eigenvalue of $B_{\tau}\circ\cdots \circ B_2\circ B_1$ is bigger than $\lambda$.
\end{lem}
\begin{rem}
In~\cite{c2}, it is presented for the constant $\lambda=1$. If $\lambda\neq 1$, then by considering $A_i'=\lambda^{-1} Id\circ A_i$ and applying the special case for the constant $1$, we can get the general statement as above.
\end{rem}
\begin{cor}\label{cor of matrix}
For any integer $d\geq 1$, $K\geq 1$, any constant $\vep>0$ and $\lambda_1<\lambda_2$, there are two integers $N$ and $\tau_0$, such that for any $A_1,\cdots,A_{\tau}$ in $GL(d,\mathbb{R})$ with $\tau\geq \tau_0$, and $max_{1\leq i\leq \tau}\{\|A_i\|,\|A_i^{-1}\|\}\leq K$, if
\begin{displaymath}
\lambda_1^{\tau}<\prod_{0\leq i< \tau/N}\|A_{(i+1)N}\cdots A_{iN+2}A_{iN+1}\|< \lambda_2^{\tau},
\end{displaymath}
then, there are $B_1,\cdots,B_{\tau}$ in $GL(d,\mathbb{R})$, with $\|B_i-A_i\|<\vep$ and $\|B_i^{-1}-A_i^{-1}\|<\vep$, for all $i=1,\cdots,\tau$, such that the maximal norm of eigenvalue of $B_{\tau}\circ\cdots \circ B_2\circ B_1$ is in the interval $(\lambda_1,\lambda_2)$.
\end{cor}
\begin{proof}
We take $\vep$ small enough such that, for any $A\in GL(d,\mathbb{R})$, if $\|A^{-1}\|\leq K$, then $B(A,\vep)\in GL(d,\mathbb{R})$, where $B(A,\vep)$ is the $\vep$ ball of $A$. By the assumption of $A_i$, we have that the maximal norm of eigenvalue of $A_{\tau}\circ\cdots \circ A_2\circ A_1$ is smaller than $\lambda_2$. By Lemma~\ref{matrix}, we can get $B^0_1,\cdots,B^0_{\tau}$ in $GL(d,\mathbb{R})$ that satisfies the conclusion for the number $\lambda_1$. We take a path $A_{i,t}|_{0\leq t\leq 1}$ contained in $B(A_i,\vep)$ that connects $A_i$ to $B^0_i$. We have that the maximal norm of eigenvalue of $B^0_{\tau}\circ\cdots \circ B^0_2\circ B^0_1$ is bigger than $\lambda_1$. Then there must be a time $0<t<1$, such that the maximal norm of eigenvalue of $A_{\tau,t}\circ\cdots \circ A_{2,t}\circ A_{1,t}$ is in the interval $(\lambda_1,\lambda_2)$. We take $B_i=A_{i,t}$ and get the conclusion.
\end{proof}
The next lemma is a generalized Frank's lemma by N. Gourmelon that preserves some pieces of invariant manifolds of hyperbolic period orbits, see~\cite{g2}.
\begin{lem}\label{f-g lemma}
Consider a constant $\vep>0$, a diffeomorphism $f\in \diff^1(M)$ and a hyperbolic periodic orbit $\mathcal{O}=orb(q)$ of $f$ with period $\tau$. Assume there is a one-parameter family of linear maps $(A_{n,t})_{n=0,1,\cdots,\tau-1;t\in[0,1]}$ in $GL(d,\mathbb{R})$, satisfying:
\begin{itemize}
\item $(1)$ $A_{n,0}=Df(f^n(q))$,
\item $(2)$ for all $n=0,1,\cdots,\tau-1$ and $t\in[0,1]$, we have $\|Df(f^n(q))-A_{n,t}\|<\vep$ and $\|Df^{-1}(f^n(q))-A^{-1}_{n,t}\|<\vep$,
\item $(3)$ $A_{\tau-1,t}\circ\cdots\circ A_{0,t}$ is hyperbolic for all $t\in [0,1]$.\\
\end{itemize}
Then, for any neighborhood $V$ of $\mathcal{O}$, any $\delta>0$, and any pair of compact sets $K^s\subset W^s_{\delta}(\mathcal{O},f)$ and $K^u\subset W^u_{\delta}(\mathcal{O},f)$ disjoint from $V$, there is a diffeomorphism $g\in \diff^1(M)$ that is $\vep$-$C^1$ close to $f$, such that:
\item $(a)$ $g$ coincides with $f$ on $\mathcal{O}$ and outside $V$;
\item $(b)$ $K^s\subset W^s_{\delta}(\mathcal{O},g)$ and $K^u\subset W^u_{\delta}(\mathcal{O},g)$;
\item $(c)$ $Dg(g^n(q))=Dg(f^n(q))=A_{n,1}$ for all $n=0,\cdots,\tau-1$.
\end{lem}
Now we give the proof of Theorem A from Theorem B.
\begin{proof}
By Theorem B, we get two constants $\lambda_0\in (0,1)$ and $m_0\in \mathbb{N}$. We prove that for any $\lambda_0<\lambda_1<\lambda_2<1$ and any $\vep>0$, there is a diffeomorphism $g$ that is C$^1$-$\vep$ close to $f$ and $g$ has a periodic orbit $orb(q)$ homoclinic related to $p_g$ such that the largest Lyapunov exponent along $E$ of $orb(q)$ is in the interval $(\log\lambda_1,\log\lambda_2)$. Then by the genericity of $f$ and Lemma 2.1 of~\cite{gy}, $f$ itself has such periodic orbits. Since $\lambda_1$ can be taken arbitrarily close to $1$, we get the conclusion of the Theorem A. Take $d=dim(M)$ and $K=max\{\|Df\|,\|Df^{-1}\|\}$. Now we fix the constants $\vep$ and $\lambda_1<\lambda_2$ in $(\lambda_0,1)$. By Corollary~\ref{cor of matrix}, we get two integers $N$ and $\tau_0$.
By Theorem B, there is a periodic orbit $orb(q)$ of $f$ with period $\tau>\tau_0$ that is homoclinically related to $orb(p)$, such that,
\begin{displaymath}
{\lambda_1}^{\tau}< \prod_{0\leq i<\tau/m} \|Df^m|_{E(f^{im}(q))}\|< {\lambda_2}^{\tau},
\end{displaymath}
where $m>m_0$ is a multiple of $N$. Denote $A_i=Df|_{f^i(q)}$ for $i=0,\cdots,\tau-1$. Since $E\oplus F$ is a dominated splitting, the two bundles $E$ and $F$ are transverse with each other, thus there is a lower bound of the angle between $E$ and $F$. By Corollary~\ref{cor of matrix}, there are $B_0,\cdots,B_{\tau-1}$ in $GL(d,\mathbb{R})$, with $\|B_i-A_i\|<\vep$ and $\|B_i^{-1}-A_i^{-1}\|<\vep$, for all $i=0,\cdots,\tau-1$, such that, $B_i$ coincides with $A_i$ along the bundle $F$ and the maximal norm of eigenvalue of $B_{\tau-1}\circ\cdots \circ B_1\circ B_0$ along the bundle $E$ is in the interval $(\lambda_1,\lambda_2)$. We take a path $A_{i,t}|_{0\leq t\leq 1}$ contained in $B(A_i,\vep)$ that connects $A_i$ to $B_i$ such that $A_{i,t}$ coincides with $A_i$ along the bundle $F$ for all $i=0,\cdots,\tau-1$ and all $t\in (0,1)$. If there is a time $t\in (0,1)$ such that $A_{\tau-1,t}\circ\cdots A_{0,t}$ is not hyperbolic, then there must be a time $t_0<t$, such that $A_{\tau-1,s}\circ\cdots A_{0,s}$ is hyperbolic for all $0\leq s\leq t_0$, and the maximal norm of eigenvalue of $A_{\tau-1,t_0}\circ\cdots A_{0,t_0}$ along the bundle $E$ is in the interval $(\lambda_1,\lambda_2)$. Otherwise, we can take $t_0=1$
Take a small constant $\delta>0$, since $orb(q)$ is homoclinically related to $orb(p)$, there exist two points $x\in W^s_{\delta}(orb(q))\pitchfork W^u(orb(p))$ and $y\in W^u_{\delta}(orb(q))\pitchfork W^s(orb(p))$. We take a pair of compact sets $K^s\subset W^s_{\delta}(orb(q))$ and $K^u\subset W^u_{\delta}(orb(q))$ such that $x\in K^s$ and $y\in K^u$. Then we take a neighborhood $V$ of $orb(q)$ such that $V\cap (K^s\cup K^u)=\emptyset$ and $V\cap (orb^-(x)\cup orb^+(y))=\emptyset$. By Lemma~\ref{f-g lemma}, considering the one-parameter family of linear maps $(A_{i,t})_{i=0,\cdots,\tau-1;t\in [0,t_0]}$, there is a diffeomorphism $g$ that is C$^1$-$\vep$ close to $f$, such that:
\quad $(a)$ $g$ coincides with $f$ on $orb(q)$ and outside $V$;
\quad $(b)$ $K^s\subset W^s_{\delta}(orb(q),g)$ and $K^u\subset W^u_{\delta}(orb(q),g)$;
\quad $(c)$ $Dg(g^i(q))=Dg(f^i(q))=A_{i,t_0}$ for all $i=0,\cdots,\tau-1$.\\
Then $x\in W^s_{\delta}(orb(q),g)\cap W^u(orb(p),g)$ and $y\in W^u_{\delta}(orb(q),g)\cap W^s(orb(p),g)$, and by another small perturbation if necessary, we can assume that the two intersections are transverse. Then the two periodic orbits $orb(q)$ and $orb(p)$ of $g$ are still homoclinically related with each other, and the largest Lyapunov exponent of $orb(q)$ along the bundle $E$ under the diffeomorphism $g$ is in the interval $(\log\lambda_1,\log\lambda_2)$. This finishes the proof of Theorem A.
\end{proof}
\section{Non-hyperbolicity implies existence of weak periodic orbits: proof of Theorem B}\label{proof of theorem b}
This section will give the proof of Theorem B. We assume that $\mathcal{R}$ is the residual set of $\diff^1(M)$ stated in Lemma~\ref{generic properties} and $f\in\mathcal{R}$ is a diffeomorphism that satisfies the hypothesis of Theorem B. Later we will assume also that $f$ belongs to another two residual subsets $\mathcal{R}_0$ and $\mathcal{R}_1$ defined below.
Since $E\oplus F$ is a dominated splitting and $dim E\leq ind(p)$, we have that: there are $\lambda_0\in (0,1)$ and $m_0\in\mathbb{N}$, such that, for any $m\geq m_0$, the splitting $E\oplus F$ is $(m,\lambda_0^2)$-dominated, and, for the hyperbolic periodic orbit $orb(p)$,
\begin{displaymath}
\|Df^{\tau(p)}|_{E(p)}\|<\lambda_0^{\tau(p)},
\end{displaymath}
where $\tau(p)$ is the period of $orb(p)$. In the following, we fix $m\geq m_0$. In order to simplify the notations, we will assume that $m=1$ and that $p$ is a fixed point of $f$, but the general case is identical.
\subsection{Existence of weak sets}
\begin{lem}\label{existence of weak sets}
For any $\lambda\in (\lambda_0,1)$, there is a $\lambda$-$E$-weak set contained in $H(p)$.
\end{lem}
\begin{proof}
Since $E$ is not contracted, there is a point $b\in H(p)$, such that, for any $n\geq 1$,
\begin{displaymath}
\prod_{i=0}^{n-1} \|Df|_{E(f^{i}(b))}\|\geq 1.
\end{displaymath}
Then the first assumption for the bundle $E$ in Lemma~\ref{selecting} is satisfied.
Assume by contradiction that there is a constant $\lambda\in (\lambda_0,1)$, such that there is no $\lambda$-$E$-weak set contained in $H(p)$. Thus the seconde assumption in Lemma~\ref{selecting} is satisfied for the bundle $E$ and the constant $\lambda$. Hence, for any $\lambda_1,\lambda_2\in (\lambda,1)$ with $\lambda_1<\lambda_2$, there is a sequence of periodic orbits $orb(q_k)$ with period $\tau(q_k)$ that are homoclinically related with each other and that converges to a subset of $H(p)$ such that for any $k\geq 0$, the following properties are satisfied:
\begin{displaymath}
{\lambda_1}^{\tau(q_k)}\leq \prod_{0\leq i<\tau(q_k)} \|Df|_{E(f^{i}(q_k))}\|\leq {\lambda_2}^{\tau(q_k)},
\end{displaymath}
Then $H(p)=H(q_k)$ by item 2 of Lemma~\ref{generic properties}, hence $q_k\in H(p)$. It is obvious that $orb(q_k)$ is a $\lambda_1$-$E$-weak set contained in $H(p)$, thus is also a $\lambda$-$E$-weak set. This contradicts the assumption that there is no $\lambda$-$E$-weak set contained in $H(p)$.
\end{proof}
\subsection{Existence of a bi-Pliss point accumulating backward to an $E$-weak set}\label{bi pliss point and weak set}
From now on, we fix any two numbers $\lambda_1<\lambda_2$ in $(\lambda_0,1)$. Then there is a $\lambda_2$-$E$-weak set contained in $H(p)$. By the domination, any $\lambda_2$-$E$-weak set $K$ is $(C,\lambda_0,F)$-expanded for some constant $C>0$ depending on $K$. By~\cite{hps}, any point $x\in K$ has a uniform local unstable manifold $W^u_{loc}(x)$ with a uniform size depending on $K$.
We extend the dominated splitting $E\oplus F$ to the maximal invariant compact set of a small neighborhood $U$ of $H(p)$ and denote it still by $E\oplus F$. We take a constant $\lambda_3\in (\lambda_2,1)$.
\begin{lem}\label{weak set and pliss point}
There are a $\lambda_2$-$E$-weak set $K$, and a $\lambda_3$-bi-Pliss point $x\in H(p)\setminus K$ satisfying: $\alpha(x)= K$.
\end{lem}
It is obvious that any compact invariant subset of a $\lambda_2$-$E$-weak set is still a $\lambda_2$-$E$-weak set. So we only have to prove that: \emph{there are a $\lambda_2$-$E$-weak set $K$, and a $\lambda_3$-bi-Pliss point $x\in H(p)\setminus K$ satisfying: $\alpha(x)\subset K$.}
\begin{proof}
By Lemma~\ref{existence of weak sets}, there exists a $\lambda_2$-$E$-weak set in $H(p)$. To prove this lemma, we consider two cases: either all the $\lambda_2$-$E$-weak sets are uniformly $E$-weak or not. More precisely, if we take the closure of the union of all $\lambda_2$-$E$-weak sets contained in $H(p)$, and denote it by $\hat K$, then there are two cases: either $\hat K$ is still a $\lambda_2$-$E$-weak set or not.
\subsubsection{The uniform case: $\hat K$ is a $\lambda_2$-$E$-weak set}
In this case, $\hat K$ is the maximal $\lambda_2$-$E$-weak set in $H(p)$ and we will take $K=\hat{K}$.
\begin{claim}
$K$ is locally maximal in $H(p)$.
\end{claim}
\begin{proof}
We prove by contradiction. Assume that $K$ is not locally maximal in $H(p)$. Take a decreasing sequence of neighborhoods $(U_n)_{n\geq 0}$ of $K$, such that $\cap_n U_n=K$. Then for any $n\geq 0$, there is a compact invariant set $K_n\subset U_n\cap H(p)$ such that $K\subsetneqq K_n$. Since $K$ is the maximal $\lambda_2$-$E$-weak set in $H(p)$, we have that $K_n$ is not a $\lambda_2$-$E$-weak set, thus there is a $\lambda_2$-$E$-Pliss point $y_n\in K_n$. Take a converging subsequence of $(y_n)$, and assume $y$ is the limit point. Then we have that $y\in K$ and $y$ is a $\lambda_2$-$E$-Pliss point. This contradicts the fact that $K$ is a $\lambda_2$-$E$-weak set.
\end{proof}
Since $K$ is locally maximal in $H(p)$, there is a neighborhood $U$ of $K$ such that $K$ is the maximal compact invariant set contained in $U\cap H(p)$. Then there is a point $z\in (U\cap H(p))\setminus K$, such that $\alpha(z)\subset K$.
\begin{claim}\label{omega z}
There exists at least one $\lambda_2$-$E$-Pliss point contained in $\omega(z)$.
\end{claim}
\begin{proof}
We proof this claim by absurd. If $\omega(z)$ contains no $\lambda_2$-$E$-Pliss points, by item $(1)$ of Corollary~\ref{cor of pliss}, $orb(z)\cup \omega(z)$ contains no $\lambda_2$-$E$-Pliss points. Then $K\cup orb(z)\cup \omega(z)$ is a $\lambda_2$-$E$-weak set, which contradicts the maximality of $\lambda_2$-$E$-weak set $K$ since $z\not\in K$. Thus $\omega(z)$ contains at least one $\lambda_2$-$E$-Pliss point.
\end{proof}
Since $K$ is a $\lambda_2$-$E$-weak set, by the domination, for any point $w\in K$, there is an integer $n_w$, such that $\prod_{i=0}^{n_w-1}\|Df^{-1}|_{F(f^{-i}(w))}\|\leq \left(\frac{\lambda_0^2}{\lambda_2}\right)^{n_w}<{\lambda_0}^{n_w}$. By item $2$ of Corollary~\ref{cor of pliss}, considering the bundle $F$, there are infinitely many $\lambda_1$-$F$-Pliss points for $f^{-1}$ on $orb^-(z)$. We take all the $\lambda_1$-$F$-Pliss points $\{f^{n_i}(z)\}$ with $n_{i+1}>n_i$ on $orb(z)$ and consider the following two cases:
\begin{itemize}
\item $\textbf{(a)}$ either the sequence $(n_i)$ has an upper bound or $(n_{i+1}-n_i)$ can be arbitrarily large;
\item $\textbf{(b)}$ the sequence $(n_i)$ has no upper bounds and $(n_{i+1}-n_i)$ is bounded.
\end{itemize}
\begin{claim}
In case \textbf{(a)}, there exists a $\lambda_2$-$E$-Pliss point $y\in H(p)$, such that, for any $\delta>0$, there is $n_i\in\mathbb{Z}$, satisfying $d(y,f^{n_i}(z))<\delta$. Thus, by taking $\delta$ small enough, we can take $x\in W^u(f^{n_i}(z))\cap W^s(y)$, such that $x$ is a $\lambda_3$-bi-Pliss point.
\end{claim}
\begin{proof}
If the sequence $\{n_i\}$ has an upper bound, we take the maximal $n_i$. That is to say, $f^{n_i}(z)$ is a $\lambda_1$-$F$-Pliss point for $f^{-1}$, and, there is no $\lambda_1$-$F$-Pliss point for $f^{-1}$ on $orb^+(f^{n_i}(z))$. By item $3$ of Lemma~\ref{property of pliss point}, we have that $f^{n_i}(z)$ is also a $\lambda_1$-$E$-Pliss point, thus $f^{n_i}(z)$ is a $\lambda_1$-bi-Pliss point. We take $x=y=f^{n_i}(z)$ in this case.
Otherwise, the sequence $\{n_i\}$ has no upper bounds but $(n_{i+1}-n_i)$ can be arbitrarily large. By item $1$ of Lemma~\ref{property of pliss point}, we can take a subsequence of $\{n_i\}$ such that $f^{n_i}(z)$ converges to a $\lambda_1$-bi-Pliss point $y\in \omega(z)$. Then for any $\delta>0$, we can take $n_i$ large enough, such that $d(y,f^{n_i}(z))<\delta$, and moreover, we can take $x\in W^u(f^{n_i}z)\cap W^s(y)$, such that, $d(f^j(x),f^j(y))<\delta$ and $d(f^{-j}(x),f^{-j}(f^{n_i}z))<\delta$, for all $j\geq 0$. Thus by taking taking $\delta$ small enough, $x$ is a $\lambda_3$-bi-Pliss point.
\end{proof}
\begin{claim}
In case \textbf{(b)}, there is a $\lambda_2$-$E$-Pliss point $y\in \omega(z)$, such that, there is $n\in\mathbb{N}$, satisfying $W^u(f^n(z))\cap W^s(y)\neq \emptyset$. Thus we can take a point $x\in W^u(f^n(z))\cap W^s(y)$, such that $orb(x)$ contains some $\lambda_3$-bi-Pliss point.
\end{claim}
\begin{proof}
In this case, there are infinitely many $\lambda_1$-$F$-Pliss points for $f^{-1}$ on $orb^+(z)$, and the time between any consecutive $\lambda_1$-$F$-Pliss points for $f^{-1}$ on $orb^+(z)$ is bounded. Then for any point $w\in\overline{orb^+(z)}$, there is an integer $n_w\in\mathbb{N}$, such that $\prod_{i=0}^{n_w-1}\|Df^{-1}|_{E(f^{-i}(w))}\|\leq {\lambda}^{n_w}$. Hence $\overline{orb^+(z)}$ is a positive invariant $F$-expanded compact set, and any point $w\in\overline{orb^+(z)}$ has a uniform unstable manifold. By Claim~\ref{omega z}, there is a $\lambda_2$-$E$-Pliss point $y\in \omega(z)$. For any $\delta>0$, there is $n\in\mathbb{N}$, such that, $d(y,f^n(z))<\delta$, and $W^u(f^n(z))\cap W^s(y)\neq \emptyset$. We take $x\in W^s(y)\cap W^u(f^n(z))$. Then $\alpha(x)=\alpha(z)$ and by item $2$ of Corollary~\ref{cor of pliss}, there are $\lambda_3$-$F$-Pliss points for $f^{-1}$ on $orb^-(x)$. Also by taking $\delta$ small enough, $d(f^i(x),f^i(y))$ can be small for all $i\geq 0$. Since $y$ is a $\lambda_2$-$E$-Pliss point, we can take $x$ to be a $\lambda_3$-$E$-Pliss point. Then, by item $2$ of Lemma~\ref{property of pliss point} there exists a $\lambda_3$-bi-Pliss point on $orb(x)$, we assume that $x$ is such a point.
\end{proof}
From the above two claims, we get a $\lambda_3$-bi-Pliss point $x\in H(p)$, such that $\alpha(x)\subset K$. We have to show that $x\not\in K$. Notice that in the two cases, we both have $\omega(x)=\omega(y)$ where $y$ is a $\lambda_2$-$E$-Pliss point. By item $1$ of Corollary~\ref{cor of pliss}, $\omega(x)$ contains some $\lambda_2$-$E$-Pliss point. Since $K$ contains no $\lambda_2$-$E$-Pliss point, we have that $x\notin K$.
\subsubsection{The non-uniform case: $\hat K$ is not a $\lambda_2$-$E$-weak set}
\begin{claim}\label{almost bi-pliss point}
In this case, for any number $L>0$ there are a $\lambda_2$-$E$-weak set $K$ and a point $z\in K$, such that $z$ is a $\lambda_1$-$F$-Pliss point for $f^{-1}$, and, $N_z>L$, where $N_z$ is taken as in Definition~\ref{weak set}.
\end{claim}
\begin{proof}
Since $\hat K$ is not a $\lambda_2$-$E$-weak set, then for any number $L>0$ there is a $\lambda_2$-$E$-weak sets $K$, and a point $z\in K$, such that $N_z>L$, that is to say, for $1\leq n\leq N_z$,
\begin{displaymath}
\prod_{i=0}^{n-1}\|Df|_{E(f^i(z))}\|\leq \lambda_2^n,
\end{displaymath}
and
\begin{displaymath}
\prod_{i=0}^{N_z-1}\|Df|_{E(f^i(z))}\|>\lambda_2^{N_z}.
\end{displaymath}
We only have to show that we can choose $z$ to be a $\lambda_1$-$F$-Pliss point for $f^{-1}$. Since $K$ is a $\lambda_2$-$E$-weak set, similarly to the arguments above, by item $2$ of Corollary~\ref{cor of pliss}, there are $\lambda_1$-$F$-Pliss points for $f^{-1}$ on $orb^-(z)$. If $z$ is not a $\lambda_1$-$F$-Pliss point for $f^{-1}$, we can take the minimal number $l\in \mathbb{N}$ such that $w=f^{-l}(z)$ is a $\lambda_1$-$F$-Pliss point for $f^{-1}$. We claim that $N_w\geq N_z+l>L$. Hence if we replace $z$ by $w$, we get the conclusion of the claim. To proof this, we only have to show that for any $1\leq n\leq l$,
\begin{displaymath}
\prod_{i=0}^{n-1}\|Df|_{E(f^i(w))}\|\leq \lambda_2^n.
\end{displaymath}
We prove this by absurd. If the above statement is not true, then there is an integer $k\in\{1,2,\cdots,l\}$, such that
\begin{displaymath}
\prod_{i=0}^{k-1}\|Df|_{E(f^i(w))}\|>\lambda_2^k,
\end{displaymath}
and for any $1\leq n< k$,
\begin{displaymath}
\prod_{i=0}^{n-1}\|Df|_{E(f^i(w))}\|\leq\lambda_2^n.
\end{displaymath}
Thus, we have, for any $1\leq n\leq k$,
\begin{displaymath}
\prod_{i=1}^{n}\|Df|_{E(f^{k-i}(w))}\|=\left(\prod_{i=0}^{k-1}\|Df|_{E(f^{i}(w))}\|\right)/\left(\prod_{i=0}^{k-n-1}\|Df|_{E(f^i(w))}\|\right)>\lambda_2^n.
\end{displaymath}
By the domination of $E\oplus F$, we have, for all $1\leq n\leq k$
\begin{displaymath}
\prod_{i=0}^{n-1}\|Df^{-1}|_{F(f^{k-i}(w))}\|\leq(\frac{\lambda_0^2}{\lambda_2})^n\leq \lambda_1^n.
\end{displaymath}
Moreover, since $w$ is a $\lambda_1$-$F$-Pliss point for $f^{-1}$, we will have, for any $n\geq 1$,
\begin{displaymath}
\prod_{i=0}^{n-1}\|Df^{-1}|_{F(f^{k-i}(w))}\|\leq\lambda_1^n.
\end{displaymath}
Thus $f^k(w)=f^{-l+k}(z)$ is a $\lambda_1$-$F$-Pliss point, contradicting the choice of $w$. This finishes the proof of Claim~\ref{almost bi-pliss point}.
\end{proof}
By taking $L$ large enough, the point $z$ in Claim~\ref{almost bi-pliss point} can be arbitrarily close to a $\lambda_2$-$E$-Pliss point $y\not\in K$. Since $z$ has a uniform local unstable manifold and $y$ has a uniform local stable manifold, when we take these two points close enough, $W^s(y)\cap W^u(z)\neq \emptyset$. Similar to the arguments in the proof of Case $\textbf{(b)}$ of \textbf{Case 1}, we can take a $\lambda_3$-bi-Pliss point $x\in W^s(y)\cap W^u(z)$ such that, $x\in H(p)\setminus K$ and $\alpha(x)\subset K$. This finishes the proof of Lemma~\ref{weak set and pliss point}.
\end{proof}
\subsection{Continuation of Pliss points}\label{choice of f}
Denote by $\mathcal{F}$ the space of all finite subsets of $M$ and by $\mathcal{M}$ the space of all compact subsets of $M$, associated with the Hausdorff topology. Denote by $\mathcal{S}$ the space of all finite subsets of $\mathcal{F}\times\mathcal{M}$ associated with the Hausdorff topology. For any positive integer $N\in\mathbb{N}$, and a diffeomorphism $g\in\diff^1(M)$, denote by $Per_N(g)$ the set of periodic points of $g$ with period less than or equal to $N$, and denote by $\mathcal{C}(q,g)$ the chain recurrence class of a periodic point $q$ of $g$. It is well known that for any $N\geq 1$, there is a dense and open subset $\mathcal{U}_N\subset\diff^1(M)$, such that, for any $g\in\mathcal{U}_N$, the set $Per_N(g)$ is a finite set and any point $q\in Per_N(g)$ is a hyperbolic periodic point.
We define a map $\Phi_{N}:\mathcal{U}_N\mapsto \mathcal{S}$, sending a diffeomorphism $g$ to the set of pairs $(q,P_{\lambda_3}(q,g))$, where $q\in Per_N(g)$, and $P_{\lambda_3}(q,g)$ is a compact set contained in $\mathcal{C}(q,g)$ defined as following:
\begin{itemize}
\item The set $P_{\lambda_3}(q,g)$ is the set of $\lambda_3$-$E$-Pliss points contained in $\mathcal{C}(q,g)$, if $\mathcal{C}(q,g)$ has a $\lambda_0^2$-dominated splitting $E\oplus F$ such that $dim(E)=ind(q)$.
\item The set $P_{\lambda_3}(q,g)=\emptyset$ if otherwise.
\end{itemize}
\begin{lem}\label{continuation of pliss points}
For each positive integer $N\in\mathbb{N}$, the set of continuity points of $\Phi_N$, denoted by $\mathcal{B}_N$, is a residual subset of $\diff^1(M)$.
\end{lem}
\begin{proof}
Assume $g\in\diff^1(M)$ and $p_g$ is a hyperbolic periodic point of $g$. There is a $C^1$-neighborhood $\mathcal{U}$ of $g$, such that, for any $h\in\mathcal{U}$, the point $p_g$ has a continuation $p_h$. For any neighborhood $V$ of $\mathcal{C}(q,g)$, there is a $C^1$-neighborhood $\mathcal{U}_1\subset\mathcal{U}$ of $g$, such that $\mathcal{C}(q,h)\subset V$ for any $h\in\mathcal{U}_1$.
If $\mathcal{C}(q,g)$ has a $\lambda_0^2$-dominated splitting, then it is a robust $\lambda_0^2$-dominated splitting. More precisely, there is a $C^1$-neighborhood $\mathcal{U}_2\subset\mathcal{U}$ of $g$, such that $\mathcal{C}(q_h,h)$ has a $\lambda_0^2$-dominated splitting for any $h\in\mathcal{U}$. Hence by the choice of $\mathcal{U}_N$, there is an open and dense subset $\mathcal{U}'_N\subset\mathcal{U}_N$, such that, for any $g\in\mathcal{U}'_N$, any $q\in Per_N(g)$, the chain recurrent class $\mathcal{C}(q,g)$ either has a robust $\lambda_0^2$-dominated splitting or has no $\lambda_0^2$-dominated splitting robustly. Moreover, if there is a sequence of diffeomorphisms $\{g_n\}_{n\geq 0}$ such that $g_n$ converges to $g$, and $g_n$ has a $\lambda_3$-$E$-Pliss point $x_n\in \mathcal{C}(q_h,h)$, then, any limit point $x$ of the sequence $\{x_n\}$ is a $\lambda_3$-$E$-Pliss point of $g$.
By the above arguments, we can see that $\Phi_N$ is an upper-semi-continuous map restricted to $\mathcal{U}'_N$. It is known that the set of continuity points of a semi-continuous map is a residual subset. Then $\mathcal{B}_N$ contained a residual subset of $\mathcal{U}'_N$. Since $\mathcal{U}'_N$ is open and dense in $\mathcal{U}_N$, we know that $\mathcal{B}_N$ is a residual subset of $\mathcal{U}_N$. Hence $\mathcal{B}_N$ is a residual subset of $\diff^1(M)$, since $\mathcal{U}_N$ is open and dense in $\diff^1(M)$.
\end{proof}
Denote by $\mathcal{R}_0=\cap_{N\geq 1}\mathcal{B}_N$, then $\mathcal{R}_0$ is a residual subset of $\diff^1(M)$. In the following we take $f\in \mathcal{R}_0\cap \mathcal{R}$.
\subsection{The perturbation to make $W^u(p)$ accumulate to $K$}
We take the $\lambda_2$-$E$-weak set $K\subset H(p)$ of $f$ obtained by Lemma~\ref{weak set and pliss point}. By proposition~\ref{asymptotic connecting 1}, one can obtain a heteroclinic orbit connecting $p$ to $K$ by a $C^1$ perturbation, since $K\subset H(p)$. Hence the set $K$ is still a $\lambda_2$-$E$-weak set if the perturbation is $C^1$ small. Moreover, using the continuation of Pliss points (Section~\ref{bi pliss point and weak set} and~\ref{choice of f}), we can guarantee that the set $K$ is contained in the chain recurrence class of $p$ after the perturbation.
\begin{lem}\label{first perturbation}
Assume $f\in\mathcal{R}_0\cap \mathcal{R}$, then for any neighborhood $\mathcal{U}$ of $f$ in $\diff^1(M)$, there are a diffeomorphism $g_1\in\mathcal{U}$ and a point $y\in M$, such that,
\begin{itemize}
\item $(1)$ $g_1$ coincides with $f$ on the set $K\cup orb(p)$, and $y\in W^u(p,g_1)$,
\item $(2)$ $\omega(y,g_1)\subset K$,
\item $(3)$ $K$ is contained in $\mathcal{C}(p,g_1)$.
\end{itemize}
\end{lem}
\begin{proof}
By Lemma~\ref{weak set and pliss point}, we obtain that, for the diffeomorphism $f$, there is a $\lambda_3$-bi-Pliss point $x\in H(p)\setminus K$ satisfying: $\alpha(x)= K$. Since $K\subset H(p)$, we have that $K\subset\overline{W^u(p)}$. By Proposition~\ref{asymptotic connecting 1}, for any neighborhood $\mathcal{U}$ of $f$ in $\diff^1(M)$, there are a point $y\in W^u(p,f)$ and a diffeomorphism $g_1\in\mathcal{U}$, such that $\omega(y,g_1)\subset K$, and $y\in W^u(p,g_1)$. Moreover, the diffeomorphism $g_1$ coincides with $f$ on the set $orb^-(x)\cup K\cup orb(p)$ and $Dg_1$ coincides with $Df$ on $orb^-(x)$. Thus items (1) and (2) are satisfied, and $x$ is a $\lambda_3$-$F$-Pliss point for $g_1^{-1}$.
Since $p$ is a hyperbolic fixed point and and $x\in P_{\lambda_3}(p,f)$, by Lemma~\ref{continuation of pliss points} and the fact that $f$ is a continuity point of $\Phi_1$, if we choose $g_1$ close enough to $f$ (by taking the neighborhood $\mathcal{U}$ small), then there is a $\lambda_3$-$E$-Pliss $x'$ close to $x$, such that $x'\in \mathcal{C}(p,g_1)$. Moreover, if $x'$ is close enough to $x$ (by taking $g_1$ close to $f$), then $W^u(x,g_1)\cap W^s(x',g_1)\neq \emptyset$. Hence $K\subset \mathcal{C}(p,g_1)$. This finishes the proof of Lemma~\ref{first perturbation}.
\end{proof}
\subsection{The perturbations to connect $p$ and $K$ by true orbits}
In this subsection, we prove that we can get heteroclinic connections between the hyperbolic fixed point $p$ and the weak set $K$ for a diffeomorphism $C^1$ close to $f$. In the former subsection, we have got a diffeomorphism $g_1$ that is $C^1$ close to $f$, and an orbit $orb(y)$ that connects $p$ to $K$. Moreover $K$ is still contained in the chain recurrence class of $p$ for $g_1$. We take two steps to get heteroclinic connections between $p$ and $K$. First, since $K\subset \mathcal{C}(p,g_1)$, by Proposition 3, we can connect $K$ by a true orbit to any neighborhood of $p$ by a $C^1$ small perturbation. Then, by the hyperbolicity of $p$, we use the uniform connecting lemma to ``push'' this orbit onto the stable manifold of $p$. We will see that in these two steps, the orbit $orb(y)$ that connects $p$ to $K$ is not changed.
\begin{lem}\label{second perturbation}
Assume $f\in\mathcal{R}_0\cap \mathcal{R}$, then for any neighborhood $\mathcal{U}$ of $f$ in $\diff^1(M)$, there are a diffeomorphism $g_2\in\mathcal{U}$ and two points $y,y'\in M$, such that,
\begin{itemize}
\item $(1)$ $y\in W^u(p,g_2)$ and $\omega(y,g_2)\subset K$,
\item $(2)$ $y'\in W^s(p,g_2)$ and $\alpha(y',g_2)\subset \omega(y,g_2)$,
\item $(3)$ $g_2$ coincides with $f$ on the set $\omega(y,g_2)\cup orb(p)$.
\end{itemize}
\end{lem}
\begin{proof}
We take several steps to prove the lemma.
\paragraph{Choice of neighborhoods.}
For any any neighborhood $\mathcal{U}$ of $f$ in $\diff^1(M)$, there are a neighborhood $\mathcal{U}_1\subset \mathcal{U}$ and three numbers $\rho>1$, $\delta_0>0$ and $N\in\mathbb{N}$ that satisfy the uniform connecting lemma (Theorem~\ref{uniform connecting}). And we can assume that the fixed point $p$ has a continuation for any $g\in\mathcal{U}_1$. For the neighborhood $\mathcal{U}_1$, there are a smaller neighborhood $\mathcal{U}'\subset \mathcal{U}_1$ of $f$ and an integer $T$ satisfying the conclusions of Proposition 3. By the hyperbolicity of periodic orbits of $f$, for the integer $T$, there is a neighborhood $\mathcal{U}_2\subset\diff^1(M)$ of $f$, such that, for any diffeomorphism $h\in\mathcal{U}_2$, any periodic point of $h$ with period less than or equal to $T$ is hyperbolic. Take a neighborhood $\mathcal{U}_3$ of $f$ in $\diff^1(M)$, such that $\overline{\mathcal{U}_3}\subset \mathcal{U}_2\cap\mathcal{U}'$.
\paragraph{The connection from $K$ to a neighborhood of $p$ by pseudo-orbits.}
By Lemma~\ref{first perturbation}, there are a diffeomorphism $g_1\in\mathcal{U}_3$ and a point $y\in M$, such that:
\begin{itemize}
\item $g_1$ coincides with $f$ on the set $K\cup orb(p)\cup orb^-(y)$,
\item $y\in W^u(p,g_1)$ and $\omega(y,g_1)\subset K\subset \mathcal{C}(p,g_1)$.
\end{itemize}
Denote $K_0=\omega(y,g_1)$.
\begin{claim}\label{connecting by pseudo orbit}
For any neighborhood $V$ of $p$, there are a $g_1$ negative invariant compact set $X$ and a point $z\in V\cap X$, satisfying that
\begin{itemize}
\item the point $p\notin X$,
\item for any $\vep>0$, there is a $g_1$-$\vep$-pseudo-orbit $Y_{\vep}=(y_0,\cdots,y_m)$ contained in $X$ such that $y_0\in K_0$ and $y_m=z$.
\end{itemize}
\end{claim}
\begin{proof}
For any neighborhood $V$ of $p$, take a smaller neighborhood $V_0$ of $p$, such that $\overline{V_0}\subset V$. For any $k\geq 1$, there is a $g_1$-$\frac{1}{k}$-pseudo-orbit $X_{k}=\{x_0^k,x_1^k,\cdots,x_{m_k}^k\}$, such that, $X_{k}\cap K_0=\{x_0^k\}$, and $X_{k}\cap V_0=\{x_{m_k}^k\}$. Take a subsequence of $\{X_{k}\}_{k\geq 1}$ if necessary, we assume $X_{k}$ converges to a compact set $X$ and $x_{m_k}^k$ converges to a point $z\in\overline{V_0}\subset V$ as $k$ goes to $+\infty$. Obviously, $X$ is a $g_1$-negative-invariant set, $p \notin X$ and $X\cap K_0\neq \emptyset$.
Now we prove that for any $\vep>0$, there is a $g_1$-$\vep$-pseudo-orbit contained in $X$ from $K_0$ to $z$. By the continuity of $g_1$, for any $\vep>0$, there is $k>\frac{3}{\vep}$, such that for all $x,y\in M$, if $d(x,y)<\frac{1}{k}$, then $d(g_1(x),g_1(y))<\frac{\vep}{3}$. Then we take a $\frac{1}{k'}$-pseudo-orbit $X_{k'}=\{x_0^{k'},x_1^{k'},\cdots,x_{m_k}^{k'}\}$, such that $x_0^{k'}\in K_0$ and $x_{m_{k'}}^{k'}\in V_0$ for a number $k'>k$. By choosing $k'$ large enough, we can assume that $d_H(X_{k'},X)<\frac{1}{k}$ and there is a point $y_0\in X\cap K_0$ such that $d(y_0,x_0^{k'})<\frac{1}{k}$ and $d(z,x_{m_{k'}}^{k'})<\frac{1}{k}$. By the assumption, for any $1\leq i\leq m_{k'}-1$, there is $y_i\in X$, such that $d(x_i^{k'},y_i)<\eta$. Denote $Y_{\vep}=(y_0,\cdots,y_{m_{k'}}=z)$, we prove that $Y_{\vep}$ is a $\vep$-pseudo-orbit of $g_1$. In fact, for any $0\leq i\leq m_{k'}-1$,
\begin{center}
$d(g_1(y_i),y_{i+1})\leq d(g_1(y_i),g_1(x_i^{k'}))+d(g_1(x_i^{k'}),x_{i+1}^{k'})+d(x_{i+1}^{k'},y_{i+1})<\frac{\vep}{3}+\frac{1}{k'}+\frac{1}{k}<\vep$.
\end{center}
Hence $Y_{\vep}\subset X$ is a $\vep$-pseudo-orbit of $g_1$ from the set $K_0$ to the point $z$.
\end{proof}
\paragraph{The perturbation to connect $K$ to a neighborhood of $p$.}
We take a local stable manifold $W^s_{loc}(p,g_1)$ of $p$, and take a compact fundamental domain $I_{g_1}$ of $W^s_{loc}(p,g_1)$. Then there is a number $\delta<\delta_0$, such that, for any point $w\in I_{g_1}$, the $N$ balls $(g_1^j(B(w,2\delta)))_{0\leq j\leq N-1}$ are each of size smaller than $\delta_0$, pairwise disjoint and disjoint with the set $K\cup orb(y,g_1)\cup orb(p)$. By the compactness of $I_{g_1}$, there are finite points $w_1,w_2,\cdots,w_L\in I_{g_1}$ such that $(B(w_i,\delta/\rho))_{1\leq i\leq L}$ is a finite open cover of $I_{g_1}$. There is a number $\eta>0$ such that, for any diffeomorphism $h\in\mathcal{U}_1$ that is $\eta$-$C^0$ close to $g_1$, we have that:
\begin{itemize}
\item $(a)$ $W^s_{loc}(p_h,h)$ is $C^0$ close to $W^s_{loc}(p,g_1)$,
\item $(b)$ $(B(w_i,\delta/\rho))_{1\leq i\leq L}$ is still a finite open cover of a fundamental domain $I_{h}$ of $W^s_{loc}(p_h,h)$
\item $(c)$ for any $1\leq i\leq L$, the $N$ balls $(h^j(B(w_i,2\delta))_{0\leq j\leq N-1}$ are each of size smaller than $\delta_0$, pairwise disjoint and disjoint with the set $K\cup orb(y,g_1)\cup orb(p,g_1)$.
\end{itemize}
Since $y\in W^u(p,g_1)$, $p\notin X$ and $X$ is negative invariant, we have that $orb(y,g_1)\cap X=\emptyset$. By the choice of $g_1$, we have that all periodic orbits of $g_1$ contained in $X$ with period less than or equal to $T$ are hyperbolic. Under all these hypothesis, $(X\setminus K_0) \cap \overline{orb(y,g_1)}=\emptyset$, then there is a neighborhood $U_0$ of $X\setminus K_0$ such that $U_0\cap \overline{orb(y,g_1)}=\emptyset$. By Proposition~\ref{asymptotic connecting}, there is a diffeomorphism $h\in \mathcal{U}_1$ which is $\eta$-$C^0$ close to $g_1$, such that $h=g_1=f|_{P\cup orb(y)\cup K_0}$, and $\alpha(z,h)\subset K_0$. Thus the above items $(a)$, $(b)$ and $(c)$ are satisfied for such a diffeomorphism $h$.
\paragraph{The perturbation to get a heteroclinic connection between $p$ and $K$.}
By the hyperbolicity of the periodic point $p$, if we take the neighborhood $V$ of $p$ small enough, then the diffeomorphism $h$ and the point $z$ chosen above would satisfy that the negative orbit of $z$ under $h$ intersect $B(w_i,\delta/\rho)$ for some $i\in \{1,2,\cdots,L\}$. Since $\alpha(z,h)\subset K_0$ and $B(w_i,\delta/\rho)\cap K_0=\emptyset$, there is a point $w=h^{-t}(z)$ for some integer $t>0$, such that $orb^-(w)\cap B(w_i,\delta/\rho)=\emptyset$ and $w$ has a positive iterate under $h$ contained in $B(w_i,\delta/\rho)$. By the item $(b)$, there is a point $y'\in W^s(p,h)$, such that $orb^+(y',h)\cap (\cup_{0\leq j\leq N-1}h^i(B(w_i,\delta/\rho)))=\emptyset$ and $y'$ has a negative iterate under $h$ contained in $B(w_i,\delta/\rho)$. By Theorem~\ref{uniform connecting}, there is a diffeomorphism $g_2\in\mathcal{U}$, such that $y'$ is on the positive iterate of $w$ under $g_2$. Moreover, $g_2=g_1$ on the set $K_0\cup orb(y)\cup orb(p)\cup orb^-(w)\cup orb^+(y')$, hence $g_2=f$ on the set $orb(p)\cup K_0$, where $K_0=\omega(y,g_1)=\omega(y,g_2)$. Thus the three items of the lemma are satisfied for $g_2$. This finishes the proof of Lemma~\ref{second perturbation}.
\end{proof}
\subsection{Last perturbation to get a weak periodic orbit}
The following lemma estimates the average contraction along the bundle $E$ on periodic orbits.
\begin{lem}\label{choose time}
Assume $f\in\mathcal{R}_0\cap\mathcal{R}$. Then for any neighborhood $\mathcal{U}$ of $f$ in $\diff^1(M)$, for any integer $L>0$, any neighborhood $U_p$ of $p$, there is $g\in\mathcal{U}$, which coincides with $f$ on $orb(p)$, satisfying that, $g$ has a periodic point $q\in U_p$ with period $\tau>L$ such that, $orb(q)$ has the $\lambda_0^2$-dominated splitting $E\oplus F$, and
\begin{displaymath}
{\lambda_1}^{\tau}\leq \prod_{0\leq i\leq \tau-1} \|Dg|_{E(g^{i}(q))}\|\leq {\lambda_2}^{\tau}.
\end{displaymath}
\end{lem}
\begin{proof}
We take several steps to prove the lemma. We take the $\lambda_2$-$E$-weak set $K\subset H(p)$ of $f$ obtained by Lemma~\ref{weak set and pliss point}. Take two numbers $\lambda_1'$ and $\lambda_2'$, such that $\lambda_1<\lambda_1'<\lambda_2'<\lambda_2$.
\paragraph{Choice of neighborhoods and constants.}
There is a neighborhood $V$ of $H(p)$ and a neighborhood $\mathcal{V}\subset \diff^1(M)$ of $f$, such that, for any $h\in\mathcal{V}$, the following properties are satisfied.
\begin{itemize}
\item The maximal invariant compact set of $h$ in $V$ has a dominated splitting which is a continuation of $E\oplus F$. To simplify the notations, we still denote this domination by $E\oplus F$.
\item The fixed point $p$ has a continuation $p_{h}\in V$ for $h$, and $\|Dh|_{E(p_{h})}\|<\lambda_0$.
\item The chain recurrence class $\mathcal{C}(p_{h},h)$ of $p_{h}$ is contained in $V$.
\end{itemize}
Moreover, since $K$ is a $\lambda_2$-$E$-weak set for $f$, there are a neighborhood $U_{K}\subset V$ of $K$ and a number $N_{K}$, such that, for any point $z$ whose orbit is contained in $V$, if the piece of orbit $(z,f(z),\cdots,f^n(z))$ is contained in $\overline{U_{K}}$ with $n\geq N_{K}$, we have:
\begin{displaymath}
\prod_{0\leq i\leq n-1}\|Df|_{E(f^i(z))}\|>{\lambda_2}^{n}.
\end{displaymath}
To simplify the proof, we just assume that $N_{K}=1$, but the general case is identical.
We can take the neighborhoods $\mathcal{V}$ and $U_p$ small, such that for any diffeomorphism $h\in\mathcal{V}$, the following additional properties are satisfied.
\begin{itemize}
\item For any point $z\in U_p$ whose orbit under $h$ is contained in $V$, we have that $\frac{\lambda_1}{\lambda_1'}<\frac{\|Dh|_{E(z)}\|}{\|Df|_{E(p)}\|}<\frac{\lambda_2}{\lambda_2'}$.
\item For any point $z\in U_{K}$ whose orbit under $h$ is contained in $V$, we have that $\|Dh|_{E(z)}\|>\lambda_2$.
\end{itemize}
We can assume more that $\overline{U_K}\cap \overline{U_p}=\emptyset$ and $\overline{U_K}\cup \overline{U_p}\subset V$. And moreover, we can assume that $\overline{\mathcal{U}}\subset\mathcal{V}$.
By Lemma~\ref{second perturbation}, there are a diffeomorphism $g_2\in\mathcal{U}$ and two points $y,y'\in M$, satisfying that:
\begin{itemize}
\item $y\in W^u(p,g_2)$ and $\omega(y,g_2)\subset K$,
\item $y'\in W^s(p,g_2)$ and $\alpha(y',g_2)\subset \omega(y,g_2)$,
\item $g_2$ coincides with $f$ on the set $\omega(y,g_2)\cup orb(p)$.
\end{itemize}
We denote $K_0=\omega(y,g_2)$. Since all periodic points of $f$ are hyperbolic and $g_2=f|_{K_0}$, then by a $C^1$ small perturbation if necessary, we can assume that $K_0$ contains no non-hyperbolic periodic point of $g_2$.
\paragraph{Choice of time.}
Now we fix the neighborhoods $U_p$ and $U_{K_0}$. Then there are two integers $l$ and $n_0$ satisfying the conclusion of Proposition~\ref{time control} for $g_2$ and the neighborhood $\mathcal{U}$. Then we take $T_{K_0}>L$ large, such that for any $h\in\mathcal{V}$, the inequality
\begin{displaymath}
m(h)^{l+n_0}{\lambda_2}^{T_{K_0}}>(\lambda_2')^{T_{K_0}+l+n_0}
\end{displaymath}
holds.
By the first item of Proposition~\ref{time control}, there is a diffeomorphism $h\in\mathcal{U}$, such that
\begin{itemize}
\item $h$ coincides with $g_2$ on $orb(p)\cup orb^-(y)\cup orb^+(y')$ and outside $U_K$;
\item the point $y'$ is on the positive orbit of $y$ under $h$, with $n_{K_0}=\sharp (orb(y,h)\cap U_{K_0})\geq T_{K_0}$ and $n_c=\sharp ((orb(y,h)\setminus (U_K\cup U_p))\leq n_0$.
\end{itemize}
Hence by the choice of $T_{K_0}$ and the neighborhoods, we have that
\begin{displaymath}
\prod_{h^i(y)\not \in U_p}\|Dh|_{E(h^i(x))}\|>(\lambda_2')^{n_{K_0}+n_c}.
\end{displaymath}
\begin{claim}\label{estimation}
There is an integer $m>0$, such that:
\begin{displaymath}
(\lambda_1')^{n_{K_0}+n_c+m+l}<\|Df|_{E(p)}\|^{l+m}\cdot\prod_{h^i(y)\not \in U_p}\|Dh|_{E(h^i(x))}\|<(\lambda_2')^{n_{K_0}+n_c+m+l}.
\end{displaymath}
\end{claim}
\begin{proof}
We assume that
\begin{displaymath}
\prod_{h^i(y)\not \in U_p}\|Dh|_{E(h^i(x))}\|=\bar{\lambda}^{n_{K_0}+n_c},
\end{displaymath}
then $\bar{\lambda}>\lambda_2'$. The inequality in the claim is equivalent to
\begin{displaymath}
\frac{(n_{K_0}+n_c)\log\frac{\bar{\lambda}}{\lambda_2'}}{\log\frac{\lambda_2'}{\|Df|_{E(p)}\|}}<l+m
<\frac{(n_{K_0}+n_c)\log\frac{\bar{\lambda}}{\lambda_1'}}{\log\frac{\lambda_1'}{\|Df|_{E(p)}\|}}.
\end{displaymath}
By the choice of $T_{K_0}$ and $n_{K_0}\geq T_{K_0}$, we have that
\begin{displaymath}
\frac{(n_{K_0}+n_c)\log\frac{\bar{\lambda}}{\lambda_2'}}{\log\frac{\lambda_2'}{\|Df|_{E(p)}\|}}>l.
\end{displaymath}
So we only need that
\begin{displaymath}
\frac{(n_{K_0}+n_c)\log\frac{\bar{\lambda}}{\lambda_1'}}{\log\frac{\lambda_1'}{\|Df|_{E(p)}\|}}-
\frac{(n_{K_0}+n_c)\log\frac{\bar{\lambda}}{\lambda_2'}}{\log\frac{\lambda_2'}{\|Df|_{E(p)}\|}}>1.
\end{displaymath}
It is equivalent to
\begin{displaymath}
(n_{K_0}+n_c)\left((\frac{1}{\log\frac{\lambda_1'}{\|Df|_{E(p)}\|}}-\frac{1}{\log\frac{\lambda_2'}{\|Df|_{E(p)}\|}})\log\bar{\lambda}
+\frac{\log\lambda_2'}{\log\frac{\lambda_2'}{\|Df|_{E(p)}\|}}-\frac{\log\lambda_1'}{\log\frac{\lambda_1'}{\|Df|_{E(p)}\|}}\right)>1.
\end{displaymath}
Since $\bar{\lambda}>\lambda_2'$, and $n_{K_0}>T_{K_0}$, it is sufficient to acquire that
\begin{displaymath}
\frac{T_{K_0}(\log\lambda_2'-\log\lambda_1')}{\log\frac{\lambda_1'}{\|Df|_{E(p)}\|}}>1.
\end{displaymath}
By taking $T_{K_0}$ large enough, the above inequality is satisfied.
\end{proof}
\paragraph{Choice of the diffeomorphism $g$.}
We take $g=h_m\in\mathcal{U}$ from item 2 of Proposition~\ref{time control}, then $g$ has a periodic orbit $O=orb(q)$, such that, $O\setminus U_p=(orb(y,h)\setminus U_p$, and $\sharp (O\cap U_p)=l+m$. Hence the period $\tau$ of $O$ equals $n_{K_0}+n_c+m+l$. By the choice of the neighborhood $\mathcal{U}$ and the constants, we have
\begin{displaymath}
\prod_{0\leq i\leq \tau-1} \|Dg|_{E(g^{i}(q))}\|=\prod_{g^i(q) \in U_p}\|Dg|_{E(g^i(q))}\|\prod_{h^i(y)\not \in U_p}\|Dh|_{E(h^i(x))}\|.
\end{displaymath}
By the choice of the neighborhoods $\mathcal{V}$ and $U_p$, and the constants $\lambda_1'$ and $\lambda_2'$, we have that
\begin{displaymath}
\left(\frac{\lambda_1}{\lambda_1'}\right)^{l+m}\|Df|_{E(p)}\|^{l+m}<\prod_{g^i(q) \in U_p}\|Dg|_{E(g^i(q))}\|<\left(\frac{\lambda_2}{\lambda_2'}\right)^{l+m}\|Df|_{E(p)}\|^{l+m}.
\end{displaymath}
Then by the estimation in Claim~\ref{estimation}, we can see that
\begin{displaymath}
{\lambda_1}^{\tau}\leq \prod_{0\leq i\leq \tau-1} \|Dg|_{E(g^{i}(q))}\| \leq {\lambda_2}^{\tau}
\end{displaymath}
This finishes the proof of Lemma~\ref{choose time}.
\end{proof}
\subsection{The genericity argument}\label{generic argument}
In this subsection, we do the genericity argument to get the conclusion of Theorem B, see like~\cite{gw}.
Take a countable basis $(V_n)_{n\geq 1}$ of $M$, and take the countable family $(U_n)_{n\geq 1}$, where each $U_n$ is a union of finitely many sets of $(V_n)_{n\geq 1}$. Take the countable pairs $(\eta_n,\gamma_n)_{n\geq 1}$ of rational numbers contained in $(\lambda_0,1)$ with $\eta_n<\gamma_n$ for each $n\geq 1$.
Let $\mathcal{H}_{n,m}$ be the set of $C^1$ diffeomorphisms $h$ such that, every $h_1$ in a $C^1$ neighborhood $\mathcal{V}\subset\diff^1(M)$ of $h$ has a hyperbolic periodic point $q\in U_n$ satisfying that the hyperbolic splitting $E^s\oplus E^u$ of $orb(q,h_1)$ is a $\lambda_0^2$-dominated splitting and
\begin{displaymath}
{\eta_m}^{\tau(q)}< \prod_{0\leq i\leq \tau(q)-1} \|Dh_1|_{E^s(h_1^{i})}\|< {\gamma_m}^{\tau(q)},
\end{displaymath}\\
where $\tau(q)$ is the period of $q$. Let $\mathcal{N}_{n,m}$ be the set of $C^1$ diffeomorphisms $h$ such that every $h_1$ in a $C^1$ neighborhood $\mathcal{V}\subset\diff^1(M)$ of $h$ has no hyperbolic periodic point $q\in U_n$ satisfying that the hyperbolic splitting $E^s\oplus E^u$ of $orb(q,h_1)$ is a $\lambda_0^2$-dominated splitting and
\begin{displaymath}
{\eta_m}^{\tau(q)}< \prod_{0\leq i\leq \tau(q)-1} \|Dh_1|_{E^s(h_1^{i})}\|< {\gamma_m}^{\tau(q)},
\end{displaymath}\\
where $\tau(q)$ is the period of $q$.
Notice that $\mathcal{N}_{n,m}=\diff^1(M)\setminus\overline{\mathcal{H}_{n,m}}$. Hence $\mathcal{H}_{n,m}\cup \mathcal{N}_{n,m}$ is $C^1$ open and dense in $\diff^1(M)$. Let
\begin{center}
$\mathcal{R}_1=\bigcap_{n\geq1,m\geq1}(\mathcal{H}_{n,m}\cup \mathcal{N}_{n,m})$.
\end{center}
Then $\mathcal{R}_1$ is a residual subset of $\diff^1(M)$, and $\mathcal{R}_0\cap\mathcal{R}_1\cap\mathcal{R}$ is also a residual subset of $\diff^1(M)$.
\begin{claim}\label{average estimation}
Assume $f\in\mathcal{R}_0\cap\mathcal{R}_1\cap\mathcal{R}$. Then for any two numbers $\lambda_1<\lambda_2\in(\lambda_0,1)$, for any neighborhood $U_p$ of $orb(p)$, and any integer $L>0$, there is a periodic point $q\in U_p$ with period $\tau>L$ such that $orb(q)$ has the $\lambda_0^2$-dominated splitting $E\oplus F$, and
\begin{displaymath}
{\lambda_1}^{\tau}\leq \prod_{0\leq i\leq \tau-1} \|Df|_{E(f^{i}(q))}\|\leq {\lambda_2}^{\tau}.
\end{displaymath}
\end{claim}
\begin{proof}
We take two rational numbers $\eta_i,\gamma_i\in(\lambda_0,1)$, such that $\lambda_1<\eta_i<\gamma_i<\lambda_2$, and take $U_j$ from the countable basis of $M$, such that $U_j\subset U_p$. Then by Lemma~\ref{choose time}, there is a diffeomorphism $g$ arbitrarily $C^1$ close to $f$, such that $g$ has a periodic point $q\in U_p$ with period $\tau>T$ such that the $\lambda_0^2$-dominated splitting $E\oplus F$ is the hyperbolic splitting on $orb(q,g)$, and
\begin{displaymath}
{\eta_i}^{\tau}\leq \prod_{0\leq i\leq \tau-1} \|Dg|_{E(g^{i}(q))}\|\leq {\gamma_i}^{\tau}.
\end{displaymath}
Then $f\notin\mathcal{N}_{j,i}$, thus $f\in\mathcal{H}_{j,i}$ and $f$ satisfies the conclusion of Claim~\ref{average estimation}.
\end{proof}
\begin{claim}
Theorem B holds for any diffeomorphisms in $\mathcal{R}_0\cap\mathcal{R}_1\cap\mathcal{R}$.
\end{claim}
\begin{proof}
Assume $f\in\mathcal{R}_0\cap\mathcal{R}_1\cap\mathcal{R}$ and $f$ satisfies the assumptions of Theorem B. By Claim~\ref{average estimation}, we get a sequence of periodic orbits $orb(q_k)$ of $f$, such that $q_k\rightarrow p$ with $\tau(q_k)\rightarrow\infty$, and
\begin{displaymath}
{\lambda_1}^{\tau(q_k)}\leq \prod_{0\leq i\leq \tau(q_k)-1} \|Df|_{E(f^{i}(q_k))}\|\leq {\lambda_2}^{\tau(q_k)}.
\end{displaymath}
Hence by the $\lambda_0^2$-domination of $E\oplus F$, we have that
\begin{displaymath}
\prod_{0\leq i\leq \tau(q_k)-1} \|Df^{-1}|_{F(f^{-i}(q_k))}\|\leq {\lambda_2}^{\tau(q_k)}.
\end{displaymath}
Then by item 2 of Lemma~\ref{cor of pliss} and item 2 of Lemma~\ref{property of pliss point}, there is a $\lambda_2$-bi-Pliss point $r_k$ on $orb(q_k)$ for each $k$. Taking a subsequence if necessary, we assume $(r_k)$ is a converging sequence. Then there is $l>0$, such that for any $m,n\geq l$, the stable and unstable manifolds of $r_m$ and $r_n$ intersect respectively, since $r_k$ has uniform stable and unstable manifolds. Hence $(orb(q_m))_{m\geq l}$ are homoclinically related together, thus $p\in H(q_k)$. By item 2 of Lemma~\ref{generic properties}, we have that $q_k\in H(p)$. This finishes the proof of the claim.
\end{proof}
The proof of Theorem B is now completed.
\section{Periodic orbits around a periodic orbit and a set: proof of Proposition 1}\label{proposition 1}
In this section, we give the proof of Proposition~\ref{time control}. To simplify the notations, we assume that $p$ is a hyperbolic fixed point of $f$, and the proof of the general case is similar. Since we want to get a periodic orbit that spends most of the time around $orb(p)$ and $K$, we just prove the proposition for $U_p$ and $U_K$ small. More precisely, we assume that $U_p\cap U_K=\emptyset$ and $x,y\notin U_K$. Moreover, by the hyperbolicity of periodic orbits in $K$, we assume that there are no periodic points with period less than or equal to $N$ contained in $U_K\setminus K$.
Taking a smaller neighborhood if necessary, we assume that the element of $\mathcal{U}$ is of the form $f\circ\phi$ with $\phi\in\mathcal{V}$, where $\mathcal{V}$ is a $C^1$ neighborhood of $Id$ and satisfies the property (F):\\
\textit{(F) For any perturbations $\phi $ and $\phi'$ of $Id$ in $\mathcal{V}$ with disjoint support, the composed perturbation $\phi\circ \phi'$ is still in $\mathcal{V}$.}\\
By the connecting lemma, there is an integer $N$ associated to the neighborhood $\mathcal U$. By the Basic perturbation lemma, there are two numbers $\theta>1$ and $r_0>0$ associated to $\mathcal U$.
Now we fix the neighborhoods $\mathcal{U}$, $U_p$ and $U_K$, and the numbers $N$, $\theta$ and $r_0$.
\subsection{The choice of $n_0$, the point $z_1$ and the perturbation domain at $z_1$.}
\begin{lem}\label{property of prec}
There is a point $z_1\in U_K \setminus K$, such that:
\begin{itemize}
\item for any neighborhood $V_{z_1}$ of $z_1$, there is $n\geq 1$ such that $f^n(x)\in V_{z_1}$;
\item $z_1\prec_{U_K} K$ and $orb^+(z_1)\subset U_K$.
\end{itemize}
\end{lem}
\begin{proof}
The proof is similar to that of Lemma~\ref{prec}. We take a smaller open neighborhood $V$ of $K$ such that $\overline{V}\subset U_K$. Since $\omega(x)\cap K\neq\emptyset$, then for any $k\geq 1$, there is $n_k\geq 1$, such that $f^{n_k}(x)\in B(K,\frac{1}{k})$. Take the smallest integer $m_k$, such that the piece of orbit $(f^{m_k}(x),f^{m_k+1}(x),\cdots,f^{n_k}(x))$ is contained in $V$. Taking a converging subsequence if necessary, we assume that the sequence $\{f^{m_k}(x)\}_{k\geq 1}$ converges to a point $z_1\in\overline{V}\subset U_K$ and the sequence $\{f^{n_k}(x)\}_{k\geq 1}$ converges to a point $z_2\in K$. Then we have that $z_1\prec_{U_K} z_2$, and the pieces of orbit that connects the neighborhoods of $z_1$ and $z_2$ are $(f^{m_k}(x),f^{m_k+1}(x),\cdots,f^{n_k}(x))_{k\geq 1}$. Since $z_2\in K$, we have that $z_1\prec_{U_K} K$. By the choice of $m_k$, we have that $f^{m_k-1}(x)\in M\setminus V$. Since $M\setminus V$ is compact, and $f^{-1}(z_1)$ is a limit point of the sequence $\{f^{m_k-1}(x)\}_{k\geq 1}$, we have that $f^{-1}(z)\in M\setminus V$. By the invariance of $K$, we have that $z_1\notin K$ and $n_k-m_k$ goes to $+\infty$. Since $(f^{m_k}(x),f^{m_k+1}(x),\cdots,f^{n_k}(x))$ is contained in $U_K$ and by the fact that $f^{m_k}(x)$ converges to $z_1$, we have that $orb^+(z_1)\subset\overline{V}\subset U_K$. Thus the second item is satisfied. The first item is a trivial fact by the choice of $z_1$.
\end{proof}
By the assumption on $U_K$, we have that $z_1$ is not a periodic point with period less than or equal to $N$. Also, since $y\in W^s(p)$, we have $z_1\notin orb(y)$. Then there are two neighborhoods $V_{z_1}\subset U_{z_1}$ of $z_1$ satisfying the conclusion of the connecting lemma for the triple $(f,\mathcal{U},N)$, and also satisfying the following conditions:
\begin{itemize}
\item $U_{z_1}\cup f(U_{z_1})\cup \cdots \cup f^N(U_{z_1}) \subset U_K\setminus K$;
\item $(U_{z_1}\cup f(U_{z_1})\cup \cdots \cup f^N(U_{z_1})) \cap orb(y)=\emptyset$.
\end{itemize}
Then there is $n_1\in \mathbb{N}$ such that $f^{n_1}(x)\in V_{z_1}$, and there is $n_2$, such that, for any $n\geq n_2$, we have $f^{-n}(y)\in U_K$. Let $n_0=n_1+n_2$.
\subsection{The choices of points and perturbation domains in $K$ and to get $h$.}\label{get h}
Take any integer $T_K$. By Lemma~\ref{property of prec}, we have that $z_1\prec_{U_K} K$, that is to say, there is a point $z_2\in K$ such that $z_1\prec_{U_K} z_2$. Now we consider two cases, depending on whether there is such a point $z_2$ that is not a periodic point with period less than or equal to $N$.
\subsubsection{The non-periodic case}
Assume that there is a point $z_2\in K$ which is not a periodic point with period less than or equal to $N$, such that $z_1\prec_{U_K} z_2$. Then there are two neighborhoods $V_{z_2}\subset U_{z_2}$ of $z_2$ satisfying the conclusion of the connecting lemma for the triple $(f,\mathcal{U},N)$ and also satisfying the following conditions:
\begin{itemize}
\item $U_{z_2}\cup f(U_{z_2})\cup \cdots \cup f^N(U_{z_2}) \subset U_K$;
\item $(U_{z_1}\cup f(U_{z_1})\cup \cdots \cup f^N(U_{z_1})) \cap (U_{z_2}\cup f(U_{z_2})\cup \cdots \cup f^N(U_{z_2}))=\emptyset$;
\item $f^{-n}(y)\notin U_{z_2}\cup f(U_{z_2})\cup \cdots \cup f^N(U_{z_2})$, for any $n\leq n_2+T_K$.
\end{itemize}
Then there is $n_3>n_2+T_K$, such that $f^{-n_3}(y)\in V_{z_2}$. Since we have the fact that $z_1\prec_{U_K} z_2$, there is a piece of orbit $(w,f(w),\cdots,f^k(w))$ contained in $U_K$, such that $w\in V_{z_1}$ and $f^k(w)\in V_{z_2}$, and by the choice of $U_{z_1}$, we have that $w\notin orb(y)$.
\paragraph{\textit{\textmd{Perturbations to get $h$ in the non-periodic case.}}} Now we do the perturbations step by step to get the conclusion.
\textbf{Step 1.} From the choice of points and neighborhoods above, we can see that the point $x$ has a positive iterate $f^{n_1}(x)\in V_{z_1}$ and the point $f^k(w)$ has a negative iterate $w\in V_{z_1}$. Then by the connecting lemma, there is a diffeomorphism $f_1\in\mathcal{U}$, such that $f_1$ coincides with $f$ outside $U_{z_1}\cup f(U_{z_1})\cup \cdots \cup f^N(U_{z_1})$ and $f^k(w)$ is on the positive orbit of $x$ under $f_1$.
\textbf{Step 2.} For the diffeomorphism $f_1$, the point $x$ has a positive iterate $f^k(w)\in V_{z_2}$, and the point $y$ has a negative iterate $f^{-n_3}(y)\in V_{z_2}$. Since $f_1$ coincides with $f$ outside $U_{z_1}\cup f(U_{z_1})\cup \cdots \cup f^N(U_{z_1})$ and $(U_{z_1}\cup f(U_{z_1})\cup \cdots \cup f^N(U_{z_1})) \cap (U_{z_2}\cup f(U_{z_2})\cup \cdots \cup f^N(U_{z_2}))=\emptyset$, then by the connecting lemma, there is a diffeomorphism $h\in\mathcal{U}$, such that $y$ is on the positive orbit of $x$ under $h$ and $h$ coincides with $f_1$ outside $U_{z_2}\cup f(U_{z_2})\cup \cdots \cup f^N(U_{z_2})$.
By the constructions above, $h$ coincides with $f$ outside $(U_{z_1}\cup f(U_{z_1})\cup \cdots \cup f^N(U_{z_1}))\cup(U_{z_2}\cup f(U_{z_2})\cup \cdots \cup f^N(U_{z_2}))$. Hence $h$ coincides with $f$ on $orb(p)\cup orb^-(x)\cup orb^+(y)$ and outside $U_K$. Moreover, $\sharp (orb(x,h)\cap U_K)\geq n_3-n_2\geq T_K$ and $\sharp (orb(x,h)\setminus (U_K\cup U_p))\leq n_1+n_2\leq n_0$.
\subsubsection{The periodic case}
Assume that any point $z_2\in K$ satisfying $z_1\prec_{U_K} z_2$ is a periodic point with period less than or equal to $N$. We take such a point $q\in K$. In this case, we can not use the connecting lemma at the point $q$ since its period is small but we can do perturbations at the stable and unstable manifolds of $q$ since it is hyperbolic. To simplify the proof, we assume that $q$ is a hyperbolic fixed point of $f$, but the general case is identical. We take a neighborhood $U_q$ of $q$ such that $\overline{U_q}\subset U_K\setminus (U_{z_1}\cup f(U_{z_1})\cup \cdots \cup f^N(U_{z_1}))$, and such that for any point $w$ satisfying $orb^+(w)\subset U_q$ (rep. $orb^-(w)\subset U_q$), we have $w\in W^s(q)$ (resp. $w\in W^u(q)$).
Since $z_1\prec_{U_K} q$, similarly to the argument of Lemma~\ref{property of prec}, we can get that, there is a point $x'\in U_q$, such that $z_1\prec_{U_K} x'$ and $orb^+(x)\subset U_q$. By the choice of $U_q$, we have that $x'\in W^s(q)$ and $x'\notin U_{z_1}\cup f(U_{z_1})\cup \cdots \cup f^N(U_{z_1})$. Then $x'$ is not a periodic point, thus $x'\notin K=\alpha(y)$. Then we can take two neighborhoods $V_{x'}\subset U_{x'}$ of $x'$ that satisfy the conclusions of the connecting lemma for the triple $(f,\mathcal{U},N)$, and also satisfy that
\begin{itemize}
\item $U_{x'}\cup f(U_{x'})\cup\cdots,\cup f^N(U_{x'})\subset U_q$;
\item $U_{x'}\cap orb(y)=\emptyset$ and $q\notin U_{x'}$.
\end{itemize}
Then, there is a piece of orbit $(w',\cdots,f^{k'}(w'))$ contained in $U_K$, such that $w'\in V_{z_1}$ and $f^{k'}(w')\in V_{x'}$. Moreover, $w'\notin orb(y)$.
Since $q\in \alpha(y)$, there is $y'\in W^u(q)\cap U_q$, such that for any neighborhood $U$ of $y'$, there is an integer $n\geq 1$, such that $f^{-n}(y)\in U$. (In fact, if $\alpha(y)=\{q\}$, we can choose $y'$ to be a negative iterate of $y$. If $\{q\}\subsetneqq \alpha(y)$, we can choose $y'$ to be contained in $\alpha(y)\cap W^u(q)$). Now, we build the perturbation domain at the point $y'$ by Lemma~\ref{basic perturbation}. More precisely, we do as the following. We take a number $r'<r_0$ small enough, such that: if we take the neighborhood $U_{y'}=f(B(f^{-1}(y'),\theta r'))$ of $y'$, then the following properties are satisfied:
\begin{itemize}
\item $U_{y'}\cup f^{-1}(U_{y'})\subset U_q\setminus (\{q\}\cup U_{x'}\cup f(U_{x'})\cup\cdots,\cup f^N(U_{x'}))$;
\item $U_{y'}\cap f^{-1}(U_{y'})=\emptyset$;
\item $\{w',\cdots,f^{k'}(w')\}\cap (U_{y'}\cup f^{-1}(U_{y'}))=\emptyset$.
\end{itemize}
Then by the $\lambda$-lemma, there is a piece of orbit $(z',f(z'),\cdots,f^{n_4}(z'))$ contained in $U_q$, such that $z'\in V_{x'}$, $f^{n_4}(z')\in B(f^{-1}(y'),r')$, $f^i(z')\notin U_{x'}$ for any $i\in \{1,2,\cdots,n_4\}$ and $n_4\geq T_K$. By the choice of $y'$, there is a negative iterate $f^{-n_5}(y)$ of $y$ contained in $B(f^{-1}(y'),r')$.
\paragraph{\textit{\textmd{Perturbations to get $h$ in the periodic case.}}}
From the above constructions, we can see that the perturbation domains are pairwise disjoint and contained in $U_K$, and the pieces of orbits that connects two perturbation domains are pairwise disjoint and disjoint from the other perturbation domains. Then we can do the perturbations step by step as in \textit{Case 1}.
\textbf{Step 1.} By the basic perturbation lemma, there is $f_1\in \mathcal{U}$, such that, $f_1$ coincides with $f$ outside $f^{-1}(U_{y'})$ and $f_1(f^{n_4}(z')=f^{-n_5+1}(y)$.
\textbf{Step 2.} For the diffeomorphism $f_1$, the point $y$ has a negative iterate $z'\in V_{x'}$, and the point $w'$ has a positive iterate $f^{k'}(w')\in V_{x'}$. Then by the connecting lemma, there is $f_2\in\mathcal{U}$, such that $f_2$ coincides with $f_1$ outside $U_{x'}\cup f(U_{x'})\cup \cdots \cup f^N(U_{x'})$, and $w'$ is on the negative orbit of $y$ under $f_2$.
\textbf{Step 3.} For the diffeomorphism $f_2$, the point $y$ has a negative iterate $w'$ in $V_{z_1}$ and the point $x$ has a positive iterate $f^{n_1}(x)\in V_{z_1}$. By the connecting lemma, there is $h\in\mathcal{U}$, such that, $h$ coincides with $f_2$ outside $U_{z_1}\cup f(U_{z_1})\cup \cdots \cup f^N(U_{z_1})$ and $y$ on the positive orbit of $x$ under $h$.
By the constructions above, the diffeomorphism $h$ coincides with $f$ outside $(U_{z_1}\cup f(U_{z_1})\cup \cdots \cup f^N(U_{z_1}))\cup (U_{x'}\cup f(U_{x'})\cup \cdots \cup f^N(U_{x'}))\cup f^{-1}(U_{y'})$. Hence $h$ coincides with $f$ on $orb(p)\cup orb^-(x)\cup orb^+(y)$ and outside $U_K$. Moreover, $\sharp (orb(x,h)\cap U_K)\geq n_4\geq T_K$ and $\sharp (orb(x,h)\setminus (U_K\cup U_p))\leq n_1+n_2\leq n_0$.
\subsection{The choice of $l$ and the perturbation domains at $x$ and $y$, and to get $h_m$.}
From the constructions in section~\ref{get h}, we get the diffeomorphism $h$ that satisfies the first item of Proposition~\ref{time control}. In this section, we do the perturbations to get the diffeomorphism $h_m$.
Assume $h^t(x)=y$. By replacing $x$ and $y$ to a negative or positive iteration, we assume that $x,y\in U_p$ and $orb^-(x,h)\cup orb^+(y,h)\subset U_p$. Assume that $\{h^n(x)\}_{1\leq n\leq t}\cap U_p=m_0$. We take a number $r<r_0$ small enough, such that, if we take the neighborhood $U_x=h(B(h^{-1}(x),\theta r))$ of $x$ and the neighborhood $U_y=B(y,\theta r)$ of $y$, then, the four sets $U_x$, $h^{-1}(U_x)$, $U_y$ and $h(U_y)$ are contained in $U_p$ and pairwise disjoint from each other and disjoint with $\{h^n(x)\}_{1\leq n\leq t}$. By the $\lambda$-Lemma, there is $l_0\in\mathbb{N}$, such that, for any $m\geq 1$, there is a piece of orbit $(h(z),h^2(z),\cdots,h^{l_0+m-1}(z))$ contained in $U_p$, such that $h(z)\in B(y,r)$, $h^{l_0+m-1}(z)\in B(h^{-1}(x),r)$ and $h^i(z)\notin U_y\cup h^{-1}(U_x)$ for any $i=2,3,\cdots,l_0+m-2$. Let $l=l_0+m_0$.
By the basic perturbation lemma and the disjointness of $U_y$, $f^{-1}(U_x)$ and $U_K$, there is $h_m\in \mathcal{U}$, such that, $h_m$ coincides with $h$ outside $U_y\cup f^{-1}(U_x)$, and $h_m(y)=h^2(z)$, $h_m(f^{l_0+m-1}(z))=h_m^{l_0+m-1}(y)=x$. Hence $h_m$ coincides with $h$ on $orb(p)$ and outside $U_p$. Moreover, the point $x$ is a periodic point of $h_m$, and denote $O=orb(x,h_m)$, we have that $O\setminus U_p=(orb(x,h)\setminus U_p$, and $\sharp (O\cap U_p)=l_0+m_0+m=l+m$.
This finishes the proof of Proposition~\ref{time control}.
\begin{rem}
We point out here that, in the non-periodic case, we can not do the same perturbations at $x'$ and $y'$ just as at the points $x$ and $y$. Because the piece of orbit $(w',\cdots,f^{k'}(w'))$ that connects the neighborhoods $V_{z_1}$ and $V_{x'}$ of $z_1$ and $x'$ respectively may enter into the neighborhood $U_{x'}$ many times before $f^{k'}(w')$. Thus if we use the basic perturbation lemma to connect $f^{k'}(w')$ to $f(z')$, the piece of orbit $(w',\cdots,f^{k'}(w'))$ may be modified and it is not clear if the negative orbit of $y$ can intersect $V_{z_1}$ after such perturbation. Thus we can not get a periodic orbit.
\end{rem}
\section{Asymptotic approximation for true orbits: proof of Proposition 2}\label{proposition 2}
In this section, we give a proof of Proposition~\ref{asymptotic connecting 1}. In fact, the proof is almost the same as the proof of Proposition 10 in~\cite{c1}. We only have to explain why the perturbations will not modify $orb^-(x)$. We assume that for any point $y\in W^u(p,f)$, we have $\omega(y)\setminus K\neq \emptyset$, otherwise there is nothing needed to prove. Also we assume that $x\notin K$. Otherwise, the proof follows exactly that of Proposition 10 in~\cite{c1}. We take two steps to get our purpose:
\begin{itemize}
\item we choose a sequence of non-periodic points $(z_n)_{n\geq 0}$, such that:
\begin{center}
$z_0\prec z_1\prec \cdots \prec K$, $z_0\in \overline{W^u(p)}$ and $z_n\notin \overline{orb^-(x)}$, for any $n\geq 0$,
\end{center}
\item then we perturb at every $z_n$ to connect all the points together and avoid $orb^-(x)$.
\end{itemize}
\bigskip
In order to prove Proposition 2, we take a decreasing sequence of $C^1$-neighborhoods $(\mathcal{U}_n)$ of $f$ that satisfies the following properties:
\begin{itemize}
\item $\overline{\mathcal{U}_0}\subset \mathcal{U}$,
\item the element of $\mathcal{U}_n$ is of the form $f\circ \phi$ with $\phi\in \mathcal{V}_n$, where $(\mathcal{V}_n)$ is a decreasing sequence of $C^1$ neighborhoods of $Id$ that satisfy the property (F) stated in Section~\ref{proposition 1}, and $\cap_n \mathcal{V}_n=\{Id\}$.\\
\end{itemize}
Then we have that $\cap_n \mathcal{U}_n=\{f\}$. The connecting lemma associates to each pair $(f,\mathcal{U}_k)$ a number $N_k$.
We need the following three lemmas for the proof of Proposition~\ref{asymptotic connecting 1}.
\begin{lem}\label{connecting points}
For any neighborhood $W$ of $K$, there is a point $z\in (W\cap\overline{W^u(p)})\setminus K$, such that, $z\prec_W K$ and $orb^+(z)\subset W$. Moreover, $z\notin orb^-(x)$.
\end{lem}
\begin{proof} The proof is similar to the proof of Lemma~\ref{property of prec}. We assume $W$ is a small neighborhood of $K$ such that $x\not\in \overline{W}$ and $p\notin W$. Take an open neighborhood $V\subset W$ of $K$, such that $\overline{V}\subset W$. Since $\overline{W^u(p)}\cap K\neq \emptyset$, then for any $k\geq 1$, there is a point $x_k\in W^u(p)$ and a positive integer $n(k)$, such that $f^{n(k)}(x_k)\in B(K,\frac{1}{k})$. For $k$ large, the set $B(K,\frac{1}{k})$ is contained in $V$. We consider the smallest integer $m_k$ such that the piece of orbit $(f^{m(k)}(x_k),\cdots,f^{n(k)}(x_k))$ is contained in $V$. By the assumption that $\omega(x_k)\setminus K\neq \emptyset$ for any $k\geq 1$, we can see that both $m(K)$ and $n(k)-m(k)$ go to infinity as $k$ goes to infinity.
Take converging subsequences if necessary, assume the sequence $\{f^{m(k)}(x_k)\}$ converges to a point $z\in \overline{V}$, and $\{f^{n(k)}(x_k)\}$ converges to a point $z'\in K$. Then $z\in \overline{W^u(p)}$, and similar to the argument of Lemma~\ref{property of prec}, we have that $z\notin K$, and $z\prec_{W} z'$, hence $z\prec_W K$. Since $f^{m(k)}(x_k),\cdots,f^{n(k)}(x_k)$ is contained in $V$ and $n(k)-m(k)$ go to infinity, we have that $orb^+(z)\subset W$. Then by the assumption $x\notin \overline{W}$, we have $z\notin orb^-(x)$.
\end{proof}
\begin{lem}\label{choose points}
There are a point $y\in W^u(p)$, a sequence of points $(z_k)_{k\geq 1}$, three sequences of neighborhoods $(U_k)_{k\geq 1}$, $(V_k)_{k\geq 1}$, $(W_k)_{k\geq 0}$ and a sequence of finite segment of orbits $Y_k=(y_k,f(y_k),\cdots,f^{m(k)}(y_k))_{k\geq 0}$, such that:
\begin{enumerate}
\item $W_k\subset W_{k+1}$ and $\cap_k W_k=K$;
\item the connecting lemma can be applied to $z_k\in V_k\subset U_k$ for the triple $(f,\mathcal{U}_k,N_k)$, and $\overline{f^n(U_k)}\subset W_k\setminus W_{k+1}$ for all $0\leq n\leq N_k$;
\item $\overline{U_k}\cap orb^-(x)=\emptyset$, for any $k\geq 1$;
\item $z_1\prec_{W_1}z_2\prec_{W_2}\cdots \prec_{W_k} z_{k+1}\prec \cdots \prec K$;
\item the points $f^{m(k)}(y_k)$ and $y_{k+1}$ are contained in $V_{k+1}$ for all $k\geq 0$ where $y_0=y$, and $orb^-(y)\cap W_1=\emptyset$;
\item $Y_k\subset W_{k}\setminus W_{k+2}$ for all $k\geq 0$ and $Y_k\cap orb^-(x)=\emptyset$, for all $k\geq 0$;
\item there is an open set $V$ containing $orb^-(x)$, such that $V\cap U_k=\emptyset$ for all $k\geq 1$.
\end{enumerate}
\end{lem}
\begin{proof} We build all the sequences by induction. Set $W_0=M$. We first choose $W_1$, $z_1$, $V_1$, $U_1$ and $Y_0$.
Since all periodic orbits in $K$ are hyperbolic, there is a neighborhood $W_1$ of $K$, such that there is no periodic points with period less than or equal to $N_1$ contained in $W_1\setminus K$. Also we can assume that $p\notin W_1$. By Lemma~\ref{connecting points}, there is $z_1\in W_1\setminus K$, such that $z_1\prec_{W_1} K$, $z_1\notin orb^-(x)$, and $z_1\in \overline{W^u(p)}$. By the choice of $W_1$, there is a neighborhood $U_1$ of $z_1$, that is disjoint from its $N_1$ first iterates. Moreover, because $z_1\notin \overline{orb^-(x)}$, we can assume $\overline{U_1}\cap (orb^-(x)\cup K)=\emptyset$. By the connecting lemma, there is $V_1\subset U_1$ associated to $(f,\mathcal{U}_1,N_1)$. Then there are a point $y\in W^u(p)\setminus W_1$ and a positive integer $m(0)$, such that $f^{m(0)}(y)\in V_1$. Moreover, by considering a negative iterate of $y$ if necessary, we can assume that $orb^-(y)\cap W_1=\emptyset$. We take $y_0=y$ and $Y_0=(y,f(y),\cdots,f^{m(0)}(y))$.
Now we construct the sequences by induction on $k$. After $W_k$, $z_k$, $V_k$, $U_k$ and $Y_{k-1}$ have been built, there is $W_{k+1}\subset W_k$ such that
\begin{itemize}
\item there is no periodic point with period less than or equal to $N_{k+1}$ contained in $W_{k+1}\setminus K$;
\item $\overline{f^n(U_k)}\cap W_{k+1}=\emptyset$, for all $1\leq n\leq N_k$;
\item $W_{k+1}\cap Y_{k-1}=\emptyset$;
\item $W_{k+1}$ is contained in $B(K,\frac{1}{k})$.\\
\end{itemize}
By Lemma~\ref{prec}, there is $z_{k+1}\in W_{k+1}\setminus K$, such that $z_k\prec_{W_k} z_{k+1}\prec_ {W_{k+1}} K$, $z_{k+1}\notin \overline{orb^-(x)}$ and $orb^+(z_{k+1})\subset W_{k+1}$. By the connecting lemma, there are neighborhoods $V_{k+1}\subset U_{k+1}$ of $z_{k+1}$ associated to $(f,\mathcal{U}_{k+1},N_{k+1})$, such that:
\begin{itemize}
\item $\overline{U_{k+1}}\cap (orb^-(x)\cup K)=\emptyset$,
\item $\overline{f^n(U_{k+1})}\subset W_{k+1}$ for all $0\leq n\leq N_{k+1}$.\\
\end{itemize}
Then there is $Y_{k}=(y_k,f(y_k),\cdots,f^{m(k)}(y_k))$, such that $y_k\in V_k$ and $f^{m(k)}(y_k)\in V_{k+1}$. Since $\overline{U_{k+1}}\cap orb^-(x)=\emptyset$, $Y_{k}$ is disjoint from $orb^-(x)$.
Since $x\notin K$, for any integer $n\in\mathbb{N}$, there is $n_k$, such that $f^{-n}(x)\notin W_{n_k}$. By item 2 and 3, there is an open neighborhood $B_n$ of $f^{-n}(x)$, such that $B_n\cap U_k=\emptyset$ for any $k\geq 1$. We take $V=\bigcup_{n\geq 0}B_n$, then item 7 is satisfied. Then we finish the proof of Lemma~\ref{choose points}.
\end{proof}
Now we fix the point $y\in W^u(p)$, the open set $V$ and the sequences $(z_k)_{k\geq 1}$, $(U_k)_{k\geq 1}$, $(V_k)_{k\geq 1}$, $(W_k)_{k\geq 0}$ and $(Y_k)_{k\geq 0}$ as in Lemma~\ref{choose points}. We have the following lemma.
\begin{lem}\label{perturbations}
There are a sequence of perturbations $(g_k)_{k\geq 0}$ of $f$ and a strictly increasing sequence of integers $(n_k)_{k\geq 0}$, such that,
\begin{enumerate}
\item $g_0=f$ and $n_0=0$;
\item there is $\phi_k\in \mathcal{V}_k$, such that $\phi_k=Id|_{M\setminus (U_k\cup \cdots \cup f^{N_k-1}(U_k))}$ and $g_k=g_{k-1}\circ \phi_k$, for $k\geq 1$;
\item for any $l=\{0,1,\cdots,k-1\}$, the piece of orbit $(g_k^{n_l}(y),g_k^{n_l+1}(y),\cdots,g_k^{n_{l+1}}(y))$ is contained in $W_l\setminus W_{l+2}$.
\end{enumerate}
\end{lem}
\begin{proof}
We build inductively the sequences $(g_k)$ and $(n_k)$ and another sequence of integers $(m_k)_{k\geq 0}$ to satisfy the conclusions and also the following properties:
\begin{itemize}
\item $m_k>n_k$ and $g_k^{m_k}(y)\in V_{k+1}$;
\item the piece of orbit $(g_k^{n_k}(y),g_k^{n_k+1}(y),\cdots,g_k^{m_k}(y))$ is contained in $W_k\setminus W_{k+2}$.
\end{itemize}
First, we take $g_0=f$ and $n_0=0$. By Lemma~\ref{choose points}, there is $m_0>0$, such that $g_0^{m_0}(y)\in V_1$ and the piece of orbit $(y=g_0^{n_0}(y),g_0(y),\cdots,g_0^{m_0}(y))$ is contained in $W_0\setminus W_2$.
Now assume that $g_k$, $n_k$ and $m_k$ have been built, we explain how to get $g_{k+1}$, $n_{k+1}$ and $m_{k+1}$. The point $g_k^{n_k}(y)$ has a positive iterate $g_k^{m_k}(y)\in V_{k+1}$, and the point $f^{m(k+1)}(y_{k+1})$ has a negative iterate $y_{k+1}\in V_{k+1}$. Since $g_k$ coincides with $f$ on the set $U_{k+1}\cup f(U_{k+1})\cup\cdots\cup f^{N_{k+1}}(U_{k+1})$, one can apply the connecting lemma to $(g_k,\mathcal{U}_{k+1},V_{k+1},U_{k+1})$ and get a diffeomorphism $g_{k+1}$, such that, the point $f^{m(k+1)}(y_{k+1})$ is on the positive orbit of $g_k^{n_k}(y)$ under iteration of $g_{k+1}$, and, moreover, the new diffeomorphism $g_{k+1}$ is of the form $g_k\circ \phi_{k+1}$, where $\phi_{k+1}=Id|_{M\setminus (U_{k+1}\cup \cdots \cup f^{N_{k+1}-1}(U_{k+1}))}$ and $f\circ \phi_{k+1}\in\mathcal{U}_{k+1}$, thus $\phi_{k+1}\in\mathcal{V}_{k+1}$.
Since $g_{k+1}=g_k|_{M\setminus (U_{k+1}\cup \cdots \cup f^{N_{k+1}-1}(U_{k+1}))}$, the piece of orbit $y,g_k(y),\cdots,g_k^{n_k}(y)$ under $g_k$ coincides with the one $y,g_{k+1}(y),\cdots,g_{k+1}^{n_k}(y)$. Moreover, the point $g_k^{n_k}(y)$ has a positive iterate $f^{m(k+1)}(y_{k+1})$ under $g_{k+1}$ contained in $V_{k+2}$. Assume $f^{m(k+1)}(y_{k+1})=g_{k+1}^{m_{k+1}}(y)$, then there is an integer $n_{k+1}$ with $n_k<n_{k+1}<m_{k+1}$, such that, the piece of orbit $(g_{k+1}^{n_k}(y),\cdots,g_{k+1}^{n_{k+1}}(y))$ is contained in $W_{k}\setminus W_{k+2}$ and the piece of orbit $(g_{k+1}^{n_{k+1}}(y),\cdots,g_{k+1}^{m_{k+1}}(y))$ is contained in $W_{k+1}\setminus W_{k+3}$. Then the conclusions are satisfied for $k+1$.
\end{proof}
\begin{proof}[End of the proof of Proposition~\ref{asymptotic connecting 1}]
Since the supports $U_i\cup\cdots\cup f^{N_i-1}(U_i)$ and $U_j\cup\cdots\cup f^{N_j-1}(U_j)$ of the perturbations $\phi_i$ and $\phi_j$ are disjoint for any $i\neq j$, and $(\mathcal{V}_n)$ satisfy the property F, then the sequence $g_k=f\circ\phi_1\circ\cdots\circ\phi_k$ converges to a diffeomorphism $g\in\mathcal{U}_0\subset \mathcal{U}$. By the constructions, $g$ coincides with $f\circ\phi_k$ in the set $U_k\cup\cdots\cup f^{N_k-1}(U_k)$ and with $f$ elsewhere, hence, by the choice of $(U_k)$ and $V$, it holds that $g$ coincides with $f$ on the set $orb(p)\cup K\cup V\cup orb^-(y)$ and $\omega(y,g)\subset K$. Moreover, since $g$ is the limit of the sequence $(g_k)$, by Lemma~\ref{perturbations}, for any $n>n_k$, $g^n(y)\in W_k$. Then we have that $\omega(y,g)\subset K$. This finishes the proof of Proposition~\ref{asymptotic connecting 1}.
\end{proof}
\section{Asymptotic approximation for pseudo-orbits: proof of Proposition 3}\label{proposition 3}
To prove Proposition 3, we use the technics of~\cite{bc,c1} to get true orbits by perturbing a pseudo-orbit. Similarly to the proof of Proposition 2, we have to perturb infinitely many times in a special neighborhood to keep some part of the initial dynamic unchanged. The proof refers a lot to~\cite{bc} and Section 3.2 of~\cite{c2}.
We take several steps for the proof. First, we choose an open set that covers all positive orbits of $X$ that are not on the local stable manifold of periodic orbits with small periods. Actually, we choose a special topological tower for $X$. Second, we construct a sequence of disjoint perturbation domains containing in their interior the special topological tower for $X$. Then, we choose an infinitely long pseudo-orbit in $X$ that goes from $z$ to $K$ and has jumps only in the perturbation domains and accumulates to $K$ in the future. Finally, we perturb in the perturbation domains to construct a true orbit which goes from $z$ to $K$ and accumulates to $K$ in the future.
We take a $C^1$ neighborhood $\mathcal{U}_0$ of the diffeomorphism $f_0$ with $\overline{\mathcal{U}_0}\subset \mathcal{U}$, such that, the element of $\mathcal{U}_0$ is of the form $f\circ\phi$ with $\phi\in\mathcal{V}_0$, where $\mathcal{V}_0$ is a $C^1$-neighborhood of $Id$ that satisfies the property (F) stated in Section~\ref{proposition 1}. Then there is a smaller $C^1$ neighborhood $\mathcal{U}'\subset\mathcal{U}_0$ of $f_0$ and an integer $N_0$ associated to $(f_0,\mathcal{U}_0)$ by the uniform connecting lemma (Theorem~\ref{uniform connecting}). Take the integer $T=10\kappa_d d N_0$ where the number $\kappa_d$ is the number given by Lemma~\ref{ttower}.
From now on, we fix the $C^1$ neighborhoods $\mathcal{U}'\subset\mathcal{U}_0$ of $f_0$ and the integer $T$. We consider a diffeomorphism $f\in\mathcal{U}'$, an invariant compact set $K$ containing no non-hyperbolic periodic point, a positive invariant compact set $X$ containing no non-hyperbolic periodic point with period less than or equal to $T$, a point $z$ that can be connected to $K$ by pseudo-orbits in $X$, and a neighborhood $U$ of $X\setminus K$. Then we consider a decreasing sequence of $C^1$-neighborhoods $(\mathcal{U}_n)_{n\geq 0}$ of $f$, such that, the element of $\mathcal{U}_n$ is of the form $f\circ \phi$ with $\phi\in \mathcal{V}_n$, where $(\mathcal{V}_n)$ is a decreasing sequence of $C^1$ neighborhoods of $Id$ that satisfy the property (F), and $\cap_n \mathcal{V}_n=\{Id\}$. The connecting lemma associates to each pair $(f,\mathcal{U}_k)$ an integer $N_k$ and we can assume that $(N_k)$ is an increasing sequence. Here the neighborhood $\mathcal{U}_0$ and the integer $N_0$ are exactly what we have chosen in the above paragraph. We assume that $z\notin K$, otherwise, there is noting to prove. For an integer $N$, denote by $Per_N(f)$ the set of periodic points of $f$ whose period is no more than $N$.
\subsection{Choice of topological towers}
In this section, we construct a family of special topological towers for the set $X$ with the properties stated in Lemma~\ref{choice of topological tower}.
\begin{lem}\label{choice of topological tower}
For any increasing sequence of integers $(L_k)_{k\geq 0}$ where $L_0=10dN_0$, for any $\delta>0$, there are a decreasing sequence of neighborhoods $(U_k)_{k\geq 0}$ of $K$, a sequence of open sets $(W_k)_{k\geq 0}$, and a sequence of compact sets $(D_k)_{k\geq 0}$, such that, denoting $X_k=X\cap \overline{(U_{k}\setminus U_{k+1})}$ for all $k\geq 0$, the following properties are satisfied.
\begin{itemize}
\item The intersection $\bigcap_{k\geq 0} U_k=K$, where $U_0=M$ and $z\notin\overline{U_1}$,
\item If we take $U_{-1}=M$, then for any $k\geq 0$,
\begin{enumerate}
\item there is no periodic orbit with period less than $\kappa_d L_{k+1}$ contained in $\overline{U_{k+1}}\setminus K$,
\item $f^i(\overline{U_{k+1}})\subset U_{k}$, for all $-4\kappa_d L_{k+1}\leq i\leq 4\kappa_d L_{k+1}$,
\item the $L_k$ sets $\overline{W_k}, f(\overline{W_k}),\cdots,f^{L_k-1}(\overline{W_k})$ are pairwise disjoint, contained in $U\setminus K$, and also contained in $U_{k-1}\setminus \overline{U_{k+2}}$,
\item for any $-4\kappa_d L_{k+2}\leq i\leq 4\kappa_d L_{k+2}$, we have that $f^i(\overline{U_{k+2}})\cap \overline{(W_0\cup\cdots\cup W_{k})}=\emptyset$,
\item for any $l<k$, any $0\leq i\leq L_l$ and any $0\leq j\leq L_{k}$, we have that $\overline{f^i(W_l)}\cap \overline{f^j(W_k)}=\emptyset$,
\item the compact set $D_k$ is contained in $W_k$, such that, any point in $X_0\setminus (\bigcup_{p\in Per_{L_0}(f)}W^s_{\delta}(p))$ has a positive iterate in $int(D_0)$ and any point in $X_k$ has a positive iterate in $int(D_{k-1}\cup D_k)$.
\end{enumerate}
\end{itemize}
\end{lem}
\begin{rem}
The set $W_0\cup \cdots \cup W_k$ can be seen as a special topological tower for $X_0\cup\cdots\cup X_k$, from the items 3, 5, and 6.
\end{rem}
\begin{proof}
We build inductively the sequences $(U_k)_{k\geq 0}$, $(W_k)_{k\geq 0}$ and $(D_k)_{k\geq 0}$ from a sequence of open sets $(W_k')_{k\geq 0}$ and a sequence of compact sets $(D_k')_{k\geq 0}$, which satisfy the following additional properties: for any $k\geq 0$,
\begin{itemize}
\item \textit{1 $'$. the set $W_k$ is contained in an arbitrarily small neighborhood of $W_k'$, and $W_k'\subset W_k$,}
\item \textit{2 $'$. the sets $\overline{W'_k}, f(\overline{W'_k}),\cdots,f^{L_k-1}(\overline{W'_k})$ are pairwise disjoint, contained in $U\setminus K$, and also contained in $U_{k-1}\setminus \overline{U_{k+2}}$,}
\item \textit{3 $'$. for all $-4\kappa_d L_{k+2}\leq i\leq 4\kappa_d L_{k+2}$, we have $f^i(\overline{U_{k+2}}))\cap \overline{(W_0'\cup\cdots\cup W_k')}=\emptyset$,}
\item \textit{4 $'$. for any $l<k$, any $0\leq i\leq L_l$ and any $0\leq j\leq L_{k}$, we have $\overline{f^i(W_l)}\cap \overline{f^j(W_k')}=\emptyset$,}
\item \textit{5 $'$. any point in $X_0\setminus (\bigcup_{p\in Per_{L_0}(f)}W^s_{\delta}(p))$ has a positive iterate in $int(D_0')$ and any point in $X_k$ has a positive iterate contained in $int(D_k')$ for any $k\geq 1$.}
\end{itemize}
\medskip
We build the sets $U_{k+3}$, $W_{k+2}'$, $D_{k+2}'$, $W_{k+1}$ and $D_{k+1}$ after the sets $U_{k+2}$, $W_{k+1}'$, $D_{k+1}',$ $W_{k}$ and $D_{k}$ have been built.
\paragraph{The sets $U_2$, $W_1'$, $D_1'$, $W_0$ and $D_0$.}
By the assumption of hyperbolicity of periodic orbits in $K$, we can take a neighborhood $U_1\subset B(K,1)$ of $K$ such that $z\notin\overline{U_1}$ and there is no periodic orbit with periodic less than $\kappa_d L_1$ in $\overline{U_1}\setminus K$. The properties \textit{1} and \textit{2} are satisfied.
By Lemma~\ref{ttower}, there are an open set $W_0'\subset U$ that is disjoint from its $L_0$ iterates and a compact set $D_0'\subset W_0'$ such that any point contained in $X_0\setminus (\bigcup_{p\in Per_{L_0}(f)}W^s_{\delta}(p))$ has a positive iterate contained in $int(D_0')$. Moreover, by the item \textit{3} of Lemma~\ref{ttower}, the open set $W_0'$ can be contained in an arbitrarily small neighborhood of $X_0\cup f(X_0)\cup\cdots\cup f^{\kappa_dL_0}(X_0)$. Hence we can assume that and $\bigcup_{i=0}^{L_0}f^i(\overline{W_0'})\subset U\setminus K$, since $X$ is positively invariant. The property \textit{5 $'$} is satisfied.
Now we take a neighborhood $U_2\subset U_1\cap B(K,\frac{1}{2})$ of $K$, such that:
\begin{itemize}
\item there is no periodic orbit with periodic less than $\kappa_d L_{2}$ in $\overline{U_{2}}\setminus K$, which is the property \textit{1},
\item $f^i(\overline{U_{2}})\subset U_{1}$, for all $-4\kappa_d L_{2}\leq i\leq 4\kappa_d L_{2}$, which is the property \textit{2},
\item $\overline{W_0'}\cap f^i(\overline{U_{2}})=\emptyset$ for all $-4\kappa_d L_{2}\leq i\leq 4\kappa_d L_{2}$, which implies the properties \textit{2 $'$} and \textit{3 $'$}.
\end{itemize}
By Lemma~\ref{ttower}, for the compact set $X_1=X\cap\overline{U_1\setminus U_2}$, there is an open set $V_{1}'$ disjoint from its $\kappa_dL_{1}$ iterates, such that any point contained in $X_{1}$ has a positive iterate contained in $V_{1}'$. Moreover, $V_{1}'$ can be contained in an arbitrarily small neighborhood of $\bigcup_{i=0}^{\kappa_dL_{1}}f^{i}(X_{1})$.
By Lemma~\ref{choose neighborhoods}, consider $W_0'$ and $V_{1}'$ as $W'$ and $V'$, we get an open set $S_{1}=W_0\cup V_{1}$, such that
\begin{itemize}
\item $\overline{S_{1}}\subset U\setminus K$,
\item $W_0$ is a small neighborhood of $W_0'$, disjoint from its $L_0$ iterates, hence the properties \textit{2 $'$} and \textit{3 $'$} imply the properties $3$ and $4$ if we choose $W_k$ small enough, and the property \textit{1 $'$} is satisfied automatically,
\item $V_{1}$ is contained in an arbitrarily small neighborhood of $V_{1}'\cup f(V_{1}')\cup\cdots\cup f^{\kappa_dL_{1}}(V_{1}')$ and disjoint from its $L_{1}$ iterates, thus we can assume that $K\cap \overline{V_{1}}=\emptyset$,
\item $V_{1}'\subset \bigcup_{i=0}^{\kappa_d L_{1}} f^{i}(S_{1})$,
\item $\overline{W_0}\cap f^{i}(\overline{V_{1}})=\emptyset$ for all $i=0,\pm 1,\cdots,\pm L_{1}$.
\end{itemize}
Then any point contained in $X_{1}$ has a positive iterate contained in $S_{1}$. By the compactness of $X_{1}$, there is a compact set $D_{1}'\subset S_{1}$ such that all such iterates are contained in $int(D_{1}')$. Set $W_{1}'=V_{1}$ and $D_0=D_0'\cup (D_{1}'\cap W_0)$. The properties \textit{6}, \textit{2 $'$}, \textit{3 $'$}, \textit{4 $'$} and \textit{5 $'$} are satisfied.
Hence we have built the sets $U_1$, $U_2$, $W_0'$, $D_0'$, $W_1'$, $D_1'$, $W_0$ and $D_0$.
\paragraph{Construction of the sets $U_{k+3}$, $W_{k+2}'$, $D_{k+2}'$, $W_{k+1}$ and $D_{k+1}$.}
For $k\geq 0$, assume the sets $U_{k+2}$, $W_{k+1}'$, $D_{k+1}',$ $W_{k}$ and $D_{k}$ have been built to satisfy the properties above for all $n\leq k$, we now explain how to get the sets $U_{k+3}$, $W_{k+2}'$, $D_{k+2}'$, $W_{k+1}$ and $D_{k+1}$.
We can take a neighborhood $U_{k+3}\subset U_{k+2}\cap B(K,\frac{1}{k+3})$ of $K$, such that:
\begin{itemize}
\item there is no periodic orbit with periodic less than $\kappa_d L_{k+3}$ in $\overline{U_{k+3}}\setminus K$, which is the property \textit{1},
\item $f^i(\overline{U_{k+3}})\subset U_{k+2}$, for all $-4\kappa_d L_{k+2}\leq i\leq 4\kappa_d L_{k+3}$, which is the property \textit{2},
\item $\overline{W_0\cup \cdots W_{k} \cup W_{k+1}'}\cap f^i(\overline{U_{k+3}})=\emptyset$ for all $-4\kappa_d L_{k+3}\leq i\leq 4\kappa_d L_{k+3}$, which implies the property \textit{3 $'$}.
\end{itemize}
By Lemma~\ref{ttower}, for the set $X_{k+2}=X\cap\overline{U_{k+2}\setminus U_{k+3}}$, there is an open set $V_{k+2}'$ disjoint from its $\kappa_dL_{k+2}$ iterates, such that any point contained in $X_{k+2}$ has a positive iterate contained in $V_{k+2}'$. Moreover, $V_{k+2}'$ can be contained in an arbitrarily small neighborhood of $\bigcup_{i=0}^{\kappa_dL_{k+2}}f^{i}(X_{k+2})$. Since $X_{k+2}\subset \overline{U_{k+2}}$, and for all $-4\kappa_d L_{k+2}\leq i\leq 4\kappa_d L_{k+2}$,we have that $f^i(\overline{U_{k+2}}))\cap \overline{(W_0\cup\cdots\cup W_{k})}=\emptyset$ and $f^i(\overline{U_{k+2}}))\subset U_{k+1}$, then we can assume that $f^i(\overline{V_{k+2}'}))\cap (\overline{W_0\cup\cdots\cup W_{k}}\cup K)=\emptyset$ and $f^i(\overline{V_{k+2}'}))\subset U_{k+1}$, for all $-2\kappa_d L_{k+2}\leq i\leq 2\kappa_d L_{k+2}$.
By Lemma~\ref{choose neighborhoods}, consider $W_{k+1}'$ and $V_{k+2}'$ as $W'$ and $V'$, we get an open set $S_{k+2}=W_{k+1}\cup V_{k+2}$, such that
\begin{itemize}
\item $\overline{S_{k+2}}\subset U\setminus K$,
\item $W_{k+1}$ is a small neighborhood of $W_{k+1}'$, disjoint from its $L_{k+1}$ iterates, hence the properties \textit{2 $'$}, \textit{3 $'$} and \textit{4 $'$} imply the properties $3$, $4$ and $5$ if we choose $W_{k+1}$ small enough, and the property \textit{1 $'$} is satisfied automatically,
\item $V_{k+2}$ is contained in an arbitrarily small neighborhood of $V_{k+2}'\cup f(V_{k+2}')\cup\cdots\cup f^{\kappa_dL_{k+2}}(V_{k+2}')$ and disjoint from its $L_{k+2}$ iterates, thus we can assume that $K\cap \overline{V_{k+2}}=\emptyset$, and for all $-\kappa_d L_{k+2}\leq i\leq \kappa_d L_{k+2}$, we have $f^i(\overline{V_{k+2}}))\cap \overline{(W_0\cup\cdots\cup W_{k})}=\emptyset$ and $f^i(\overline{V_{k+2}}))\subset U_{k+1}$,
\item $V_{k+2}'\subset \bigcup_{i=0}^{\kappa_d L_{k+2}} f^{i}(S_{k+2})$,
\item $\overline{W_{k+1}}\cap f^{i}(\overline{V_{k+2}})=\emptyset$ for all $i=0,\pm 1,\cdots,\pm L_{k+2}$.
\end{itemize}
Then any point contained in $X_{k+2}$ has a positive iterate contained in $S_{k+2}$. By the compactness of $X_{k+2}$, there is a compact set $D_{k+2}'\subset S_{k+2}$ such that all such iterates are contained in $int(D_{k+2}')$. Set $W_{k+2}'=V_{k+2}$ and $D_{k+1}=D_{k+1}'\cup (D_{k+2}'\cap W_{k+1})$. The properties \textit{6}, \textit{2 $'$}, \textit{3 $'$}, \textit{4 $'$} and \textit{5 $'$} are satisfied.
Then we have built the sets $U_{k+3}$, $W_{k+2}'$, $D_{k+2}'$, $W_{k+1}$ and $D_{k+1}$. The first property that $\bigcap_{k\geq 0} U_k=K$ and $z\notin\overline{U_1}$ are obviously satisfied by the choice of $U_k$. This finishes the proof of Lemma~\ref{choice of topological tower}.
\end{proof}
\subsection{Construction of perturbation domains}
Now we take $L_k=10dN_k$ for all $k\geq 0$, and take a small number $\delta>0$, such that for any two different hyperbolic periodic points $q_1,q_2\in Per_{N_0}(f)\cap X$, we have $W^{\sigma_1}_{\delta}(q_1)\cap W^{\sigma_2}_{\delta}(q_2)=\emptyset$, where $\sigma_i\in \{u,s\}$. By Lemma~\ref{choice of topological tower}, we get the sequences $(U_k)_{k\geq 0}$, $(W_k)_{k\geq 0}$ and $(D_k)_{k\geq 0}$. We still denote $X_k=X\cap \overline{(U_{k-1}\setminus U_k)}$ for all $k\geq 0$.
Now we build the perturbation domains for the family $(X_k)$. The technics are mainly from Section 4.1 and 4.2 of~\cite{bc}. First, we build the perturbation domains that covers the points which are not on the local stable manifolds of periodic orbits with period less than or equal to $N_0$. The proof is essentially due to Corollaire 4.1 of~\cite{bc}. They deal with a family of perturbation domains with the same order, thus the union forms a perturbation domain. Here we have a sequence of perturbation domains with different orders. The construction of each perturbation domain can be separated.
\begin{lem}\label{perturbation domains 1}
There is a perturbation domain $B_k$ of order $N_k$ for $(f,\mathcal{U}_k)$ for each $k\geq 0$, such that the sequence $({B}_k)_{k\geq 0}$ satisfies the following properties.
\begin{enumerate}
\item The support of the perturbations domains $B_k$ are pairwise disjoint, contained in $U$, and also contained in $U_{k-1}\setminus \overline{U_{k+2}}$.
\item Any point of $X_0\setminus (\bigcup_{p\in Per_{N_0}(f)}W^s_{\delta}(p))$ has a positive iterate in the interior of one tile of one perturbation domain of ${B}_0$ and any point of $X_k$ has a positive iterate in the interior of one tile of one perturbation domain of $B_{k-1}\cup B_k$.
\end{enumerate}
In consequence, for any $k\geq 0$, there is a finite family of tiles $\mathcal{C}_k$ associated to ${B}_k$, and a family of compact sets $\mathcal{D}_k$ contained in the interior of tiles of $\mathcal{C}_k$, such that:
\begin{itemize}
\item each tile of $\mathcal{C}_k$ contains exactly one element of $\mathcal{D}_k$, for all $k\geq 0$ and each element of $\mathcal{D}_k$ is contained in a tile of $\mathcal{C}_k$,
\item any point of $X_0\setminus (\bigcup_{p\in Per_{N_0}(f)}W^s_{\delta}(p))$ has a positive iterate in the interior of one element of $\mathcal{D}_0$ and any point of $X_k$ has a positive iterate in the interior of one element of $\mathcal{D}_{k-1}\cup \mathcal{D}_k$.
\end{itemize}
\end{lem}
\begin{proof}
By Lemma~\ref{choice of topological tower}, we get a sequence of open sets $(W_k)_{k\geq 0}$ and a sequence of compact sets $(D_k)_{k\geq 0}$. Moreover, the diameters of components of each $W_k$ can be chosen small enough such that all their first $L_k$ iterates are contained in a perturbation domain of order $L_k$ by Lemma~\ref{existence of perturbation domain}.
Assume $W$ is a component of $W_k$, and denote $D=D_k\cap W$. By assumption, $W$ is contained in a chart of perturbation $\varphi:W\rightarrow \mathbb{R}^d$. We can tile $W$ with tiles of proper size such that any cube that intersects $\varphi(D)$ is contained in $\varphi(W)$. We do the same thing for all other components of $W_k$ that has non-empty intersection with $D_k$ and we get a finite family $\mathcal{P}_0$ of perturbation domains, each of them being an open set, pairwise disjoint, contained in $W_k$, and the union of their closure contains $D_k$ in its interior. Denote $\Phi_0$ the family of perturbation charts in the construction of $\mathcal{P}_0$.
Repeat the construction for $f^{2iN_k}(W_k)$ and $f^{2iN_k}(D_k)$, $i\in \{1,\cdots,5d-1\}$, and we get the families $\mathcal{P}_i$ of perturbation domains contained in $f^{2iN_k}(W_k)$, pairwise disjoint and the union of their closure contains $f^{2iN_k}(D_k)$ in its interior. Denote $\Phi_i$ the family of perturbation charts corresponding to $\mathcal{P}_i$. Consider the family $f^{-2iN_k}(\mathcal{P}_i)$ contained in $W_k$. The union of the closure of all cubes of $f^{-2iN_k}(\mathcal{P}_i)$ contains $D_k$ in its interior. By a $C^1$ small perturbation of $\Phi_i$, we can suppose that a point in $D_k$ can only be contained on the boundary of at most $d$ different cubes of all cubes contained in $\cup_{i=0}^{5d-1}f^{-2iN_k}(\mathcal{P}_i)$ \footnote{In~\cite{bc}, they call the sets of $\cup_{i=0}^{5d-1}f^{-2iN_k}(\mathcal{P}_i)$ on general position. To simplify, we do not introduce this definition. One can refer to Section 3.3 of~\cite{bc} for more details}. Since there are at least $5d$ families of cubes, we get that any point of $D_k$ is contained in the interior of at least $4d$ families of such cubes.
We replace every cube in $\mathbb{R}^d$ by another one with the same center and homothetic with rate $\rho<1$ close to $1$. Then we get the families $\mathcal{P}_{i,\rho}$ of perturbation domains whose closures are pairwise disjoint. If we choose $\rho$ close enough to $1$, then any point of $D_k$ is still contained in the interior of a cube of at least $4d$ families of $(f^{-2dN_k}(\mathcal{P}_{i,\rho}))_{0\leq k\leq 5d-1}$. By the compactness of $D_k$, for each $i$, there is a finite family $\Gamma_i$ of tiles of the domains $f^{-2iN_k}(\mathcal{P}_{i,\rho})$, such that the union $\Sigma_i$ of the tiles of $\Gamma_i$ satisfies: any point of $D_k$ is contained in the interior of at least $4d$ compact $(f^{-2iN_k}(\Sigma_i))_{0\leq k\leq 5d-1}$.
By another $C^1$ small perturbation of $\Phi_i$, we can suppose that any point of $D_k$ is contained on the boundary of the tiles of at most $d$ families of $(f^{-2iN_k}(\Gamma_i))_{0\leq k\leq 5d-1}$. Any point is contained in at least $4d$ families of tiles, hence any point is contained in the interior of at least one of these tiles. Define $B_k$ and $\mathcal{C}_k$ to be the union of the families $\mathcal{P}_{i,\rho}$ and the union of the families $\Gamma_i$ respectively.
Then the compact set $D_k$ is covered by the interior of the tiles of the family $f^{-2iN_k}(\Gamma_i)$. We can take all the components of the intersection of $f^{2iN_k}(D_k)$ and the elements of the family $\Gamma_i$, and this is the family $\mathcal{D}_k$.
Finally, by the assumption that $L_k=10dN_k$ and the choice of $W_k$ in Lemma~\ref{choice of topological tower}, the supports of perturbation domains $(B_k)_{k\geq 0}$ are pairwise disjoint and are contained in $U$. Moreover, the support of the perturbation domain $B_k$ is also contained in $U_{k-1}\setminus \overline{U_{k+2}}$. This finishes the proof of Lemma~\ref{perturbation domains 1}.
\end{proof}
We also have to construct perturbation domains that cover the stable and unstable manifolds of periodic orbits contained in $X_0\cap Per_{N_0}(f)$. By the assumption of hyperbolicity of periodic orbits, $X_0\cap Per_{N_0}(f)$ is a finite set. By Proposition 4.2 in~\cite{bc}, we can construct in the following way.
\begin{lem}\emph{(Proposition 4.2 of~\cite{bc})}\label{perturbation domains 2}
For any periodic orbit $Q\subset X\cap Per_{N_0}(f)$, any neighborhood $V$ of $Q$, there are a neighborhood $W$ of $Q$, two perturbation domains $B_s$ and $B_u$ of order $N_0$ for $(f,\mathcal{U}_0)$, two finite families of tiles $\mathcal{C}_s$ and $\mathcal{C}_u$ associated to $B_s$ and $B_u$ respectively, two finite families of compact sets $\mathcal{D}_s$ and $\mathcal{D}_u$, and an integer $n_0$, such that:
\begin{enumerate}
\item $V$ contains $\overline{W}$ and $\bigcup_{0\leq i\leq N_0-1} f^i(B_s \cup B_u)$.
\item $f^i(B_s) \cap f^j(B_u)=\emptyset$ for all $0\leq i,j\leq N_0-1$,
\item each element of $\mathcal{D}_s$ is contained in the interior of an element of $\mathcal{C}_s$, and each element of $\mathcal{D}_u$ is contained in the interior of an element of $\mathcal{C}_u$. Moreover, each tile of $\mathcal{C}_s$ and $\mathcal{C}_u$ contains exactly an element of $\mathcal{D}_s \cup \mathcal{D}_u$
\item for any two pairs $D_s\in \mathcal{D}_s$ and $D_u\in \mathcal{D}_u$, there is $n\in \{0,\cdots,n_0\}$, such that $f^n(D_s)\cap D_u\neq \emptyset$.
\item for any point $z\in W\setminus W^s_{loc}(Q)$, there is $n>0$ and $D\in \mathcal{D}_u$, such that $f^n(z)\in int(D)$ and $f^i(z)\in V$ for all $0\leq i\leq n$. Moreover, if $f(z)\not \in W$, then $n\leq n_0$.
\item for any point $z\in W\setminus W^u_{loc}(Q)$, there is $n>0$ and $D\in \mathcal{D}_s$, such that $f^{-n}(z)\in int(D)$ and $f^{-i}(z)\in V$ for all $0\leq i\leq n$. Moreover, if $f^{-1}(z)\not \in W$, then $n\leq n_0$.
\end{enumerate}
\end{lem}
\subsection{Choice of a pseudo-orbit}
By Lemma~\ref{perturbation domains 1}, we have the sequences of perturbation domains $(B_k)_{k\geq 0}$, tiles $(\mathcal{C}_k)_{k\geq 0}$ and families of compact sets $(\mathcal{D}_k)_{k\geq 0}$. Since there are only finitely many periodic orbits contained in $Per_{N_0}(f)\cap X$, we can take for each periodic orbit $Q\subset Per_{N_0}(f)\cap X$ an open neighborhood $V(Q)\subset U$ that are pairwise disjoint, disjoint from $\overline{U_1}$ and disjoint from $f^i(B_k)$ for any $0\leq i\leq N_k-1$ and any $k\geq 0$. By Lemma~\ref{perturbation domains 2}, we have for each $Q$ the open set $W(Q)$, the perturbation domains $B_s(Q)$ and $B_u(Q)$, the families of tiles $\mathcal{C}_s(Q)$ and $\mathcal{C}_u(Q)$, the families of compact sets $\mathcal{D}_s(Q)$ and $\mathcal{D}_u(Q)$ and the number $n_0(Q)$. By the choice of $V(Q)$, we have that $f^i(B_{\sigma}(Q))\cap f^j(B_k)=\emptyset$ for any $\sigma=s,u$, any $0\leq i\leq N_k-1$ and any $k\geq 0$.
We take the union of $(B_s(Q),\mathcal{C}_s(Q),\mathcal{D}_s(Q))$, $(B_u(Q),\mathcal{C}_u(Q),\mathcal{D}_u(Q))$ and $(B_0,\mathcal{C}_0,\mathcal{D}_0)$, and to simplify the notations, we still denote the union by $(B_0,\mathcal{C}_0,\mathcal{D}_0)$. By Remark~\ref{union of perturbation domains}, we know the modified $(B_0,\mathcal{C}_0,\mathcal{D}_0)$ is still a perturbation domain of order $N_0$ for $(f,\mathcal{U}_0)$. Denote $D'_k$ the union of the compact sets of the family $\mathcal{D}_k$ for each $k\geq 0$. Similar to the analysis of section 4.3 in~\cite{bc}, we assume that $z$ is not in any of the perturbation domains that we have choose.
Recall that the support of the perturbation domain $B_k$ is $supp(B_k)=\bigcup_{0\leq n\leq N_k-1}f^n(B_k)$. From the above constructions, the supports of the perturbation domains $(B_k)_{k\geq 0}$ are pairwise disjoint and are contained in $U$. Moreover, we have that $supp(B_k)\subset U_{k-1}\setminus \overline{U_{k+2}}$ for any $k\geq 0$.
\begin{lem}\label{choice of pseudo-orbit}
There is an infinitely long pseudo-orbit $Y=(y_0,y_1,\cdots)$ for $f$ contained in $X$ that has jumps only in tiles of $(\mathcal{C}_k)_{k\geq 0}$ with $y_0=z$ and $d(y_n,K)\rightarrow 0$ as $n\rightarrow \infty$. Moreover, for each $k\geq 0$, there is a minimal number $l_k$, such that $y_i\in U_k$ for all $i\geq l_k$.
\end{lem}
\begin{proof}
By the former constructions, any point $x\in X_0$ has a positive iterate contained in the union of the interior of the compact set $D'_0$ and the open sets $W(Q)$ for all periodic orbits $Q\subset X\cap Per_{N_0}(f)$. Any point $x\in X_k$ has a positive iterate contained in the union of the interior of compact sets $D'_{k-1}\cup D'_k$ for $k\geq 1$. By the compactness of the sets $X_k$, there are integers $T_k$, compact sets $\tilde{D_k}\subset D_k'$, and compact sets $\tilde{W}(Q)\subset W(Q)$, such that
\begin{itemize}
\item all points $x\in X_0$ will enter the union of $\tilde{D_0}$ and $\tilde{W}(Q)$ for all $Q\subset X\cap Per_{N_0}(f)$ in time bounded by $T_0$,
\item all points $x\in X_k$ will enter in $\tilde{D_{k-1}}\cup\tilde{D_k}$ for $k\geq 1$ in time bounded by $T_k$.
\end{itemize}
We can assume that $T_0$ is larger than $n_0(Q)$, for any $Q\subset X\cap Per_{N_0}(f)$.
\paragraph{Setting of the constants.}
For $k\geq 0$, set $\eta_k$ to be smaller than half of the minimum of the distances between a point in $(\bigcup_{Q\subset X\cap Per_{N_0}(f)}\tilde{W}(Q))\cup (\bigcup_{0\leq i\leq k}\tilde{D_i})$ and a point in the completement of $(\bigcup_{Q\subset X\cap Per_{N_0}(f)}W(Q))\cup (\bigcup_{0\leq i\leq k}D'_i)$. Moreover, we also assume that $\eta_k$ is smaller than half of the minimum of the distances between a point in $f(\overline{M\setminus U_k})$ to a point in $U_{k+1}$, and smaller than the minimum of the distances between a point in a compact set $D\in \mathcal{D}_k$ and a point on the boundary of the tile $C\in \mathcal{C}_k$ that contains $D$. Then for any $k\geq 0$, there is a number $0<\vep_k<\eta_k$, such that for any $\vep_k$-pseudo-orbit $(x_0,\cdots,x_{T_k})$, we have $d(x_i,f^i(x_0))<\frac{1}{2}\eta_k$, and $d(x_i,f^{i-T_k}(x_{T_k}))<\frac{1}{2}\eta_k$ for all $0\leq i\leq T_k$. For each $\vep_k$, there is a number $\delta_k\in(0,\frac{1}{3}\vep_k)$, such that, for any two points $x,y\in M$, if $d(x,y)<\delta_k$, then $d(f(x),f(y))<\frac{1}{3}\vep_k$. Without loss of generality, we can assume that the sequences $(\eta_k)_{k\geq 0}$, $(\vep_k)_{k\geq 0}$ and $(\delta_k)_{k\geq 0}$ are strictly decreasing sequences.
\paragraph{The sets $\tilde{X_k}$ and the pseudo-orbits $Z_k$.}
Now we take a finite $\delta_k$-dense set $\tilde{X}_k$ of $X_k$ for any $k\geq 0$, such that $z\in \tilde{X_0}$. For any $k\geq 0$, take a $\delta_k$-pseudo-orbit $(y_1^k,\cdots,y_{m_k}^k)$ in $X\setminus K$, such that $y_1^k=z$ and $d(y_{m_k}^k,K)<\delta_k$. Then we project this pseudo-orbit to the set $\bigcup_{i\geq 0}\tilde{X}_i$: if $y_j^k\in X_i\setminus X_{i+1}$, then there is $z_j^k\in \tilde{X_i}$, such that $d(y_j^k,z_j^k)<\delta_i$. Then the pseudo-orbit $Z_k=(z_1^k,\cdots,z_{m_k}^k)$ is a pseudo-orbit contained in $\bigcup_{i\geq 0}\tilde{X}_i$ that connects $z$ to $K$, where $z_1^k=z$. Moreover, if $y_j^k,y_{j+1}^k\in X_i$, we have $d(f(z_j^k),z_{j+1}^k)<d(f(z_j^k),f(y_j^k))+d(f(y_j^k),y_{j+1}^k)+d(y_{j+1}^k,z_{j+1}^k)<\frac{1}{3}\vep_i+\delta_k+\delta_i<\frac{2}{3}\vep_i+\frac{1}{3}\vep_k$. Hence $d(f(z_j^k),z_{j+1}^k)<\vep_i$ when $k\geq i$. By cutting some part of $Z_k$, we can assume that $z_j^k\neq z_l^k$ for any $j\neq l$ and any $k\geq 0$. Then for any $k\geq 0$, there is a minimal integer $l(m,k)$, such that $z_i^k\in U_m$ for all $i> l(m,k)$.
\paragraph{The infinitely long pseudo-orbit $Z$.}
Since $\tilde{X_k}$ is a finite set for any $k\geq 0$, one can extract a subsequence $(Z^1_k)$ of $(Z_k)$, such that all pseudo-orbits in this subsequence have the same piece before staying in $U_1$, that is to say, $(z_1^k,\cdots,z_{l(1,k)}^k)$ are equal to each other for any $Z_k\in \{Z^1_k\}$. Similarly, there is a subsequence $(Z^2_k)$ of $(Z^1_k)$, such that all pseudo-orbits in this subsequence have the same piece before staying in $U_2$. We can continue this process, and finally, by taking the limit, we can get an infinitely long pseudo-orbit $Z=(z_1,z_2,\cdots)$ such that $z_1=z$, $d(z_n,K)\rightarrow 0$ as $n\rightarrow \infty$. Moreover, if $z_j,z_{j+1}\in X_i$, then $d(f(z_j),z_{j+1})<\vep_i$, since $Z$ is a limit set of $(Z_k)$.
By the analysis of Lemma $4.6$ in~\cite{bc}, the pseudo-orbit $Z=(z_1,z_2,\cdots)$ has the property stated in the following claim. We omit the proof here since it follows exactly the proof of Lemma $4.6$ in~\cite{bc}.
\begin{claim}\label{bound of time}
There is a strictly increasing sequence $t_0=1,t_1,\cdots$, such that for $j>0$, $z_{t_j}$ is contained in a compact set $E_j$ of $\bigcup_{k\geq 0}\mathcal{D}_k$. Moreover, for any $j\geq 0$,
\begin{itemize}
\item if $E_j\in \mathcal{D}_0$, then either $t_j-t_{j-1}<T_1$ or there is $Q\subset X\cap Per_{N_0}(f)$, such that $E_{j-1}\in \mathcal{D}_s(Q)$ and $E_{j}\in \mathcal{D}_u(Q)$,
\item if $E_j\in \mathcal{D}_k$ for some $k\geq 1$, then $t_j-t_{j-1}<T_k$.
\end{itemize}
\end{claim}
\paragraph{Construction of the pseudo-orbit $Y$ from $Z$.}
Now we replace some part of $Z$ to get an infinitely long pseudo-orbit that connects $\tilde{U}$ to $K$, accumulates to $K$ in the future, and has jumps only in the tiles of the perturbation domains. Using Claim~\ref{bound of time}, we construct $Y$ as the following.
\begin{itemize}
\item If $E_j\in \mathcal{D}_0$ and $t_j-t_{j-1}<T_1$ or if $E_j\in \mathcal{D}_k$ where $k\geq 1$, we replace the piece of pseudo-orbit $(z_{t_{j-1}},\cdots,z_{t_j})$ by the piece of true orbit $(z_{t_{j-1}},f(z_{t_{j-1}}),\cdots,f^{t_j-t_{j-1}}(z_{t_{j-1}}))$.
\item If $E_j\in \mathcal{D}_0$ and $t_j-t_{j-1}\geq T_1$, we have that there is $Q\subset X\cap Per_{N_0}(f)$, such that $E_{j-1}\in \mathcal{D}_s(Q)$ and $E_{j}\in \mathcal{D}_u(Q)$. By Lemma~\ref{perturbation domains 2}, there is a piece of true orbit $(x,f(x),\cdots,f^t(x))$ such that $x\in E_{j-1}$, $f^t(x)\in E_{j}$ and $t<T_0$. Then we replace the piece of pseudo-orbit $(z_{t_{j-1}},\cdots,z_{t_j})$ by the piece of true orbit $(x,f(x),\cdots,f^t(x))$.
\end{itemize}
Then we get a new pseudo-orbit $Y=(y_0,y_1,\cdots)$. We can see that $Y$ has jumps only in tiles of $(\mathcal{C}_k)_{k\geq 0}$ with $y_0=z$ and $d(y_n,K)\rightarrow 0$ as $n\rightarrow \infty$. Moreover, there is a minimal number $l_k$, such that $y_i\in U_k$ for all $i\geq l_k$ and all $k\geq 0$. This finishes the proof of Lemma~\ref{choice of pseudo-orbit}.
\end{proof}
\subsection{The connecting processes}
We take the infinitely long pseudo-orbit $Y=(y_0,y_1,\cdots)$ with $y_0=z$ contained in $X$ from Lemma~\ref{choice of pseudo-orbit}. Then $Y$ has jumps only in tiles of $(\mathcal{C}_k)_{k\geq 0}$ and $d(y_n,K)\rightarrow 0$ as $n\rightarrow \infty$. Moreover, for each $k\geq 0$, there is a minimal number $l_k$, such that $y_i\in U_k$ for all $i\geq l_k$. For the pseudo-orbit $Y=(y_0,y_1,\cdots)$ and the sequence of integers $(l_k)_{k\geq 0}$, we have the following lemma.
\begin{lem}\label{sequence of diffeomorphisms}
For each $k\geq 0$, there are a diffeomorphism $f_k$, an infinitely long pseudo-orbit $Y_k=(y_0^k,y_1^k,\cdots)$ of $f_k$ with $y_0^k=z$, and two sequences of positive integers $(m_k)_{k\geq 0}$ and $(n_k)_{k\geq 0}$, and if we denote $f=f_{-1}$, then for any $k\geq 0$, the following properties are satisfied.
\begin{enumerate}
\item There is $\phi_k\in\mathcal{V}_k$, such that $\phi_k=Id|_{M\setminus supp(B_k)}$, and $f_{k}=f_{k-1}\circ \phi_k$.
\item The integer $m_{k-1}$ is the smallest positive integer, such that $f_{k}^{m_{k-1}}(z)\in U_{k-1}$. Moreover, we have $m_{k-1}<m_{k}$.
\item The piece of pseudo-orbit $(y_0^{k},y_1^{k},\cdots,y_{m_{k-1}-1}^{k})$ for $f_{k}$ equals $(z,f_{k}(z),\cdots,f_{k}^{m_{k-1}-1}(z))$.
\item $n_k\leq l_{k+2}$, and $y_{n_{k}+m}^k=y_{l_{k+2}+m}$, for all $m\geq 0$.
\item The pseudo-orbit $Y_k$ of $f_k$ has only jumps in the tiles $\{\mathcal{C}_{k+1},\mathcal{C}_{k+2},\cdots\}$.
\end{enumerate}
\end{lem}
\begin{proof}
We build the sequences by induction. We construct $f_{k+1}$, $Y_{k+1}$, $n_{k+1}$ and $m_k$ after $f_{k}$, $Y_{k}$, $n_{k}$ and $m_{k-1}$ has been built.
\paragraph{The constructions for $n=0$.} Consider $(f,\mathcal{U}_0,B_0)$. By Definitions~\ref{perturbation domain} and~\ref{jumps}, and the fact that the pseudo-orbit $Y$ of $f$ has jumps only in tiles of $(\mathcal{C}_k)_{k\geq 0}$, there are a diffeomorphism $f_0\in \mathcal{U}_0$, and an infinitely long pseudo-orbit $Y_0=(y_0^0,y_1^0,\cdots)$ of $f_0$ with $y_0^0=z$, which connects some pieces of $Y$, satisfying the following three properties.
\begin{itemize}
\item The diffeomorphism $f_0$ coincides with $f$ outside $supp(B_0)$, hence there is $\phi_0\in\mathcal{V}_0$, such that $\phi_0=Id|_{M\setminus supp(B_0)}$, and $f_0=f\circ \phi_0$, which is the property \textit{1}.
\item The pseudo-orbit $Y_0$ has only jumps in the tiles $\{\mathcal{C}_1,\mathcal{C}_2,\cdots\}$, which is the property \textit{5}.
\end{itemize}
Since $supp(B_0)\cap \overline{U_2}=\emptyset$, and for all $i\geq l_2$, the point $y_i\in U_2$, then there is a positive integer $n_0$, such that $y_{n_0+m}=y_{l_2+m}$, for all $m\geq 0$. Moreover, from Definition~\ref{perturbation domain}, we can see that $n_0\leq l_2$. Hence the property \textit{4} is satisfied. We do not have to check the properties \textit{2} and \textit{3} for $n=0$.
\paragraph{The constructions for $n=k+1$.}
For $k\geq 0$, assume that $f_n$, $Y_k$, $m_{k-1}$ and $n_k$ have been built for all $n\leq k$. The infinitely long pseudo-orbit $Y_k=(y_0^k,y_1^k,\cdots)$ of $f_k$ with $y_0^k=z$, has only jumps in the tiles $\{\mathcal{C}_{k+1},\mathcal{C}_{k+2},\cdots\}$. Moreover, the piece of the pseudo-orbit $(y_0^k,y_{1}^k,\cdots,y_{m_{k-1}-1}^k)$ coincides with $(z,f_k(z),\cdots,f_k^{m_{k-1}-1}(z))$ and the piece of the pseudo-orbit $(y_{n_{k}}^k,y_{n_{k}+1}^k,\cdots)$ coincides with $(y_{l_{k+2}},y_{l_{k+2}+1},\cdots)$. Then by Definitions~\ref{perturbation domain} and~\ref{jumps} and by the fact that $supp(B_{k+1})\cap\overline{U_{k+3}}=\emptyset$, there is a diffeomorphism $f_{k+1}$, satisfying the following properties.
\begin{itemize}
\item The diffeomorphism $f_{k+1}$ is of the form $f_{k+1}=f_k\circ\phi_{k+1}$, where $\phi_{k+1}\in\mathcal{V}_{k+1}$ coincides with $Id$ outside $supp(B_{k+1})$, which is the property \textit{1}.
\item There is a pseudo-orbit $Y_{k+1}=(y_0^{k+1},y_1^{k+1},\cdots)$ of $f_{k+1}$ with $y_0^{k+1}=z$, which connects some pieces of $Y_k$ and has jumps only in the tiles $\{\mathcal{C}_{k+2},\mathcal{C}_{k+3},\cdots\}$, which is the property \textit{5}.
\item There is a positive integer $n_{k+1}\leq l_{k+3}$, such that the piece of the pseudo-orbit $(y_{n_{k+1}}^{k+1},y_{n_{k+1}+1}^{k+1},\cdots)$ coincides with $(y_{l_{k+3}},y_{l_{k+3}+1},\cdots)$, which is the property \textit{4}.
\end{itemize}
Then we take the smallest integer $m_{k}$, such that $f_{k+1}^{m_{k+1}}(z)\in U_{k+1}$. Particularly, we take $m_{k}$ as the following:
\begin{itemize}
\item we take $m_0=1$,
\item when $k\geq 1$, we take $m_k$ such that $f_{k+1}^{m_{k+1}}(z)\in U_{k+1}$, and for all $0\leq i<m+{k+1}$, we have $f_{k+1}^{i}(z)\notin U_{k}$.
\end{itemize}
Since $supp(B_{k+1})\subset U_{k}\setminus\overline{U_{k+3}}$, the diffeomorphism $f_{k+1}$ coincides with $f_k$ on the piece of orbit $(z,f_k(z),\cdots,f_k^{m_{k-1}-1}(z))$. By the item \textit{2} of Lemma~\ref{choice of topological tower}, we have that $m_{k-1}<m_k$, which is the property \textit{2}. The property \textit{3} is satisfied since the new pseudo-orbit $Y_{k+1}$ has no jumps in $\mathcal{C}_{k+2}$. This finishes the proof of Lemma~\ref{sequence of diffeomorphisms}.
\end{proof}
\begin{proof}[End of the proof of Proposition~\ref{asymptotic connecting}]
Now we consider the sequences $(f_k)_{k\geq 0}$, $(Y_k)_{k\geq 0}$, $(m_k)_{k\geq 0}$ and $(n_k)_{k\geq 0}$ from Lemma~\ref{sequence of diffeomorphisms}.
By the choice of $(\mathcal{U}_k)_k$ (see $Property(F)$), the sequence of diffeomorphism $f_k=f\circ\phi_0\circ\cdots\circ\phi_k$ converges to a diffeomorphism $g\in\mathcal{U}$. And since the diameters of the pairwise disjoint perturbations domains can be chosen arbitrarily small by Lemma~\ref{ttower}, we can take $g$ to be arbitrarily C$^0$-close to $f$. Moreover, since the supports of all perturbation domains of $(B_k)_{k\leq 0}$ are contained in $U$, we have that $g=f|_{M\setminus U}$.
Since $supp(B_{k+1})\subset U_{k}\setminus \overline{U_{k+3}}$, by the items \textit{2} and \textit{3} of Lemma~\ref{sequence of diffeomorphisms}, the piece of orbit $(z,f_k(z),\cdots,f_k^{m_{k-1}-1}(z))$ is also a piece of orbit of $f_n$, when $n\geq k+1$. This implies that the limit of the sequence of pseudo-orbits $Y_k$ is the positive orbit of $z$ under $g$ since the sequence $(m_k)_{k\geq 0}$ is strictly increasing. By the item \textit{4} of Lemma~\ref{sequence of diffeomorphisms}, we can see that $orb^+(z,g)$ has only finitely many points outside $U_k$ for any $k\geq 0$ (bounded by $n_k$), hence $\omega(z,g)\subset K$. This finishes the proof of Proposition~\ref{asymptotic connecting}.
\end{proof}
\section{Proofs of the applications}\label{applications}
In this section, we give the proofs of the applications of the main theorem.
\subsection{Structural stability and hyperbolicity}
To prove Corollary~\ref{stuctually stable}, we use some of the results in~\cite{sv,ww,wgw,wenx}. We take two steps: first, we prove that the statement is true for a residual subset of $\diff^1(M)$, and then we prove it for all diffeomorphisms in $\diff^1(M)$.
Assume that $H(p)$ is the homoclinic class of a hyperbolic periodic point $p$ of a diffeomorphism $f\in\diff^1(M)$. We state two properties as follows:
\begin{itemize}
\item $(P1)$ There are $m\in\mathbb{N}$, $C>0$ and $0<\lambda<1$, such that $H(p)$ admits an $(m,\lambda)$-dominated splitting $T_{H(p)}M=E\oplus F$ with $dim(E)=ind(p)$. And for any periodic point $q$ homoclinically related to $p$, denote by $\tau(q)$ the period of $q$, then the followings are satisfied:
\begin{displaymath}
\prod_{0\leq i<\tau(q)/m} \|Df^m|_{E(f^{im}(q))}\|< C{\lambda}^{\tau(q)},
\end{displaymath}
\begin{displaymath}
\prod_{0\leq i<\tau(q)/m} \|Df^{-m}|_{F(f^{-im}(q))}\|< C{\lambda}^{\tau(q)}.
\end{displaymath}
\item $(P2)$ $H(p)$ is shadowable and every periodic pseudo-orbit can be shadowed by a periodic orbit.
\end{itemize}
Now we state the following two Lemmas, whose proofs will be omitted.
\begin{lem}\emph{(Theorem 1.1 of~\cite{wenx})}\label{no weak periodic orbit}
Assume that $f$ is a diffeomorphism in $\diff^1(M)$. If a homoclinic class $H(p)$ is structurally stable, then the property $(P1)$ is satisfied for $H(p)$.
\end{lem}
\begin{lem}\emph{(Proposition 4.1 of~\cite{wenx})}\label{wxdot}
Assume that $f$ is a diffeomorphism in $\diff^1(M)$ and $p$ is a hyperbolic periodic point. If the two properties $(P1)$ and $(P2)$ are satisfied, then $H(p)$ is hyperbolic.
\end{lem}
\begin{lem}\emph{(Proposition 2.4 of~\cite{wgw} and Proposition 3.3 of~\cite{wgw})}\label{stable chain class}
The conclusions of Lemma~\ref{no weak periodic orbit} and Lemma~\ref{wxdot} are also valid for the chain recurrence class $C(p)$.
\end{lem}
\begin{proof}[Proof of Corollary~\ref{stuctually stable}]
By Theorem B, for any diffeomorphism $f$ contained in a residual subset $\mathcal{B}\subset \diff^1(M)$, if the property $(P1)$ is satisfied for a homoclinic class $H(p)$ of $f$, then $H(p)$ is hyperbolic. We can take $\mathcal{B}$ such that $\mathcal{B}\subset\mathcal{R}$, where $\mathcal{R}$ is the residual subset in Lemma~\ref{generic properties}. Hence by Lemma~\ref{no weak periodic orbit}, for any diffeomorphism $f\in\mathcal{B}$, if a homoclinic class $H(p)$ is structurally stable, then it is hyperbolic.
Now we assume that $f$ is an arbitrarily diffeomorphism in $\diff^1(M)$. If a homoclinic class $H(p)$ of $f$ is structurally stable, then there is a $C^1$ neighborhood $\mathcal{U}$ of $f$, such that, for any $h\in\mathcal{U}$, there is a homeomorphism $\phi:H(p)\rightarrow H(p_h)$, satisfying $\phi\circ f|_{H(p)}=h\circ\phi|_{H(p)}$. Since $\mathcal{B}$ is residual in $\diff^1(M)$, we can take a diffeomorphism $g\in\mathcal{B}\cap \mathcal{U}$. Therefore, $H(p_g)$ is structurally stable, where $\mathcal{U}$ is the neighborhood of $g$ in the definition of structurally stable. Then $H(p_g)$ is hyperbolic by the argument above, hence $H(p_g)$ satisfies the property $(P2)$. It is easy to see that the property $(P2)$ is unchanged under conjugacy, thus is satisfied by $H(p)$ since $f\in\mathcal{U}$. The property $(P1)$ is satisfied by $H(p)$ by Lemma~\ref{no weak periodic orbit}. Then by Lemma~\ref{wxdot}, we have that $H(p)$ is hyperbolic. This finishes the proof for homoclinic classes.
For chain recurrence classes of a hyperbolic periodic point, we only have to show that Corollary~\ref{stuctually stable} is valid for $f\in\mathcal{B}$, and then with the same argument as above, we can get the conclusion. Assume $f\in\mathcal{B}$ and $p$ is a hyperbolic periodic point of $f$. By item 2 of Lemma~\ref{generic properties}, $C(p)=H(p)$. By Lemma~\ref{stable chain class}, the property $(P1)$ is satisfied for $C(p)$ and hence for $H(p)$. Thus $C(p)=H(p)$ is hyperbolic. This finishes the proof of Corollary~\ref{stuctually stable}.
\end{proof}
\subsection{Partial hyperbolicity}
Now we give the proofs of Corollary~\ref{application 1} and Corollary~\ref{application 2}.
\begin{proof}[Proof of Corollary~\ref{application 1}]
We just prove when $dim(E)$ is smaller than the smallest index of periodic orbits contained in $H(p)$, thus $dim(E)<ind(p)$. Assume $E$ is not contracted, by the conclusion of the main theorem, we can get a sequence of periodic orbits $orb(q_n)\subset H(p)$ with arbitrarily long periods such that $ind(q_n)=dim(E)$, which contradicts to the assumption that $dim(E)<ind(q_n)$.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{application 2}]
We assume that $f$ satisfies the properties in Lemma~\ref{generic properties} and that of the main theorem. For a homoclinic class $H(p)$ of $f$, we denote by $j\geq 1$ and $l\leq d-1$ the smallest and the largest index of periodic point contained in $H(p)$. From~\cite{abcdw}, one has that for any $j\leq i\leq l$, the hyperbolic periodic points with index $i$ are dense in $H(p)$. By Theorem A of~\cite{w1}, $H(p)$ admits a dominated splitting with index $i$. By Remark~\ref{bundle of ds}, we have that $H(p)$ admits a dominated splitting $T_{H(p)}M=E^{cs}\oplus E^c_1\oplus\cdots\oplus E^c_n\oplus E^{cu}$, where $dim(E^{cs})=j$, $n=l-j$, and $dim(E^c_i)=1$ for all $1\leq i\leq n$. Since $H(p)$ contains hyperbolic periodic points with index $i$ for all $j\leq i\leq l$, we can see easily that the central bundle $E^c_i$ is neither contracted nor expanded, for any $1\leq i\leq n$.
Now we consider the bundles $E^{cs}$ and $E^{cu}$. If the bundle $E^{cs}$ is contracted and $E^{cu}$ is expanded, then we get the partially hyperbolic splitting and the smallest or largest index of periodic orbits contained in $H(p)$ also satisfy the conclusion. Let us assume otherwise that $E^{cs}$ is not contracted. By our main theorem, there are periodic orbits with arbitrarily long period and index $j$, whose largest Lyapunov exponent along $E^{cs}$ converges to 0. Moreover, such periodic orbits form a dense set in $H(p)$. Since $f$ is far away from tangency, the other Lyapunov exponents of such periodic orbits along $E^{cs}$ are uniformly controlled by the largest one. Thus by the Franks' Lemma, there are periodic orbits with index $j-1$ in any neighborhood of $H(p)$ by C$^1$ small perturbations. Then by a genericity argument like in Section~\ref{generic argument}, for the diffeomorphism $f$, $H(p)$ can be accumulated by periodic orbits with index $j-1$. Therefore, by Theorem A of~\cite{w1} and Remark~\ref{bundle of ds}, $E^{cs}$ has a dominated splitting $E^{cs}=E^s\oplus E^c_0$ with $dim(E^c_0)=1$. By Corollary~\ref{application 1}, we have that $E^s$ is contracted. With similar argument to $E^{cu}$, we get that $H(p)$ admits a partially hyperbolic splitting $T_{H(p)}M=E^s\oplus E^c_1\oplus\cdots\oplus E^c_k\oplus E^u$ with each central bundle of dimension 1 and neither contracted nor expanded. Moreover, the minimal index of periodic points contained in $H(p)$ is $dim(E^s)$ or $dim(E^s)+1$, and the maximal one is $d-dim(E^u)$ or $d-dim(E^u)-1$. By the main theorem, there exist periodic orbits contained in $H(p)$ with arbitrarily long periods with a Lyapunov exponent along $E^c_i$ arbitrarily close to $0$. This finishes the proof.
\end{proof}
\subsection{Lyapunov stable homoclinic classes}
Now we give the proofs of Corollary~\ref{application 3} and Corollary~\ref{application 4}.
\begin{proof}[Proof of Corollary~\ref{application 3}]
From Corollary~\ref{application 1}, we only have to prove the case when $dim(E)$ equals to the largest index of periodic orbits contained in $H(p)$. The idea of the proof follows from~\cite{po} and Section 3 of~\cite{acco}. We just give an explanation here and for more details, please refer to~\cite{po} and Section 3 of~\cite{acco}.
Assume $f\in\mathcal{R}$ where $\mathcal{R}$ is the residual set in Lemma~\ref{generic properties}. There is a neighborhood $\mathcal{U}$ of $f$, such that the items 3, 4 and 6 stated in Lemma~\ref{generic properties} are satisfied for $(f,H(p),\mathcal{U})$. We can assume that $p$ has the largest index among the periodic points contained in $H(p)$, hence $dim(E)=ind(p)$. Assume that the bundle $F$ is not expanded, then by the conclusion of the Theorem A, we can get a sequence of periodic orbits $orb(q_n)$ homoclinically related to $orb(p)$ with arbitrarily long period such that the smallest Lyapunov exponent of $orb(q_n)$ along the bundle $F$ can be arbitrarily close to 0. By Lemma 2.3 of~\cite{gy}, we can assume that all the eigenvalues of $\|Df\|$ along $orb(q_n)$ are real. Then by Theorem 1 of~\cite{g2} (Theorem 2.5 in~\cite{acco}) and a proper construction of a path of diffeomorphism (see~\cite{acco}), there is a diffeomorphism $g\in\mathcal{U}$ and a periodic point $q$ of $g$ with index larger than $dim(E)$ such that $W^s(q)\cap W^u(p_g)\neq\emptyset$. By $C^1$ small perturbation, we can assume that $W^s(q)$ intersects $W^u(p_g)$ transversely. This property is persistent under $C^1$ perturbation, since $ind(q)+ind(p_g)> dim(M)$. Hence there is a neighborhood $\mathcal{V}\subset\mathcal{U}$ of $g$, such that for any $h\in\mathcal{V}$, we have $W^s(q_h)\pitchfork W^u(p_h)\neq\emptyset$. Take a diffeomorphism $h\in\mathcal{V}\cap\mathcal{R}$, then $H(p_h)$ is Lyapunov stable by the item 6 of Lemma~\ref{generic properties}, and hence $q_h\in H(p_h)$. This contradicts the item 4 of Lemma~\ref{generic properties} by the choice of $\mathcal{U}$, since $ind(q_h)>ind(p_h)=ind(p)$.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{application 4}]
We assume that the second item does not happen. By Lemma~\ref{generic properties}, all periodic orbits contained in $H(p)$ have the same index. By~\cite{po}, $H(p)$ has a dominated splitting $T_{H(p)}M=E\oplus F$ such that $dim(E)=ind(p)$. By Corollary~\ref{application 3}, we have that the bundle $F$ is expanded. With the same argument to $f^{-1}$ and the bundle $E$, we get that $E$ is contracted for $f$. Hence the splitting $T_{H(p)}M=E\oplus F$ is hyperbolic. Then $H(p)$ is a hyperbolic chain recurrence class by item $2$ of Lemma~\ref{generic properties}. Hence by a standard argument using the shadowing lemma, $H(p)$ is an isolated chain recurrence class. By Theorem 5 of~\cite{abd}, since $M$ is connected, the homoclinic class $H(p)$ is in fact the whole manifold, hence $f$ is Anosov.
\end{proof}
\section*{Acknowledgments} I am very grateful to Prof. Lan Wen and Prof. Sylvain Crovisier, who have given me great help and encouragement. Professor Sylvain Crovisier gave me many useful suggestions both in solving the problem and in the writing of the paper, and Professor Lan Wen carefully listened to the proof and gave me many useful comments. I would also like to thank Shaobo Gan, Dawei Yang, Rafael Potrie, Xiao Wen and Nicolas Gourmelon for listening to the proof and for useful discussions. Dawei Yang pointed out to me that one can prove that structurally stable homoclinic classes are hyperbolic with the conclusion of the main theorem. This work was done when I was in University Paris-Sud 11 as a joint PhD student under the supervision of Lan Wen and Sylvain Crovisier, and I would like to thank University Paris-Sud 11 for the hospitality and China Scholarship Council (CSC) for financial support (201306010008).
|
{
"timestamp": "2015-04-14T02:14:06",
"yymm": "1504",
"arxiv_id": "1504.03153",
"language": "en",
"url": "https://arxiv.org/abs/1504.03153"
}
|
\section{Introduction}
Mode coupling plays a fundamental role in cosmology. A long-wavelength scalar
density fluctuation modifies the formation of small-scale structure via gravitational
evolution (see \cite{bernardeau/etal:2001} for a review), and possibly also
through the physics of inflation. This effect of long-wavelength modes manifests
itself through a dependence of observables, e.g., the $n$-point statistics and
the halo mass function, on the local long-wavelength overdensity, or equivalently,
the position in space (see \cite{chiang/etal:2014,wagner/etal:2014,wagner/etal:2015}
for the $n$-point statistics and \cite{cole/kaiser:1989,mo/white:1996}
for the mass function). Measurements of spatially-varying, ``position-dependent''
observables capture the effects of mode coupling, and can be used to test our
understanding of gravity and the physics of inflation. A similar idea of measuring
the shift of the peak position of the baryonic acoustic oscillation in different
environments has been studied in ref.~\cite{roukema/etal:2014}.
In this paper, we focus on the position-dependent two-point function. Consider
a galaxy redshift survey. Instead of measuring the two-point function of
galaxy pairs within the entire survey volume, we divide the survey volume into many
subvolumes, within which we measure the two-point function of galaxy
pairs. These two-point functions vary spatially from subvolume to subvolume, and the
variation is correlated with the mean overdensities of the subvolumes
with respect to the entire survey volume. As we show later in detail, this correlation
measures an integral of the three-point function, which represents the
response of the small-scale clustering of galaxies (as measured by the
position-dependent two-point functions) to the long-wavelength density
perturbation (as measured by the mean overdensities of the subvolumes) \cite{chiang/etal:2014}.
Not only is the position-dependent correlation function conceptually straightforward
to interpret, but the computational requirement for measuring three-point statistics
is also largely alleviated. The usual three-point correlation function measurements
rely on finding particle triplets with the naive algorithm scaling as $N_{\rm par}^3$
where $N_{\rm par}$ is the number of particles. Current galaxy redshift surveys contain
roughly a million galaxies, and we need 50 times as many random samples as the galaxies
for characterizing the survey window function accurately (see, e.g., \cite{chiang/etal:2013}).
Counting triplets thus becomes computationally challenging. Similarly, the measurement
of the three-point function in Fourier space, i.e., the bispectrum, requires counting
of all possible triangle configurations formed by different Fourier modes, which is also
computationally demanding. This explains why only few measurements of the three-point
function of the large-scale structure have been reported in the literature
\cite{scoccimarro/etal:2000,verde/etal:2001,kayo/etal:2004,nishimichi/etal:2006,
mcbride/etal:2010a,mcbride/etal:2010b,marin/etal:2013,gilmarin/etal:2014b,guo/etal:2014}.
The computational requirement is alleviated for the position-dependent
correlation function technique because we explore a subset of triplets
corresponding to the ``squeezed configurations'' of the three-point function;
namely, two short-wavelength modes correlated with one long-wavelength mode.
We only need to count particle pairs for measuring the two-point function
in subvolumes, which scales as $N_{\rm par}^2$. The scaling is further
improved because the number of particles in each subvolume decreases by
the number of the subvolumes $N_s$; hence, it scales as
$N_s(N_{\rm par}/N_s)^2=N_{\rm par}^2/N_s$.
The position-dependent correlation function technique is thus particularly
efficient for extracting the information of the squeezed-limit bispectrum,
which we shall demonstrate in \refapp{fisher}, while it is relatively
insensitive to the bispectrum in other configurations.
In this paper, we report on the first measurement of the three-point function
with the position-dependent correlation function from the SDSS-III Baryon
Oscillation Spectroscopic Survey Data Release 10 (hereafter BOSS DR10) CMASS
sample \cite{ahn/etal:2013,anderson/etal:2013}. We compare this measurement
with those from the PTHalos mock catalogs
\cite{scoccimarro/sheth:2001,manera/etal:2012,manera/etal:2014}.
While the mocks were designed to reproduce the global two-point function of
the BOSS DR10 CMASS sample, it is not guaranteed that they can reproduce the
three-point function as measured by the position-dependent correlation function.
We shall show that the position-dependent correlation functions from the real
data and the mocks are consistent with each other. Finally, we use tree-level
perturbation theory to predict the position-dependent correlation function as
a function of the galaxy bias parameters and the cosmological parameters, and
determine the quadratic nonlinear bias parameter of the BOSS DR10 CMASS sample
by combining the constraints from the position-dependent correlation function,
the global two-point function, and the weak lensing signal.
The rest of the paper is organized as follows. In \refsec{theory}, we define the
position-dependent correlation function and the integrated three-point function,
and describe the tree-level perturbation theory prediction for the integrated
three-point function in redshift space. In \refsec{mock} and \ref{sec:data},
we apply the position-dependent correlation function technique to the mocks
and the BOSS DR10 CMASS sample, respectively. The cosmological interpretation
of the measurements is given in \refsec{interpretation}. We conclude in \refsec{conclusion}.
In \refapp{gaussian}, we test our estimator using Gaussian realizations.
In \refapp{test}, we study the effects of using extended models of the
bispectrum with the effective $F_2$ and $G_2$ kernels and a tidal bias.
In \refapp{iz_zevolve}, we compare the mocks and BOSS DR10 CMASS samples in
different redshift bins.
In \refapp{fisher}, we use the Fisher matrix calculation to demonstrate
the information content of the position-dependent correlation function.
Throughout the paper we adopt the cosmology of the
mocks as our fiducial cosmology, i.e., a flat $\Lambda$CDM cosmology
with $\Omega_m=0.274$, $\Omega_b h^2=0.0224$, $h=0.7$, $\sigma_8=0.8$, and $n_s=0.95$.
\section{Position-dependent correlation function and the integrated
three-point function in redshift space}
\label{sec:theory}
\subsection{Position-dependent correlation function}
\label{sec:pos_dep_xi}
Consider a density fluctuation field, $\delta({\bf r})$, in a survey (or simulation)
volume $V_r$. The mean overdensity of this volume vanishes by
construction, i.e., $\bar\delta=\frac{1}{V_r}\int_{V_r}d^3r~\delta({\bf r})=0$.
The global two-point function is defined as
\begin{equation}
\xi(r)=\langle\delta(\v{x})\delta(\v{x}+\vr)\rangle ~,
\end{equation}
where we assume that $\delta({\bf r})$ is statistically homogeneous and isotropic,
so $\xi(r)$ depends only on the separation $r$. As the ensemble average
cannot be measured directly, we estimate the global two-point function as
\begin{equation}
\hat{\xi}(r)=\frac{1}{V_r} \int \frac{d^2\hat{\vr}}{4\pi}
\int_{\v{x},\v{x}+\vr\in V_r}d^3x~\d(\vr+\v{x})\d(\v{x})\,.
\label{eq:hat_xi_g}
\end{equation}
The ensemble average of \refeq{hat_xi_g} is not equal to $\xi(r)$. Specifically,
\ba
\langle\hat\xi(r)\rangle=\:&\frac{1}{V_r}\int\frac{d^2\hat{r}}{4\pi}
\int_{\v{x},\v{x}+\vr\in V_r}d^3x~\langle\d(\vr+\v{x})\d(\v{x})\rangle
=\xi(r)\frac{1}{V_r}\int\frac{d^2\hat{r}}{4\pi}\int_{\v{x},\v{x}+\vr\in V_r} d^3x~.
\label{eq:ensavg_hat_xi_g}
\ea
The second integral in \refeq{ensavg_hat_xi_g} is $V_r$ only if $\vr=0$, and
the fact that it departs from $V_r$ is due to the finite boundary of $V_r$.
We shall quantify this boundary effect later in \refeq{ensavg_hat_xi}.
We now identify a subvolume $V_L$ centered at $\vr_L$, and compute the mean
overdensity and the correlation function within $V_L$. The mean overdensity is
\begin{equation}
\bar\delta(\vr_L)=\frac{1}{V_L}\int_{V_L}d^3r~\delta({\bf r})=\frac{1}{V_L}\int d^3r~\delta({\bf r}) W(\vr-\vr_L) ~,
\end{equation}
where $W(\vr)$ is the window function. Throughout this paper, we use a
cubic window function given by
\begin{equation}
W(\vr)=W_L(\vr)=\prod_{i=1}^3\:\theta(r_i), \quad
\theta(r_i) = \left\{
\begin{array}{cc}
1, & |r_i|\le L/2, \\
0, & \mbox{otherwise}~,
\end{array}\right.
\end{equation}
where $L$ is the side length of $V_L$.
The results are not sensitive to the exact choice of the window
function, provided that the separation between galaxy pairs is much smaller
than $L$.
While $\bar\delta=0$, $\bar\delta(\vr_L)$ is non-zero in general. In other words, if $\bar\delta(\vr_L)$
is positive (negative), then this subvolume is overdense (underdense) with respect
to the mean density in $V_r$.
Using the same window function, we define the position-dependent correlation
function in the subvolume $V_L$ centered at $\vr_L$ as
\ba
\hat\xi(\vr,\vr_L)=\:&\frac{1}{V_L}\int\displaylimits_{\v{x},\vr+\v{x} \in V_L}d^3x~\d(\vr+\v{x})\d(\v{x}) \nonumber\\
=\:&\frac{1}{V_L}\int d^3x~\d(\vr+\v{x})\d(\v{x})W_L(\vr+\v{x}-\vr_L)W_L(\v{x}-\vr_L) ~.
\ea
This is essentially an estimator for a local two-point function.
In this paper we shall consider only the angle-averaged position-dependent
correlation function (i.e.,~the monopole) defined by
\begin{equation}
\hat\xi(r,\vr_L)=\int\frac{d^2\hat{r}}{4\pi}~\hat\xi(\vr,\vr_L)
=\frac{1}{V_L}\int\frac{d^2\hat{r}}{4\pi}\int d^3x~
\d(\vr+\v{x})\d(\v{x})W_L(\vr+\v{x}-\vr_L)W_L(\v{x}-\vr_L) ~.
\label{eq:hat_xi}
\end{equation}
Similarly to that of the global two-point function, the ensemble average
of \refeq{hat_xi} is not equal to $\xi(r)$.
Specifically,
\ba
\langle\hat\xi(r,\vr_L)\rangle=\:&\frac{1}{V_L}\int\frac{d^2\hat{r}}{4\pi}\int d^3x~
\langle\d(\vr+\v{x})\d(\v{x})\rangle W_L(\vr+\v{x}-\vr_L)W_L(\v{x}-\vr_L) \nonumber\\
=\:&\xi(r)\frac{1}{V_L}\int\frac{d^2\hat{r}}{4\pi}\int d^3x'~
W_L(\vr+\v{x}')W_L(\v{x}')\equiv\xi(r)f_{\rm bndry}(r) ~,
\label{eq:ensavg_hat_xi}
\ea
where $f_{\rm bndry}(r)$ is the boundary effect due to the finite size of
the subvolume. While $f_{\rm bndry}(r)=1$ for $r=0$, the boundary effect
becomes larger for larger separations. The boundary effect can be computed
by the five-dimensional integral in \refeq{ensavg_hat_xi}. Alternatively,
it can be evaluated by the ratio of the number of the random particle pairs
of a given separation in a finite volume to the expected random particle pairs
in the shell with the same separation in an infinite volume.
We have evaluated $f_{\rm bndry}(r)$ in both ways, and the results are in
an excellent agreement.
As the usual two-point function estimators based on pair counting (such as
Landy-Szalay estimator which will be discussed in \refsec{sub_quan}) or grid
counting (which will be discussed in \refapp{gaussian}) do not contain the
boundary effect, when we compare the measurements to the model which is
calculated based on \refeq{hat_xi}, we shall divide the model by $f_{\rm bndry}(r)$
to correct for the boundary effect.
\subsection{Integrated three-point function}
\label{sec:int_xi}
The correlation between $\hat\xi(r,\vr_L)$ and $\bar\delta(\vr_L)$ is given by
\ba
\langle\hat\xi(r,\vr_L)\bar\delta(\vr_L)\rangle=\:&
\frac{1}{V_L^2}\int\frac{d^2\hat{r}}{4\pi}\int d^3x_1\int d^3x_2
~\langle\d(\vr+\v{x}_1)\d(\v{x}_1)\d(\v{x}_2)\rangle \nonumber\\
& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \times
W_L(\vr+\v{x}_1-\vr_L)W_L(\v{x}_1-\vr_L)W_L(\v{x}_2-\vr_L) \nonumber\\
=\:&\frac{1}{V_L^2}\int\frac{d^2\hat{r}}{4\pi}\int d^3x_1\int d^3x_2
~\zeta(\vr+\v{x}_1+\vr_L,\v{x}_1+\vr_L,\v{x}_2+\vr_L) \nonumber\\
& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \times
W_L(\vr+\v{x}_1)W_L(\v{x}_1)W_L(\v{x}_2) ~,
\label{eq:iz}
\ea
where $\zeta(\vr_1,\vr_2,\vr_3)\equiv\langle\d(\vr_1)\d(\vr_2)\d(\vr_3)\rangle$
is the three-point correlation function. Because of the assumption of homogeneity
and isotropy, the three-point function depends only on the separations $|\vr_i-\vr_j|$
for $i\neq j$, and so $\langle\hat\xi(r,\vr_L)\bar\delta(\vr_L)\rangle$ is independent
of $\vr_L$. Furthermore, as the right-hand-side of \refeq{iz} is an integral
of the three-point function, we will refer to this quantity as the ``integrated
three-point function,'' $i\zeta(r)\equiv\langle\hat\xi(r,\vr_L)\bar\delta(\vr_L)\rangle$.
$i\zeta(r)$ can be computed if $\zeta(\vr_1,\vr_2,\vr_3)$ is known. For example,
standard perturbation theory (SPT) with the local bias model at the tree level
in real space gives
\begin{equation}
i\zeta(r)=b_1^3i\zeta_{\rm SPT}(r)+b_1^2b_2i\zeta_{b_2}(r)\,,
\end{equation}
where $i\zeta_{\rm SPT}$ and $i\zeta_{b_2}$ are given below. Here, $b_1$ and $b_2$
are the linear and quadratic (nonlinear) bias parameters, respectively. Because
of the high dimensionality of the integral, we use the Monte Carlo integration
routine in the GNU Scientific Library to numerically evaluate $i\zeta(r)$. The
first term, $i\zeta_{\rm SPT}$, is given by \cite{jing/borner:1996,barriga/gaztanaga:2001}
\ba
\zeta_{\rm SPT}(\vr_1,\vr_2,\vr_3)=\:&\frac{10}{7}\xi_l(r_{12})\xi_l(r_{23})
+\mu_{12,23}[\xi_l'(r_{12})\phi_l'(r_{23})+\xi_l'(r_{23})\phi_l'(r_{12})] \nonumber\\
\:&+\frac{4}{7}\Bigg\lbrace-3\frac{\phi_l'(r_{12})\phi_l'(r_{23})}{r_{12}r_{13}}
-\frac{\xi_l(r_{12})\phi_l'(r_{23})}{r_{23}}-\frac{\xi_l(r_{23})\phi_l'(r_{12})}{r_{12}} \nonumber\\
&~~~~~~~~ +\mu_{12,23}^2\left[\xi_l(r_{12})+\frac{3\phi_l'(r_{12})}{r_{12}}\right]
\left[\xi_l(r_{23})+\frac{3\phi_l'(r_{23})}{r_{23}}\right]\Bigg\rbrace \nonumber\\
\:&+2~{\rm cyclic} ~,
\label{eq:zeta_spt}
\ea
where $r_{12}=|\vr_1-\vr_2|$, $\mu_{12,23}$ is the cosine between $\vr_{12}$
and $\vr_{23}$, $'$ refers to the spatial derivative, and
\begin{equation}
\xi_l(r)\equiv\int\frac{dk}{2\pi^2}~k^2P_l(k){\rm sinc}(kr)\\, ~~~~~~
\phi_l(r)\equiv\int\frac{dk}{2\pi^2}~P_l(k){\rm sinc}(kr)\\,
\label{eq:xil_phil}
\end{equation}
with $P_l(k)$ being the linear matter power spectrum, and ${\rm
sinc}(x)=\sin(x)/x$. The subscript $l$ denotes the quantities in the
linear regime.
The second term, $i\zeta_{b_2}$, is the nonlinear local bias three-point function. Since halos
(galaxies) are biased tracers of the underlying matter density field, the local
bias prescription yields the density field of the biased tracers as
$\d_h(\vr)=b_1\d_m(\vr)+\frac{b_2}{2}\d_m^2(\vr)+...$, where $b_1$ and $b_2$ are
the linear and nonlinear biases, respectively, and $\d_m(\vr)$ is the matter density field
\cite{fry/gaztanaga:1992}. The nonlinear bias three-point function is then
\begin{equation}
\zeta_{b_2}(\vr_1,\vr_2,\vr_3)=\xi_l(r_{12})\xi_l(r_{23})+2~{\rm cyclic} ~.
\label{eq:zeta_b2}
\end{equation}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\textwidth]{iz_norm_z=0.pdf}
\caption{Normalized $i\zeta_{\rm SPT}$ (solid) and $i\zeta_{b_2}$ (dashed) for $L=100~h^{-1}~{\rm Mpc}$ (red),
$200~h^{-1}~{\rm Mpc}$ (green), and 300$~h^{-1}~{\rm Mpc}$ (blue) at $z=0$.}
\label{fig:iz_norm}
\end{figure}
\refFig{iz_norm} shows the scale-dependencies of $i\zeta_{\rm SPT}$ and $i\zeta_{b_2}$ at
$z=0$ with $P_l(k)$ computed by CLASS \cite{lesgourgues:2011}.
We normalize $i\zeta(r)$ by $\sigma_{L,l}^2$, where
\begin{equation}
\sigma_{L,l}^2\equiv\langle\bar\delta_l(\vr_L)^2\rangle=\frac{1}{V_L^2}\int\frac{d^3k}{(2\pi)^3}~P_l(k)|W_L(\v{k})|^2
\label{eq:sigmalL2}
\end{equation}
is the variance of the linear density field in the subvolume $V_L$. The choice of
this normalization will become clear in \refsec{squeezed} where we discuss $i\zeta$
in the squeezed limit, i.e., $r\ll L$. We find that the scale-dependencies of
$i\zeta_{\rm SPT}(r)$ and $i\zeta_{b_2}(r)$ are similar especially on small scales.
This is because the scale-dependence of the bispectrum in the squeezed limit is
(see e.g. the appendix of ref.~\cite{chiang/etal:2014})
\begin{equation}
B_{\rm SPT}\to\left[\frac{68}{21}-\frac13\frac{d\ln k^3P_l(k)}{d\ln k}\right]P_l(k)P_l(q) ~,~~
B_{b_2}\to2P_l(k)P_l(q) ~,
\end{equation}
where $k$ and $q$ are the short- and long-wavelength modes, respectively. For
a power-law power spectrum without features, the squeezed-limit $B_{\rm SPT}$
and $B_{b_2}$ have exactly the same scale dependence and cannot be distinguished.
This results in a significant residual degeneracy between $b_1$ and $b_2$, and will
be discussed in \refsec{interpretation}. When $r$ is small, $i\zeta(r)/\sigma_{L,l}^2$
becomes independent of the subvolume size. We derive this feature when we discuss
the squeezed limit in \refsec{squeezed}.
\subsection{Connection to the integrated bispectrum}
\label{sec:iz_to_ib}
Fourier transforming the density fields, the integrated three-point function
can be written as
\ba
i\zeta(\vr) \:&=
\frac{1}{V_L^2}\int\frac{d^3q_1}{(2\pi)^3}\cdots\int\frac{d^3q_6}{(2\pi)^3}~(2\pi)^9
\delta_D(\v{q}_1+\v{q}_2+\v{q}_3)\delta_D(\v{q}_1+\v{q}_2+\v{q}_4+\v{q}_5)\delta_D(\v{q}_3+\v{q}_6) \nonumber\\
\:&~~~~~~~~~~~~~~~~~~\times B(\v{q}_1,\v{q}_2,\v{q}_3)W_L(\v{q}_4)W_L(\v{q}_5)W_L(\v{q}_6)
e^{i[\vr\cdot(\v{q}_1+\v{q}_4)-\vr_L\cdot(\v{q}_4+\v{q}_5+\v{q}_6)]} \nonumber\\
\:&=\int\frac{d^3k}{(2\pi)^3}~iB(\v{k})e^{i\vr\cdot\v{k}} ~,
\label{eq:iz_ft_d}
\ea
where $B(\v{q}_1,\v{q}_2,\v{q}_3)$ is the bispectrum of the tracers,
and
\begin{equation}
iB(\v{k})\equiv\frac{1}{V_L^2}\int\frac{d^3q_1}{(2\pi)^3}\int\frac{d^3q_3}{(2\pi)^3}~
B(\v{k}-\v{q}_1,-\v{k}+\v{q}_1+\v{q}_3,-\v{q}_3)W_L(\v{q}_1)W_L(-\v{q}_1-\v{q}_3)W_L(\v{q}_3)\,,
\label{eq:ib}
\end{equation}
is the integrated bispectrum as defined in eq.~(2.7) of
ref.~\cite{chiang/etal:2014}. Eq.~\eqref{eq:iz_ft_d} shows that the integrated
three-point function is the Fourier transform of the integrated bispectrum.
Similarly, the angle-averaged integrated three-point function is related to the
angle-averaged integrated bispectrum, $iB(k)\equiv(4\pi)^{-1}\int{d^2\hat k}~iB(\v{k})$, as
\ba
i\zeta(r)=\int\frac{k^2dk}{2\pi^2}~iB(k)\,{\rm sinc}(kr)\,.
\label{eq:iz_ib_ang_avg}
\ea
\subsection{Squeezed limit}
\label{sec:squeezed}
In the squeezed limit, where the separation of the position-dependent correlation
function is much smaller than the size of the subvolume ($r\ll L$), the
integrated three-point function has a straightforward physical
interpretation \cite{chiang/etal:2014}. In this case, the mean
density in the subvolume acts effectively as a constant ``background''
density.
Consider the position-dependent correlation function, $\hat\xi(\vr,\vr_L)$,
measured in a subvolume with overdensity $\bar\delta(\vr_L)$. If the overdensity
is small, we may Taylor expand $\hat\xi(\vr,\vr_L)$ in orders of $\bar\delta$ as
\begin{equation}
\hat\xi(\vr,\vr_L) = \left.\xi(\vr)\right|_{\bar\delta=0}+
\left.\frac{d\xi(\vr)}{d\bar\delta}\right|_{\bar\delta=0}\bar\delta+\mathcal O(\bar\delta^2) ~.
\label{eq:sep_uni}
\end{equation}
The integrated three-point function in the squeezed limit is then, at leading order
in the variance $\langle\bar\delta^2\rangle$ (dropping $\bar\delta=0$ in the subscript of
the derivative term for clarity), given by
\begin{equation}
i\zeta(\vr)=\langle\hat\xi(\vr,\vr_L)\bar\delta(\vr_L)\rangle
=\frac{d\xi(\vr)}{d\bar\delta}\langle\bar\delta^2\rangle+\mathcal O(\bar\delta^3) ~.
\label{eq:iz_squeezed_1}
\end{equation}
As $\langle\bar\delta^2\rangle=\sigma_L^2$\footnote{If $\bar\delta=\bar\delta_l$ then $\sigma_L^2=\sigma_{L,l}^2$.
But $\bar\delta$ can in principle be nonlinear or the mean overdensity of the biased tracers,
so here we denote the variance to be $\sigma_L^2$.},
$i\zeta(\vr)$ normalized by $\sigma_L^2$ is $d\xi(\vr)/d\bar\delta$ at leading order,
which is the linear response of the correlation function to the overdensity.
Note that in \refeq{iz_squeezed_1} there is no dependence on the subvolume
size apart from $\sigma_L^2$, as shown also by the asymptotic behavior of
the solid lines in \reffig{iz_norm} for $r\to0$.
As $i\zeta(r)$ is the Fourier transform of $iB(k)$, the response of the
correlation function, $d\xi(r)/d\bar\delta$, is also the Fourier
transform of the response of the power spectrum, $dP(k)/d\bar\delta$.
For example, we can calculate the response of the linear matter
correlation function, $d\xi_l(r)/d\bar\delta$, by Fourier transforming
$dP_l(k)/d\bar\delta=[68/21-(1/3)d\ln k^3P_l(k)/d\ln k]P_l(k)$
\cite{chiang/etal:2014}. In \reffig{iz_norm_squ}, we compare the normalized
$i\zeta_{\rm SPT}(r)$ with $d\xi_l(r)/d\bar\delta$. Due to the large dynamic range of the
correlation function, we divide all the predictions by $\xi(r)$. As expected,
the smaller the subvolume size, the smaller the $r$ for $i\zeta_{\rm SPT}(r)$ to
be close to $[1/\xi_l(r)][d\xi_l(r)/d\bar\delta]$, i.e., reaching the squeezed limit.
Specifically, for $100~h^{-1}~{\rm Mpc}$, $200~h^{-1}~{\rm Mpc}$, and $300~h^{-1}~{\rm Mpc}$ subvolumes,
the squeezed limit is reached to 10\% level at $r\sim10~h^{-1}~{\rm Mpc}$,
$18~h^{-1}~{\rm Mpc}$, and $25~h^{-1}~{\rm Mpc}$, respectively.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\textwidth]{iz_norm_z=0_squeezed.pdf}
\caption{The linear response function $[1/\xi_l(r)][d\xi_l(r)/d\bar\delta]$ (black
solid) and the normalized $i\zeta_{\rm SPT}(r)$ for $L=100~h^{-1}~{\rm Mpc}$ (red dotted),
$200~h^{-1}~{\rm Mpc}$ (green dashed), and 300$~h^{-1}~{\rm Mpc}$ (blue dot-dashed). The light and
dark bands correspond to $\pm5\%$ and $\pm10\%$ of the predictions, respectively.}
\label{fig:iz_norm_squ}
\end{figure}
\subsection{Bispectrum in redshift space}
\label{sec:rsd}
To model $i\zeta(r)$ in redshift space, we need a model for the bispectrum in
redshift space. SPT in redshift space at the tree level predicts
the halo bispectrum with local bias as \cite{scoccimarro/couchman/frieman:1999}
\begin{equation}
B_{z,{\rm tree-level}}(\v{k}_1,\v{k}_2,\v{k}_3)=2[Z_2(\v{k}_1,\v{k}_2)Z_1(\v{k}_1)Z_1(\v{k}_2)
P_l(k_1)P_l(k_2)+2~{\rm cyclic}]\,,
\label{eq:treeredshift}
\end{equation}
with
\ba
Z_1(\v{k}_i)\:&=(b_1+f\mu_i^2)\,, \nonumber\\
Z_2(\v{k}_1, \v{k}_2)\:&=b_1F_2(\v{k}_1,\v{k}_2)+f\mu^2G_2(\v{k}_1,\v{k}_2)+
\frac{f\mu k}{2}\Big[\frac{\mu_1}{k_1}(b_1+f\mu_2^2)
+\frac{\mu_2}{k_2}(b_1+f\mu_1^2)\Big]+\frac{b_2}{2} ~,
\ea
where $F_2$ and $G_2$ are the standard kernels of SPT
\cite{bernardeau/etal:2001}, $f=d\ln D/d\ln a$ is the logarithmic growth rate,
$\mu\equiv\hat{k}\cdot\hat{r}_{\rm los}$,
$\mu_i\equiv\hat{k}_i\cdot\hat{r}_{\rm los}$, and
$\v{k}\equiv\v{k}_1+\v{k}_2$.
The integrated three-point function is the Fourier transform of the
integrated bispectrum. Thus, we can evaluate $i\zeta(r)$ by using
\refeq{treeredshift} in \refeq{ib} and averaging over the angle of
${\bf k}$ as in \refeq{iz_ib_ang_avg}. This operation requires a
nine-dimensional integral. On the other hand, if we have an expression
for the three-point function in configuration space, such as
\refeq{zeta_spt}, we can use \refeq{iz} to evaluate $i\zeta(r)$, which
requires an eight-dimensional integral. We do not always
have an analytical expression for the three-point function in
configuration space; thus, we in general need to perform the
nine-dimensional integral to obtain $i\zeta(r)$ from the bispectrum.
Nevertheless, to check the precision of numerical integration, we compare the
results from the eight-dimensional integral in \refeq{iz} with
\refeq{zeta_spt}, and the nine-dimensional integral in \refeq{ib} and
\refeq{iz_ib_ang_avg} with $B({\bf k}_1,{\bf k}_2,{\bf
k}_3)=2[F_2({\bf k}_1,{\bf k}_2)P_l(k_1)P_l(k_2)+2~\rm{cyclic}]$. As the
latter gives a noisy result, we apply a Savitzky-Golay filter (with window size
9 and polynomial order 4) six times. We find that, on the scales of interest
($30~h^{-1}~{\rm Mpc}\le r\le78~h^{-1}~{\rm Mpc}$, which we will justify in \refsec{mock_r}),
both results are in agreement to within 2\%. We repeat the same test
for $i\zeta_{b_2}$ (\refeq{zeta_b2}), finding a similar result. As the
current uncertainty on the measured integrated correlation function
presented in this paper is of order 10\%, we
conclude that our numerical integration yields sufficiently
precise results.
\subsection{Shot noise}
\label{sec:shot}
If the density field is traced by discrete particles, $\d_d(\vr)$, then the three-point
function contains a shot noise contribution given by
\ba
\langle\d_d(\vr_1)\d_d(\vr_2)\d_d(\vr_3)\rangle\:&=\langle\d(\vr_1)\d(\vr_2)\d(\vr_3)\rangle \nonumber\\
\:&+\left[\frac{\langle\d(\vr_1)\d(\vr_2)\rangle}{\bar n(\vr_3)}\d_D(\vr_1-\vr_3)+2~{\rm cyclic}\right]
+\frac{\delta_D(\vr_1-\vr_2)\delta_D(\vr_1-\vr_3)}{\bar n(\vr_2)\bar n(\vr_3)} ~,
\label{eq:shot}
\ea
where $\bar n(r)$ is the mean number density of the discrete particles. The shot noise
can be safely neglected for the three-point function because it only contributes when
$\vr_1=\vr_2$, $\vr_1=\vr_3$, or $\vr_2=\vr_3$. On the other hand, the shot noise of
the integrated three-point function can be computed by inserting \refeq{shot} into
\refeq{iz}, which yields
\ba
i\zeta_{\rm shot}(r)=\xi(r)\frac1{V_L^2}\int\frac{d^2\hat r}{4\pi}\int d^3x~
\left[\frac1{\bar n(\v{x}+\vr+\vr_L)}+\frac1{\bar n(\v{x}+\vr_L)}\right]W_L(\v{x}+\vr)W_L(\v{x}) ~,
\ea
where we have assumed $r\neq0$.
If we further assume that the mean number
density is constant, then the shot noise of the integrated three-point function can be
simplified as
\begin{equation}
i\zeta_{\rm shot}(r)=2\xi(r)\frac1{V_L\bar n}f_{\rm bndry}(r) ~.
\end{equation}
For the measurements of PTHalos mock catalogs and the BOSS DR10 CMASS sample, the shot
noise is subdominant (less than 7\% of the total signal on the scales of interest).
\section{Application to PTHalos mock catalogs}
\label{sec:mock}
Before we apply the position-dependent correlation function technique to
the real data, we apply it to the 600 PTHalos mock galaxy catalogs of the
BOSS DR10 CMASS sample in the North Galactic Cap (NGC). From now on, we
refer to the real and mock BOSS DR10 CMASS samples as the ``observations''
and ``mocks'', respectively.
We use the redshift range of $0.43<z<0.7$, and each realization of mocks
contains roughly 400,000 galaxies. We convert the positions of galaxies
in RA, DEC, and redshift to comoving distances using the cosmological
parameters of the mocks. The mocks have the same observational conditions
as the observations, and we correct the observational systematics by weighting
each galaxy differently. Specifically, we upweight a galaxy if its nearest
neighbor has a redshift failure ($w_{\rm zf}$) or a missing redshift due
to a close pair ($w_{\rm cp}$). We further apply weights to correct for
the correlation between the number density of the observed galaxies and
stellar density ($w_{\rm star}$) and seeing ($w_{\rm see}$).
We apply the same weights as done in the analyses of the BOSS collaboration,
namely FKP weighting, $w_{\rm FKP}=[1+P_w\bar n(z){\rm comp}]^{-1}$
\cite{feldman/kaiser/peacock:1993}, where $P_w=20000~h^{-3}~{\rm Mpc}^3$,
and $\bar n(z)$ and ``${\rm comp}$'' are the expected galaxy number density
and the survey completeness, respectively, provided in the catalogs.
Therefore, each galaxy is weighted by
$w_{\rm BOSS}=(w_{cp}+w_{zf}-1)w_{\rm star}w_{\rm see}w_{\rm FKP}$.
In this section, we present measurements from mocks in real space in
\refsec{mock_r} and redshift space in \refsec{mock_z}. The application
to the CMASS DR10 sample is the subject of \refsec{data}.
\subsection{Dividing the subvolumes}
\label{sec:division}
We use SDSSPix\footnote{SDSSPix: \url{http://dls.physics.ucdavis.edu/~scranton/SDSSPix}}
to pixelize the DR10 survey area. In short, at the lowest resolution (res=1)
SDSSPix divides the sphere equally into $n_x=36$ longitudinal slices across
the hemisphere (at equator each slice is 10 degrees wide), and each slice is
divided into $n_y=13$ pieces along constant latitudes with equal area. Thus,
for res=1 there are $n_x\times n_y=468$ pixels. In general the total number
of pixels is $n_x'\times n_y'=({\rm res}~n_x)\times({\rm res}~n_y)=(\rm res)^2\times468$,
and in this paper we shall set res=1024. After the pixelization, the $i^{\rm th}$
object (a galaxy or a random sample) has the pixel number $(i_x,i_y)$.
We use two different subvolume sizes. To cut the irregular survey volume into
subvolumes with roughly the same size, we first divide the random samples at
all redshifts into 10 and 20 slices across longitudes with similar numbers of
random samples; we then divide the random samples in each slice into 5 and 10
segments across latitudes with similar numbers of random samples. \refFig{ran_div}
shows the two resolutions of our subvolumes before the redshift cuts. (Note that
this resolution is different from the resolution of SDSSPix, which we always set
to res=1024.) Each colored pattern extends over the redshift direction. Finally,
we divide the two resolutions into three ($z_{\rm cut}=0.5108$, 0.5717) and five
($z_{\rm cut}=0.48710$, 0.52235, 0.55825, 0.60435) redshift bins.
As a result, there are 150 and 1000 subvolumes for the low and high resolution
configurations, respectively. The sizes of the subvolumes are approximately
$V_L^{1/3}=220~h^{-1}~{\rm Mpc}$ and $120~h^{-1}~{\rm Mpc}$, respectively\footnote{The shapes of the
subvolumes are not exactly cubes. For example, for the high resolution, the ratios
of square root of the area to the depth, $\sqrt{L_xL_y}/L_z$, are roughly 0.78,
1.42, 1.51, 1.28, and 0.71, from the lowest to the highest redshift bins. The
results are not sensitive to the exact shape of the subvolumes, as long as the
separation of the position-dependent correlation function that we are interested
in is sufficiently smaller than $L_x$, $L_y$, and $L_z$.}. The fractional differences
between the numbers of the random samples in subvolumes for the low and high
resolutions are within $^{+0.68\%}_{-0.58\%}$ and $^{+1.89\%}_{-1.83\%}$,
respectively. Since the number of random samples represents the effective volume,
all subvolumes at a given resolution have similar effective volumes. We assign
galaxies into subvolumes following the division of random samples.
\begin{figure}[t]
\centering
\includegraphics[width=0.495\textwidth]{random_ny=10_nx=5.jpg}
\includegraphics[width=0.495\textwidth]{random_ny=20_nx=10.jpg}
\caption{Division of random samples into subvolumes with two resolutions
in the RA-DEC plane. Each colored pattern extends over the redshift direction.}
\label{fig:ran_div}
\end{figure}
\subsection{Estimators in the subvolumes}
\label{sec:sub_quan}
In the $i^{\rm th}$ subvolume, we measure the mean overdensity with respect
to the entire NGC, $\bar\delta_i$, and the position-dependent correlation function,
$\hat\xi_i(r)$. The mean overdensity is estimated by comparing the total weighted
galaxies to the expected number density given by the random samples, i.e.,
\begin{equation}
\bar\delta_i=\frac{1}{\alpha}\frac{w_{g,i}}{w_{r,i}}-1\,, ~~~~~
\alpha\equiv\frac{\sum_{i=1}^{N_s}w_{g,i}}{\sum_{i=1}^{N_s}w_{r,i}}
=\frac{w_{g,{\rm tot}}}{w_{r,{\rm tot}}} ~,
\end{equation}
where $w_{g,i}$ and $w_{r,i}$ are the total weights ($w_{\rm BOSS}$) of galaxies
and random samples in the $i^{\rm th}$ subvolume, respectively, and $N_s$ is the
number of subvolumes.
We use the Landy-Szalay estimator \cite{landy/szalay:1993} to estimate
the position-dependent correlation function as
\begin{equation}
\hat\xi_{{\rm LS},i}(r,\mu)=\frac{DD_i(r,\mu)}{RR_i(r,\mu)}
\left( \frac{[\sum_r w_{r,i}]^2-\sum_r w_{r,i}^2}{[\sum_g w_{g,i}]^2-\sum_g w_{g,i}^2}\right)
-\frac{DR_i(r,\mu)}{RR_i(r,\mu)}\frac{([\sum_r w_{r,i}]^2-\sum_r w_{r,i}^2)}{\sum_g w_{g,i} \sum_r w_{r,i}}+1 ~,
\label{eq:ls_xi_est}
\end{equation}
where $DD_i(r,\mu)$, $DR_i(r,\mu)$, and $RR_i(r,\mu)$ are the weighted numbers
of galaxy-galaxy, galaxy-random, and random-random pairs within the $i^{\rm th}$
subvolume, respectively, and $\mu$ is the cosine between the line-of-sight vector
and the vector connecting galaxy pairs ($\vr_1-\vr_2$). The summations such as
$\sum_{r}w_{r,i}$ and $\sum_{g}w_{g,i}$ denote the sum over all the random samples
and galaxies within the $i^{\rm th}$ subvolume, respectively. The angular average
correlation function is then $\hat\xi_{{\rm LS},i}(r)=\int_0^1d\mu~\hat\xi_{{\rm LS},i}(r,\mu)$.
\refEq{ls_xi_est} estimates the correlation function assuming that the density
fluctuation is measured relative to the ${\it local}$ mean. However, the position-dependent
correlation function defined in \refsec{theory} uses the density fluctuation
relative to the {\it global} mean. These two fluctuations can be related by
$\d_{\rm global}=(1+\bar\delta)\d_{\rm local}+\bar\delta$ with $\bar\delta=\bar n_{\rm local}/\bar n_{\rm global}-1$.
Thus, the position-dependent correlation function, $\hat\xi_i(r)$, is related
to the Landy-Szalay estimator as
\begin{equation}
\hat\xi_i(r)=(1+\bar\delta_i)^2\hat\xi_{{\rm LS},i}(r)+\bar\delta_i^2 ~.
\label{eq:xi_corr}
\end{equation}
To compute the average quantities over all subvolumes, we weight by $w_{r,i}$
in the corresponding subvolume. For example, for a given variable $g_i$ in the
$i^{\rm th}$ subvolume, the average over all subvolumes, $\bar g$, is defined by
\begin{equation}
\bar g=\frac{1}{w_{r,{\rm tot}}}\sum_{i=1}^{N_s}g_iw_{r,i} ~.
\label{eq:ens_avg}
\end{equation}
Since the number of random samples in each subvolume represents the effective
volume, the average quantities are effective-volume weighted. \refEq{ens_avg}
assures that the mean of the individual subvolume overdensities is zero,
\begin{equation}
\bar\delta=\frac{1}{w_{r,{\rm tot}}}\sum_{i=1}^{N_s}\bar\delta_iw_{r,i}
=\frac{1}{w_{r,{\rm tot}}}\sum_{i=1}^{N_s}\left[\frac{1}{\alpha}w_{g,i}-w_{r,i}\right]
=\frac{\alpha}{\alpha}-1=0 ~.
\end{equation}
We also confirm that $\bar{\hat{\xi}}(r)$ from \refeq{xi_corr} agrees with the
two-point function of all galaxies in the entire survey, on scales smaller than
the subvolume size.
integrated three-point function in the subvolume of size $L$ as
\begin{equation}
i\zeta(r)=\frac{1}{w_{r,{\rm tot}}}\sum_{i=1}^{N_s}\left[\hat\xi_i(r)\bar\delta_i
-2\bar{\hat\xi}(r)\frac{(1+\alpha)}{\alpha}\frac{\sum_rw^2_{r,i}}{\sum_r\bar n_{r,i}{\rm comp}_{r,i}w^2_{r,i}}
\left(\sum_r\frac{1}{\bar n_{r,i}{\rm comp}_{r,i}}\right)^{-1}\right]w_{r,i} ~,
\end{equation}
where the second term in the parentheses is the shot noise contribution, and
$\bar n_{r,i}$ and ${\rm comp}_{r,i}$ are the expected galaxy number density
and the survey completeness, respectively, of the random samples. Similarly,
we estimate the shot-noise-corrected variance of the fluctuations in the
subvolumes of size $L$ as
\begin{equation}
\sigma_L^2=\frac{1}{w_{r,{\rm tot}}}\sum_{i=1}^{N_s}
\left[\bar\delta_i^2-\frac{(1+\alpha)}{\alpha}\frac{\sum_rw^2_{r,i}}
{\sum_r\bar n_{r,i}{\rm comp}_{r,i}w^2_{r,i}}
\left(\sum_r\frac{1}{\bar n_{r,i}{\rm comp}_{r,i}}\right)^{-1}\right]w_{r,i} ~,
\end{equation}
where the second term in the parentheses is the shot noise contribution.
We find that the shot noise is subdominant (less than 10\%) in both $i\zeta(r)$
and $\sigma_L^2$.
\subsection{Measurements in real space}
\label{sec:mock_r}
\refFig{mock_r} shows the measurements of the two-point function
$\xi(r)$ from the entire survey (top left) and the normalized integrated
three-point functions (bottom panels), $i\zeta(r)/\sigma_L^2$,
for the subvolumes of two sizes ($220~h^{-1}~{\rm Mpc}$ in the bottom-left and $120~h^{-1}~{\rm Mpc}$
in the bottom-right panels). The gray lines show individual realizations, while
the dashed lines show the mean.
\begin{figure}[t]
\centering
\includegraphics[width=0.495\textwidth]{xi_N_Pw=20000_mock_r.pdf}
\includegraphics[width=0.495\textwidth]{chi2_N_ny=10_nx=5_zcut=3_ny=20_nx=10_zcut=5_Pw=20000_mock_r.pdf}
\includegraphics[width=0.495\textwidth]{iz_norm_N_ny=10_nx=5_zcut=3_Pw=20000_mock_r.pdf}
\includegraphics[width=0.495\textwidth]{iz_norm_N_ny=20_nx=10_zcut=5_Pw=20000_mock_r.pdf}
\caption{(Top left) $\xi(r)$ of the mocks in real space. The gray lines show
individual realizations, while the dashed line shows the mean. The black solid
line shows the best-fitting model. (Top right) $\chi^2$-histogram of the 600
mocks jointly fitting the models to $\xi(r)$ and $i\zeta(r)/\sigma_L^2$ in real
space. The dashed line shows the $\chi^2$-distribution with d.o.f.=36. (Bottom
left) $i\zeta(r)/\sigma_L^2$ of the mocks in real space for $220~h^{-1}~{\rm Mpc}$ subvolumes.
(Bottom right) Same as the bottom left panel, but for $120~h^{-1}~{\rm Mpc}$ subvolumes.}
\label{fig:mock_r}
\end{figure}
We now fit models of $\xi(r)$ and $i\zeta(r)/\sigma_L^2$ to the measurements in
$30~h^{-1}~{\rm Mpc}\le r\le78~h^{-1}~{\rm Mpc}$. We choose this fitting range because there are less
galaxy pairs at larger separations due to the subvolume size, and the nonlinear
effect becomes too large for our SPT predictions to be applicable at smaller
separations. For the two-point function, we take the Fourier transform of
\cite{crocce/scoccimarro:2008}
\begin{equation}
P_g(k)=b_1^2[P_l(k)e^{-k^2\sigma_v^2}+A_{\rm MC}P_{\rm MC}(k)]~,
\end{equation}
where $b_1$ is the linear bias, $P_l(k)$ is the linear power spectrum,
$A_{\rm MC}$ is the mode coupling constant, and
\begin{equation}
P_{\rm MC}(k)=2\int\frac{d^3q}{(2\pi)^3}~P_l(q)P_l(|\v{k}-\v{q}|)[F_2(\v{q},\v{k}-\v{q})]^2 ~.
\label{eq:xi_model}
\end{equation}
Hence, $\xi_g(r)=b_1^2[\xi_{l,\sigma_v}(r)+A_{\rm MC}\xi_{\rm MC}(r)]$ with
\begin{equation}
\xi_{l,\sigma_v}(r)=\int\frac{d^3k}{(2\pi)^3}~P_l(k)e^{-k^2\sigma_v^2}e^{i\v{k}\cdot\vr}\,, ~~~~~
\xi_{\rm MC}(r)=\int\frac{d^3k}{(2\pi)^3}~P_{\rm MC}(k)e^{i\v{k}\cdot\vr} \ .
\label{eq:xi_model_2}
\end{equation}
We use a fixed value of $\sigma_v^2=20.644~h^{-2}~{\rm Mpc}^2$. Varying it has
only small effect on the other fitted parameters. For the integrated three-point
function, we use the SPT calculation
\begin{equation}
\frac{i\zeta_g(r)}{\sigma_L^2}
=\frac{b_1i\zeta_{\rm SPT}(r)+b_2i\zeta_{b_2}(r)}{\sigma_{L,l}^2}\frac1{f_{\rm bndry}(r)} ~,
\label{eq:iz_model}
\end{equation}
where $i\zeta_{\rm SPT}(r)$ and $i\zeta_{b_2}(r)$ are computed from \refeq{iz}
with eqs.~\eqref{eq:zeta_spt} and \eqref{eq:zeta_b2}, respectively, and
$\sigma_{L,l}^2$ is computed from \refeq{sigmalL2}, using the subvolume
sizes of $L=220$ and $120~h^{-1}~{\rm Mpc}$ and the redshift of $z=0.57$. Note that
the size of the subvolumes affects the values of $\sigma_{L,l}^2$. We
determine $L$ by first measuring $b_1^2$ using the real-space two-point
function of the entire survey, and then find $L$ such that $b_1^2\sigma_{L,l}^2=\sigma_L^2$
assuming the cubic top-hat window function\footnote{In principle, the
shape of the window function also affects $\sigma_{L,l}^2$, but we ignore
this small effect.}. We find that these values ($L=220$ and $120~h^{-1}~{\rm Mpc}$)
agree well with the cubic root of the total survey volume divided by the
number of subvolumes, to within a few percent.
We fit the models to $\xi(r)$ and $i\zeta(r)/\sigma_L^2$ of both subvolumes
simultaneously by minimizing
\begin{equation}
\chi^2=\sum_{ij}C^{-1}_{ij}(D_i-M_i)(D_j-M_j)\,,
\label{eq:chi2}
\end{equation}
where $C^{-1}$ is the inverse covariance matrix computed from the 600 mocks,
$D_i$ and $M_i$ are the data and the model in the $i^{\rm th}$ bin, respectively.
The models contain three fitting parameters $b_1$, $b_2$, and $A_{\rm MC}$.
The models computed with the mean of the best-fitting parameters of 600 mocks
are shown as the black solid lines in \reffig{mock_r}.
The best-fitting parameters are $b_1=1.971\pm0.076$, $b_2=0.58\pm0.31$, and
$A_{\rm MC}=1.44\pm0.93$, where the error bars are 1-$\sigma$ standard deviations.
The agreement between the models and the mocks is good, with a difference much
smaller than the scatter among 600 mocks. Upon scrutinizing, the difference in
$\xi(r)$ is larger for larger separations because the fit is dominated by the
small separations with smaller error bars. On the other hand, for $i\zeta(r)/\sigma_L^2$
the agreement is good for both sizes of subvolumes at all scales of interest.
This indicates that the SPT calculation is sufficient to capture the three-point
function of the mocks in real space.
The data points in \reffig{mock_r} are highly correlated. To quantify the quality
of the fit, we compute the $\chi^2$-histogram from 600 mocks, and compare it with
the $\chi^2$-distribution with the corresponding degrees of freedom (d.o.f.). There
are 13 fitting points for each measurement ($\xi(r)$ and two sizes of subvolumes
for $i\zeta(r)/\sigma_L^2$) and three fitting parameters, so d.o.f.=36. The top right
panel of \reffig{mock_r} shows the $\chi^2$-histogram. The dashed line shows the
$\chi^2$-distribution with d.o.f.=36. The agreement is good, and we conclude that
our models well describe both $\xi(r)$ and $i\zeta(r)/\sigma_L^2$ of the mocks in real
space.
Our $b_1$ is in good agreement with the results presented in figure 16 of
ref.~\cite{gilmarin/etal:2014b}, whereas our $b_2$ is smaller than theirs,
which is $\simeq 0.95$, by 1.2$\sigma$. This may be due to the difference
in the bispectrum models. While we restrict to the local bias model and
the tree-level bispectrum, ref.~\cite{gilmarin/etal:2014b} includes a
non-local tidal bias \cite{mcdonald/roy:2009,baldauf/etal:2012,sheth/chan/scoccimarro:2012}
and uses more sophisticated bispectrum modeling using the effective $F_2$
kernel \cite{gilmarin/etal:2011,gilmarin/etal:2014a}. In \refapp{test},
we show that using the effective $F_2$ kernel and the non-local tidal
bias in the model increases the value of $b_2$, but the changes are well
within the 1-$\sigma$ uncertainties. Also, the differences of the goodness
of fit for various models are negligible.
The fitting range as well as the shapes of the bispectrum may also affect
the results: the integrated correlation function is sensitive only to the
squeezed configurations, whereas ref.~\cite{gilmarin/etal:2014b} includes
more equilateral and collapsed triangle configurations. Understanding this
difference merits further investigations.
\subsection{Measurements in redshift space}
\label{sec:mock_z}
\refFig{mock_z} shows the measurements of $\xi(r)$ (top left) and $i\zeta(r)/\sigma_L^2$
($220~h^{-1}~{\rm Mpc}$ in the bottom-left and $120~h^{-1}~{\rm Mpc}$ in the bottom-right panels) of
the mocks in redshift space. The gray lines show individual realizations, while
the dashed lines show the mean. Similar to the analysis in \refsec{mock_r}, we
fit the models in redshift space to the measurements in $30~h^{-1}~{\rm Mpc}\le r\le78~h^{-1}~{\rm Mpc}$.
In this section, we use General Relativity to compute the growth rate,
$f(z)\approx\Omega_m(z)^{0.55}$, which yields $f(z=0.57)=0.751$. We shall allow
$f$ to vary when interpreting the measurements in the actual data.
\begin{figure}[t]
\centering
\includegraphics[width=0.495\textwidth]{xi_N_Pw=20000_mock_z.pdf}
\includegraphics[width=0.495\textwidth]{chi2_N_ny=10_nx=5_zcut=3_ny=20_nx=10_zcut=5_Pw=20000_mock_z.pdf}
\includegraphics[width=0.495\textwidth]{iz_norm_N_ny=10_nx=5_zcut=3_Pw=20000_mock_z.pdf}
\includegraphics[width=0.495\textwidth]{iz_norm_N_ny=20_nx=10_zcut=5_Pw=20000_mock_z.pdf}
\caption{Same as figure~\ref{fig:mock_r} but in redshift space.}
\label{fig:mock_z}
\end{figure}
Since there is no baryonic acoustic oscillation feature on the scales we are
interested in, we model the redshift-space two-point correlation function as
\begin{equation}
\xi_{g,z}(r)=b_1^2\left[\xi_{l,\sigma_v}(r)+A_{\rm MC}\xi_{\rm MC}(r)\right]K_p ~,
\label{eq:xiz_model}
\end{equation}
where $\xi_{l,\sigma_v}(r)$ and $\xi_{\rm MC}(r)$ are given in \refeq{xi_model_2} and
\begin{equation}
K_p\equiv 1+\frac23\beta+\frac15\beta^2 ~,
\label{eq:kaiser}
\end{equation}
is the Kaiser factor with $\beta\equiv f/b_1$ \cite{kaiser:1987}.
As we do not include the subdominant term proportional to $b_2$ in the two-point
function, it only gives a constraint on $b_1$, which we can then use to break the
degeneracy with $b_2$ in the integrated three-point function. We find that this
simple modeling yields unbiased $b_1$ and fulfills the demand.
We calculate the redshift-space normalized integrated three-point function using
SPT at the tree level, as described in \refsec{rsd},
and then correct for the boundary effect.
The $\sigma_L^2$ of the mocks in redshift space agrees with $b_1^2 K\,\sigma_{L,l}^2$
to percent level. The redshift-space models thus contain, as before in real space,
the three fitting parameters, $b_1$, $b_2$, and $A_{\rm MC}$. We then simultaneously
fit $\xi(r)$ and $i\zeta(r)/\sigma_L^2$ of both subvolumes by minimizing \refeq{chi2}.
\refFig{covred} shows the correlation matrix ($C_{ij}$ in $\chi^2$, normalized by
$\sqrt{C_{ii}C_{jj}}$) estimated from the 600 mocks in redshift space. Because we
normalize the integrated three-point function by $\sigma_L^2$, the covariance between
$i\zeta(r)/\sigma_L^2$ and $\sigma_L^2$ is negligible. On the other hand, the covariances
between $i\zeta(r)/\sigma_L^2$ and $\xi(r)$, between $\xi(r)$, and between $i\zeta(r)/\sigma_L^2$
for two sizes of subvolumes are significant.
\begin{figure}[t]
\centering
\includegraphics[width=0.415\textwidth]{cov_N_ny=10_nx=5_zcut=3_ny=20_nx=10_zcut=5_Pw=20000.pdf}
\caption{Correlation matrix estimated from 600 mocks in redshift space. The
figure shows $\sigma_L^2$ and $i\zeta(r)/\sigma_L^2$ of $220~h^{-1}~{\rm Mpc}$ subvolumes
from bin 0 to 13, $\sigma_L^2$ and $i\zeta(r)/\sigma_L^2$ of $120~h^{-1}~{\rm Mpc}$ subvolumes
from bin 14 to 27, and $\xi(r)$ from bin 28 to 40.}
\label{fig:covred}
\end{figure}
The models computed with the mean of the best-fitting parameters of 600
mocks are shown as the thick solid lines in \reffig{mock_z}.
The best-fitting parameters are $b_1=1.931\pm0.077$, $b_2=0.54\pm0.35$,
and $A_{\rm MC}=1.37\pm0.82$. The agreement between the models and the
measurements in redshift space is as good as in real space.
Again, our $b_1$ is in good agreement with the results presented in figure 16 of
ref.~\cite{gilmarin/etal:2014b}, whereas our $b_2$ is smaller than theirs, which is
$\simeq 0.75$, but still well within the 1-$\sigma$ uncertainty. As noted in \refsec{mock_r},
the adopted models of the bispectrum are different. In \refapp{test}, we show that
using the effective $F_2$ and $G_2$ kernels and the non-local tidal bias in the model
increases the value of $b_2$. However, the changes are within the uncertainties,
and the goodness of the fit is similar for different models. Thus, in this paper
we shall primarily use the SPT at the tree level with local bias for simpler
interpretation of the three-point function, but also report the results for the
extended models.
\section{Measurements of the BOSS DR10 CMASS sample}
\label{sec:data}
\begin{figure}[t]
\centering
\includegraphics[width=0.495\textwidth]{xi_N_Pw=20000_data.pdf}
\includegraphics[width=0.495\textwidth]{chi2_N_ny=10_nx=5_zcut=3_ny=20_nx=10_zcut=5_Pw=20000_data.pdf}
\includegraphics[width=0.495\textwidth]{iz_norm_N_ny=10_nx=5_zcut=3_Pw=20000_data.pdf}
\includegraphics[width=0.495\textwidth]{iz_norm_N_ny=20_nx=10_zcut=5_Pw=20000_data.pdf}
\caption{Measurements of the BOSS DR10 CMASS sample (black solid lines).
The gray lines show individual mocks in redshift space and the dashed line
shows the mean of mocks. (Top left) $\xi(r)$, (Bottom left) $i\zeta(r)/\sigma_L^2$
for $220~h^{-1}~{\rm Mpc}$ subvolumes, and (Bottom right) $i\zeta(r)/\sigma_L^2$ for
$120~h^{-1}~{\rm Mpc}$ subvolumes. (Top right) $\chi^2$-histogram of the 600 mocks
jointly fitting the three amplitudes to $\xi(r)$ and $i\zeta(r)/\sigma_L^2$
in redshift space. The dashed line shows the $\chi^2$-distribution with
d.o.f.=38. The $\chi^2$ value measured from the BOSS DR10 CMASS sample is 46.4.}
\label{fig:data}
\end{figure}
We now present measurements of the position-dependent correlation function from
the BOSS DR10 CMASS sample\footnote{Catalogs of galaxies and the random samples
can be found in \url{http://www.sdss3.org}.} in NGC. The detailed description
of the observations can be found in refs.~\cite{ahn/etal:2013,anderson/etal:2013}.
Briefly, the sample contains 392,372 galaxies over 4,892 deg$^2$ in the redshift
range of $0.43<z<0.7$, which corresponds to the comoving volume of approximately
$2~h^{-3}~{\rm Gpc}^3$. We also weight the galaxies by $w_{\rm BOSS}$ to correct
for the observational systematics. We follow \refsec{division} to divide the
observations into subvolumes. However, the observations have their own set of
random samples, which are different from the ones of the mocks (the random samples
of the mocks have slightly higher $\bar n$ and different $\bar n(z)$), so we adjust
the redshift cuts to be $z_{\rm cut}=0.5108$, 0.5717 and $z_{\rm cut}=0.48710$,
0.52235, 0.55825, 0.60435 for the two resolutions, respectively. The resulting
properties of subvolumes of the observations and mocks are similar.
The mocks are constructed to match the two-point function of the observed galaxies,
but not for the three-point function. Hence there is no guarantee that the three-point
function of mocks agrees with the observations. We can test this using our measurements.
\begin{table}[t]
\centering
\begin{tabular}{ | c | c c c | }
\hline
& avg$[\sigma_{L,\rm mock}^2]$ & var$[\sigma_{L,\rm mock}^2]$ & $\sigma_{L,\rm data}^2$ \\
\hline
$220~h^{-1}~{\rm Mpc}$ & $4.6\times10^{-3}$ & $5.6\times10^{-4}$ & $4.9\times10^{-3}$ \\
$120~h^{-1}~{\rm Mpc}$ & $2.4\times10^{-2}$ & $1.3\times10^{-3}$ & $2.5\times10^{-2}$ \\
\hline
\end{tabular}
\caption{Measurements of $\sigma_L^2$ of the mock catalogs and the BOSS DR10 CMASS sample.}
\label{tab:sigma2}
\end{table}
The measurements of $\xi(r)$ and $i\zeta(r)/\sigma_L^2$ from the observations are
shown as the solid lines in \reffig{data}; the measurements of $\sigma_L^2$ is
summarized in \reftab{sigma2}. The measurements are consistent visually
with the mocks within the scatter of the mocks\footnote{These measurements of
$i\zeta(r)/\sigma_L^2$ are done for one effective redshift. We compare $i\zeta(r)/\sigma_L^2$
of the observations and mocks in different redshift bins in \refapp{iz_zevolve},
finding that the observations and mocks are consistent at all redshift bins to
within the scatter of the mocks.}, and we shall quantify the goodness of fit
using $\chi^2$ statistics later.
To quantify statistical significance of the detection of $i\zeta(r)/\sigma_L^2$
and the goodness of fit, we use the mean of the mocks as the model (instead
of the model based on perturbation theory used in section~\ref{sec:mock_z}),
and fit only the amplitudes of $i\zeta(r)/\sigma_L^2$, $\xi(r)$, and $\sigma_L^2$
to the observations and the 600 mocks by minimizing \refeq{chi2}. Specifically,
we use $O_i(r) = A_i\,O_i^{\rm mock}(r)$ as the model, where $O_1(r) = i\zeta(r)/\sigma_L^2$,
$O_2(r)=\xi(r)$, and $O_3 = \sigma_L^2$, with the amplitudes $A_1,\,A_2,\,A_3$.
\begin{table}[t]
\centering
\begin{tabular}{ | c | c c c | }
\hline
& $A_1$ & $A_2$ & $A_3$ \\
\hline
1-$\sigma$ error & 0.12 & 0.03 & 0.04 \\
best-fit (DR10) & 0.89 & 1.02 & 1.08 \\
\hline
\end{tabular}
~~~~~~~
\begin{tabular}{ | c | c c c | }
\hline
& $(A_1,A_2)$ & $(A_1,A_3)$ & $(A_2,A_3)$ \\
\hline
corr & 0.34 & 0.09 & 0.36 \\
\hline
\end{tabular}
\caption{Results of fitting the amplitudes: $A_1$ is $i\zeta(r)/\sigma_L^2$, $A_2$
is $\xi(r)$, and $A_3$ is $\sigma_L^2$. (Left) The 1-$\sigma$ uncertainties of
the amplitudes estimated from the mocks, and the best-fitting amplitudes of BOSS
DR10 CMASS sample with respect to the mean of the mocks. (Right) The correlation
coefficients of the amplitudes.}
\label{tab:fit_amp}
\end{table}
\refTab{fit_amp} summarizes the fitted amplitudes. The 1-$\sigma$ uncertainties
and the correlations are estimated from the 600 mocks. Since we normalize $i\zeta(r)$
by $\sigma_L^2$, the correlation between $A_1$ and $A_3$ is small. On the other hand,
$A_2$ and $A_3$ are correlated significantly because $\sigma_L^2$ is an integral of
the two-point function [\refeq{sigmalL2}].
Comparing the BOSS DR10 CMASS sample to the mean of the mocks, we find that
$i\zeta(r)/\sigma_L^2$ is 1-$\sigma$ lower,
$\xi(r)$ is unbiased (by construction of the mocks), and $\sigma_L^2$ is 2-$\sigma$
higher. The result of $A_1$ for the data is driven by the correlation between
different separations of $i\zeta(r)/\sigma_L^2$.
On the other hand, the result of $A_3$ is driven by the positive correlation
between $\xi(r)$ and $\sigma_L^2$.
While $\sigma_L^2$ of the data for two subvolumes are larger than that of the mocks
but still at the boundary of the variances (see \reftab{sigma2}), it requires an even
higher $A_3$ to minimize $\chi^2$ when we jointly fit the three amplitudes.
The fact that $A_3$ is larger than $A_2$ is also possibly due
the contributions to $\sigma_L^2$ from small separations (including
stochasticity at zero separations), where the mocks were not optimized.
We find $A_1=0.89\pm 0.12$, i.e., a 7.4$\sigma$ detection of the
integrated three-point function of the BOSS DR10 CMASS sample.
In order to assess the goodness of fit, we use the distribution of $\chi^2$,
a histogram of which is shown in the top right panel of \reffig{data}. In total
there are 41 fitting points (13 fitting points for $\xi(r)$ and two sizes of
subvolumes for $i\zeta(r)/\sigma_L^2$, and two fitting points for $\sigma_L^2$)
with three fitting parameters, so d.o.f.=38. The $\chi^2$ value of the observations
is 46.4, and the probability to exceed this $\chi^2$ value is more than 16\%.
Given the fact that the mocks are constructed to match only the two-point function
of the observations, this level of agreement for both the two-point and integrated
three-point correlation functions is satisfactory.
\section{Cosmological interpretation of the integrated three-point function}
\label{sec:interpretation}
What can we learn from the measured $i\zeta(r)/\sigma_L^2$? In section~\ref{sec:mock_z},
we show that the prediction for $i\zeta(r)/\sigma_L^2$ based on SPT at the tree-level
in redshift space provides an adequate fit to the mocks to within the scatter of
the mocks; thus, we can use this prediction to infer cosmology from $i\zeta(r)/\sigma_L^2$.
Note that any unmodeled effects in the integrated three-point function such as
nonlinearities of the matter density, nonlocal bias parameters, and redshift-space
distortions beyond the Kaiser factor, will tend to bias our measurement of cosmological
parameters based on $i\zeta(r)$. We will discuss caveats at the end of this section.
Since the linear two-point and the tree-level three-point functions are proportional
to $\sigma_8^2$ and $\sigma_8^4$, respectively, and $\sigma_L^2$ is proportional to $\sigma_8^2$,
the scaling of the redshift-space correlation functions is
\ba
\xi_{g,z}(r)\:&=b_1^2K\left[\xi^{\rm fid}_{l,\sigma_v}(r)\left(\frac{\sigma_8}{\sigma_{8,\rm fid}}\right)^2
+A_{\rm MC}\xi^{\rm fid}_{\rm MC}(r)\left(\frac{\sigma_8}{\sigma_{8,\rm fid}}\right)^4\right]\,, \nonumber\\
\frac{i\zeta_{g,z}(r)}{\sigma_L^2}\:&=
\frac{i\zeta^{\rm fid}_{g,z}(r)}{b_1^2\sigma_{L,l}^2K_p}\left(\frac{\sigma_8}{\sigma_{8,\rm fid}}\right)^2
\frac1{f_{\rm bndry}(r)} ~,
\label{eq:obs_zspace}
\ea
where ``fid'' denotes the quantities computed with the fiducial value of $\sigma_8$.
Note that $\xi_{\rm MC}(r)$ is proportional to $\sigma_8^4$ because it is an integral
of two linear power spectra (see \refeq{xi_model}). Since $\xi_{l,\sigma_v}(r)$
dominates the signal, the parameter combinations $b_1\sigma_8$ and $K=1+2\beta/3+\beta^2/5$
are degenerate in the two-point function. That is, the amplitude of the two-point
function measures only $(b_1\sigma_8)^2+\frac{2}{3}(b_1\sigma_8)(f\sigma_8)+\frac{1}{5}(f\sigma_8)^2$.
This degeneracy can be lifted by including the quadrupole of the two-point function
in redshift space. See refs.~\cite{samushia/etal:2013,tojeiro/etal:2014,sanchez/etal:2013,beutler/etal:2013}
for the latest measurements using the BOSS DR11 sample.
As for the three-point function, \reffig{iz_norm} shows that the $b_1^3$ and
$b_1^2 b_2$ terms are comparable for $b_1\approx b_2$. This means that, at the
three-point function level, the nonlinear bias appears in the leading order,
so the amplitude of the three-point function measures a linear combination of
$b_1$ and $b_2$. This provides a wonderful opportunity to determine $b_2$. The
challenge is to break the degeneracy between $b_2$, $b_1$, $f$, and $\sigma_8$.
For this purpose, we combine our results with the two-point function in redshift
space and the weak lensing measurements of BOSS galaxies. We take the constraints
on $b_1\sigma_8(z=0.57)=1.29\pm0.03$ and $f(z=0.57)\sigma_8(z=0.57)=0.441\pm0.043$ from table~2
in ref.~\cite{samushia/etal:2013}. To further break the degeneracy between $b_1$,
$f$, and $\sigma_8$, we take the constraint on $\sigma_8=0.785\pm0.044$ from ref.~\cite{miyatake/etal:2013,more/etal:2014},
where they jointly analyze the clustering and the galaxy-galaxy lensing using the
BOSS DR11 CMASS sample and the shape catalog from Canada France Hawaii Telescope
Legacy Survey.
We assume Gaussian priors on $b_1\sigma_8$, $f\sigma_8$, and $\sigma_8$ with the known covariance
between $b_1\sigma_8$ and $f\sigma_8$. The cross-correlation coefficient between $b_1\sigma_8$
and $f\sigma_8$ is $-0.59$, as shown in figure~6 of ref.~\cite{samushia/etal:2013}.
We then run the Markov Chain Monte Carlo with the Metropolis-Hastings algorithm
to fit the model \refeq{obs_zspace} to the observed $i\zeta(r)/\sigma_L^2$.
We find $b_2=0.41\pm0.41$, and the results for the extended models are summarized
in \reftab{b2_data}.
\begin{table}[t]
\centering
\begin{tabular}{ | c | c c c c | }
\hline
& baseline & eff kernel & tidal bias & both \\
\hline
$b_2$ & $0.41\pm0.41$ & $0.51\pm0.41$ & $0.48\pm0.41$ & $0.60\pm0.41$ \\
\hline
\end{tabular}
\caption{Best-fitting $b_2$ and their uncertainties for BOSS DR10 CMASS sample
for the extended models. The detailed description of the extended models is
in \refapp{test}.}
\label{tab:b2_data}
\end{table}
The value of $b_2$ we find is lower than the mean of the mocks, $b_2^{\rm mock}=0.54\pm0.35$.
The difference is mainly due to two reasons. First, the amplitude of the integrated
three-point function of the observations is lower than that of the mocks by 10\%
($A_1=0.89\pm0.12$).
Second, the priors from the correlation function and lensing constraint $b_1$ to
be close to 2.18, which is larger than that of the mocks, $b_1^{\rm mock}=1.93$.
Thus, it requires a smaller $b_2$ to fit the three-point function.
The argument is similar for the extended models. Note, however, that the nonlinear
bias of the data is still statistically consistent with the mocks.
Let us conclude this section by listing three caveats regarding our cosmological
interpretation of the measured integrated three-point function.
\begin{enumerate}
\item The models we use, \refeq{obs_zspace}, are based on tree-level perturbation
theory, the lowest order redshift-space distortion treatment, as well as
on the local bias parametrization. While this simple model describes the
mocks well, as shown in \refsec{mock_r} and \ref{sec:mock_z}, we discuss
in \refapp{test} that using the effective $F_2$ and $G_2$ and the non-local
tidal bias brings $b_2$ closer to that of ref.~\cite{gilmarin/etal:2014b}.
We, however, find similar goodness of fit for various models, and thus
we cannot distinguish between these models.
\item Covariances between the integrated three-point function, monopole and quadrupole
two-point function, and weak lensing signals are ignored in our treatment.
This can and should be improved by performing a joint fit to all the observables.
\item The cosmology is fixed throughout the analysis, except for $f$ and $\sigma_8$.
In principle, marginalizing over the cosmological parameters is necessary
to obtain self-consistent results, although the normalized integrated three-point
function is not sensitive to cosmological parameters such as $\Omega_m$ as
shown in figure~6 of ref.~\cite{chiang/etal:2014}.
\end{enumerate}
These caveats need to be addressed in the future work.
\section{Conclusions}
\label{sec:conclusion}
In this paper, we have reported on the first measurement of the three-point
function with the position-dependent correlation function from the SDSS-III
BOSS DR10 CMASS sample. The correlation between the position-dependent correlation
function measured within subvolumes and the mean overdensities of those subvolumes
is robustly detected at 7.4$\sigma$. This correlation measures the integrated
three-point function, which is the Fourier transform of the integrated bispectrum
introduced in ref.~\cite{chiang/etal:2014}, and is sensitive to the bispectrum
in the squeezed configurations.
Both the position-dependent correlation function and the mean overdensity are easier
to measure than the three-point function. The computational expense for the two-point
function is much cheaper than the three-point function estimator using the triplet-counting
method. In addition, for a fixed size of the subvolume, the integrated three-point
function depends only on one variable (i.e., separation), unlike the full three-point
function which depends on three separations. This property allows for a useful compression
of information in the three-point function in the squeezed configurations, and makes
physical sense because the integrated three-point function measures how the small-scale
two-point function, which depends only on the separation, responds to a long-wavelength
fluctuation \cite{chiang/etal:2014}. As there are only a small number of measurement
bins, the covariance matrix of the integrated three-point function is easier to estimate
than that of the full three-point function from a realistic number of mocks. We have
demonstrated this advantage in the paper.
Of course, since this technique measures the three-point function with one long-wavelength
mode (mean overdensity in the subvolumes) and two relatively small-wavelength modes
(position-dependent correlation function), it is not sensitive to the three-point
function of other configurations, which were explored by ref.~\cite{gilmarin/etal:2014b}.
We have used the mock galaxy catalogs, which are constructed to match the two-point
function of the SDSS-III BOSS DR10 CMASS sample in redshift space, to validate our
method and theoretical model. We show that in both real and redshift space, the
integrated three-point function of the mocks can be well described by the tree-level
SPT model. However, the nonlinear bias which we obtain from the mocks is higher than
that reported in ref.~\cite{gilmarin/etal:2014b}. This is possibly due to the differences
in the scales and configurations of the three-point function used for the analyses.
As discussed in \refsec{interpretation}, any unmodeled nonlinear effects in the redshift-space
integrated three-point function of CMASS galaxies will tend to bias $b_2$, and will
bias this parameter differently than the measurement of ref.~\cite{gilmarin/etal:2014b}.
Taking the mean of the mocks as the model, and treating the amplitudes of two- and
three-point functions as free parameters, we find the best-fit amplitudes of
$i\zeta(r)/\sigma_L^2$, $\xi(r)$, and $\sigma_L^2$ of the CMASS sample. With respect to
the mean of the mocks, the observations show a somewhat smaller $i\zeta(r)/\sigma_L^2$
($A_1=0.89\pm 0.12$) and larger $\sigma_L^2$, while the ensemble two-point function
$\xi(r)$ matches the mocks. Given that the mocks are generated to match specifically
the two-point function of the BOSS DR10 CMASS sample within a certain range of
separations, the level of agreement between the observations and mocks is satisfactory.
Finally, by combining the integrated three-point function and the constraints
from the anisotropic clustering ($b_1\sigma_8$ and $f\sigma_8$ in \cite{samushia/etal:2013})
and from the weak lensing measurements ($\sigma_8$ in \cite{more/etal:2014}), we break
the degeneracy between $b_1$, $b_2$, $f$, and $\sigma_8$. We find $b_2=0.41\pm0.41$
for the BOSS DR10 CMASS sample. The caveat of this result is that our model,
\refeq{obs_zspace}, relies on a rather simple model in redshift space as well
as on the local bias parametrization. We leave the extension of the model to
improved bias and redshift-space distortion modeling (especially in light of the
comparison with the results in ref.~\cite{gilmarin/etal:2014b}) for future work.
In summary, we have demonstrated that the integrated three-point function is
a new observable which can be measured straightforwardly from galaxy surveys
using basically the existing and routinely applied machinery to compute the
two-point function, and has the potential to yield a useful constraint on the
quadratic nonlinear bias parameter. Moreover, since the integrated three-point
function is most sensitive to the bispectrum in the squeezed configurations,
it is sensitive to primordial non-Gaussianity of the local type (parametrized
by $f_{\rm NL}$), thereby offering a probe of the physics of inflation. We plan
to extend this work to search for the signature of primordial non-Gaussianity
in the full BOSS galaxy sample.
\acknowledgments
We would like to thank Marc Manera for sharing the mock catalogs, and Lado Samushia
for computing the correlation between $b_1\sigma_8$ and $f\sigma_8$. We would also
like to thank Masahiro Takada, Shun Saito, and Surhud More for useful discussions.
We would like to thank an anonymous referee for useful comments.
Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the
Participating Institutions, the National Science Foundation, and the U.S. Department
of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/.
SDSS-III is managed by the Astrophysical Research Consortium for the Participating
Institutions of the SDSS-III Collaboration including the University of Arizona,
the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon
University, University of Florida, the French Participation Group, the German
Participation Group, Harvard University, the Instituto de Astrofisica de Canarias,
the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University,
Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics,
Max Planck Institute for Extraterrestrial Physics, New Mexico State University,
New York University, Ohio State University, Pennsylvania State University,
University of Portsmouth, Princeton University, the Spanish Participation Group,
University of Tokyo, University of Utah, Vanderbilt University, University of
Virginia, University of Washington, and Yale University.
|
{
"timestamp": "2015-07-29T02:08:59",
"yymm": "1504",
"arxiv_id": "1504.03322",
"language": "en",
"url": "https://arxiv.org/abs/1504.03322"
}
|
\section{Introduction} \label{sec:introduction}
Magnetic reconnection is arguably one of the most important energy conversion and plasma transport processes in solar and space plasmas. Among other effects, it determines the energy entry from the solar wind into Earth's magnetosphere, and it enables energy transport and dissipation therein \citep{dungey61a}. At Earth's magnetopause, reconnection proceeds asymmetrically between magnetosheath plasmas, namely solar wind plasmas compressed by Earth's bow shock, and magnetospheric plasmas.
The magnetosheath side has a typical magnetic field strength $\sim 20$ nT, density $\sim$ 15 cm$^{-1}$ and plasma-$\beta \sim 2$; The magnetosphere side has magnetic field strength $\sim 60$ nT, density $\sim$ 0.5 cm$^{-1}$ and plasma-$\beta \sim 0.1$ (e.g., \citet{phan96a}).
The magnetic field shear can be an arbitrary angle $\phi$.
Considering a planar current sheet, the x-line could develop at any angle from 0 to $\phi$, where the fields in the plane normal to this orientation have opposite signs, as suggested by \citet{cowley76a}. It is unclear if there is a simple principle to determine the orientation of the x-line in a three-dimensional (3D) system.
The first attempt to address this fundamental problem was by \citet{sonnerup74a} (also independently by \citet{gonzalez74a}), who suggested that reconnection will occur in a plane where the guide field is uniform.
Motivated by \citet{cowley76a}, the angle that bisects the total shear has been employed in global modeling \citep{moore02a,borovsky08a,sibeck09a}.
More recently, other sophisticated models based on maximizing various physics quantities were proposed.
\citet{swisdak07a} suggested the plane in which the reconnection outflow jets have a maximum speed. \citet{schreier10a} pointed out another possibility by maximizing the reconnection electric field (equivalent to the reconnection rate), where the formulation in \citet{cassak07b} for asymmetric reconnection could be used. Based on 2D simulations at different oblique reconnection planes, \citet{hesse13a} further proposed that the x-line orientation should be determined by maximizing the peak reconnection electric field, which was found to be proportional to the product of available magnetic energy density at both sides. The maximum of the peak reconnection electric field was shown to bisect the total magnetic shear angle $\phi$.
The principle that determines the orientation of the x-line in this simple planar current sheet could potentially guide us to find the location of reconnection in a more realistic magnetopause geometry. Global magnetospheric MHD simulations were recently performed \citep{komar15a} to compare these models, along with other ideas that predict the locations first in the global geometry with the ``orientation'' being the resulting locus that connects these locations. These predictions include maximizing the total magnetic shear angle \citep{trattner07a}, the total current density \citep{alexeev98a} and the divergence of the Poynting flux \citep{papadopoulos99a}. Another scenario suggested the x-line to be the magnetic separator simply resulting from the vacuum superposition of Earth's dipolar and solar wind magnetic fields \citep{cowley73a,siscoe01a,dorelli07a}, which was also tested \citep{komar13a}.
Observationally, the location and orientation of x-lines have been inferred from patterns of accelerated flows \citep{dunlop11b, phan06b, pu07a, scurry94a}, and patterns of precipitating ion dispersions \citep{trattner07a} during quasi-steady reconnections. Statistical studies of the flux transfer events (FTEs) generated by bursty reconnection \citep{fear12a, dunlop11a, wild07a, kawano05a}, and the global distribution of streaming energetic ion anisotropies \citep{daly84a} also provide clues. In addition, methods for locally reconstructing the reconnection geometry \citep{teh08a, shi05a, denton12a} were developed, some of these methods \citep{shi05a, denton12a} could potentially take advantage of satellite clusters that are deployed closely, such as NASA's Magnetospheric Multiscale Mission (MMS) \citep{burch09a}. These {\it in-situ} observations constrain theoretical modeling, however, it has been difficult to use them to distinguish between the detailed predictions of these models. To accurately determine the x-line orientation in observation is still a challenging and active research area in space study.
In this paper, we use 3D simulations to study the orientation of x-lines in a given asymmetric planar geometry. In a similar work, \citet{schreier10a} used 3D Hall-MHD simulations, where the x-line orientation is concluded to be consistent with that of the maximized reconnection outflow speed \citep{swisdak07a} or reconnection electric field (rate) \citep{cassak07b}.
However, a study using 3D fully kinetic simulations does not exist, where kinetic effects such as the particle streaming could be potentially important.
Furthermore, we develop a way to clearly test the orientation of a single x-line in 3D systems that develops in a controlled fashion. For reconnections that develops from a long current sheet without a perturbation, or with a perturbation similar to that in the GEM-challenge \citep{birn01a}, the x-line orientation may be strongly affected and even selected by oblique flux ropes arising from tearing instabilities in the linear phase (e.g., \citet{yhliu13a}).
To avoid this, we use a spatially localized perturbation to induce a single x-line.
This localized perturbation prevents the linear tearing instability before the development of a single x-line, and does not pre-select the orientation of x-line. The single x-line that develops with sufficient freedom appears to bisect the total magnetic shear angle for larger mass ratios. This is consistent with that suggested in \citet{hesse13a}.
The layout of this paper is the following. Section 2 describes the setup of our particle-in-cell simulations. Section 3 shows the measurements of x-line orientations in 3D simulations with $m_i/m_e=25$ and $m_i/m_e=1$ plasmas. Section 4 compares the results with 2D simulations, and the mass ratio dependency of x-line orientation is investigated. Section 5 contains the summary and discussions.
\section{Simulation setup}
The asymmetric configuration employed \citep{hesse13a, aunai13b, pritchett08a} has the magnetic profile, ${\bf B}=B_0(0.5+S)\hat{\bf x}_0+B_{y0}\hat{\bf y}_0$ where $S=\alpha_1\mbox{tanh}(z/\lambda)$. This corresponds to a shear angle $\phi=180^\circ-\mbox{tan}^{-1}[(B_{y0}/B_0)/(0.5+\alpha_1)]-\mbox{tan}^{-1}[(B_{y0}/B_0)/(0.5-\alpha_1)]$ across the sheet. The plasma has density $n=n_0[1-\alpha_2(S+S^2)/3]$ and an uniform total temperate $T=3B_0^2/(8\pi n_0 \alpha_2)$. We choose $\alpha_1=\alpha_2=1$, then the resulting $B_{2x0}=1.5B_0$, $B_{1x0}=0.5B_0$ and $n_2=n_0/3$, $n_1=n_0$. Here the subscripts ``1'' and ``2'' indicate the magnetosphere and the magnetosheath sides respectively. We use a uniform guide field $B_{y0}=B_0$ then the total magnetic shear agnle $\phi\sim 82.87^\circ$. The temperature ratio is $T_i/T_e=5$, and the ratio of electron plasma to gyro-frequency is $\omega_{pe}/\Omega_{ce}=4$. Here, $\omega_{pe}\equiv(4\pi n_0 e^2/m_e)^{1/2}$ and $\Omega_{ce}\equiv eB_0/m_e c$.
In this paper, fully kinetic simulations were performed using the particle-in-cell code -{\it VPIC} \citep{bowers09a}. Densities are normalized by density $n_0$, time is normalized by the ion gyro-freqency $\Omega_{ci}$, velocities are normalized by Alfv\'enic speed $V_A\equiv B_0/(4\pi n_0 m_i)^{1/2}$, and spatial scales are normalized by the inertia length $d_j\equiv c/\omega_{pj}$, where $j=e, i$ for electron or ion respectively.
For the rest of this paper, the x-line orientation will be quantified using the angle $\theta$ respect to the ${\bf y}_0$-axis.
The simulation box can be rotated to $\hat{\bf x}=\mbox{cos}\theta_{box}\hat{\bf x}_0+\mbox{sin}\theta_{box}\hat{\bf y}_0$ and $\hat{\bf y}=\mbox{sin}\theta_{box}\hat{\bf x}_0-\mbox{cos}\theta_{box}\hat{\bf y}_0$. In a 2D system, this machinery allows us study the reconnection with a pre-selected x-line orientation $\theta=\theta_{box}$. The in-plane magnetic field vanishes at $z_{n}=\lambda \mbox {tanh}^{-1}\{-[0.5+(B_{y0}/B_0)\mbox{tan}\theta_{box}]/\alpha_1\}$.
The primary 3D run (case {\bf k} in Table 1) discussed in detail uses $m_i/m_e=25$ and has a domain size of $L_x \times L_y \times L_z=64d_i \times 64d_i \times 16d_i$ with $1024 \times 512 \times 256$ cells. The simulation domain is rotated to $\theta=+10^\circ$, this does not affect the conclusions in this paper. The boundary conditions are periodic both in the x- and y-directions, while in the z-direction are conducting for fields and reflecting for particles. We use $150$ particles per cell. The half-thickness of the initial sheet is $\lambda= 0.8 d_i$. In addition to 3D simulations, 2D runs with $m_i/m_e=1, 4, 25, 100$ and $256$ are also conducted to study the mass ratio dependency. These runs are listed in Table 1.
To study the simplest situation with a single x-line, we want to avoid the development of tearing instabilities before a well-defined x-line forms. We use a perturbation localized in the x-direction since the tearing mode is more stable in a short current sheet. In addition, a perturbation being uniform in the y-direction might pre-select the orientation. Therefore, we further localize the perturbation in the y-direction so that the single x-line can develop with sufficient freedom. The perturbation used in $m_i/m_e=25$ cases is illustrated in Fig.~\ref{pert}.
The perturbation has the functional form $\tilde{B}_z\propto \mbox{cos}[\pi (z-z_p)/L_z]\times\mbox{sin}(2\pi x/L_x) \mbox{exp}(-|x|/L_{p1}) \times f(y)$, where $f(y)=\mbox{tanh}[(y+L_{p2})/L_{p3}]-\mbox{tanh}[(y-L_{p2})/L_{p3}]$. $\tilde{B}_x$ is derived using $\nabla\cdot \tilde{\bf B}=0$ and $\tilde{B}_y=0$. The peak value of the perturbation is $\delta B_z=0.05 B_0$. For runs with mass ratio $m_i/m_e=1$, $4$, $25$ and $256$, we choose $L_{p1}=L_x/40$, $L_x/8$, $L_x/20$ and $L_x/20$; $L_{p2}/d_i=1$, $0.5$, $1$ and $1$; $L_{p3}/d_i=2$, $3$, $2$ and $2$ respectively. To simplify the comparison, we fix $z_p=-0.5493 L$ for all cases. This is the location where the in-plane magnetic vanishes at the $\theta=0^\circ$ plane.
It may be argued that the orientation of these single x-lines will be constrained to meet the resonant condition imposed by periodic boundaries, i.e., $(L_y/L_x)\mbox{tan}(\theta+\theta_{box})$ being a rational number, which is equivalent to the safety factor in fusion Tokamaks (e.g., \citet{beidler11a}). An extensive study on the effect of periodic boundary using mass ratio $m_i/m_e=4$ is performed, but not shown here. We find that the x-line in a large enough simulation box develops into the same orientation even with different box aspect ratio and box orientation. We conclude that the localization of x-line as described mitigates this effect from periodic boundaries.
\section{3D simulation results}
With the localized perturbation, a well-defined x-line emerges near the center of the simulation box. Figure \ref{local_mime25_10d_3D} shows the total current density $|{\bf J}|$ of the primary case at time $60/\Omega_{ci}$, when a single x-line with large scale outflows has developed. The planes are cuts at $z=0$ and $y=0$. In order to measure the orientation of the x-line, we focus on the $x-y$ plane (top-view) in Fig.~\ref{local_mime25_10d}. The current density is shown in panel (a) and a black-dotted line of $\theta=-13^\circ$ is overlaid for comparison. To avoid a potential dependency on the choice of the $x-y$ plane, the 3D iso-surface of $|{\bf J}|=2$ is plotted in Fig.~\ref{local_mime25_10d}(b), which further justifies the measurement of this angle. The reconnected magnetic field $B_z$ is shown in Fig.~\ref{local_mime25_10d}(c). The region with $B_z=0$ indicates the topological separator and follows the same black-dotted line. Note that the average magnitude of $B_z$ is around $\sim O(0.1 B_0)$, as expected in the nonlinear stage of reconnection. Figure.~\ref{local_mime25_10d}(d) depicts the non-ideal electric field $E_\|$, which traces the diffusion region of magnetic reconnection, also shows the same orientation.
Fig. \ref{local_mime25_10d}(b) and (d) suggest the x-line extension $\approx 20 d_i=100 d_e$, and interestingly the x-line does not appear to extend much longer at later time. A similar finite extension was observed in symmetric reconnection simulations \citep{shay03a}.
To evaluate the global reconnection rate, we apply the general magnetic reconnection theory (GMR) \citep{schindler88a, hesse88a, hesse93a} on this three-dimensionally localized x-line. GMR theory points out the importance of evaluating the integration of the parallel electric field $E_\|$ along magnetic field lines, $\Xi\equiv\int{E_\| ds}$, especially for field lines that thread the ideal region ($E_\|=0$) through the non-ideal region ($E_\| \neq 0$) to the ideal region at another end. The maximum value, $\Xi_{max}\equiv \mbox{max}[\Xi(x,z)]$, is the global reconnection rate. This will be an accurate measure of reconnection rate since the net contribution of electrostatic component in $E_\|$, that is not directly relevant to reconnection, will vanish in this integration. The integration reduces the 3D system to a 2D map of $\Xi$, as shown in Fig.~\ref{potential}(a). This $\Xi$ map in the $y=0$ plane is generated by integrating $E_\|$ along field lines for $30 d_i$ arc-length at both sides of the $y=0$ plane.
We can then identify the location of $\Xi_{max}$ on this 2D map and trace the magnetic field line from this seed point (yellow). This magnetic field line that carries $\Xi_{max}$ is expected to be tangential to the x-line locally around the diffusion region if the diffusion region is quasi-2D for a reasonably long extension.
For comparison, we also trace 15 field lines seeded evenly along the z-direction at the same x and y coordinate of $\Xi_{max}$. These sample field lines with positive (negative) $B_x$ are colored in red (blue).
Figure \ref{potential}(b) shows the top-view of these field lines overlaid with the iso-surface of $E_\|=0.08V_AB_0/c$ (green).
The field line with $\Xi_{max}$ (yellow) appears to pass through the non-ideal region and is tangential to the black-dashed line with orientation $\theta=-13^\circ$. This orientation approximately bisects the total magnetic shear angle across the current sheet (i.e., the angle between the red and blue field lines). It may be argued that the field line behavior may be sensitive to the choice of seed points due to the chaotic nature of magnetic field lines \citep{boozer12a}. Hence, to get a more conclusive measurement, we also seed 100 points evenly distributed inside a sphere of radius $0.1d_e$ centered at the location of $\Xi_{max}$. These field lines traced from these seeds are shown in Fig.~\ref{potential}(c) in yellow. They align with orientation $\theta\approx-13^\circ$ inside the non-ideal region (green), then separate quickly from each other outside the non-ideal region. The global reconnection rate is $\Xi_{max} \approx 4.8 V_AB_0d_e/c$. Divided by the length of the x-line $\approx 100 d_e$, the spatially-averaged 2D rate is roughly $0.048V_AB_0/c$, quantitatively similar to the peak $E_\|$ measured in the corresponding 2D simulation at this orientation (Fig.~\ref{2D_rates}(c)). This further justifies that this 3D x-line is at its nonlinear phase.
In Fig.~\ref{local_mime25_10d}(b), theoretical predictions of x-line orientation \citep{sonnerup74a, swisdak07a, cassak07b, schreier10a, birn10a, hesse13a} are plotted as dashed lines using different colors. Note that a prediction based on the reconnection electric field in \citet{birn10a} is also presented here, where a more accurate energy equation is considered to improve the Cassak-Shay formula \citep{cassak07b}. The ratio of specific heats $5/3$ is used. The closest prediction is the angle of bisection with $\theta=-14.87^\circ$ \citep{hesse13a, moore02a, borovsky08a, sibeck09a}. Using this same asymmetric configuration with $m_i/m_e=25$, \citet{hesse13a} found a relation between the peak reconnection electric field and the available magnetic energy for reconnection $E_{rec}\propto B^2_{1,rec} B^2_{2,rec}$. The orientation that bisects the total magnetic shear angle maximizes this $E_{rec}$.
While the agreement between this 3D simulation and the theoretical prediction is excellent, it is important to test if this bisection orientation is generic. We perform a similar 3D simulation in electron-positron plasmas with mass ratio $m_i/m_e=1$ to test the mass ratio dependency. The measurement of $|{\bf J}|$, $B_z $ and $E_\|$ displayed in Fig.~ \ref{local_mime1_10d} consistently suggest an angle $\approx -28^\circ$, which is larger than the bisection angle. However, all existing analytical predictions \citep{sonnerup74a, swisdak07a, cassak07b, schreier10a, birn10a, hesse13a} remain the same as indicated in Fig.~\ref{local_mime1_10d}(b), since they do not have the mass ratio dependency. This is investigated further in the following section.
\section{2D modeling and predictions}
To understand the difference with a lower mass ratio, we go back to 2D simulations at oblique planes (i.e., $\theta_{box}\neq 0$). Unlike 3D, the advantage of using 2D simulation is that we can choose the orientation of the x-line, which is out of the 2D plane. We study the evolution of the reconnection rate $R_0\equiv \left<\partial_t \psi/\partial t\right> /(V_A B_0)$ at different orientations in Fig.~\ref{2D_rates}. Here $\psi$ is the difference of the flux function $A_y$ between the primary x- and o-points, which are the saddle point and local maximum of $A_y$ respectively.
As shown in Fig.~\ref{2D_rates}(c), with $m_i/m_e=25$ the orientation that maximizes the peak reconnection rate are consistent with the bisection angle with $\theta=-14.87^\circ$ \citep{hesse13a}. However, with a lower mass ratio the orientation that maximizes the peak rate shifts to a larger angle as shown in Fig.~\ref{2D_rates}(a)-(b). The suggested angle from these 2D simulations with electron-positron plasmas ($m_i/m_e=1$) is $\theta \approx -28^\circ$, consistent with the orientation measured in the 3D system (Fig.~\ref{local_mime1_10d}). This further suggests that 2D models are sufficient to capture the physics that determines the x-line orientation in 3D systems, and this orientation maximizes the peak reconnection rate among these 2D oblique planes.
With a larger mass ratio, the bisection angle persists to maximizes the peak rate in 2D simulations, as shown in Fig.~\ref{2D_rates} (d) with $m_i/m_e=256$. While a 3D simulation similar to that of Fig.~\ref{local_mime25_10d_3D} with a realistic mass ratio $m_i/m_e=1836$ is impossible with current computational capability, the consistency between 3D and 2D simulations demonstrated here suggests that the x-line may still bisect the magnetic shear with real mass ratio $m_i/m_e=1836$. This prediction is directly relevant to the reconnection events at Earth's magnetopause.
To explain the mass ratio dependency, we notice that secondary plasmoids are generated to cause fluctuations in the reconnection rates shown in Fig.~\ref{2D_rates}(a)-(b), even though we used the localized perturbation and a thicker initial sheet. In contrast, the induced singe x-line in plasmas of higher mass ratios (e.g., $m_i/m_e=25, 256$ in Fig.~\ref{2D_rates}(c)-(d)) does not generate secondary plasmoids, presumably because the Hall effect arising from the mass ratio difference prevents the opened reconnection exhaust from collapsing \citep{shay99a, stanier15a}, and hence makes tearing modes more stable \footnote{However, the reconnection rate has the same order regardless the difference in mass ratio.}.
This further motivates us to conjecture that the physics of tearing instability may play some role, that is more apparent with a lower mass ratio. The tearing instability is driven by the filamentation tendency of current sheet. In principle, the nonlinear current sheet of the single x-line could still be subject to the same filamentation tendency.
To investigate this, we use a current sheet of $d_e$-scale thick. The half-thickness $\lambda=1.36d_e$ is taken to be the mean value of the inertial lengths at both sides (i.e., $d_e$ and $\sqrt{3}d_e$). This thickness mimics the current sheet scale observed in the nonlinear stage of reconnection. In Fig.~\ref{tearing} (a), we show the tearing modes that spontaneously grow in this $d_e$-scale current sheets without any perturbation. At the time of measurement, the tearing mode amplitude is still small, $\delta B_z/B_0\sim O(10^{-3})$, and hence justifies its linear stage. The dominant tearing modes, presumably the fastest growing tearing mode, has a similar orientation $\theta\approx -28^\circ$ as that of the single x-line observed in Fig.~\ref{local_mime1_10d}. This supports our conjecture on the role of the tearing instability.
In addition, we do the same experiment with higher mass ratio $m_i/m_e=25$. Interestingly, the dominant tearing modes manifest an angle $\theta\approx -13^\circ$ as shown in Fig.~\ref{tearing}(b), that is also consistent with the x-line orientation in Fig~\ref{local_mime25_10d}. These results imply that tearing modes may have some relation to the peak reconnection rate measured in 2D simulations (Fig.~\ref{2D_rates}), and hence the bisection solution in the large mass ratio limit.
\section{Summary and Discussion}
We demonstrate that in the large mass ratio limit the x-line bisects the total magnetic shear angle across the current sheet, at least in the 3D simulations presented here. The orientation can generally be predicted by scanning through a series of 2D simulations to find the orientation that maximizes the peak reconnection electric field.
This result serves as a practical prediction to reconnection events at Earth's magnetopause.
The fact that $d_e$-scale tearing modes share the same orientation as a nonlinear single x-line may have profound implications.
The tearing instability is driven by the filamentation tendency of the current sheet. In principle, the nonlinear current sheet of the single x-line could still be subject to the same tendency, and consequently develop into a state that is marginally stable to the tearing instability. The linear tearing mode hence may provide predictions on some properties of the x-line, for instance, the orientation shown here and maybe the spatial scale of the x-line \citep{yhliu14a}.
To study these $d_e$-scale tearing modes in 3D simulations with a larger mass ratio is still computationally feasible, since we only need to check the linear phase and the tearing instability's growth rate is large for a $d_e$-scale current sheet. In terms of ion gyro-frequency, the growth rate \citep{yhliu13a,daughton11a} is $\gamma/\Omega_{ci} \sim (d_e/\lambda)^3(m_i/m_e)(\rho_e/d_e)$. In this Harris-type equilibrium $\beta\sim O(1)$, then $\rho_e/d_e \sim O(1)$, hence the growth rate is proportional to the mass ratio $m_i/m_e$ in a $d_e$-scale current sheet. Therefore, simulations like Fig.~\ref{tearing} could also serve as a useful indicator in predicting the x-line orientation.
Some caveats and limitations need to be kept in mind.
First, at late times, periodic boundaries may start to affect and secondary flux ropes (i.e., 3D version of plasmoids) develop along the separatrix \citep{daughton11a} of these single x-lines. These oblique flux ropes intertwine with each other and complicate the current sheet as seen in Fig.~\ref{Bz_evolution}(d), where a definite measurement of the x-line orientation becomes difficult. However, the orientation of the primary topological separator in Fig.~\ref{Bz_evolution}(d) appears to remain similar. Second, for multiple x-lines that develop from periodic tearing modes in a current sheet without a localized perturbation, the orientation could be strongly affected by the nonlinear flux-ropes \citep{yhliu13a}. Third, this study employs one possible asymmetric configuration where the stabilization by the diamagnetic drift \citep{swisdak10a, phan10a} is weak. The difference of plasma-$\beta$ between both sides is $\Delta\beta \sim 2(\delta/d_i)\mbox{tan}(\phi/2) \sim 2$ if the current sheet thickness $\delta \sim d_i$ is assumed. Future work will explore the regime with $\Delta\beta \gg 2(\delta/d_i)\mbox{tan}(\phi/2)$ to demonstrate the effects of diamagnetic drifts on the development of x-lines in 3D systems.
\begin{acknowledgments}
Y. -H. Liu thanks for helpful discussions with W. Daughton, D. G. Sibeck, C. M. Komar, N. Bessho, J. C. Dorelli, P. Cassak, N. Aunai, L. -J Chen, D. Wendel, M. L. Adrian, I. Honkonen and L. B. Wilson III. We are grateful for support from NASA through the NASA Postdoctoral Program and MMS mission. Simulations were performed with LANL institutional computing and NASA Advanced Supercomputing.
\end{acknowledgments}
|
{
"timestamp": "2015-04-14T02:17:53",
"yymm": "1504",
"arxiv_id": "1504.03300",
"language": "en",
"url": "https://arxiv.org/abs/1504.03300"
}
|
\section{Introduction}
\label{Sec:Intro}
One of direct consequences of the kinetic theory is that real substances manifest transport properties, which leads to existence of dissipative mechanisms like viscosity and thermal conductivity. Cosmological fluids should not be an exception. Although perfect fluids in local thermal equilibrium (LTE) have been successfully used to model the matter content of the universe, it is dissipation that drives the fluid towards equilibrium: a fluid without appropriate transport properties will never reach this state. Consequently, dissipation must have played an important role in the early universe.
\par
A brief review of the traditional theories of dissipative thermodynamics can be found in~\cite{Maartens1996}; see references therein for applications to cosmology. At the present moment, the most advanced of these approaches is the second-order Israel-Stewart~(IS) theory~\cite{Israel1976,Israel1979}, often referred to as transient, or causal, thermodynamics. An alternative formalism proposed by Carter~\cite{Carter1991}, see~\cite{Andersson2007} for a review, has been shown to be essentially equivalent to the IS theory~\cite{Priou1991}.
\par
The simplified, or "truncated", version of the IS theory is obtained by omitting certain divergence terms in the transport equations of the full theory in a way that does not violate the crucial features of causality and stability. As pointed out by Maartens~\cite{Maartens1996}, the truncated IS theory can be considered as a relativistic generalization of the Navier-Stokes equations for a fluid, in the sense that the values of the transport coefficients are equal to their local equilibrium values. Due to this property, the truncated IS theory is sometimes treated as an independent approach rather than an approximation to the full theory.
\par
Maartens \cite{Maartens1995} has shown that the truncated IS theory can yield solutions which deviate significantly from those of the full theory. Furthermore, Zimdahl \cite{Zimdahl1996a} has demonstated for the case of scalar dissipation that the full and truncated versions are equivalent only in a few special cases. Recently, Shogin et al. \cite{Shogin2015} have discovered that the truncation of the IS transport equations can produce solutions with pathological dynamical features in anisotropic spacetimes. However, despite these results, the truncated theory is still widely used in cosmology, see e.\,g. references in the review by Brevik and Gr{\o}n in~\cite{Travena2013}.
\par
The standard Friedmann-Robertson-Walker cosmological models have been studied using the truncated \cite{Coley1995,Zimdahl1997a} and the full \cite{Maartens1995, Coley1996} versions of the IS theory. In particular, the processes of inflation and reheating, where dissipative effects play a significant role, have been discussed respectively in~\cite{Maartens1995} and~\cite{Zimdahl1997a}.
\par
However, the geometry of the spatially homogeneous and isotropic Friedmann-Robertson-Walker model does not allow for any dissipative mechanisms different from bulk viscosity. Therefore, anisotropic backgrounds must be considered when modelling more realistic dissipation. For an early application of causal thermodynamics to anisotropic cosmological models, see Belinskii et al.\,~\cite{Belinskii1979}. Van den Hoogen and Coley \cite{Hoogen1995} have considered Bianchi type~V cosmological models using the truncated IS theory. Heuristical methods have been used to find out that, in contrast to the first-order Eckart theory, the shear viscous stresses play a major role in determining the dynamics of the model; namely, depending on the coefficients of shear viscosity, the spacetime may, or may not, isotropize in the future. Recently, Shogin et al. \cite{Shogin2015} studied Bianchi type IV and V models in the framework both of the full and the truncated versions of the IS theory. The existence of effectively anisotropic solutions, as well as the importance of shear viscosity, was confirmed; in addition, it was demonstrated that even the full IS theory can break down in cosmological applications, as the viscous stresses can drive the fluid far away from LTE.
\par
In this paper, we investigate spatially homogeneous plane symmetric Bianchi type~I cosmological models. By this choice we intentionally exclude the effects of spatial curvature to concentrate on determining the limits of applicability of the IS theory, and, in addition, finding the source of the singular behaviour of the solutions of the system in the case when the transport equations are truncated.
\par
We apply a dynamical systems approach~\cite{Wainwright1997} to describe the future attractors of the system of differential equations governing the dynamics of the cosmological model and perform a stability analysis of the fixed points of this system. Special attention is paid to the late-time behaviour of the relative dissipative fluxes, the magnitude of which represents the deviation of the fluid from LTE, and, by this, determines the applicablity of the IS theory in this particular case.
\par
The paper is organized as follows. In section~\ref{Sec:Model} we briefly discuss the chosen fluid model. Then, the Hubble-normalized field equations are written down in section~\ref{Sec:Equations}. The future asymptotic states of the cosmological model, as well as the dynamics of the relative dissipative fluxes, are discussed in sections~\ref{Sec:BVSV}--\ref{Sec:SVonly}. Conclusions, together with some possible directions for the further work are presented in section~\ref{Sec:Conslusions}.
\section{The fluid model}
\label{Sec:Model}
We consider the cosmological model to incorporate a non-tilted~\cite{King1973} dissipative fluid, neglecting the effects of thermal conductivity. The energy-momentum tensor of the fluid is then given by
\begin{equation}
T_{\alpha \beta} = (\rho + p + \pi)u_\alpha u_\beta + (p+\pi)g_{\alpha \beta}+\tau_{\alpha \beta},
\end{equation}
where~$\pi$ denotes the bulk viscous pressure, $u_\alpha$ is the four-velocity of the fluid, and~$\tau_{\alpha \beta}$ stands for the shear viscous stress tensor, with~$\tau_{\alpha \beta}u^\beta=\tau_{[\alpha \beta]}=\tau_\alpha^{~\alpha}=0$.
We are working in the so-called Eckart frame, so the variables $\rho$ and $p$ refer to the local equilibrium values of energy density and pressure of the fluid, and $u_\alpha$ is defined as the {\it particle} four-velocity of the fluid.
\par
Since our purpose is to perform a qualitative investigation, we consider a {\it mathematical} fluid, for which $\rho$ and~$p$ are connected by a linear barotropic equation of state, widely known as the~$\gamma$-law:
\begin{equation}
p=(\gamma-1)\rho, \qquad 0<\gamma<2.
\end{equation}
Note that the equations of state for {\it physical} radiative fluids are somewhat more complicated, see e.\,g. \cite{Weinberg1971,Straumann1976}, involving temperature and particle number density as independent variables\footnote{A realistic fluid model may, however, lead to simplifications in other aspects of the analysis.}. A simple $\gamma$-fluid with $1<\gamma<4/3$ has been used as a {\it very} naive model of a matter/radiation mixture.
\par
The dissipative properties of the fluid are described by the IS theory, the transport equations of the full version\footnote{The transport equation describing heat conduction is omitted.} (in a spatially homogeneous background) being formulated as\footnote{Dots represent derivation with respect to proper time~$t$ of observers comoving with the fluid.}:
\begin{align}
\tau_0\dot{\pi}+\pi &= -3\zeta H-\frac{1}{2}\tau_0\pi \left[3H+\frac{\dot{\tau}_0}{\tau_0}-\frac{\dot{\zeta}}{\zeta}-\frac{\dot{T}}{T} \right],\label{Eq:Model:IS-Full-0-Bulk} \\
\tau_2\dot{\tau}_{ab}+\tau_{ab} &= -2\eta \sigma_{ab}-\frac{1}{2}\tau_2\tau_{ab} \left[3H+\frac{\dot{\tau}_2}{\tau_2}-\frac{\dot{\eta}}{\eta}-\frac{\dot{T}}{T} \right]. \label{Eq:Model:IS-Full-2-Shear}
\end{align}
Here~$T$ is the local equilibrium value for the temperature,~$H$ is the Hubble rate, and~$\sigma_{ab}$ is the geometric rate of shear tensor. In causal thermodynamics, the relaxation times $\tau_0$ and~$\tau_2$ of the dissipative processes are finite and related to the bulk and the shear viscosity coefficients~$\zeta$ and~$\eta$ by
\begin{equation}
\tau_0=\zeta \beta_0, \qquad \tau_2=2\eta \beta_2, \label{Eq:Model:RelaxationTimes}
\end{equation}
where~$\beta_0, \beta_2\geq 0$ are the thermodynamic coefficients for scalar and tensor contributions to the entropy density~\cite{Maartens1996}.
\par
The transport equations of the truncated IS theory are obtained by dropping the terms in the square brackets on the right-hand sides of~(\ref{Eq:Model:IS-Full-0-Bulk}) and~(\ref{Eq:Model:IS-Full-2-Shear}), which results in:
\begin{align}
\tau_0\dot{\pi}+\pi &= -3\zeta H,\label{Eq:Model:IS-Tr-0-Bulk} \\
\tau_2\dot{\tau}_{ab}+\tau_{ab} &= -2\eta \sigma_{ab}, \label{Eq:Model:IS-Tr-2-Shear}
\end{align}
while the relations (\ref{Eq:Model:RelaxationTimes}) still hold.
\par
To complete the model, the transient and the transport coefficients must be specified. Following \cite{Hoogen1995} and \cite{Shogin2015}, we assume a barotropic form of the transport coefficients and the relaxation times, the values of the exponents being uniquely determined from dimensional analysis:
\begin{align}
\label{Eq:Model:ViscIndices}
\begin{split}
\zeta & \propto \rho^{1/2}, \qquad \frac{1}{\beta_0} \propto \rho; \\
\eta & \propto \rho^{1/2}, \qquad \frac{1}{\beta_2} \propto \rho.
\end{split}
\end{align}
Note that for a {\it realistic} fluid these coefficients are more complicated functions of the fluid variables and are derived from radiative thermodynamics \cite{Schweizer1982, Udey1982}. The barotropic approximation may be reasonable in some cases, but here we use it purely for the sake of technical convenience.
\par
An artefact of these widely adopted models is the need for a separate equation of state for the temperature of the mathematical fluid. For a {\it physical} radiative fluid, no special "temperature model" is needed, as temperature is already incorporated in the equations of state \cite{Schweizer1982,Udey1982}. The two most common choices are \cite{Maartens1996}:
\begin{enumerate}
\item
The barotropic temperature model, described by
\begin{equation}
T\propto \rho^{(\gamma-1)/\gamma},
\end{equation}
and used as an approximation for radiation-dominated mixtures ($\gamma$ close to $4/3$);
\item
The ideal-gas temperature model, with~$p=nT$,
where~$n$ is the (conserved) number density of the fluid obeying
\begin{equation}
\label{Eq:Model:PNumberCons}
\nabla_\alpha(nu^\alpha)=0.
\end{equation}
In Bianchi type~I spacetimes, tensor equation~(\ref{Eq:Model:PNumberCons}) can be rewritten in terms of scalars as
\begin{equation}
\dot{n}+3Hn=0.
\end{equation}
Note that the physical relevance of this approximation is not proven, with a possible exception for ultra-relativistic and highly relativistic ideal gases.
\end{enumerate}
We use both models in the full version of the IS theory. In the truncated version, however, a particular equation of state for the temperature will not have any influence on the cosmological dynamics, unless a heat-conductive fluid is involved.
\par
The relative dissipative fluxes caused by the bulk and the shear viscous stresses, respectively, are introduced by:
\begin{equation}
x_\pi=\left \vert \frac{\pi}{p} \right \vert, \qquad x_\tau= \frac{{(\tau_{ab}\tau^{ab})}^{1/2}}{p}.
\end{equation}
The underlying assumption of the IS theory is that the fluid is close to LTE, which implies that these dissipative fluxes are small:
\begin{equation}
\label{Eq:Model:Applicability}
x_\pi,\,x_\tau<<1.
\end{equation}
When condition (\ref{Eq:Model:Applicability}) is violated, the IS theory breaks down, and the resulting solutions cannot be considered physically relevant, although they are mathematically consistent and dynamically stable. We do {\it not} conjecture that the IS theory can be "extrapolated" to produce relevant results when applied to fluids far away from LTE; hence, it is extremely important to keep track of the dissipative fluxes in all the solutions obtained using the IS theory.
\par
Finally, it is assumed that the mean interaction time~$t_i$ of the fluid particles is much shorter than the characteristic timescale for macroscopic processes, i.\,e. $t_i<<H^{-1}$, for the hydrodynamic description to be applicable to the
considered matter model.
\section{The system of equations}
\label{Sec:Equations}
We work in the orthornormal frame~\cite{Elst1997} and introduce the scale-independent variables using the common notations of the field, see e.\,g. \cite{Wainwright1997,Ellis2012,Lim2004a}. The normalized energy density~$\Omega$ and the dimensionless geometric shear~$\Sigma_{ab}$ are defined by:
\begin{equation}
\frac{\rho}{3H^2}=\Omega, \qquad \frac{\sigma_{ab}}{H}=\Sigma_{ab}=\text{diag}(-2\Sigma_+,\Sigma_+,\Sigma_+).
\end{equation}
The bulk and the shear viscous stresses, respectively, are normalized by:
\begin{equation}
\frac{\pi}{3H^2}=\Pi, \qquad \frac{\tau_{ab}}{H^2}=\mathcal{T}_{ab}=\text{diag}(-2\mathcal{T}_+,\mathcal{T}_+,\mathcal{T}_+).
\end{equation}
For convenience, we consider Bianchi type~I spacetimes with planar symmetry, for which both~$\Sigma_{ab}$ and~$\mathcal{T}_{ab}$ take the diagonal form. We shall omit the index~"+" in the present paper. Then, the Einstein field equations are\footnote{Primes denote derivation with respect to {\it dimensionless} time $\tilde{t}$, introduced by ${\rm d}\tilde{t}/{\rm d}t=H.$}:
\begin{align}
\Sigma^\prime &= (q-2)\Sigma+\mathcal{T},\\
\Omega^\prime &= (2q+2-3\gamma)\Omega-3\Pi-2\Sigma\mathcal{T},
\end{align}
where
\begin{equation}
q=-\frac{\dot{H}}{H^2}-1=-\frac{H^\prime}{H}-1=2\Sigma^2+\left(\frac{3}{2}\gamma-1 \right)\Omega+\frac{3}{2}\Pi
\end{equation}
is the deceleration parameter. In addition, the Hamiltonian constraint binds~$\Sigma$ and~$\Omega$ algebraically:
\begin{equation}
\label{Eq:Eqs:Hamilton}
1=\Sigma^2+\Omega.
\end{equation}
The system is completed by the dimensionless transport equations, where the bulk and the shear viscosity parameters~$a_0, b_0$ and~$a_2, b_2,$ respectively, are introduced by:
\begin{align}
\begin{split}
\frac{1}{H^2}\cdot\frac{1}{\beta_0} &= a_0 \Omega, \qquad \frac{1}{H}\cdot \frac{1}{\zeta \beta_0}=b_0\sqrt{\Omega},\\
\frac{1}{H^2}\cdot\frac{1}{\beta_2} &= a_2 \Omega, \qquad \frac{1}{H}\cdot \frac{1}{2\eta \beta_2}=b_2\sqrt{\Omega}
\end{split}
\end{align}
and treated as positive constants. The bulk viscosity parameter~$a_0$ is in addition restricted from above by~$a_0<3\gamma(2-\gamma)$, providing that bulk viscous perturbations propagate at finite speeds \cite{Maartens1996}.
\par
The full transport equations depend on the particular choice of a temperature model. We consider the following options:
\begin{enumerate}
\item
For the barotropic temperature model, the full transport equations are
\begin{align}
\Pi^\prime &= \left[ -\frac{3}{2}+\frac{1+q}{\gamma}-b_0\sqrt{\Omega}+\frac{2\gamma-1}{2\gamma}\frac{\Omega^\prime}{\Omega}\right]\Pi-a_0\Omega, \label{Eq:System:Baro1} \\
\mathcal{T}^\prime &= \left[ -\frac{3}{2}+\frac{1+q}{\gamma}-b_2\sqrt{\Omega}+\frac{2\gamma-1}{2\gamma}\frac{\Omega^\prime}{\Omega}\right]\mathcal{T}-2a_2\Sigma.\label{Eq:System:Baro2}
\end{align}
\item
For the ideal-gas temperature model, the full transport equations take the form
\begin{align}
\Pi^\prime &= \left[ -b_0\sqrt{\Omega}+\frac{\Omega^\prime}{\Omega}\right]\Pi-a_0\Omega, \\
\mathcal{T}^\prime &= \left[ -b_2\sqrt{\Omega}+\frac{\Omega^\prime}{\Omega}\right]\mathcal{T}-2a_2\Sigma.
\end{align}
\item
Finally, the transport equations of the truncated IS theory are:
\begin{align}
\Pi^\prime &= \left[2(q+1)-b_0\sqrt{\Omega}\right]\Pi-a_0\Omega, \\
\mathcal{T}^\prime &= \left[2(q+1)-b_2\sqrt{\Omega}\right]\mathcal{T}-2a_2\Sigma.
\end{align}
\end{enumerate}
Note that the evolution equations for the temperature~$T$ and the particle number density~$n$ decouple from the main system. Hence, the dynamics of these fluid variables is trivial and follows directly from the equation of state and/or particle number conservation. This would not be the case in a model incorporating a realistic radiative fluid.
\par
Note also that the energy density~$\Omega$ and the shear stress~$\Sigma$ are bounded variables, as follows from the restriction~$\Omega \geq 0$ and the Hamiltonian constraint~(\ref{Eq:Eqs:Hamilton}). However, the state space as a whole is unbounded, since no mathematical restriction is imposed on~$\Pi$ and~$\mathcal{T}$. The dimension of the physical state space is three for the fluids with both bulk and shear viscosity, and two in case when one of these dissipative mechanisms is not taken into account. The state vector then belongs to a subspace of~$S^2\times \mathbb{R}^2$ or~$S^2\times \mathbb{R}$, respectively.
\par
In each case, we investigate the full system of equations both analytically and numerically. We determine the fixed points of the system and perform an analysis of their {\it local} stability in the future\footnote{All the fixed points of the considered dynamical systems are not listed in this paper; only those which can be locally stable in the future are discussed.}. This is done by standard analytical methods, see e.\,g. \cite{Hoogen1995, Hervik2005, Shogin2014}, while multiple numerical runs at different sets of model parameters and initial conditions are used to make a conjecture about the {\it global} attractor of the system. In all the numerical simulations, the Hamiltonian constraint is chosen to be initially satisfied.
\par
The dissipative fluxes, which determine the applicability of the IS theory, are studied by keeping track of the quantities
\begin{equation}
x=\left \vert \frac{\Pi}{\Omega} \right \vert=\vert \gamma-1 \vert x_\pi, \qquad y= \left \vert \frac{\mathcal{T}}{\Omega} \right \vert=3\vert \gamma-1 \vert x_\tau.
\end{equation}
\section{Solutions for bulk/shear viscous fluids}
\label{Sec:BVSV}
In all the three models under consideration, the only locally stable future attractor is described by
\begin{equation}
\label{Eq:BVSV:Attractor}
[\Sigma,\mathcal{T}, \Omega,\Pi, q]=[0,0,1,\bar{\Pi},\bar{q}],
\end{equation}
with~$\bar{\Pi}<0$. This state describes an izotropizing cosmological model, which is dominated by the bulk viscous fluid at late times. The sign of~$\bar{q}$ is determined by a model-specific relation between~$\gamma$ and the bulk viscosity parameters. Thus, depending on this sign, the spacetime can end up in a state of decelerated, uniform, or accelerated expansion; the latter case represents bulk viscous inflation. Note that anisotropic solutions, which are present in anisotropic cosmological models with non-zero spatial curvature \cite{Hoogen1995,Shogin2015}, are not obtained in Bianchi type~I backgrounds.
\par
While the numerical runs reveal that all the solutions obtained with the full IS equations tend asymptotically to the state~(\ref{Eq:BVSV:Attractor}), the situation is different when the transport equations are truncated. For a wide range of values of the viscosity parameters, the solutions behave unphysically in the truncated IS theory, running into singularities. We consider this in details in section~\ref{Ssec:BVSV:Trunk}.
\par
For the non-singular solutions, the relative dissipative flux caused by the shear viscous stresses decays, while that caused by the bulk viscous stresses tends to a negative constant value. In general, this value is not small. Thus, the bulk viscosity can, and generally {\it does}, prevent the fluid from approaching LTE at late times, which is quite typical for similar systems where this dissipative mechanism is involved~\cite{Shogin2015}.
\subsection{The full transport equations, barotropic temperature}
The constants in~(\ref{Eq:BVSV:Attractor}) are given by
\begin{align}
\begin{split}
\label{Eq:BVSV:Constants:Baro}
\bar{\Pi} &= \frac{1}{3}\left[ \gamma b_0 -\sqrt{\gamma^2 b_0^2+6\gamma a_0}\right],\\
\bar{q} &= \frac{1}{2} \left[ 3\gamma-2+\gamma b_0 -\sqrt{\gamma^2 b_0^2+6\gamma a_0}\right].
\end{split}
\end{align}
Calculating the eigenvalues corresponding to the state~(\ref{Eq:BVSV:Attractor}) yields
\begin{align}
\begin{split}
\lambda_1 &= -\frac{\sqrt{\gamma^2 b_0^2+6\gamma a_0}}{\gamma},\\
\lambda_{2,3} &= \frac{3\bar{\Pi}(\gamma+1)+3\gamma(\gamma-2)-2\gamma b_2 \pm \sqrt{\left[3\bar{\Pi}(\gamma-1)+3\gamma(\gamma-2)+2\gamma b_2 \right]^2-32\gamma^2 a_2}}{4},
\end{split}
\end{align}
the rational part of~$\lambda_{2,3}$ being negative. As~$\lambda_1$ is negative, the local stability of the future attractor requires~$\text{Re}(\lambda_{2,3})<0.$ This yields
\begin{equation}
a_2>-\frac{3}{8\gamma}(3\bar{\Pi}-2\gamma b_2)(\bar{\Pi}+\gamma-2),
\end{equation}
which is fulfilled, since the right-hand side is negative.
\subsection{The full transport equations, ideal-gas temperature}
In this case, the future attractor~(\ref{Eq:BVSV:Attractor}) is specified by
\begin{align}
\begin{split}
\label{Eq:BVSV:constants:IG}
\bar{\Pi} &= -\frac{a_0}{b_0},\\
\bar{q} &= \frac{3}{2}\left(\gamma-\frac{2}{3}-\frac{a_0}{b_0}\right),
\end{split}
\end{align}
and the corresponding eigenvalues are given by
\begin{align}
\begin{split}
\lambda_1 &= -b_0,\\
\lambda_{2,3} &=\frac{3(\bar{\Pi}+\gamma-2)-2b_2\pm \sqrt{\left[3(\bar{\Pi}+\gamma-2)+2b_2 \right]^2-32a_2} }{4}.
\end{split}
\end{align}
Again, $\lambda_1$ is negative; the future stability requires~$\text{Re}(\lambda_{2,3})<0$, which leads to
\begin{equation}
a_2>\frac{3}{4}b_2(\bar{\Pi}+\gamma-2).
\end{equation}
The right-hand side is negative, while~$a_2>0$. The future attractor is locally stable for the whole range of values of the model parameters.
\subsection{The truncated transport equations}
Now the constants in~(\ref{Eq:BVSV:Attractor}) are
\label{Ssec:BVSV:Trunk}
\begin{align}
\begin{split}
\label{Eq:BVSV:constants:Trunk}
\bar{\Pi} &= \frac{1}{6}\left[-3\gamma+b_0-\sqrt{(3\gamma-b_0)^2+12a_0}\right],\\
\bar{q} &= -1+\frac{1}{4}\left[3\gamma+b_0-\sqrt{(3\gamma-b_0)^2+12a_0}\right],
\end{split}
\end{align}
the corresponding eigenvalues being
\begin{align}
\begin{split}
\lambda_1 &= -\sqrt{(3\gamma-b_0)^2+12a_0},\\
\lambda_{2,3} &= \frac{3(3\bar{\Pi}+3\gamma-2)-2b_2\pm \sqrt {\left[ 3(\bar{\Pi}+\gamma+2-2b_2) \right]^2-32a_2} }{4}.
\end{split}
\end{align}
The local stability of the future asymptotic state requires~$\text{Re}(\lambda_{2,3})<0;$ the solution of this inequality yields
\begin{align}
\begin{split}
\label{Eq:BVSV:Trunk:Stability}
b_2 &> \frac{3}{2}(3\gamma-2)+9\bar{\Pi},\\
a_2 &> \frac{3}{4} \left[ b_2(\bar{\Pi}+\gamma-2)+6(\bar{\Pi}+\gamma)-3(\bar{\Pi}+\gamma)^2 \right].
\end{split}
\end{align}
The shear viscosity coefficients turn out to play a crucial role in the truncated IS theory. For a wide range of values of these parameters, the requirements~(\ref{Eq:BVSV:Trunk:Stability}) are not fulfilled. In this case, no fixed point of the system is locally stable in the future; numerical simulations show that all the solutions obtained end up in a singularity. The nature of this singularity is that the energy density variable~$\Omega$ crosses the vacuum boundary~($\Omega=0$) and becomes negative. The transport equations are non-defined in this region, and the solutions break down. This is not found to happen in the full IS theory, but singularities of the same type have been reported in more advanced Bianchi models with a dissipative mathematical fluid described by the truncated IS theory~\cite{Shogin2015}.
\begin{figure}[ht!]
\begin{minipage}[ht!]{0.45\linewidth}
\includegraphics[width=0.8\linewidth]{PT-Omega-VB.eps}
\end{minipage}
\hfill
\begin{minipage}[ht!]{0.45\linewidth}
\includegraphics[width=0.8\linewidth]{PT-Sigma-VB.eps}
\end{minipage}
\caption{The dynamics of~$\Omega(\tau)$ (left) and~$\Sigma(\tau)$ (right) in a Bianchi type~I cosmological model with a bulk/shear viscous fluid described by the truncated Israel-Stewart theory. The plots illustrate that the dynamical behaviour of the solutions depends strongly on the initial conditions.\\
At~$\Sigma(0)=-0.2,~\Omega(0)=0.96$ (solid lines), the solution asymptotically approaches the state~(\ref{Eq:BVSV:Attractor}), as might be expected. At~$\Sigma(0)=-0.5,~\Omega(0)=0.75$ (dashed lines), the solution ends up in a singularity, as the energy density variable crosses the vacuum boundary ($\Omega=0$). \\
The model parameters are~$\gamma=6/5,~ a_0=0.1,~b_0=5.0,~a_2=0.5,~b_2=3.0$, so the inequalities~(\ref{Eq:BVSV:Trunk:Stability}) are satisfied. }
\label{Fig:PT}
\end{figure}
\par
Moreover, the system with the truncated transport equations is found to be sensitive to alterations of the initial conditions. That is, it is possible to obtain singular solutions even if the viscosity parameters {\it do} satisfy the inequalities~(\ref{Eq:BVSV:Trunk:Stability}). An example demonstrating this feature is shown in Figure~\ref{Fig:PT}.
\par
For the cosmological model considered, such singular solutions do not exist in the full IS theory. Thus, it is the truncation of the IS equations which allows the energy density to manifest unphysical behaviour.
\section{Solutions for bulk viscous fluids with vanishing shear viscosity}
\label{Sec:BVonly}
In the absence of shear viscosity the future attractor retains the form~(\ref{Eq:BVSV:Attractor}), assumed~$\mathcal{T}\equiv 0$. The expressions for $\bar{\Pi}$ and~$\bar{q}$, obtained in section~\ref{Sec:BVSV}, also hold in this case. However, the instabilities of the truncated IS theory described in section~\ref{Ssec:BVSV:Trunk} are removed.
\par
Although the future attractor is essentially the same, the dynamics of the approach to it is changed, as the corresponding eigenvalues are different from those calculated in section~\ref{Sec:BVSV}.
\par
It turns out that the shear viscous stresses play only a minor role in the full IS theory, provided the bulk viscosity is non-zero; namely, they affect neither the future asymptotic state nor its stability. On the other hand, the role of shear viscosity in the truncated version of the IS theory is crucial, which is consistent with the results of \cite{Hoogen1995}.
\par
The relative dissipative flux caused by the bulk viscous stresses tends to a finite constant at late times: in general, the fluid does not asymptotically approach LTE.
\subsection{The full transport equations, barotropic temperature}
The future asymptotic values of~$\Pi$ and $q$ are given by~(\ref{Eq:BVSV:Constants:Baro}); the corresponding eigenvalues are
\begin{equation}
\lambda_1 = -\frac{\sqrt{\gamma^2b_0^2+6\gamma a_0}}{\gamma}, \qquad \lambda_2 = \frac{3}{2}\left[ \bar{\Pi}+(\gamma-2) \right].
\end{equation}
Since~$\bar{\Pi}<0$ and~$\gamma<2$, both eigenvalues are negative reals, and the asymptotic state is locally stable in the future.
\subsection{The full transport equations, ideal-gas temperature}
The values of~$\bar{\Pi}$ and~$\bar{q}$ are provided by~(\ref{Eq:BVSV:constants:IG}), the eigenvalues being
\begin{equation}
\lambda_1 = -b_0, \qquad \lambda_2 = \frac{3}{2}\left[ \bar{\Pi}+(\gamma-2) \right].
\end{equation}
Again, the eigenvalues are negative real numbers, and the future attractor is locally stable.
\subsection{The truncated transport equations}
The constants in~(\ref{Eq:BVSV:Attractor}) are given by~(\ref{Eq:BVSV:constants:Trunk}). The corresponding eigenvalues are
\begin{equation}
\lambda_1 = -\sqrt{(3\gamma-b_0)^2+12a_0}, \qquad \lambda_2 = \frac{3}{2}\left[ \bar{\Pi}+(\gamma-2) \right].
\end{equation}
In contrast to the case of section~\ref{Ssec:BVSV:Trunk}, the eigenvalues are now real and negative. The instability of the future attractor is removed. The singular solutions, which appear in the case of nonvanishing shear viscosity, cannot be obtained in this case.
\par
Hence, the pathological dynamical features of solutions in the truncated IS theory originate from the evolution of the shear viscous stresses.
\section{Solutions for shear viscous fluids with vanishing bulk viscosity}
\label{Sec:SVonly}
The only future stable stationary point of the system is given by
\begin{equation}
\label{Eq:SV:Attractor}
[\Sigma, \mathcal{T}, \Omega, q]=[0,0,1,\bar{q}],
\end{equation}
with
\begin{equation}
\bar{q}=\frac{1}{2}(3\gamma-2).
\end{equation}
The asymptotic value of the deceleration parameter depends on~$\gamma$ only and is positive for typical $\gamma$-fluids. Hence, the shear viscosity does not contribute to accelerating the expansion of the universe; however, spatial anisotropy is eliminated at late times.
\par
For the full IS transport equations, the eigenvalues are either negative reals or complex conjugates with a negative real part; this provides the local stability of the future attractor. For the truncated transport equations, the situation is different: the negativity of~$\text{Re}(\lambda_{1,2})$ can be violated, which makes the state~(\ref{Eq:SV:Attractor}) unstable in the future. The corresponding solutions are found to end up in a singularity, which is discussed in section~\ref{Ssec:SV:Trunk} below.
\par
For all the non-singular solutions, the relative dissipative fluxes caused by the shear viscous stresses decay in the future, and the fluid approaches LTE at late times. This is the only case in the considered cosmological model when the underlying assumption~(\ref{Eq:Model:Applicability}) of the IS theory is not violated and the solutions of the system can be considered fully reasonable.
\subsection{The full transport equations}
The eigenvalues corresponding to the state~(\ref{Eq:SV:Attractor}) are the same for both temperature models:
\begin{equation}
\lambda_{1,2}=\frac{(3\gamma-2b_2-6) \pm \sqrt{(3\gamma+2b_2-6)^2-32a_2}}{4},
\end{equation}
where the rational part is negative. The local stability requirement~$\text{Re}(\lambda_{1,2})<0$ yields
\begin{equation}
a_2>\frac{3}{4}b_2(\gamma-2),
\end{equation}
which is fulfilled automatically by~$a_2>0$ and~$\gamma<2$.
\subsection{The truncated transport equations}
\label{Ssec:SV:Trunk}
For the truncated transport equations, the eigenvalues are given by
\begin{equation}
\lambda_{1,2}=\frac{ (9\gamma-2b_2-6)\pm \sqrt{(3\gamma-2b_2+6)^2-32a_2}}{4}.
\end{equation}
An algebraic analysis of the local stability requirement~$\text{Re}(\lambda_{1,2})<0$ results in the following system of inequalities:
\begin{align}
\begin{split}
\label{Eq:SV:Restrict}
b_2 &> \frac{3}{2}(3\gamma-2),\\
a_2 &> \frac{3}{4}(\gamma-2)(b_2-3\gamma).
\end{split}
\end{align}
If the values of~$a_2$ and~$b_2$ do not satisfy these conditions, the corresponding solutions behave unphysically and end up in a singularity of the same nature as described in section~\ref{Ssec:BVSV:Trunk}. Moreover, similarly to the case with nonvanishing bulk and shear viscosity, the stability of the future attractor depends crucially on the initial conditions; it is possible to obtain singular solutions, even if the inequalities~(\ref{Eq:SV:Restrict}) are satisfied. As might be expected, this is not a property of the system with the full transport equations.
\par
This confirms our assumption that it is the evolution equations for the shear viscous stresses that lead to unacceptable properties of the solutions obtained using the truncated IS theory.
\section{Conclusions}
\label{Sec:Conslusions}
We have used the dynamical systems approach to investigate the future attractors and their stability conditions for viscous mathematical fluids in Bianchi type~I spacetimes. We have studied the properties of cosmological solutions obtained with the full and the truncated versions of the IS~theory, having used two simple temperature models in the full version. Also, we have determined the asymptotic future of the relative dissipative fluxes to find out when the near-equilibrium conditions are violated and the IS~theory breaks down.
\par
All the solutions obtained using the full IS~theory are found to be non-singular in future and stable under alterations of the initial conditions. On the contrary, the truncated IS transport equations, if applied to a fluid with nonvanishing shear viscosity, allow the energy density to cross the vacuum boundary, which leads to unphysical, singular behaviour of the solutions already in the simplest anisotropic spacetimes.
\par
Another pathological feature of the truncated IS equations discovered in the present work is the extreme sensitivity of the resulting system to the choice of the initial conditions. When the truncated IS theory is applied to a $\gamma$-fluid with nonvanishing shear viscosity, the solutions can run into a singularity even if there exists a stable future asymptotic state.
\par
The solutions of the full IS theory describe an izotropizing universe dominated by the dissipative fluid in the asymptotic future. The shear viscous stresses, if present, decay at late times, while the bulk viscous stress, if present, freezes into a negative constant value. In these solutions, the bulk viscosity eliminates spatial anisotropy and, in addition, can accelerate the expansion of the universe. The shear viscosity does not contribute to accelerating the expansion, but effectively eliminates the Bianchi type~I anisotropy even when bulk viscosity is zero. Anisotropic solutions, which exist in more complicated spacetimes~\cite{Hoogen1995,Shogin2015}, cannot be obtained in Bianchi type~I cosmological models.
\par
The full IS theory provides a completely reasonable description {\it only} for shear viscous mathematical fluids with vanishing bulk viscosity: the relative dissipative flux caused by the shear viscous stresses decays exponentially in the future, and the fluid is driven towards LTE.
\par
The full IS theory yields stable solutions for bulk viscous fluids. However, these solutions are not fully consistent with the underlying assumptions of the IS theory, since generally the fluid neither is close to LTE during its evolution nor approaches thermal equilibrium in the asymptotic future. The near-equilibrium conditions can be violated by the bulk viscosity: the corresponding relative dissipative flux is asymptotically a negative constant, which is not small in general. This departure from LTE is, of course, a physical result, which may depend on the fluid model adopted, and per se not a failure of the IS theory.
\par
Large deviations from LTE caused by the shear viscous stresses, which have been discovered in more general cosmological models~\cite{Shogin2015}, are not present in Bianchi type~I spacetimes.
\par
The future asymptotic states of the Bianchi type~I spacetimes are of similar form for the two temperature models considered in the full IS~theory. Still, the solutions can behave differently in the future; for example, these models under the same initial conditions can result in opposite signs of the asymptotic value of the deceleration parameter~$q$. Also, the dynamical character of the solutions (monotoneous or oscillatory behaviour) can differ between them.
\par
The results obtained suggest that in solving real physical problems, the full IS theory should be preferred over its truncated version. For a mathematical fluid with barotropic transport coefficients, the bulk viscosity creates finite, non-decaying relative dissipative fluxes, which lead to a possible breakdown of the full IS theory; hence, consistent non-linear thermodynamical theories, which can describe fluids substantially away from LTE, are potentially important. However, for realistic radiative fluids with transient and transport coefficients derived from the kinetic theory, the dynamics of the variables may be essentially different. We leave considering physical fluids in anisotropic cosmological backgrounds for the further investigations.
|
{
"timestamp": "2016-07-08T02:10:09",
"yymm": "1504",
"arxiv_id": "1504.03472",
"language": "en",
"url": "https://arxiv.org/abs/1504.03472"
}
|
\section{Introduction}
Kinetic schemes are widely used for studying the thermodynamic, dynamic, and stochastic
properties of macromolecules
\cite{Jackson}.
These schemes are usually selected to be as simple as possible, such as
the 2-state schemes for the bound and unbound states of enzymes or receptors
and the open and closed states of ion channels.
Nevertheless, they can also be rather sophisticated (e.g., 8-state
inositol trisphosphate receptors \cite{Fall}, the 10-state hemoglobin \cite{Blatz}, and the 56-state chloride channels \cite{Blatz}).
The selection of kinetic schemes is mainly determined by the desired accuracy and the measurable quantities \cite{Hille,Keizer}.
Since a low-dimensional scheme can usually be contracted from higher-dimensional ones,
there exists a cascade of hierarchical Markovian network models suitable for
describing the time evolution of the populations of a macromolecule's functional states \cite{Noe}.
These networks are anticipated to have indistinguishable kinetics,
exhibiting identical mean trajectories after being projected to the low-dimensional network space.
However, models with indistinguishable means do not necessarily have indistinguishable fluctuations.
A question that arises is that which schemes will give more relevant fluctuations to a real system and under which conditions unique fluctuation features can be obtained
from different levels of contracted schemes?
These issues are essential for the reliability of various biological properties derived in terms of the fluctuations of a selected kinetic scheme, such as chemoreception \cite{Bialek_1,Wolde}, membrane conductance \cite{Chen5}, and ion channel density \cite{Sigworth}.
The inter-network fluctuation relations arise from a comparison between different coarse-grained dynamical systems.
It resembles the comparison between different rate equations in the lumping analysis,
widely used in systems biology and general chemical engineering
\cite{Wei_1,Li_1,Toth,Okino,Gorban}.
A central issue in that analysis is finding the lumping conditions for eliminating unimportant events
or time scales in a large network, of typically over $10^{4}$ species in systems biology, to reduce its complexity \cite{Liao}.
Interestingly, this contraction is mathematically analogous to merging experimentally indistinguishable states to obtain
simple transition networks for the conformational change of a macromolecule.
For instance, the Hodgkin-Huxley potassium ion channel has $16$ configurations depending on whether its individual four gates are open or closed \cite{Hille}.
However, this channel is often regarded as a 2-state system,
described by whether or not ions can pass through it in a patch-clamp recording.
The contraction from a 16-state to a 2-state model is
because the gating current recording is incapable of resolving the detailed structure of the channel configuration.
In terms of lumping analysis, this contraction is an approximate lumping \cite{Wei_2}.
Despite that correspondence, the original lumping analysis focuses on the relations between mean dynamics and is not concerned with fluctuations.
To extract this stochastic component, we generalize the lumping theory from original rate equations (RE) to chemical master equations (CME) and stochastic differential equations (SDE) and study kinetically equivalent (KE) and thermodynamically equivalent (TE) hierarchical kinetic schemes, under intrinsic and extrinsic noises.
The results go beyond the conventional assumption of ``fast relaxations" and contribute to our understanding of why a kinetic system can be contracted.
In the case of extrinsic noise, different kinetic schemes can give different fluctuations even when their average trajectories are the same.
This opens a possibility of identifying a correct kinetic model by observing fluctuations.
Notably, lumping conditions here are used for generating complex KE or TE networks from simple networks, in opposite to their original goal of reducing complex networks to simple networks.
Furthermore, for the conformational change of macromolecules discussed below, it is sufficient to focus on linear REs and linear lumping transformations.
\section{Lumping rate equations}
Let system $A$ be an $n$-dimensional kinetic scheme
described by the linear RE,
\begin{equation}\label{}
\frac{d{\bf N}}{dt}={\bf MN}\;\;
\mbox{ or }\;\; \frac{dN_i}{dt}=\sum_{j=1}^n k_{ji}N_j-k_{ij}N_i,
\end{equation}
where $N_i$ is the population of the $i$-th state and may represent the mean dynamics of some stochastic processes discussed later, ${\bf M}$ denotes the matrix of rate constants $k_{ij}$ from states $i$ to $j$, with $k_{ii}\equiv 0$,
and ${\bf N}\equiv[N_1,N_2,...,N_n]^T$ represents a state vector, in which the superscript $T$ stands for
the transpose of a vector.
If ${\bf U}$ is an $n'\times n$ full rank lumping matrix ($n'<n$), ${\bf N}$ can be contracted into
an $n'$-dimensional vector ${\bf N'}=[N'_1,N'_2,...,N'_{n'}]^T$ via
\begin{equation}\label{}
{\bf N'}={\bf U}{\bf N},
\end{equation}
which is the state vector of some reduced system $A'$.
If each column of ${\bf U}$ is a standard unit vector, ${\bf U}$ denotes a proper lumping (see the example in S1 \cite{Supplemental}).
Since all lumpings in the following discussions are ``proper," this term will be neglected below.
The RE which ${\bf N'}$ satisfies is generally an integral-differential equation
with a memory kernel \cite{Keizer}.
If that kernel vanishes, the RE has a simple autonomous form as (1),
\begin{equation}\label{}
\frac{d{\bf N'}}{dt}={\bf M'N'}\;\;
\mbox{ or }\;\; \frac{dN'_a}{dt}=\sum_{b=1}^n k'_{ba}N'_b-k'_{ab}N'_a,
\end{equation}
with $k_{aa}'\equiv 0$, and network $A$ is called ``exactly lumpable.''
Exact lumping makes the contracted system of an autonomous system
again autonomous, self-contained, and not having a memory kernel.
If the memory kernel does not vanish but is small, $A$ is called ``approximately lumpable,''
which has a broad practical application \cite{Wei_2}.
Exact lumping is the limiting case of all approximate lumpings when the memory effect tends to zero.
Equations (2) and (3) together constitute
the KE condition between $A$ and $A'$, or the condition for which $A$ can be exactly lumped into $A'$.
Notice that (2) alone is insufficient for this condition,
because any ${\bf U}$ can lump ${\bf N}$ into some ${\bf N'}$, which is not necessarily self-contained.
Quantitatively, the KE condition between $A$ and $A'$ can be expressed by their rate constant matrices
\begin{equation}\label{}
{\bf UM}={\bf M'U},
\end{equation}
which implies ${\bf U}e^{{\bf M}t}=e^{{\bf M'}t}{\bf U}$ \cite{Wei_1}.
When ${\bf U}$ is used to lump $A$ into $A'$, the $n$ states in $A$ are first partitioned into $n'$
sets $S_a$, with $a = 1, ..., n'$, by the row vectors of ${\bf U}$ (see S1 \cite{Supplemental}).
Then, all states in $S_a$ are merged as the state $a$ in $A'$ and termed ``the internal states" of $a$.
Using the same procedure to merge all states in $S_a$ on both sides of (1),
one obtains the KE condition in terms of rate constants
\begin{equation}
k'_{ab}=\sum_{j\in S_b}k_{ij},
\end{equation}
for any $a$, $b\in\{1,2,...,n'\}$ with $a\neq b$ and any $i\in S_a$,
in analogy to that known for finite Markov chains \cite{Kemeny}.
Notice that the KE condition is fulfilled only when (5) is satisfied for all $i\in S_a$.
In brief, the KE condition can be expressed as (4) or (5), or equivalently as (2) together with (3).
Since (5) does not demand fast relaxations between the internal states in $S_a$,
the existence of fast variables or large $k_{ij}$ is not the prerequisite for exact lumpability.
However, lumping analysis can also eliminate fast variables,
as the quasi-equilibrium or quasi-steady-state approximations do \cite{Okino, Liao}.
Given a ${\bf U}$, whether $A$ described by (1) can be exactly lumped into $A'$ by ${\bf U}$
is decided by whether $A'$ has an autonomous RE (3), as discussed above.
If two autonomous $A$ and $A'$ are given first instead, whether $A$ can be lumped into $A'$ is decided by whether
some ${\bf U}$ can be found to connect them by (5).
If such ${\bf U}$ exists, ${\bf N'}$ of $A'$ and ${\bf N}$ of $A$ are indistinguishable,
in that the trajectories ${\bf N'}$ and ${\bf UN}$ are identical.
\section{Lumping master equations}
To extract the fluctuation relations of intrinsic noises between
hierarchical networks, we extend the lumping analysis from the RE (1) to its CME.
Suppose a macromolecule has $n$ conformational states whose
transition network $A$ obeys the kinetic equation (1).
If a system consists of $N$ macromolecules, its CME \cite{Hill_2,Chen_1},
\begin{eqnarray}\label{}
\frac{d{\bf P}}{dt}&=&{\bf LP} \mbox{ or }\\
\frac{dP_{\bf \tilde{N}}(t)}{dt}&=&\sum_{i,j=1}^nk_{ij}\left[(\tilde{N}_i+1)P_{{\bf \tilde{N}}-\boldsymbol{\omega}_{ij}}(t)-\tilde{N}_iP_{\bf \tilde{N}}(t)\right], \nonumber
\end{eqnarray}
describes the evolution of the joint probability $P_{\bf \tilde{N}}(t)$ of finding the state vector
${\bf \tilde{N}}\equiv[\tilde{N}_1,\tilde{N}_2,...,\tilde{N}_n]^T$ at time $t$,
where $\tilde{N}_i\geq 0$ is the number of macromolecules in the $i$-th state and $\sum_{i=1}^n \tilde{N}_i= N$.
Therein, ${\bf \tilde{N}}$ is related to the ${\bf N}$ in (1) by
$\sum_{{\bf \tilde{N}}} \tilde{N}_i P_{\bf \tilde{N}}(t)=N_i$,
where the sum runs over all accessible ${\bf \tilde{N}}$.
The vector $\boldsymbol{\omega}_{ij}$ has values $-1$ and $+1$ in its $i$-th and $j$-th components, respectively,
and $0$ elsewhere.
It stands for the change of molecule numbers in different states during the reaction
shifting one molecule from $i$ to $j$.
Notice that ${\bf P}$ is a vector whose ``${\bf \tilde{N}}$-th" component is the probability $P_{\bf \tilde{N}}(t)$,
just as ${\bf N}$ in (1) is a vector whose $i$-th component is $N_i$.
For each lumping matrix ${\bf U}$, which contracts ${\bf N}$ of $A$ into ${\bf N'}={\bf U}{\bf N}$ of $A'$,
there exists an associated lumping operator ${\bf \widehat{U}}$,
which contracts ${\bf P}$ into a reduced vector
\begin{equation}\label{}
{\bf P'}={\bf \widehat{U}}{\bf P},
\end{equation}
whose ${\bf \tilde{N}'}$-th component is (see S2 \cite{Supplemental})
\begin{equation}\label{}
P'_{\bf \tilde{N}'}(t)=\sum_{{\bf \tilde{N}}}P_{\bf \tilde{N}}(t)\prod_{c=1}^{n'}
\delta\left(\tilde{N}_c'-\sum_{k\in S_c}\tilde{N}_k\right),
\end{equation}
where sets $S_c$ are partitioned by ${\bf U}$ as explained in the text that follows (4)
and $\delta(X'-X)$ is a Kronecker delta whose value is one when $X'=X$ and zero elsewhere.
If ${\bf U}$ is arbitrary, ${\bf N'}$ does not necessarily obey a simple RE as (3) and
$P'_{\bf \tilde{N}'}(t)$ does not necessarily satisfy any CME of the same form as (6).
However, if ${\bf U}$ can exactly lump $A$ into $A'$, ${\bf N'}$ does follow (3) and
$P'_{\bf \tilde{N}'}(t)$ indeed obeys a simple lumped CME
\begin{eqnarray}\label{}
\frac{d{\bf P'}}{dt}&=&{\bf L'P'} \mbox{ or } \\
\frac{dP'_{\bf \tilde{N}'}(t)}{dt}&=&\sum_{a,b=1}^{n'} k'_{ab}\left[(\tilde{N}'_a+1)P'_{{\bf \tilde{N}'}-\boldsymbol{\omega}_{ab}}(t)-\tilde{N}'_aP'_{\bf \tilde{N}'}(t)\right], \nonumber
\end{eqnarray}
which turns out to be the CME of $A'$ (see S2 \cite{Supplemental}).
Alternatively, suppose the REs of $A$ and $A'$ are (1) and (3) and some ${\bf U}$ can exactly lump $A$ into $A'$ through (2).
Then their ${\bf P}$ and ${\bf P'}$ in (6) and (9) are related by (7) and thus indistinguishable from each other,
which is the exact lumpability in terms of joint probabilities.
Just as (2) and (3) form the KE condition between two REs, (7) and (9) constitute the KE condition on the level of CME.
With the same argument as for (4), the lumping condition for the CME is
\begin{equation}\label{}
{\bf \widehat{U}}{\bf L}={\bf L'}{\bf \widehat{U}}.
\end{equation}
Notably, the exactly lumped CME (9) via the KE condition is distinct from the reduced CME entirely based
on the time scale separation \cite{Roussel_2}.
The above argument indicates that the exact lumpability in RE (1) implies the exact lumpability in its CME (6)
and vice versa (see S2 \cite{Supplemental}).
Therefore, the KE condition is a rather strong condition for systems under intrinsic noises.
It not only conveys the original meaning of identical first moments, ${\bf UN}$ and ${\bf N'}$,
but also the identities of all other moments, owing to the identity of probabilities, ${\bf \widehat{U}}{\bf P}={\bf P'}$, (see (2.13) in S2 \cite{Supplemental}).
Physically it indicates that experimentally measured fluctuations cannot be used for
judging whether a state has internal states,
if the fluctuations are caused by small numbers of macromolecules.
Among all moments, of special interest are the indistinguishable second moments,
\begin{equation}\label{}
\boldsymbol{\tilde{\sigma}'}={\bf U}\boldsymbol{\tilde{\sigma}}{\bf U}^T,
\end{equation}
where
$\tilde{\sigma}_{ij}\equiv\langle\delta N_i\delta N_j\rangle$
($\tilde{\sigma}'_{ab}\equiv\langle \delta N'_a\delta N'_b\rangle$)
is an average over the probability $P_{\bf \tilde{N}}(t)$ ($P'_{\bf \tilde{N}'}(t)$)
and $\delta N_i=\tilde{N}_i-N_i$ ($\delta N_a'=\tilde{N}_a'-N_a'$) is the fluctuation
around the mean $N_i$ of $\tilde{N}_i$ ($N'_a$ of $\tilde{N}_a'$) defined in (6).
In Fig. 1, the indistinguishable
variances,
$\tilde{\sigma}_{ii}$ and ${\bf \tilde{\sigma}'}_{aa}$, induced by the intrinsic noises of two KE networks,
are numerically confirmed.
Besides the strict KE condition, a kinetic scheme may be selected merely because it is TE to the real system
\cite{Wales}.
If two kinetic networks $A$ and $A'$ are TE to each other,
the stationary states ${\bf N'}^s$ and ${\bf N}^s$ of their REs are related
by ${\bf N'}^s={\bf UN}^s$ via some lumping matrix ${\bf U}$.
The stationary solution of the CME is the multinomial distribution,
\begin{equation}\label{}
P^s_{\bf \tilde{N}}=\frac{N!}{\prod_{i=1}^n\tilde{N}_i!}\prod_{j=1}^n\left(\frac{N_j^s}{N}\right)^{\tilde{N}_j},
\end{equation}
where $N_i^s$ is the $i$-th component of ${\bf N}^s$ \cite{Hill_2}.
Let ${\bf P}^s$ be the vector whose ${\bf \tilde{N}}$-th component is the ${P}^s_{\bf \tilde{N}}$ of $A$ and
${\bf P'}^s$ be the vector whose ${\bf \tilde{N}'}$-th component is the ${P'}^s_{\bf \tilde{N}'}$ of a TE system $A'$
of $A$.
One can show that ${\bf P}^s$ and ${\bf P'}^s$ are related by (see S3 \cite{Supplemental})
\begin{equation}\label{}
{\bf P'}^s={\bf \widehat{U}}{\bf P}^s
\end{equation}
irrespective of whether $A'$ is KE to $A$ or not.
More precisely, (13) is sufficient and necessary for the TE condition ${\bf N'}^s={\bf UN}^s$,
or is the TE condition on the level of stationary joint probability (see S3 \cite{Supplemental}).
While under the KE condition the contracted probability $P'_{\bf \tilde{N}'}(t)$ must satisfy
(9) at any $t$, under the TE condition it must only obey the form (12)
at $t=\infty$.
An arbitrary network $A$ does not always have a reduced KE system.
However, it usually has infinitely many reduced TE systems $A'$'s, which are TE to one another.
An interesting indication from (7) and (13) is that
if $A$ and $A'$ are TE, but not KE, to each other, their initially distinguishable ${\bf P}$ and ${\bf P'}$
will become indistinguishable as $t\rightarrow\infty$, irrespective of which ${\bf U}$ is
used to contract $A$ to $A'$ (Fig. 2).
Therefore, the lumpability between the probabilities of TE systems is similar to the Lyapunov function
for quantifying entropy production, where Kullback-Leibler divergence may be a proper lumpability measure.
\section{Lumping stochastic differential equations}
Another frequently used approach for exploring fluctuations is the SDE,
\begin{equation}\label{}
\frac{d{\bf {\hat N}}}{dt}={\bf M}{\bf {\hat N}}+{\bf f},
\end{equation}
where ${\bf {\hat N}}={\bf N}+\delta {\bf N}$ is a real-valued random variable with the fluctuations $\delta {\bf N}$
about the ensemble mean ${\bf N}$, which satisfies a deterministic equation as (1).
Here, ${\bf f}$ is a Gaussian white noise with $\langle {\bf f}(t)\rangle={\bf 0}$,
$\langle {\bf f}(t'){\bf f}^T(t)\rangle={\bf \Gamma}\delta(t-t')$,
and $\langle {\bf f}(t'){\bf \hat{N}}^T(t)\rangle={\bf 0}$ for $t<t'$,
where the covariance matrix ${\bf \Gamma}$ is symmetric, positive semi-definite, and generally time-dependent.
The solution of (14), ${\bf \hat{N}}(t)=e^{{\bf M}t}{\bf \hat{N}}(0)+\int_0^te^{{\bf M}\tau }{\bf f}(t-\tau )\,d\tau$,
is also a Gaussian random variable.
The conditional covariance of $\delta{\bf N}$ is
$\boldsymbol{\sigma}\equiv\left\langle\delta{\bf N}\delta{\bf N}^T\right\rangle
=\int_0^t e^{{\bf M}\tau }{\bf \Gamma} \left(e^{{\bf M}\tau }\right)^T\,d\tau$,
which is symmetric and has the time derivative
$d{\boldsymbol{\sigma}}/dt={\bf M}{\boldsymbol{\sigma}}+{\boldsymbol{\sigma}}{\bf M}^T
+{\bf \Gamma}$.
This equation is reduced to the fluctuation-dissipation theorem (FDT) when the system reaches equilibrium
as $t\rightarrow\infty$, where $d{\boldsymbol{\sigma}}/dt$ vanishes \cite{Keizer}.
If ${\bf f}$ represents an intrinsic noise, $\boldsymbol{\sigma}$ will be the
$\boldsymbol{\tilde{\sigma}}$ in (11), when the system is close to the thermodynamic limit.
Together with the given ${\bf M}$ it uniquely determines ${\bf \Gamma}$ via the FDT.
The ${\bf \Gamma}$ in the chemical Langevin equation in Ref. \cite{Gillespie1} belongs to this category.
If ${\bf f}$ is an extrinsic noise, $\boldsymbol{\sigma}$ and ${\bf \Gamma}$ can be freely tuned as long as they
comply with the FDT.
Let $A$ and $A'$ be two network models approaching a real system,
where $A$ is described by (14) and $A'$ satisfies
\begin{equation}\label{}
\frac{d{\bf \hat{N}'}}{dt}={\bf M'}{\bf \hat{N}'}+{\bf f'}.
\end{equation}
Here ${\bf {\hat N}'}={\bf N'}+\delta {\bf N'}$ and ${\bf f'}$ has statistical properties analogous to ${\bf f}$.
If $A$ and $A'$ are KE to each other, they are connected by some ${\bf U}$ via ${\bf N'}={\bf UN}$
(notably not ${\bf \hat{N}'}={\bf U\hat{N}}$).
The covariance of the fluctuations of ${\bf U\hat{N}}$ is
${\bf U}\boldsymbol{\sigma}{\bf U}^T=\left\langle{\bf U}\,\delta{\bf N}\,\delta{\bf N}^T{\bf U}^T\right\rangle
=\int_0^t {\bf U}e^{{\bf M}\tau }{\bf \Gamma} \left(e^{{\bf M}\tau }\right)^T{\bf U}^T\,d\tau
=\int_0^t e^{{\bf M'}\tau }{\bf U}{\bf \Gamma}{\bf U}^T \left(e^{{\bf M'}\tau }\right)^T\,d\tau$,
where the exchange relation implied by (4) has been used to obtain the last equality.
Since ${\bf U}\boldsymbol{\sigma}{\bf U}^T$ is indistinguishable from $\boldsymbol{\sigma}$,
the distinguishiability between the covariances $\boldsymbol{\sigma'}$ and $\boldsymbol{\sigma}$
of two KE systems $A'$ and $A$ can be determined by the difference
\begin{equation}\label{}
\boldsymbol{\sigma}_{\rm diff}\equiv\boldsymbol{\sigma'}-{\bf U}\boldsymbol{\sigma}{\bf U}^T
=\int_0^t e^{{\bf M'}\tau }{\bf \Gamma}_{\rm diff}\left(e^{{\bf M'}\tau }\right)^T\,d\tau,
\end{equation}
where ${\bf \Gamma}_{\rm diff}\equiv{\bf \Gamma'}-{\bf U}{\bf \Gamma}{\bf U}^T$ is a time-dependent
symmetric matrix.
While (16) tells us that ${\bf \Gamma}_{\rm diff}={\bf 0}$ implies $\boldsymbol{\sigma}_{\rm diff}={\bf 0}$,
its time derivative,
$d\boldsymbol{\sigma}_{\rm diff}/dt=e^{{\bf M'}t}{\bf \Gamma}_{\rm diff}\left(e^{{\bf M'}t}\right)^T$,
implies the opposite, since $e^{{\bf M'}t}$ is an invertible matrix.
Thus, $\boldsymbol{\sigma}_{\rm diff}={\bf 0}$ if and only if
\begin{equation}\label{}
{\bf \Gamma}_{\rm diff}={\bf 0}\mbox{, or equivalently } {\bf \Gamma'}={\bf U}{\bf \Gamma}{\bf U}^T.
\end{equation}
This relation was already known for ${\bf U}$ replaced by invertible transformations ((8.2.39) in Ref. \cite{Keizer}),
for which the argument is more straightforward than that for (17).
Relation ${\bf \Gamma}_{\rm diff}={\bf 0}$ in (17) is a weak condition, under which $A$ and $A'$ have only ``statistically" indistinguishable ${\bf \hat{N}}$ and ${\bf \hat{N}'}$.
A plausible stronger condition is
\begin{equation}\label{}
{\bf f'}={\bf U}{\bf f},
\end{equation}
which fulfills (17) and generates indistinguishable individual stochastic trajectories ${\bf \hat{N}}$ and ${\bf \hat{N}'}$.
Both (17) and (18) lead to indistinguishable covariances and variances of fluctuations of ${\bf \hat{N}}$ and ${\bf \hat{N}'}$.
Together with the indistinguishable means of the KE condition, it yields the indistinguishable
Gaussian distributions of ${\bf \hat{N}}$ and ${\bf \hat{N}'}$.
Although (17) shows that $\boldsymbol{\sigma}_{\rm diff}={\bf 0}$ if and only if
${\bf \Gamma}_{\rm diff}={\bf 0}$, it
does not reveal whether two KE systems should have $\boldsymbol{\sigma}_{\rm diff}={\bf 0}$ or not.
For intrinsic noises, the indistinguishable covariances in (11) from the CME approach lead to the expectation that
$\boldsymbol{\sigma}_{\rm diff}={\bf 0}$ in the SDE approach, because the SDE can describe
CME fluctuations near the thermodynamic limit.
According to (17), this expectation would be true
if ${\bf \Gamma}_{\rm diff}={\bf 0}$,
which indeed can be proved (see $\Gamma_{ij}$ of ion channels below and S4 \cite{Supplemental}).
For extrinsic noises, ${\bf \Gamma}$ is not decided by ${\bf N}$
and distinct ${\bf \Gamma}$'s will generate different fluctuations.
Let ${\bf V}_{\rm diff}$ be a variance matrix whose diagonal terms
are the same as those of $\boldsymbol{\sigma}_{\rm diff}$ and zero elsewhere.
For two KE systems $A$ and $A'$, (16) implies the simple ordering rule for the variances of their state fluctuations
at any $t$:
\begin{equation}\label{}
{\bf \Gamma}_{\rm diff}\geq {\bf 0} (\leq {\bf 0}, = {\bf 0}) \Rightarrow {\bf V}_{\rm diff}\geq {\bf 0} (\leq {\bf 0}, = {\bf 0}),
\end{equation}
where $\geq {\bf 0}$ ($\leq {\bf 0}$) and $={\bf 0}$ stand for positive (negative) semi-definite and null matrices, respectively.
In practice, which of (17), (18), and (19) is the correct relation between two KE models $A$ and $A'$ of
a real macromolecule depends on what we study.
For intrinsic noises, the ${\bf \Gamma}$ and ${\bf \Gamma'}$ of $A$ and $A'$
can be analytically derived and must be related by (17).
For extrinsic noises, if $A$ and $A'$ are to approach the same experimental data,
their covariances should obey (18).
If $A$ and $A'$ are to approach two individually measured experimental data of the same macromolecule,
their fluctuations may have diverse orderings (19), because environmental noises in different
experiments are likely different.
Yet, if the noises are statistically the same, the covariance relation is (17), as for intrinsic noises.
Experimentally, fluctuations have been measured to predict the ion channels density, e.g., in nerve fibers of {\it Rana pipiens} \cite{Sigworth}.
To model this experiment with SDE (14), one considers a variety of channels,
each of which can stochastically transit between $n$ conformational states, with transition probabilities
given by the rate constants in (1).
According to the canonical theory \cite{Keizer} or the linear noise approximation \cite{van_Kampen_1},
the stochastic force ${\bf f}$ in (14) has the covariance
$\Gamma_{ij}=\sum_{k=1}^n(k_{ki}N_k+k_{ik}N_i)\delta_{ij}-(k_{ij}N_i+k_{ji}N_j)$,
where $N_i$ is the probability of finding a channel in the $i$-th state and
$\delta_{ij}$ denotes the Kronecker delta.
This covariance depends on the evolution of the mean value $N_i$ and thus varies with time.
If the channel is modeled by a two-state (open/closed) system,
the $(1,1)$ entry of its equilibrium covariance \cite{Fall}, $\Gamma_{11}^e=k_{12}N_1^e+k_{21}N_2^e$,
complies with Onsager's statistical theory of equilibrium ensembles \cite{Keizer}.
If the channel is modeled by two KE systems of different dimensions with the same form as $\Gamma_{ij}$,
they fulfill $\boldsymbol{\Gamma}_{\rm diff}={\bf 0}$ in (17) (see S4 \cite{Supplemental})
and then $\boldsymbol{\sigma}_{\rm diff}={\bf V}_{\rm diff}={\bf 0}$.
Therefore, the indistinguishability $\boldsymbol{\sigma}_{\rm diff}={\bf 0}$ from the Gaussian probability
in the SDE approach coincides with the indistinguishability (11) from the joint probability in the CME approach.
\section{Conclusion}
Theoretically we generalized the lumping theory from deterministic dynamics to stochastic processes.
It allows us to compare stochastic properties between hierarchical networks,
such as networks of small systems, which are sensitive to
external noises, or large networks whose species contain small number of copies.
In applications, we introduced lumping techniques from systems biology to molecular biology
to explore the fluctuation relations of experimentally indistinguishable kinetic schemes of biomolecules
and the legitimacy of estimating macromolecular fluctuations by low-dimensional schemes.
These findings are a kind of contractions beyond the widely discussed ones based on ``fast relaxations" and are useful for extracting correct kinetic models by observing extrinsic noise induced fluctuations.
The analytical results derived from exact lumping here provide limiting properties
for networks connected by all kinds of approximate lumping conditions.
They further give insights into more general fluctuation relations in other contraction theories,
which usually utilize similar block-triangular matrices to reduce systems
\cite{Liao}, such as Keizer's memoryless contraction \cite{Keizer}
and hierarchical Volterra equations in the Zwanzig-Mori formalism \cite{Berne}.
For further study, one may take into account more subtle issues, such as the approximate lumping for non-Markovian networks \cite{Wei_2} and the deformation of hidden complexity of free energy surfaces \cite{Krivov}.
\section*{Acknowledgments}
We thank Tetsuya J. Kobayashi, Jun Ohkubo, Jung-Hsin Lin, and Lee-Wei Yang
for useful discussions, the National Center for Theoretical Sciences at Taiwan for its supports, and the support of the Ministry of Science and Technology of Taiwan through Grant No. NSC 102-2112-M-009-012.
|
{
"timestamp": "2015-04-14T02:12:08",
"yymm": "1504",
"arxiv_id": "1504.03092",
"language": "en",
"url": "https://arxiv.org/abs/1504.03092"
}
|
\section{Introduction}
Clusters of galaxies are the most massive vi\-ria\-li\-zed structures in the Universe. Hence, they are excellent laboratories to study the physics of baryonic and dark matter at large scales in bound objects \citep{Voit05,Pratt09,Arnaud10,Giodini13}. Numerical si\-mu\-la\-tions show that massive clusters are formed from the mer\-ging of smaller structures in the hierarchical structure formation \citep[see review,][]{Kravtsov12}. Therefore, the study of low X-ray galaxy clusters could shed light on the assembly processes and environmental e\-ffects on their galaxy population, since these systems are likely to be e\-vol\-ving by sub-structure interations and accretion. In these systems, velocity dispersions are lower than in massive cluster ($\lesssim 800$\,km\,s$^{-1}$), favoring the interactions and mergers between the galaxy members. Thus, morphological transformations are more frequent in these clusters. Also, low mass clusters are more common than rich clusters due to the steepness of the cluster mass function. However, at the same time these systems are fainter and cooler, which makes them more difficult to detect and distinguish from background. Hence, these clusters have not been extensively studied compared to massive, luminous X-ray systems.
The evolution of galaxy clusters has been probed to be determined by cosmological parameters. In particular, the cluster mass function provides observational constraints to cosmology, given its sensitivity on the cosmological pa\-ra\-me\-ters \citep[e.g.,][]{Mandelbaum07,Rozo09,Vikhlinin09,Allen11,Planck14}. The main limitation in the use of this mass function is the practical determination of the masses. Weak and strong gavitational lensing probe the projected mass distribution of clusters, with strong lensing confined to the central regions of clusters, whereas weak lensing can yield mass measurements for larger radii. Mass estimations from gravitational lensing is affected by substructure, triaxiality, large-scale structure and the po\-ssi\-ble presence of multiple haloes along the line-of-sight
\citep{Oguri05,Sereno07,Corless09,Meneghetti10,Sereno10,Sereno11,Giocoli12,Sereno12,Spinelli12}. However, other me\-thods such as the caustic technique employing spec\-tros\-co\-pic measurements of galaxies velocity \citep{Rines06}, might be expensive in telescope time. Besides, radial mass distribution of clusters could be determined u\-sing X-ray surface brightness under the assumption of hydrostatic e\-qui\-li\-brium \citep{LaRoque06,Donahue14}. Nevertheless, deviations from the equilibrium could highly affect the estimations. Therefore, gravitational len\-sing is an excellent and a fairly clean technique for mass cluster determinations.
Galaxy clusters and groups are expected to fo\-llow simple relations linking the total mass with another physical quantities \citep{Kaiser86}. Given the difficulties of determining the mass of these systems, the study of these relations are important since they are suitable to convert simple observables into mass estimates. In particular, the X-ray luminosity of groups and clusters can be considered a good tracer of halo masses with approximately 20$\%$ scatter in the M-L$_{X}$ relation \citep{Stanek06, Maughan07, Pratt09, Rozo08,Rykoff08,Vikhlinin09b}. The main advantage in its use is that X-ray luminosity can be accurately measured at high redshifts, re\-qui\-ring only previous cluster detection and redshift information. Weak lensing provides a suitable technique to study the M-L$_{X}$ relation and it has been recently a\-pplied in several works \citep{Bardeau07,Hoekstra07,Rykoff08,Leauthaud10,Okabe10}. In this sense, three studies s\-pa\-nning from low \mbox{X-ray} luminosity clusters to groups \citep{Rykoff08,Leauthaud10,Kettula14} show a single relation with a well defined slope \citep{Foex12}, in agreement with those of massive clusters.
This work is the third in a series of papers aimed to understand the processes involved in the formation and evolution of low X-ray luminosity galaxy clusters at in\-ter\-me\-dia\-te redshifts. The first paper of the series \citep[][hereafter Paper I]{PaperI} contains the main goals, sample selection, and details of observations and data reduction for both, photometry and spectroscopy. The second paper \citep[][hereafter Paper II]{PaperII}, presents photometric properties of seven low X-ray luminosity observed with Gemini telescopes. As the redshift increases, an increment of blue galaxies and a decline in the fraction of lenticulars is observed, while the early-type fraction remains almost constant. These results are in agreement with those for high mass clusters. At lower redshifts, the presence of a well-defined cluster red sequence extending by more than 4 magnitudes showed that these intermediate mass clusters had reached a relaxed stage.
In this oportunity we present the weak lensing analysis of eight galaxy clusters of the low X-ray luminosity sample. The paper is organizated as follow: In Sec.\,\ref{sec:sample}, we describe the sample of clusters, and the acquisition and reduction of the images. In Sec.\,\ref{sec:method} we give the details of the weak lensing analysis for the mass determination. In Sec.\,\ref{sec:results} we present and discuss the estimated mass, and compared them with X-ray luminosity. Finally, in Sec.\,\ref{sec:conclusions} we summarise the main results of this work. We adopt when necessary a standard cosmological model $H_{0}$\,=\,70\,km\,s$^{-1}$\,Mpc$^{-1}$, $ \Omega_{m} $\,=\,0.3, and $ \Omega_{\Lambda} $\,=\,0.7.
\section{GALAXY CLUSTERS, OBSERVATIONS AND DATA REDUCTION}
\begin{table*}
\caption{Low X-ray luminosity Galaxy Cluster sample}
\label{table1}
\begin{tabular}{crrrrclccc}
\hline
[VMF98] & $\alpha$ & $\delta$ & $L_X$ & $L_X$ &z & Program & g$^{\prime}$ & r$^{\prime}$ & i$^{\prime}$\\
& & & [0.5--2.0] keV & [0.1--2.4] keV & & & & & \\
Id. &(J2000) & (J2000) & ($h^{-2}_{70}$10$^{43}$ cgs)& ($h^{-2}_{70}$10$^{43}$ cgs) & & Id. & & & \\
\hline
001 & 00 30 33.2 & +26 18 19 & 26.1 & 30.7 & 0.500 & GN-2010B-Q-73 & -- &15$\times$300 & 15$\times$150 \\
022 & 02 06 23.4 & +15 11 16 & 3.6 & 3.8 & 0.248 & GN-2003B-Q-10 & -- & 4$\times$300 & 4$\times$150 \\
093 & 10 53 18.4 & +57 20 47 & 1.4 &1.6 & 0.340 & GN-2011A-Q-75 & -- & 5$\times$600 & 4$\times$150 \\
097 & 11 17 26.1 & +07 43 35 & 6.4 & 7.7 & 0.477 & GS-2003A-SV-206 & 12$\times$600 & 7$\times$900 & -- \\
102 & 11 24 13.9 & -17 00 11 & 8.1 & 9.3 &0.407 & GS-2003A-SV-206 & -- & 5$\times$600 & -- \\
119 & 12 21 24.5 & +49 18 13 & 42.7 &53.6 & 0.700 & GN-2011A-Q-75 & -- & 7$\times$190 & 4$\times$120 \\
124 & 12 52 05.4 & -29 20 46 & 3.4 & 3.4 &0.188 & GS-2003A-SV-206 & 5$\times$300 & 5$\times$600 & -- \\
148 & 13 42 49.1 & +40 28 11 & 16.2& 21.4 &0.699 & GN-2011A-Q-75 & -- & 7$\times$190 & 5$\times$120 \\
\hline
\end{tabular}
\medskip
\begin{flushleft}
\textbf{Notes.} Columns: (1), the cluster identification; (2) and (3), the equatorial
coordinates of the X-ray centre; (4), the X-ray
luminosity in the [0.5 - 2.0] keV energy band obtained from \citet{Vikhlinin98}; (5), shows the X-ray luminosity in the [0.1 - 2.4] keV energy band calculated using $L_{500}$ from the MCXC catalogue \citep[\textit{Meta-Catalogue of X-ray Detected Clusters of Galaxies,}][]{Piffaretti11}; (6), the mean redshift for each cluster from \citet{Mullis03}; (7), the Gemini Program identification;
(8), (9) and (10), the number of exposures and individual exposure time in
seconds for each passband.
\end{flushleft}
\end{table*}
\subsection{Sample description}
\label{sec:sample}
The studied sample of low X-ray luminosities was selected from the catalogue of extended X-ray sources by \citet{Mullis03}. This catalogue is a revised version of the 223 galaxy clusters serendipitously detected in the ROSAT PSPC pointed observations by \citet{Vikhlinin98}. Our galaxy cluster sample comprises a random selection of 19 systems from the total sample of 140 galaxy clusters with X-ray luminosities in the [0.5--2.0]\,keV energy band (rest frame), close to the detection limit of the ROSAT PSPC survey ranging from $10^{42}$ to $\sim 50 \times 10^{43}$ erg s$^{-1}$. The redshift range of our selection is 0.16 to 0.70 and a full description of the project and sample can be found in Paper\,I.
The galaxy clusters subsample studied in this work is mainly based on the clusters optically analized in Paper II: 7 galaxy clusters with X-ray luminosity ranging from 1.4 to 26.1 $\times$10$^{43}$ ergs$^{-1}$ in the [0.5--2.0]\,keV energy band, and redshifts between 0.185 to 0.7. We add to this sample with observed colours, the galaxy cluster [VMF98]102 located at $ z\sim 0.401$, observed only in $r'$ passband. In Table\,\ref{table1} we summarize the main characteristics of the clusters. The mean X-ray luminosity in [0.5--2.0] keV band is 13.4 $\times$ 10$^{43}$ erg s$^{-1}$ , an intermediate/low luminosity when compared to $\sim$10$^{42}$ erg s$^{-1}$ for groups with extended X-ray emission or the larger values than 5 $\sim$ 10$^{44}$ erg s$^{-1}$ of rich clusters. $L_{X}$ in [0.1--2.4]\,keV band are used for further analysis and comparison with other workes (see Section\,\ref{lmrelation})
\subsection{Observations}
Photometric observations for the eight galaxy clusters were obtained with Gemini North (GN) and South (GS) telescopes, during the system verification process (SVP) and specific programs with Argentinian time allocation. Seven clusters were observed using the Gemini Multi-Object Spectrograph \citep{Hook04} in the image mode, in the $r'$ and $g'$ or $i'$ passbands with an array of three EEV CCDs of $2048\times4608$ pixels and only one ([VMF98]102) in the $r'$ passband. Using a $2\times2$ binning, the pixel scale is 0.1454 arcsec per pixel which corresponds to a FOV (\textit{Field of View}) of $5.5\times5.5$ arcmin$^{2}$ in the sky.
All images were observed under excellent photometric conditions, with mean seeing values of 0.75, 0.66 and 0.74 arcsec in the $g'$ , $r'$ and $i'$ filters, respectively. Some observations were made under exceptional weather conditions, such as those made to the galaxy cluster [VMF98]001, with a median seeing of about 0.485 in the $r'$ image. Further details about these observations are given in Paper\,II. Columns 6 to 9 in Table\,\ref{table1} show a summary of the photometric observations.
All observations were processed with the Gemini IRAF package v1.4 inside IRAF\footnote{IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.} \citep{Tody93} . The images were bias/overscan-subtracted, trimmed and flat-fielded. The final processed images were registered to a common pixel position and then combined.
\section{WEAK LENSING ANALYSIS}
\label{sec:method}
We developed a pipeline based on Python Language (version 2.7. Available at http://www.python.org) to make the lensing analysis. The pipeline computes the shear profile and fits a model to estimate the mass of a galaxy cluster, taking as input the observed image of the cluster. In the next subsections, we describe in detail the implemented weak lensing analysis pipeline and the results of the application on simulated data to test its performance.
\subsection{Object detection and classification}
The first step in the lensing analysis is the detection and classification of the sources in stars and galaxies. To perform the detection and photometry of the sources we implement SExtractor \citep{Bertin96}. From SExtractor output, we use for the analysis the parameters: MAG\_BEST, as the magnitude in each filter; MU\_MAX, defined as the central surface brightness of the objects ($\mu_{MAX}$); FLUX\_MAX as the peak flux above background; FWHM as the gaussian full width at half maximum; CLASS\_STAR as the stellarity index and FLAG, which corresponds to the notes generated by SExtractor in the detection and measurement processes.
SExtractor is run twice (in a two-pass mode): A first run is made to detect bright objects in order to estimate the seeing and the saturation level of each image, and a second run to do the final detection. The first run of SExtractor is made with a detection level of 5$ \sigma $ above the background. The seeing is estimated using the average FWHM of the point-like objects selected from the FWHM/MAG\_BEST diagram, since for these objects the FWHM is independent of the magnitude. Determining the seeing is important for the star-galaxy classification, given that SExtractor uses it to compute the stellarity index. The saturation level is estimated as 0.8 times the maximum value of the FLUX\_MAX parameter. These parameters, \textit{seeing} and \textit{saturation level}, are taken into account in the SExtractor configuration file for the second run, with a lower threshold detection limit of 1.5$ \sigma $. A second run is made in dual mode, detecting objects on the $r'$ image, while astrometric and photometric pa\-ra\-me\-ters are measured on all individual images.\\
\begin{figure}
\centering
\includegraphics[width=.45\textwidth]{mu_mag2.eps}~\\
\includegraphics[width=.45\textwidth]{mag_fwhm2.eps}
\caption{Classification of objects detected in the $r'$ image of the galaxy cluster [VMF98]102. Here stars are represented by triangles, galaxies by points, and artifacts by cruxes. Upper pannel shows $\mu_{MAX}/r'$ plane, where stars are situated in the region marked by the solid line $\pm$ 0.4 magnitudes, and in the lower pannel we show $r'/FWHM$ plane.}
\label{sources}
\end{figure}
Sources are classified according to their position in the magnitude/central flux diagram, the FWHM respect to the seeing and the stellarity index, following \citet{Bardeau05}, in stars, galaxies and false detections. In Figure\,\ref{sources} we show, as an example, $\mu_{MAX}$ as a function of the $r'$ magnitude (u\-pper panel) and the $r'$ magnitude against the FWHM (lower panel), for all objects dected by SExtractor in the cluster [VMF098]102. Objects that are more sharply peaked than the \textit{Point Spread Function} (from now on PSF), thus with FWHM $<$ \textit{seeing} - 0.5 pixel, and with FLAG parameter $>$ 4, are considered as false detections. As the light distribution of a point source scales with magnitude, objects on the line magnitude/central flux, $\pm$ 0.4 magnitudes, FWHM\,$<$\,\textit{seeing}\,+\,1\,pixel and CLASS\_STAR $>$ 0.8 are considered as stars. The rest of the objects are considered as galaxies. \\
The first step in the pipeline ends generating two catalogues, one for the objects classified as stars and another for the galaxies.
\subsection{Shape measurements}
Measurements of galaxy shape are central in this analysis, given that galaxy ellipticities are used for the shear estimations and therefore to estimate cluster masses. It is important to take into account the roundness effects of the atmosphere as well as the distortions caused by the telescope optics, all together included in the PSF, which is convolved with the galaxy intensity distribution.
For the shape measurements we use IM2SHAPE \citep{Bridle02}. This code computes the shape parameters modeling the object as a mixture of Gaussians, convolved with a PSF which is also a sum of Gaussians. For simplicity both, the PSF and the object, are modeled with a single elliptical Gaussian profile.
The PSF field across the image is estimated from the shape of the stars, since they are intrinsically point-like objects. We only used objects with a measured ellipticity smaller than 0.2 to remove most of the remaining false detections and faint galaxies present in the catalogue. Looking at the 5 nearest stars at each position, we have also removed those that differ by more than 2$\sigma$ from the local average shape. Then, we linearly interpolate the local PSF at each object position by averaging the shapes of the five closest stars. After PSF determination, we use again IM2SHAPE to measure the galaxy shapes, and the result is a catalogue of the galaxies with its intrinsic shape parameters.
\subsection{Shear radial profiles}
\label{sec:profile}
Gravitational lensing maps the unlensed image in the source plane, specified by coordinates $(\beta^{1},\beta^{2})$, to the lensed image $(\theta^{1},\theta^{2})$ in the image plane, using a matrix transformation:
\begin{equation*}
\left( {\begin{array}{c}
\delta\beta^1 \\
\delta\beta^2 \\
\end{array} } \right)
=
\left( {\begin{array}{cc}
1-\kappa-\gamma_1 & -\gamma_2 \\
-\gamma_2& 1-\kappa+\gamma_1
\end{array} } \right)
\left( {\begin{array}{c}
\delta\theta^1 \\
\delta\theta^2
\end{array} } \right)
;
\end{equation*}
where $\gamma_1$ and $\gamma_2$ are the components of the complex shear
$\gamma = \gamma_1 + i \gamma_2$.
This can also be expressed as:
\begin{equation*}
\left( {\begin{array}{c}
\delta\beta^1 \\
\delta\beta^2 \\
\end{array} } \right)
=
(1-\kappa)
\left( {\begin{array}{cc}
1- g_1 & - g_2 \\
- g_2 & 1 + g_1
\end{array} } \right)
\left( {\begin{array}{c}
\delta\theta^1 \\
\delta\theta^2
\end{array} } \right)
;
\end{equation*}
where $g_{1}$ and $g_{2}$ are the components of the reduced shear:
\begin{equation}
g=\dfrac{\gamma}{1-\kappa}
\end{equation}
which is a nonlinear function of the two lensing functions: the complex shear, $\gamma$, and the convergence, $\kappa$, related to the projected mass density.
If lensing is weak, the image of a circular source with ratio \textit{r}, appears elliptical, with axis given by
\begin{equation*}
a=\dfrac{r}{1-\kappa-|\gamma|},\,\,\,\,\,\,\,\,\,\,b=\dfrac{r}{1-\kappa+|\gamma|}
\end{equation*}
Defining the ellipticity as
\begin{equation*}
e=\dfrac{a-b}{a+b}=\dfrac{|\gamma|}{1-\kappa}\approx|\gamma|
\end{equation*}
where \textit{g} becomes the normal shear, $\gamma$, since $\kappa \ll 1$, which generally holds in the weak lensing regime for clusters, and will be assumed henceforth here.\\
If the source has an intrinsic ellipticity $\bmath{e_{s}}$, the observed ellipticity in the weak lensing limit will be:
\begin{equation*}
\bmath{e}=\bmath{e_{s}}+\gamma
\end{equation*}
Assuming that unlensed galaxies are randomly oriented on the sky plane ($\langle \bmath{e_{s}} \rangle = 0$ ) and averaging over sufficiently many sources:
\begin{equation}
\langle \bmath{e} \rangle=\langle \gamma \rangle
\end{equation}
Hence, in the weak lensing approximation, we get an unbiased estimator of the reduced shear by averaging the shape of background galaxies in concentric annuli around the cluster centre. Spherical symmetry also implies that the average in annular bins of the tangential component ellipticity of the lensed galaxies, defined as the E-mode, traces the reduced shear. On the other hand, the average in annular bins of the component tilted at $\pi/4$ relative to the tangential component, the B-mode, should be exactly zero for the case of perfect symmetry \citep[e.g.][Sec.\,4]{Bartelmann01}.
Because of the random orientation of the galaxies in the source plane, the error in the observed galaxy ellip\-ti\-ci\-ties and thus, on the estimated shear, will depend on
the number of galaxies averaged together to measure the shear \citep{Schneider00}. Thus, the errors in the measured shear can be estimated as:
\begin{equation} \label{eq:err}
\sigma_{\gamma}\approx\dfrac{\sigma_{\epsilon}}{\sqrt{N}}
\end{equation}
where $\sigma_{\epsilon}$ is the dispersion of the intrinsic ellipticity distribution ($\sigma_{\epsilon} \approx 0.3$) and $N$ is the number of objects in the annular bin.
We have adopted the brightest cluster galaxy in $r'$ filter as the cluster centre, a criterium commonly used for lensing masses determinations \citep{Okabe10,Hoekstra11,Foex12}. Shear profiles were computed using nonoverlapping logarithmic annuli, in order to have similar signal-to-noise ratio (S/N) in each annuli. We have tested different annuli sizes but the final mass results have not showed a strong dependence on this parameter. We have fixed the size for the one we obtained lowest errors for the SIS and NFW profile fits. The profiles were fitted from the inner part were the signal becomes significantly positive, to reduce the impact of miscentering, up to the bin with highest number of ga\-la\-xies ($\sim$\,3\,arcmin for most of the clusters, which roughly corresponds to 0.8-1.4 Mpc). Our profiles were mainly limited by the FOV of the images. With these limits, 4-6 points were available in the shear profiles.
\subsection{Background Galaxies selection and redshift distribution}
\label{back}
To perform the shear estimation, background galaxies were selected as those with $r'$ magnitudes between $m_{P}$ and $m_{max} + 0.5$. $m_{P}$ is defined as the faintest magnitude where the probability that the galaxy is behind the cluster is higher than 0.7 and $m_{max}$ corresponds to the peak of the magnitude distribution of galaxies in the $r'$ passband. Keeping galaxies brighter than $m_{max} + 0.5$ ensures that we are not taking into account too faint galaxies with higher uncertainties in the shape measurements. We have also restricted the objects to those with good S/N and with a good pixel sampling by using only the galaxies with $\sigma_{e} < 0.2$ ($\sigma_{e}$ is defined as the quadratic sum of the errors $\sigma_{e1}$ and $\sigma_{e2}$ given by IM2SHAPE) and with FWHM $>$ 5 pixels.
Once we obtain a catalogue for the background ga\-la\-xies, we average the components of the ellipticities (E-mode and B-mode) in nonoverlapping annuli. The average E-mode components corresponds to the shear value which depends on the geometrical factor \mbox{$ \beta = D_{LS}/ D_{S} $}, where $D_{LS}$ is the angular diameter distance from the lens to the background source galaxy, and $D_{S}$ is the distance from the observer to the background galaxy. A galaxy at the same radial distance from the centre of the cluster but at a different background redshift is sheared differently. This variation is taken into account once we fit the profiles by $\langle\beta\rangle$.
To estimate $m_{P}$ and $\langle\beta\rangle$ we used the catalogue of photometric redshifts computed by \citet{Coupon09}, based on the public release Deep Field 1 of the Canada-France-Hawaii Telescope Legacy Survey, which is complete up to $m_{r} = 26.$. We compute the fraction of galaxies with $z > z_{cluster}$ in magnitude bins of 0.25 magnitudes for the $r'$ filter, and then we chose $m_{P}$ as the lowest magnitude for which the fraction of galaxies was greater than 0.7. Then we applied the photometric selection criteria to the catalogue ($m_{P} < m_{r} < m_{max} + 0.5$) and we computed $\beta$ for the whole distribution of galaxies. To take into account the contamination by foreground galaxies given our selection criteria, we set $\beta(z_{phot} < z_{cluster}) = 0$ which outbalances the dilution of the shear signal by these unlensed galaxies. Deep Field\,1 covers a sky region of 1\,degree$^2$, thus to estimate the cosmic variance, we divide the field in 25 non-overlapping areas of $\sim$\,140\,arcmin$^2$ and we compute $m_{P}$ and $\langle \beta \rangle$ at $z_{cluster}=0.5$ for each area. The uncertainties due to the cosmic variance were estimated as the dispersion of the values obtained for each area, obtaining $\sim 0.3$ for $m_{P}$ and $\sim 0.01$ for $\langle \beta \rangle$. Given that the errors in $\langle \beta \rangle$ are lower than the $3\%$, which represents an error of the $\sim\,5\%$ in the mass, we did not consider these uncertanties in the estimation of the masses errors since the uncertainty due to the intrinsic shape of field galaxies is much bigger.
\begin{figure}
\centering
\includegraphics[scale=0.45]{./distrib.eps}
\caption{Fraction of galaxies with $z > 0.5$ ($N(z>0.5)/N_{tot}$), for different magnitudes in filter $r'$ and colours \textit{r'-i'}, computed using photometric redshifts given by \citet{Coupon09}, used to compute the weight for the shear estimation. The vertical line idicates the $m_{max}$ position (see text for its definition).}
\label{weigh}
\end{figure}
In order to take into account the contamination of foreground galaxies in the catalogue, we weighted the estimated shear, $\langle \gamma \rangle$, with the probability that the galaxy was behind the cluster. We compute this probability using \citeauthor{Coupon09} catalogue, from the fraction of galaxies with $z > z_{cluster}$ for each bin in magnitude, $r'$, and colour (\textit{g' - r'} and \textit{r' - i'}), see Figure\,\ref{weigh}. Hence, given the magnitude and the colour of each galaxy, we assigned to it a weigh, \textit{w}, as the fraction of galaxies with $z > z_{cluster}$ in that bin. For [VMF98]102 we have only one image in the filter $r'$, therefore for weighing the shear profile we take into account the probability that each galaxy was behind the cluster given the magnitude of that galaxy (we did not take into account the colours for computing this probability, as in the other clusters).
\subsection{Fitting the profiles}
\begin{figure*}
\includegraphics[scale=0.4]{map2.eps}~\hfill
\includegraphics[scale=0.4]{map_correc2.eps}
\caption{PSF treatment applied to stars of one of the images of the DES
simulation: Mayor semiaxis ($a$ cos $\theta$, $a$ sin $\theta$) before (\textit{left}) and after (\textit{right}) the PSF deconvolution in the CCD. Notice that the semiaxis are more randomly distributed and the scale (given by the first thicker segment in the upper-left corner and which corresponds to 3 pix) is much more smaller after the taking into account the PSF. }
\label{PSF}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[scale=0.44]{shear_profile_LN2.eps}\hfill
\includegraphics[scale=0.44]{shear_profile_HN2.eps}
\caption{Shear profiles obtained for the Low (\textit{left}) and High (\textit{right}) Noise PSF Applied File, from the DES cluster simulation. The dashed curve shows the SIS profile for the input value of $\sigma_{V}$ and the solid one the fitted profile. E and B modes are represented by full circles and crosses, respectively.}
\label{DES}
\end{figure*}
We finally estimate the M$_{200}$ mass, defined as \mbox{M$_{200}\equiv\,$M$\,(\,<\,$R$_{200})\,=200\rho_{crit}(z)\dfrac{4}{3}\pi\,r_{200}^{3}$}, where $R_{200}$ is the radius that encloses a mean density equal to 200 times the critical density ($\rho_{crit} \equiv 3 H^{2}(z)/8 \pi G$; $H(z)$ is the redshift dependent Hubble parameter and $G$ is the gravitational constant). In order to do that we fit the shear data with the singular isothermal sphere (SIS) and the NFW profile \citep{Navarro97} using $\chi^{2}$ minimization. These density profiles are the standard parametric models used in lensing analysis to characterize the lenses. Following, we explain briefly the lensing formulae for these two profiles:
\subsubsection{SIS profile}
The SIS mass model is the simplest one for describing a relaxed massive sphere with a constant and isotropic ve\-lo\-ci\-ty dispersion. This is mainly described by the density distribution:
\begin{equation*}
\rho(r) = \dfrac{\sigma_{V}^{2}}{2 \pi G r^{2}}
\end{equation*}
This model corresponds to a distribution of self-gravitating particles where the velocity distribution at all radii is a Maxwellian with one dimensional velocity dispersion, $\sigma_{V}$. From this equation, we can get the critical Einstein radius for the source sample as:
\begin{equation}\label{eq:SIS}
\theta_{E} = \dfrac{4 \pi \sigma_{V}^{2}}{c_{vel}^{2}} \frac{1}{\langle \beta \rangle}
\end{equation}
where $c_{vel}$ is the speed of light, in terms of wich one obtains:
\begin{equation}
\kappa_{\theta} = \gamma_{\theta} = \dfrac{\theta_{E}}{2 \theta}
\end{equation}
where $\theta$ is the distance to the cluster centre. Hence, fitting the shear for different radius, we can estimate the Einstein radius, and from that, we can obtain an estimation of the mass M$_{200}$ as \citep{Leonard10}:
\begin{equation}\label{eq:MSIS}
M_{200} = \dfrac{2 \sigma_{V}^{3} }{\sqrt{50} G H(z)}
\end{equation}
\subsubsection{NFW profile}
The NFW profile is derived from fitting the density profile
of numerical simulations of cold dark matter halos \citep{Navarro97}. This profile depends on two parameters, the virial radius, R$_{200}$, and a dimensionless concentration parameter, \textit{c}:
\begin{equation*}
\rho(r) = \dfrac{\rho_{c} \delta_{c}}{(r/r_{s})(1+r/r_{s})^{2}}
\end{equation*}
where $r_{s}$ is the scale radius, $r_{s} = $R$_{200}/c$ and $\delta_{c}$ is the cha\-rac\-te\-ris\-tic overdensity of the halo,
\begin{equation*}
\delta_{c} = \frac{200}{3} \dfrac{c^{3}}{\ln(1+c)-c/(1+c)}
\end{equation*}
We used the lensing formulae for the spherical NFW density profile from \citet{Wright00}. If we fit the shear for different radius we can have an estimation of the pa\-ra\-me\-ters $c$ and R$_{200}$. Once we obtain R$_{200}$ we can compute the M$_{200}$ mass. Nevertheless, there is a well-known degeneracy between the parameters R$_{200}$ and $c$ when fitting the shear profile in the weak lensing regime. This is due to the lack of information on the mass distribution near the cluster centre and only a combination of strong and weak lensing can raise it and provide useful constraints on the concentration parameter. Since we do not have strong lensing modeling for the clusters in the sample, we decided to fix the concentration parameter, $c_{200} = 4$, according to the predicted concentrations given by \citet{Duffy11} for a relaxed cluster with M$ = 1 \times 10^{14} M_{\odot} h_{70}^{-1}$ placed at $z \sim 0.4$. Thus, we fit the mass profile with only one free parameter, R$_{200}$.
\subsection{Testing the pipeline with simulated data}
\label{sec:sim}
To check the performance of our weak lensing analysis pipeline, we tested it on the DES cluster simulation images publically available \citep{Gill09}. This simulation consists of a sets of images, with different grades of difficulty, of sheared galaxies due to the presence of a SIS profile with a velocity dispersion of 1250\,km\,s$^{-1}$. This is a suitable test for our pipeline given that the idea is to apply it to real clusters of galaxies. We applied our pipeline to three of the available image files, High Noise File, High Noise PSF Applied File and Low Noise PSF Applied File.
For the PSF Applied files, we checked that our IM2SHAPE implementation can recover point-like objects by applying the PSF correction to each star. Figure\,\ref{PSF} shows the results of the shape parameters measurements for these stars, with and wihtout taking into account the PSF in the shape measurement: the size distribution is dominated by point sources, and the orientation is more uniformly distributed after the PSF correction.
The images contain only the sheared galaxies, hence all the galaxies detected were considered as background galaxies at $z=0.8$, which is the average redshift of the galaxies. We cut the catalogue discarding the galaxies with \mbox{FWHM $<$ 5} and with $\sigma_{e} > 0.2$. Shear profiles are shown in Figure\,\ref{DES}. For the most complex image that we treated (high noise image of sheared galaxies convolved with a PSF), we obtained a deviation parameter of 1.3, defined as the number of $\sigma$ that the result is away from the input value of $\sigma_{V} = 1250$ km/sec, i.e. $\sigma = \dfrac{result - input}{error}$ , where the e\-rror was estimated according to the root mean square e\-rror of the Einstein radius. Given these results, we conclude that our weak lensing pipeline is able to reproduce the input shear signal, thus it could be applied to real observations to extract the lensing signal and to estimate the masses of cluster of galaxies.
\begin{table*}
\caption{Main results of the weak lensing analysis}\label{tab:esp}
\label{table:2}
\begin{tabular}{@{}crrrrccccrrcr@{}}
\hline
\rule{0pt}{1.05em}%
[VMF\,98] & $\alpha$ & $\delta$ & $\rho_{back}$ & $m_{P}$ & $m_{max}$ & $\langle\beta\rangle$&$\sigma_{V}^{spec}$ & \multicolumn{2}{c}{SIS} & \multicolumn{2}{c}{NFW} \\
Id. & (J2000) & (J2000) & & & & & & $\sigma_{V}$ & M$_{200}$ & R$_{200}$ & M$_{200}$ \\
\hline
\rule{0pt}{1.05em}%
001 & 00 30 34.0 & +26 18 10 & 56 & 23.0 & 26.1 & 0.41 & - & 780 $\pm$ 100 & 3.4 $\pm$ 1.3 & 1.3 $^{+0.2}_{-0.2}$ & 4.0$^{+2.2}_{-2.0}$ \\
022 & 02 06 21.2 & +15 11 01 & 18 & 20.7 & 25.1 & 0.61 & 508 & 570 $\pm$ 100 & 1.5 $\pm$ 0.8 & 1.1 $^{+0.2}_{-0.2}$ & 2.1$^{+1.2}_{-1.1}$ \\
093 & 10 53 18.9 & +57 20 45 & 8 & 22.3 & 24.0 & 0.48 & - & 750 $\pm$ 140 & 3.4 $\pm$ 1.9 & 1.4 $^{+0.4}_{-0.4}$ & 4.0$^{+3.6}_{-3.1}$ \\
097 & 11 17 26.1 & +07 43 35 & 40 & 23.0 & 26.0 & 0.43 & 775 & 720 $\pm$ 100 & 2.7 $\pm$ 1.1 & 1.1 $^{+0.3}_{-0.2}$ & 2.8$^{+1.9}_{-1.7}$ \\
102 & 11 24 05.8 & -17 00 50 & 40 & 22.7 & 25.9 & 0.49 & 675 & 650 $\pm$ 120 & 2.1 $\pm$ 1.2 & 1.2 $^{+0.3}_{-0.2}$ & 2.7$^{+1.9}_{-1.7}$ \\
119 & 12 21 29.3 & +49 18 40 & 13 & 24.5 & 25.4 & 0.29 & - & 1000 $\pm$ 160 & 6.3 $\pm$ 3.1 & 1.4 $^{+0.2}_{-0.2}$ & 7.3$^{+3.8}_{-3.4}$ \\
124 & 12 52 04.1 & -29 20 29 & 33 & 19.5 & 25.7 & 0.71 & 700 & 430 $\pm$ 60 & 0.7 $\pm$ 0.3 & 0.8 $^{+0.3}_{-0.2}$ & 0.8$^{+0.8}_{-0.7}$ \\
148 & - & - & 26 & 24.5 & 25.9 & 0.29 & - & - & - & - & \\
\hline
\end{tabular}
\medskip
\begin{flushleft}
\textbf{Notes.} Columns: (1) shows the cluster identification; (2) and (3), the coordinates of the centre adopted for the lensing analysis; (4), the density of background galaxies (galaxies\,arcmin$^{-2}$); (5) and (6), the brightest and faintest magnitude limits considered for the galaxy
background selection (see Section\,\ref{back}); (7), the geometrical factor; (8), the line-of-sight spectroscopic velocity dispersion from Paper\,I; (9) and (10) the results from the SIS profile fit, the velocity dispersion and M$_{200}$ (see Equations\,\ref{eq:SIS} and \ref{eq:MSIS}); (11) and (12), the results from the NFW profile fit, R$_{200}$ and M$_{200}$. The velocity dispersion, M$_{200}$ and R$_{200}$ are in units of km\,s$^{-1}$,$10^{14} M_{\odot} h_{70}^{-1}$ and Mpc\,$h_{70}^{-1}$, respectively.
\end{flushleft}
\end{table*}
\section{RESULTS}
\label{sec:results}
From our weak lensing analysis, we estimated the mass of seven clusters in the sample. Due to its low signal to noise, for cluster [VMF98]148 it was not possible to derive a reliable mass estimate from our lensing measurements. The results of the analysis are shown in Table\,\ref{table:2}. Errors in $\sigma_{V}$, $R_{200}$ and the masses were computed according to the $\chi^{2}$ dispersion. Errors in M$_{200}^{NFW}$ are higher than M$_{200}^{SIS}$, given the big uncertainties in the R$_{200}$ parameter . Nevertheless, both estimations are consistent being the NFW masses sys\-te\-ma\-ti\-ca\-lly larger by a $\sim 20 \%$ ($\langle$ M$_{200}^{NFW} / $ M$_{200}^{SIS} \rangle$ = 1.21 $\pm$ 0.13, where the uncertainty corresponds to the scatter around the mean), in excellent agreement with the result presented by \citet{Okabe11} for the virial masses. Shear profiles obtained for the galaxy clusters are shown in Figures\,\ref{shear-profile1} and \ref{shear-profile2} with the reduced $\chi^{2}$ for each fit. We include both fits, SIS (solid line) and NFW (dashed line) models. Points and crosses represent the E and B modes averaged in annular bins, respectively. All profiles are well fitted by both mo\-dels. In the next subsections we discuss our results and we study the relation between the mass derived and the cluster X-ray luminosities.
\begin{figure*}
\centering
\includegraphics[scale=0.6]{shear_profile_0012.eps}
\includegraphics[scale=0.6]{shear_profile_0222.eps}
\includegraphics[scale=0.6]{shear_profile_0932.eps}
\includegraphics[scale=0.6]{shear_profile_0972.eps}
\includegraphics[scale=0.6]{shear_profile_1022.eps}
\caption{Shear radial profiles as a function of cluster-centric projected distance (in arcsec and Mpc) obtained for the $r'$ images of sample of clusters. The solid and the dashed lines represent the best fit of SIS and NFW profiles, respectively, with the fitted parameters given in the box. The points and crossings show the E and B modes profiles averaged in annular bins, respectively. Error bars are computed according to Equation\,\ref{eq:err}.}
\label{shear-profile1}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[scale=0.6]{shear_profile_1192.eps}
\includegraphics[scale=0.6]{shear_profile_1242.eps}
\caption{Shear radial profiles as a function of cluster-centric projected distance (in arcsec and Mpc) obtained for the $r'$ images of sample of clusters. The solid and the dashed lines represent the best fit of SIS and NFW profiles, respectively, with the fitted parameters given in the box. The points and crossings show the E and B modes profiles averaged in annular bins, respectively. Error bars are computed according to Equation\,\ref{eq:err}.}
\label{shear-profile2}
\end{figure*}
\subsection{Properties of individual clusters}
\subsubsection{[VMF]001}
For the galaxy cluster [VMF98]001 we obtained a shear signal consistent with a velocity dispersion of $\sim$\,800\,km\,s$^{-1}$. There is a big offset between the position BCG, adopted as the centre
for the lensing analysis, and the X-ray luminosity peak from ROSAT ($\sim$\,110\,kpc), not observed in the X-ray countours obtained with XMM-Newton (see Figure 6, from Paper I). Thus, given the lower resolution of ROSAT observations, the X-ray peak might be poorly determinated leading to unrealistic offsets. Further evidence of this fact is the absence of the shear profile signal centred at the X-ray position. We argue that the centre of the gravitational potential should be close to the BCG position.
\subsubsection{[VMF]022}
The galaxy cluster [VMF98]022 shows an elongated distribution of galaxies in the NE-SW direction. The cluster is dominated by a bright elliptical galaxy, which presents a shift of $\sim$12$^{\prime\prime}$ in the south-west direction with respect to the X-ray peak emission (for further details about the cluster morphology, see Section\,3.4 in Paper II). We compute the shear profile taking this bright elliptical as the centre of the cluster. This system presents a shear profile signal consistent with a velocity dispersion of 540\,km\,s$^{-1}$, in good agreement with the velocity dispersion fitted from the redshift distribution (see Sec.\,4 in Paper\,I).
\subsubsection{[VMF]093}
For the cluster [VMF98]093, in spite of the low density of background galaxies, we obtain a significant signal consistent with a velocity dispersion of 750\,km\,s$^{-1}$. As evidence of the relaxed state of this cluster, we observe a dominant population of red galaxies as well as concentric X-ray countours centred in the BCG (Figure 6 in Paper I).
\subsubsection{[VMF]097}
The galaxy cluster [VMF98]097 was previously a\-na\-ly\-sed by \citet{Carrasco07}, using the same set of ima\-ges. They obtained a large discrepancy between mass estimates, where the X-ray mass exceeds by more than a factor three the weak lens derived estimate. Moreover, they found a large degree of substructure, as also seen in the redshift distribution presented in Paper I (Figure 11). However, substructure cannot explain the defect in the weak lensing mass, given that substructure in the surroundings would tend to dilute the tangential shear leading to mass under-estimation \citep{Meneghetti10,Giocoli12,Giocoli14}. We improve the profile \citep[see Figure\,9 from][p. 11]{Carrasco07}, adding a new constraint for the E-mode and obtain a profile consistent with zero for the B-mode. Nevertheless, our weak lensing mass estimate is consistent with that obtained by \citet{Carrasco07}, corresponding to a velocity dispersion of $\sim$\,700\,km\,s$^{-1}$ and also, with the velocity dispersion from the redshift distribution of 775\,km\,s$^{-1}$ (see Sec.\,4 in Paper\,I).
\begin{figure*}
\centering
\includegraphics[scale=0.78]{LX_M.eps}
\caption{Weak lensing masses versus X-ray luminosities for the sample of clusters (diamonds), combined with the stacked measurement by \citet{Leauthaud10} (open triangles), EXCPRES clusters by \citet{Foex12} (open circles) and low-mass from the CFHTLS (open squares) by \citet{Kettula14}. Dashed, pointed, and solid lines represent the fit obained by \citet{Leauthaud10}, \citet{Foex12} and \citet{Kettula14}, respectively.}
\label{lumass}
\end{figure*}
\subsubsection{[VMF]102}
The results from cluster [VMF98]102 give a velocity dispersion of 640\,km\,s$^{-1}$ , in good agreement with the spectroscopic value obtained in Paper I. In this case, the profile was built adopting a centre between the X-ray peak and the second brightest galaxy member. This was selected af\-ter trying to fit the profile taken the centre as the second brightest galaxy and then, as the X-ray peak, without ge\-tting enough signal-noise ratio to fit the profile. The second brightest galaxy in this case is close to the other bright cluster members and, unlike the brightest galaxy, it is an elliptical galaxy, so it is a more adequate guess for the cluster centre in this case. This cluster presents irregular X-ray contours (Figure\,6 in Paper\,I) and, based in spectroscopic information, we found a non related group of galaxies in the line of sight (Figure\,11, in Paper\,I). Also, there is a big offset between the X-ray peak and the centre adopted for the lensing analysis ($\sim$\,220\,kpc), however we could not confirm this offset with higher resolution observations.
\subsubsection{[VMF]119}
The cluster [VMF98]119 is one of the highest redshift clusters ($ z \sim 0.7 $) in our sample. Even with a very low density of background galaxies, it shows a significant shear signal according with a velocity dispersion of 1000\,km\,s$^{-1}$. The centre was placed at the brightest galaxy member, $\sim$\,1' from the ROSAT X-ray peak. Using X-ray observations from CHANDRA Data Archive, we built the X-ray contours and the peak is displaced $\sim$0.9' from the ROSAT centre, but still $\sim$0.4' ($\sim$170\,kpc) displaced from the BCG. Also, the B-modes do not follow a null flat profile, which could be suggesting a large deviation from the spherical symmetry. This can also be seen in the distribution of member galaxies (Figure\,12 in Paper\,II).
\subsubsection{[VMF]124}
Finally, for the cluster [VMF98]124, the centre from the X-ray data using XMM-Newton contours (Figure\,6 in Paper\,I) agrees with the BCG position. Besides, there is no evidence of another group in redshift space (see Figure\,11 from Paper\,I) and we observed a dominant red galaxy population (see Paper\,II), which indicates the relaxed state of this system. This cluster presents a low shear signal consistent
with 430\,km\,s$^{-1}$. There is a large difference between the velocity dispersion obtained by the lensing analysis and that derived from the redshift distribution (700\,km\,s$^{-1}$, Sec.\,4 in Paper I). We notice, however, the high uncertainty in this value given the small number of available redshifts.
\subsection{$M - L_{X}$ relation}
\label{lmrelation}
We have also investigated the relation between the estimated mass and the X-ray luminosity, which is
a diagnostic of the halo baryon fraction and the entropy structure of the intracluster gas \citep{Rykoff08}. The $L_{X} - M$ relation has been extensively studied, mainly at low redshifts ($z \lesssim 0.1$) using X-ray data \citep{Markevitch98, Arnaud02,Reiprich02,Popesso05, Morandi07,Pratt09,Vikhlinin09}. The main conclusion was that the relation follows a power-law, but with a slope and amplitude that differ from the self-similar prediction of $M \propto L_{X}^{3/4}$. Instead, they found a flatter slope, $\alpha = 0.56 - 0.63$. Physical mechanism ruling the baryonic content of clusters, could strongly affect the X-ray luminosity, and so on the $L_{X} - M$ relation, causing deviations from a simple gravitational model. Simulations combining the gravitational evolution of dark matter structures together with the hydrodynamical behaviour \citep{Borgani04,Kay04,Borgani08} favor a lower slope value.
Figure\,\ref{lumass} shows the $M-L_{X}$ relation for the galaxy clusters studied in this paper with masses estimated from the weak lensing analysis, together with those derived by other studies, the $M - L_{X}$ relation based on 12 low mass clusters from the CFHTLS by \citet{Kettula14}; 11 X-ray bright clusters selected and 206 stacked galaxy groups in the COSMOS field by \citet{Leauthaud10}, and the $L_{X} - M$ relation obtained from the EXCPRES sample by \citet{Foex12}. In principle, the slopes from $M - L_{X}$ ($\beta$) and $L_{X} - M$ ($\alpha$) could be easily compared ($\alpha = 1/\beta$) assuming that the halo mass function is locally a power-law \citep{Leauthaud10}. For comparison with other authors estimates, we used the NFW masses showed in Table\,\ref{table:2}. Given that \citet{Kettula14} derived core-excised luminosities, they are systematically lower than the rest of the plotted luminosities for a given mass. We notice that our mass determinations are in very good agreement with \citet{Leauthaud10} fit. The largest deviation from this fit corresponds to the two lowest X-ray luminosity clusters. Besides, [VMF98]093 contains a very low density of background galaxies which affects the precision of the shear estimates, and in the field of [VMF98]124 there is a star with X-ray emition, which could bias high the quoted X-ray luminosity of the cluster.
\section{Summary and conclusions}
\label{sec:conclusions}
In this work we presented the weak lensing analysis of eight low X-ray luminosity galaxy cluster. We described the pipeline for determining weak lensing masses of clusters using ground-based images. The analysis consisted in: the detection and classification of the sources, the shape measurements on the $r'$ images taking into account the PSF, the galaxy background selection, the computation of shear profiles weighing the ellipticities according to the $r'$ magnitude and the colour of the galaxy, and finally, the fit of the mass density distribution models (SIS and NFW profiles). We have tested it succesfully on simulated data and then, we have applied it to a sample of low X-ray luminosity clusters.
From this analysis we could estimate the mass of seven low X-ray luminosity galaxy clusters. One of these clusters ([VMF98]097) was previously analysed with a similar approach \citep{Carrasco07}. We improved the shear fit and we obtained a mass consistent with the previous result. For the other clusters in the sample, we estimated the mass for the first time.
The velocity dispersions obtained from the SIS fit, are in general agreement with the spectroscopic values available for four of the clusters in the sample. Masses obtained were compared to the X-ray luminosities. Our results are mostly in good agreement with previous analysis of the $M - L_{X}$ relation, in particular with \citet{Leauthaud10} result. In this work we provide further constraints for the $M - L_{X}$ relation, in low-intermediate X-ray luminosity galaxy clusters, by increassing the number of observables.
We plan in future works to include different models to fit the shear profiles, in order to include non-spherical models. Also, we plan to extend the pipeline to analyse low-massive galaxy systems employing stacking techniques.
\section*{Acknowledgments}
This work was partially supported by the Consejo Nacional de Investigaciones Cient\'{\i}ficas y T\'ecnicas (CONICET, Argentina)
and the Secretar\'{\i}a de Ciencia y Tecnolog\'{\i}a de la Universidad Nacional de C\'ordoba (SeCyT-UNC, Argentina).\\
This research has made use of NASA's Astrophysics Data System and Cornell University arXiv repository.\\
Based on observations obtained at the Gemini Observatory processed using the Gemini IRAF package, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Minist\'{e}rio da Ci\^{e}ncia, Tecnologia e Inova\c{c}\~{a}o (Brazil) and Ministerio de Ciencia, Tecnolog\'{i}a e Innovaci\'{o}n Productiva (Argentina). \\
We made an extensively use of the following python libraries: http://www.numpy.org/, http://www.scipy.org/, http://roban.github.com/CosmoloPy/ and http://www.matplotlib.org/.\\
JLNC acknowledges the financial support from the Dirección de Investigación de la Universidad de La Serena (DIULS, ULS) and the Programa de Incentivo a la Investigación Académica (PIA-DIULS). Also acknowledges partial support from the postdoctoral fellow ALMA/CONICYT N 31120026.\\
Part of this research was developed while MJdLDR was postdoctoral researcher on the
\href{Center for Gravitational Wave Astronomy at The University of Texas in Brownsville}{http://cgwa.phys.utb.edu/}.
\bibliographystyle{mn2e}
|
{
"timestamp": "2015-04-16T02:10:36",
"yymm": "1504",
"arxiv_id": "1504.03364",
"language": "en",
"url": "https://arxiv.org/abs/1504.03364"
}
|
\section{Introduction}
\label{sec: intro}
Consider a target probability distribution $\mu$ defined
on a possibly infinite dimensional separable Hilbert space $\mathcal H$.
It is of interest to sample from this probability measure
and assumed that there is a density of $\mu$
w.r.t. a Gaussian reference measure $\mu_0$ on $\mathcal H$
given by
\begin{equation} \label{equ:mu}
\frac{\d\mu}{\d\mu_0}(u) = \frac{1}{Z} \exp(-\Phi(u)), \qquad u\in \mathcal H.
\end{equation}
Here $\Phi \colon \mathcal H \to \mathbb{R}_+$
is a measurable function and $Z= \int_{\mathcal H} \exp(-\Phi(u))\, \mu_0(\d u)$
the normalizing constant.
Such probability measures $\mu$ arise as
posterior distributions in Bayesian inference with $\mu_0$ as a Gaussian prior.
Common examples
in infinite dimensional spaces are inferring spatially
distributed properties of porous media
or stock prices.
Unfortunately,
the fact that the normalizing constant $Z$ is typically
unknown and that $\Phi$ is only available in the form of function evaluations
makes it difficult to sample $\mu$ directly.
But Markov chains and in particular Metropolis-Hastings (MH) algorithms are applicable
for approximate sampling.
These algorithms consist of a proposal and an acceptance/rejection step.
A state is proposed by
a proposal kernel
but it is only accepted with a certain probability
which depends on $\frac{\d \mu}{\d \mu_0}$.
The authors of \cite{BeskosEtAl2008} suggested
a modification of a
Gaussian random walk proposal which is $\mu_0$-reversible.
The latter property leads to a well-defined MH algorithm in
infinite dimensional Hilbert spaces, see also \cite{Ti98}.
This proposal was later \cite{CotterEtAl2013}
referred to as \emph{preconditioned Crank-Nicolson (pCN)} proposal.
Remarkably, the Markov chain of the
resulting pCN Metropolis algorithm
has
\emph{dimension-independent sampling efficiency}, see \cite{CotterEtAl2013},\cite{HaStVo14}.
This is a significant advantage compared to earlier,
popular MH algorithms whose performance usually
deteriorates with increasing state space dimension \cite{CotterEtAl2013},\cite{HaStVo14},\cite{RobertsRosenthal2001}.
We extend the pCN proposal to incorporate information about the target measure $\mu$.
Such an adaption might account for
the anisotropy of the covariance of $\mu$ or the local curvature of $\Phi$.
Intuitively, the resulting Markov chain has on average a larger step size and, thus, explores the state space faster.
This idea is not entirely new.
It is already mentioned in \cite{Ti94} where it is suggested to choose the covariance of the proposal adapted to the target measure.
Later in \cite{GirolamiCalderhead2011} the authors explain how to propose new states using general local metric tensors.
Moreover, in \cite{MartinEtAl2012} the Hessian
of the negative log density $\Phi$ of $\mu$ is
employed as local curvature information to design
a stochastic Newton MH method in finite dimensions
and in \cite{CuiEtAl2014},\cite{Law2014}
a Gauss-Newton variant for capturing global curvature in an
infinite dimensional setting is outlined.
Our approach for adapting the proposal to the target measure $\mu$
has a similiar motivation as the proposals considered
in \cite{CuiEtAl2014},\cite{Law2014}. It comes from a
local linearization of the unknown-to-observable map in Bayesian inverse problems.
This suggests a particular form for approximating the
covariance of the
target measure,
namely $(C + \Gamma)^{-1}$,
where $C$ denotes the covariance of the
reference measure $\mu_0$ and $\Gamma$ is a
suitable self-adjoint and positive operator.
We then consider the class of Gaussian proposals with covariance $C_\Gamma = (C + \Gamma)^{-1}$.
By enforcing $\mu_0$-reversibility
we derive our class of generalized pCN (gpCN) proposal kernels $P_\Gamma$.
In a numerical simulation the resulting Metropolis algorithm seems to perform
independent of dimension and variance.
Here variance independence refers to the variance of the observational noise, which affects
the covariance of the target distribution $\mu$.
Particularly, if the variance of the noise decreases the measure
$\mu$ becomes more concentrated.
Our numerical experiments also indicate that other popular MH
or random walk algorithms perform worse, i.e., variance dependent.
Moreover, we present a convergence result
for the gpCN Metropolis based on spectral gaps.
It is well known, see \cite{RoRo97},
that for Markov chains with reversible transition kernels $K$ a
strictly positive spectral gap, denoted
$\gap(K)>0$, is equivalent to a form of geometric ergodicity.
The latter roughly means that, in an appropriate setting,
the distribution of the $n$th step
of a Markov chain converges exponentially fast to its stationary measure.
We refer to Section~\ref{sec:MC} for precise definitions and further details.
Our main theoretical result, stated in Theorem~\ref{theo:gpCN_conv}, is as follows.
Let us assume that the transition kernel $M_0$ of the pCN algorithm
has a positive spectral gap, i.e. $\gap(M_0)>0$.
Then, for any
$\varepsilon > 0$
there is
an explicitly given probability measure $\mu_R$ such that
\[
\norm{\mu-\mu_R}{\text{tv}} \leq \varepsilon
\quad
\text{ and }
\quad
\gap(M_{\Gamma,R})>0
\]
where $M_{\Gamma,R}$ denotes the transition kernel of the gpCN Metropolis algorithm
targeting the measure $\mu_R$ and
$\norm{\cdot}{\text{tv}}$ is the total variation
distance, see \eqref{eq:tv_definition}.
The key for the proof is a new comparison theorem for spectral gaps of Markov chains generated by MH algorithms.
In order to apply this comparison argument we show that the proposal kernels of the pCN and gpCN Metropolis are equivalent and that their Radon-Nikodym derivative
belongs to an $L_p$-space for a $p>1$.
We note that in \cite{HaStVo14} under additional assumptions on the density function $\frac{\d \mu}{\d \mu_0}$
it is proven that there exists a strictly positive spectral gap of the pCN Metropolis.
Thus, in this setting the gpCN Metropolis algorithm targeting $\mu_R$ also converges exponentially.
The remainder of the paper is organized as follows.
In Section \ref{sec:pre} we state the precise framework, recall preliminary facts,
and give a brief introduction to Markov chain Monte Carlo
and MH algorithms including the pCN Metropolis algorithm.
The gpCN Metropolis algorithm is motivated and defined in Section \ref{sec:gpCN}.
Particularly, in Section \ref{subsec: numerics}
we illustrate its superior
performance compared to other popular MH algorithms.
In Section \ref{sec:Conv} we state a general result for comparing spectral gaps of MH algorithms and then apply it to the gpCN and pCN Metropolis.
Section~\ref{subsec: loc_gpCN} provides an outlook to gpCN algorithms in
infinite dimensions which use
Gaussian proposals with state-dependent covariance.
For the convenience of the reader we
recall some facts about Gaussian measures
in Appendix~\ref{sec:Gaussian} and relegate
more technical proofs to Appendix~\ref{sec:proofs}.
\section{Preliminaries} \label{sec:pre}
\noindent
Let $\mathcal H$ be a separable Hilbert space with
inner-product and norm denoted
by $\langle \cdot, \cdot \rangle$ and $\|\cdot\|$.
By $\mathcal{B}(\mathcal H)$ we denote the corresponding Borel $\sigma$-algebra
and by $\mathcal L(\mathcal H)$ the set of all bounded, linear operators
$A\colon \mathcal H \to \mathcal H$.
Further, we have a Gaussian measure $\mu_0 = N(0, C)$
on $(\mathcal H,\mathcal{B}(\mathcal H))$.
Here and in the remainder of the
paper $C\colon \mathcal H \to \mathcal H$ denotes a nonsingular \emph{covariance operator}
on $\mathcal H$, i.e., a bounded, self-adjoint and positive trace class operator with $\ker C= \{0\}$.
By $\mu$ we denote the probability measure
of interest on $(\mathcal H,\mathcal{B}(\mathcal H))$ given through
the density defined in \eqref{equ:mu}.
Typically, the desired distribution is complicated and the density only
known up to a constant,
which makes direct sampling from
$\mu$ difficult.
This is the reason why Markov chains are
used for approximate sampling according to $\mu$.
\subsection{Markov chains and spectral gaps}\label{sec:MC}
We give a short introduction to Markov chains and Markov chain Monte Carlo (MCMC)
methods on general state spaces.
We call a mapping $K \colon \mathcal H \times \mathcal{B}(\mathcal H) \to [0,1]$ a \emph{transition kernel},
if $K(x,\cdot)$ is a probability measure on $(\mathcal H,\mathcal{B}(\mathcal H))$
for each $x\in \mathcal H$ and $K(\cdot,A)$ is a measurable function for each $A\in\mathcal{B}(\mathcal H)$.
Then, a \emph{Markov chain with transition kernel $K$}
is a sequence of random variables $(X_n)_{n\in \mathbb{N}}$, mapping from some probability
space $(\Omega,\mathcal{F},\mathbb{P})$ to $(\mathcal H,\mathcal{B}(\mathcal H))$,
satisfying
\[
\mathbb{P}(X_{n+1} \in A \mid X_1,\dots,X_n)
= \mathbb{P}(X_{n+1} \in A \mid X_n)
= K(X_n,A)
\]
almost surely for all $A\in \mathcal{B}(\mathcal H)$.
Most properties of a Markov chain can be expressed as properties of its
transition kernel.
For example,
we say the transition kernel $K$ is \emph{$\mu$-reversible} if
\begin{equation} \label{eq: det_balance_gen}
K(u, \d v) \, \mu(\d u) = K(v, \d u) \, \mu(\d v)
\end{equation}
in the sense of measures on $\mathcal H\times \mathcal H$.
This property is also known as the
\emph{detailed balance condition} and it implies
that the distribution $\mu$ is a stationary
or invariant probability measure of a
Markov chain with transition kernel $K$, i.e., if $X_1 \sim \mu$ then also $X_{2} \sim \mu$.
Each $\mu$-reversible transition kernel $K$ on $(\mathcal H,\mathcal{B}(\mathcal H))$
induces a \emph{Markov operator}, which we shall also denote by $K$, given by
\[
Kf(u) = \int_{\mathcal H} f(v)\, K(u,\d v), \quad f\in L_2(\mu),
\]
where
\[
L_2(\mu) = \left\{ f\colon \mathcal H \to \mathbb{R}
\mid \| f \|_{2,\mu}
:= \left( \int_{\mathcal H} |f(u)|^2\,\mu(\d u) \right)^{1/2} <\infty \right\},
\]
is the Hilbert space of measurable, square integrable functions with respect to
$\mu$.
By the $\mu$-reversibility we have that $K\colon L_2(\mu)\to L_2(\mu)$ is a
bounded and self-adjoint linear operator.
We also introduce the closed subspace
\[
L_2^0(\mu) = \left\{ f\in L_2(\mu) \mid \int_{\mathcal H} f(u)\, \mu(\d u)=0 \right\}
\]
of $L_2(\mu)$
and the operator norm
\[
\norm{K}{\mu} =
\sup_{f\in L_2^0(\mu),\, f\not =0}
\frac{\norm{Kf}{2,\mu}}{\norm{f}{2,\mu}}
\]
for $K: L_2^0(\mu) \to L_2^0(\mu)$.
Let $\spec(K\,|\, L_2^0(\mu))$ denote
the spectrum of $K$ on $L_2^0(\mu)$. Then, we also have
\[
\norm{K}{\mu} = \sup\{\abs{\lambda}\colon \lambda \in \spec(K\,|\, L_2^0(\mu))\}.
\]
We define the \emph{spectral gap of $K$ (w.r.t. $\mu$) by} $\gap(K) = 1- \norm{K}{\mu}$.
This is an important quantity which can be used to formulate
conditions ensuring an exponentially fast convergence of the distribution of $X_n$ to $\mu$.
To be more precise, we introduce the total variation distance of two
probability measures $\nu_1,\nu_2$ on $(\mathcal H,\mathcal{B}(\mathcal H))$
by
\begin{equation} \label{eq:tv_definition}
\norm{\nu_1 - \nu_2}{\text{tv}}
:= \sup_{A\in\mathcal{B}(\mathcal H)} \abs{\nu_1(A)-\nu_2(A)}.
\end{equation}
Let $\nu$ be the initial distribution of our Markov chain, i.e., $X_1\sim \nu$.
Then,
with
\[
K^{n}(u,A) = \int_{\mathcal H} K^{n-1}(v, A)\,K(u, \d v),\quad A\in \mathcal B(\mathcal H),
\]
for $n\in \mathbb{N}$, the distribution of $X_{n+1}$ is given by
\[
\nu K^n(A) = \int_{\mathcal H} K^n(u,A)\, \nu(\d u).
\]
In the setting above it is well known, see \cite[Proposition~2.2]{RoRo97},
that $\norm{K}{\mu}<1$, or equivalently $\gap(K)>0$,
holds, iff the transition kernel is $L_2(\mu)$-geometrically ergodic.
Here by $L_2(\mu)$-geometric ergodicity we mean that,
there exists a number $r\in [0,1)$ such that
for any probability measure $\nu$, which has a density
$\frac{\d\nu}{\d\mu}\in L_2(\mu)$ w.r.t $\mu$,
there is a constant $C_\nu<\infty$ such that
\[
\norm{\nu K^n - \mu}{\text{tv}} \leq C_\nu\, r^n, \qquad n\in\mathbb{N}.
\]
If the distribution of $X_{n}$ converges to $\mu$, then the Markov chain $(X_n)_{n\in\mathbb{N}}$
can be used for approximate sampling from
$\mu$.
This leads to Markov chain Monte Carlo methods for the computation of expectations.
The mean $\mathbb{E}_\mu(f)$ of a function $f\colon \mathcal H \to \mathbb{R}$ w.r.t $\mu$
can then be approximated by the time average
\[
S_{n,n_0}(f) = \frac{1}{n} \sum_{j=1}^n f(X_{j+n_0})
\]
where $n$ is the sample size and $n_0$ a burn-in parameter to decrease the influence of the initial distribution.
The spectral gap of $K$ of the Markov chain $(X_n)_{n\in \mathbb{N}}$ can then be applied to assess the error of the time average $S_{n,n_0}(f)$.
We assume $\gap(K)>0$ and mention two results.
The first is rather classical and due to Kipnis and Varadhan \cite{Kipnis86}.
If the initial distribution is $\mu$ and $f\in L_2(\mu)$, then the error
$\sqrt{n}(S_{n,n_0}(f)-\mathbb{E}_{\mu}(f))$ converges weakly to $N(0,\sigma_{f,K}^2)$ with
\[
\sigma_{f,K}^2
= \langle (I+K)(I-K)^{-1}(f-\mathbb{E}_\mu(f)),(f-\mathbb{E}_\mu(f)) \rangle_{\mu}
\leq \frac{2 \norm{f}{2,\mu}^2}{\gap(K)}
\]
where $\langle \cdot,\cdot \rangle_{\mu}$ denotes the inner-product in $L_2(\mu)$.
The second result is more recent and provides a non-asymptotic bound for
the mean square error. We have
\[
\sup_{\norm{f}{4}\leq 1} \mathbb{E}\abs{S_{n,n_0}(f)-\mathbb{E}_\mu(f)}^2
\leq \frac{2}{n \cdot \gap(K)} + \frac{C_\nu \norm{K}{\mu}^{n_0}}{n^2 \cdot \gap(K)^2}
\]
with $\norm{f}{4} = \left( \int_{\mathcal H} \abs{f(u)}^4\, \mu(\d u)\right)^{1/4}$ and a number $C_\nu\geq 0$ depending on the initial distribution $\nu$.
We refer to \cite{Ru12} for
details.\par
This shows that $\gap(K)$ is a crucial quantity in the study of Markov chains and the numerical analysis of MCMC methods.
\subsection{Metropolis algorithm with pCN proposal} \label{sec: pCN}
In this work we focus on Markov chains generated by
the Metropolis algorithm.
This algorithm employs a transition kernel on $(\mathcal H,\mathcal B(\mathcal H))$ for proposing new states which we shall denote by $P$ and call \emph{proposal kernel}.
Moreover, let $\alpha\colon \mathcal H \times \mathcal H \to [0,1]$ be a measurable function denoting the \emph{acceptance probability}.
Then, a transition of a Markov chain $(X_n)_{n\in \mathbb{N}}$ generated by the Metropolis algorithm can be represented in algorithmic form:
\begin{enumerate}
\item
Given the current state $X_n = u$, draw independently a sample $v$ of a random variable $V\sim P(u,\cdot)$ and a sample $a$ of a random variable $A\sim \text{Unif}[0,1]$.
\item
If $a < \alpha(u,v)$, then set $X_{n+1} = v$, otherwise set $X_{n+1} = u$.
\end{enumerate}
The transition kernel of such a Markov chain is then
\begin{equation} \label{eq: metro_kern}
M(u, \d v) = \alpha(u,v) P(u,\d v)
+\delta_u(\d v) \,\int_{\mathcal H} (1 - \alpha(u, w)) \, P(u,\d w)
\end{equation}
and we call it \emph{Metropolis kernel}.
It is well known, see \cite{Ti98}, that $M$ is reversible w.r.t.
$\mu$ if $\alpha(\cdot,\cdot)$ is chosen as
\begin{align} \label{al: acc_prob}
\alpha(u,v) = \min\left\{ 1 , \frac{\d \eta^\bot}{\d \eta}(u,v)\right\}, \qquad u,v\in \mathcal H,
\end{align}
where $\frac{\d \eta^\bot}{\d \eta}$ denotes the Radon-Nikodym derivative of the measures
\begin{align*}
\eta(\d u,\d v) & := P(u, \d v) \, \mu(\d u) \qquad \mbox{and} \qquad \eta^\bot (\d u,\d v) := P(v, \d u) \, \mu(\d v),
\end{align*}
which we assume to exist.
For finite dimensional state spaces
the condition of absolute continuity
of $\eta^\bot$ w.r.t.
$\eta$ is often easily satisfied.
However, for infinite dimensional state spaces
this becomes a real issue, since there measures tend to be mutually singular.
As pointed out in \cite{BeskosEtAl2008},\cite{CotterEtAl2013} a possible way to ensure the existence of $\frac{\d \eta^\bot}{\d \eta}$ is to choose a proposal kernel $P$ which is $\mu_0$-reversible, i.e.,
\begin{equation} \label{eq: det_balance}
P(u, \d v) \, \mu_0(\d u) = P(v, \d u) \, \mu_0(\d v).
\end{equation}
Then, due to the fact that $\frac{\d \mu}{\d \mu_0}$ and $\frac{\d \mu_0}{\d \mu}$ exist,
see \eqref{equ:mu}, it follows that
\begin{align}
\frac{\d \eta^\bot}{\d \eta}(u,v) =
\frac{\d \mu}{\d\mu_0}(v) \frac{\d \mu_0}{\d\mu}(u)
= \exp( \Phi(u) - \Phi(v))
\end{align}
and, hence, $\alpha(u,v) = \min\left\{1, \exp( \Phi(u) - \Phi(v))\right\}$.
We next introduce
the Metropolis algorithm with the \emph{preconditioned Crank-Nicolson} (pCN) proposal,
see also \cite{CotterEtAl2013} for details.
The pCN proposal kernel arises from a discretization of an Ornstein-Uhlenbeck process with invariant measure $\mu_0$
and takes the form
\begin{align}\label{al: pCN}
P_{0}(u,\cdot) = N(\sqrt{1-s^2}u,s^2 C)
\end{align}
where $s\in[0,1]$ denotes a variance or stepsize parameter.
It is straightforward to verify that $P_{0}$ is $\mu_0$-reversible.
Namely, by applying \eqref{equ:Gaussian_affine} from
Appendix \ref{sec:Gaussian} we deduce
\[
P_{0}(u, \d v) \, \mu_0(\d u)
= N\left( \begin{bmatrix} 0\\ 0 \end{bmatrix}, \begin{bmatrix} C & \sqrt{1-s^2}C \\ \sqrt{1-s^2}C & C \end{bmatrix} \right)
= P_{0}(v, \d u) \, \mu_0(\d v).
\]
In the following we call the resulting Metropolis algorithm with proposal $P_{0}$
simply pCN Metropolis algorithm or
pCN Metropolis and denote its Metropolis kernel by $M_{0}$.
Next, we generalize the pCN Metropolis algorithm to allow for proposal kernels which employ a different covariance structure than the covariance of $\mu_0$.
\section{Metropolis with gpCN proposals} \label{sec:gpCN}
In recent years many authors have proposed and pursued the idea to construct proposals
which try to exploit certain geometrical features of the target measure, see for example \cite{GirolamiCalderhead2011},\cite{MartinEtAl2012},\cite{Law2014},\cite{CuiEtAl2014}.
\par
We consider generalized pCN (gpCN) proposals
which aim to adapt to the covariance structure of the target measure $\mu$.
We motivate our gpCN proposal, show that it is well-defined in function spaces
and illustrate its superior
performance in a simple but common setting.
\subsection{Motivation from Bayesian inference} \label{subsec: Motiv}
We briefly recall the Bayesian framework for inverse problems and refer to \cite{ErnstEtAl2015} for an overview
and to \cite{Stuart2010} for a comprehensive introduction to the topic.
Assume $X$ is a random variable on $(\mathcal H,\mathcal{B}(\mathcal H))$
with distribution $\mu_0 = N(0,C)$.
Here $\mu_0$ is called the \emph{prior} distribution and describes our initial uncertainty
about $X$.
Let $Y$ be a random variable on $\mathbb{R}^m$ given by
\begin{equation} \label{eq: bay_model}
Y = G(X) + \varepsilon
\end{equation}
with a continuous map $G\colon \mathcal H\to \mathbb{R}^m$
and $\varepsilon \sim N(0,\Sigma)$, independent of $X$, with $\Sigma \in \mathbb{R}^{m\times m}$.
The variable $Y$ models an observable quantity depending on
$X$ via the map $G$ which is perturbed by additive noise $\varepsilon$.
Then, given some observation $y\in \mathbb{R}^m$ of $Y$
we want to infer $X$, i.e., we are interested in the conditional distribution of $X$ given the event $Y=y$.
We denote this conditional distribution by $\mu$ and call it \emph{posterior} distribution.
In particular, in this setting $\mu$
admits a representation of the form \eqref{equ:mu} with
\begin{equation}\label{equ:Phi}
\Phi(u) = \frac 12 |y- G(u)|^2_{\Sigma^{-1}}
\end{equation}
where $\abs{x}_{\Sigma^{-1}}^2 = x^T \Sigma^{-1} x $ for $x\in \mathbb{R}^m$.
A special situation appears if $G(u)=L u + b$ with a linear mapping
$L\colon \mathcal H \to \mathbb{R}^m$ and $b\in \mathbb{R}^m$.
Then, it is known from \cite{Mandelbaum1984} that $\mu= N(m,\widehat C)$ with
\begin{equation} \label{eq: target_normal}
m = C L^*(LCL^*+\Sigma)^{-1} (y-b),
\qquad \widehat C=(C^{-1}+L^*\Sigma^{-1}L)^{-1},
\end{equation}
where $L^*$ denotes the adjoint operator of $L$.
If we want to sample approximately from a Gaussian
target measure $\mu = N(m,\widehat C)$ by Metropolis
algorithms with Gaussian proposals, it seems beneficial to
employ $s^2 \widehat C$ as proposal covariance,
see for example \cite{Ti94},\cite{RobertsRosenthal2001},\cite{Law2014}.
Intuitively, since then the Gaussian proposal possesses the same principal directions and the same ratio of variances as the Gaussian target measure, the proposed states should be accepted more often than for other proposals. See also Figure~\ref{Fig: gauss_prop} for an illustration.
This leads to a higher average acceptance probability
and, thus, a faster exploration of the state space.
\begin{figure}[htb]
\centering
\def\target{(0,0) ellipse (3 and 8)}
\def\targetZ{(0,0) ellipse (4.5 and 8*4.5/3)}
\def\targetZZ{(0,0) ellipse (6 and 8*6/3)}
\def\propball{(0,8) ellipse (2 and 2)}
\def\propballZ{(0,8) ellipse (3 and 3)}
\def\propballZZ{(0,8) ellipse (4 and 4)}
\def\proptarg{(0,8) ellipse (3/2.45 and 8/2.45)}
\def\proptargZ{(0,8) ellipse (3/1.63 and 8/1.63)}
\def\proptargZZ{(0,8) ellipse (3/1.23 and 8/1.23)}
\tikzstyle{P_1} = [draw,black,ultra thick]
\tikzstyle{P_21} = [draw,blue!100,ultra thick]
\tikzstyle{P_22} = [draw,blue!70,ultra thick]
\tikzstyle{P_23} = [draw,blue!40,ultra thick]
\tikzstyle{P_31} = [draw,red!100,ultra thick]
\tikzstyle{P_32} = [draw,red!70,ultra thick]
\tikzstyle{P_33} = [draw,red!40,ultra thick]
\tikzset{
pat1/.style={pattern=horizontal lines,pattern color=#1},
pat1/.default=red
}
\tikzset{
pat2/.style={pattern=vertical lines,pattern color=#1},
pat2/.default=black
}
\begin{minipage}{0.49\textwidth}
\centering
(a)\\
\begin{tikzpicture}[scale=0.2,>=latex']
\begin{scope}[rotate=-110]
\fill[fill=black!10] \targetZZ;
\fill[fill=black!25] \targetZ;
\fill[fill=black!50] \target;
\fill(0,8)circle(12pt);
\node at (0.7,8.8) {$u$};
\path[P_1] \target;
\path[P_1] \targetZ;
\path[P_1] \targetZZ;
\path[P_21] \propball;
\path[P_22] \propballZ;
\path[P_23] \propballZZ;
\end{scope}
\end{tikzpicture}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
(b)\\
\begin{tikzpicture}[scale=0.2,>=latex']
\begin{scope}[rotate=-110]
\fill[fill=black!10] \targetZZ;
\fill[fill=black!25] \targetZ;
\fill[fill=black!50] \target;
\fill(0,8)circle(12pt);
\node at (0.6,8.9) {$u$};
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\path[P_1] \targetZ;
\path[P_1] \targetZZ;
\path[P_31] \proptarg;
\path[P_32] \proptargZ;
\path[P_33] \proptargZZ;
\end{scope}
\end{tikzpicture}
\end{minipage}
\caption{\label{Fig: gauss_prop}
For a Gaussian target measure
$\mu=N(m,\widehat{C})$
and current state $u$ the
region of acceptance $\{v: \alpha(u,v) = 1\}$
(dark grey region) as well as two regions of possible
rejection $\{v: \underline{p} \leq \alpha(u,v) < \overline{p} \leq 1\}$ (lighter grey regions) are displayed.
Moreover, we present the contour lines (blue and red, resp.)
of Gaussian proposals $N(u,s^2C)$ with covariance $C=I$
in part (a) and target covariance $C=\widehat C$ in part (b).
}
\label{fig:motiv_gpCN}
\end{figure}
The affine case indicates how we can construct good Gaussian proposal kernels if the map $G$ is nonlinear but smooth.
For a fixed $u_0 \in \mathcal H$ local linerization leads to
\[
G(u) = G(u_0) + \nabla G(u_0)\,(u-u_0) + r(u)
\]
with a remainder
term $r(u) \in \mathbb{R}^m$.
For a sufficiently smooth $G$ the remainder
$r$ is small (in a neighborhood of $u_0$), so that
\[
\widetilde G(u) = G(u_0) + \nabla G(u_0)\,(u-u_0)
\]
is close to $G(u)$ (in a neighborhood of $u_0$).
The substitution of $G$ by $\widetilde G$
in the model \eqref{eq: bay_model} leads to a
Gaussian target measure $\widetilde \mu = N(\widetilde m, \widetilde C)$
with covariance
\[
\widetilde C = (C^{-1} + L^* \Sigma^{-1} L)^{-1},
\qquad
L = \nabla G(u_0).
\]
By the fact that $G$ and $\widetilde G$ are close,
we also have that the measures $\mu$ and $\widetilde \mu$ are close as well.
Then, it is reasonable to use $\widetilde C$ in the covariance operator of the proposal in a Metropolis algorithm.
Of course, there might be other choices besides
a simple linearization of $G$ at one point.
For example, averaging linearizations at several points $u_1,\dots,u_n \in \mathcal H$ leads to
\[
\widetilde C = \Big(C^{-1} + \frac 1N \sum_{n=1}^N L_n^* \,
\Sigma^{-1}L_n\Big)^{-1}, \qquad L_n = \nabla G(u_n).
\]
Natural candidates for the points $u_1,\dots,u_N$ are
samples according to the prior or
samples taken from a short run of a preliminary
Markov chain with the posterior as stationary measure,
cf. the adaptive method in \cite[Section 3.4]{CuiEtAl2014}.
One could also think of a state-dependent covariance $C(u)$.
This motivates the study of
proposals which use covariances of the form
$
C_{\Gamma} = (C^{-1} + \Gamma)^{-1}
$
for suitably chosen operators $\Gamma$.
\subsection{Well-defined gpCN proposals} \label{sec:well_gpCN}
In this section we introduce the gpCN proposal kernel
and prove that the Metropolis algorithm with this proposal is well-defined
in the sense that it leads to a $\mu$-reversible transition kernel.
For this we introduce the set $\mathcal L_+(\mathcal H)$ of all bounded,
self-adjoint and positive linear operators $\Gamma: \mathcal H \to \mathcal H$.
We define the operators
\begin{equation} \label{equ:tildeC}
C_\Gamma := (C^{-1} + \Gamma )^{-1}, \qquad \Gamma \in \mathcal L_+(\mathcal H),
\end{equation}
motivated in Section~\ref{subsec: Motiv}, where $C$ denotes the covariance operator of the prior measure $\mu_0 = N(0,C)$, for which we also use the equivalent representation
\begin{equation} \label{equ:tildeC_2}
C_\Gamma = C^{1/2} \; ( I + H_\Gamma)^{-1} \; C^{1/2},
\qquad H_\Gamma:=C^{1/2} \Gamma C^{1/2}.
\end{equation}
In the following we prove that $C_\Gamma$ can
be considered as covariance operator.
\begin{propo} \label{propo:tildeC}
Let $C$ be a nonsingular covariance operator
on $\mathcal H$, $\Gamma\in \mathcal L_+(\mathcal H)$
and $C_\Gamma$ with $H_\Gamma$ given as in \eqref{equ:tildeC_2}.
Then $H_\Gamma \in \mathcal L_+(\mathcal H)$ is trace class and $C_\Gamma$ is also a nonsingular covariance operator on $\mathcal H$.
\end{propo}
\begin{proof}
That $H_\Gamma \in \mathcal L_+(\mathcal H)$ follows by construction.
Furthermore, since $H_\Gamma$ is a composition of two Hilbert-Schmidt
and one bounded operator, $C^{1/2}$ and $\Gamma$, respectively,
it is trace class \cite[Proposition~1.1.2]{DaPrato2004}.
Since $H_\Gamma$ is
selfadjoint and compact, we have from Fredholm operator theory that the
operator $I+H_\Gamma$ is invertible iff ${\rm ker}\,H_\Gamma = \{0\}$.
The latter is the case since $H_\Gamma$ is positive which implies $\langle (I+H_\Gamma)u, u\rangle \geq \langle u,u \rangle$.
Hence, the inverse $(I+H_\Gamma)^{-1}$ exists and, moreover, $(I+H_\Gamma)^{-1} \in \mathcal L_+(\mathcal H)$ with $\|(I+H_\Gamma)^{-1}\|\leq 1$.
The self-adjointness and positivity of $C_\Gamma$ follows immediately and since $C_\Gamma$ is a composition of two nonsingular Hilbert-Schmidt operators and a nonsingular bounded operator, $C^{1/2}$ and $( I + H)^{-1}$, respectively, it is trace class and nonsingular as well.
\end{proof}
By Proposition \ref{propo:tildeC} we can use the covariance operator $C_\Gamma$ for constructing proposal kernels. Specifically, we consider
\begin{equation} \label{equ:gpCN_ansatz}
P(u, \cdot) = N( Au, s^2 C_\Gamma), \qquad s \in [0,1), \; \Gamma \in \mathcal L_+(\mathcal H),
\end{equation}
where $A\colon \mathcal H \to \mathcal H$ denotes a suitably chosen bounded linear operator on $\mathcal H$.
Here $A$ should be chosen such that $P$ is $\mu_0$-reversible,
which means that a Metropolis kernel with proposal $P$ is $\mu$-reversible,
see Section~\ref{sec: pCN}.
By applying \eqref{equ:Gaussian_affine} we obtain in this setting
\[
P(u, \d v)\, \mu_0(\d u)
= N\left( \begin{bmatrix} 0\\ 0 \end{bmatrix}, \begin{bmatrix} C
& CA^* \\ AC & ACA^* + s^2C_\Gamma\end{bmatrix} \right)
\]
and
\[
P(v, \d u)\, \mu_0(\d v)
= N\left( \begin{bmatrix} 0\\ 0 \end{bmatrix},
\begin{bmatrix} ACA^* + s^2 C_\Gamma & AC \\ CA^* & C\end{bmatrix} \right).
\]
Thus, for satisfying \eqref{eq: det_balance} we need to choose $A$
so that
\begin{equation} \label{equ:A_cond}
AC = CA^*, \qquad ACA^* +s^2 C_\Gamma = C.
\end{equation}
By straightforward calculation we obtain as the formal solution to \eqref{equ:A_cond}
\begin{equation} \label{equ:A}
A = A_{\Gamma} = C^{1/2} \; \sqrt{I - s^2 \left( I + H_\Gamma \right)^{-1} } C^{-1/2}.
\end{equation}
The following lemma ensures that this choice of
$A$ yields a well-defined bounded linear operator on $\mathcal H$.
\begin{lem} \label{lem:A}
Let the assumptions of Proposition \ref{propo:tildeC} be satisfied and let $s\in[0,1)$.
Then \eqref{equ:A} defines a bounded linear operator $A_{\Gamma}:\mathrm{Im}\, C^{1/2} \to \mathcal H$.
\end{lem}
The well-definedness of $A_{\Gamma}:\mathrm{Im}\, C^{1/2} \to \mathcal H$ follows rather easily whereas its boundedness is not trivial. Namely, one easily can construct a bounded $B \in \mathcal L(\mathcal H)$ such that $C^{1/2}B C^{-1/2}$ is unbounded on $\mathrm{Im}\, C^{1/2}$.
Since the proof of Lemma \ref{lem:A} is rather technical, it is postponed to Appendix \ref{sec:proof_lem:A}.
Lemma \ref{lem:A} allows us now to extend $A_\Gamma$ to $\mathcal H$ by continuation, because the Cameron-Martin space $\mathrm{Im}\, C^{1/2}$ is a dense subspace of $\mathcal H$.
For simplicity we denote this continuous extension again by $A_\Gamma:\mathcal H\to\mathcal H$.
\begin{defi}[gpCN proposal] \label{def:gpCN_proposal}
For $s \in [0,1)$ and $\Gamma \in \mathcal L_+(\mathcal H)$
the \emph{generalized pCN proposal kernel} is given by
\begin{equation} \label{equ:gpCN}
P_{\Gamma}(u, \cdot) := N(A_{\Gamma} u, s^2 C_\Gamma).
\end{equation}
\end{defi}
For the zero operator $\Gamma = 0$ we recover
the pCN proposal.
By Lemma~\ref{lem:A} and the arguments given
in Section~\ref{sec: pCN}
we obtain the following important result.
\begin{cor} \label{cor:gpCN}
Let $\mu_0 = N(0,C)$
and $\mu$ be given by \eqref{equ:mu}.
Let the assumptions of Lemma \ref{lem:A} be satisfied.
Then, a gpCN proposal kernel $P_{\Gamma}$ given by \eqref{equ:gpCN}
and an acceptance probability
$\alpha(u,v) = \min\left\{1, \exp( \Phi(u) - \Phi(v))\right\}$
induce a $\mu$-reversible Metropolis kernel denoted by $M_{\Gamma}$.
\end{cor}
For simplicity we also call the Metropolis algorithm with transition kernel $M_{\Gamma}$ just gpCN Metropolis.
There are connections of the gpCN Metropolis to other
recently developed Metropolis algorithms for general Hilbert spaces which also use more sophisticated choices for the proposal than the pCN proposal.
The following two remarks address these connections.
\begin{remark}
The gpCN proposals form a subclass of the
\emph{operator weighted proposals} introduced in \cite{CuiEtAl2014},\cite{Law2014}.
The particular form of the gpCN proposal
allows us to derive properties such as boundedness of the ``proposal mean operator''
$A_\Gamma$ and the convergence of
the resulting Markov chain, see Section \ref{sec:Conv}.
These issues were left open in \cite{CuiEtAl2014},\cite{Law2014}.
\end{remark}
\begin{remark}
In \cite{PinskiEtAl2014} the authors compute a Gaussian measure $\mu_* = N(m_*, C_*)$ which comes closest
to $\mu$ w.r.t. the Kullback-Leibler distance.
The admissible class of Gaussian measures
considered there is closely related to our
parametrized proposal covariances $C_\Gamma$,
although their class of Gaussian measures is slightly larger.
The measure $\mu_*$ is then used to construct a proposal kernel
$P_*(u,\cdot) = N(m_* + \sqrt{1-s^2}(u-m_*),s^2C_*)$ for Metropolis algorithms.
Note that $P_*$ is not $\mu_0$-reversible but $\mu_*$-reversible,
since it is a pCN proposal given
the prior $\mu_*$.
In order to obtain a $\mu$-reversible Metropolis
kernel the authors need to adapt the acceptance probability
by including terms of $\frac{\d \mu_*}{\d \mu_0}$, cf. Section \ref{subsec: loc_gpCN}.
Thus, the authors of \cite{PinskiEtAl2014} also use a different covariance operator than the prior covariance in a pCN proposal in order to increase the efficiency of the Metropolis algorithm.
The difference to our approach is the way they ensure the $\mu$-reversibility of the algorithm.
They keep the mean of the original pCN proposal and modify the acceptance probability whereas we modify also the mean of the proposal to maintain its $\mu_0$-reversibility and, therefore, can leave the acceptance probability unchanged.
\end{remark}
\subsection{Numerical illustrations} \label{subsec: numerics}
We illustrate the gpCN Metropolis
algorithm for approximating samples of a posterior distribution in Bayesian inference.
In particular, we compare different Metropolis algorithms
and investigate which of these
perform independently of the state space dimension and of the variance of the involved noise.
We consider the same setting and inference problem as in \cite[Section 6.1]{PinskiEtAl2014}.
Assume noisy observations $y_j = p(0.2j) + \varepsilon_j$ with $j=1,\ldots,4$, of the solution $p$ of
\begin{equation} \label{equ:PDE}
\frac \d{\d x}\left( \e^{u(x)}\, \frac \d{\d x}p(x)\right) = 0, \qquad p(0) = 0,\; p(1) = 2,
\end{equation}
on $D=[0,1]$ are given and we want to infer $u$.
Here the $\varepsilon_j$ are independent realizations of
the normal distribution $N(0, \sigma^2_\varepsilon)$.
We place a Gaussian prior $N(0,\Delta^{-1})$
with $\Delta = \frac {\d^2}{\d x^2}$ on the completion $\mathcal H_c$ of $H_0^1(D) \cap H^2(D)$ in $L^2(D)$.
Recall that $(\Omega,\mathcal{F},\mathbb{P})$ is a probability space
and let $U\colon\Omega \to \mathcal H_c \subset L^2(D)$
be a random function with distribution $N(0,\Delta^{-1})$.
This allows us to represent the random function $U$ as
\begin{equation} \label{equ:KLE}
U(\omega)(x)
= \frac{\sqrt 2}{\pi} \sum_{k=1}^\infty \xi_k(\omega) \sin(k \pi x),
\qquad \xi_k \sim N(0,k^{-2}),
\end{equation}
$\mathbb{P}$-a.s. where all random variables $\xi_k$ are independent.
Thus, inference on $u$ is equivalent to inference on $\boldsymbol{\xi} = (\xi_k)_{k\in\mathbb{N}}$.
This leads to the prior $\mu_0$ for $\boldsymbol{\xi}$ on $\mathcal H := \ell^2(\mathbb{R})$ given by $\mu_0 = N(0,C)$ with
$C=\mathrm{diag}\{k^{-2}: k\in\mathbb{N}\}$.
Further, we denote by $\mu$ the resulting conditional distribution of $\boldsymbol{\xi}$
given the observed data $y_1,\dots,y_4$.
The measure $\mu$ is given by a density of the form \eqref{equ:mu} with $\Phi$ as in \eqref{equ:Phi} where $\Sigma = \sigma^2_\varepsilon I$ and $G(\boldsymbol{\xi})$ is the mapping
\[
\boldsymbol{\xi} \mapsto u(\cdot, \boldsymbol{\xi}) \mapsto p(\cdot, \boldsymbol{\xi}) \mapsto (p(0.2j, \boldsymbol{\xi}))_{j=1}^4.
\]
We test the performance of $\mu$-reversible Metropolis algorithms for computing expectations w.r.t. $\mu$ of a function
$f\colon\ell^2(\mathbb{R})\to\mathbb{R}$.
We consider four Metropolis algorithms denoted by RW, pCN, GN-RW and
gpCN with different proposal kernels:
\begin{itemize}
\item
RW: Gaussian random walk proposal $P_1(\boldsymbol{\xi}, \cdot) = N(\boldsymbol{\xi}, s^2 C)$,
\item
pCN: pCN proposal $P_2(\boldsymbol{\xi}, \cdot) = N(\sqrt{1-s^2}\boldsymbol{\xi}, s^2 C)$,
\item
GN-RW: Gauss-Newton random walk proposal $P_3(\boldsymbol{\xi}, \cdot) = N(\boldsymbol{\xi}, s^2 C_\Gamma)$,
\item
gpCN: gpCN proposal $P_4(\boldsymbol{\xi}, \cdot) = N(A_{\Gamma}\boldsymbol{\xi}, s^2 C_\Gamma)$.
\end{itemize}
Here we choose $\Gamma = \sigma_{\varepsilon}^{-2} LL^\top$
with $L = \nabla G(\boldsymbol{\xi}_\mathrm{MAP})$ and
\[
\boldsymbol{\xi}_\mathrm{MAP} =
\operatornamewithlimits{argmin}_{\xi\in \Image C^{1/2}}
\left(\sigma_\varepsilon^{-2}
\abs{\mathbf y - G(\boldsymbol{\xi})}^2 + \| C^{-1/2} \boldsymbol{\xi}\|^2 \right).
\]
The solution of \eqref{equ:PDE} is given by
$p(x) = 2 S_x(\e^{-u})/S_1(\e^{-u})$ with $S_x(f) = \int_0^x f(y) \d y$ and, thus, the gradient $\nabla G(\boldsymbol{\xi})$ can be easily computed by differentiating the explicit formula for $p$ w.r.t. $\boldsymbol{\xi}$.\footnote{In general elliptic PDEs can be solved in a weak sense by variational methods.
Then adjoint methods known from PDE constrained optimization and parameter identification can be employed to compute $\nabla G(\boldsymbol{\xi})$,
see \cite[Chapter 6]{Vogel2002} for details.}
Furthermore, we apply the Levenberg-Marquardt algorithm to solve the above optimization problem for the MAP estimator $\boldsymbol{\xi}_\mathrm{MAP}$.
For all Metropolis algorithms we tune $s$ such that the average acceptance rate is about $0.25$\footnote{
The empirical performance of each algorithm was best for this particular tuning.}.
As a metric for comparison we consider and estimate
the \emph{effective sample size}
\[
\text{ESS} = \text{ESS}(n,f,(\boldsymbol{\xi}_k)_{k\in\mathbb{N}}) = n\left[1+2\sum_{k\geq 0}\gamma_f(k)\right]^{-1}.
\]
Here $n$ is the number of samples taken from a Markov chain $(\boldsymbol{\xi}_k)_{k\in\mathbb{N}}$
with, say, a Metropolis transition kernel $M$
and $\gamma_f$ denotes the autocorrelation
function $\gamma_f(k) = \mathrm{Corr}(f(\boldsymbol{\xi}_{n_0}), f(\boldsymbol{\xi}_{n_0+k}))$
for a quantity of interest $f$.
The value of $\text{ESS}$ corresponds to the number of
independent samples w.r.t. $\mu$
which would approximately yield the same mean squared error as the MCMC estimator $S_{n,n_0}(f)$ for computing $\mathbb{E}_\mu(f)$.
This can be justified under the assumption that $\boldsymbol{\xi}_{n_0}\sim\mu$, since then
by virtue of \cite[Proposition~3.26]{Ru12}
we have
\begin{align*}
\lim_{n\to \infty} n\cdot \mathbb{E}\abs{S_{n,n_0}(f)-\mathbb{E}_\mu(f)}^2
& = \sigma_{f,M}^2,\\
1+2\sum_{k\geq 0}\gamma_f(k) & = \frac {\sigma_{f,M}^2}{\mathbb{E}_\mu(f^2)-\mathbb{E}_\mu(f)^2}
\end{align*}
where $\sigma_{f,M}^2$ denotes the asymptotic variance
of the estimator $S_{n,n_0}(f)$ as in Section \ref{sec:MC}.
For numerical simulations we use an uniform
discretization of $[0,1]$ with $\Delta x = 2^{-9}$
and apply the trapezoidal rule for evaluating integrals w.r.t. $\d x$.
Furthermore, we truncate the expansion \eqref{equ:KLE}
after $N$ terms where we vary
$N$ in order to test the Metropolis algorithms for dimension independent performance.
The noise-free
observations are generated by $u(x) = 2\sin(2\pi x)$.
We also consider different
noise levels $\sigma_\varepsilon$
to examine the effect of smaller variances $\sigma_\varepsilon^2$,
leading to more concentrated posterior distributions $\mu$, to the performance of the Metropolis algorithms.
In all cases we take $n_0=10^5$ as burn-in length and $n=10^6$ as sample size.
We use
$f(\boldsymbol{\xi}) := \int_0^1 \e^{u(x,\boldsymbol{\xi})}\d x$ as the quantity of interest\footnote{
We also studied other functions such as
$f(\boldsymbol{\xi}) = \xi_1$, $f(\boldsymbol{\xi}) = \max_x \e^{u(x,\boldsymbol{\xi})}$ and $f(\boldsymbol{\xi}) = p(0.5,\boldsymbol{\xi})$
but the results of the comparison were essentially the same.}.
To estimate the $\text{ESS}$ we use the initial monotone sequence estimators\footnote{We also estimated the $\text{ESS}$ by batch means ($100$ batches of size $10^4$) to control our simulations. This lead to similar results.},
for details we refer to \cite[Section 3.3]{Geyer1992}.
The results of the simulations are illustrated in Figure~\ref{fig:acrf} and Figure~\ref{fig:ess}.
The former displays the estimated
autocorrelation functions $\gamma_f$
resulting from the four Metropolis algorithms for $N=50$ and $\sigma_\varepsilon=0.1$ in (a), for $N=50$ and $\sigma_\varepsilon=0.01$ in (b),
for $N=400$ and $\sigma_\varepsilon=0.1$ in (c) and for $N=400$ and $\sigma_\varepsilon=0.01$ in (d).
In Figure~\ref{fig:ess} we display the estimated $\text{ESS}$ for varying $\sigma_\varepsilon= 0.1, 0.05, 0.025, 0.01$ with fixed $N = 100$ in (a) and varying $N = 50, 100, 200, 400, 800$ with fixed $\sigma_\varepsilon = 0.1$ in (b).
We see in both figures that the performance of pCN and gpCN is independent of the dimension
and only GN-RW and gpCN perform robustly w.r.t. the noise variance.
Thus, the gpCN Metropolis seems to be the only algorithm with both desirable properties.
Intuitively, the variance independent performance might come from the fact that
our choice of $C_\Gamma$ incorporates the noise covariance $\sigma_\varepsilon^2 I$
in a way as the posterior covariance might depend on.
Thus, the smaller $\sigma_\varepsilon$ becomes, i.e., the more pronounced the change from prior to posterior is, the more pronounced is also the adaptation in the proposal covariance by $C_{\Gamma} = (C^{-1} + \sigma_{\varepsilon}^{-2} LL^\top)^{-1}$.
Moreover, the gpCN performs best among the four algorithms also in absolute terms of the $\text{ESS}$.
\begin{figure}[h]
\hfill
\begin{minipage}{0.49\textwidth}
\centering(a) \\
\includegraphics[width = \textwidth]{ACRF_Q1_N50_sigma01}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering(b) \\
\includegraphics[width = \textwidth]{ACRF_Q1_N50_sigma001}
\end{minipage}
\vspace*{1ex}
\hfill
\begin{minipage}{0.49\textwidth}
\centering(c) \\
\includegraphics[width = \textwidth]{ACRF_Q1_N400_sigma01}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering(d) \\
\includegraphics[width = \textwidth]{ACRF_Q1_N400_sigma001}
\end{minipage}
\hfill
\caption{Autocorrelation of $f$ given
samples generated by the four Metropolis algorithms
denoted by RW, pCN, GN-RW and gpCN for:
(a) state dimension $N = 50$ and noise standard deviation $\sigma_\varepsilon = 0.1$;
(b) $N = 50$ and $\sigma_\varepsilon = 0.01$;
(c) $N = 400$ and $\sigma_\varepsilon = 0.1$;
(d) $N = 400$ and $\sigma_\varepsilon = 0.01$.}
\label{fig:acrf}
\end{figure}
\begin{figure}[h]
\hfill
\begin{minipage}{0.49\textwidth}
\centering(a) \\
\includegraphics[width = \textwidth]{ESS_sigma_IES}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering(b) \\
\includegraphics[width = \textwidth]{ESS_N_IES}
\end{minipage}
\hfill
\caption{Dependence of empirical $\text{ESS}$
for each Metropolis algorithm RW, pCN, GN-RW and gpCN w.r.t.:
(a) noise variance with fixed state dimension $N=100$; (b) state dimension with fixed noise variance $\sigma^2_\varepsilon = 0.01$.}
\label{fig:ess}
\end{figure}
\section{Qualitative comparison of gpCN Metropolis} \label{sec:Conv}
In this section we develop qualitative comparison arguments
for Metropolis algorithms in a general setting
and apply those results to the gpCN Metropolis algorithms.
In particular, we
relate the existence of a spectral gap
for the gpCN to the existence of a spectral gap of the pCN Metropolis.
Here it is worth mentioning
that in \cite{HaStVo14} sufficient conditions
for the latter were proven under additional regularity assumptions on the function $\Phi$ in \eqref{equ:mu}.
With our approach we do not need to rely on those conditions and will benefit from any improvement of the results stated in
\cite{HaStVo14}.
We start with stating a general comparison result
for the spectral gaps of Metropolis algorithms with equivalent proposals.
We then
verify the corresponding assumptions
for the gpCN Metropolis: positivity and equivalence to the pCN proposal.
In order to derive our main theorem, we consider in Section \ref{sec:restrict}
restrictions of the target measure $\mu$ to arbitrary $R$-balls in $\mathcal H$
and prove convergence of the gpCN Metropolis to these
restricted measures.
\subsection{Comparison of spectral gaps} \label{subsec: comp_conduct}
Let $K$ be a $\mu$-reversible transition kernel on $(\mathcal H,\mathcal B(\mathcal H))$,
i.e., the associated Markov operator $K\colon L_2(\mu) \to L_2(\mu)$ is self-adjoint.
Let the largest element of the spectrum $\spec(K\,|\, L_2^0(\mu))$ be given by
\[
\Lambda(K) := \sup\{ \lambda \colon \lambda \in \spec(K\,|\, L_2^0(\mu)) \}
\]
and define the \emph{conductance of $K$} (w.r.t. $\mu$) by
\[
\varphi(K) := \inf_{\mu(A)\in (0,1/2]} \frac{\int_A K(u,A^c)\mu(\d u)}{ \mu(A)}.
\]
Under the assumptions above the Cheeger inequality for Markov operators,
see \cite{LawlerSokal1988}, given by
\begin{equation} \label{eq:Cheeger}
\frac{\varphi(K)^2}{2} \leq 1-\Lambda(K) \leq 2 \varphi(K)
\end{equation}
provides a
useful relation between $\Lambda(K)$ and
the conductance $\varphi(K)$.
Let us assume that $M_1$ and $M_2$ are $\mu$-reversible transition kernels
of Metropolis algorithms with the same acceptance probability $\alpha$ and
proposals $P_1$ and $P_2$, respectively.
Then, we obtain the following result.
\begin{lem} \label{lem:conductance}
Let $\mu$ be a probability measure on $(\mathcal H, \mathcal B(\mathcal H))$
and for $i=1,2$ let
\[
M_i(u, \d v) = \alpha(u,v) P_i(u,\d v) +\delta_u(\d v) \,\int_{\mathcal H} (1 - \alpha(u, w)) \, P_i(u,\d w)
\]
be
Metropolis kernels.
Assume that for any $u\in\mathcal H$ the
Radon-Nikodym derivative of
$P_1(u,\d v)$
w.r.t.
$P_2(u,\d v)$ exists, i.e.,
the proposal kernels admit a density
\[
\rho(u, v) = \frac{\d P_1(u)}{ \d P_2(u)}(v),\qquad u,v\in\mathcal H.
\]
If for a number $p> 1$ we have
\begin{equation} \label{eq: kappa}
\kappa_p := \sup_{\mu(A)\in (0,1/2]} \frac{\int_A \int_{A^c} \rho(u,v)^{p}
P_2(u,\d v)\,\mu(\d u)}{\mu(A)} < \infty,
\end{equation}
then
\[
\varphi(M_1) \leq \kappa_p^{1/p}\, \varphi(M_2)^{(p-1)/p}.
\]
\end{lem}
\begin{proof}
Let $A\in \mathcal B(\mathcal H)$ with $\mu(A) \in (0,1/2]$.
Further, let $q=p/(p-1)$ such that $1/q + 1/p=1$.
Then
\begin{align*}
\int_A M_1(u,A^c) \,\d\mu( u)
& = \int_{\mathcal H} \int_{\mathcal H} \mathbf{1}_{A^c}(v)\mathbf{1}_{A}(u)\, \alpha(u,v)\, P_1(u,\d v)\, \d\mu( u)\\
& = \int_{\mathcal H} \int_{\mathcal H} \mathbf{1}_{A^c}(v)\mathbf{1}_{A}(u)\, \alpha(u,v)\,\rho(u;v)\, P_2(u,\d v)\, \d\mu( u).
\end{align*}
Note that $P_2(u,\d v)\mu(\d u)$ is a probability measure
on $(\mathcal H\times \mathcal H,\mathcal B(\mathcal H\times \mathcal H))$
and we can apply H\"older's inequality according to this measure with parameters $p$ and $q$.
Thus, by using $\alpha(u,v) = \alpha(u,v)^{1/q} \alpha(u,v)^{1/p}$ we obtain
\begin{align*}
& \int_A M_1(u,A^c)\, \d\mu(u) \\
&\leq \left( \int_A M_2(u,A^c)\, \d \mu(u)\right)^{1/q} \left( \int_A \int_{A^c} \rho(u,v)^p \alpha(u,v)\, P_2(u,\d v)\,\d\mu(u)\right)^{1/p}\\
& \leq \left( \int_A M_2(u,A^c)\, \d \mu(u)\right)^{1/q} \left( \int_A \int_{A^c} \rho(u,v)^{p} P_2(u,\d v)\,\d\mu(u)\right)^{1/p}
\end{align*}
Dividing by $\mu(A)$, applying $\mu(A)^{-1} = \mu(A)^{-1/q}\,\mu(A)^{-1/p}$ and taking the infimum yields
\[
\varphi(M_1) \leq \varphi(M_2)^{1/q} \kappa_p^{1/p}.
\]
\end{proof}
Employing comparison inequalities in terms of the conductance
is not an entirely new idea, see for example \cite[Proof of Theorem~4]{LeeLat2014}.
There the authors obtained a conductance inequality for transition
kernels with bounded Radon-Nikodym derivatives w.r.t. each other.
An immediate consequence of Lemma \ref{lem:conductance}
and \eqref{eq:Cheeger} is the following theorem.
\begin{theo}[Spectral gap comparison] \label{theo:comparison_gap}
Let the assumptions of Lemma \ref{lem:conductance}
be satisfied and let the Markov operators associated with
$M_1$ and $M_2$ be positive and self-adjoint on $L_2(\mu)$.
Then
\[
\left( \frac{\gap(M_1)}{2} \right)^p \leq \kappa_p\, (2\,\gap(M_2))^{(p-1)/2}.
\]
\end{theo}
We apply Theorem \ref{theo:comparison_gap}
to prove our convergence result for
the gpCN Metropolis.
We therefore verify in the following section the condition
that the corresponding Markov operator is positive.
\subsection{Positivity of Metropolis with Gaussian proposals}
Recall that $\langle f,g \rangle_\mu = \int_{\mathcal H} f g\, \d\mu$
denotes the inner-product of $L_2(\mu)$ and that a
Markov operator $K\colon L_2(\mu) \to L_2(\mu)$ is positive
if $\langle Kf, f\rangle_{\mu} \geq 0$
for all $f\in L_2(\mu)$.
\begin{lem}[Positivity of proposals] \label{propo:GaussianRW_positiv}
Let $\mu_0 = N(0,C)$ be a Gaussian measure
on a separable Hilbert space $\mathcal H$ and
let $P(u,\cdot) = N(A u, Q)$ be a $\mu_0$-reversible proposal kernel with a bounded,
linear operator $A: \mathcal H\to\mathcal H$.
If there exists a bounded, linear operator $B: \mathcal H \to \mathcal H$ such that
\[
B^2 = A, \qquad BC = CB^*,
\]
and $D:=C - BCB^*$
is positive and trace class, then,
the Markov operator associated with the proposal $P$ is positive on $L_2(\mu_0)$.
\end{lem}
\begin{proof}
Because of the assumptions on $B$ and $D$ we obtain that the proposal kernel $P_1(u,\cdot) = N(Bu,D)$ is well-defined.
Further, since $BCB^* + D=C$ we derive
\[
P_1(u,\d v) \mu_0(\d u)
= N\left(
\begin{bmatrix} 0\\ 0 \end{bmatrix},
\begin{bmatrix} C & CB^* \\ BC & C\end{bmatrix} \right),
\]
which leads by $BC=CB^*$ to the $\mu_0$-reversibility of $P_1$ and, thus,
to the self-adjointness of its associated Markov operator in $L_2(\mu_0)$.
It remains to prove that $P_1^2 = P$ holds for
the associated Markov operators which then immediately yields the assertion.
The equality of the Markov operators is equivalent to the equality of the measures $P_1^2(u,\cdot)$ and $P(u,\cdot)$ for all $u\in \mathcal H$.
In order to show that $P_1^2(u,\cdot) = P(u,\cdot)$ for all $u\in\mathcal H$, we take $(\xi_n)_{n\in\mathbb{N}}$ to be an i.i.d. sequence with $\xi_1\sim N(0,D)$ and construct an auxiliary Markov chain by
\[
X_{n+1} = B X_{n} + \xi_n, \quad n\geq 1,
\]
where $X_1=u$ for an arbitrary $u\in \mathcal H$.
The transition kernel of the chain $(X_n)_{n\in\mathbb{N}}$ is the kernel $P_1$.
In particular, for $G\in \mathcal{B}(\mathcal H)$ holds $\mathbb{P}[X_3 \in G] = P_1^2(u,G)$.
By
\[
X_3 = B X_2 + \xi_2 = B^2 u + B \xi_1 + \xi_2
\]
and $B\xi_1 + \xi_2 \sim N(0,BDB^*+D)$ we obtain $X_3\sim N(B^2u,BDB^*+D)$.
Due to the assumptions we have $B^2 = A$ and
\[
BDB^*+D = B(C-BCB^*)B^* + C-BCB^* = C-ACA^*.
\]
The last step $C-ACA^* = Q$ follows by the assumed $\mu_0$-reversibility of $P$, because we know from Section \ref{sec:well_gpCN} that $P$ being $\mu_0$-reversible is equivalent to $A$ and $Q$ satisfying $AC = CA^* $ and $ACA^* + Q = C$.
We thus arrive at $X_3 \sim N(Au, Q)$ which proves $P_1^2(u,\cdot) = P(u,\cdot)$.
\end{proof}
The next lemma extends the previous result
to Markov operators associated with Metropolis algorithms.
The proof follows by the same line of arguments as developed
in \cite[Section~3.4]{RuUl13} and is therefore omitted.
\begin{lem}[Positivity of Metropolis kernels] \label{propo:MH_positiv}
Let $\mu$ be a measure on $\mathcal H$ given by \eqref{equ:mu} and let $P$ be a $\mu_0$-reversible proposal kernel whose associated Markov operator is positive on $L_2(\mu_0)$.
Then the Markov operator associated with
a $\mu$-reversible Metropolis kernel
\[
M(u,\d v) = \alpha(u,v) P(u,\d v)
+ \delta_{u}(\d v) \int_{\mathcal H} (1-\alpha(u,w))P(u,\d w)
\]
with $\alpha(u,v) = \min\{1,\frac{\d\mu}{\d\mu_0}(v) \frac{\d\mu_0}{\d\mu}(u) \}$
is positive on $L_2(\mu)$.
\end{lem}
The previous two lemmas lead to the following result about the gpCN Metropolis.
\begin{theo}[Positivity of gpCN Metropolis] \label{theo:gpCN_positiv}
Let $\mu_0 = N(0,C)$ and $\mu$ as in \eqref{equ:mu} and let $M_{\Gamma}$ denote the gpCN Metropolis kernel as in Corollary \ref{cor:gpCN}. Then the associated Markov operator $M_{\Gamma}$ is self-adjoint and positive on $L_2(\mu)$.
\end{theo}
\begin{proof}
It is enough to verify the assumptions of Lemma~\ref{propo:GaussianRW_positiv} for the gpCN proposal.
Recall that $P_{\Gamma}(u,\cdot) = N(A_{\Gamma}u, s^2C_\Gamma)$
which is $\mu_0$-reversible by construction with bounded
$A_{\Gamma} = C^{1/2} \sqrt{I-s^2 (I+H_\Gamma)^{-1}} C^{-1/2}$.
By choosing
\[
B := C^{1/2} \sqrt[4]{I-s^2 (I+H_\Gamma)^{-1}} C^{-1/2},
\]
we obtain $B^2 = A_{\Gamma}$ and $BC = CB^*$.
Moreover,
\[
D= C-BCB^* = C^{1/2} (I-\sqrt{I-s^2(I+H_\Gamma)^{-1}}) C^{1/2}.
\]
The eigenvalues of $I-\sqrt{I-s^2(I+H_\Gamma)^{-1}}$ take the form
$1 - \sqrt{1-\frac{s^2}{1+\lambda}} \geq 0$ with $\lambda\geq0$ being an eigenvalue of $H_\Gamma$.
Thus, $I-\sqrt{I-s^2(I+H_\Gamma)^{-1}}$
is positive and bounded which yields $D$ being positive and trace class since $D$ is then a product of two Hilbert-Schmidt and one bounded operator.
Thus, the conditions of Lemma \ref{propo:GaussianRW_positiv} are satisfied and the assertion follows.
\end{proof}
\subsection{Density between pCN and gpCN proposal} \label{sec:gpCN_density}
In this section we show
that for any state $u\in \mathcal H$ the gpCN proposal
is equivalent to the pCN proposal in the sense of measures.
Moreover, we will also derive an integrability result for the corresponding density.
For proving the equivalence we need the following technical result.
\begin{lem}\label{lem:Delta_Gamma}
Let the assumptions of Corollary \ref{cor:gpCN} be satisfied and define the bounded, linear operator $\Delta_\Gamma:\mathcal H \to \mathcal H$ by
\begin{equation}\label{equ:Delta_Gamma}
\Delta_\Gamma := A_{0}-A_{\Gamma} = \sqrt{1-s^2} I - C^{1/2} \; \sqrt{I - s^2 \left( I + H_\Gamma \right)^{-1} } C^{-1/2}.
\end{equation}
Then $\mathrm{Im}\, \Delta_\Gamma \subseteq \mathrm{Im}\, C^{1/2}$, i.e., $C^{-1/2}\Delta_\Gamma$ is a bounded operator on $\mathcal H$.
\end{lem}
The proof of this lemma can be found in Appendix \ref{sec:proof_lem:Delta_Gamma}.
It is similar to the proof of Lemma \ref{lem:A} and again rather technical.
However, Lemma \ref{lem:Delta_Gamma}
ensures that we can apply the Cameron-Martin theorem,
Theorem~\ref{thm: Cameron_Martin_form} in Appendix \ref{sec:Gaussian},
in the proof of the following result.
The other main tool for
deriving the next theorem is a
variant of the Feldman-Hajek theorem
as stated in Theorem \ref{thm: Feldman_form} in Appendix \ref{sec:Gaussian}.
\begin{theo}[Density of pCN w.r.t. gpCN] \label{theo:gpCN_density}
With the notation and assumptions of Corollary \ref{cor:gpCN} holds the following.
\begin{enumerate}
\item
The measures $\mu_0 = N(0,C)$
and $\mu_\Gamma = N(0,C_\Gamma)$ are equivalent with
\begin{equation} \label{eq: density_fh}
\pi_{\Gamma}(v) := \frac {\d \mu_{0}}{\d \mu_{\Gamma}}(v)
= \frac {\exp\left( \frac 12 \langle \Gamma v, v \rangle\right)}{\sqrt{\det(I+H_\Gamma)}}.
\end{equation}
\item
For $u\in\mathcal H$ the measures $P_{0}(u,\cdot)$
and $P_{\Gamma}(u,\cdot)$ are equivalent with
\begin{equation} \label{eq: density_fh_cm}
\frac {\d P_{0}(u)}{\d P_{\Gamma}(u)}(v)
= \pi_{\mathrm{CM}}\Big( \Delta_\Gamma u,
\frac {1}s(v - A_{\Gamma}u) \Big) \,
\pi_{\Gamma}\Big( \frac {1}s(v - A_{\Gamma}u) \Big)
\end{equation}
where $\Delta_\Gamma$ as in \eqref{equ:Delta_Gamma} and
\begin{equation} \label{eq: density_cm}
\pi_{\mathrm{CM}}(h,v) := \exp\left( - \frac 12 \|C^{-1/2} h\|^2 + \langle C^{-1} h, v \rangle \right).
\end{equation}
(The subscript in $\pi_{\mathrm{CM}}$ indicates the Cameron-Martin formula.)
\end{enumerate}
\end{theo}
\begin{proof}
We prove \eqref{eq: density_fh} by verifying the assumptions of Theorem \ref{thm: Feldman_form} from Appendix \ref{sec:Gaussian}.
We observe
\[
I - C^{-1/2} C_\Gamma C^{-1/2} = I - (I+H_\Gamma)^{-1}
\]
and set $T_\Gamma :=I - (I+H_\Gamma)^{-1}$.
The eigenvalues $(t_n)_{n\in\mathbb{N}}$ of the self-adjoint operator $T_\Gamma$ are given by
\[
t_n = 1 - \frac 1{1+\lambda_n}= \frac {\lambda_n}{1+\lambda_n} < 1
\]
where $(\lambda_n)_{n\in\mathbb{N}}$ are the eigenvalues of the positive
trace class operator $H_\Gamma$.
Thus, $T_\Gamma$ is also trace class and satisfies $\langle T_\Gamma u, u\rangle < \|u\|^2$ for any $u\in\mathcal H$.
Then, the assertion follows by Theorem \ref{thm: Feldman_form} and
\[
T_\Gamma (I-T_\Gamma)^{-1} = \left( I - (I+H_\Gamma)^{-1} \right) (I+H_\Gamma) = H_\Gamma
\]
as well as
\[
\langle H_\Gamma \, C^{-1/2}v, C^{-1/2}v \rangle = \langle \Gamma v, v \rangle \qquad \forall v \in \mathcal H.
\]
To show the equivalence of $P_{0}(u,\cdot)$ and $P_{\Gamma}(u,\cdot)$ for any $u\in\mathcal H$ we introduce the auxiliary kernel $K_{\Gamma}(u,\cdot) = N(A_{\Gamma} u, s^2C)$.
The first assertion and a simple change of variables, see Lemma \ref{lem: change_of_variables} in the appendix, lead to
\[
\frac {\d K_{\Gamma}(u)}{\d P_{\Gamma}(u)}(v)
= \pi_{\Gamma}\left( \frac 1s \left[v - A_{\Gamma}u\right] \right), \qquad u,v\in\mathcal H.
\]
Thus, it remains to prove the equivalence of $K_{\Gamma}(u,\cdot)$ and $P_{0}(u,\cdot)$ for any $u\in \mathcal H$.
By the Cameron-Martin formula, see Theorem \ref{thm: Cameron_Martin_form} in Appendix \ref{sec:Gaussian},
this holds iff
\[
\Image(A_{\Gamma} - \sqrt{1-s^2} I) \subseteq \Image(C^{1/2})
\]
which was shown in Lemma \ref{lem:Delta_Gamma}.
Now Theorem \ref{thm: Cameron_Martin_form}
combined with a change of variables, see Lemma \ref{lem: change_of_variables}, then yields
\[
\frac{\d P_{0}(u)}{\d K_{\Gamma}(u)}(v)
= \pi_{\mathrm{CM}}\left( [ \sqrt{1-s^2} I-A_{\Gamma}] u, \frac{1}{s}(v - A_{\Gamma}u) \right)
\]
and the assertion follows by
\[
\frac{\d P_{0}(u)}{\d P_{\Gamma}(u)}(v)
=
\frac{\d P_{0}(u)}{\d K_{\Gamma}(u)}(v)
\frac{\d K_{\Gamma}(u)}{\d P_{\Gamma}(u)}(v).
\qedhere
\]
\end{proof}
Note that Theorem \ref{theo:gpCN_density} implies that for any $\Gamma_1, \Gamma_2\in\mathcal L_+(\mathcal H)$ there exists a density between the two gpCN proposals $P_{\Gamma_1}(u)$ and $P_{\Gamma_2}(u)$.
However, for the application
of Theorem \ref{theo:comparison_gap} we
still have to verify condition \eqref{eq: kappa}.
This is partly addressed in the following result.
\begin{theo}[Integrability of gpCN density] \label{theo:int_rho}
Let the assumptions of Lemma \ref{lem:Delta_Gamma} be satisfied and set
\[
\rho_{\Gamma}(u, v) := \frac{\d P_{0}(u)}{\d P_{\Gamma}(u)}(v), \qquad u,v\in\mathcal H.
\]
Then, for any $0 < p < 1 + \frac {1}{2\|H_\Gamma\|}$
there exist constants $c = c(p, H_\Gamma) < \infty$
and $b = b(p, \|C^{-1/2}\Delta_\Gamma\|) < \infty$ such that
\[
\int_{\mathcal H} \rho^{p}_{\Gamma}(u, v) \, P_{\Gamma}(u, \d v)
\leq c \, \exp\left(\frac{b}{2} \|u\|^2\right).
\]
\end{theo}
\begin{proof}
We employ the same notation as in Theorem \ref{theo:gpCN_density}, i.e.,
let $\mu_0 = N(0,C)$
and $\mu_\Gamma = N(0,C_\Gamma)$ as well as $\pi_{\Gamma}$ and $\pi_{\mathrm{CM}}$ be as in \eqref{eq: density_fh} and \eqref{eq: density_cm}, respectively.
By Theorem \ref{theo:gpCN_density} we know
\[
\rho_{\Gamma}(u, v) = \pi_{\mathrm{CM}}\Big( \Delta_\Gamma u, \frac {1}s(v - A_{\Gamma}u) \Big)\;
\pi_{\Gamma}\Big( \frac {1}s(v - A_{\Gamma}u) \Big).
\]
By first applying a change of variables, see Lemma \ref{lem: change_of_variables}, and then the
Cauchy-Schwarz inequality we obtain
\begin{align*}
\int_{\mathcal H} \rho_{\Gamma}^{p}(u, v) \, P_{\Gamma}(u, \d v)
& = \int_{\mathcal H} \pi^{p}_{\mathrm{CM}}(\Delta_\Gamma u, v) \; \pi^{p}_{\Gamma}(v) \, \mu_\Gamma(\d v)\\
& = \int_{\mathcal H} \pi^{p}_{\mathrm{CM}}(\Delta_\Gamma u, v) \; \pi^{p-1}_{\Gamma}(v) \, \mu_0(\d v)\\
& \leq \left( \int_{\mathcal H} \pi^{2p}_{\mathrm{CM}}(\Delta_\Gamma u, v) \mu_0(\d v) \right)^{1/2} \;
\left( \int_{\mathcal H} \pi^{2p-2}_{\Gamma}(v) \mu_0(\d v) \right)^{1/2}.
\end{align*}
Furthermore, we have by applying \eqref{equ:int_Wu} from Appendix \ref{sec:Gaussian}
\begin{align*}
\int_{\mathcal H} \pi^{2p}_{\mathrm{CM}}\Big( \Delta_\Gamma u, v\Big) \, \mu_0(\d v)
& = \int_{\mathcal H} \e^{- \frac {2p}2 \|C^{-1/2}\Delta_\Gamma u \|^2} \e^{2p\; \langle C^{-1} \Delta_\Gamma u, v\rangle} \, \mu_0(\d v)\\
& = \exp\left((2p^2-p) \|C^{-1/2} \Delta_\Gamma u\|^2\right).
\end{align*}
We apply
$\|C^{-1/2} \Delta_\Gamma u\| \leq \|C^{-1/2} \Delta_\Gamma\|\, \|u\|$
and set
\[
b := (2p^2-p)\,\|C^{-1/2} \Delta_\Gamma\|.
\]
Note, that $b\leq 0$ for $p\leq \frac 12$.
Due to the assumptions on $p$ we have
\[
\langle (2p-2)H_\Gamma v, v\rangle < \frac {\langle H_\Gamma v, v\rangle}{\|H_\Gamma\|} \leq \|v\|^2, \qquad v\in \mathcal H.
\]
Thus, we can apply \eqref{equ:int_Tu} from Appendix \ref{sec:Gaussian} and get
\begin{align*}
\int_{\mathcal H} \pi^{2p-2}_{\Gamma}(v) \mu_0(\d v)
& = \int_{\mathcal H} \frac {\exp\left( \frac 12 \langle (2p-2)H_\Gamma \, C^{-1/2}v, C^{-1/2}v \rangle\right)}{\det(I+H_\Gamma)^{(2p-2)/2}} \, \mu_0(\d v)\\
& = \left( \det(I-(2p-2)H_\Gamma) \; \det(I+H_\Gamma)^{2p-2}\right)^{-1/2}\\
& =: c^2.
\end{align*}
Since $H_\Gamma$ is positive and trace class, $\det(I+H_\Gamma)$
is well-defined (see Appendix \ref{sec:Gaussian}) and $\det(I+H_\Gamma) \in [1,\infty)$.
Furthermore, due to $\langle (2p-2)H_\Gamma v, v\rangle < \|v\|^2$,
the eigenvalues of $(2p-2)H_\Gamma$ lie within $[0,1)$
which ensures that $\det(I-(2p-2)H_\Gamma)>0$ and, hence $0 < c^2<\infty$.
This proves the assertion.
\end{proof}
Thus, the above theorem allows
us to estimate the integral in \eqref{eq: kappa}.
We obtain for $0 < p < 1 + 1/(2\|H_\Gamma\|)$ that
\begin{align*}
\int_A \int_{A^c} \rho_{\Gamma}(u;v)^{p}
P_{\Gamma}(u,\d v)\,\mu(\d u)
\leq c\, \int_A \exp\left(\frac {b}2\, \|u\|^2\right) \mu(\d u).
\end{align*}
Unfortunately, if we divide the right-hand side by $\mu(A)$
and take the supremum over all $\{A: 0 < \mu(A) \leq 0.5\}$ this is
unbounded.
In the next section we introduce restrictions of the target measure for which we
can circumvent this problem.
\subsection{Restrictions of the target measure} \label{sec:restrict}
In order to show boundedness of $\kappa_p$ from \eqref{eq: kappa}
for the gpCN proposal we consider restrictions of the target measure to bounded sets.
For appropriately chosen sets, the restricted measures become arbitrarily close to the target measure.
Let $R\in (0,\infty]$ and set
\[
\mathcal H_R := \{ u\in \mathcal H \colon \norm{u}{}< R \}.
\]
\begin{defi}[Restricted measure] \label{defi:mu_R}
Let $\mu$ be a probability measure on $(\mathcal H, \mathcal B(\mathcal H))$ and $R\in (0,\infty]$.
We define its restriction to $\mathcal H_R$ as the probability measure $\mu_R$ on $\mathcal H$ given by
\begin{equation}\label{equ:mu_R}
\mu_R(\d u) := \frac{1}{\mu(\mathcal H_R)} \mathbf{1}_{\mathcal H_R}(u) \mu(\d u).
\end{equation}
\end{defi}
For sufficiently large $R$ the measure $\mu_R$ is close to $\mu$, because
\[
\| \mu_R - \mu \|_{\text{tv}}
= \int_{\mathcal H} \left|\frac{\d \mu_R}{\d \mu}(u) - 1 \right| \d\mu( u)
= \mu(\mathcal H_R^c) + 1 - \mu(\mathcal H_R)
= 2\mu(\mathcal H_R^c)
\]
and since $\mu$ is a probability
measure on $(\mathcal H,\mathcal{B}(\mathcal H))$
there exists for any $\varepsilon>0$ a number $R>0$
such that $2\mu(\mathcal H_R^c)< \varepsilon$.
Let us mention here that restricted measures appear,
for example, also in \cite[Equation (3.5)]{Bou-Rabee13}
and in the recent work \cite{HuYaLi2015},
in order to analyze the convergence of Metropolis-Hastings based algorithms.
We ask now whether good convergence properties of a $\mu$-reversible transition kernel $K$ are inherited on a
suitably modified $\mu_R$-reversible
transition kernel $K_R$.
\begin{defi}[Restricted transition kernel] \label{defi:K_R}
Let $K$ be a transition kernel on $\mathcal H$ and $R\in (0,\infty]$.
We define its restriction to $\mathcal H_R$ as the following transition kernel
$K_R\colon \mathcal H \times \mathcal B(\mathcal H) \to [0,1]$
given by
\begin{equation}\label{equ:K_R}
K_R(u,\d v) := \mathbf{1}_{\mathcal H_R}(v)\, K(u, \d v) + K(u, \mathcal H_R^c) \, \delta_u(\d v).
\end{equation}
\end{defi}
Note that if $K$ is $\mu$-reversible, then $K_R$ is $\mu_R$-reversible and if $K$ is of Metropolis form \eqref{eq: metro_kern}, then so is $K_R$.
\begin{propo} \label{propo: K_R_reversible}
Let $\mu$ be a probability measure on $(\mathcal H, \mathcal B(\mathcal H))$ and $K$ be a $\mu$-reversible transition kernel.
Then for any $R>0$ the transition kernel $K_R$ given in \eqref{equ:K_R} is
$\mu_R$-reversible with $\mu_R$ as in \eqref{equ:mu_R}.
Moreover, for a Metropolis kernel $M$ of the form \eqref{eq: metro_kern}
the corresponding restricted kernel $M_R$ is again a Metropolis kernel
\[
M_R(u, \d v) = \alpha_R(u,v)P(u,\d v) + \delta_u(\d v)\left(1-\int_{\mathcal H} \alpha_R(u,w)P(u,\d w)\right)
\]
with $\alpha_R(u,v) := \mathbf{1}_{\mathcal H _R}(v) \alpha(u,v)$.
\end{propo}
\begin{proof}
Recall that $K$ is $\mu$-reversible iff
\[
\int_A K(u,B)\,\d\mu(u) = \int_B K(u,A)\, \d \mu(u), \qquad \forall A,B\in \mathcal{B}(\mathcal H).
\]
Let $A,B \in \mathcal B(\mathcal H)$. We have
\begin{align*}
& \int_A K_R(u,B) \, \d\mu_R(u)
= \int_A K(u,B\cap \mathcal H_R)\,\d\mu_R(u)
+ \int_{A\cap B} K(u,\mathcal H_R^c)\, \d\mu_R(u)\\
& \qquad =
\frac{1}{\mu(\mathcal H_R)} \int_{A\cap \mathcal H_R} K(u,B\cap \mathcal H_R)\, \d\mu(u)
+\int_{A\cap B} K(u,\mathcal H_R^c)\,\d \mu_R(u).
\end{align*}
Because of the $\mu$-reversibility of $K$ we can interchange $A$ and $B$ which leads to the first assertion.
The second statement follows by
\begin{align*}
M_R(u,\d v)
& = \mathbf{1}_{\mathcal H_R}(v) M(u,\d v) + \delta_u(\d v) M(u,\mathcal H_R^c) \\
& = \mathbf{1}_{\mathcal H_R}(v) \alpha(u,v) P(u,\d v) \\
& \qquad + \delta_u(\d v)\left(1-\int_{\mathcal H} \alpha(u,w)P(u,\d w) + \int_{\mathcal H_R^c} \alpha(u,w)P(u,\d w) \right)\\
& = \mathbf{1}_{\mathcal H_R}(v) \alpha(u,v)P(u,\d v) +
\delta_u(\d v)\left(1-\int_{\mathcal H_R} \alpha(u,w)P(u,\d w)\right).
\end{align*}
\end{proof}
Now we ask whether a spectral gap of $K$ on $L_2(\mu)$ implies a spectral gap
of the Markov operator associated with
$K_R$ on $L_2(\mu_R)$.
Note that
\[
K_R f(u) = \int_{\mathcal H} f(v)\, K_R(u,\d v)
= \int_{\mathcal H_R} f(v)\, K(u,\d v) + f(u)\,K(u,\mathcal H_R^c).
\]
We have the following relation between $\|K_R \|_{\mu_R}$ and $\| K \|_{\mu}$.
\begin{lem} \label{lem:K_R_gap}
With the notation and assumptions from above holds
\begin{equation} \label{eq: rel_spec_gap}
\| K_R \|_{\mu_R}
\leq \| K \|_{\mu} + \sup_{u\in \mathcal H_R} K(u,\mathcal H_R^c).
\end{equation}
Furthermore, if the Markov operator $K$ is positive on $L_2(\mu)$, then $K_R$ is also positive on $L_2(\mu_R)$.
\end{lem}
\begin{proof}
For $f \in L_2(\mu_R)$ let
\[
(Ef) (u) := \mathbf 1_{\mathcal H_R}(u) f(u) \in L_2(\mu).
\]
Note that
$\|f\|_{2,\mu_R} =\frac{1}{\sqrt{\mu(\mathcal H_R)}}\, \|Ef\|_{2,\mu}$ and for
$\int_{\mathcal H_R} f \, \d \mu_R =0$ follows $\int_{\mathcal H} Ef \, \d \mu = 0$.
Further, for any $f\in L_2(\mu_R)$ we have
\begin{align*}
\| K_R f \|^2_{2,\mu_R}
& = \int_{\mathcal H_R} \left| \int_{ \mathcal H_R} f(v)\, K(u,\d v)
+ f(u)\,K(u, \mathcal H_R^c)\right|^2 \d\mu_R(u)\\
& = \int_{\mathcal H_R} \left| \int_{\mathcal H} Ef(v)\, K(u,\d v) + Ef(u) \,K(u,\mathcal H_R^c) \right |^2 \d\mu_R(u)\\
& = \| K(Ef) + g\, Ef \|^2_{2,\mu_R}
\end{align*}
with $g(u) := \mathbf 1_{\mathcal H_R}(u) \, K(u,\mathcal H_R^c)$.
Then
\begin{align*}
\frac{\| K_R f \|_{2,\mu_R}}{\| f \|_{2,\mu_R}}
& = \frac{ \| K(Ef) + g\,Ef \|_{2,\mu_R}}{\| Ef \|_{2,\mu_R}}
= \frac{ \| E(K(Ef)) + g\,Ef \|_{2,\mu}}{\| Ef \|_{2,\mu}}\\
& \leq \frac{ \| K(Ef) \|_{2,\mu} + \| g\,Ef \|_{2,\mu}}{\| Ef \|_{2,\mu}}\\
& \leq \frac{ \| K(Ef) \|_{2,\mu}}{\| Ef \|_{2,\mu}} + \sup_{u\in \mathcal H_R} K(u,\mathcal H_R^c),
\end{align*}
where we applied $\|Ef\|_{2,\mu} \leq \|f\|_{2,\mu}$ in the first inequality.
By taking the supremum over all $f \in L^0_2(\mu_R)$ and because of $E(L^0_2(\mu_R)) \subseteq L^0_2(\mu)$ the first assertion follows.
Moreover, we have for $f \in L_2(\mu_R)$ that
\begin{align*}
\langle K_R f, f\rangle_{\mu_R}
& = \int_{\mathcal H} K_R f(u) \, f(u) \, \mu_R(\d u)\\
& = \int_{\mathcal H} \left( \int_{\mathcal H_R} f(v)\, K(u,\d v) + f(u)\,K(u,\mathcal H_R^c)\right) \, f(u) \, \mu_R(\d u)\\
& = \int_{\mathcal H} \int_{\mathcal H} (Ef)(v)\, K(u,\d v) \; (Ef)(u) \, \frac{\mu(\d u)}{\mu(\mathcal H_R)}\\
& \qquad + \int_{\mathcal H} f^2(u)\,K(u,\mathcal H_R^c) \, \mu_R(\d u).
\end{align*}
The second term is always positive since $f^2(u)\,K(u,\mathcal H_R^c) \geq 0$ for all $u\in\mathcal H$ and the first term coincides with $\langle K(Ef), Ef\rangle_{\mu} \, /\mu(\mathcal H_R)$.
Thus, the second statement is proven.
\end{proof}
Lemma \ref{lem:K_R_gap} tells us that there exists an absolute spectral gap of $K_R$ if there exists an absolute spectral gap of $K$ and $\sup_{u\in \mathcal H_R} K(u,\mathcal H_R^c)$ is sufficiently small.
Indeed, we can apply this result to the pCN Metropolis algorithm.
\begin{theo}[Spectral gap of restricted pCN Metropolis] \label{theo:Gap_restricted}
Let $\mu$ be as in \eqref{equ:mu} and let $M_{0}$ denote the $\mu$-reversible pCN Metropolis kernel.
If there exists a spectral gap of $M_{0}$ in $L_2(\mu)$,
then for any $\varepsilon>0$ there exists a number $R\in(0,\infty)$ such that $M_{0,R}$ possesses a spectral gap in $L_2(\mu_R)$, i.e.,
\[
{\rm gap}(M_{0,R}) = 1 - \| M_{0,R} \|_{\mu_R} \geq \gap(M_{0}) - \varepsilon,
\]
where $\mu_R$ as in \eqref{equ:mu_R} and $M_{0,R}$ according to Definition \ref{defi:K_R}.
\end{theo}
\begin{proof}
Given the results of Proposition \ref{propo: K_R_reversible} and Lemma \ref{lem:K_R_gap} it suffices to prove that for any $\varepsilon>0$ there exists an $R>0$ such that $\sup_{u\in \mathcal H_R} M_{0}(u,\mathcal H_R^c) \leq \varepsilon$.
We recall that the proposal kernel of $M_{0}$ is $P_{0}(u,\cdot) = N(\sqrt{1-s^2}u, s^2 C)$
and obtain with $\mu^s := N(0, s^2C)$ that
\begin{align*}
\sup_{u\in \mathcal H_R} M_{0}(u,\mathcal H_R^c)
& \leq \sup_{u\in \mathcal H_R} P_{0}(u,\mathcal H_R^c)
= \sup_{u\in \mathcal H_R} \int_{\| \sqrt{1-s^2} u + v\| \geq R} \d \mu^s(v)\\
& \leq \sup_{u\in \mathcal H_R} \int_{\| \sqrt{1-s^2} u\| + \|v\| \geq R} \d\mu^s(v)\\
& = \sup_{u\in \mathcal H_R} \int_{\|v\| \geq R - \sqrt{1-s^2}\|u\|} \d\mu^s(v)\\
& \leq \int_{\|v\| \geq (1 - \sqrt{1-s^2}) R} \d\mu^s(v)
= \mu_0( \mathcal H_{R_s}^c)
\end{align*}
where $R_s = \frac{1 - \sqrt{1-s^2}}{s}R$ and $\mu_0 = N(0, C)$.
Again, since $\mu_0$ is a probability measure on $\mathcal H$ we know that there exists a number $R$, such
that $\mu_0( \mathcal H_{R_s}^c)\leq \varepsilon$.
\end{proof}
\subsection{Spectral gap of restricted gpCN Metropolis}
Now, we are able to formulate and to prove our main convergence result.
\begin{theo}[Convergence of restricted gpCN Metropolis] \label{theo:gpCN_conv}
Let $\mu$ be as in \eqref{equ:mu} and assume that the pCN Metropolis kernel possesses a spectral gap in $L_2(\mu)$, i.e., $\gap(M_{0})>0$.
Then,
for any $\Gamma \in \mathcal L_+(\mathcal H)$ and any $\varepsilon \in (0,\gap(M_{0}))$
there exists a number $R_0 = R_0(\varepsilon)\in(0,\infty)$ such that for any $R\geq R_0$ holds
\[
\norm{\mu-\mu_R}{\text{tv}} < \varepsilon
\quad \text{and} \quad
\gap(M_{\Gamma,R})>0
\]
where $\gap(M_{\Gamma,R}) = 1 - \| M_{\Gamma,R} \|_{\mu_R}$ denotes the spectral gap of $M_{\Gamma,R}$ in $L_2(\mu_R)$.
\end{theo}
\begin{proof}
By Theorem~\ref{theo:Gap_restricted} we have that for any $\varepsilon \in (0,\gap(M_0))$ there exists a number $R_0\in (0,\infty)$ such that for any $R\geq R_0$ holds
\[
\norm{\mu-\mu_R}{\text{tv}} \leq \varepsilon
\quad \text{and} \quad
\gap(M_{0,R}) >0.
\]
Moreover, Proposition~\ref{propo: K_R_reversible}, Theorem~\ref{theo:Gap_restricted}
and Theorem~\ref{theo:gpCN_positiv} yield
that for any $\Gamma \in \mathcal L_+(\mathcal H)$ the Markov operator associated
to $M_{\Gamma,R}$ is self-adjoint and positive on $L_2(\mu_R)$.
In particular, $M_{\Gamma,R}$
is again a Metropolis kernel with proposal $P_\Gamma$ and acceptance probability $\alpha_R$.
Thus, in order to apply Theorem \ref{theo:comparison_gap} to $M_{0,R}$ and $M_{\Gamma,R}$
it remains to verify that there exists a $p>1$ so that
\[
\kappa_{p,R} := \sup_{\mu_R(A)\in (0,1/2]} \frac{\int_A \int_{A^{c}}
\rho_{\Gamma}(u,v)^p\,P_{\Gamma}(u,\d v)\, \d\mu_R(u)}{\mu_R(A)} < \infty
\]
where $\rho_{\Gamma}(u, v) = \frac{\d P_{0}(u)}{\d P_{\Gamma}(u)}(v)$.
By Theorem \ref{theo:int_rho} we have for any $p < 1 + \frac {1}{2\|H_\Gamma\|}$ that
\begin{align*}
\kappa_{p,R}
& \leq \sup_{\mu_R(A)\in (0,1/2]} \frac{\int_A c \exp\left(\frac {b}2\, \|u\|^2\right) \d\mu_R(u)}{\mu_R(A)}
\leq c \exp\left(\frac {b}2\, R^2\right) <\infty.
\end{align*}
Hence, Theorem~\ref{theo:comparison_gap} leads to
\[
\gap(M_{\Gamma,R})^{(p-1)/2} \geq \frac{1}{2^{(3p-1)/2}} \; \frac{\gap(M_{0,R})^{p}}{\kappa_{p,R}} > 0
\]
which proves the assertion.
\end{proof}
Theorem~\ref{theo:gpCN_conv} tells us that the
corresponding restricted gpCN Metropolis converges exponentially
fast to any, arbitrarily close, restriction $\mu_R$ of $\mu$ whenever
the pCN Metropolis has a spectral gap, e.g., under the conditions of \cite[Theorem~2.14]{HaStVo14}.
In particular, Theorem~\ref{theo:gpCN_conv} is a statement about the inheritance of geometric convergence
from the pCN to the restricted gpCN Metropolis.
We emphasize that a quantitative comparison of their spectral gaps is
not proven.
We provide a lower bound for the spectral gap of $\gap(M_{\Gamma,R})$ in nonlinear terms of the spectral gap of the pCN Metropolis.
Additionally, the stated estimate behaves rather poor in $R$, more precise, it decays exponentially as $R\to \infty$.
Although we argued in the above theorem with restrictions of $\mu$
in order to bound $\kappa_p$ from Theorem~\ref{theo:comparison_gap},
let us mention that, in simulations when $R$ is sufficiently large
one cannot distinguish between $\mu$ and $\mu_R$ as well as between Markov
chains with transition kernels $M_\Gamma$ and $M_{\Gamma,R}$.
Moreover,
we conjecture that the gpCN Metropolis targeting $\mu$ has a strictly positive
spectral gap whenever the pCN Metropolis has one.
Recalling the results of the numerical simulations
in Section~\ref{subsec: numerics} we even conjecture that
the spectral gap of the gpCN Metropolis with suitably chosen $\Gamma \in \mathcal L_+(\mathcal H)$ is much larger than the one of the pCN Metropolis.
\section{Outlook on gpCN proposals with state-dependent covariances} \label{subsec: loc_gpCN}
In this section we comment on state-dependent proposal covariances as they are a natural extension of the idea behind the gpCN proposal.
The advantage of such a state-dependent approach is that
the resulting Metropolis algorithm might be even better adapted to the
target measure by allowing locally different proposal covariances.
For an illustrative motivation of state-dependent proposal
covariances we refer to \cite{GirolamiCalderhead2011},\cite{MartinEtAl2012}
and for recent positive and negative theoretical results we refer to \cite{Livingstone15}.
In the Hilbert space setting we are now able to define MH algorithms
by means of Theorem \ref{theo:gpCN_density}.
Consider the proposal kernel
\begin{equation} \label{equ:Ploc}
P_\mathrm{loc}(u,\cdot) = N(A_{\Gamma(u)} u, s^2 C_{\Gamma(u)})
\end{equation}
where we assume
that for $u\in \mathcal H$ we have $\Gamma(u)\in\mathcal L_+(\mathcal H)$
and that the corresponding mapping $u \mapsto \Gamma(u) $
is measurable.
Further,
by $A_{\Gamma(u)}$ and $C_{\Gamma(u)}$ we denote
the components of the gpCN proposal for $\Gamma = \Gamma(u)$.
Following the heuristic presented
in Section \ref{subsec: Motiv} for Bayesian inference problems
where $\Phi$ in \eqref{equ:mu} is of the form \eqref{equ:Phi},
we could chose for instance
\begin{equation} \label{eq: local_G}
\Gamma(u) = \nabla G(u)^*\, \Sigma^{-1}\, \nabla G(u).
\end{equation}
When considering the measure
$\eta_\mathrm{loc}(\d u , \d v) = P_\mathrm{loc}(u,\d v)\mu_0(\d u)$
we notice that $\eta_\mathrm{loc}$ is no longer a Gaussian measure due to the dependence of $\Gamma$ on $u$.
However, to construct a $\mu$-reversible Metropolis kernel
with the proposal $P_\mathrm{loc}$ above,
we can apply the same trick as in \cite[Theorem 4.1]{BeskosEtAl2008}.
Namely, with $\rho_{\Gamma}(u,v) = \frac{\d P_0(u)}{\d P_\Gamma (u)}(v)$ as given in Theorem \ref{theo:gpCN_density} we obtain
\begin{align*}
P_\mathrm{loc}(u,\d v)\mu_0(\d u)
& = \frac 1{\rho_{\Gamma(u)}(u, v)} \, P_0(u, \d v) \mu_0(\d u)\\
& = \frac {1}{\rho_{\Gamma(u)}(u, v)} \, P_0(v, \d u) \mu_0(\d v)\\
& = \frac {\rho_{\Gamma(v)}(v, u)}{\rho_{\Gamma(u)}(u, v)} \, P_\mathrm{loc}(v, \d u) \mu_0(\d v),
\end{align*}
where we used the $\mu_0$-reversibility of the pCN proposal $P_0$.
Hence, according to the general Metropolis kernel
construction outlined in Section \ref{sec: pCN},
we have that a Metropolis kernel $M_\mathrm{loc}$
with proposal $P_\mathrm{loc}$ and acceptance probability
\begin{equation}\label{equ:aloc}
\alpha_\mathrm{loc}(u,v) = \min \left\{1, \exp(\Phi(u)-\Phi(v)) \; \frac {\rho_{\Gamma(u)}(u, v)}{\rho_{\Gamma(v)}(v, u)} \right\}
\end{equation}
is $\mu$-reversible.
Note, that the same construction can analogously be applied to proposals
of the form
\begin{equation}\label{equ:Ploc2}
P'_\mathrm{loc}(u, \cdot ) = N(\sqrt{1-s^2}u, s^2 C_{\Gamma(u)}),
\end{equation}
where the modified acceptance probability is then given by
\begin{equation}\label{equ:aloc2}
\alpha'_\mathrm{loc}(u,v) = \min \left\{1, \exp(\Phi(u)-\Phi(v)) \;
\frac {\pi_{\Gamma(u)}(\frac 1s [v-A_0u])}{\pi_{\Gamma(v)}(\frac 1s [u-A_0v])}
\right\}
\end{equation}
with $\pi_\Gamma$ as stated in Theorem \ref{theo:gpCN_density}.
The arguments above show that this type of algorithms are well-posed
in infinite dimensions.
Of course, the question arises if the
additional computational costs of evaluating $\Gamma(u)$
and $\rho_{\Gamma(u)}$ or $\pi_{\Gamma(u)}$ in each
step pay off in a significantly higher statistical efficiency.
Related to this concern, one could think of
substituting $\nabla G(u)$ in \eqref{eq: local_G} by
a cheaper approximation in order to reduce the computational work.
This might help to make MH algorithms
with local proposal covariances feasible.
Unfortunately, the tools and results developed and presented in
Section~\ref{sec:Conv} are not sufficient to prove spectral gaps
of these MH algorithms with state-dependent proposals.
The main reason for this is the missing reversibility of the proposals w.r.t. $\mu_0$.
This condition played a key role in Theorem \ref{theo:comparison_gap}
and is the main reason why the analysis of Section~\ref{sec:Conv} is not applicable.
We leave this open for future research.
\subsection*{Acknowledgement}
We thank Oliver Ernst and Hans-J\"org Starkloff for fruitful discussions and valuable
comments. D.R. was supported by the DFG priority program 1324
and the DFG Research training group 1523. B.S. was supported by the DFG priority program 1324.
|
{
"timestamp": "2016-10-13T02:03:12",
"yymm": "1504",
"arxiv_id": "1504.03461",
"language": "en",
"url": "https://arxiv.org/abs/1504.03461"
}
|
\section{Introduction} \label{sec1}
Glitches in pulsars are events in which the rotation period and its derivative (and possibly higher
derivatives as well) undergo an abrupt change in value, on time scales of less than minutes, often
followed by a recovery to approximately the pre glitch values, over time scales of days to tens of
days, or even much longer; see \citep{Shemar1996, Lyne2000, Wong2001, Espinoza2011} for details of
pulsar glitches, their history and their relevance. Glitches are important to study because they are
probably one of the very few methods available to study the internal structure of neutron stars
\citep{Baym1969}; see also \citep{Ruderman1998} and references therein. Glitches are rare events;
in the Crab pulsar (PSR B0531+21 or J0534+2200) they occur once in $\approx 1.6$ years
\citep{Espinoza2011}. A typical glitch in the Crab pulsar involves a very small fractional change
of rotation period (or alternately, rotation frequency) of about $10^{-7}$ to $10^{-9}$
\citep{Wong2001, Espinoza2011}. Coupled with the sudden
onset of a glitch, this implies that frequent and regular pulsar timing observations are required
to study the glitch phenomenon. The very low rotation period of the Crab pulsar ($P \approx 33.5$
ms, or alternately very high rotation frequency $\nu \approx 29.851$ Hz, at the middle of the year
2000) necessitates timing observations at least twice a day, over a period of one year, to properly
analyze a glitch; see \citep{Manchester1977, Backer1986, Lyne2006} for pedagogical reviews of
pulsar timing in general, and analysis of pulsar glitches in particular.
Clearly a dedicated telescope is required to study pulsar glitches. At radio frequencies
this has been done (and continues to be done) for the Crab pulsar by the Jodrell Bank
observatory \citep{Lyne1993}. Over the last $\approx 40$ years, they have accumulated
pulse timing information of the Crab pulsar at 610 and 1400 MHz radio frequencies,
observing it daily, and have published (and continue to update) the monthly timing
ephemeris\footnote{{http://www.jb.man.ac.uk/pulsar/crab.html}} of the Crab pulsar; see
\citep{Lyne1993} for details; {also see \citep{Lyne2015} for the 45 year rotation history
of the Crab pulsar.} The Crab pulsar has also been observed daily by the Green Bank
Telescope, at 327 and 610 MHz radio frequencies \citep{Backer2000, Wong2001}. {Ideally
such work should have been carried out at xray energies, which are not affected by
problems associated with propagation through the interstellar medium, that radio
signals are susceptible to \citep{Lyne1993, Backer2000}. However, xray telescopes are
difficult to build and expensive, in comparison to radio telescopes.} RXTE is one of
the few xray observatories that can time the arrival
of pulses from pulsars to the accuracy required for timing analysis. However RXTE is not a
dedicated pulsar timing observatory, and most Crab pulsar observations of RXTE are spaced,
on the average, two weeks apart. Fortunately, during the period late 1999 to late 2000,
three sets of very closely spaced RXTE observations of the Crab pulsar were available, one
of them being just after glitch of 15 July 2000. It is mainly these three clusters of
observations, and the existence of observations immediately after the glitch, that have
motivated this work.
This work is organized as follows. Section~\ref{sec2} and section~\ref{sec3} describe HEXTE
data and the method of analysis. Section~\ref{sec4} presents phase coherent timing results
for the July 2000 glitch of the Crab pulsar. Section~\ref{sec5} contains discussion and
describes the behavior of some Crab pulsar parameters before and after the glitch.
\section{Observations} \label{sec2}
The HEXTE instrument \citep{Rothschild1998} of RXTE consists of two independent clusters of
detectors, labeled clusters 0 and 1. Each cluster contains four NaI(Tl)/CsI(Na) phoswich
scintillation photon counters, and has a field of view of one degree in the sky. For
practical purposes this instrument is sensitive to photons in the 15 to 240 keV range, and
each photon's arrival time is measured with an accuracy of $\approx 7.6$ $\mu$sec (see
``The ABC of XTE'' guide on the RXTE
website\footnote{heasarc.gsfc.nasa.gov/docs/xte/data\_analysis.html}). During normal
operation, the two clusters switch between the source and a background region of the sky,
such that when one cluster is pointed at the source, the other is pointed at the background,
and vice versa. For pulsar observations another mode of observation is also used, in which
both clusters dwell only on the source. The data used in this work consist of both modes,
but predominantly of the latter kind.
The first observation used in this work was obtained on 18 Dec 1999, and the last on 24 Dec
2000; the corresponding observation identification numbers (ObsID) for the data are
40090-01-01-00 and 50804-01-14-00, respectively. The epoch of the glitch is MJD $51740.656
\pm 0.002$ \citep{Espinoza2011}, which is at $\approx$ 15:45 UTC on 15 July 2000. The pre
glitch data is relatively more frequently observed during the months Dec 1999 and Jan 2000,
but not later. It extends up to {14 May 2000} only (ObsID 50099-01-26-00), which implies that
no observations exist for the two month duration just prior to the glitch. Fortuitously, the
first post glitch observation is on 17 July 2000 at $\approx$ 00:45 UTC (ObsID 50098-01-01-00),
which is just $\approx 1.4$ days after the glitch. From then onward the data is well sampled
(frequently observed) up to 31 Jul 2000 (ObsID 50099-01-02-00), after which the data is under
sampled until 5 Dec 2000 (ObsID 50099-01-11-00), which is for most of the post glitch duration.
Then again the data is well sampled until 24 Dec 2000 (ObsID 50804-01-14-00). There are $50$
ObsID during this period, out of which $2$ were not useful; during ObsID 50100-01-01-05 the
Crab pulsar was completely occulted by the {Earth} (ELV $< 0^\circ$, {which is the instantaneous
angle between the Earth's limb and the astronomical source}); and ObsID 50099-01-05-00F
was obtained on 11 Sept 2000, during the week when there were up to $1^\circ$ errors in the
spacecraft attitude. Six of the remaining 48 ObsIDs have more than $10$\% data
gaps in them, but they have sufficient useful data for our purpose. Thus the Crab pulsar is
under sampled (from the timing point of view) for most of the one year duration under
consideration, except for three brief periods of well sampled data, one at the very beginning
of the pre glitch duration, one just after the glitch, and one at the very end of the
observations used in this work.
In fact the pre and post glitch duration were chosen based partly on the availability of
closely spaced (frequently observed) data. The other reason was for the data to be
sufficiently isolated in time from the previous and the next Crab pulsar glitches, so that
the timing analysis is not corrupted by their residual effects. The previous Crab pulsar
glitch (a small glitch) occurred at $\approx$ 00:29 UTC on 1 Oct 1999, and the next glitch
(a large one) occurred at $\approx$ 01:44 UTC on 24 Jun 2001 \citep{Espinoza2011}; {the
very small glitch at $\approx$ 18:00 UTC on 17 Sept 2000 is ignored} because the data of
this work is not sensitive to it. The first observation of this work is 78 days after the
previous glitch, which is several times the {longest of the short decay timescales for the
Crab pulsar \citep{Lyne2000, Wong2001, Wang2012}}, {so one expects} that the pre glitch
timing solution in this work is not corrupted by the decay phenomenon of the previous
glitch. The last observation of this work is six months before the next big Crab pulsar
glitch.
\section{Data Processing} \label{sec3}
All data used in this work have been acquired in the \textbf{Event List} mode (operating
modes E\_8us\_256\_DX0F, E\_8us\_256\_DX1F, etc.), in which the photon arrival times have
accuracy $\approx 7.6$ $\mu$sec (the best possible for HEXTE), and also the highest energy
resolution (256 channels in the energy range 0 to 250 keV); see ``Reduction and Analysis
of HEXTE data" on the RXTE
website\footnote{heasarc.gsfc.nasa.gov/docs/xte/recipes/cook\_book.html}. Throughout this
analysis, data of each cluster are analyzed separately.
The first step in data processing is the creation of the so called Good Time Intervals
(GTI), which are time duration identifying the useful data, not corrupted by instrumental
and extraneous factors. The GTI that account for system provided checks are created by
the tool \textbf{maketime} using the filter file available in the directory STDPROD, and
using the screening criterion (1) ELV $> 10^\circ$, (2) {the difference between the source
position and the pointing of the satellite} (OFFSET) $< 0.02^\circ$, (3) {the time since the
peak of the last South Atlantic Anomaly passage} (TIME\_SINCE\_SAA) $> 30$ min or
TIME\_SINCE\_SAA $< 0$ min.
\subsection{Standard Processing of HEXTE Data} \label{sec31}
The next step is to create additional GTI based on the light curves of the observation;
for example, abrupt changes of large magnitude in photon count rates should be excluded
from the analysis.
For that, each data file for each cluster is processed by the tool \textbf{hxtback} to
separate the data pertaining to source and sky background regions; for timing analysis only
the source data are used. Then light curves are obtained using the tool \textbf{seextrct},
screening the data using the GTI available in each file, as well as the GTI file created
in the previous section. The light curves are binned at the telemetry interval DELTAT (16
sec), and photons are selected from energy channels 15 to 240 keV. The light curves are
corrected for dead time using the tool \textbf{hxtdead}, using the appropriate house keeping
file. Although all four detectors are chosen for both clusters while running the
{tools}, the third detector of cluster 1 lost ability to assign energy
information to a photon after 6 March 1996; photons of this detector fall in the first two
energy channels irrespective of their actual energy. Therefore our choice of 15 to 240 keV
energy range essentially filters out photons from this detector, even though they have
valid arrival time information; see ``The XTE Technical Appendix'' on RXTE
website\footnote{heasarc.gsfc.nasa.gov/docs/xte/appendix\_f.html}.
Light curves for both clusters are plotted, and the range of count rates, within which the
data appears good, are chosen. These light curves and count rate limits are used to create
a second set of GTI files, one for each cluster, using the tool \textbf{maketime}. These
GTI are then merged with the earlier GTI, using the tool \textbf{mgtime} with the AND
option, to yield the final GTI, which are then used to filter the individual data files,
using the tool \textbf{fselect}. By this stage, one has screened the data for all system
provided checks, as well as for user provided count rate limits. Finally, the photon arrival
times are referred to the solar system barycenter using the tool \textbf{faxbary}, using the
orbit file for the given ObsID, and the Crab pulsar's coordinates ($83.6332208^\circ$ for
right ascension and $22.0144611^\circ$ for declination, for the epoch J2000,
{\citep{McNamara1971}}, taken from the online pulsar catalog of
ATNF\footnote{www.atnf.csiro.au/research/pulsar/psrcat/}).
\subsection{Obtaining the Period of Crab Pulsar} \label{sec32}
The next step is to obtain the best period of the Crab pulsar for each data file of each
cluster. The fundamental frequency in the power spectrum of the data gives the first
approximation to the period, which is obtained using the tool \textbf{powspec}, with bin
size $0.67$ ms and data length $2^{20} = 1048576$ bins for most files, but half or
quarter of that for shorter files. This is done for each data file of each cluster, not
only to check for consistency of the period among all files of a single ObsID, but also
to check the health of the data. The second approximation to the period is obtained by
folding the data over a range of 600 periods centered on the first approximation period,
and searching for the maximum $\chi^2$, using the tool \textbf{efsearch}, with an
increment of $10^{-8}$ sec in period, and zero period derivative. The third approximation
to the period is obtained by doing a finer search centered on the second approximation
period, using an increment of $0.2 \times 10^{-8}$ sec in period, and a nominal period
derivative of $420 \times 10^{-15}$ sec per sec {(explained later)}, over a range of 128
periods. A Gaussian is fit to the $\chi^2$ as a function of period, to obtain the
centroid. The final period is obtained by folding the earlier and later portions of
the data at the third approximation period, then cross correlating the two integrated
profiles, then measuring
the shift (if at all) versus time between the two profiles. Folding is done using the
tool \textbf{efold}, while cross correlation is done using independently developed
software. The accuracy of the final period was typically $7$ nano sec; for ObsID with
long duration observations it could be as small as fraction of a nano sec.
\subsection{Obtaining the Epoch of the Main peak of Crab Pulsar} \label{sec33}
The next step is to combine data files of each ObsID to get the integrated profile of
the Crab pulsar separately for each cluster, with a resolution of 128 bins per period,
using the best period of the epoch, and a nominal period derivative of $420 \times
10^{-15}$ sec per sec, {that is obtained by fitting the periods versus epochs for the
48 ObsIDs; this value is also consistent with the mean period derivative of the Crab
pulsar for the duration Dec 1999 to Dec 2000, as estimated from radio data, given in
the Jodrell monthly ephemeris.}
The zero phase of the integrated profile is set to the
arrival time of the first valid photon in the data, after conversion to MJD by adding
the MJDREF available in the data files. For validity of this procedure, it must be
ensured that the keyword RADECSYS in the barycenter corrected data files is set to FK5,
the keyword TIMEZERO is set to $0$, and the keyword CLOCKAPP is set to T; see ``A Time
Tutorial" in ``The ABC of XTE'' guide, and also ``RXTE Absolute Timing Accuracy" on RXTE
website\footnote{heasarc.gsfc.nasa.gov/docs/xte/abc/time.html}.
The final step is to obtain the epoch of arrival of the peak of the main pulse of Crab
pulsar, which is taken as the fiducial point in its integrated profile. For this the
epoch of the zero phase of the integrated profile (described above) should be added to
the position of the peak of the main pulse. This position is found by three independent
means, as done in \citep{Rots2004} but with some difference -- (1) fitting a Gaussian
to the main peak data, (2), fitting a
Lorentzian to the main peak data, and (3) finding the first moment of the main peak
data higher than $80$\% of the peak value. The first difference with \citep{Rots2004}
is that they fit a parabola instead of a Gaussian in method (1). The second difference
is that they have 200, 400 and 800 bins in the integrated profile for the three methods,
respectively, whereas the number of bins in this work are 128 for all three methods,
because several ObsID do not have sufficient exposure to obtain sufficient signal to
noise ratio with higher number of bins. The disadvantage of having lower number of
bins in the integrated profile is that one has to be cautious while fitting the
Gaussian and the Lorentzian; one has to choose as much of the main peak data as
possible, to maximize the sensitivity of the fit, but should not include the asymmetric
parts of the peak in its wings. {This fitting is done for each cluster for each ObsID.}
All three methods give consistent timing results for
the Crab pulsar glitch of July 2000. The typical accuracy of the fit is $\approx 1$
milli period. Figure ~\ref{fig1} shows the integrated profile of the Crab pulsar for
ObsID 40090-01-01-00 for the combined data of both clusters.
\begin{figure}[h]
\epsscale{1.0}
\plotone{crab_glitch_2000_rev2_fig1.plt}
\caption
{\small
Integrated profile of Crab pulsar obtained by combining data of ObsID
40090-01-01-00 from both clusters. The epoch of the peak of the main pulse of
cluster 0 data is used as reference phase for the tool \textbf{efold}, and
data is folded at the period of this epoch (0.0335046788 sec). The average
counts per sec (used to normalize the ordinate) is 176.477783. \label{fig1}
}
\end{figure}
\section{Phase Coherent Timing Solution for the July 2000 Glitch} \label{sec4}
The timing solution for Crab pulsar presented in this work was obtained using the
Gaussian fitting method described earlier, with three variations of epoch of
arrival of the peak of the main pulse (henceforth referred to as pulse arrival epoch)
-- (1) epochs from data of cluster 0 only, (2) epochs from data of both clusters, but
separately, and (3) epochs from combined data of both clusters. Methods (1) and (3)
both have 48 epochs, while method (2) has twice the number. The epochs from method
(3) are expected to be the most reliable due to enhanced number of photons in the
integrated profile by the average factor 1.75 (cluster 1 gathers $25$\% less photons
due to excluding detector 3). All three methods give consistently similar results.
The results presented in this work are derived using method (3). As mentioned earlier,
timing solutions obtained using method (3) along with fitting a Lorentzian, and
finding the first moment, all give results similar to those presented in
Table~\ref{tbl2} in this work. In addition, compared to the Gaussian fit method, the
first moment method gave a mean departure of $1 \pm 2$ milli periods for the 48
epochs; for the Lorentzian method the mean departure was $0.3 \pm 0.4$ milli periods.
Thus it is concluded that pulse arrival epochs are consistent among the three peak
finding methods.
\begin{figure}[h]
\epsscale{1.0}
\plotone{crab_glitch_2000_rev2_fig2.plt}
\caption
{\small
Result of using TEMPO2 on the 14 pre glitch pulse arrival epochs, fitting for $\nu$,
$\dot \nu$ and $\ddot \nu$, using the epoch of the first data at -210.221 (MJD
51530.4349129857) as the reference epoch for phase zero in TEMPO2; the results are
given in Table~\ref{tbl1}. The origin of the abscissa is at the epoch of the glitch
(MJD 51740.656). Note the four closely spaced epochs at the start of the data, and
the lack of any observation for two months prior to the glitch. \label{fig2}
}
\end{figure}
\begin{table}[h]
\begin{center}
\caption{\small TEMPO2 best fit parameters to the pre glitch data of Figure~\ref{fig2}.
$\nu$ is the rotation frequency of the Crab pulsar at the epoch MJD 51530.4349129857,
which is the first epoch in our data; $\dot \nu$ and $\ddot \nu$ are the first and
second time derivatives of $\nu$, respectively, at the same epoch. The errors in
brackets are in the last digit of each result. \label{tbl1}}
\begin{tabular}{|l|l|}
\tableline
Parameter & Value \\
\tableline
$\nu$ (Hz) & $29.846592902(2)$ \\
\tableline
$\dot \nu$ ($10^{-10}$ s$^{-2}$) & $-3.745962(8)$ \\
\tableline
$\ddot \nu$ ($10^{-20}$ s$^{-3}$) & $0.94(1)$ \\
\tableline
\end{tabular}
\end{center}
\end{table}
Figure ~\ref{fig2} shows the result of using TEMPO2 \citep{Hobbs2006} on the 14 pre
glitch pulse arrival epochs. {In the absence of glitches and timing noise, pulsars
obey a simple slowdown model that is adequately represented, at least for short
duration of several months, by three parameters -- rotation frequency $\nu$, its
time derivative $\dot \nu$ and its second derivative $\ddot \nu$; see Equation 1 of
\citep{Espinoza2011}. The three fitted parameters are shown in Table~\ref{tbl1},}
and are consistent with the interpolated values from the Jodrell monthly ephemeris
for the Crab pulsar for that epoch. The formal one standard deviation
error on the pulse arrival epoch in Figure~\ref{fig2} is typically $0.87$ milli
periods. An independent estimate of the error is obtained by the difference in
pulse arrival epochs for clusters 0 and 1 for the same ObsID. Often these differ
by $\approx 16$ sec or a few multiples of it, but on rare occasions can differ
by hours. Ideally these relatively short duration differences should be equivalent
to almost integer number of pulse cycles, since the estimated period of rotation
of the Crab pulsar would be very accurate for closely spaced epochs; the departure
from integer values will give us an idea of the errors involved in pulse arrival
epochs in Figure~\ref{fig2}. The mean value of the departure from integer number
of cycles turns out to be $\approx 1.0$ milli period. Therefore all error bars in
Figure ~\ref{fig2} are set to one milli period. The rms residual of the 14 pre
glitch arrival times after the TEMPO2 fit is $3.2$ milli periods.
TEMPO2 is used with the parameters of Table~\ref{tbl1} as constant input (i.e.,
without any fitting) for the 34 post glitch pulse arrival epochs; the results are
shown in Figure~\ref{fig3}. These post glitch residuals $\Delta \phi$ (in sec) are
fit to a {modified version} of the glitch model of \citep{Shemar1996},
\begin{eqnarray}
\Delta \phi & = & -\frac{1}{\nu_0} \int_{\epsilon}^t dt \left [ \Delta \nu_p +
\Delta \dot \nu_p t + \Delta \nu_n \exp \left ( -\frac{t}{\tau_d} \right )
\right ] \nonumber \\
& \approx & -\Delta \phi_0 - \frac{\Delta \nu_p}{\nu_0} t
- \frac{\Delta \dot \nu_p}{\nu_0} \frac{t^2}{2} \nonumber \\
& & - \frac{\tau_d \Delta \nu_n}{\nu_0}
\left ( 1 - \exp \left ( - \frac{t}{\tau_d} \right ) \right ),
\end{eqnarray}
\noindent
where $t$ {is the time elapsed since the glitch epoch} (in seconds), $\nu_0$ is the
rotation frequency of the Crab pulsar at the epoch of the glitch (in Hz), $\Delta
\nu_p$ and $\Delta \nu_n$ are the permanent and exponentially decaying parts of
the step change in rotation frequency at the epoch of the glitch, $\Delta \dot
\nu_p$ is the permanent step change in the time derivative of the rotation
frequency, and $\tau_d$ is the decay time scale (in sec) of the step frequency
change. {The parameter $\Delta \phi_0$ accounts for any uncertainty
$\epsilon$ in the epoch of the glitch, which is assumed to be much smaller
than $\tau_d$.
The equation above differs from Equation 1 of \citep{Shemar1996} in having a single
exponential only; as mentioned in their paper, only occasionally one requires more
than one transient component, and the decay times of these additional transients
are typically hundreds of days.} The negative sign in Equation 1 accounts for the
fact that the phase residuals after a glitch increase in the negative direction
for a positive step change in rotation frequency \citep{Shemar1996}.
Table~\ref{tbl2} gives the minimum $\chi^2$ fit values of the parameters in
Equation 1. The rms residual of the 34 post glitch arrival times after the fit
is $8.6$ milli periods.
\begin{figure}[h]
\epsscale{1.0}
\plotone{crab_glitch_2000_rev2_fig3.plt}
\caption
{\small
Result of using TEMPO2 on the 34 post glitch pulse arrival epochs, using the values
of Table~\ref{tbl1} as input parameters and without fitting. The origin of the abscissa
is at the epoch of the glitch (MJD 51740.656). The vertical dashed line is at epoch
after which one phase cycle was subtracted for all subsequent epochs, using the command
PHASE $-1$ in the input file to TEMPO2. Note the clusters of closely spaced epochs, one
just after the glitch epoch, and the other at the end of the data. \label{fig3}
}
\end{figure}
\begin{table}[h]
\begin{center}
\caption{\small Minimum $\chi^2$ parameters obtained by fitting Equation 1 to the post glitch
data of Figure~\ref{fig3}. \label{tbl2}}
\begin{tabular}{|l|l|}
\tableline
Parameter & Value \\
\tableline
$\Delta \phi_0$ (ms) & $3.3 \pm 0.5$ \\
\tableline
$\Delta \nu_p$ ($10^{-6}$ Hz) & $0.180 \pm 0.003$ \\
\tableline
$\Delta \dot \nu_p$ ($10^{-13}$ s$^{-2}$) & $-0.350 \pm 0.006$ \\
\tableline
$\Delta \nu_n$ ($10^{-6}$ Hz) & $0.71 \pm 0.08$ \\
\tableline
$\tau_d$ (days) & $4.7 \pm 0.5$ \\
\tableline
\end{tabular}
\end{center}
\end{table}
The epoch of the glitch is set to the published value of MJD $51740.656$, since
our data is unable to verify this number. \citep{Espinoza2011} obtain this by
using trial glitch epochs as reference epochs in TEMPO2 for the pre and post
glitch timing solutions, then comparing the two solutions for a match in phase
at the reference epoch (glitch epoch). This is not possible in this work because
of under sampling of data; the one and only pre glitch solution available
(Table~\ref{tbl1}) was with {the reference epoch used}, because close to that the
data was well sampled. {An alternate method of formulating the parameter
$\Delta \phi_0$ is to take the integral in Equation 1 from the limits $0$ to
$t - \epsilon$, instead of from $\epsilon$ to $t$. It can be shown that the two
methods are equivalent as long as
$\epsilon / \tau_d \ll 1$, and the derived parameters $\Delta \nu_p$, $\Delta
\dot \nu_p$ and $\Delta \nu_n$ have the relative orders of magnitude as derived
in Table 2.}
The total step change in rotation frequency at the glitch, as a fraction of the
pre glitch frequency, is $(\Delta \nu_p + \Delta \nu_n) / \nu_0 = (30 \pm 3)
\times 10^{-9}$, {which is not too different from} the value of $(25.1 \pm 0.3)
\times 10^{-9}$ published by \citep{Espinoza2011}. The actual step change of $\Delta \nu_p +
\Delta \nu_n \approx 0.89 \pm 0.08$ $\mu$Hz compares well with the top panel of
Figure 5 of \citep{Espinoza2011}. The fraction of frequency recovery $Q$ is $0.71
/ (0.71 + 0.18) \approx 0.80 \pm 0.11$, which is very high, as evident from Figure
5 of \citep{Espinoza2011}, although the decay of frequency in their figure (top
panel) does not appear to be entirely exponential, {most probably due to the
very small glitch that has been ignored in this work.}
The total step in frequency derivative at the glitch as a fraction of the pre glitch
frequency derivative is $(\Delta \dot \nu_p - \Delta \nu_n / (\tau * 86400) ) / \dot
\nu_0 = (4.8 \pm 0.6) \times 10^{-3}$, {which is also not too different from} the value
of $(2.9 \pm 0.1) \times 10^{-3}$ of \citep{Espinoza2011}. The fraction of recovery
of the frequency derivative is $-17.48 / (-17.48 - 0.35) \approx 0.98 \pm 0.16$,
which is consistent with Figure 5 of \citep{Espinoza2011}. However, the step change
in $\dot \nu_0 \approx (-18 \pm 2) \times 10^{-13}$ estimated here is inconsistent
with the bottom panel of Figure 5 of \citep{Espinoza2011}, in which it is more like
$\approx -5 \times 10^{-13}$. However, the value expected from their work is
$\approx 2.9 \times 10^{-3} \times -3.744255 \times 10^{-10} \approx -11 \times
10^{-13}$, which is in between the above two numbers.
\citep{Espinoza2011} do not quote a value for the decay time scale $\tau_d$\ for this
glitch. By expanding and gridding their Figure 5, and reading off values of the peak
and the first decay point, one can obtain an approximate value for $\tau_d$. This
turns out to be $\approx 13 \pm 3$ days, for the decay of both the frequency and
its derivative (top and bottom panels respectively of that figure). The number
derived in this work, $\tau_d$\ $= 4.7 \pm 0.5$ days, is a factor of $\approx
2.8$ smaller, although it is in the right range of decay time scales for Crab pulsar
\citep{Lyne2000}. If it turns out that the correct value of $\tau_d$\ is indeed $\approx
13$ days, then the reason for the factor of $2.8$ underestimate in this work is most
probably on account of lack of observations very close to the glitch epoch, to which
the estimate of $\tau_d$\ is very sensitive.
\section{Discussion} \label{sec5}
{In section~\ref{sec4} we concluded} that the hard xray timing of the Crab pulsar
glitch of 15 July 2000, using HEXTE/RXTE data, is consistent with the results
obtained at radio frequencies by the Jodrell Bank observatory. This is, {to the best
of our knowledge}, the first time that a glitch of the Crab pulsar has been analyzed
using xray data, resulting in what can be considered to be the closest to absolute
timing of the Crab pulsar at hard xray energies.
\citep{Rots2004} found that {the main peak of the xray pulse profile of the Crab
pulsar (the peak at phase 0 in Figure~\ref{fig1}) leads the main peak of the radio
pulse profile (just after the radio precursor; see Figure 6a of \citep{Kuiper2003})}
by $344 \pm 40$ $\mu$sec. Figure~\ref{fig4} shows the result of fitting the data of
Figure~\ref{fig3} combined with the Jodrell data at radio frequencies, obtained
from their monthly ephemeris. {It shows the combined pre and post glitch residuals
after subtracting the pre and post glitch models, respectively. The pre glitch
model is given in Table 1. The post glitch model is obtained by including
additionally in Equation 1, permanent step changes in the second and third time
derivatives of the rotation frequency, $\Delta \ddot \nu_p$ and $\Delta \nu_p^{\! \! \! \! \cdots}$,
respectively. This solution gave the lowest rms residual of $3.2$ milli periods
for the 34 post glitch arrival times; the values of the rest of the parameters
obtained in this fit are consistent with those in Table 2.} The mean pre glitch
separation of the Jodrell data with respect to the xray data is $411 \pm
167$ $\mu$sec. Although this result has $\approx 4.2$ times larger error than
that quoted by \citep{Rots2004}, it is consistent with their result, and also
with the results of others \citep{Kuiper2003, Carrillo2012}. The post glitch
differences with Jodrell data are difficult to analyze, due to the baseline
varying in a quasi periodic manner. However, after accounting for these variations,
the last three post glitch Jodrell data certainly support the above number.
\begin{figure}[h]
\epsscale{1.0}
\plotone{crab_glitch_2000_rev2_fig4.plt}
\caption
{\small
Pre and post glitch xray residuals (this work, dots; {see text for the corresponding
models}) along with eleven radio residuals from Jodrell Bank data (stars), which lie
consistently above xray residuals, except for the residual of 15 Aug 2000, which
has a very large error bar ($4$ ms). \label{fig4}
}
\end{figure}
Several properties of the Crab pulsar can be studied as a function of pre and post glitch
epochs. Three parameters were plotted as a function of epoch -- (1) The separation of the
two peaks in the Crab pulsar's integrated profile, (2) the ratio of their peaks, and (3)
integrated energy in the pulse profile. {The first two showed no variation worth
reporting} (see also \citep{Rots2004}). Figure~\ref{fig5} shows the dead time corrected
photon count variation of the Crab pulsar as a function of epoch.
\begin{figure}[h]
\epsscale{1.0}
\plotone{crab_glitch_2000_rev2_fig5.plt}
\caption
{\small
Normalized and dead time corrected on pulse energy of Crab pulsar during the year
2000. \label{fig5}
}
\end{figure}
Each point in Figure~\ref{fig5} is obtained by first estimating the average off pulse
counts in the integrated profile, then subtracting it from the on pulse counts before
integrating them, then dividing the result by the average off pulse counts; this is
done for every DELTAT (16 sec) of data for each ObsID. Off pulse counts are obtained
by integrating in the phase range $0.61$ to $0.79$ in Figure~\ref{fig1}; the rest of
phase range represents the on pulse. Although this procedure will exclude any small
off pulse emission from the Crab pulsar \citep{Tennant2001}, it has the advantage of
correcting for dead time, which is not available for Barycenter corrected data, and
also for data that has been filtered in energy range, which is the situation with our
data; see ``The XTE Technical Appendix'' on RXTE
website\footnote{heasarc.gsfc.nasa.gov/docs/xte/appendix\_f.html}. Finally all $48$
energies are normalized by their mean value, which lies at the value $1.0$ in
Figure~\ref{fig5}. The assumption made here is that dead time correction is, to a
large order of accuracy, the same for the on and off pulse phases of a pulsar, even
for bright pulsars such as the Crab. Dead time is the duration immediately after the
arrival of an xray photon or high energy particle, during which the HEXTE detectors
are unable to process any more photons. For the HEXTE detector there are two sources
of dead time. One is due to the arrival of an xray photon, after which the HEXTE
detectors are "dead" for $16$ to $30$ $\mu$sec. The much larger effect is due to
arrival of high energy particles, after which the detectors are dead for $2500$
$\mu$sec; these are known as XULD events. Even for a luminous pulsar such as the
Crab and its nebula, the normal photon count rate is typically $< 400$ photons per
sec per cluster, while the normal XULD event rate is typically $150$ particles per
sec per detector (there are $4$ detectors per cluster). Taking the mid value of
$23$ $\mu$sec, it can be easily seen that the ratio of the photon to XULD
contribution to the dead time is $\le 0.6$\%. Since the XULD event rate is the
same for on and off pulse regions, the above method of dead time correction is
justified. If required, more refined dead time correction can be done by using the
integrated pulse profile of the pulsar.
In Figure~\ref{fig5}, the standard deviation of the spread in values is $\approx
16$\% of the mean value; the corresponding spread in the uncorrected fluxes is
$\approx 49$\% of the mean value. This technique of dead time correction for
pulsars significantly improves the precision of the estimate of integrated xray
flux, and may therefore prove useful to find out if the Crab pulsar xray flux
varies on time scales of decades; for the crab nebula this has already been
observed at several xray energies (see \citep{Wilson-Hodge2011} and references
therein).
{In Figure~\ref{fig5} there is no strong evidence for the xray flux of
the Crab pulsar to be affected by the glitch}. The mean pre and post glitch xray
fluxes are $1.06$ and $0.98$, respectively, while the standard error on these
means is $0.03$. The mean xray fluxes differ by $0.08 \pm 0.04$, {which
can not be considered a strong result. Moreover, this decrease appears to be mainly
due to the data well after the glitch in Figure~\ref{fig5}, making it less likely
to be related to the glitch itself. One instrumental feature that can simulate
such a result is a variation of the average off pulse counts as function of epoch
in Figure~\ref{fig5}; however this does not appear to be the case.} Although this
can not be considered a strong result, it may be interesting to speculate on the
possible causes
of glitch related xray flux variations in rotation powered pulsars. Such studies
have not been done so far. Although glitch related xray flux enhancements have
been reported in some Magnetars (see \citep{Espinoza2011} and references therein),
it is generally believed that in Magnetars and AXPs, glitches are not always
associated with xray flux variations \citep{Dib2008}. For the Crab pulsar glitch
under consideration here, the post glitch permanent changes in rotation frequency
$\Delta \nu_p$ and its derivative $\Delta \dot \nu_p$ can contribute at most to
a fractional increase of $\approx 0.01$\% in the post glitch xray flux, while
Figure~\ref{fig5} shows a decrease of $8$\%; so simple energy loss rate of the
Crab pulsar may not be the explanation. The glitch model of \citep{Ruderman2009}
provides a causal connection between a glitch in a rotation powered pulsar and
changes in its surface magnetic field. In this model, a glitch is caused by excessive
stresses in the neutron star crust, that are built up by super fluid vortices
moving outwards due to the spin down of the neutron star, dragging with them
magnetic flux tubes. The crust cracks at the instant of the glitch, and then
adjusts itself to a new configuration, leading to a corresponding reconfiguration
of the surface magnetic field, presumably both in terms of its field strength
as well as in terms of its field line structure (curvature of field lines,
direction of the opening bundle of field lines at the surface, presence of
higher magnetic multipoles, etc). There are
several quantitative uncertainties in the predictions from this model. However
it may be worth exploring if a small change in surface magnetic field structure
of the Crab pulsar can lead to a non-linearly large change (decrease in the present
case, but maybe an increase in other glitches) in the post glitch xray flux.
{
It is well known that the dispersion measure of the Crab pulsar varies due to
radio propagation within the Crab nebula \citep{Lyne1993}, and that the radio
pulses suffer significant and variable refractive and scattering effects
within the Crab nebula (see \citep{Backer2000} and references therein). Such
effects can cause errors in the arrival times of the radio pulses, with
consequent errors on the derived glitch parameters. The above interstellar
effects are frequency dependent, and are absent for pulses at xray energies.
The consistency between the radio and hard xray timing results of this work
imply that the effect of radio pulse propagating through the Crab
nebula has not been significant during the glitch of July 2000.
}
{
Glitches are one of the two timing irregularities observed in rotation powered
pulsars, the other being timing noise, which manifests observationally as
random wandering of timing residuals. It is currently believed that timing
noise is due to instabilities in the pulsar magnetosphere \citep{Lyne2010};
see also \citep{Arons2009} and references therein for a possible theoretical
explanation in terms of magnetic reconnection. Now, the radio and xray emitting
regions in the Crab pulsar's magnetosphere are supposed to be identical (the
outer gaps), except for the radio precursor, which is supposed to arise in the
polar gap, while for other rotation powered pulsars (eg. Vela) the radio and
xray emitting regions are supposed to be different; see \citep{Harding2009}
and references therein. Therefore simultaneous radio and xray studies of
timing noise in the Crab pulsar should show similar timing noise properties (
rms residuals, time scales of wandering, etc), but probably not in Vela like pulsars.
}
{
Glitches, on the other hand, are supposed to be due to the steady differential
spin down of the super fluid core and outer crust of the neutron star; see
\citep{Ruderman2009} and references therein. At the instant of the glitch,
the outer crust speeds up, thereby also speeding up the pulse emitting regions
in the magnetosphere, which are firmly anchored into the crust by means of the
magnetic field. Simultaneous timing observations of pulsar glitches at radio
and xray wavelengths can probably be used to find out if the emission regions
at different energies are anchored equally firmly onto the surface of the
neutron star. For example in the case of the Crab pulsar one would expect the
glitch behavior to be almost identical, while for the Vela like pulsars one
may or may not notice differences.
}
\acknowledgments
{\small
I thank the anonymous referee for detailed comments to improve this manuscript.
This research made use of data obtained from the High Energy Astrophysics Science
Archive Research Center Online Service, provided by the NASA-Goddard Space Flight
Center.
}
{\small
{\it Facilities:} \facility{RXTE (HEXTE)}.
}
|
{
"timestamp": "2015-04-14T02:11:44",
"yymm": "1504",
"arxiv_id": "1504.03081",
"language": "en",
"url": "https://arxiv.org/abs/1504.03081"
}
|
\section{Introduction}\label{sec:intro}
Several promising architectures for quantum computers are based on the superconducting qubits~\cite{Makhlin-RMP01,Devoret-Martinis,Nori-PT05} based on Josephson junctions (JJs). These designs utilize either charge~\cite{Nakamura-Nature99}, phase~\cite{Martinis-PRL02} or flux~\cite{Friedman-Nature00,Mooij-Science00} degrees of freedom. These systems have made tremendous progress in recent years in realizing increasingly sophisticated quantum states, measurements and operations with high fidelity~\cite{Devoret-Review}. Superconducting quits are also attractive technologically because they can be naturally integrated into large-scale quantum circuits~\cite{BarendsPRL2013,Schoelkopf-Girvin}.
However, this main advantage of superconducting qubits brings a substantial challenge at the same time since strong coupling also implies a substantial interaction between the qubits and their environment, which can break quantum coherence. Understanding the limiting factors of qubit operation is of fundamental and practical importance. Previously, a separation of various contributing factors to the qubit relaxation and decoherence has been achieved~\cite{Astafiev-PRB04,Bertet-PRL05,Martinis-PRL05,Neeley-PRB08}. Abrikosov vortex is one important example of such environment. Dissipation caused by vortices has also been studied in a superconducting resonator~\cite{SongPRB2009}. Recently, vortices have been shown to trap quasiparticles in superconducting resonator and qubit, leading to the increase in the quality factor of resonator~\cite{NsanzinezaPRL2014} and the relaxation time of qubit~\cite{wang_measurement_2014}.
One interesting theoretical possibility of drastically improving quantum coherence in qubits is to couple them to Majorana fermions~\cite{beenakker2013}. A qubit based on Majorana states is expected to exhibit especially long coherence times. One approach to create Majorana states is to deposit a superconductor onto a topological insulator and to create vortices in the superconductor. In this case Majorana states can nucleate in the vortex core~\cite{beenakker2013}. Thus a study of qubits coupled to vortices is needed, in order to determine whether a qubit coupled to a vortex can preserve its quantum coherence and for how long. Here we use Meisser qubits to quantify dissipation effects produced by single vortices.
A major advance in the superconducting qubit performance became possible after the invention of the \textit{transmon qubit}~\cite{koch2007PRA}. When combined with the three-dimensional circuit-quantum electrodynamics (cQED) platform developed in Ref.~\cite{Paik2011PRL}, the transmon has shown huge improvements of the relaxation time, up to several hundreds microseconds~\cite{Rigetti2012PRB,Oliver2013MRS}. Like the common transmon our device involves a capacitance linked by a nonlinear kinetic inductance (JJs). The main difference is that our qubit is coupled to the Meissner current and the supercurrents generated by vortices. Yet the relaxation time of such device, designed to probe the environment is rather large, namely about 50 $\mu$s in the best case. We argue that the limiting factor was the Purcell effect, thus the relaxation time can be made even longer if necessary.
Because in our qubit design the Meissner current is allowed to flow, partially, into the qubit, a significant amplification of the magnetic field effect is demonstrated. The qubit transition frequency is periodically modulated by the applied magnetic field, but the period is much smaller compared to the value estimated by dividing the flux quantum by the qubit loop geometric area.
The Meissner qubit allows a strong coupling to the vortices in the leads. We perform a detailed study of the radiation-free decoherence effects produced by the vortex cores. It should be stressed that the qubit relaxation time may be shortened by the presence of a vortex in the superconducting film, due to the Bardeen-Stephan viscous vortex flow. No quantitative study has been done so far to test how qubit quantum states relax due to coupling to Abrikosov vortices. Our key finding is that vortices can remain, over many microseconds, in quantum superposition states generated because the Lorentz force experiences strong quantum fluctuations. The relaxation rate added to the qubit by each single vortex was measured and appears to be surprisingly low, of the order of 10 kHz. We propose a semi-quantitative model which allows us to estimate this radiation-free relaxation rate caused by viscous flow of vortices. Up until now it was well established that classical supercurrents can generate heat through viscous flow of vortices~\cite{SongPRB2009}. Now we establish that quantum superposition currents, such as those existing in qubits and characterized by zero expectation value, can also generate heat through the same mechanism. Such heat dissipation occurs through the spread of the wavefunction of the vortex center followed by a collapse of this smeared wave function.
\begin{figure}[t]
\begin{subfloat}{\label{fig:setup_a}}
\end{subfloat}
\begin{subfloat}{\label{fig:setup_b}}
\end{subfloat}
\begin{subfloat}{\label{fig:setup_c}}
\end{subfloat}
\begin{subfloat}{\label{fig:setup_d}}
\end{subfloat}
\centering
\includegraphics[width=0.98\columnwidth]{setup}
\caption{(a) Optical image of the Meissner transmon qubit fabricated on a sapphire chip, which is mounted in the copper cavity. (b) A zoomed-in optical image of the qubit. Two rectangular pads marked A1 and A2 act as an RF antenna and shunt capacitor. (c) Scanning electron microscope (SEM) image of the electrodes marked E1 and E2, and a pair of JJs. (d) Schematics of the Meissner qubit. The X, Y and Z denote the width, the distance between the electrodes, and the distance between two JJs, which are indicated by $\times$ symbols. The red dot and circular arrow around it in the bottom electrode represent a vortex and vortex current flowing clockwise, respectively. $\Theta_\text{v}$ is a polar angle defined by two dashed lines connecting the vortex and two JJs. The orange rectangular loop on the boundary of the bottom electrode indicates the Meissner current circulating counterclockwise. }\label{fig:setup}.
\end{figure}
\section{Experimental Results}\label{sec:results}
\subsection{Qubit frequency modulation}
\begin{figure}[t]
\begin{subfloat}{\label{fig:TransmVsB}}
\includegraphics[width=\columnwidth]{TransmVsB}
\end{subfloat}
\begin{subfloat}{\label{fig:EffArea}}
\includegraphics[width=\columnwidth]{EffArea}
\end{subfloat}
\caption{\label{fig:Bdep}(Color online) (a) Periodic heterodyne voltage oscillation (``HV-oscillation'') as a function of magnetic field. Four different colors represent the separate measurement runs where the magnetic field swept either round-trip or one-way. $\Delta B$ shows the period of the modulation. The arrows marked $B_4$ and $B_5$ indicate the positions of two adjacent sweet spots, characterized by $df_{01}/dB$=0. Each sweet spot is equivalent to $B$=0 state. The actual B=0 sweet spot is outside the range of the plot. The mark ``S'' indicates the field at which the periodic signal was shifted to the right, revealing some hysteresis caused by vortex entrance to the electrodes. This hysteretic behavior can be seen in the second (red) segment of the B-field sweep ranging from about 1.45 to 2 Gauss. The mark ``J'' shows the moment when a vortex (or a small group of vortices) entered the electrodes during the forward-sweeping magnetic field. (b) Comparison of theoretical ($A^\text{\tiny th}_\text{\tiny eff}$) and experimental ($A^\text{\tiny ex}_\text{\tiny eff}$) effective areas of five samples. The dashed line depicts the ideal case of $A^\text{\tiny th}_\text{\tiny eff}$=$A^\text{\tiny ex}_\text{\tiny eff}$. See text for the definition of these quantities.}
\end{figure}
The design of our devices is shown in Fig.~\hyperref[fig:setup]{\ref{fig:setup}}, while details of the fabrication and measurement techniques are described at length in the Appendix~\hyperref[sec:device]{\ref{sec:device}}. In brief, the qubits have been placed inside a three-dimensional (3D) microwave cavity made of Cu (Fig.~1a). The state of the qubit has been determined by measuring the transmission of the 3D cavity. First, we investigated the magnetic field dependence of the qubits, anticipating to observe periodic SQUID-type oscillations. The transmission versus the magnetic field (``B-field'') varied as shown in Fig.~\hyperref[fig:TransmVsB]{\ref{fig:TransmVsB}}. This plot represents the heterodyne voltage, produced by mixing a microwave signal passing through the cavity containing the qubit and a reference signal. During this measurement the qubit remains in its ground state. Yet the cavity input power is chosen such that the transmission of the cavity is the most sensitive to the qubit transition from the ground to the excited state, i.e., the maximum-contrast power was used [\onlinecite{Reed2010PRL}]. The probing microwave frequency for this measurement equals the bare cavity frequency. Four segments in different color (color online) represent four separate measurement runs. The magnetic field was swept round-trip (up and down) in the first three segments (black, red and green), while it was swept one-way (up) in the last segment (blue). The modulation of the transmission at low $B$-field arises from the change of the onset power---the lowest power at which the cavity starts to show a sharp increase in transmission, which was termed ``bright state'' (near-unity transmission) in Ref.~\onlinecite{Reed2010PRL}. The onset power depends on the difference between the qubit transition frequency ($f_{01}$) and the bare cavity frequency ($f_\text{c}$). The key point is that $f_{01}$ is modulated periodically by the applied magnetic field, because the qubit includes a SQUID-like loop formed by the two JJs and the two electrodes, marked in Fig.1c as E1 and E2. The heterodyne voltage $V_\text{H}$ is proportional to the microwave transmission. Thus, as magnetic field was increased, we observed periodic or quasi-periodic heterodyne voltage oscillation (``HV-oscillation''). The voltage changes reproducibly and periodically with magnetic field, up to the first critical field of the electrodes, $B_{c1}\approx$1.6 Gauss. This is the critical field at which Abrikosov vortices begin to enter the electrodes. The period of the HV-oscillation, $\Delta B$, can be defined as the distance between the adjacent principle maxima, as is illustrated in Fig.~\hyperref[fig:Bdep]{\ref{fig:TransmVsB}} by the horizontal arrow. Equivalently, the period can be defined as the separation between the so-called \textit{sweet spots}. Some of these sweet spots, namely $B_4$ and $B_5$ are indicated by the vertical arrows in Fig.~\hyperref[fig:Bdep]{\ref{fig:TransmVsB}}. The sweet spots are the points equivalent to zero magnetic field. It should be reminded here that if the device is tuned to a sweet spot then it is insensitive, in the first order, to the flux noise, because $df_{01}/dB$=0 [see Fig.~\hyperref[fig:spec]{\ref{fig:spec}}] and $dV_\text{H}/dB$=0 [see Fig.~\hyperref[fig:Bdep]{\ref{fig:TransmVsB}}].
At sufficiently low fields, when there are now vortices in the electrodes, the sweet spots occur periodically because the critical current of the SQUID loop changes periodically with magnetic field.
In our case the design is such that the phase gradient created by vortices present in the electrodes couples to the SQUID loop. Therefore the exact periodicity of the sweet spots becomes broken when vortices begin to enter the electrodes at $B>B_{c1}$. Thus our device acts as a vortex detector. Entrapment of vortices inside the electrodes makes the transmission hysteretic with magnetic field up-down sweeps. An example of such hysteresis is clearly seen at the position marked ``S'' in Fig.~\hyperref[fig:TransmVsB]{\ref{fig:TransmVsB}}. In addition to the period increases and the hysteresis, we also observe abrupt jumps in the transmission. One such jump is marked ``J'' in Fig.~\hyperref[fig:TransmVsB]{\ref{fig:TransmVsB}}. The jumps indicate that the vortex enters in the near proximity of the qubit loop and thus should be strongly coupled to the qubit state.
\begin{table}[t]
\caption{\label{tab:table1} Comparison of the experimental and theoretical periods for five samples. $\Delta B$ is the measured period of oscillation. $\Delta B_{\text{\tiny YZ}}$, $\Delta B_{\text{\tiny YZ+cXZ}}$, and $\Delta B_{\text{\tiny kYZ+cXZ}}$ are the theoretical periods calculated using three different effective areas denoted by the subscripts.}
\begin{ruledtabular}
\begin{tabular}{lcccllll}
Sample & X &Y & Z & $\Delta B$& $\Delta B_{\text{\tiny YZ}}$ & $\Delta B_{\text{\tiny YZ+cXZ}}$ & $\Delta B_{\text{\tiny kYZ+cXZ}}$\\
& ($\mu$m) &($\mu$m) &($\mu$m) & (G) & (G) & (G) & (G) \\
\hline
N1 & 10 & 1 & 5 & 0.38 & 4 & 0.49 & 0.30\\
N2 & 10 & 2 & 5 & 0.3 & 2 & 0.43 & 0.29\\
N3 & 15 & 1 & 5 & 0.43 & 4 & 0.34 & 0.22\\
N6 & 10 & 2 & 8 & 0.16 & 2.5 & 0.30 & 0.19\\
N7 & 10 & 2 & 8 & 0.2 & 1.25 & 0.27 &0.18 \\
\end{tabular}
\end{ruledtabular}
\end{table}
In our qubits, the effective Josephson energy is modulated by the external magnetic field, but the period is set differently as compared to the typical SQUID-type device~\cite{Schreier2008PRB}. Unlike in usual SQUIDs, the modulation of the critical current in the present case is driven mostly by the Meissner currents in the electrodes, and to a much lesser extent by the magnetic flux through the SQUID loop. This can be seen from the fact (See Table~\hyperref[tab:table1]{\ref{tab:table1}}) that the experimental period, $\Delta B$, is much smaller than the period computed using the area of the superconducting loop, $\Delta B_{\text{\tiny YZ}}$.
This is why the sensitivity of the Meissner qubit energy to the external field is higher compared to the usual split-junction transmon~\cite{Schreier2008PRB}.
One can understand this new period by considering the phase constraint~\cite{Hopkins2005science,Pekker2005PRB}:
\begin{equation}\label{eq:phaseconstraint}
\theta_1-\theta_2+2\delta (B)=2\pi n_\text{v},
\end{equation}
where $\theta_{1,2}$ is the phase difference across each JJ, $\delta (B)$ the phase difference generated in the thin-film electrodes by the Meissner current and defined as the phase difference between the entrance points of of the JJ bridges, i.e., between the bridges in which JJs are created. Here $n_\text{v}$ is the vorticity. Since the current-phase relationship of JJs is single-valued, and the inductance of the JJ bridges and the electrodes is very small, the vorticity is always zero, $n_\text{v}=0$, in our SQUID-type devices, just like in common SQUIDs. The field-dependent phase accumulation is $\delta (B)$=$\int{\vec{\nabla}\varphi(B)\cdot d \vec{l}}$, where $\nabla\varphi (B)$ is the phase gradient of the order parameter in the electrodes. The gradient originates from the Meissner (screening) current in the \textit{electrodes}, if there are no vortices. At sufficiently high fields, at which vortices enter the electrodes, an additional contribution to the total phase gradient occurs due to the vortices. Following the Ref.~\onlinecite{Pekker2005PRB}, the magnetic period can be estimated as
\begin{equation}\label{eq:period}
\Delta B = \left[ \left(\frac{\Phi_0}{cXZ}\right)^{-1}+\left(\frac{\Phi_0}{YZ}\right)^{-1}\right]^{-1},
\end{equation}
where the numerical coefficient $c=(8/\pi^2)\sum^{\infty}_{n=0}(-1)^n/(2n+1)^2\approx 0.74$ can be found by solving appropriate boundary problem for the Laplace equation. We cautiously notice that the Eq.~\eqref{eq:period} may not be strictly applicable to our case because it was derived for mesoscopic electrodes, where $\lambda_\perp$ is much larger than $X$, the width of the electrode. In our samples $\lambda_\perp\approx550$ nm $<X=10\text{~or~ } 15$ $\mu$m. Meissner currents are stronger in the case of a relatively small perpendicular magnetic length, therefore this model may still provide a semi-quantitative estimate. To achieve a satisfactory agreement we will have to introduce corrections related to the field focusing effect.
We emphasize that unlike a regular SQUID the period is not set by the area $YZ$ enclosed by SQUID loop only, but rather by a much larger effective area $YZ+cXZ$; the dimensions re indicated in Fig.~\hyperref[fig:setup]{\ref{fig:setup_d}}~\cite{Hopkins2005science, Pekker2005PRB}. To compare those two effective areas, we plot in Fig.~\hyperref[fig:EffArea]{\ref{fig:EffArea}} the theoretical effective areas $A^{\text{\tiny th}}_\text{\tiny eff}$ versus the ``experimental'' effective area $A^{\text{\tiny ex}}_\text{\tiny eff}=\Phi_0/\Delta B$ for five samples. Here $\Delta B$ is the low-field, vortex-free period of the HV-oscillation for each sample. The black squares and the red triangles represent, respectively, the theoretical effective area calculated as $A^{\text{\tiny th}}_\text{\tiny eff}$=$YZ$ (geometric SQUID loop area approach) and as $A^{\text{\tiny th}}_\text{\tiny eff}$=$YZ+cXZ$ (Meissner current phase gradients approach). The dashed curve represents the ideal case, $A^{\text{\tiny th}}_\text{\tiny eff}$=$A^{\text{\tiny ex}}_\text{\tiny eff}$. The red triangles appear much closer to the ideal dashed line. Therefore the qubit energy is controlled mostly by the Meissner currents, which produce a strong phase bias of the SQUID loop.
Another possible contribution to the observed amplification of the magnetic sensitivity is the focusing of the magnetic field into the SQUID loop area by the superconducting electrodes. Such focusing is also due to Meissner effect, which is strong due the relatively small $\lambda_{\perp}$. This field focusing effect enhances the magnetic field by a factor of $\kappa=B_1/B_0 >1$. The ratio of the field $B_1$, enhanced by the field-focusing, to the applied field $B_0$ is estimated in Appendix~\hyperref[sec:kappa_cal]{\ref{sec:kappa_cal}}. Thus the effective area set by the SQUID loop increases by a factor of $\kappa$. To incorporate the field focusing effect, we replaced $YZ$ with $\kappa YZ$ in Eq.~(\ref{eq:period}). The result is plotted in Fig.~\hyperref[fig:Bdep]{\ref{fig:EffArea}}, showing an improved and now quite satisfactory agreement with the experimental results.
\begin{figure}[t]
\begin{subfloat}{\label{fig:spec}}
\includegraphics[width=0.7\columnwidth]{spec}
\end{subfloat}
\begin{subfloat}{\label{fig:specfit}}
\includegraphics[width=0.7\columnwidth]{specfit}
\end{subfloat}
\caption{\label{fig:Spec}(Color online) (a) Spectroscopy of Meissner transmon (N1) as a function of applied magnetic field. This is raw data. (b) The parabola-like dashed line shows a phenomenological fit to the qubit transition frequency $f_{01}$ versus magnetic field $B$.}
\end{figure}
Now, we turn to the magnetic field dependence of the qubit energy. For the spectroscopy, the qubit was excited with 2 $\mu$s long saturation pulse, which was immediately followed by a few microsecond readout pulse. The excitation frequency was swept up---low to high frequency---with a fixed step size at a fixed magnetic field, and this process was repeated for equally spaced magnetic field. Fig.~\hyperref[fig:spec]{\ref{fig:spec}} shows the 2D color plot of the transmission as a function of the excitation frequency and external magnetic fields. The color represents the heterodyne voltage of the transmission of the cavity. The dashed line is a fit to the qubit spectrum with the following fit function: $f_{01}$=$f_{0}\sqrt{|\cos (\pi (B-B_0)A_\text{\tiny eff}/\Phi_0)|} $, where $f_0$, $B_0$ and $A_\text{\tiny eff}$ are the fitting parameters. We used the approximate relation for $f_{01}$=$\sqrt{8E_\text{J} E_\text{C}}/h$, where $E_\text{J}$=$\hbar I_\text{c}/2e$, and $I_\text{c}(B)$=$2I_\text{c1}|\cos (\pi\Phi/\Phi_0)|$=$2I_\text{c1}|\cos (\pi BA_\text{\tiny eff}/\Phi_0)|$. $B_0$ and $A_\text{\tiny eff}$ are offset field to account for the residual magnetic field and the effective area, respectively. The best fit values of $B_0$ and $f_0$ were $-4.5$ mG and 6.583 GHz. The best fit effective area was $A_\text{\tiny eff}=55.2$ $\mu$m$^2$, which is consistent with the $A_\text{\tiny eff}=55$ $\mu$m$^2$ determined from the periodic oscillations of Fig.~\hyperref[fig:Bdep]{\ref{fig:TransmVsB}}.
\subsection{Time domain measurement: low magnetic field}
\begin{figure}[t,b]
\begin{subfloat}{\label{fig:Timedomain_T1_inset}}
\includegraphics[width=\columnwidth]{Timedomain_T1_inset}
\end{subfloat}
\begin{subfloat}{\label{fig:Timedomain_ramsey_inset}}
\includegraphics[width=\columnwidth]{Timedomain_ramsey_inset}
\end{subfloat}
\begin{subfloat}{\label{fig:Timedomain_echo_inset}}
\includegraphics[width=\columnwidth]{Timedomain_echo_inset}
\end{subfloat}
\caption{\label{fig:Timedomain} (Color online) Time domain measurements of the N7 sample at $B =7.5$ mG. (a) Relaxation time measurement ($T_1=51$ $\mu$s) (b) Ramsey fringe experiment ($T_2^*=18$ $\mu$s) (c) Hahn spin echo experiment ($T_2=27$ $\mu$s). The red solid lines are the fits to the data. See the main text for the fitting functions.}
\end{figure}
Now we will look into the conventional time-domain measurements under the applied magnetic field. The time-domain measurements shown in Fig.~\hyperref[fig:Timedomain]{\ref{fig:Timedomain}} were performed to measure three time scales: relaxation time ($T_1$), phase coherence time ($T^*_2$) by Ramsey fringe, and phase coherence time ($T_2$) by Hahn spin echo.
For the relaxation time measurement, we applied $\pi$ pulse (100 ns) first and then read out the qubit state after the time interval $\Delta t$. For the Ramsey measurement, we applied two $\pi/2$ pulses (50 ns) separate by pulse separation time $\Delta t_\text{R}$, and then readout was performed immediately after the second $\pi/2$ pulse. Similarly, in spin echo measurement, the measurement protocol was as the Ramsey protocol, except that an additional $\pi$ pulse was inserted right in the middle of the two $\pi/2$ pulses. The separation between the two $\pi/2$ pulses is denoted by $\Delta t_\text{e}$ [see the inset in Fig.~\hyperref[fig:Timedomain]{\ref{fig:Timedomain_echo_inset}}].
Three time scales, $T_1$, $T^*_2$ and $T_2$ were extracted by fitting data with exponential decay function $\exp(-t/t_0)$ for $T_1$ and $T_2$, and sine-damped function $\exp(-t/t_0)\sin\left(2\pi f_{\text{\tiny R}}t+\varphi_0\right)$ for $T^*_2$. In Fig.~\hyperref[fig:lowB]{\ref{fig:lowB}}, we present $f_{01}$, $T_1$, $T^*_2$ and $T_2$ versus magnetic field measured in a vicinity of the sweet spot at $B$=0. The used weak magnetic field, up to $\sim$50 mG, is much weaker than the field needed to drive vortices into the electrodes. Thus here we discuss the vortex-free regime.
\begin{figure}[t]
\makebox[\linewidth][c]{
\begin{subfloat}{\label{fig:lowB_N1}}
\centering
\includegraphics[width=0.5\columnwidth]{lowB_N1}
\end{subfloat}
\begin{subfloat}{\label{fig:lowB_N7}}
\includegraphics[width=0.5\columnwidth]{lowB_N7}
\end{subfloat}
}
\caption{\label{fig:lowB}(Color online) (a) Magnetic field dependence of the qubit frequency ($f_{01}$), three measured time scales ($T_1$,$T^*_2$, and $T_2$) and two calculated time scales ($T_\text{P}$ and $T_1^\text{cal}$) at low magnetic field much smaller than the SQUID oscillation period for the sample N1. $T_\text{P}$ (Purcell time) was calculated by $T_\text{P}=1/\Gamma_\text{P}$---inverse of Purcell rate, and $T_1^\text{cal}$ by $1/T_1^\text{cal} =1/T_\text{NP}+1/T_\text{P}$ (see texts) (b) The qubit frequency and three measured time scales for N7.}
\end{figure}
We first examined the energy relaxation time, $T_1$, for both samples. They were measured separately in
the same cavity which has its loaded lowest order mode at $\sim$8.42 GHz with a loaded (measured) quality factor $Q_\text{L}=5000$. As Fig.~\hyperref[fig:lowB]{\ref{fig:lowB_N1}} and \hyperref[fig:lowB]{\ref{fig:lowB_N7}} show, at zero field the relaxation time $T_1$ was substantially larger for the sample N7 than for N1. Furthermore, when a small magnetic field was applied, the energy relaxation time for N1 increased, while hardly any change
was observed for N7. Both of these effects can be understood as consequences of the Purcell effect in
which the rate of spontaneous emission is increased when the cavity mode to which the qubit couples lies
close by in frequency. The excitation frequency of N7 (4.97 GHz) was further from the cavity frequency
than was the excitation frequency of N1 (6.583 GHz) and this almost completely determines the difference
in $T_1$. To see this we first compare the measured ratio $\Gamma_\text{N1}/\Gamma_\text{N7}$ and the calculated ratio of Purcell rates for the two devices. (Since the next higher cavity resonance is more than 11 GHz above the fundamental, we ignore its contribution to the Purcell relaxation rate.) For the Purcell relaxation rate we use $\Gamma_\text{P}=\kappa_1(g/\Delta)^2$, where $\kappa_1=\omega_c/Q_\text{L}$ is the cavity power decay rate, $g$ is the qubit-cavity coupling rate, and $\Delta$ is the qubit-cavity
detuning $\Delta=|\omega_{01}-\omega_c|=2\pi |f_{01}-f_\text{c}|$. The ratio of the Purcell rates depends only on the qubit-cavity frequency differences which are easy to measure. We find that the ratio of the qubits measured lifetimes for samples N1 and N7 is $T_\text{1,N1}/T_\text{1,N7}=13$ $\mu\text{s}/44$ $ \mu\text{s}=0.3$, while the ratio of the calculated Purcell times, $T_\text{P}=1/\Gamma_\text{P}$ is
$(\Delta_\text{N1}/\Delta_\text{N7})^2=0.29$. Since the measured and the estimated ratios are very close to each other, one can conclude that the relaxation is Purcell limited.
To confirm this conclusion we analyze the field dependence of the relaxation time. This is done using the qubit-cavity coupling, $g=130$ MHz (see Appendix~\hyperref[sec:device]{\ref{sec:device}}) and the formula for the Purcell rate $\Gamma_\text{P}=\kappa_1(g/\Delta)^2$. The increase in relaxation time for N1 with the applied B-field can be understood as a consequences of the Purcell effect. Indeed $\Gamma_\text{P}\sim1/\Delta^2$ decreases with $B$ because the qubit frequency decreases with increasing B-field thus causing the detuning $\Delta$ to increase. This makes a
measurable difference in $\Gamma_\text{P}$ for N1. For N1 we plot the relaxation time versus the B-field calculated as $1/T_1(B)=\Gamma_\text{P} (B)+\Gamma_\text{NP}$ [Fig.~\hyperref[fig:lowB_N1]{\ref{fig:lowB_N1}}]. Here $\Gamma_\text{NP}=2\pi\times 3.5$ kHz is a constant Non-Purcell rate estimated at zero magnetic field. A good agreement between the data and the fit is observed, confirming that the qubit is Purcell limited. Note that sample N7 does not show a noticeable dependence on the B-field because its detuning value is high and therefore the Purcell effect is relatively weak.
Finally we can estimate the internal relaxation rate of our devices assuming that the Purcell effect is eliminated by making the difference between the qubit frequency and the cavity difference sufficiently large. Such internal relaxation is represented by $\Gamma_\text{NP}$, as explained above. For N1 we obtain, at the sweet spot, $\Gamma_\text{P}=2\pi\times 8.5$ kHz and $\Gamma_\text{NP}=2\pi\times 3.5$ kHz, and, correspondingly $T_\text{NP}= 45$ $\mu\text{s}$. For N7 we estimate, again at the zero field sweet spot, $\Gamma_\text{P}=2\pi\times 2.4$ kHz and $\Gamma_\text{NP}=2\pi\times 1.2$ kHz, and, therefore $T_\text{NP}=132$ $ \mu\text{s}$. This analysis reveals that the relaxation time could be above 100 $\mu$s if it were not Purcell-limited, indicating that the coupling to energy absorbing defects in the circuit and qubit is low.
We now consider the spin echo coherence time $T_2$ at $B=0$ which measures the phase coherence attainable in a qubit process. $T_2$ is related to $T_1$ by the constitutive relation $1/T_2=1/(2T_1)+ 1/T_{\varphi}$, where $T_{\varphi}$ is the dephasing time due to random fluctuations of the phase evolution rate of the qubit wave function. N1 had a much shorter $T_2$ than N7. Specifically,
$T_\text{2,N1}=4.2$ $\mu\text{s}$ and $T_\text{2,N7}=39$ $ \mu\text{s}$, almost ten times as large. Using the measured values of $T_1$ we obtain
$T_{\varphi,\text{N1}}=3$ $\mu\text{s}$ and $T_{\varphi,\text{N7}}=70$ $\mu\text{s}$. We attribute the much longer dephasing time for N7 to the fact that the testing conditions were different. For measurements of N7, base-temperature copper powder filters were added to the input and output ports of the cavity. These are known to reduce stray-photon noise by providing attenuation at low temperature. Such photon noise may be responsible for the significantly lower dephasing time seen in the N1 measurements. The photon noise can induce dephasing because of a strong ac-Stark shift~\cite{Sears2012PRB}. Meanwhile, the $T_2$ of sample N7 became shorter as magnetic field was applied. Such behavior can be expected because an application of even a small $B$-field leads to a shift from the magnetic sweet spot. Thus the qubit becomes
more susceptible to dephasing caused by flux noise. This effect is less visible in the sample N1 because $T_2$ is already strongly suppressed by the stray-photon noise in this sample.
\subsection{Time domain measurement: high magnetic field}
Now we consider a different regime where sufficiently high magnetic fields creates vortices on the electrodes. We investigate the effect of Abrikosov vortices on the coherence times. Since the samples were zero-field-cooled there were no vortices in the electrodes to begin with. We managed to gradually increase the number of vortices by sweeping up the external perpendicular magnetic field. All measurements presented below have been made at the sweet spots, which occur periodically or approximately periodically with the magnetic field.
In contrary to a single transmon, the Meissner transmons have advantage to allow us to detect the entrance of a vortex (or multiple vortices) into the electrodes. In actual measurements, we ramped up the magnetic field until the next sweet spot was reached [see Fig.~\hyperref[fig:Bdep]{\ref{fig:TransmVsB}}] and then performed the next series of time-domain measurements. Upon the event of vortex entrance, we observed two signatures: hysteresis of the transmission plotted versus the magnetic field and a shift of the next sweet spot to higher magnetic field than would be expected if the pattern was exactly periodic. This happens because each vortex in the electrodes adds a phase gradient opposite to the one generated by the Meissner current [Fig.~\hyperref[fig:Bdep]{\ref{fig:TransmVsB}}].
\begin{figure}[b]
\makebox[\linewidth][c]{%
\begin{subfloat}{\label{fig:highB_N1}}
\includegraphics[width=0.5\columnwidth]{highB_N1}
\end{subfloat}
\begin{subfloat}{\label{fig:highB_N7}}
\includegraphics[width=0.5\columnwidth]{highB_N7}
\end{subfloat}
}
\caption{\label{fig:highB}(Color online) The qubit transition frequencies ($f_{01}$) and three times scales ($T_1$,$T^*_2$, and $T_2$) were measured at the sweet spots over the wide range of magnetic field for the N1 (a) and N7 (b). }
\end{figure}
In Fig.~\hyperref[fig:highB]{\ref{fig:highB_N1}}, we show the magnetic field dependence of three times scales and the qubit frequency for sample N1. It is observed that $T_1$ was enhanced from 10 $\mu$s to 14 $\mu$s as the magnetic field was increased. The trend was observed up to about 2 Gauss, while at higher fields the trend was reversed. In the case of sample N7 [Fig.~\hyperref[fig:highB]{\ref{fig:lowB_N7}}], all three time scales stayed almost constant up to about 2 Gauss and started to drop as the field was increasing further. This is explained by the fact that vortices begin to penetrate into the electrodes at the first critical field $B_\text{c1}\approx$2 G. They provide a radiation-free dissipation source and thus suppress the relaxation time significantly [see Fig.~\hyperref[fig:highB]{\ref{fig:highB}}]. Meanwhile, the coherence time $T_2$ (and $T^*_2$) also decreased, mainly due to the reduction of $T_1$. Since the measurements were carried out at the sweet spots, the dephasing caused by non-zero dispersion of $f_{01}(B)$ was negligible.
The entrance of vortices is confirmed by a comparison to a theoretical model. In Ref.~\onlinecite{Marksimov1997} the first critical field is approximated as $B_\text{c1}$=$\Phi_0/\left[2\pi\xi\sqrt{2\lambda_{\perp} X}\right]$, where $\lambda_{\perp}$=$2\lambda(0)^2/d$ represents the penetration depth of a thin film in perpendicular field. Using the relations, $\lambda(0)$=$ \lambda_\text{L}\sqrt{1+\xi_0/l}$ and $\xi$=$\sqrt{\xi_0 l}$, we estimate $B_\text{c1}$=5.9 G which is similar to the measured value, 2 G. Here $\xi$ is the coherence length, $\xi_0$ is the clean limit coherence length, $\lambda (0)$ is the bulk penetration depth, $ \lambda_\text{L}$ is the clean-limit penetration depth for Al. The parameter values are: $\xi_0=1600$ nm, $l=16.7$ nm, $ \lambda_\text{L}=16 $ nm, $\xi$=163 nm, $\lambda (0)=158$ nm, $\lambda_{\perp}=552$ nm, $X=10\,\mu$m. The electronic mean free path $l$ is calculated from the measured resistivity of the Al films forming the electrodes, $\rho_\text{n}=2.4\times 10^{-8}$, according to Ref.~\cite{Romijn1982PRB}, using $l\rho_\text{n}=4\times10^{-16}\,\Omega$m$^2$.
\begin{figure}[t,b]
\makebox[\linewidth][c]{%
\begin{subfloat}{\label{fig:T1rate_N1}}
\includegraphics[width=0.5\columnwidth]{T1rate_N1}
\end{subfloat}
\begin{subfloat}{\label{fig:T1rate_N7}}
\includegraphics[width=0.5\columnwidth]{T1rate_N7}
\end{subfloat}
}
\caption{\label{fig:T1rate}(Color online) The relaxation rates $\Gamma$=$1/T_1$ versus magnetic field were plotted for both N1 (a) and N7 (b). All measurements have been done at the sweet spots. The linear fits (red solid lines) were shown over the magnetic field range where the vortices were present in the electrodes. }
\end{figure}
Our goal now is to achieve a quantitative characterization of the non-radiative relaxation process caused by vortices. For this, we plot the relaxation rate $\Gamma=1/T_1$ versus magnetic field, $B$, in Fig.~\hyperref[fig:T1rate]{\ref{fig:T1rate}}. The relaxation rate remains approximately constant at $B<B_\text{c1}$ and increases approximately linearly at $B>B_\text{c1}$. The observed increase of the relaxation rate per Gauss ($d\Gamma/dB$) was 78.5 kHz/G for sample N1 and 43.7 kHz/G for sample N7. In what follows we suggest a model of non-radiative decay, which can explain these values.
We suggest that the energy relaxation of the qubits is mainly due to the energy dissipation originating from the vortex viscous motion. The motion of vortices is initiated be the Lorentz force, which, in turn, is due to the currents generated by the qubit itself. The motion of vortices is overdamped due to the viscous drag force, which, per unit vortex length, is $\textbf{f}_v=-\zeta\textbf{v}$, where $\zeta$ is a viscous drag coefficient for a vortex of unit length, and $\textbf{v}$ is the vortex velocity~\cite{Tinkhambook}. This process is a non-radiative relaxation in which the qubit energy is dissipated as heat. We estimate this relaxation rate semi-classically, using the Bardeen-Stephen model~\cite{Tinkhambook}. According to their model, the viscosity $\zeta$ per unit length is $\zeta=\Phi_0B_\text{c2}/\rho_\text{n}$, where $B_{c2}=\Phi_0/(2\pi\xi^2)$, and $\xi=\sqrt{\xi_0 l}$. Here $B_{c2}$ is the second critical field of the thin-film Al electrodes and $\rho_\text{n}$ is their normal-state resistivity.
Let us estimate the energy relaxation rate, $\Gamma_\text{v}$, caused by one vortex via viscous damping. The rate of the energy dissipation---dissipated power---is $P=-(\textbf{f}_\text{v}\cdot\textbf{v}) d=f_\text{v}^2d / \zeta$, where $d$ is the thickness of film. Thus the energy relaxation rate of a transmon by a vortex can be evaluated as $\Gamma_\text{v}$=$P/(\hbar\omega_{01})$=$\left(f_\text{v}^2/\zeta\right)d/\hbar\omega_{01}$, where $\hbar\omega_{01}$ is the energy stored in the first excited state of the qubit. The vortex is driven by the Lorentz force $\textbf{f}_L$=$\textbf{J}\times\Phi_0$ (this is the force per unit length), where $\textbf{J}$ is a supercurrent density. The supercurrent density is proportional to the total current, $J=I/Xd$, i.e., the current density magnitude is approximated by the total current $I$ divided by the cross section area of the electrode. A naive first guess might be that vortices in the electrodes should not move at all since the expectation value of the current generated by the qubit is zero, $\langle J\rangle =0$, both in the ground and in the exited state of the qubit. Yet we will see soon that the dissipation is proportional to $\langle J^2\rangle$, which is greater than zero.
The next step is to set $f_\text{v}=f_\text{L}$ based on a reasonable assumption that the effective mass of the vortex and the pinning force are negligible. Consequently, we obtained the energy relaxation rate per vortex,
\begin{equation}\label{eq:dissipationrate}
\Gamma_\text{v}=\frac{J^2\Phi^2_0 d}{\zeta \hbar\omega_{01}},
\end{equation}
Of course for the quantum states of the qubit the current and the current density are quantum variables which do not have definite values but should be viewed as quantum mechanical operators. The probability amplitudes of these quantities are defined by the wave function of the qubit. Thus under $J^2$ we understand the mean square of the current density, $J^2=\left<1|\hat{I}^2|1\right>/(Xd)^2$, where the averaging is done for the excited quantum state of the qubit. Here $\hat{I}$ is the operator of the current in the qubit and $X$ and $d$ represent the width and the thickness of the electrodes.
The model outlined above leads to the following estimates for the relaxation rate induced on the qubit by a single vortex: $\Gamma_\text{v}=89$ kHz/vortex and $\Gamma_\text{v}=48$ kHz/vortex, for samples N1 and N7 correspondingly. The following set of sample-specific parameters has been used for sample N1, $\sqrt{\left<I^2\right>}=29.5$ nA, $J=32.8$ kA/m$^2$, $\omega_{01}/2\pi=6.583$ GHz, $\zeta=1.1\times 10^{-9}$ N$\cdot$s/m$^2$, $B_{c2}=12.2$ mT, $\rho_\text{n}=2.4\times 10^{-8}$ $\Omega\cdot$m, $\xi =163$ nm, $\xi_0=1600$ nm, $X=10$ $\mu$m, $d=90$ nm, and $l =16.7$ nm. For sample N7, all the parameters are the same, with the exception of $\sqrt{\left<I^2\right>}=18.9$ nA, $J=21.0$ kA/m$^2$, and $\omega_{01}/2\pi=4.972$ GHz.
\begin{figure}[t,b]
\includegraphics[width=0.8\columnwidth]{vortexcount}
\begin{subfloat}{\label{fig:vortexcount_a}}
\end{subfloat}
\begin{subfloat}{\label{fig:vortexcount_b}}
\end{subfloat}
\begin{subfloat}{\label{fig:vortexcount_c}}
\end{subfloat}
\caption{(Color online)(a) The magnetic fields $B_n$ at sweet spots (black open circle) are depicted as a function of $n$---index of the sweet spots, for sample N7. The blue and red solid lines represent a linear fit to the data for $n\leq6$ and $n\geq7$, respectively. The blue fitted line is extended for $n\geq7$ to show the expected $B_n$ when no vortex penetration is assumed. (b) The difference $\Delta B_n$=$B_n-B_{n-1}$ is shown on the left axis, while the number of vorticies $\Delta N_n$ (defined in the text) provides the scale for the right axis. (c) The total number of vorticies is calculated by summation of $\Delta N_n$. The dashed lines are periodically spaced. They provide a guide-to-the-eye to emphasize the stepwise characteristics of the increasing number of vortices in the electrodes.}\label{fig:vortexcount}
\end{figure}
To compare the relaxation rate $\Gamma_\text{v}$, computed per a single vortex (see above), with the experimental relaxation rates, $d\Gamma/dB$, measured ``per Gauss'', we need to estimate the average rate of the vortex entrance, $dN/dB$. Then one can use a formula $d\Gamma/dB=\Gamma_\text{v}(dN/dB)$, which assumes that the relaxation rates of all vortices simply add up.
A detailed analysis of various possibilities to estimate the vortex entrance rate are given in Appendix~\hyperref[sec:vortex_count]{\ref{sec:vortex_count}}. Here we briefly outline the most intuitive estimate. First, we define $B_n$ as the sequence of magnetic fields corresponding to the sequence of the sweet spots, indexed by the integer $n$=0, 1,$\ldots$, 27. Here 27 is the total number of the observed sweet spots [see Fig.~\hyperref[fig:vortexcount]{\ref{fig:vortexcount_a}}].
At low fields $B_n$ (black open circle) increase linearly with $n$, due to exact periodicity of the HV-oscillation in the vortex-free regime. The linear fit [blue line in Fig.~\hyperref[fig:vortexcount]{\ref{fig:vortexcount_a}}] gives the value of the period, $\Delta B$ =0.2 G. For $n>7$ the period becomes larger because vortices begin to penetrate. The new slope, and, correspondingly, the new period is $\Delta B+\Delta B_\text{v}=0.278$ G, for sample N7. This best-fit value is obtained from the linear fit represented by the red solid line in Fig.~\hyperref[fig:vortexcount]{\ref{fig:vortexcount_a}}. The enlargement of the period happens because vortices compensate, to some extent, the strength of the Meissner current.
Now we calculate the difference between the consecutive sweet spot fields $\Delta B_n$=$B_n-B_{n-1}$. The result is plotted in Fig.~\hyperref[fig:vortexcount]{\ref{fig:vortexcount_b}}. The result was then converted into some effective change of the vortex number, $\Delta N \sim (B_n-B_{n-1})-\Delta B$, where $\Delta B$ is the distance between the sweet spots in the vortex-free low-field regime. Thus defined $\Delta N$ should be considered as a function proportional to the number of vortices coupled to the qubit. But since the conversion factor is not well known, $\Delta N$ should not be considered as equal to the number of relevant vortices. Finally, we integrate $\Delta N_{n}$ with respect to $n$. The result is shown in Fig.~\hyperref[fig:vortexcount]{\ref{fig:vortexcount_c}}. This integrated function exhibits clear steps, which we interpret as vortices entering the sensitivity area of the qubit. The steps are made more noticeable by placing the horizontal dashed lines. The spacing between the lines is made constant and they serve as guide to the eye. The total number of effectively coupled vortices equals the number of steps, i.e. equals 6. These 6 vortices have entered over the interval of 5.6 Gauss. Thus the effective entrance rate is estimated as $dN/dB\approx$ 1.07 vortex/G.
Remember that the experimental relaxation rates per Gauss $d\Gamma/dB$, obtained from Fig.~\hyperref[fig:T1rate]{\ref{fig:T1rate}}, are 78.5 and 43.7 kHz/G for samples N1 and N7. Now these values need to be divided by 1.07 vortex/G, which is the rate of the vortex entrance. Thus we conclude that the experimental relaxation rates are $\Gamma_\text{v}=73$ kHz/vortex and $\Gamma_\text{v}=41$ kHz/vortex, for samples N1 and N7, calculated using $\Gamma_\text{v}=(d\Gamma/dB)/(dN/dB)$. These values are in excellent agreement with the theoretical estimates 89 kHz and 48 kHz/vortex.
To understand the significance of the obtained results for the macroscopic quantum physics, it is instructive to compare the relaxation rates generated by individual vortices to the theory of localization caused by dissipative environment. According to the Caldeira and Leggett (CL) theory~\cite{Caldeira1981PRL}, the particle wave function should be exponentially localized with the localization length scale estimated as $x_\text{CL}^2\simeq h/\eta$, where $\eta$ is the viscous drag coefficient of the macroscopic quantum particle coupled to the environment. This theory was later generalized to the case of periodic potentials in Refs.~\cite{SchmidPRL1983,BulgadaevJTEPL1984}. The conclusion of these papers was that if the period of the potential is larger than $x_\text{CL}$ then the particle becomes localized in one of the wells in the limit of zero temperature. On the other hand, if the period is smaller than $x_\text{CL}$ then the particle can tunnel from one minimum to the next one even at zero temperature. It is important that the scale of the environmental localization, $x_\text{CL}$, is independent of the amplitude of the modulation of the potential energy. Thus it can provide a useful estimate even if the potential is approximately flat, which is the case for the Abrikosov vortex in the Al film in our devices. The viscous drag coefficient for a single vortex is $\eta=d\zeta=9.9\times 10^{-17}$ N$\cdot$s/m, where $d=90$ nm is the vortex length, which is set by the film thickness $d$. Then the Caldeira-Leggett (CL) localization scale is $x_\text{CL}=\sqrt{h/\eta}=2.6\times 10^{-9}$ m = 2.6 nm. This is the scale up to which the wave function of a vortex can spread. If the spread is larger than $x_\text{CL}$ then the coupling to the environment causes the wave function collapse.
The CL localization scale should be compared to the estimated smearing of the wave function of the center-of-mass of the vortex, generated by fluctuations of the supercurrent and the corresponding fluctuations of the Lorentz force. The smearing of the wave function can be estimated as follows. The root-mean-square (rms) value of the Lorentz force is $F_\text{L}=d\cdot f_\text{L}=6.1\times 10^{-18}$ N. (Here we consider the example of sample N1. All estimates for sample N7 are very similar.). Then, assuming viscous motion, the rms velocity is $v=F_\text{L}/\eta=6.2\times10^{-2}$ m/s. Therefore, the rms quantum fluctuations of the vortex center position, $x_\text{v}$, can be estimated as $x_\text{v}=v/\omega_{01}$. This relation would be exact if the motion of the vortex would be described by a classical trajectory, in response to a harmonic drive, such that the deviations from its point of equilibrium would be proportional to $\sin(\omega_{01}t)$. In the case considered the vortex is not described by a classical trajectory, since it should act as a quantum particle at time scales shorter than the relaxation time of the qubit and also because the force is generated by a quantum superposition of currents with opposite polarities. But we assume that the relationship between the quantum fluctuation of the position and the velocity is approximately the same as in the case of a classical harmonic motion. Such assumption is motivated by the natural expectation that the vortex should behave as a damped quasi-classical particle. Thus we can now evaluate the rms smearing, $x_\text{v}$, of the wave function of the vortex center induced by the quantum fluctuations of the Lorentz force. The result is $x_\text{v}=(F_\text{L}/\eta)/\omega_{01}=1.5\times 10^{-12}$ m = 1.5 pm. Here we have used the qubit frequency $f_{01}=\omega_{01}/2\pi=6.58$ GHz for sample N1. The conclusion is that the quantum uncertainty of the vortex position is much smaller than the CL localization scale, $x_\text{v} \ll x_\text{CL}$, namely $x_\text{CL}/x_\text{v}=1700$. Such uncertainty is achieved within one period, $T_{01}=2\pi/\omega_{01}$.
The CL scale provide the maximum value of the rms smearing of the wave function. When such level of smearing is achieved, the wave function collapses and the qubit experiences a decoherence event. The number of complete phase revolutions of the qubit, $N_\text{dch}$, which is needed to achieve the CL scale, at which the probability of decoherence becomes of the order of unity, can be estimated assuming that the wave function of the vortex spreads similar to a diffusive random walk. Then $N_\text{dch}=x_\text{CL}^2/x^2=3.0\times 10^6$. Thus the corresponding decoherence time can be estimated as $t_\text{dch}\sim N_\text{dch}T_{01}$. This heuristic argument can be made more precise using Eq.~\eqref{eq:dissipationrate}. From that equation, using the relations listed above, one obtains $t_\text{dch}=(1/2\pi)^2 N_\text{dch}T_{01}=12\,\mu$s. Finally, the estimate for the relaxation rate, added to the system due to one vortex coupled to the qubit, is $\Gamma_\text{v}=1/t_\text{dch}=83$ kHz. This analysis leads to a conclusion that the relaxation rate induced by one vortex can be linked to the ratio of the smearing of its wave function and the Caldeira-Leggett dissipative localization scale. The above estimate shows that the wave function collapse occurs typically at the time when the wave function smearing reaches the Caldeira-Leggett localization scale.
\section{Summary}\label{sec:summary}
We demonstrate the operation of Meissner qubit, which is a transmon qubit strongly coupled to the Meissner current in the electrodes. The periodicity in magnetic field was set mainly by the width of the electrodes rather than the SQUID loop area. Both the frequency and the time domain measurements were performed using the circuit-QED architecture. The three time scales of $T_1$, $T_2^*$ and $T_2$ were measured as functions of the applied magnetic field. The increase of the relaxation rates was attributed to the radiation-free dissipation associated with the viscous motion of Abrikosov vortices pushed by fluctuations of the Lorentz force. Each vortex can exist in a quantum superposition of different position states for $\sim$10-100 $\mu$s, but eventually causes the wave function of the coupled qubit to collapse. The collapse happens when the smearing of the vortex center becomes of the order of the Caldeira-Leggett dissipative localization scale. The presented Meissner qubit provides an effective and controlled coupling of the qubit to Abrikosov vortices. Such coupling provides a new tool to study vortices, which can eventually be applied to vortices harboring Majorana states.
\begin{acknowledgments}
This work was supported by the National Science Foundation under the Grant No. ECCS-1408558 (A.B.) and ECCS-1407875 (A.L.).
\end{acknowledgments}
|
{
"timestamp": "2015-05-14T02:05:06",
"yymm": "1504",
"arxiv_id": "1504.03360",
"language": "en",
"url": "https://arxiv.org/abs/1504.03360"
}
|
\section{Relation between the robustness and the principal eigenvalue}
\noindent
In this section we derive the relation $r = \lambda/(d(a-1))$, first given in \cite{van N}. First, we define the phenotype robustness $r$ as a weighted average over genotype robustness; second, we define it as the extent to which mutation off the neutral network does not deplete the growth due to fitness.
Readers who are less interested in the biological motivation may skip this section.
\emph{Genotype and phenotype robustness.}
Consider a neutral network $P$, and let its adjacency matrix be $A$. The genotype robustness $r_i$ \cite{wagner} of a genotype $g_i$ is the probability of a mutation being neutral: the number of neutral edges incident to $g_i$ (i.e. edges which do not lead to a different phenotype)
divided by the total number of incident edges $d(a-1)$. The genotype robustness can therefore be written as
\begin{align}
\label{geno}
r_i = \frac{\sum_j A_{ij}}{d(a-1)}.
\end{align}
For a neutral network, let $n(t)$ be its population vector at time $t$, with the $i$th component $n_i(t)$ corresponding to the population on genotype $g_i$. The normalized population is distributed according to $n(t)/\sum_i n_i(t)$. Suppose for now that in the limit $t\to\infty$, the normalized population is distributed according to a unique distribution. Then we define the phenotype robustness $r$ to be the long-time population-weighted average of the genotype robustnesses $r_i$:
\begin{align}
\label{pheno}
r = \frac{\sum_i n_i(\infty) r_i}{\sum_i n_i(\infty)}.
\end{align}
It is the fraction of the population flux that is neutral. We will now determine this limit.
\emph{Mutational flux and fitness.} Mutation induces a population flux across neighbouring genotypes. If the mutation rate per letter is $\mu$, the mutational flux is $1-(1-\mu)^{d(a-1)} \approx \mu d(a-1)$ for $\mu d(a-1) \ll 1$. It is the fraction of a population that mutates per generation.
Some of this mutational flux will also cross phenotypic boundaries when neighbouring genotypes lie in two different phenotypes. That which does not cross phenotypic boundaries is neutral. The fitness $f$ is the raw reproductive rate of the phenotype. After $t$ generations, the total population of a neutral network will have changed by a factor of $f^t$, in the absence of mutations.
{\it Mutation matrix}.
The action of mutation on the population distribution over a single generation can be expressed by the mutation matrix $M$:
\begin{align}
M = (1-\mu d(a-1)) I + \mu A.
\label{K}
\end{align}
The first term is the probability that no mutation occurs and the second the probability of mutating.
Being symmetric, $A$ can be diagonalised by an orthonormal set of eigenvectors $x_i$:
\begin{align}
M = (1-\mu d(a-1))\sum_i x_i x_i^\intercal + \mu \sum_i x_i \nu_i x_i^\intercal,
\label{mutation_matrix}
\end{align}
where the $x_i$ satisfy the eigenvalue equation $A x_i = \nu_i x_i$. The population vector $n(t)$ is obtained by transforming an initial vector $n_0$ by $M^t$ and multiplying it by $f^t$:
\begin{align}
\label{population_distribution}
n(t) = f^t M^t n(0) = \sum_i x_i^\intercal n(0) f^t \left(1-\mu d(a-1) \left(1-\frac{\nu_i}{d(a-1)}\right)\right)^t x_i.
\end{align}
Let $\nu_1$ be the largest (principal) eigenvalue of $A$, denoted hereafter $\lambda$.
Since $\lvert \nu_i \rvert \leq d$, all terms $i>1$ decay exponentially with respect to the first for $\mu>0$.
In the large time limit the sum is dominated by the first term, whose eigenvalue $\nu_1 \equiv \lambda$ is largest:
\begin{align}
n_t \approx x_1^\intercal n(0) f^t \left(1-\mu d(a-1) \left(1-\frac{\lambda}{d(a-1)}\right)\right)^t x_1.
\label{n_t_large_time_limit}
\end{align}
We now show that defining the robustness as $\lambda/(d(a-1))$ agrees with the definition of phenotype robustness in (\ref{pheno}). Indeed, plugging (\ref{geno}) into (\ref{pheno}),
\begin{align*}
r &= \frac{\sum_{ij} A_{ij}n_i(\infty)}{d(a-1)\sum_i n_i(\infty)} \\
&= \frac{\sum_{j}\lambda n_j(\infty)}{d(a-1)\sum_i n_i(\infty)} \\
&= \frac{\lambda}{d(a-1)}.
\end{align*}
The quantity $r$ therefore measures how well the shape of the neutral network can reduce the rate of deleterious mutation acting on the population as a whole. We see from (\ref{n_t_large_time_limit}) that at large time $t$ at every generation, a fraction $\mu d(a-1)(1-r)$ of the population mutates off the neutral network, and the growth rate $(1 - \mu d (1-r)) f$ is the fitness that can be usefully employed to increase the population and not spent replenishing population lost to deleterious mutations incurred at the boundary. The steady state distribution of the population depends only on the shape of the neutral network and on neither the mutation rate $\mu$ nor the fitness $f$.
\section{Neutral networks with large eigenvalues}
\noindent
\emph{Bricklayer's graphs.}
Just how robust a phenotype can be---or how large an eigenvalue a neutral network can have---has remained an open question.
For short sequences ($d \leq 4, a=2$), we found from exhaustive enumeration that the most robust neutral networks are themselves hypercubes
or interpolations between them, illustrated in Figure 1.
Computational sampling for slightly longer sequences ($5 \le d \le 9, a=2$) agrees with this.
We generalize the sequence of graphs and interpolations between them in Figure 1 for $a>2$ as follows:
suppose all vertices $\{q\}$ in $H_{d,a}$ are labelled as integers from 0 to $a^d-1$, and two vertices share an edge if their base $a$ representations differ in exactly one digit.
Then $G_{n,a}$ is the subgraph induced by the vertices $q < n$.
We call these graphs $G_{n,a}$ ``bricklayer's graphs'' because they form the sequence by which a bricklayer would instinctively fill in the Hamming graph $H_{d,a}$.
For the remainder of this Note we set the alphabet size $a=2$, so we are only concerned with hypercubes and their subgraphs.
For simplicity we denote $G_{n,2}$ by $G_n$.
We conjecture an extension of our main result for general $a$ in the Conclusion.
For neutral networks on sequences of short length $d$, the bricklayer's graphs $G_n$ are the most robust; they have maximal principal eigenvalues.
For longer lengths $d$, however, a surprise is in store: the $G_n$ are \emph{not} the most robust neutral networks.
In particular, we discovered the following counterexamples for $d=19$ and above.
Let $S_n$ be the star graph: a tree with one internal vertex and $n$ leaf vertices.
The principal eigenvalue of $S_n$ is readily found to be $\sqrt{n}$.
Now let us compare the eigenvalue of a star of $n$ vertices, $S_{n-1}$, to the eigenvalue $\lambda_n$ of $G_n$.
As we prove below, $\lambda_n \leq \log_2 n$.
For the bricklayer's graphs $G_n$ to win, we need $\sqrt{n-1} < \lambda_n$, implying
\begin{align*}
\sqrt{n-1} < \log_2 n.
\end{align*}
However, this not true for $n \ge 20$. It is an open question as to what shape does maximize the eigenvalue for larger graphs. In the concluding remarks of \cite{friedman}, the authors consider the possibility that Hamming balls (graphs consisting of all points that are most a given distance from a point) are asymptotic maximizers of hypercube subgraphs, but then provide some evidence that they are not.
\emph{Our main result.}
In this Note we prove that the principal eigenvalue $\lambda_{n}$ of the bricklayer's graph $G_{n}$ satisfies $\lambda_n \leq \log_2 n$.
Our general approach is to show by a geometric staircase argument that for $d \ge 3$, a slightly stronger inequality ($\lambda_n < \log_2 (n-1)$) holds for most $n$; it will then suffice to examine the cases where $n = 2^d \pm 1$, using polynomials that have $\lambda_{2^d \pm 1}$ as roots.
\begin{theorem*}
\label{bricklayerclustertheorem}
For all graphs $G_n$, we have $\lambda_n \leq \log_2 n$, with equality if and only if $n$ is a power of 2.
\end{theorem*}
Equality is attained in the Theorem if $n$ is a power of $2$ since $\lambda_n$ must lie between the mean and maximum vertex degree \cite{Brouwer}, and for $n$ a power of 2, all vertices are of degree $\log_2 n$.
We wish to show that if $n$ is not a power of $2$, there is strict inequality. We make two observations. Observation 1: Since the principal eigenvalue of a proper subgraph of a connected graph is less than the principal eigenvalue of the graph itself, it follows that if $n < m$ then $\lambda_n < \lambda_m$. Observation 2: Since $G_{2n} = G_{n} \Box K_2$, and the spectrum of a Cartesian product of graphs is the sum of their individual spectra \cite{Brouwer}, it follows that if $\lambda_n < \log_2 n$ then $\lambda_{2n} = \lambda_n + 1 < \log_2 2n$. Using these observations, we claim:
\begin{lemma}\label{reducetoedgecases}
The Theorem is true for all $n$ if for some $k$,
\begin{align}
\label{strongercondition}
\lambda_n < \log_2 (n-1) \textnormal{ for $2^k + 2 \le n \le 2^{k+1} - 1$},
\end{align}
and also
\begin{align}
\label{minusapointtheorem}
&\textnormal{$\lambda_{2^d-1} < \log_2 (2^d-2)$, $d \geq 5$, and} \\
\label{plusapointtheorem}
&\textnormal{$\lambda_{2^d+1} < \log_2 (2^d+\tfrac12)$, $d \geq 3$}.
\end{align}
\end{lemma}
\begin{proof}
The ``staircase" argument is illustrated in Figure \ref{staircase_figure}. We verify numerically that the Theorem is true for $n \leq 16$ and (\ref{strongercondition}) holds for $k=3$. Now if (\ref{strongercondition}) is true for some $k$, then by Observation 2,
\begin{align}
\label{overlappinginequalities}
\lambda_{2n} < \log_2 (2n-2) \textnormal{ for $2^k + 2 \leq n \leq 2^{k+1} - 1$}.
\end{align}
By Observation 1 we have the expansion
\begin{align}
\label{overlappinginequalitiesexpansion}
\lambda_{2n-2} < \lambda_{2n-1} < \lambda_{2n} < \log_2 (2n-2) < \log_2 (2n-1) < \log_2 2n
\end{align}
for $2^k + 2 \leq n \leq 2^{k+1} - 1$, so that $\lambda_{m} < \log_2 m$ for $2^{k+1} + 2 \leq m \leq 2^{k+2} - 2$. Conditions (\ref{minusapointtheorem}) and (\ref{plusapointtheorem}) then ensure that $\lambda_{m} < \log_2 m$ for $m = 2^{k+1} + 1$ and $m=2^{k+2} - 1$ as well, proving the Theorem for $2^{k+1} \le n \le 2^{k+2}$.
Finally, note that (\ref{overlappinginequalitiesexpansion}) together with (\ref{minusapointtheorem}) and (\ref{plusapointtheorem}) imply that (\ref{strongercondition}) holds with $k$ replaced by $k+1$, so we may repeat our induction indefinitely.
\end{proof}
\begin{figure}[b!]
\begin{center}
\includegraphics[width=0.6\columnwidth]{staircase_figure.pdf}
\caption{\label{staircase_figure}
The ``staircase" argument for the proof of Lemma \ref{reducetoedgecases}. Because each bricklayer's graph is a subgraph of its successor, knowing the principal eigenvalue of a bricklayer's graph immediately places restrictions on higher-dimensional bricklayer's graphs (as demonstrated by the staircase figures of each color).
}
\end{center}
\end{figure}
Therefore, the Theorem reduces to (\ref{minusapointtheorem}) and (\ref{plusapointtheorem}), which we will prove by looking at polynomials that have the eigenvalues of our desired graphs as roots.
We will make use of the following standard theorem in linear algebra:
$\newline$
(Cauchy's Interlacing Theorem \cite{Hwang}).
\textit{Let $A$ be an $n \times n$ symmetric nonnegative matrix with eigenvalues $a_1 \leq \dots \leq a_n$, and let $B$ be an $m \times m$ principal submatrix of $A$ with eigenvalues $b_1 \leq \dots \leq b_m$. Then for all $j < m+1$, $a_j \leq b_j \leq a_{n-m+j}$.}
\section{Polynomials with eigenvalues $\lambda_{2^d \pm 1}$ as roots}
\noindent
Let $\chi_{n}$ be the characteristic polynomial of the adjacency matrix of $G_{n}$. Enumerating the hypercube spectrum, we find
\[
\chi_{2^d} (x) = \prod_{i=0}^d (x - (d - 2i))^{\binom{d}{i}},
\]
so we may define the polynomial
\begin{align}
\label{P2ddefinition}
P_{2^d} (x) = \frac{\chi_{2^d} (x)}{\prod_{i=1}^{d-1} (x - (d - 2i))^{\binom{d}{i}-1}} = \prod_{i=0}^d (x - (d - 2i)).
\end{align}
By applying Cauchy's Interlacing Theorem to the adjacency matrices of $G_{2^{d}+1}$ and $G_{2^d}$ (thus ``sandwiching" the spectrum of $\chi_{2^{d}+1}$ by that of $\chi_{2^{d}}$), we see from the multiplicity of the eigenvalues (of $\chi_{2^d}$) that $\chi_{2^d+1} (x)$ has the factor $\prod_{i=1}^{d-1} (x - (d - 2i))^{\binom{d}{i}-1}$. A similar argument shows that $\chi_{2^d-1} (x)$ has the same factor. So we may define the polynomials
\[
P_{2^d \pm 1} (x) = \frac{\chi_{2^{d} \pm 1} (x)}{\prod_{i=1}^{d-1} (x - (d - 2i))^{\binom{d}{i}-1}}.
\]
Furthermore, we see that $\chi_{2^d-1}$ has at most $d$ simple roots, all of which are also roots of the $d$-degree polynomial $P_{2^d-1}$.
The reason we use the polynomials $P$ is related to the fact that $G_{2^d-1}$ is a Hamming ball of radius $d-1$ in $H_{d,2}$. We define $B_{d,r}$, the $d$-dimensional ball of radius $r$, as the set of points in $H_{d,2}$ that are Hamming distance at most $r$ from the ``origin'' (the point labelled ``0'' according to the labelling scheme specified in the first paragraph of section 2). We determine recursive equations that give $\lambda$, the principal eigenvalue of $B_{d,r}$. Consider the corresponding eigenvector $w$, and note that by symmetry the component of $w$ corresponding to a given vertex depends only on the distance of the vertex from the origin. Therefore, let $w_k$ be the value of the component of $w$ corresponding to a vertex of distance $k$ from the origin. By matrix multiplication, we find that
\begin{align}
\label{lambda1}
\lambda w_0 &= d w_1 \\
\label{lambda2}
\lambda w_k &= kw_{k-1} + (d-k)w_{k+1} ~\textnormal{for $1 \leq k < r$} \\
\label{lambda3}
\lambda w_r &= rw_{r-1}.
\end{align}
By setting $w_0 = 1$ and following the equations above for each fixed $r$, we find that the principal eigenvalue of $B_{d,r}$ is a root of the polynomial $p_r (\lambda)$, where $p_0 = \lambda$, $p_1 = \lambda^2 - d$, and
\begin{align*}
p_r &= \lambda p_{r-1} - r(d-r+1)p_{r-2} \textnormal{ for $r \geq 2$.}
\end{align*}
Applying this to $\lambda_{2^k-1}$, we can generate polynomials in $\lambda$ with coefficients in $d$ (say $f_{k}(d,\lambda)$) such that when $k$ is substituted for $d$, the resulting polynomial in $\lambda$ has $\lambda_{2^k-1}$ as a root. Then $f_{1}(d,\lambda) = \lambda$,
$f_{2}(d,\lambda) = \lambda^2 - d$ and, in general,
\begin{align}
\label{frecursiverelation}
f_{k}(d,\lambda) = \lambda f_{k-1}(d,\lambda)- (k-1)(d-k+2)f_{k-2}(d,\lambda),
\end{align}
and $f_{k}(k,\lambda)$ has $\lambda_{2^k-1}$ as a root. In fact, $f_{k}(k,\lambda)$ has every simple eigenvalue of $G_{2^k-1}$ as a root (by the reasoning of the derivation). Since the degree of $f_{k}(k,\lambda)$ as a polynomial in $\lambda$ is $d$, and we have from above that $\chi_{2^d-1}$ has at most $d$ simple roots (all of which are also roots of the $d$-degree polynomial $P_{2^d-1}$), it must be the case that
\begin{align}
\label{pequalsf}
P_{2^k-1}(\lambda) = f_{k}(k,\lambda).
\end{align}
\section{Bounding $\lambda_{2^d-1}$}
\noindent
Rewriting the right side of (\ref{minusapointtheorem}) by applying the Taylor expansion with Lagrange remainder gives
\begin{align*}
\log_2(2^d - 2) = d + \log_2 \left(1-\frac{2}{2^d} \right) >d + \frac{1}{\log 2} \left(-\frac{2}{2^d} - \frac{2^2}{2^{2d-1}} \right).
\end{align*}
Therefore, for $d \geq 5$,
\begin{align}
\label{minusapointlogbound}
\log_2(2^d - 2) > d - \frac32 \, \frac{2}{2^d}.
\end{align}
Now we deal with the left side of (\ref{minusapointtheorem}).
\begin{lemma}
\label{minusapointnonexplicitbound}
$\lambda_{2^d-1} < d - P_{2^d-1}(d)/P'_{2^d-1}(d)$.
\end{lemma}
\begin{proof}
Note that the function $P_{2^d-1}(x)$ is convex on $x \geq \lambda_{2^d-1}$. To see this, let $f(x) = \prod_{i=1}^n (x-r_i)$ be any monic polynomial with all real roots and observe that
\begin{align*}
f''(x)
= 2 \sum_{\substack{j_1,j_2=1 \\ j_1 < j_2}}^n \left[ \prod_{i=1,i \neq j_1, i \neq j_2}^n (x-r_i) \right],
\end{align*}
which is always nonnegative if $x$ is at least the largest root of $f$. Now since the tangent linear approximation of a convex function is an underestimate, we obtain $P_{2^d-1}(d) + P'_{2^d-1}(d) \cdot (\lambda_{2^d-1}-d) < 0$, which implies the lemma.
\end{proof}
We evaluate the desired values of $P_{2^d-1}$ and its derivative using (\ref{pequalsf}) and the recursive relation in (\ref{frecursiverelation}).
\begin{lemma}
\label{minusapointp}
$P_{2^d-1}(d) = d!$
\end{lemma}
\begin{proof}
Recall the definition of $f_k(d,\lambda)$ from the previous section and the fact that $P_{2^k-1} (\lambda) = f_k(k,\lambda)$. We seek to prove that $f_{k}(k,k) = k!$, and to do this we will prove a stronger claim that for $i, k \in \mathbb{N}^+$,
\[
f_{k}(i,i) = (i)_k,
\]
where $(i)_k = i (i-1) (i-2) \cdots (i-k+1)$ is the Pochhammer symbol.
We use induction on $k$. For $k = 1$ we have $f_{1}(i,i) = i$ and $f_{2}(i,i) = i^2 - i = i(i-1)$. Supposing the claim is true for $f_{k-2}(i,i)$ and $f_{k-1}(i,i)$, we find from (\ref{frecursiverelation}) that
\begin{align*}
f_{k}(i,i) &= i f_{k-1}(i,i)- (k-1)(i-k+2)f_{k-2}(i,i) \\
&= i (i)_{k-1} - (k-1)(i-k+2) (i)_{k-2} \\
&= (i)_k.
\qedhere
\end{align*}
\end{proof}
\begin{lemma}
\label{minusapointpderivative}
\[
P'_{2^d-1}(d) = d! \sum_{j=0}^{d-1} \frac{2^j}{j+1}.
\]
\end{lemma}
\begin{proof}
With respect to the polynomials $f_{k}(d,\lambda)$, let $f'_{k}(d,\lambda)$ denote $\frac{\partial }{\partial \lambda} f_{k}(d,\lambda)$. Then $P'_{2^k-1}(\lambda) = f'_{k}(k,\lambda)$, and $f'_{1}(d,\lambda) = 1$, $f'_{2}(d,\lambda) = 2\lambda$, and for $k \geq 3$, from (\ref{frecursiverelation}),
\begin{align*}
f'_{k}(d,\lambda) &= f_{k-1}(d,\lambda) + \lambda f'_{k-1}(d,\lambda) - (k-1)(d-k+2)f'_{k-2}(d,\lambda).
\end{align*}
We seek to prove that $f'_{k}(k,k) = k! \sum_{j=0}^{k-1} \frac{2^j}{j+1}$, and to do this we will prove a stronger claim that for integers $k > 0, i \ge 0$,
\begin{align*}
f'_{k}(k+i,k+i) = k! \sum_{j=0}^{k-1} \binom{k-j+i-1}{i} \frac{2^{j}}{j+1}.
\end{align*}
We use induction on $k$. For the base cases $k = 1$ and $k = 2$ we find
that $f'_{1}(1+i,1+i) = 1$ and $f'_{2}(2+i,2+i) = 2(2+i)$, as desired. Now supposing the claim is true for $f'_{k-2}(k+i,k+i)$ and $f'_{k-1}(k+i,k+i)$, it follows that
\begingroup
\allowdisplaybreaks
\small
\begin{align*}
f'_{k}(k+i,k+i) &= f_{k-1}(k+i,k+i) + (k+i) f'_{k-1}(k+i,k+i) - (k-1)(i+2)f'_{k-2}(k+i,k+i) \\
&= (k+i)_{k-1} + (k+i) (k-1)! \sum_{j=0}^{k-2} {\textstyle \frac{(k-j+i-1)!}{(i+1)!(k-j-2)!} \frac{2^{j}}{j+1} - (k-1)!} \sum_{j=0}^{k-3} \textstyle \frac{(k-j+i-1)!}{(i+1)!(k-j-3)!} \frac{2^{j}}{j+1} \\
&= {\textstyle \frac{(k+i)!}{(i+1)!} + (k-1) \frac{(k+i)!}{(i+1)!}} + \sum_{j=0}^{k-2} \left[ \textstyle \frac{(k-1)!}{(k-j-2)!} \frac{(k-j+i)!}{(i+1)!} + j \frac{(k-1)!}{(k-j-2)!} \frac{(k-j+i-1)!}{(i+1)!} \right] {\textstyle \frac{2^{j}}{j+1}} -(k-1)(k-2) {\textstyle \frac{(k+i-1)!}{(i+1)!}} - \sum_{j=2}^{k-2} \textstyle \frac{(k-1)!}{(k-j-2)!} \frac{2^{j-1}}{j} \frac{(k-j+i)!}{(i+1)!} \\
&= {\textstyle \frac{(k+i)!}{(i+1)!} + (k-1) \textstyle \frac{(k+i)!}{(i+1)!} + (k-1)(k-2)\frac{(k+i-1)!}{(i+1)!}} + \sum_{j=2}^{k-2} \left[ \textstyle \frac{(k-1)!}{(k-j-2)!} \frac{2^j}{j+1} + (j-1) \frac{(k-1)!}{(k-j-1)!} \frac{2^{j-1}}{j} \right] {\textstyle \frac{(k-j+i)!}{(i+1)!}} \\
&\phantom{=} + (k-2)(k-1)! {\textstyle \frac{2^{k-2}}{k-1} -(k-1)(k-2)\frac{(k+i-1)!}{(i+1)!}} - \sum_{j=2}^{k-2} {\textstyle \frac{(k-1)!}{(k-j-2)!} \frac{2^{j-1}}{j} \frac{(k-j+i)!}{(i+1)!} } \\
&= k {\textstyle \frac{(k+i)!}{(i+1)!}} + \sum_{j=2}^{k-2} \textstyle \left[ \textstyle \frac{(k-1)!}{(k-j-2)!} \left( \frac{2^j}{j+1} - \frac{2^{j-1}}{j} \right) + (j-1) \frac{(k-1)!}{(k-j-1)!} \frac{2^{j-1}}{j} \right] \frac{(k-j+i)!}{(i+1)!} + [2(k-1)(k-1)! - k!] \textstyle \frac{2^{k-2}}{k-1} \\
&= k {\textstyle \frac{(k+i)!}{(i+1)!}} + \sum_{j=2}^{k-2} \textstyle \frac{k!}{(k-j-1)!} \left( \frac{2^j}{j+1} - \frac{2^{j-1}}{j} \right) \frac{(k-j+i)!}{(i+1)!} + k! \left( \frac{2^{k-1}}{k} - \frac{2^{k-2}}{k-1} \right) \\
&= k! \sum_{j=0}^{k-1} \left[ \textstyle \frac{(k-j+i)!}{(i+1)!(k-j-1)!} \frac{2^j}{j+1} - \frac{(k-j-1)(k-j+i-1)!}{(i+1)!(k-j-1)!} \frac{2^j}{j+1} \right] \\
&= k! \sum_{j=0}^{k-1} \textstyle \frac{(k-j+i-1)!}{i!(k-j-1)!} \frac{2^{j}}{j+1}.
\qedhere
\end{align*}
\normalsize
\endgroup
\end{proof}
Now substituting the results of Lemmas \ref{minusapointp} and \ref{minusapointpderivative} into Lemma \ref{minusapointnonexplicitbound},
\begin{align}
\label{minusapointbound}
\lambda_{2^d-1} < d - \frac{1}{\sum_{j=0}^{d-1} \frac{2^j}{j+1}}.
\end{align}
A simple induction with base cases $1 \le d \le 3$ shows that the sum in the denominator of the second term of (\ref{minusapointbound}) on the right-hand side can be bounded by
\begin{align}
\label{sumbound}
\sum_{j=1}^{d} \frac{2^j}{j} < 3 \, \frac{2^d}{d}.
\end{align}
Therefore,
\begin{align}
\label{minusapointsumbound}
d - \frac{1}{\sum_{j=0}^{d-1} \frac{2^j}{j+1}} < d - \frac{2}{3} \, \frac{d}{2^d}.
\end{align}
Since we may combine the bounds (\ref{minusapointsumbound}) and (\ref{minusapointlogbound}) for $d \geq 5$, we have proved (\ref{minusapointtheorem}).
\section{Bounding $\lambda_{2^d+1}$}
\noindent
Rewriting the right side of (\ref{plusapointtheorem}) in much the same way as the previous section, we find that for $d \geq 3$,
\begin{align}
\label{plusapointlogbound}
\log_2(2^d + \tfrac12) > d + \frac{\tfrac12}{2^d}.
\end{align}
As for the left side of (\ref{plusapointtheorem}), we have a computational shortcut:
\begin{lemma}
\label{Ge}
Let $G$ be a graph, and let $e$ be a bridge of $G$ (i.e., $e$ is an edge such that removing it would increase the number of connected components of $G$). Let $G^{*}$ be the graph $G$ with $e$ removed, and $G^{**}$ be the graph $G$ with $e$ and its endpoints removed.
Then $\chi_{G} = \chi_{G^{*}} - \chi_{G^{**}}$, where $\chi_{G}$ is the characteristic polynomial of the adjacency matrix of $G$.
\end{lemma}
\begin{proof}
This is an extension of Lemma 1 in \cite{Lovasz}, which states the result when $G$ is a forest and $e$ is any edge, and is also Theorem 1.3 in \cite{stevanovic}. The proof involves expanding the matrix whose determinant is $\chi_{G}$, using Laplacian expansion and linearity of the determinant.
\end{proof}
\begin{corollary}
\label{prelation}
$\chi_{2^d+1}(\lambda) = \lambda \chi_{2^d}(\lambda) - \chi_{2^d-1}(\lambda)$, and so dividing by $\prod_{i=1}^{d-1} (x - (d - 2i))^{\binom{d}{i}-1}$,
\[
P_{2^d+1}(\lambda) = \lambda P_{2^d}(\lambda) - P_{2^d-1}(\lambda).
\]
\end{corollary}
\begin{lemma}
\label{plusapointnonexplicitbound}
$\lambda_{2^d+1} < d - P_{2^d+1}(d)/P'_{2^d+1}(d)$.
\end{lemma}
\begin{proof}
From the preceding corollary,
\begin{align*}
P''_{2^d+1}(x) = 2P'_{2^d}(x) + x P''_{2^d}(x) - P''_{2^d-1}(x).
\end{align*}
We wish to show that this is nonnegative on $x \geq d$. From the argument of the proof of Lemma \ref{minusapointnonexplicitbound}, $P'_{2^d}(x) \geq 0$ for $x \geq d$, and from the equation for the second derivative of a polynomial there, $P''_{2^d}(x) > P''_{2^d-1}(x)$ for $x \geq d$ since the roots of $P_{2^d-1}$ interlace those of $P_{2^d}$ by Cauchy's Interlacing Theorem. So $P_{2^d+1}(x)$ is convex on $x \geq d$, and so by linear approximation, $P_{2^d+1}(d) + P'_{2^d+1}(d) \cdot (\lambda(G_{2^d+1})-d) < 0$, which implies the lemma.
\end{proof}
Now as in the previous section, we evaluate the desired values of $P_{2^d+1}(d)$ and its derivative.
\begin{lemma}
\label{plusapointp}
$P_{2^d+1}(d) = -d!$.
\end{lemma}
\begin{proof}
Using Corollary \ref{prelation} and Lemma \ref{minusapointp}, $P_{2^d+1}(d) = d p_{2^d}(d) - p_{2^d-1}(d) = 0 - d!$.
\end{proof}
\begin{lemma}
\label{plusapointpderivative}
\[
P'_{2^d+1}(d) = d! \left( d \, 2^d - \sum_{j=0}^{d-1} \frac{2^j}{j+1} \right).
\]
\end{lemma}
\begin{proof}
Differentiating (\ref{P2ddefinition}) and then plugging in $d$ gives $P'_{2^d}(d) = 2^d d!$.
Then using Corollary \ref{prelation} and Lemma \ref{minusapointpderivative},
\begin{align*}
P'_{2^d+1}(d) &= P_{2^d}(d) + d P'_{2^d}(d) - P'_{2^d-1}(d) \\
&= 0 + d \, 2^d d! - d! \sum_{j=0}^{d-1} \frac{2^j}{j+1} \\
&= d! \left( d \, 2^d - \sum_{j=0}^{d-1} \frac{2^j}{j+1} \right).
\qedhere
\end{align*}
\end{proof}
Substituting the results of Lemmas \ref{plusapointp} and \ref{plusapointpderivative} into Lemma \ref{plusapointnonexplicitbound},
\begin{align}
\label{plusapointbound}
\lambda_{2^d+1} < d + \frac{1}{d \, 2^d - \sum_{j=0}^{d-1} \frac{2^j}{j+1}}.
\end{align}
From (\ref{sumbound}), we obtain that for $d > 1$,
\begin{align}
\label{plusapointsumbound}
d + \frac{1}{d \, 2^d - \sum_{j=0}^{d-1} \frac{2^j}{j+1}} < d + \frac{1}{(d - \frac{3}{2d})2^d}.
\end{align}
Since we may combine the bounds (\ref{plusapointsumbound}) and (\ref{plusapointlogbound}) for $d \geq 5$, we have proved (\ref{plusapointtheorem}).
This concludes the proof of the Theorem.
\section{Conclusion}
\noindent
As an additional remark, if we bound $\log_2 (2^d -N)$ for general $N$ in the manner of (\ref{minusapointlogbound}), we can deduce an asymptotic result: for any $N$, there exists $D$ so large that $\lambda_{2^d-1} < \log_2 (2^d - N)$ for all $d > D$. Similarly generalizing (\ref{plusapointlogbound}) leads to the result that for any $\epsilon > 0$, there exists $D$ so large that $\lambda_{2^d+1} < \log_2 (2^d + \epsilon)$ for all $d > D$.
Throughout most of this paper, we have set $a=2$.
We conjecture an extension of the Theorem for general $a$:
\begin{conjecture*}
For all graphs $G_{n,a}$, we have $\lambda_{n,a} \le (a-1) \log_a n$, with equality if and only if $n$ is a power of $a$.
\end{conjecture*}
There are several interesting and potentially important questions that we have not considered here, which merit further investigation.
We prove that the form of the hypercube subgraph with maximal eigenvalue is a bricklayer's graph for small $d$ but the general form of the maximizers is unknown.
Indeed it is an open avenue of study to find even non-trivial bounds on the eigenvalue of a hypercube subgraph in terms of its number of vertices.
While for small dimension $d$ the bricklayer's graphs are optimal, for $s \ge 20$ and $d \ge s-1$, Hamming balls of radius 1 are superior.
For large $s$ and $d\ll s$, Hamming balls of larger radius may eventually dominate, but this is unproven.
How these transitions extend to larger values of alphabet size $a$ is also an open question, though it seems that the critical dimension separating bricklayer's graphs and balls grows with $a$.
We hope that further research by others will shed light on these questions.
|
{
"timestamp": "2015-11-17T02:07:11",
"yymm": "1504",
"arxiv_id": "1504.03065",
"language": "en",
"url": "https://arxiv.org/abs/1504.03065"
}
|
\section{Introduction and preliminaries}
\subsection{The basic setting}
Let $\mb{C}[z] = \mb{C}[z_1, \ldots, z_d]$ denote the ring of complex polynomials in $d$ variables.
Every $d$-tuple $T = (T_1, \ldots, T_d)$ of commuting operators on a Hilbert space $H$ defines an action of $\mb{C}[z]$ on $H$ via
\[
p\cdot h = p(T) h \quad, \quad p \in \mb{C}[z], h \in H.
\]
This gives $H$ the structure of a module over $\mb{C}[z]$, and we say that $H$ is a {\em Hilbert module} over $\mb{C}[z]$. The Hilbert module $H$ is said to be {\em essentially normal} if the commutator $[T_i, T_j^*]:= T_i T_j^* - T_j^* T_i$ is compact for all $i,j$.
If $p \geq 1$ and $[T_i,T_j^*] \in \mathcal{L}^p$ (meaning that $|[T_i, T_j^*]|^p$ is trace class), then we say that $H$ is {\em $p$-essentially normal}.
One of the interesting ways in which Hilbert modules arise is as follows. Assume that an inner product is given on $\mb{C}[z]$ such that the multiplication operators $S_1, \ldots, S_d$ given by
\[
(S_i f)(z) = z_i f(z)
\]
are bounded. Let $H$ denote the completion of $\mb{C}[z]$ with respect to the given inner product.
Then $H$ becomes a Hilbert module. Examples includes all analytic Hilbert modules on bounded domain in $\mathbb C^d$ \cite{CG}. In the case where (1) monomials of different degrees are orthogonal, and (2) the row operator $[S_1, \ldots, S_d] : H^{(d)} \rightarrow H$ has closed range, these types of modules were referred to as {\em graded completions} of $\mb{C}[z]$ in \cite{Arv07}.
Moreover, if $r \in \mb{N}$, then $H \otimes \mb{C}^r$ is also a Hilbert module in a natural way --- such modules were referred to as {\em standard Hilbert modules} in \cite{Arv07}. Finally, if $M$ is a closed submodule of $H \otimes \mb{C}^r$, then $M$ and $(H \otimes \mathbb{C}^r)/M$ can also be given a natural Hilbert module structure.
\subsection{Homogeneous and quasi homogeneous submodules}
The algebra of polynomials has a natural grading by degree
\[
\mb{C}[z] = \mb{H}_0 \oplus \mb{H}_1 \oplus \mb{H}_2 \oplus \ldots.
\]
This induces a direct sum decomposition
\[
H = \mb{H}_0 \oplus \mb{H}_1 \oplus \mb{H}_2 \oplus \ldots .
\]
A vector valued polynomial $h \in H \otimes \mb{C}^r$ is said to be {\em homogeneous} if $h \in \mb{H}_n \otimes \mb{C}^r$ for some $n$. A module is said to be {\em homogeneous} if it is generated by homogeneous polynomials.
Let ${\bf n} = (n_1, \ldots, n_d) \in \mb{N}^d$, where $n_i > 0$ for all $i$. A vector valued polynomial $h \in \mb{C}[z] \otimes \mb{C}^r$ is said to be {\em ${\bf n}$-quasi homogeneous of degree $m$}, denoted $deg_{\bf n}(h) = m$, if for every monomial $z^\alpha \otimes v_\alpha$ appearing in $h$ it holds that $n_1 \alpha_1 + \ldots n_d \alpha_d = m$.
A module $M \subseteq \mb{C}[z] \otimes \mb{C}^r$ is said to be {\em ${\bf n}$-quasi homogeneous} if it is generated by ${\bf n}$-quasi homogeneous polynomials.
Any ${\bf n}$-quasi homogeneous submodule of a graded completion $H$ (including $H$ itself) also decomposes as a direct sum of ${\bf n}$-quasi homogeneous summands.
\subsection{Stable division and approximate stable division}
\begin{definition}
A submodule $M \triangleleft \mb{C}[z] \otimes \mb{C}^r\subseteq H\otimes\mathbb C^r$ is said to have the {\em approximate stable division property} if there are elements $f_1, \ldots, f_k \in M$ and a constant $A$ such that for every $\epsilon > 0$ and for every $h \in M$, one can find polynomials $g_1, \ldots, g_k \in \mb{C}[z]$ such that
\begin{equation}\label{eq:sum_app}
\|h - \sum_{i=1}^k g_i f_i\| \leq \epsilon ,
\end{equation}
together with the norm constraint
\begin{equation}\label{eq:stab_app}
\sum_{i=1}^k \|g_i f_i \| \leq A \|h\| .
\end{equation}
If $\epsilon$ can be chosen $0$, then we say that $M$ has the {\em stable division property}. The set $\{f_1, \ldots, f_k\}$ is said to be an {\em (approximate) stable generating set}.
\end{definition}
\subsection{The Hilbert spaces $\mathcal{H}_d^{(t)}$}
In this paper we will let $\mathcal{H}_d^{(t)}$ ($t \geq -d$) be the reproducing kernel Hilbert space on the unit ball $\mb{B}_d$ with kernel
\[
k(z,w) = \frac{1}{(1-\langle z, w \rangle)^{d+t+1}} .
\]
This is the completion of $\mb{C}[z]$ with respect to the inner product making all monomials orthogonal, and for which
\begin{equation}\label{eq:tnorm}
\|z^\alpha\|_t^2 = \frac{\alpha!}{\Pi_{i=1}^{|\alpha|} (d+t+i)} .
\end{equation}
When $t = 0$ and $t=-1$, respectively, we get the Bergman space and the Hardy space, respectively, on the unit ball (see \cite{ZhuBook}).
When $t = -d$ then we get the {\em Drury-Arveson space}, denoted $H^2_d$ (see \cite{Arv98, ShalitSurvey}).
Note that if $|\alpha| = n$, then
\begin{equation}\label{eq:quotient_norm}
\frac{\|z^\alpha\|_t^2}{\|z^\alpha\|_{H^2_d}^2} = \frac{n!}{\Pi_{i=1}^{n} (d+t+i)} = \frac{\Gamma(n+1)\Gamma(d+t+1)}{\Gamma(d+t+n+1)} .
\end{equation}
Thus for all $f \in \mb{H}_n$ we have
\[
\|f\|^2_t = c_{n,t}\|f\|^2_{H^2_d} ,
\]
where $c_{n,t}$ denotes the right hand side of (\ref{eq:quotient_norm}).
We see that the $c_{n,t}$ are less than one, decreasing, and tend to $0$ with $n\rightarrow \infty$ for all $t > -d$.
In fact, Arveson showed in the appendix of \cite{Arv07} that if $H$ is any standard Hilbert module which has {\em maximal symmetry} (in the sense that the inner product in $H$ is invariant under the action of the unitary group of $\mb{C}^d$), then any homogeneous polynomial $f \in \mb{H}_n$ has norm
\[
\|f\|^2_H = \gamma_n\|f\|^2_{H^2_d}.
\]
If $f = \sum_{i=k}^m f_i$ is a polynomial with homogeneous parts $f_k, \ldots, f_m$, then
\[
\|f\|_H^2 = \sum_{i=k}^m \|f_i\|_H^2 \leq \sum_{i=k}^m \gamma_i\|f_i\|_{H_d^2}^2 \leq \max_{k \leq i \leq m}\gamma_i \|f\|_{H^2_d}^2 .
\]
Likewise $\|f\|^2_H \geq \min_{k \leq i \leq m} \gamma_i \|f\|^2_{H^2_d}$.
For the spaces $\mathcal{H}_d^{(t)}$ we get
\begin{equation}\label{eq:equiv_norm}
c_{m,t}\|f\|^2_{H^2_d} \leq \|f\|^2_t \leq c_{k,t}\|f\|^2_{H^2_d} \leq \|f\|^2_{H^2_d}.
\end{equation}
\subsection{What this paper is about}
Arveson conjectured \cite{Arv02} that every quotient of $H^2_d \otimes \mathbb{C}^r$ by a homogneous submodule $M$ is $p$-essentially normal for all $p>d$; later Douglas refined this conjecture \cite{Dou06b} to include the range $p > \dim (M)$.
The Arveson-Douglas conjecture drew several mathematicians to work on essential normality (see, e.g., \cite{Arv05,Arv07,Dou06a,DS11,Esc11,GW08,Ken15,KenSha12,KenSha15,Sha11}), and positive results showing that the conjecture holds for certain classes of ideals were obtained.
With time it has come to be believed that the {\em homogeneity} assumption should not be of central importance.
Moreover, the conjecture is believed to hold for $\mathcal{H}_d^{(t)}$ for all $t \geq -d$.
In fact, it is a folklore result that in the homogeneous setting the validity of the conjecture in one of these spaces is equivalent to its validity in all these spaces.
In two important papers \cite{DW11} and \cite{FX13} it was proved that the closure in some of the spaces $\mathcal{H}_d^{(t)}$ of a principal ideal $I$ is $p$-essentially normal for all $p>d$ (see also \cite{FX15} and \cite{GZ13} which followed).
An even bigger breakthrough occurred with the appearance of the two papers \cite{DTY14} and \cite{EngEsc13}, in which the conjecture was confirmed under natural smoothness and transversality conditions.
In a different effort, the paper \cite{Sha11} suggested an approach that was based on stable division; in that paper it was shown that if a homogeneous submodule $M \triangleleft \mathbb{C}[z] \otimes \mathbb{C}^r \subset H^2_d \otimes \mathbb{C}^r$ has the stable division property, then it fully satisfies the Arveson-Douglas conjecture.
This was used to obtain new proofs of the conjecture (and also to give some explanation of it) for classes of ideals for which a stable generating set was shown to exist (monomial ideals and ideals in two variables).
Our goal in this paper is two-fold.
First, we wish to show that the closure in $\mathcal{H}_d^{(t)}$ of an ideal that has the stable division property is $p$-essentially normal for all $p>d$; in fact, only the approximate stable division property is required.
We only obtain this for $t>-3$. See Theorems \ref{thm:stab_EN} and \ref{thm:stab_EN2} for precise statements.
This improves on the result from \cite{Sha11} in that homogeneity is not required.
The key tools we use are the main results as well as some auxiliary results and techniques from \cite{FX13,FX15}, which say that principal ideals are $p$-essentially normal, together with the main result from \cite{Ken15}, which allows us to promote the result from principal ideals to general ideals.
Our second goal is to show that quasi homogeneous ideals in $\mathbb{C}[x,y]$ have the stable division property with respect to any one of the $\mathcal{H}_d^{(t)}$ norms; see Theorem \ref{thm:stab_div_quasi}.
Theorems \ref{thm:stab_EN} (or \ref{thm:stab_EN2}) and \ref{thm:stab_div_quasi} combine to give a new proof that the closure of any quasi homogeneous ideal in $\mathcal{H}_2^{(t)}$ is $p$-essentially normal for all $p>2$ and all $t \geq -2$.
This result was already obtained using different methods in \cite[Theorem 4.2]{DS11} and \cite[Corollary 1.3]{GZ13}.
In fact, using results from \cite{FX15}, one can easily get more --- we can show $p$-essential normality for every ideal in two variables; this is obtained in Theorem \ref{thm:two_dim}.
However, we think that the application of stable division in a new setting --- even to obtain a known result --- is an important development, and this urges us to continue to look for stable division (or lack of) in other classes of ideals.
\section{Stable division and essential normality in the Hilbert spaces $\mathcal{H}_d^{(t)}$}
\begin{lemma}\label{lem:T}
Let $M_1, \ldots, M_k$ be linear subspaces of a Hilbert space $H$.
Let $M = M_1 + \ldots + M_k$ denote the algebraic sum of these spaces inside $H$, and let $\overline{M_1} \oplus \ldots \oplus \overline{M_k}$ denote the (disjoint) orthogonal sum formed by them.
Consider the map
\[
T : \overline{M_1} \oplus \ldots \oplus \overline{M_k} \longrightarrow H
\]
given by
\[
T(m_1, \ldots, m_k) = m_1 + \ldots + m_k.
\]
Then the following are equivalent:
\begin{enumerate}
\item T has closed range.
\item There exists a constant $C$ such that for every $h \in M$ and every $\epsilon > 0$, there are $m_1 \in M_1, \ldots, m_k \in M_k$ satisfying
\begin{equation}\label{eq:msum}
\|h - (m_1 + \ldots + m_k)\| \leq \epsilon ,
\end{equation}
and
\begin{equation}\label{eq:mnorm}
\sum \|m_i\|^2 \leq C \|h\|^2.
\end{equation}
\end{enumerate}
In case that all the subspaces $M_i$ are closed, then (1) implies (2) with $\epsilon = 0$ as well.
\end{lemma}
\begin{proof}
Let $K$ be the kernel of $T$, and consider the induced map
\[
\tilde{T} : G:= (\overline{M_1} \oplus \ldots \oplus \overline{M_k})/K \longrightarrow H.
\]
If $T$ has closed range, then $\tilde{T}$ is a bounded linear bijection of $G$ onto the Hilbert space $\overline{M}$, hence $\tilde{T}^{-1}$ is bounded.
Say $\|\tilde{T}^{-1}\| \leq c$. Given $h \in M$, let $g = \tilde{T}^{-1}(h)$. Then $\|g\| \leq c \|h\|$, which means that there is an element $(m_1, \ldots, m_k)$ in $\overline{M_1} \oplus \ldots \oplus \overline{M_k}$ satisfying (\ref{eq:msum}) and (\ref{eq:mnorm}) with $\epsilon = 0$ and any $C > c^2$.
In the case where the subspaces $M_i$ are not closed we may replace this tuple by $(m_1, \ldots, m_k)$ in $M_1 \oplus \ldots \oplus M_k$ to obtain (\ref{eq:msum}) and (\ref{eq:mnorm}) with $\epsilon > 0$ arbitrarily small and some constant $C > c^2$.
Conversely, assume that condition (2) above holds, and let $h \in \overline{M}$. Let $\{h^{(n)}\}$ be a sequence in $M$ converging to $h$ with $\|h^{(n)}\| \leq \|h\|$ for all $n$. For all $n$, we find $(m^{(n)}_1, \ldots, m^{(n)}_k) \in M_1 \oplus \ldots \oplus M_k$ satisfying
\[
\|h^{(n)} - (m_1^{(n)} + \ldots + m^{(n)}_k)\| \leq 1/n
\]
and
\[
\sum \|m^{(n)}_i\|^2 \leq C \|h\|^2.
\]
By weak compactness of the unit ball we find $(m_1, \ldots, m_k)$ in $\overline{M_1} \oplus \ldots \oplus \overline{M_k}$ satisfying (\ref{eq:mnorm}) and (\ref{eq:msum}) with $\epsilon = 0$. In particular, the range of $T$ is equal to $\overline{M}$, and is therefore closed.
\end{proof}
\begin{remark}\label{rem:notsquare}
Note that, given $k$, condition \eqref{eq:mnorm} is equivalent to
\begin{eqnarray*}
\sum \|m_i\| \leq C' \|h\|.
\end{eqnarray*}
\end{remark}
The relevance of Lemma \ref{lem:T} to the problem of stable division is that $\{f_1, \ldots, f_k\}$ is an approximate stable generating set for $I$ if and only if the subspaces $M_i = \langle f_i \rangle$ satisfy condition (2) of the lemma.
\begin{theorem}\label{thm:stab_EN}
Let $t> -3$ and let $I$ be an ideal in $\mb{C}[z]$.
Assume that $I$ has the approximate stable division property with respect to the $\mathcal{H}_d^{(t)}$ norm.
Then the submodule $\overline{I}$ of $\mathcal{H}_d^{(t)}$ is $p$-essentially normal for all $p>d$.
\end{theorem}
\begin{proof}
Let $\{f_1, \ldots, f_k\}$ be an approximate stable generating set for $I$, and for all $i=1,\ldots, k$ let $N_i$ denote the closure of the principal ideal $\langle f_i \rangle$.
By Theorem 1.1 in \cite{FX13} and Theorem 1.2 in \cite{FX15}, every one of the submodules $N_i \subseteq \mathcal{H}_d^{(t)}$ is $p$-essentially normal for all $p>d$.
By Lemma \ref{lem:T} above, the algebraic sum $N_1 + \ldots + N_k$ is closed; it follows that it must be equal to the closure of $I$,
\[
\overline{I} = N_1 + \ldots + N_k.
\]
In the terminology of \cite{Ken15}, this means that $\overline{I}$ is $p$-essentially decomposable (meaning simply that $\overline{I}$ is the algebraic sum of closed $p$-essentially normal submodules of $\mathcal{H}^{(t)}_d$).
By Theorem 3.3 of \cite{Ken15}, $\overline{I}$ is also $p$-essentially normal.
To be precise, \cite[Theorem 3.3]{Ken15} is stated for submodules of $H^2_d = \mathcal{H}_d^{(-d)}$, but the proof of this result is a direct application of \cite[Theorem 4.4]{Arv07}, which is stated in the generality of abstract Hilbert modules which are $p$-essentially normal, and therefore applies here.
\end{proof}
The following variant of Theorem \ref{thm:stab_EN} highlights somewhat different techniques.
Note that now the assumption is that there is stable division with respect to the $\mathcal{H}_d^{(t+1)}$ norm, and the proof of the previous theorem does not seem to apply.
\begin{theorem}\label{thm:stab_EN2}
Let $t> -3$ and let $I$ be an ideal in $\mb{C}[z]$.
Assume that $I$ has the stable division property with respect to the $\mathcal{H}_d^{(t+1)}$ norm.
Then the submodule $\overline{I}$ of $\mathcal{H}_d^{(t)}$ is $p$-essentially normal for all $p>d$.
\end{theorem}
\begin{proof}
Denote $N = \overline{I\,}^t$ (meaning the closure of $I$ with respect to the $\mathcal{H}_d^{(t)}$ norm).
We will apply the techniques developed for the proof of Theorem 1.1 in \cite{FX13}.
One of the standard approaches for showing that a closed submodule $N$ of $H \otimes \mb{C}^r$ is $p$-essentially normal is to show that
\[
S^*_i P_N - P_N S^*_i = S^*_i P_N - P_N S^*_i P_N = P_N^\perp S^*_i P_N \in \mathcal{L}^{2p} ,
\]
(where $P_N$ denotes the orthogonal projection of $H$ onto $N$ and $P_N^\perp = I - P_N$).
See, e.g., \cite[Proposition 4.2]{Arv07}.
To show that $P_N^\perp S^*_i P_N$ is in $\mathcal{L}^{2p}$, we will use the following result from \cite[Proposition 4.2]{FX13}, which we reformulate slightly for our needs.
Recall that $\mathcal{H}_d^{(t)}$ is always contained in $\mathcal{H}_d^{(t+1)}$.
\begin{proposition}[Fang-Xia]\label{prop:FX}
Let $I$ and $N$ be as above, let $T$ be a linear operator on $\mathcal{H}_d^{(t)}$ and suppose that there is some $K$ such that
\[
\|Tg\|_t \leq K\|g\|_{t+1}
\]
for all $g \in I$. Then $T P_N \in \mathcal{L}^{2p}$ for all $p>d$.
\end{proposition}
By Fang and Xia's proposition and the above remarks, it suffices to show that there is a constant $K$ such that
\[
\|P_N^\perp S^*_j h\|_t \leq K \|h\|_{t+1} \,\, , \,\, h \in I.
\]
Let $f_1, \ldots, f_k$ be a stable generating set for $I$ in the $\mathcal{H}_d^{(t+1)}$ norm.
Denote by $Q^{(t)}_i$ the orthogonal projection onto the complement in $\mathcal{H}_d^{(t)}$ of the ideal generated by $f_i$.
In the proof of \cite[Theorem 1.1]{FX13} (see p. 3007 there) it is shown that there is an integer $\ell_i$ and a constant $C_i$ such that
\begin{equation}\label{eq:ineq_QSqf}
\|Q_i^{(t)}S^*_j g f_i \|_{t} \leq C_i \|g f_i \|_{t+1}
\end{equation}
for all $g$ satisfying $\partial^\alpha g(0) = 0 $ when $|\alpha|\leq \ell_i$
(for the range $t \in (-3,-2]$ we require the proof of \cite[Theorem 1.2]{FX15}, see Proposition 4.4 --- in particular Equation (4.10) --- and Proposition 5.10 there).
Our goal now is to show that the inequality \eqref{eq:ineq_QSqf} holds for all $g\in\mathbb{C}[z]$, with a perhaps larger constant $C_i$.
This follows from the fact that ${\overline E_i}^{t+1}$ has finite codimension in ${\overline {\langle f_i \rangle}}^{t+1}$, together with Lemma \ref{lem:gen} in the appendix below, but we wish to give the argument in full detail.
First, we note that the ideal
\[
E_i = \{g f_i : \partial^\alpha g(0) = 0 \textrm{ for all } |\alpha|\leq \ell_i \}
\]
is finite codimensional in $\langle f_i \rangle$.
Let $R_i = \mbox{span}\{z^\alpha f_i : |\alpha| \leq \ell_i\}$. Since the equivalence classes $[z^\alpha f_i],\,|\alpha| \leq \ell_i$, form a basis for $\langle f_i \rangle/ E_i$, we have $\langle f_i \rangle = E_i \dot{+} R_i$, direct sum as linear subspaces.
Next, we claim that ${\overline E_i}^{t+1} \cap R_i = \{0\}$, where ${\overline E_i}^{t+1}$ is the closure of $E_i$ in the $\mathcal H_d^{(t+1)}$ norm.
To see this, fix $g f_i\in R_i$ for some polynomial $g$ with $\deg (g) \leq l_i$.
Let $z^{\beta_i}$ be the monomial of least degree, with respect to graded lexicographic ordering, that appears with a non-zero coefficient in the expression of $f_i$.
Likewise, let $z^\alpha$ be a monomial of least degree that appears with a non-zero coefficient in the expression of $g$.
The coefficient of $z^{\alpha + \beta_i}$ in the expression of $g f_i$ is then nonzero.
Since every monomial that appears in any polynomial in $E_i$ is strictly greater than $z^{\alpha + \beta_i}$ in the aforementioned ordering, and since monomials are orthogonal in $\mathcal{H}_d^{(t+1)}$, we have that $E_i \subseteq \{\mathbb C z^{\alpha + \beta_i}\}^{\perp}$ (where $\{\mathbb C z^{\alpha + \beta_i}\}^{\perp}$ denotes the orthogonal complement in $\mathcal{H}_d^{(t+1)}$ of the one dimensional subspace spanned by $z^{\alpha + \beta_i}$).
Thus
$$
{\overline E_i}^{t+1}\subseteq \{\mathbb C z^{\alpha + \beta_i}\}^{\perp}.
$$
But as $\langle g f_i, z^{\alpha + \beta_i} \rangle \neq 0$, our claim that ${\overline E_i}^{t+1} \cap R_i = \{0\}$ follows.
Let $M = E_i$, $N = R_i$ considered as subspaces of $\mathcal{H}_d^{(t+1)}$, let $K = \mathcal{H}_d^{(t)}$, and let $T$ be the restriction of $Q^{(t)}S_j^*$ to $\mathbb{C}[z]$.
Lemma \ref{lem:gen} then tells us that \eqref{eq:ineq_QSqf} implies that $Q^{(t)}S_j^*$ is bounded from $\langle f_i \rangle = E_i + R_i\subset \mathcal{H}_d^{(t+1)}$ to $\mathcal{H}_d^{(t)}$, meaning that \eqref{eq:ineq_QSqf} holds for all $g \in \mathbb{C}[z]$, perhaps with a bigger constant $C_i$.
Now let $C = \max{C_i}$.
Since $P_N^\perp \leq Q^{(t)}_i$, then for every $g \in \mathbb{C}[z]$, every $f_i$ in the stable generating set, and all $j$,
\begin{equation*}
\|P_N^\perp S^*_j g f_i \|_{t} \leq C \|g f_i \|_{t+1} .
\end{equation*}
For $h \in I$,
let $g_1, \ldots, g_k \in \mathbb{C}[z]$ satisfy \eqref{eq:sum_app} and \eqref{eq:stab_app} with respect to the $\mathcal{H}_d^{(t+1)}$ norm and with $\epsilon = 0$.
Then we have
\begin{align*}
\|P_N^\perp S^{*}_j h \|_{t} &\leq \sum_{i=1}^k \|P_N^\perp S^{*}_j g_i f_i \|_{t}\\ &\leq \sum_{i=1}^k C \|g_i f_i \|_{t+1}\\ &\leq AC \|h\|_{t+1} .
\end{align*}
By Proposition \ref{prop:FX}, $P_N^\perp S^*_j P_N$ is in $\mathcal{L}^{2p}$ for all $p>d$.
By the remarks at the beginning of the proof, we are done.
\end{proof}
\begin{remark}
Here is another proof of the claim ${\overline E_i}^{t+1} \cap R_i = \{0\}$ that works in any analytic Hilbert module in the sense of \cite{CG} where monomials are not necessarily orthogonal.
Following \cite[Theorem 2.1.7(2)]{CG}, one could write
\begin{equation}\label{eq:charac}
E_i = \{q\in \langle f_i \rangle : p(D)q|_{0} = 0\, \text{ for all } p\in E_{i0}\} ,
\end{equation}
where $E_{i0}$ is the characteristic space of $E_i$ at $0$ and $p(D)$ denotes the differential operator $\sum_{\alpha}a_{\alpha}\frac{\partial^{\alpha}}{{\partial z}^\alpha}$ for any polynomial $p = \sum_{\alpha}a_{\alpha}z^\alpha$. This is because the characteristic spaces of $E_i$ and $\langle f_i \rangle$ are same at any $\lambda\neq 0$ and different at $\lambda = 0$. The first assertion follows from the fact that $E_i = \mathfrak M_0^{l_i+1}\langle f_i \rangle$, $\mathfrak M_0$ is the maximal ideal of $\mathbb C[z]$ at $0$, and by repeated use of \cite[Proposition 1.3]{DG} while the last follows as at $\lambda = 0$, one observes that
$$
p(D)\{z^\alpha f_i\}|_0 = \frac{\partial^{\alpha}p}{{\partial z}^\alpha}(D)f_i|_0 ,
$$
and hence
$$
E_{i0} = \{q\in\mathbb C[z] : \frac{\partial^{\alpha}q}{{\partial z}^\alpha}\in\langle f_i \rangle_{0} \mbox{~for~all~}\alpha \mbox{~such~that~}|\alpha| = l_i+1\} ,
$$
(where $\langle f_i \rangle_{0}$ is the characteristic space of $\langle f_i \rangle$ at $0$).
Having establsihed \eqref{eq:charac}, we obtain that
$$
{\overline E_i}^{t+1} \subset \{f\in \mathcal{H}_d^{(t+1)} : p(D)f|_{\lambda} = 0,\, \mbox{~for~all~}p\in E_{i0}\},
$$
and this combined with \eqref{eq:charac} ensures that ${\overline E_i}^{t+1} \cap R_i = \{0\}$.
\end{remark}
\section{Stable division for quasi homogeneous ideals in $\mb{C}[x,y]$}
Theorems \ref{thm:stab_EN} and \ref{thm:stab_EN2} motivate us to find new examples of modules which have the (approximate) stable division property.
We will show that quasi homogeneous ideals in $\mb{C}[x,y]$ have the stable division property (it is convenient to use the notation $\mb{C}[x,y]$ for the case $d=2$).
Our discussion is an improvement to \cite[Section 2.2]{Sha11}, where it was proved that {\em homogeneous} ideals in $\mb{C}[x,y]$ have the stable division property.
Before proceeding, we record the following proposition, which shows that when the generating set consists of quasi homogeneous polynomials, the stable division and the approximate stable division properties are the same.
Since this result is not needed we omit the simple proof.
\begin{proposition}
Let $M$ be an ${\bf n}$-quasi homogeneous module in $\mb{C}[z] \otimes \mb{C}^r$.
Then an approximate stable generating set consisting of ${\bf n}$-quasi homogeneous polynomials is a stable generating set.
\end{proposition}
\begin{lemma}\label{lem:stab_div_diff_norms}
Let $M$ be a quasi homogeneous submodule in $\mb{C}[z] \otimes \mb{C}^r$ and let $t>-d$.
Then a generating set for $M$ consisting of quasi homogeneous polynomials is a stable generating set with respect to the $H^2_d$ norm if and only if it is a stable generating set with respect to the $\mathcal{H}_d^{(t)}$ norm.
\end{lemma}
\begin{proof}
Suppose that $M$ be an ${\bf n}$-quasi homogeneous submodule, where ${\bf n} = (n_1, \ldots, n_d)$.
Denote $n = \max_{1 \leq i \leq d} n_i$.
Let $f_1, \ldots, f_k$ be a generating set for $M$ such that every $f_i$ is quasi homogeneous.
Let $h$ be a quasi homogeneous element of $M$, $deg_{\bf n}(h) = m$.
If $z^\alpha$ is some monomial appearing in $h$, then $|\alpha| = \alpha \cdot {\bf 1} \leq \alpha \cdot {\bf n} = deg_{\bf n}(z^\alpha) = m$;
while on the other hand $m = \alpha \cdot {\bf n} \leq n |\alpha|$.
Thus $h = \sum_{i=\floor{m/n}}^m h_i$ where every $h_i$ is homogeneous, so by (\ref{eq:equiv_norm})
\[
c_{m,t}\|h\|^2_{H^2_d} \leq \|h\|^2_t \leq c_{\floor{m/n},t}\|h\|^2_{H^2_d} .
\]
Now suppose that $h = \sum_{j=1}^k a_j f_j$ and that $deg_{\bf n}(a_j f_j) = m$ for all $j$. Assume that $\sum_j \|a_j f_j\|^2_{H^2_d} \leq C \|h\|^2_{H^2_d}$. Then
\begin{align*}
\sum_j \|a_j f_j\|_t^2 &\leq \sum_j c_{\floor{m/n},t}\|a_j f_j\|^2_{H^2_d}\\ &\leq C c_{\floor{m/n},t}\|h\|^2_{H^2_d} \\ &\leq C \frac{c_{\floor{m/n},t}}{c_{m,t}} \|h\|^2_t .
\end{align*}
Since
\begin{equation}\label{eq:NEEDS_CHECKING
\lim_{m \rightarrow \infty} \frac{c_{\floor{m/n},t}}{c_{m,t}} \in [1, \infty),
\end{equation}
we see that stable division with respect to the $H^2_d$ norm implies stable division with respect to the $\mathcal{H}_d^{(t)}$ norm.
The converse is proved in the same way.
\end{proof}
We now recall some facts from computational algebraic geometry.
A standard reference for this is \cite{CLS92}. Let ${\bf n} \in \mb{N}^2$ be a weight vector.
We fix an ordering on the monomials in $\mb{C}[x,y]$ that is determined by ${\bf n}$: if $a$ and $b$ are non-zero scalars, then we say that $b x^m y^n$ is {\em greater than} $a x^k y^l$, denoted $a x^k y^l < b x^m y^n$, if and only if
\[
(k n_1+ l n_2 < m n_1 + n n_2) \textrm{ or } (k n_1+ l n_2 = m n_1 + n n_2 \textrm{ and } k < m).
\]
The {\em leading term} of a polynomial $f$, denoted $LT(f)$, is the largest monomial appearing in $f$. We say that $b x^m y^n$ is {\em divisible} by $a x^k y^l$ (where $a,b \in \mathbb{C} \setminus \{0\}$) if and only if $k \leq m$ and $l \leq n$, and then we have $(b x^m y^n) / (a x^k y^l) := (b/a) x^{m-k}y^{n-l}$.
Let us remind the reader of the {\em division algorithm} for polynomials in several variables (we need it only in two variables).
Given $h, f_1, \ldots, f_k \in \mb{C}[x,y]$ we may divide $h$ by $f_1, \ldots, f_k$ with remainder, meaning that we find $a_1, \ldots, a_k, r \in \mb{C}[x,y]$ (with $LT(r)$ not divisible by any $LT(f_i)$) such that
\begin{equation}\label{eq:div_w_r}
h = \sum a_i f_i + r .
\end{equation}
The decomposition \eqref{eq:div_w_r} is obtained using the following algorithm.
We start with
\[
r = a_1 = \ldots = a_k = 0 ,
\]
and define a temporary polynomial $p$ which is set at the start to $p=h$.
We now start iterating over the terms of $p$ in decreasing order.
If $LT(p)$ is divisible by $LT(f_i)$ for some $i$, we replace $a_i$ by $a_i + LT(p)/LT(f_i)$ and replace $p$ with $p - \left(LT(p)/LT(f_i)\right) f_i$.
In principal one is free to choose which $i$ to use, but we will always choose the {maximal $i$ possible}.
If $LT(p)$ is not divisible by any of the $LT(f_i)$s, we put $r = r + LT(p)$ and replace $p$ with $p - LT(p)$.
We continue this way until $p=0$.
Even if $h$ is in the ideal generated by $f_1, \ldots, f_k$, the above division algorithm might terminate with a non-trivial remainder.
However, for every ideal $I$ there exists a {\em Groebner basis} $\{f_1, \ldots, f_k\}$, which is a generating set for $I$ with the property that the above algorithm, when run with $h \in I$ and $f_1, \ldots, f_k$ as input, terminates with $r = 0$ \cite[Corollary 2, p. 82]{CLS92}.
\begin{lemma}\label{lem:stab_div_r}
Let ${\bf n} = (n_1, n_2)$ be a weight, and let $f_1, \ldots, f_k$ be ${\bf n}$-quasi homogeneous polynomials in $\mb{C}[x,y]$ with $deg_{\bf n}(f_i) = m$ for $i=1, \ldots, k$.
There exists a constant $C$ such that for every ${\bf n}$-quasi homogeneous polynomial $h \in \mb{C}[x,y]$ there are $a_1, \ldots, a_k, r \in \mb{C}[x,y]$ such that
\[
h = \sum_{i=1}^k a_i f_i + r
\]
with
\begin{equation}\label{eq:stab_div_r}
\sum_{i=1}^k \|a_i f_i\|^2_{H^2_d} \leq C\left( \|h\|^2_{H^2_d} + \|r\|^2_{H^2_d} \right) .
\end{equation}
In fact, the $a_i$s and $r$ can be found by running the division algorithm.
\end{lemma}
\begin{remark}
Note that we will use the same weight ${\bf n}$ determining quasi homogeneity as the weight determining the monomial ordering.
In fact, we choose the order on monomials according to type of quasi homogeneous polynomials we wish to deal with.
In the case that we are dealing with homogeneous ideals then the monomial order is the graded lexicographic order, as in \cite[Lemma 2.3]{Sha11} --- on which the following proof is modeled.
\end{remark}
\begin{proof}
In the proof we assume without loss of generality that $n_1 \geq n_2$.
We will show, by induction on $k$, that there is a constant $C$ such that when running the division algorithm in an appropriate manner to divide $h$ by $f_1, \ldots, f_k$, one has the estimate (\ref{eq:stab_div_r}).
If $k=1$, then this is obvious from the triangle inequality, so assume $k>1$.
Suppose that $LT(f_1) > LT(f_2) > \ldots > LT(f_k)$. Let
\[
f_i = \sum_{j}a_{ij}x^{\frac{m-n_2j}{n_1}}y^j
\]
where the sum is on all $j$ starting from $j_i$ until the last such $j$ for which there is some non-negative integer $l$ satisfying $m = l n_1 + j n_2$.
Because of our assumptions we have $j_1 < j_2 < \ldots < j_k$.
By compactness considerations, we may assume that $deg_{\bf n}(h) = n > 4m$.
Now initiate the algorithm as in the description above by setting $p = h$.
At a certain iteration of the division algorithm we have
\[
LT(p) = b_t x^{\frac{n - tn_2}{n_1}}y^t.
\]
When $t<j_1$, then $LT(p)$ is not divisible by any $LT(f_i)$, so we move $LT(p)$ to the remainder.
This happens at most $j_1-1$ iterations.
We will use $f_1$ to divide $p$ at most $j_2 - j_1$ iterations, only when $j_1 \leq t < j_2$.
After $t$ becomes greater than $j_2$ we will never have to use $f_1$ again, and we can use the inductive hypothesis.
All there is to check is that there is some constant $C$, independent of $h$, such that $\|p\|^2 \leq C \|h\|^2$ and $\|a_1 f_1 \|^2 \leq C\|h\|^2$ after the last iteration when the algorithm used $f_1$.
When $t<j_1$, $\|p\|$ only decreases and $a_i$ does not change.
Then there are at most $j_2 - j_1$ iterations in which $a_1 f_1$ is modified to $(a_1 + LT(p)/LT(f_1))f_1$ and $p$ is modified to $p - LT(p)/LT(f_1) f_1$.
Thus we will be done once we show that there is some constant $C$ such that
\[
\|LT(p)/LT(f_1) f_1\|^2 \leq C\|p\|^2.
\]
We compute
\[
\frac{LT(p)}{LT(f_1)} f_1 = \frac{b_t}{a_{1j_1}} \sum a_{1j}x^{\frac{n-n_2(t+j-j_1)}{n_1}}y^{t+j-j_1}.
\]
Therefore
\[
\left\|\frac{LT(p)}{LT(f_1)} f_1\right\|^2 = \left|\frac{b_t}{a_{1j_1}}\right|^2 \sum |a_{1j}|^2 \frac{(\frac{n-n_2(t+j-j_1)}{n_1})!(t+j-j_1)!}{(\frac{n+ (n_1 - n_2)(t+j-j_1)}{n_1})!} .
\]
Now consider the integer $s = t+j-j_1$ occurring in the above expression.
Because $j$ runs from $j_1$ to the highest power of $y$ appearing in $f_1$, we have that $t \leq s \leq 2m$. Thus
\begin{align*}
\frac{(\frac{n-n_2s}{n_1})!s!}{(\frac{n+ (n_1 - n_2)s}{n_1})!} & \leq \frac{(\frac{n-n_2s}{n_1})!(2m)!}{(\frac{n+ (n_1 - n_2)t}{n_1})!}\\ &\leq \frac{(\frac{n-n_2t}{n_1})!(2m)!}{(\frac{n+ (n_1 - n_2)t}{n_1})!} \\
& \leq (2m)! \frac{(\frac{n-n_2t}{n_1})!t!}{(\frac{n+ (n_1 - n_2)t}{n_1})!}\\ & = \frac{(2m)!}{|b_t|^2} \|LT(p)\|^2.
\end{align*}
Thus we have the bound
\[
\|LT(p)/LT(f_1) f_1\|^2 \leq \frac{(2m)!}{|a_{1j_1}|^2} \sum |a_{1j}|^2 \|p\|^2,
\]
and since $C = \frac{(2m)!}{|a_{1j_1}|^2} \sum |a_{1j}|^2$ is independent of $h$, this completes the proof.
\end{proof}
\begin{theorem}\label{thm:stab_div_quasi}
Let $I$ be an ${\bf n}$-quasi homogeneous ideal in $\mb{C}[x,y]$.
For all $t \geq -2$, $I$ has the stable division property in the $\mathcal{H}_2^{(t)}$ norm.
If $\{f_1, \ldots, f_k\}$ is a Groebner basis for $I$ consisting of ${\bf n}$-quasi homogeneous polynomials of the same degree, then it is a stable generating set.
\end{theorem}
\begin{proof}
One can always construct a Groebner basis for $I$ consisting of ${\bf n}$-quasi homogeneous polynomials (Exercise 3 on p. 377, \cite{CLS92}).
By multiplying basis elements by monomials one then obtains a Groebner basis consisting of ${\bf n}$-quasi homogeneous polynomials of the same degree $m$, for the finite co-dimensional subideal $I_m \oplus I_{m+1} \oplus \ldots$ where $m$ is sufficiently large.
Thus, we concentrate on proving the second assertion.
Let $\{f_1, \ldots, f_k\}$ be a Groebner basis for $I$ consisting of ${\bf n}$-quasi homogeneous polynomials of the same degree $m$.
By Lemma \ref{lem:stab_div_diff_norms}, it suffices prove the assertion for $t=-2$, that is, to show that $\{f_1, \ldots, f_k\}$ is a stable generating set for $I$ in the $H^2_2$ norm.
Let $h \in I$.
We may assume $h$ is quasi homogeneous, otherwise decompose it into an orthogonal sum of quasi homogeneous polynomials.
We divide $h$ by $f_1, \ldots, f_k$ and obtain $h = \sum a_i f_ i + r$ with (\ref{eq:stab_div_r}).
But since $\{f_1, \ldots, f_k\}$ is a Groebner basis, $r = 0$ (Corollary 2 on p. 82, \cite{CLS92}).
Thus $\{f_1, \ldots, f_k\}$ is a stable generating set, and $I$ has the stable division property.
\end{proof}
Putting together the previous theorem with either Theorem \ref{thm:stab_EN} or Theorem \ref{thm:stab_EN2} we obtain the following corollary.
\begin{corollary}\label{cor:ess_norm_quasi}
Let $t\geq -2$ and let $I$ be a quasi homogeneous ideal in $\mb{C}[x,y]$.
Then the submodule $\overline{I}$ of $\mathcal{H}_2^{(t)}$ is $p$-essentially normal for all $p>2$.
\end{corollary}
The above corollary was proved by Douglas and Sarkar by completely different methods, see \cite[Theorem 4.2]{DS11} (another proof appeared in \cite[Corollary 1.3]{GZ13}).
In fact, using Fang and Xia's result we may prove a stronger result, which applies to not necessarily quasi homogeneous ideals.
This has been observed in \cite[Corollary 3.1]{GZ13} for the case $t>-2$.
\begin{theorem}\label{thm:two_dim}
Let $I$ be an ideal in $\mb{C}[x,y]$.
Then for all $t \geq -2$ and all $p>2$, the submodule $\overline{I}$ of $\mathcal{H}_2^{(t)}$ is $p$-essentially normal.
\end{theorem}
\begin{proof}
Let $p = \operatorname{gcd}(I)$.
Then $I= p J$, where $J$ is a finite codimensional ideal in $\mb{C}[x,y]$ \cite[Lemma 2.2.9, p.28]{CG}.
Thus, $I$ is finite codimensional in $\langle p \rangle$, so $\overline{I}$ is finite codimensional in $\overline{\langle p \rangle }$.
By \cite[Theorem 1.1]{FX13} and \cite[Corollary 1.3]{FX15}, $\overline{\langle p \rangle}$ is $p$-essentially normal for all $p>2$, and from finite codimensionality it follows that $\overline{I}$ is, too.
\end{proof}
\begin{remark}
The factorization of $I$ as $pJ$ above is known as the {\em Beurling form} of $I$.
The idea of using the Beurling form to reduce the essential normality of an ideal to that of a principle ideal has appeared several places in the literature (for example in \cite{GW08,GZ13}) but we could not trace its precise origin.
Admittedly, the above proof is much shorter than the proof we provide for Corollary \ref{cor:ess_norm_quasi}.
On the other hand, it does require implicitly some results in algebraic geometry that are less intuitive than what is used in the stable division proof.
\end{remark}
Since the conjectures of Arveson and Douglas treat also submodules with multiplicity, it is natural to ask whether Lemma \ref{lem:stab_div_r} extends to vector valued modules.
The following example shows that the lemma fails in the vector valued case.
However, we will also show that the following example is {\em not} a counter example to the vector valued version of the first assertion of Theorem \ref{thm:stab_div_quasi}.
\begin{example}
Consider the module generated by $f_1 = (x, 0, y)$, $f_2 = (0,x,y)$, and let $h = (xy^n, -xy^n, 0)$.
Then
\[
h = y^nf_1 - y^n f_2 ,
\]
and there is no other way to write $h$ as an element in the module spanned by $f_1$ and $f_2$.
Therefore the output of the division algorithm is $a_1 = -a_2 = y^n$.
However $\|h\|^2 \sim 1/n$, while $\|a_if_i\|^2 \sim 1$.
Thus, the conclusion of Lemma \ref{lem:stab_div_r} fails.
On the other hand, we now show that the basis $\{f_1 - f_2, f_2\}$
is a stable generating set for the module spanned by $f_1$ and $f_2$.
To see this, consider $h = pf_1 + q f_2 = (px, qx, (p+q)y)$, where $p,q \in \mathbb{C}[x,y]$.
Then
\[
\|h\|^2 = \|px\|^2 + \|qx\|^2 + \|(p+q)y\|^2 .
\]
Write $h = p(f_1 - f_2) + (p+q)f_2$.
Then we compute the norm of each term:
\[
\|p(f_1 - f_2)\|^2 = \|(px, -px, 0)\|^2 = 2\|px\|^2 \leq 2 \|h\|^2 ,
\]
and
\begin{align*}
\|(p+q)f_2\| &= \|(0,(p+q)x,(p+q)y)\| \\
&\leq \|px\| + \|qx\| + \|(p+q)y\| \leq \sqrt{3} \|h\|.
\end{align*}
Thus $\{f_1 - f_2, f_2\}$ is a stable generating set.
\end{example}
We stress that it is still an open problem whether there exists an ideal that does not have the stable division property.
Examples as above show why it is so hard to settle the problem of whether or not every (homogeneous) ideal has the stable division property.
On the one hand, it is not too hard to cook up an ideal with a generating set which is not a stable generating set.
On the other hand, in all examples that we know, after making a few changes to the generating set it becomes a stable generating set.
\section{Appendix}
In the proof of Theorem \ref{thm:stab_EN2} we required the following elementary result, the proof of which we include for completeness.
\begin{lemma}\label{lem:gen} Let $M$ and $N$ be two linear subspaces in a Hilbert space $H$ such that $\overline{M}\cap N =\{0\}$.
Let $T$ be a linear operator defined on $M + N$ mapping to another Hilbert space, say $K$, and assume that
(i) $T$ is bounded on $M$,
(ii) $N$ is finite dimensional.
Then $T$ is bounded on $M+N$.
\end{lemma}
\begin{proof}
By induction, it suffices to prove for the case that $N$ is one dimensional, say $N = \operatorname{span} \{v\}$ where $v$ is a unit vector. We will show in fact that $T$ extends to a bounded operator from $\overline{M} + N$ into $K$.
Now, $T$ extends to a bounded operator on $\overline{M}$.
Moreover, $T$ is bounded on $N$ because $N$ is finite dimensional.
Denote $C = \max\{\|T\big|_N\|, \|T\big|_M\|\}$. Since $v \notin \overline{M}$, $v$ has a positive angle from $\overline{M}$ in the sense that $c:=\sup \{|\langle v, x \rangle | : x \in M, \|x\|=1\} < 1$.
It follows that for $m \in M$ and $n \in N$,
\[
\|m + n\|^2 \geq \|m\|^2 + \|n\|^2 - 2 c \|m\| \|n\| \geq (1-c)(\|m\|^2+ \|n\|^2) .
\]
Thus, for $m \in M$ and $n \in N$, we have
\begin{eqnarray*}\|T (m + n) \| &\leq & \|T m\| + \|T n\| \leq C (\|m\| + \|n\|)\\ &\leq & C \sqrt{2} (\|m\|^2 + \|n\|^2)^{1/2}\\ & \leq & C \sqrt{2} (1-c)^{-1/2} \|m + n\|.\end{eqnarray*}
\end{proof}
\subsection*{Acknowledgement} The authors are grateful to an anonymous referee for spotting a couple of mistakes in a previous version of this paper, and for several other helpful remarks that improved our presentation.
The authors also wish to thank Guy Salomon for providing useful feedback.
\bibliographystyle{amsplain}
|
{
"timestamp": "2016-01-26T02:08:16",
"yymm": "1504",
"arxiv_id": "1504.03465",
"language": "en",
"url": "https://arxiv.org/abs/1504.03465"
}
|
\section{Introduction}
Majorana bound states (MBSs) which reside in topological superconducting systems are drawing much attention both theoretically and experimentally\cite{kanermp,aliceareview,beenakkerreview,kitaev,law09,green,
kanefu,chliu,sarma1d,oppen1d,kouwenhoven,pergescience,flensberg,wang,beenakker13,lawnc,mengcheng}. The so-called topological superconductivity in this context is the spinless p-wave superconductivity, in which a Majorana number can be defined by the Pffafian of the Bogoliubov-de Gennes Hamiltonian\cite{kitaev}. MBSs are zero energy quasiparticles in these topological superconductors. They are localized at the ends of one dimensional systems\cite{kitaev}, or the core area of the superconducting vortices in two dimensional systems\cite{green}.
Electrons in natural systems always have spin, therefore, MBSs are predicted in artificial structures such as the interface of a three dimensional topological insulator and a superconductor\cite{kanefu,chliu}, the spin-orbit coupling nanowire in proximity to a superconductor\cite{sarma1d,oppen1d,kouwenhoven}, and the ferromagnetic atom chain on top of a superconductor\cite{pergescience}.
These systems have a common character that the spins of the electrons near the fermi surface are effectively eliminated by the spin-orbit coupling and the Zeeman energy. Then, the effective spinless superconductivity, {\it i.e.} the topological superconductivity, is achieved through the proximity effect.
MBSs in topological superconductors are interesting because of their non-Abelian exchange statistics\cite{ivanov,aliceanphy}. They provide a realistic platform for investigating a simplest example of non-Abelian particles.
Besides, MBSs are proposed to be useful in quantum computation\cite{kanermp}. They can construct Majorana qubits, which resist to local perturbations and can store quantum information for a long time.
Furthermore, the Majorana qubits can be rotated by the braiding of MBSs\cite{ivanov}. Importantly, these rotation operations are topologically protected. The effect of the operation is determined by the topology of the braiding, not the
detailed path. This topological quantum processing provides a possible scheme for the topological quantum computation. However, the braiding operations are not sufficient to realize universal quantum gates\cite{aliceareview}. They must be supplemented by non-topological operations on Majorana qubits, which have been proposed by using quantum dots\cite{flensberg,wang}, superconducting qubits\cite{jiang,bonderson}, or microwave cavities\cite{pekker}.
\begin{figure}[b]
\includegraphics[clip=true,width=1\columnwidth]{fig1.eps}
\caption{(Color online). (a) Schematic setup of the Majorana qubit, which consists of an rf SQUID with a tunneling barrier junction on the topological nanowire. Four Majorana bound states are localized at the ends of the wire and the tunneling barrier. (b) The energy spectrum of the Majorana qubit as a function of magnetic flux $\Phi$, where the eigenstates are up-spin (solid line) and down-spin (dashed line) in the psudo-spin representation.}
\end{figure}
One promising approach for operating qubits is the Landau-Zener-St\"{u}ckelberg (LZS) interference\cite{nori}. The LZS interference
is a standard quantum phenomena in quantum mechanics. It occurs in quantum two-level systems in which the two levels exhibit avoided level crossings. At the two sides of the crossing point, the physical properties of the eigenstates are exchanged. Therefore, non-adiabatic Landau-Zener transitions between the two levels may happen when the system transverses the avoided crossing under a time varied control parameter\cite{landau,zener}. The interference between these transitions is called LZS interference\cite{stueckelberg}.
The LZS interference is very useful for coherent quantum operation, since it is robust to certain noises and possible to implement high fidelity quantum gates\cite{nori,GGC}. In Ref. [27], the LZS interference has been experimentally
achieved for realistic quantum control on quantum dot qubits.
In this work, we propose to use the LZS interference to implement quantum operations on Majorana qubits. For this purpose, we adopt the system sketched in Fig. 1, which consists of an rf SQUID with a topological nanowire Josephson junction. The nanowire hosts MBSs near the tunneling barrier and the ends of the wire.
These MBSs form a Majorana qubit, which is correlated with the superconducting phase difference across the junction. Because of this correlation, the Majorana qubit is described by the Josephson energy of the junction, which is a $2\times2$ Hamiltonian with avoided level crossings.
If a magnetic flux pulse is applied, the Josephson Hamiltonian will evolve and
transverse through these crossings. We consider a triangular pulse, for which the maximum flux value is tuned around the superconducting flux quanta $\Phi_0=2e/\hbar$. Within one pulse, the Josephson Hamiltonian transverses through the same crossing point twice, and then come back to the initial Hamiltonian. In contrast, the Majorana qubit state does not come back to the initial state after the pulse.
Landau-Zener transitions happen at the avoided crossings, and the resulted LZS interference rotates the Majorana qubit, where the rotating angle is controlled by changing the time length of the pulse. Our work provides a one-qubit gate on the Majorana qubit, which is important for realizing topological quantum computation.
We organize this work as follows. The model for the system is presented in section II, then we study the LZS interference analytically under this model in section III. We present a numerical simulation in section IV. Finally we give discussions and a conclusion in section V.
\section{Model}
The system illustrated in Fig. 1a is constructed by a nanowire Josephson junction which hosts four MBSs. Two of them ($\gamma_1$, $\gamma_4$) are localized at the two ends of the wire, and the other twos ($\gamma_2$, $\gamma_3$) at the two sides of the tunneling barrier. The Majorana qubit built by the two MBSs near the tunneling barrier is directly described by the Josephson energy of the junction,
which is different from conventional junctions in two aspects. First, it is actually a matrix for a Majorana junction\cite{kitaev}, rather than a pure number for conventional junction. The basis states of the matrix are the Majorana qubit states. In this sense, it is more accurate to be named as Josephson Hamiltonian. Second, the Josephson Hamiltonian has the $4\pi$ period components in its diagonal elements\cite{kitaev,aliceareview}, different from the $2\pi$ period Josephson energy in conventional junctions. With these insights, the Josephson Hamiltonian for this Majorana junction has been given as\cite{aliceareview},
\begin{equation}
H = iE_J\gamma_2\gamma_3\cos({\pi \Phi }/{\Phi_0}) + i\delta_L\gamma_1\gamma_2 + i\delta_R\gamma_3\gamma_4,
\end{equation}
where $\gamma_{1,2,3,4}$ are the four MBSs, $E_J$ is the Josephson energy due to the coupling of the two MBSs around the tunneling barrier; $\delta_{L,R}$ represent the coupling between the distant MBSs at the left and the right side of the barrier, respectively; $\Phi$ is the applied magnetic flux through the SQUID, which controls the phase difference across the junction in the vanishing inductance regime under current consideration.
We define two fermionic operators $f^\dagger_1 = (\gamma_2 + i\gamma_3)/2$ and $f^\dagger_2 = (\gamma_4 + i\gamma_1)/2$ to construct a fermionic representation for the MBSs. Then the Hamiltonian is transformed to,
\begin{equation}
\begin{aligned}
H=& E_J(1 - 2f^\dagger_1f_1)\cos{({\pi \Phi}/{\Phi_0})} + \delta_+(f^\dagger_2f_1 + f^\dagger_1f_2)\\
&+ \delta_-(f^\dagger_2f^\dagger_1 + f_1f_2).
\end{aligned}
\end{equation}
where $\delta_{\pm} = \delta_L \pm \delta_R$.
We find that the Josepshon Hamiltonian is depending on the occupation states of the two fermionic operators, which is nothing but the parity states built by the four MBSs\cite{pekker}. We are studying the quantum coherent evolution of the Majorana states, which occurs at a low temperature where the superconducting energy gap becomes large enough. Thus we can ignore the quasiparticle poisoning from high energy quasiparticles which strongly suppressed by the superconducting gap at low temperature.
In this case,
the total fermionic parity is conserves in this system. The Hilbert space of the MBSs can be divided into two disconnected subspace with even and odd total fermionic parities. Without losing generality, we choose the even total parity subspace. In this subspace, we can define psudo-spin states $\mid \uparrow \rangle = |0\rangle$ and $\mid \downarrow \rangle = f^\dagger_1 f^\dagger_2|0 \rangle$, with $\mid 0\rangle$ the vacuum states for the two fermionic operators.
These two psudo-spin states represent the two eigenstates of the Majorana qubit. Therefore, a spin representation for the Majorana qubit is established. In this representation, the Hamiltonian of the Majorana qubit is rewritten as,
\begin{equation}\label{spin}
H = E_J\cos\left({\pi\Phi}/{\Phi_0}\right)\sigma_z + \delta \sigma_x,
\end{equation}
where $\sigma_{x,z}$ are Pauli matrices acting on the pusdo-spin states, and $\delta=\delta_-$. The two eigenvalues of $H$ are $E_\pm = \pm \sqrt{E^2_J \cos^2(\frac{\pi\Phi}{\Phi_0}) + \delta^2}$ , which are depicted in Fig. 1b as a function of magnetic flux $\Phi$.
This Hamiltonian is a typical two-level qubit system with avoided level crossings. Away from the crossings, the coupling between spin-up and spin-down state is small compared to the energy difference. Then these two spin states are the eigenstates of the qubit. The qubit dynamics is adiabatic. However, the energy difference between the two spin states vanishes at the crossing points, then the spin coupling $\delta$ dominates the dynamics of the qubit states. When the system traverses through the crossings under a time varying magnetic flux, interesting physics occurs: non-adiabatic Landau-Zener transitions may happen, and the qubit state will experience a rotation on the Bloch sphere representation\cite{nori}.
This Landau-Zener transition has been extensively studied in the literature\cite{nori}, with realistic examples achieved in atomic systems, quantum dot systems, and superconducting systems.
\section{Landau-Zener-St\"{u}ckelberg Interference}
The Landau-Zener transition can significantly modulate the quantum state of the Majorana qubit. However, the system will not restore to its original Hamiltonian after one Landau-Zener transition. The Hamiltonian
will be added with magnetic flux, which brings an obstacle for the realistic quantum control on the Majorana qubit.
One solution to this obstacle is to consider a magnetic flux pulse. During the pulse, the flux first increases and the system transverses through an avoided crossing point, then the flux decreases and the system transverses back through the same crossing point. Finally, the flux vanishes after the pulse and the system will restore to the original Hamiltonian. The Majorana qubit transverses through the same crossing point twice within one pulse. Two Landau-Zener transition happens and the LZS interference will rotate the Majorana qubit.
A flux pulse with a triangular shape is a natural choice for this purpose. It is one of the simplest pulse shapes, and provides a constant increasing and decreasing speed for the magnetic flux. This linear dependence simplifies the analytic solutions. We consider an triangular pulse with a function of,
\begin{equation}
\Phi(t) = \Phi_0 \times
\begin{cases}
\omega_1 t, & 0 < t <\frac{A}{\omega_1}\\
A - \omega_2 (t - \frac{A}{\omega_1}), & \frac{A}{\omega_1} < t < \frac{A}{\omega_1} + \frac{A}{\omega_2}\\
0, & others
\end{cases}
\end{equation}
where the amplitude $A$ and the velocity $\omega_{1,2}$ are positive. As shown in Fig. 2a, this pulse starts from time zero, increases with a constant speed $\omega_1$, reaches a maximum value of $A\Phi_0$, then decrease with a constant speed of $\omega_2$, finally vanishes at time ${A}/{\omega_1} + {A}/{\omega_2}$. This triangular pulse is asymmetric for general case. It becomes symmetric under the special condition of $\omega_1 = \omega_2$. In the following calculations, we find that the asymmetric pulse is more suitable for our purpose to control the Majorana qubit.
\begin{figure}[t]
\includegraphics[clip=true,width=1\columnwidth]{fig2.eps}
\caption{(Color online). (a) The asymmetric triangular flux pulse as a function of time. (b) The time evolution of the two diagonal terms of the Hamiltonian Eq. (\ref{spin}). (c) The time evolution of the Majorana qubit sate represented by the two components $|\psi_1(t)|^2$ (solid line) and $|\psi_2(t)|^2$ (dashed line). The parameters are $A = 1$, $\delta/E_J = 0.001$, $\omega_1/\delta = 0.2$ and $\omega_2/\delta = 0.025$.}
\end{figure}
Landau-Zener transitions appear in this system only when the pulse amplitude $A$ is greater than $1/2$. For simplicity, we consider a pulse amplitude of $1/2 < A < 3/2$. Then only one avoided crossing point is involved during the pulse. In the following analysis, we denote $t_1 = {1}/{(2\omega_1)}$ and $t_2 = t_1 + (A - 1/2)/\omega_2$ as the crossing time when the system transverses across the avoided crossing $\Phi = \Phi_0/2$.
Near $t_1$ and $t_2$, the diagonal part of the Hamiltonian is small and can be linearized as\cite{nori},
\begin{equation}
\epsilon(t) = E_J \cos(\pi \Phi(t) /\Phi_0) \approx \pm v_{1,2} (t - t_{1,2}),
\end{equation}
where
\begin{equation}
v_{1,2} = E_J \left[\frac{d}{dt}\cos({\pi\Phi(t)}/{\Phi_0})\right]_{t = t_{1,2}} = \pi E_J \omega_{1,2}.
\end{equation}
Then the effective Hamiltonian is linearized near the crossing times,
\begin{equation}
H_{1,2}(t) = v_{1,2} (t - t_{1,2}) \sigma_z + \delta \sigma_x.
\end{equation}
These linearized Hamiltonian $H_{1,2}$ are typical examples of Landau-Zener problem. The Landau-Zener transition probability for a single crossing is given by\cite{landau,zener}
\begin{equation}
P_{1,2} = \exp(-2\pi\beta_{1,2}),
\end{equation}
where $\beta_{1,2} = \delta^2/2v_{1,2}$. If the system starts from the lower level initially, then the Landau-Zener transition probability $P_{1,2}$ describes the probability of upper level occupation after one crossing.
As shown in Ref. [28], these Landau-Zenner transitions can be described approximately by an unitary evolution matrix $\hat{N}_{1,2}$\cite{nori,nori2},
\begin{equation}
\begin{aligned}
&\hat{N}_{1,2} = \left(\begin{array}{cc}
\sqrt{P_{1,2}} & \sqrt{1 - P_{1,2}}e^{i\tilde{\varphi}_{1,2}}\\
-\sqrt{1 - P_{1,2}}e^{-i\tilde{\varphi}_{1,2}} & \sqrt{P_{1,2}}
\end{array}\right),
\end{aligned}
\end{equation}
where $\tilde{\varphi}_{1,2} = -\frac{\pi}{2} + \beta_{1,2}(\ln\beta_{1,2} - 1) + arg \Gamma(1 - i\beta_{1,2})$, $\Gamma$ is the gamma function and the phase $\tilde\varphi_{1,2}$ is monotonous function changes from 0 in the adiabatic limit ($P_{1,2} \rightarrow 0$) to $\pi/4$ in the diabatic limit ($P_{1,2} \rightarrow 1$)\cite{nori}.
These two evolution matrices connect the wave function of the system before and after the Landau-Zener transitions,
which reads $\Psi(t_{1,2}+0) = \hat{N}_{1,2}\Psi(t_{1,2}-0)$ with $\Psi= \psi_1 \mid \uparrow \rangle + \psi_2 \mid \downarrow \rangle$.
Now we consider the LZS interference between the two Landau-Zener transitions, and obtain the finial Majorana qubit state. The system evolution between these two Landau-Zener transitions is approximated by an adiabatic evolution matrix\cite{nori,nori2},
\begin{equation}
\hat{U}(t',t) = \left(\begin{array}{cc}
e^{-i\zeta(t',t)} & 0 \\
0 & e^{i\zeta(t',t)}
\end{array}\right),
\end{equation}
where $\zeta(t',t) = \frac{E_J}{\hbar}\int^{t'}_t \cos{\frac{\pi \Phi(\tau)}{\Phi_0}} d\tau$ are adiabatic evolution phase.
The quantum state rotation induced by the LZS interference is written as\cite{GGC},
\begin{equation}\label{final_Psi}
\Psi (t) \approx \hat{U}(t,t_2)\hat{N}_2\hat{U}(t_2,t_1)\hat{N}_1\hat{U}(t_1,t_0)\Psi_0,
\end{equation}
where $\Psi_0 = (0,1)^T$ is the initial state for the Majorana qubit, and $\Psi(t)= [\psi_1(t),\psi_2(t)]^T$ is the final state.
We take $\hat{U}(t_1,t_0)$ as identity matrix with appropriate choice of the phase for the basis states. Then the final qubit state reads,
\begin{eqnarray}
\psi_1(t) && = [\sqrt{P_1(1 - P_2)} e^{i\zeta(t_2,t_1) + i\tilde{\varphi_2}} \\\nonumber
&&+ \sqrt{(1 - P_1)P_2} e^{-i\zeta(t_2,t_1) + i\tilde{\varphi_1}}]e^{-i\zeta(t,t_2)}.
\end{eqnarray}
We immediately find that the Majorana qubit state has been rotated. The phase of $\psi_1(t)$ experiences the Larmor precession, therefore will rotate after the pulse horizontally in the Bloch sphere. In principle, any horizontal rotation angle for the Majorana qubit can be achieved with appropriate precession time\cite{GGC}. Here, we concentrate on the longitudinal rotation of the Majorana qubit, which is given by the amplitude of the final wave function,
\begin{eqnarray}\label{upper}
|\psi_1|^2 &&= (P_1 + P_2 - 2P_1 P_2) \\\nonumber
&&+ 2\sqrt{P_1 P_2 (1 - P_1)(1 - P_2)} \cos \chi,
\end{eqnarray}
where $\chi = 2\zeta(t_2,t_1) - \tilde{\varphi}_1 + \tilde{\varphi}_2$ comes from the adiabatic phase evolution, which is
sensitive to the parameters of the pulse\cite{nori}. The first term in Eq. (\ref{upper}) provides the rotation angle of the Majorana qubits from $(0,1)$ to $[\psi_1(t), \psi_2(t)]$. The rotation angle is fully determined by the probability of the two Landau-Zener transitions $P_1$ and $P_2$, therefore is controlled by the flux varying speed $\omega_1$ and $\omega_2$.
The second term can be treated as
the uncertainty for the rotation operation, since the phase factors are uncontrollable.
We could reduce this uncertainty by modulating $P_1$ or $P_2$, making one of them approaches $0$ or $1$. This regime can be achieved only for the asymmetric triangular pulse, since the symmetric triangular pulse leads to $P_1 = P_2$. Then the uncertainty term will always be significant.
\begin{figure}[tb]
\includegraphics[clip=true,width=1\columnwidth]{fig3.eps}
\caption{(Color online) The time evolution of the Majorana qubit sate represented by the two components $|\psi_1(t)|^2$ (solid line) and $|\psi_2(t)|^2$ (dashed line), with (a) $\omega_2/\delta = 0.025$, (b) $\omega_2/\delta = 0.01$, (c) $\omega_2/\delta = 0.004$, and (d) $\omega_2/\delta = 0.002$. Other parameters are $A = 1$, $\delta/E_J = 0.001$, and $\omega_1/\delta = 0.5$}
\end{figure}
\section{Numerical simulation}
Now we study the system described by Eq. (3) numerically to obtain quantitative results.
First we setup the equations for numerical simulations. We expand the Hilbert space of the Majorana qubit system with the psudo-spin states $|\Psi (t) \rangle = \psi_1(t) \mid\uparrow \rangle + \psi_2 (t) \mid \downarrow \rangle$. Then the Schr\"{o}dinger equation for the Majorana qubit is written explicitly as,
\begin{equation}
i\frac{d}{dt} \left(\begin{array}{c}
\psi_1\\
\psi_2
\end{array}\right) = \left(\begin{array}{cc}
E_J\cos\frac{\pi\Phi(t)}{\Phi_0} & \delta\\
\delta& -E_J\cos\frac{\pi\Phi(t)}{\Phi_0}
\end{array}\right)
\left(\begin{array}{c}
\psi_1\\
\psi_2
\end{array}\right).
\end{equation}
Numerical simulations for this Schr\"{o}dinger equation are performed with standard finite difference method, in which the evolution operation is linearized within each small segment of time. We consider the asymmetric triangular magnetic flux pulse described in Eq. (4). The signal starts from $t=0$ and ends at $t=A/\omega_1 + A/\omega_2$, as shown in Fig. 2a. This pulse increases the flux through the SQUID from zero to $A\Phi_0$ rapidly, then reduces back to zero slowly. During this process, the two diagonal terms of the Hamiltonian, which is shown in Fig. 2b, variate and cross with each other twice.
Around these two special crossing points, the diagonal part vanishes, thus the off-diagonal part dominate the system and the qubit experiences Landau-Zener transitions. We calculate the wave function evolution of the Majorana qubit starting from $\Psi=(0,1)^T$, and show the results in Fig. 2c. The qubit is staying at the initial state before the pulse. After $t=0$, the pulse is applied and the Hamiltonian begins evolution. We find that the system stays at the initial state away from the crossing points, because
the energy difference between the diagonal part of the Hamiltonian is big enough to induce adiabatic dynamics. When the flux increases and the system approaches the avoided crossing, the evolution becomes non-adiabatic, and the qubit
will have a probability of psudo-spin rotation. This is the Landau-Zener transition as expected in theoretical analysis. The increasing speed of the flux is very quick for this pulse. Therefore, the evolution is in the diabatic limit. The psudo-spin rotation is small.
When the flux decreases, the second Landau-Zener transition happens. This time, the decreasing speed of the flux is moderate. The LZS interference between these two transitions then rotates the qubits, which is clearly demonstrated with the two components of the wave function in Fig. 2c.
This qubit rotation is controllable by modulating the time scale of the magnetic pulse. More specifically, it is controlled by the flux decreasing speed of the pulse, since we increase the flux fast enough to achieve the extreme diabatic behavior. In this scenario, the error term due to the phase accumulation in the adiabatic region is reduced.
We show the results with different rotation angles for the Majorana qubit in Fig. 3. The rotation angle is determined by the flux decreasing speed of the pulse $\omega_2$.
We find that the rotation can be as large as an inversion from $(0,1)$ state to $(1,0)$ state.
With these rotation operations, we achieve a one-qubit gate for the Majorana qubit simply by applying a magnetic flux pulse.
This one-qubit gate is a good supplement to the braiding operations, and should be important for realizing universal quantum gates in topological quantum computation.
\section{Discussions and Conclusion}
In this work, we exploit the asymmetric triangular pulse to implement the one-qubit control. The triangular pulse has the advantage of being simple in theory, yet is experimentally easy to achieve. However, a symmetric pulse causes a large error as shown in Eq. (\ref{upper}), and the operation on the qubit will be extremely sensitive to the parameters of the pulse. Therefore,
it is difficult to achieve a realistic control on the Majorana qubit. In contrary, the asymmetric pulse can induce well controlled one-qubit gate, in which the qubit rotation angle is fully determined by the decreasing speed of the pulse. Therefore, it is more applicable for realistic quantum gates.
Finally, we discuss the orders of the physical quantities in our work. The topological superconductivity is achieved through the proximity effect, with an approximate critical temperature of $T_c \sim $ 10 K. The Josephson energy $E_J$ should be much smaller than the superconducting gap, with a typically value of $E_J \sim $ 10 GHz\cite{pekker}. The coupling energy between distant Majorana bound states are exponentially small for long wire, which can be reasonably taken as $\delta \sim $ 10 MHz. Then the pulse length for operation Majorana qubits should be in the range of $\omega_{1,2} \sim$ (1-10) MHz. With these parameters, we estimate the time scale of the operations to be around (0.1-1) ms, which establishes a high-speed quantum gate on Majorana qubits.
In summary, we analyze the Landau-Zener-St\"{u}ckelberg interference of a Majorana qubit in a topological rf SQUID. An asymmetric triangular magnetic flux pulse is applied in the SQUID to drive the system. In one pulse period, the system transverses through the same avoided crossing point twice with different speed, and two Landau-Zener transitions happen. The Landau-Zener-St\"{u}ckelberg interference between these two transitions induces a rotation on the Majorana qubit. Importantly, the rotation angle can be controlled by the time scale of the pulse. Therefore, the Landau-Zener-St\"{u}ckelberg interference can achieve a one-qubit gate for the Majorana qubit. This quantum gate might be useful for topological quantum computation.
\section*{Acknowledgment}
This work was supported by NSFC-11304400, NSFC-61471401 SRFDP-20130171120015, 985 Project of Sun Yat-Sen University, and State Key Laboratory of Optoelectronic Materials and Technologies.
D.X.Y. is supported by NSFC-11074310, NSFC-11275279, SRFDP-20110171110026, NBRPC-2012CB821400, and Fundamental Research Funds for the Central Universities of China.
|
{
"timestamp": "2015-04-14T02:10:51",
"yymm": "1504",
"arxiv_id": "1504.03055",
"language": "en",
"url": "https://arxiv.org/abs/1504.03055"
}
|
\section{Introduction}
$\beta$-delayed fission ($\beta$DF) is a two-step process whereby the fissioning nucleus could be created in an excited state after $\beta$ decay of a precursor. Since the excitation energy of the fissioning daughter product is limited by the $Q_{\beta}$ value for $\beta$ decay of the parent, $\beta$DF provides a unique tool to study low-energy fission of nuclei far from stability, especially for those not fissioning spontaneously. Figure \ref{fig:bdfSchem} provides a schematic representation of this process, for nuclides on the neutron-deficient side of the nuclear chart. Recent experiments at ISOLDE-CERN \cite{Andreyev2010,Elseviers2013,Liberati2013,Ghys2014} and SHIP-GSI \cite{Andreyev2013, Lane2013} have studied this exotic decay mode in several short-lived neutron-deficient isotopes in the lead region. The fission-fragment mass and energy distributions resulting from $\beta$DF have established a new region of asymmetric fission around $^{178,180}\mathrm{Hg}$ \cite{Andreyev2010,Liberati2013} and indicated multimodal fission in $^{194,196}$Po and $^{202}$Rn \cite{Ghys2014}. A recent review of the $\beta$DF process is given in \cite{Andreyev2013a}, in which a total of 27 $\beta$DF cases, both on the neutron-rich and neutron-deficient sides, were summarized.
\begin{figure}
\includegraphics[width=0.5\columnwidth]{Figure1.png}
\caption{(Color online) Schematic representation of the $\beta$DF process on the neutron-deficient side of the nuclear chart. The $Q_{EC}$ value of the parent (A,Z) nucleus is indicated, while the curved line shows the potential energy of the daughter (A,Z-1) nucleus with respect to nuclear elongation, displaying also the fission barrier $B_f$. The color code on the right-hand side represents the probability for excited states, with excitation energies close to $B_f$, to undergo fission; the darker colors correspond to higher probabilities.}
\label{fig:bdfSchem}
\end{figure}
It is furthermore believed that $\beta$DF could, together with neutron-induced and spontaneous fission, influence the fission-recycling in r-process nucleosynthesis \cite{Panov2005,Petermann2012}. Therefore, a reliable prediction of the relative importance of $\beta$DF in nuclear decay, often expressed by the $\beta$DF probability $P_{\beta \mathrm{DF}}$, is needed. $P_{\beta \mathrm{DF}}$ is defined as
\begin{equation}
P_{\beta \mathrm{DF}} = \frac{N_{\beta \mathrm{DF}}}{N_{\beta}},
\label{eq:Pbdf}
\end{equation}
where $N_{\beta DF}$ and $N_{\beta}$ are respectively the number of $\beta$DF and $\beta$ decays of the precursor nucleus.
An earlier comparison of $P_{\beta \mathrm{DF}}$ data in a relatively narrow region of nuclei in the vicinity of uranium showed a simple exponential dependence with respect to $Q_{\beta}$ \cite{Shaughnessy2000,Shaughnessy2002}. It was assumed that fission-barrier heights $B_{ \rm f}$ of the daughter nuclei do not vary greatly in this region \cite{Britt1980} ($B_{ \rm f}\sim4$\,--\,6\,MeV) and thus have a smaller influence on $P_{\beta \mathrm{DF}}$ as compared to $Q_{\beta}$ values ($Q_{\beta}\sim3$\,--\,6\,MeV). In addition, these nuclei have a typical $N$/$Z$ ratio around $\sim$\,1.4\,--\,1.5, which is close to that of traditional spontaneous fission of heavy actinides.\\
The aim of this paper is to further explore such systematic features by including the newly obtained data in the neutron-deficient lead region whose $\beta$DF nuclides have significantly different N/Z ratios ($\sim$\,$1.2$\,--\,1.3), $B_{\rm f}$ ($\sim$\,7\,--\,10\,MeV) and $Q_{\beta}$ values ($\sim$\,9\,--\,11\,MeV) as compared to those in the uranium region.\\
However, from an experimental point of view, the dominant $\alpha$-branching ratio ($\gtrsim90$\,$\%$) in most $\beta$DF precursors in the neutron-deficient lead region \cite{Nubase2012} makes precise determination of $N_{\beta}$ in equation (\ref{eq:Pbdf}) difficult. Therefore, the partial $\beta$DF half-life $T_{\rm 1/2p,\beta DF}$, as proposed in \cite{Andreyev2013a}, is discussed in the present study. By analogy with other decay modes, $T_{\rm 1/2p,\beta DF}$ is defined by
\begin{equation}
T_{\rm 1/2p,\beta DF} = T_{1/2}\frac{N_{\mathrm{dec,tot}}}{N_{\beta \mathrm{DF}}},
\label{eq:Tbdf}
\end{equation}
where $T_{1/2}$ represents the total half-life and $N_{\mathrm{dec,tot}}$ the number of decayed precursor nuclei. The relation between $T_{\rm 1/2p,\beta DF}$ and $P_{\beta \mathrm{DF}}$ can be derived from equations (\ref{eq:Pbdf}) and (\ref{eq:Tbdf}) as
\begin{equation}
T_{\rm 1/2p,\beta DF} = \frac{T_{1/2}}{b_{\beta}P_{\beta \mathrm{DF}}},
\label{eq:PbdfvsTbdf}
\end{equation}
with $b_{\beta}$ denoting the $\beta$-branching ratio. If the $\alpha$-decay channel dominates, as is often the case in the neutron-deficient lead region, one can safely approximate $N_{\mathrm{dec,tot}}$ in equation (\ref{eq:Tbdf}) by the amount of $\alpha$ decays $N_{\mathrm{\alpha}}$.\\
This work shows an apparent exponential dependence of $T_{\rm 1/2p,\beta DF}$ on ($Q_{\beta}-B_{\rm f}$) for certain sets of calculated fission-barrier energies. Such relation may arise naturally by simple phenomenological approximations of the $\beta$-strength function of the precursor and the fission-decay width of excited states in the daughter nucleus. These assumptions may be justified considering the scale of the systematic trend discussed here, spanning $T_{\rm 1/2p,\beta DF}$ values over several orders of magnitude. Deviations lower than one order of magnitude are thus acceptable.
\section{Theoretical considerations}
\label{sec:Theo}
Following \cite{Gangrsky1980,Klapdor1983,Kuznetsov1999}, the expression for $P_{\beta \mathrm{DF}}$ is given by
\begin{equation}
P_{\beta \mathrm{DF}} = \frac{\int_0^{Q_{\beta}}S_{\beta}(E)F(Q_{\beta}-E)\frac{\Gamma_{\rm f}(E)}{\Gamma_{\text{tot}}(E)}dE}{\int_0^{Q_{\beta}}S_{\beta}(E)F(Q_{\beta}-E)dE},
\label{eq:PBdf}
\end{equation}
whereby the $\beta$-strength function of the parent nucleus is denoted by $S_{\beta}$ and the Fermi function by $F$. The excitation energy is here, and further, given by E. The fission and total decay widths of the daughter, after $\beta$ decay, are respectively given by $\Gamma_{\text{f}}$ and $\Gamma_{\text{tot}}$. Equation (\ref{eq:PbdfvsTbdf}) can be combined with equation (\ref{eq:PBdf}) to deduce the decay constant of $\beta$DF, defined as $\lambda_{\beta \mathrm{DF}} = \mathrm{ln}(2)/T_{\rm 1/2p,\beta DF}$, as
\begin{equation}
\lambda_{\beta \mathrm{DF}} = \int_0^{Q_{\beta}}S_{\beta}(E)F(Q_{\beta}-E)\frac{\Gamma_{\rm f}(E)}{\Gamma_{\text{tot}}(E)}dE.
\label{eq:lambdaBdf}
\end{equation}
This section will be devoted to the derivation of an analytical expression for $\lambda_{\beta \mathrm{DF}}$ by approximating $S_{\beta}$, $F$ and $\Gamma_{\rm f}/\Gamma_{\text{tot}}$. Since most of the reliable experimental data on $\beta$DF are recorded on the neutron-deficient side of the nuclear chart (see Table \ref{tab:Tbdf} and \cite{Andreyev2013a}), only EC/$\beta^+$-delayed fission will be considered here.
\subsection{Approximations}
\label{approx}
A first simplification in equation (\ref{eq:lambdaBdf}) is to approximate $S_{\beta}$ by a constant $C_1$, as proposed in previous studies (see for example \cite{Kratz1973,Hornshoj1974}). Possible resonant structures in $S_{\beta}$, considered in e.g. \cite{Klapdor1983, Izosimov2011}, are thus ignored, thereby assuming a limited sensitivity of $T_{\rm 1/2p,\beta DF}$ on $S_{\beta}$ with respect to the scale of the systematic trend discussed here. This approximation is further supported by the study in \cite{Veselsky2012}, which shows a limited influence of $S_{\beta}$ in the calculation of $P_{\beta \mathrm{DF}}$. Furthermore, $C_1$ was taken equal for all isotopes listed in Table \ref{tab:Tbdf}, thereby neglecting possible variations of $C_1$ with respect to the nuclear properties of the $\beta$DF precursors - such as mass, proton number, isospin, spin and parity.\\
The Fermi function $F$ can be fairly well described by the function $C_2(Q_{EC}-E)^2$ \cite{Habs1978,Hall1992,logft} for EC decay. The prefactor $C_2$ was again considered element independent, thereby ignoring its slight dependence on the atomic number $Z$ \cite{logft}.
According to \cite{Firestone1999,logft}, EC decay is dominant for transition energies below 5\,MeV if $Z$ exceeds 80. Since $Q_{\beta}$ values of $\beta$DF precursors in the uranium region are typically smaller than 5\,MeV (see Table \ref{tab:Tbdf}), $\beta^+$ decay can be disregarded here. $Q_{\beta}$ values in the neutron-deficient lead region can however reach 10\,--\,11\,MeV, implying a relatively high $\beta^+$ over EC decay ratio to the ground or a low-lying excited state in the daughter. However, since $\beta$DF should primarily happen at excitation energies which are only a few MeV below $Q_{\beta}$ \cite{Moller2012}, EC-delayed fission should dominate over $\beta^+$ delayed fission in the full region of the nuclear chart (see further).\\
The prompt decay of an excited state in a nucleus can, in the most general case, happen through fission, emission of a $\gamma$ ray, proton, $\alpha$ particle or neutron. The total decay width is thus given by $\Gamma_{\text{tot}} = \Gamma_{\rm f} + \Gamma_{\gamma} + \Gamma_{\rm p} + \Gamma_{\alpha} + \Gamma_{\rm n}$.\\
For the $\beta$DF precursors considered in Table \ref{tab:Tbdf}, the neutron separation energies exceed the $Q_{\beta}$ value by at least several MeV \cite{AME2012} and charge particle-emission is strongly hindered due to the large Coulomb barrier. Therefore, the de-excitation of states below $Q_{\beta}$ is mostly dominated by $\gamma$ decay, which makes that $\Gamma_{\text{tot}} \simeq \Gamma_{\gamma}$ \cite{NRV,Veselsky2012}.
In addition, $\Gamma_{\gamma}$ can be approximated by a constant (see for example \cite{Veselsky2012}). Reference \cite{NRV} provides a calculation of $\Gamma_f$ with respect to the excitation energy $E$ by including the fission-barrier penetrability and the influence of level densities at the ground state and saddle point. This calculation shows that $\Gamma_{\text{f}}$ seems well approximated by a single exponential behavior $\Gamma_{\text{f}} \sim e^{-X(B_{ \rm f}-E)}$ at excitation energies around $B_{\rm f}$. For the fissile nuclei listed in Table \ref{tab:Tbdf}, the decay constant adopts a value $X\approx4$ MeV$^{-1}$ \cite{NRV}. The ratio $\Gamma_{\rm f}/\Gamma_{\rm tot}$ is thus approximated by
\begin{equation}
\frac{\Gamma_{\rm f}}{\Gamma_{\rm tot}}(E) \simeq \frac{\Gamma_{\rm f}}{\Gamma_{\gamma}}(E) \approx C_3e^{-X(B_{ \rm f}-E)}.
\label{ratioG}
\end{equation}
The constants $C_3$ and $X$ are assumed to adopt the same value for all isotopes of interest. At excitation energies $E$ moderately above $B_{\rm f}$, de-excitation by fission should dominate and $\Gamma_{\rm f}$/$\Gamma_{tot}(E)$ will thus be close to unity. Since the $Q_{\beta}$ value of most known $\beta$DF precursors (see Table \ref{tab:Tbdf}) does not exceed $B_{\rm f}$ of the daughter by more than a few MeV, it is further assumed that equation (\ref{ratioG}) remains valid for excitation energies in the daughter nucleus close to $Q_{\beta}$. \\
Using the above approximations and taking $C=C_1C_2C_3$, the right-hand side of equation (\ref{eq:lambdaBdf}) reduces to
\begin{equation}
\lambda_{\beta \rm DF} = C\int_0^{Q_{\beta}}(Q_{\beta}-E)^2e^{-X(B_{\rm f}-E)}dE.
\label{eq:l_calc}
\end{equation}
\subsection{Calculating $\lambda_{bdf}$}
Equation (\ref{eq:l_calc}) can be rewritten, in order to isolate the exponential dependance on $(Q_{\beta}-B_{\rm f})$, as
\begin{equation}
\lambda_{\beta \rm DF} = Ce^{X(Q_{\beta}-B_{\rm f})}{\int_0^{Q_{\beta}}(Q_{\beta}-E)^2e^{-X(Q_{\beta}-E)}}dE.
\label{eq:l_calc2}
\end{equation}
The integrand in equation (\ref{eq:l_calc2}) is thus proportional to the $\beta$DF probability at a given $E$ of the daughter nucleus. This function, plotted in Figure \ref{fig:prob} for different values of $X$ around the deduced value $X\approx4$\,$\mathrm{MeV}^{-1}$ from \cite{NRV}, shows that $\beta$DF primarily happens at energy levels 0\,--\,2\,MeV below $Q_{\beta}$. Moreover, since all $Q_{\beta}$ values of the neutron-deficient $\beta$DF precursors listed in Table \ref{tab:Tbdf} exceed $\sim$\,2\,MeV, the value of the integral in equation (\ref{eq:l_calc2}) is little dependent on the precise value of $Q_{\beta}$. As a consequence, $\lambda_{\beta \rm DF}$ primarily depends on the difference $(Q_{\beta}-B_{\rm f})$.
\begin{figure}
\includegraphics[width=0.5\columnwidth]{Figure2.png}
\caption{Plot showing the integrand of equation (\ref{eq:l_calc2}), which is proportional to the $\beta$DF probability, for $X$ equal to 3,4 or 5.}
\label{fig:prob}
\end{figure}
In order to prove latter statement analytically, a substitution with $u=X(Q_{\beta}-E)$ and adjustment of integration borders in equation (\ref{eq:l_calc2}) is performed:
\begin{equation}
\lambda_{\beta \rm DF} = \frac{Ce^{X(Q_{\beta}-B_{\rm f})}}{X^3}{\int_0^{XQ_{\beta}}u^2e^{-u}du}.
\label{eq:l_calc3}
\end{equation}
The integral in equation (\ref{eq:l_calc3}) is similar to the mathematical form of the so-called normalized upper incomplete Gamma function, defined as
\begin{equation}
\Gamma(s,x) = \frac{1}{\Gamma(s)}\int^x_0t^{s-1}e^{-t}dt,
\label{GammaFI}
\end{equation}
whereby $\Gamma(s)$ is
\begin{equation}
\Gamma(s) = \int^{+\infty}_0t^{s-1}e^{-t}dt.
\label{GammaFC}
\end{equation}
Equation (\ref{eq:l_calc3}) thus transforms into
\begin{equation}
\lambda_{\beta \rm DF}=\frac{Ce^{X(Q_{\beta}-B_{\rm f})}}{X^3}\Gamma(3)\Gamma(3,XQ_{\beta}).
\label{eq:l_calc4}
\end{equation}
Table \ref{tab:Tbdf} shows that all $Q_\beta$ values of the neutron-deficient $\beta$DF precursors exceed 3\,MeV, while the fitted values for $X$ in Table \ref{tab:FitResults}, as well as the theoretical estimate from \cite{NRV} ($X\approx4$\,$\mathrm{MeV}^{-1}$), are all greater than 1.7\,$\mathrm{MeV}^{-1}$. The value $XQ_{\beta}$ thus exceeds 5 in all discussed cases, implying that, as shown in Figure \ref{fig:gamma}, one can thus safely approximate $\Gamma(3,XQ_{\beta})\simeq 1$ in equation (\ref{eq:l_calc4}).
\begin{figure}
\includegraphics[width=0.5\columnwidth]{Figure3.png}
\caption{The normalized incomplete Gamma function $\Gamma(3,XQ_{\beta})$, needed for the calculation of the integral under the $\beta$DF probability curves shown in Figure \ref{fig:prob}.}
\label{fig:gamma}
\end{figure}
In this simple picture, it is thus found that $\text{ln}(\lambda_{\beta DF})$ depends linearly on ($Q_{\beta}-B_{\rm f}$). In terms of the partial $\beta$DF half-life $T_{\rm 1/2p,\beta DF}$ one finds the relation
\begin{equation}
\mathrm{log_{10}}(T_{\rm 1/2p,\beta DF}) = C' - X\mathrm{log_{10}}(e)(Q_{\beta}-B_{\rm f}),
\label{eq:l_def}
\end{equation}
with the constant $C'$ given by
\begin{equation}
C'=\mathrm{ln}\left(\frac{\mathrm{ln(2)}X^3}{C\Gamma(3)}\right)\mathrm{log_{10}}(e).
\end{equation}
\section{Systematic comparison of experimental data}
\label{sec:data}
This section aims at verifying equation (\ref{eq:l_def}) by using experimental $\beta$DF partial half-lives and theoretical values for ($Q_{\beta}-B_{\rm f}$), summarized in Table \ref{tab:Tbdf} and Figure \ref{fig:Tbdf}. Tabulated fission barriers from four different fission models were used, of which three are based on a macroscopic-microscopic and one a mean-field approach. The latter model is based on the Extended Thomas-Fermi plus Strutinsky Integral (ETFSI) method \cite{Mamdouh2001}, but tabulated barriers for the most neutron-deficient isotopes in Table \ref{tab:Tbdf} are absent in literature. The microscopic-macroscopic approaches all rely on shell corrections from \cite{Moller1995} and describe the macroscopic structure of the nucleus by either a Thomas-Fermi (TF) \cite{Myers1999}, liquid-drop (LDM) \cite{NRV} or the Finite-Range Liquid-Drop Model (FRLDM) \cite{Moller2015}. The $Q_{\beta}$ values were taken from the 2012 atomic mass evaluation tables \cite{AME2012} and are derived from the difference between the atomic masses of parent $M_P(Z,A)$ and daughter $M_D(Z',A)$ nuclei as
\begin{equation}
Q_{\beta} = c^2[M_P(Z,A) - M_D(Z',A)].
\label{eq:Qvalue}
\end{equation}
About half of these values are known from experiments, while the others are deduced from extrapolated atomic masses. In latter cases, the difference of the $Q_{\beta}$ values from \cite{AME2012} with the theoretical values from \cite{Moller1995} or \cite{Myers1996} is always lower than 0.4 MeV.\\
\begingroup
\squeezetable
\begin{table*}
\caption{List of all precursors for which $\beta$DF was observed. The measured half-life $T_{1/2}$, $\beta$-branching ratio $b_{\beta}$, $\beta$DF probability $P_{\beta \mathrm{DF}}$, ratio of observed $\beta$DF to $\alpha$ decays $N_{\beta \mathrm{DF}}/N_{\alpha}$ and calculated $\beta$DF partial half lives $T_{1/2p,\beta\mathrm{DF}}$ are listed. Reliable values for $T_{1/2p,\beta\mathrm{DF}}$, as evaluated by the criteria in \cite{Andreyev2013a}, are indicated in bold. ($Q_{\beta} - B_{\rm f}$) is tabulated for fission barriers from four different fission models : Thomas-Fermi (TF) \cite{Myers1999}, Finite Range Liquid Drop (FRLDM) \cite{Moller2015}, Liquid Drop (LDM) \cite{NRV} and the Extended Thomas-Fermi plus Strutinsky Integral (ETFSI) model \cite{Mamdouh2001}. $Q_{\beta}$ values were taken from \cite{AME2012} and are defined by equation (\ref{eq:Qvalue}).}
\label{tab:Tbdf}
\begin{ruledtabular}
\begin{tabular}{lccccccc D{,}{\times}{-1} D{,}{\times}{-1} D{,}{\times}{-1} c}
& & & \multicolumn{4}{c}{$Q_{\beta}-B_{\rm f}$ (MeV)} & & & & &\\
\cline{4-7}
precursor & $T_{1/2}$ (s) & $Q_{\beta}$ (MeV) & TF & FRLDM & LDM & ETFSI & $b_{\beta}$ & \multicolumn{1}{c}{$P_{\beta \mathrm{DF}}$} & \multicolumn{1}{c}{$N_{\beta \mathrm{DF}}/N_{\alpha}$} & \multicolumn{1}{c}{$T_{1/2p,\beta\mathrm{DF}}$ (s)} & ref. \\
\hline
\multicolumn{12}{l}{\it $\beta^+$/EC-delayed fission in the neutron-deficient lead region} \\
$^{178}\mathrm{Tl}$ & 0.25(2) &11.5 & 2.5 & 2.2 & 3.0 & & 0.38(2)
& 1.5(6), 10^{-3} & & \mathbf{4(2)},\mathbf{10^2} & \cite{Liberati2013} \\
$^{180}\mathrm{Tl}$ & 1.09(1) &11.0 & 1.4 & 1.2 & 2.6 & & 0.94(4)
& 3.2(2), 10^{-5} & & \mathbf{3.6(3)},\mathbf{10^4}
& \cite{Elseviers2013} \\
$^{186g,m}\mathrm{Bi}$ & 0.012(3) \footnote{\label{foot:T}Value extracted according to equation (\ref{eq:T1/2}) by using evaluated experimental data from \cite{Nubase2012}.} &11.6 & 2.8 & 2.0 & 3.1 & & $\sim 0.006$
\footnote{\label{foot:btheo}Calculated $\beta$-branching ratio from \cite{Moller1997}.} & & 2.2(13), 10^{-4} & \multicolumn{1}{c}{$56(35)$} & \cite{Lane2013} \\
$^{188g,m}\mathrm{Bi}$ & 0.16(10) \textsuperscript{\ref{foot:T}} &10.6 & 0.9 & 0.3 & 1.2 & & $\sim 0.03$
\textsuperscript{\ref{foot:btheo}} & & 3.2(16),10^{-5} & 5(4),10^3 & \cite{Lane2013} \\
$^{192g,m}\mathrm{At}$ & 0.05(4) \textsuperscript{\ref{foot:T}} &11.0 & 4.2 & 2.8 & 4.2 & & $\sim 0.03$
\textsuperscript{\ref{foot:btheo}} & & 4.2(9), 10^{-3} & \multicolumn{1}{c}{$12(9)$} & \cite{Andreyev2013} \\
$^{194g,m}\mathrm{At}$ & 0.28(3) \textsuperscript{\ref{foot:T}} &10.3 & 2.5 & 0.8 & 2.7 & & $\sim 0.08$ \textsuperscript{\ref{foot:btheo}} & & 5.9(4), 10^{-5} & 4.8(6),10^2 & \cite{Ghys2014} \\
$^{196}\mathrm{At}$ & 0.358(5) &9.6 & 0.3 & -0.7 & 1.1 & & $0.026(1)$
& 9(1), 10^{-5} & 2.3(2), 10^{-6} & \mathbf{1.5(2)},\mathbf{10^5} & \cite{Ghys2014,Truesdale2014} \\
$^{200}\mathrm{Fr}$ & 0.049(4) \textsuperscript{\ref{foot:T}} &10.2 & 3.3 & 1.5 & 3.7 & & $<0.021(4)$
& >3.1(17), 10^{-2} & 7^{+5}_{-3} ,10^{-4} & \mathbf{7^{+6}_{-3}},\mathbf{10} & \cite{Ghys2014}\\
$^{202g,m}\mathrm{Fr}$ & 0.33(4) \footnote{Value extracted according to equation (\ref{eq:T1/2}) by using experimental data from \cite{Kalaninova2014}.} &9.4 & 0.8 & -0.9 & 0.7 & & $\sim 0.007$ \textsuperscript{\ref{foot:btheo}} & & 7.3(8),10^{-7} & 4.5(8),10^4 & \cite{Ghys2014} \\
\\
\multicolumn{11}{l}{\it $\beta^+$/EC-delayed fission in the neutron-deficient uranium region} \\
$^{228}\mathrm{Np}$ & 61(1) &4.4 & 0.0 & -0.8 & 0.3 & & $0.60(7)$
& 2.0(9), 10^{-4} & & \mathbf{5.1(2)},\mathbf{10^5} & \cite{Kreek1994} \\
$^{232}\mathrm{Am}$ & 79(2) &4.9 & 1.3 & 1.7 & 0.5 & & $\sim 0.96$ \textsuperscript{\ref{foot:btheo}} & 6.9(10), 10^{-4} & & \mathbf{1.2(2)},\mathbf{10^{5}} & \cite{Hall1990} \\
$^{234}\mathrm{Am}$ & 139(5) &4.1 & 0.0 & 0.3 & -0.3 & -0.1 & $\sim 1.00$ \textsuperscript{\ref{foot:btheo}} & 6.6(18), 10^{-5} & & \mathbf{2.1(6)},\mathbf{10^6} & \cite{Hall1990a} \\
$^{238}\mathrm{Bk}$ & 144(5) &4.8 & 1.1 & -0.2 & 0.4 & -0.1 & $\sim 0.95$ \textsuperscript{\ref{foot:btheo}} & 4.8(20), 10^{-4} & & \mathbf{3.2(13)},\mathbf{10^5} & \cite{Kreek1994a} \\
$^{240}\mathrm{Bk}$ & 252(48) &3.9 & -0.3 & -1.9 & -0.8 & -1.6 & $\sim 1.00$ \textsuperscript{\ref{foot:btheo}} & 1.3^{+1.8}_{-0.7}, 10^{-5}& & \mathbf{1.9^{+2.3}_{-1.1}},\mathbf{10^{7}} & \cite{Galeriu1983} \\
$^{242}\mathrm{Es}$ & 11(3) &5.4 & 1.8 & -0.7 & 1.2 & -0.1 & $0.57(3)$
\footnote{$\beta$-branching ratio from \cite{Antalic2009}.} & 6(2), 10^{-3} & & \mathbf{3(1)},\mathbf{10^3} & \cite{Shaughnessy2000} \\
$^{244}\mathrm{Es}$ & 37(4) \textsuperscript{\ref{foot:T}} &4.5 & 0.2 & -2.2 & -0.3 & -1.7 & $0.96(3)$ \footnote{\label{foot:bexp}Evaluated $\beta$-branching ratio from \cite{Nubase2012}.} & 1.2(4), 10^{-4} & & \mathbf{3(1)},\mathbf{10^5} & \cite{Shaughnessy2002} \\
$^{246}\mathrm{Es}$ & 462(30) &3.8 & -0.8 & -3.4 & -1.7 & -2.7 & $0.901(18)$ \textsuperscript{\ref{foot:bexp}} & 3.7^{+8.5}_{-3.0}, 10^{-5}& & \mathbf{1.4^{+5.9}_{-1.0}},\mathbf{10^{7}} & \cite{Shaughnessy2001} \\
$^{248}\mathrm{Es}$ & $1.4(2)\times 10^{3}$ &3.1 & -1.9 & -4.2 & -2.8 & -3.6 & $0.997(3)$ \textsuperscript{\ref{foot:bexp}} & 3.5(18), 10^{-6} & & \mathbf{4.0(21)},\mathbf{10^8} & \cite{Shaughnessy2001} \\
$^{246m2}\mathrm{Md}$ & \multicolumn{1}{c}{$4.4(8)$} &5.9 & 2.1 & -0.2 & 1.6 & 0.0 & $>0.77$
& \multicolumn{1}{c}{$> 0.1$} & & \multicolumn{1}{c}{$<57$} & \cite{Antalic2009} \\
$^{250}\mathrm{Md}$ & $52(6)$ \textsuperscript{\ref{foot:T}} &4.6 & -0.3 & -2.7 & -1.0 & -2.1 & $0.93(3)$ \textsuperscript{\ref{foot:bexp}} & 2^{+2}_{-1}, 10^{-4} & & 3^{+3}_{-1},10^5 & \cite{Gangrsky1980} \\
\\
\multicolumn{11}{l}{\it $\beta^-$-delayed fission in the neutron-rich uranium region} \\
$^{228}\mathrm{Ac}$ & $2.214(7)\times 10^4$ \textsuperscript{\ref{foot:T}} &2.1 & -4.0 & -4.4 & -4.4 & -4.3 & $\sim 1.00$ \textsuperscript{\ref{foot:btheo}} & 5(2), 10^{-12} & & 4(2),10^{15} & \cite{Yanbing2006}\\
$^{230}\mathrm{Ac}$ & $122(3)$ \textsuperscript{\ref{foot:T}} &3.0 & -3.4 & -2.7 & -3.7 & -3.8 & $\sim 1.00$ \textsuperscript{\ref{foot:btheo}} & 1.19(40), 10^{-9} & & 1.0(3), 10^{10} & \cite{Shuanggui2001} \\
$^{234g}\mathrm{Pa}$ & $2.41(2)\times 10^4$ \textsuperscript{\ref{foot:T}} &2.2 & -3.4 & -2.7 & -3.8 & -2.6 & $\sim 1.00$ \textsuperscript{\ref{foot:btheo}} & 3, 10^{-(12\pm1)} & & 8, 10^{(15\pm 1)} & \cite{Gangrsky1978} \\
$^{234m}\mathrm{Pa}$ & $69.54(66)$ \textsuperscript{\ref{foot:T}} &2.2 & -3.4 & -2.7 & -3.8 & -2.6 & $0.9984(4)$
& \multicolumn{1}{c}{$10^{-(12\pm 1)}$} & & 7,10^{(13\pm 1)} & \cite{Gangrsky1978} \\
$^{236}\mathrm{Pa}$ & $546(6)$ \textsuperscript{\ref{foot:T}} &2.9 & -2.9 & -2.1 & -3.2 & -2.3 & $\sim 1.00$ \textsuperscript{\ref{foot:btheo}} & \multicolumn{1}{c}{$10^{-9\pm 1}$} & & 5, 10^{(11\pm 1)} & \cite{Gangrsky1978} \\
$^{238}\mathrm{Pa}$ & $138(6)$ \textsuperscript{\ref{foot:T}} &3.6 & -2.3 & -2.0 & -3.2 & -2.1 & $\sim 1.00$ \textsuperscript{\ref{foot:btheo}} & <2.6,10^{-8} & & >5.3,10^9 & \cite{Baas-May1985} \\
$^{256m}\mathrm{Es}$ & $2.7\times 10^4$ \textsuperscript{\ref{foot:T}} &1.7 & -2.3 & -3.4 & -3.2 & -3.8 & $\sim 1.00$ \textsuperscript{\ref{foot:btheo}} & \sim 2,10^{-5} & & \sim 1,10^9 & \cite{Hall1989} \\
\end{tabular}
\end{ruledtabular}
\end{table*}
\endgroup
$T_{\rm 1/2p,\beta DF}$ values were extracted from reported $P_{\beta \mathrm{DF}}$ values using equation (\ref{eq:PbdfvsTbdf}), if the precursor nucleus has a significant $\beta$-decay branch ($b_{\beta} \gtrsim 10$\,\%). When multiple measurements on $P_{\beta \mathrm{DF}}$ were performed, only the reliable value, as evaluated by \cite{Andreyev2013a}, or the most recent value was tabulated. In case of a dominant $\alpha$-decay branch ($b_{\beta} \lesssim 10$\,\%), $T_{\rm 1/2p,\beta DF}$ was calculated by equation (\ref{eq:Tbdf}), whereby $N_{\mathrm{dec,tot}}$ was approximated by the observed amount of $\alpha$ decays $N_{\alpha}$, corrected for detection efficiency.\\
Since the isotopes $^{186,188}\mathrm{Bi}$, $^{192,194}\mathrm{At}$ and $^{202}\mathrm{Fr}$ have both a ground and a low-lying alpha-decaying isomeric state with comparable half-lives, only an overall $N_{\beta \mathrm{DF}}/N_{\alpha}$ value could be extracted with present experimental techniques. We refer the reader for a detailed discussion of this issue to \cite{Lane2013,Andreyev2013,Ghys2014}. Therefore, these precursors have been excluded from the fit in Figure \ref{fig:Tbdf}. Nonetheless, as a first approximation the value for $T_{1/2p,\beta\mathrm{DF}}$ was extracted by defining the half-life $T_{1/2}$, shown in table \ref{tab:Tbdf}, as the unweighted average
\begin{equation}
T_{1/2} = \frac{T_{\rm 1/2,g} + T_{\rm 1/2,m}}{2}.
\label{eq:T1/2}
\end{equation}
where the respective half-lives for ground and isomeric states are denoted by $T_{\rm 1/2,g}$ and $T_{\rm 1/2,m}$. The uncertainty $\Delta T_{1/2}$ is conservatively taken as
\begin{equation}
\Delta T_{1/2} = \frac{|T_{\rm 1/2,g} - T_{\rm 1/2,m}|}{2}.
\label{eq:DT1/2}
\end{equation}
Figure \ref{fig:Tbdf} shows $\mathrm{log_{10}}(T_{\rm 1/2p,\beta DF})$ against ($Q_{\beta}-B_{\rm f}$) for the fission barriers from the four different models under consideration. Using the same evaluation criteria as proposed in \cite{Andreyev2013a} for $P_{\beta \mathrm{DF}}$ measurements, 13 reliable $T_{1/2p,\beta\mathrm{DF}}$ values, marked in bold in Table \ref{tab:Tbdf}, were selected. These data points, represented by the closed symbols, are fitted by a linear function. An equal weight to all fit points is given because the experimental uncertainties on $\mathrm{log_{10}}(T_{\rm 1/2p,\beta DF})$ are in most cases much smaller than the deviation of the data points with the fitted line, of which the extracted parameters are summarized in Table \ref{tab:FitResults}. The remaining data points from Table \ref{tab:Tbdf} are shown by open symbols and were excluded from the fit. The color code discriminates between the neutron-deficient lead region (red), neutron-deficient (black) and neutron-rich (blue) uranium region. \\
\begin{figure*}
\includegraphics[width=1.0\columnwidth]{Figure4.png}
\caption{(Color online) Plots of $T_{\rm 1/2p,\beta DF}$ versus ($Q_{\beta} - B_{\rm f}$) for different fission models as listed in Table \ref{tab:Tbdf}. The closed symbols, representing reliable values for $T_{1/2p,\beta\mathrm{DF}}$ in Table \ref{tab:Tbdf} are used for a linear fit with equal weights to the data points. Other data from Table \ref{tab:Tbdf} are indicated by the open symbols. The color code represents the different regions of the nuclear chart for which $\beta$DF has been experimentally observed : the neutron-deficient lead region (red), the neutron-deficient (black) and neutron-rich (blue) uranium region.}
\label{fig:Tbdf}
\end{figure*}
Figure \ref{fig:Tbdf} illustrates a linear dependence of $\mathrm{log_{10}}(T_{\rm 1/2p,\beta DF})$ on $(Q_{\beta}-B_{\rm f})$ for TF and LDM barriers for over 7 orders of magnitude of $T_{1/2p,\beta\mathrm{DF}}$. In addition, Table \ref{tab:FitResults} shows a relatively small root-mean-square deviation (RMSD) of the 13 reliable experimental $\mathrm{log_{10}}(T_{\rm 1/2p,\beta DF})$ values (represented by the closed symbols in Figure \ref{fig:Tbdf}) to the corresponding values extracted from the fit. The dependence is somewhat less pronounced for the FRLDM model, as evidenced by a larger RMSD value. A similar linear trend is observed for the ETFSI model, but the lack of tabulated fission barriers in the neutron-deficient region, especially in the lead region, prohibits drawing definite conclusions.\\
Moreover, Table \ref{tab:FitResults} shows that the four fitted values for $X$ are similar to each other as well as to the theoretical estimate $X \approx 4$\,$\mathrm{MeV}^{-1}$ \cite{NRV}. The extracted values for the offset $C'$ are also found to be comparable.\\
\begin{table}
\caption{Results of the fits, corresponding to four different fission models, shown in Figure \ref{fig:Tbdf}. The values for the parameters $X$ and $C'$ in equation (\ref{eq:l_def}) are listed. Also the root-mean-square deviations (RMSD) of the reliable experimental $\mathrm{log_{10}}(T_{\rm 1/2p,\beta DF})$ values (represented by the closed symbols in Figure \ref{fig:Tbdf}) to the fit are given.}
\label{tab:FitResults}
\begin{ruledtabular}
\begin{tabular}{cccc}
Model & X ($\mathrm{MeV}^{-1}$) & C' (MeV) & RMSD\\
\hline
TF & 3.0(2) & 6.2(1) & 0.47\\
FRLDM & 1.7(4) & 4.9(3) & 1.19\\
ETFSI & 2.1(7) & 5.0(6) & 1.10\\
LDM & 2.2(2) & 5.8(2) & 0.62\\
\end{tabular}
\end{ruledtabular}
\end{table}
In contrast to a rather good agreement for most neutron-deficient nuclei, all models show a larger systematical deviation from this linear trend for the neutron-rich $\beta$DF precursors $^{228}\mathrm{Ac}$ and $^{234,236}\mathrm{Pa}$. In \cite{Andreyev2013a}, concerns were raised on the accuracy of the $P_{\beta \mathrm{DF}}$ values measured in this region, which could explain this deviation. Note also that the precursors in this region of the nuclear chart undergo $\beta^{-}$ decay in contrast to the EC-delayed fission on the neutron-deficient side for which equation (\ref{eq:l_def}) was deduced, influencing the numeric value of the offset $C'$. In particular, the Fermi function for $\beta^{-}$ decay is approximately proportional to $(Q_{\beta}-E)^5$ \cite{Hall1992,Kuznetsov1999}, in contrast to the quadratic dependence on $(Q_{\beta}-E)$ for EC decay. The parameter $X$ should however remain unchanged, because equation (\ref{ratioG}) approximating $\Gamma_{\rm f}/\Gamma_{\rm tot}$ remains valid as long as the neutron-separation energy $S_{\rm n}$ is larger than $Q_{\beta}$. Since, at excitation energies higher than $S_{\rm n}$, de-excitation through neutron emission is favored over decay by $\gamma$-ray emission, thus implying $\Gamma_{\text{tot}} \simeq \Gamma_{\rm n} \gg \Gamma_{\gamma},\Gamma_{\rm f}$ \cite{Bohr1939,NRV}. For all nuclei mentioned in Table \ref{tab:Tbdf} however, $Q_{\beta}$ is below $S_{\rm n}$. An approximation of $T_{1/2p,\beta\mathrm{DF}}$, similar to equation (\ref{eq:l_def}), can thus also be derived for neutron-rich $\beta$DF precursors by taking into account above considerations. However, considering the limited experimental information on $\beta$DF in the neutron-rich region, a detailed derivation is omitted in this paper.
\section{Conclusions}
Recent experiments have measured the $\beta$DF of 9 precursor nuclei in the neutron-deficient lead region. Because of the dominant $\alpha$-decay branch in most of these nuclei, $\beta$DF probabilities are extracted with large experimental uncertainties. In contrast, the partial half-life for $\beta$DF can be determined with a better accuracy. In addition, $T_{\rm 1/2p,\beta DF}$ can be easily derived from the $\beta$DF probability by using equation (\ref{eq:PbdfvsTbdf}).\\
A systematical evaluation of $\beta$DF partial half-lives was performed by using fission barriers deduced from four different models for a broad range of nuclei in the lead and uranium regions. A linear relation between $\mathrm{log_{10}}(T_{\rm 1/2p,\beta DF})$ and ($Q_{\beta}-B_{\rm f}$) was observed for neutron-deficient precursor nuclei, when using tabulated fission barriers from the TF or LDM approach, and to a lesser extent for FRLDM and ETFSI barriers. This linear trend persists for values of $T_{\rm 1/2p,\beta DF}$ spanning over 7 orders of magnitude and a wide variety of precursor nuclei going from $^{178}\mathrm{Tl}$ to $^{248}\mathrm{Es}$ with $N$/$Z$ ratios of 1.20 and 1.51 respectively. This observation may help to assess $\beta$DF branching-ratios in very neutron-rich nuclei, which are inaccessible by present experimental techniques but might play a role in the fission-recycling mechanism of the r-process nucleosynthesis.\\
\\
This work has been funded by FWO-Vlaanderen (Belgium), by the Slovak Research and Development Agency (Contract No. APVV-0105-10), by the UK Science and Technology Facilities Council (STFC), by the Slovak grant agency VEGA (contract No. 1/0576/13), by the Reimei Foundation of JAEA, and by the European Commission within the Seventh Framework Programme through I3-ENSAR (Contract No. RII3-CT-2010-262010).
\FloatBarrier
|
{
"timestamp": "2015-04-15T02:06:01",
"yymm": "1504",
"arxiv_id": "1504.03443",
"language": "en",
"url": "https://arxiv.org/abs/1504.03443"
}
|
\section{Introduction}
We study a generalization of the discrete Toda lattice parametrized by a triple of integers $(n,m,k)$,
which corresponds to a network on a torus with
$n$ horizontal wires, $m$ vertical wires and $k$ {\em shifts} at the horizontal boundary. An example of the kind of toric network that we consider is given in Figure \ref{fig:toda3}.
\begin{figure}[h!]
\begin{center}
\scalebox{0.8}{\input{toda3.pstex_t}}
\end{center}
\caption{The toric network for $n=3$, $m=4$ and $k=0$.}
\label{fig:toda3}
\end{figure}
The phase space $\mM$ of our system is the space of parameters $q_{ij} \in \C$ assigned to each of the crossings of the wires (a $3 \times 4 = 12$ dimensional space for the example in Figure \ref{fig:toda3}).
We construct two families of commuting discrete time evolutions acting on the parameters $q_{ij}$, together generating an action of $\Z^m \times \Z^N$, where $N := \gcd(n,k)$. These time evolutions act as birational transformations of the phase space. The purpose of this paper is to study the algebro-geometrical structure of these maps and to solve the corresponding initial value problem.
\medskip
The rational transformations of the phase space come from the {\it {affine geometric $R$-matrix}}, acting on the parameters of two adjacent parallel wires, either horizontal or vertical. The affine geometric $R$-matrix arises in the theory of affine geometric crystals \cite{BK, KNO1,KNO2}, being a birational lift of the {\it {combinatorial $R$-matrix}} of certain tensor products of Kirillov-Reshetikhin crystals for $U_q({\widehat{\mathfrak{sl}}_n})$ \cite{KKMMNN}. Geometric $R$-matrices also arises independently in the study of Painlev\'e equations \cite{KNY} and total positivity \cite{LP}; see also \cite{Ki,Et}.
The geometric $R$-matrix satisfies the Yang-Baxter relation and we show in Theorem \ref{thm:dynamics} that the action of the geometric $R$-matrix generates commuting birational actions of two affine Weyl groups $W$ and $\tW$: one swapping horizontal wires, and one swapping vertical wires. The commutativity was proved in \cite{KNY} for the case $k=0$; we extend it here to arbitrary $k$.
The $\Z^m \times \Z^N$ time-evolutions of our dynamical system come from the subgroup of translation elements in the corresponding affine Weyl group. In the case $(n,m,k) = (n,2,n-1)$, a rational map given by one of the $\Z^{m=2}$ actions corresponds to the discretization of the well-known $n$-periodic Toda lattice equation, studied in \cite{HTI}.
The dynamics that we study come from positive birational maps, and they tropicalize to the box-ball system \cite{TS} and the combinatorics of jeu-de-taquin (see \S \ref{sec:tab}).
\medskip
To study the dynamics of our generalized discrete Toda lattice, we construct a spectral map
\begin{align*}
\phi:\mM &\longrightarrow \{\text{plane algebraic curve}\} \times \Pic^g(C_f) \times \mathcal{S}_f \times R_O \times R_A \\
(q_{ij}) &\longmapsto (C_f, \D, (c_1,\ldots,c_M), O , A)
\end{align*}
where $C_f$ is a spectral curve, $\D \in \Pic^g(C_f)$ is degree $g$ divisor, and the remaining data is explained in \S \ref{sec:eigenvector}. We show that when appropriately restricted, the spectral map is an injection. Such spectral data is frequently encountered in the theory of integrable systems, and our approach follows that of van Moerbeke and Mumford \cite{vMM}. Van Moerbeke and Mumford used similar spectral data to study periodic difference operators.
The heart of our work is the calculation of the double affine geometric $R$-matrix action in terms of the spectral data: we show that the translation subgroup $\Z^m \times \Z^N$ acts as constant motions on the Jacobian of $C_f$, while the symmetric subgroups of $W$ and of $\tW$ act by permuting the additional data $R_O$ and $R_A$ (which are certain special points on $C_f$).
We explicitly invert (for $N = 1$) the spectral map $\phi$ using Riemann theta functions, and give a solution to the initial value problem. This extends work of Iwao \cite{Iwao07,Iwao10}, who studied the initial value problem of the $\Z^m$ action in the $(n,m,n-1)$ and $(n,m,0)$ cases. We relate our theta function solutions to the octahedron recurrence \cite{Spe} via Fay's trisecant identity\cite{Fay73}. We also give an interpretation of our dynamics in terms of dimer model transformations \cite{GonchaKenyon13}.
\medskip
\noindent {\bf Outline.}
The outline of this paper is as follows:
in \S \ref{sec:netw}, we introduce a family of commuting actions
on the toric network, generalizating the
discrete Toda lattice.
We briefly summarize the main results of this paper in \S \ref{subsec:main}, and give some examples and applications in \S \ref{subsec:example}.
In \S \ref{sec:Lax}, we introduce a space $\mL$ of Lax matrices for our system, and construct the spectral curves $C_f$, where $f \in \C[x,y]$. We also study certain special points on our spectral curves, and analyze some of the singular points.
In \S \ref{sec:eigenvector}, we study the spectral map which sends a point in the phase space to the spectral curve $C_f$, a divisor $\D$ on $C_f$, and some additional data. We modify the strategy in \cite{vMM} to fit our situation. In \S \ref{sec:proofLax} and \S \ref{sec:proofeigenvector}
we present the proofs of results in \S \ref{sec:Lax} and
\S \ref{sec:eigenvector} respectively. In particular, the coefficients of the spectral curves (which are integrals of motion) are described explicitly in terms of the combinatorics of these networks.
In \S \ref{sec:actions}, we study the vertical $\Z^m$ actions, the horizontal $\Z^N$ actions, and the snake path actions.
We show that all the actions preserve the spectral curve,
and the commuting $m+N$ actions induce constant motions on
the Picard group of $C_f$, through the map $\phi$.
In \S \ref{sec:theta}, we solve, for the case of $N=1$, the initial value problem
for the commuting actions by constructing the inverse of $\phi$
in terms of the Riemann theta function.
Our method extends the strategy in \cite{Iwao10}. For $N > 1$, a solution is given that relies on a technical condition.
In \S \ref{sec:fay}, we show that the theta function solution for the system satisfies the octahedron recurrence,
by specializing Fay's trisecant identity for the Riemann theta function. In \S \ref{sec:transpose}, we study the symmetry between
the network and its transposition obtained by swapping the roles of the
vertical and the horizontal wires.
In \S \ref{sec:cluster}, we realize the $R$-matrix transformation
on a toric network in terms of transformations of the honeycomb dimer model on a torus~\cite{GonchaKenyon13}. An explicit interpretation of the $R$-matrix as a cluster transformation will be the subject of future work \cite{ILP}, so we do not elaborate on the relation to cluster structures here.
\medskip
\noindent
{\bf Future directions.}
There are several systematic ways to construct integrable rational maps
using combinatorial objects on surfaces, such as directed networks, electrical networks, and bipartite graphs~\cite{LP,GonchaKenyon13,GSTV}. In particular, the $R$-matrix of the present work has an ``electrical" analogue \cite{LPElec}. It would be interesting to study these discrete-time dynamical systems from the view point of algebraic geometry (Cf.~\cite{FockMarsha14}).
In all these cases, the rational transformations are additionally positive, and can be tropicalized to piecewise-linear maps that generate a discrete-time and discrete-space dynamical system. It is natural to consider the tropical counterpart of our work using tropical geometry as in\cite{InoueTakenawa}, where
tropical curves and tropical theta functions are used to study
the piecewise-linear map arising from the $(n,m,k) = (n,2,n-1)$ case.
\medskip
\noindent
{\bf Acknowledgments.} We thank Rick Kenyon for helpful discussions.
We also thank the anonymous referee for kind comments which improved
the manuscript.
\section{Dynamical system from a toric network}
\label{sec:netw}
We study the action of the geometric $R$-matrix on an array of variables. The geometric $R$-matrix generates an action of a product of commuting (extended) affine symmetric groups, acting as birational transformations.
\subsection{The affine geometric $R$-matrix} \label{sec:Rmatrix}
For a vector $\aa = (a_1,\ldots,a_n)$, let $\aa^{(i)}:= (a_{i+1},a_{i+2},\ldots,a_n,a_1,\ldots,a_{i})$.
Let $\aa = (a_1,\ldots,a_n)$ and $\bb= (b_1,\ldots,b_n)$ be two vectors. Define the {\it {energy}} $E(\aa,\bb)$ to be
$$E(\aa,\bb) = \sum_{i=0}^{n-1} \left(\prod_{j=1}^i b_{j} \prod_{j=i+2}^{n} a_{j} \right).$$
Define the affine geometric $R$-matrix to be the transformation $R:(\aa,\bb) \mapsto (\bb',\aa')$, given by
$$b_{i}' = b_{i}\frac{E(\aa^{(i)}, \bb^{(i)})}{E(\aa^{(i-1)}, \bb^{(i-1)})} \qquad \text{and} \qquad
a_{i}' = a_{i}\frac{E(\aa^{(i-1)}, \bb^{(i-1)})}{E(\aa^{(i)}, \bb^{(i)})}.$$
We will usually think of $R$ as a birational transformation of $\C^n \times \C^n$.
It is easy to see
\begin{align}\label{eq:q1q2}
\prod_{i=1}^n a_{i} = \prod_{i=1}^n a_{i}'
\qquad \text{and} \qquad
\prod_{i=1}^n b_{i} = \prod_{i=1}^n b_{i}'.
\end{align}
\subsection{Description of dynamics}\label{sec:dynamics}
The $R$-matrix can be used to act on a rectangular array of variables, by acting on consecutive pairs of rows or of columns.
Let $\a,\b,\c$ be three positive integers, and let $0 \leq \d <\c$ be a nonnegative integer satisfying $\gcd(\c,\d)=1$. Note that we allow $\d=0$ if and only if $\c=1$.
Let $\d^{-1}$ be the unique number in the range $0 \leq \d^{-1} <\c$ such that $\d\d^{-1}=1 \mod \c$. For a positive integer $a$, we define $[a] := \{1,2,\ldots,a\}$.
We consider an array of $\a \times \b \times \c$ variables $\{q_{a,b,c}\}_{a \in [\a], b \in [\b], c \in [\c]}$. We consider two distinct ways to arrange them in a two-dimensional rectangular array:
\begin{align}\label{eq:q-barq}
q_{i,j} = q_{a,b,c} \text { for } i=a, \; j=b+\b(c-1)
\end{align}
and
\begin{align}\label{eq:q-tildeq}
\tilde q_{i,j} = q_{a,b,c} \text { for } i=\b+1-b, \; j=\a \d^{-1} c - a +1 \mod \a\c.
\end{align}
By convention, $q_{i,j}$ denotes the entry in the {\it $i$-th column} and {\it $j$-th row}.
We call these arrays $Q$
and $\tilde Q$. The first array $Q$ has dimensions $\b\c \times \a$ and the second array $\tilde Q$ has dimensions $\a\c \times \b$.
Note that changing $\a \mapsto \b$, $\d \mapsto \d^{-1}$, $q_{a,b,c} \mapsto q_{1-b,1-a,\d^{-1}c}$ swaps $Q$ and $\tilde Q$.
\begin{example}\label{ex:2221}
Let $(\a,\b,\c,\d)= (2,2,2,1)$. The two arrays are as follows.
\begin{align}
Q := \left(\begin{array}{cc}
q_{1,1,1} & q_{2,1,1} \\
q_{1,2,1} & q_{2,2,1} \\
q_{1,1,2} & q_{2,1,2}\\
q_{1,2,2} & q_{2,2,2}
\end{array} \right),
\qquad
\tilde Q := \left(\begin{array}{cc}
q_{2,2,1} & q_{2,1,1}\\
q_{1,2,1} & q_{1,1,1}\\
q_{2,2,2} & q_{2,1,2}\\
q_{1,2,2} & q_{1,1,2}
\end{array} \right).
\end{align}
\end{example}
\begin{example}\label{ex:3232}
Let $(\a,\b,\c,\d)= (3,2,3,2)$. The two arrays are as follows.
\begin{align}
Q := \left(\begin{array}{ccc}
q_{1,1,1} & q_{2,1,1} & q_{3,1,1}\\
q_{1,2,1} & q_{2,2,1} & q_{3,2,1}\\
q_{1,1,2} & q_{2,1,2} & q_{3,1,2}\\
q_{1,2,2} & q_{2,2,2} & q_{3,2,2}\\
q_{1,1,3} & q_{2,1,3} & q_{3,1,3}\\
q_{1,2,3} & q_{2,2,3} & q_{3,2,3}
\end{array} \right),
\qquad
\tilde Q := \left(\begin{array}{cc}
q_{3,2,1} & q_{3,1,1}\\
q_{2,2,1} & q_{2,1,1}\\
q_{1,2,1} & q_{1,1,1}\\
q_{3,2,3} & q_{3,1,3}\\
q_{2,2,3} & q_{2,1,3}\\
q_{1,2,3} & q_{1,1,3}\\
q_{3,2,2} & q_{3,1,2}\\
q_{2,2,2} & q_{2,1,2}\\
q_{1,2,2} & q_{1,1,2}
\end{array} \right).
\end{align}
\end{example}
The arrays $Q$ and $\tilde Q$ correspond to a network on a torus in
the following way.
Let $G$ a network embedded into a torus, as illustrated in Figure~\ref{fig:networkG}.
Each rectangle of red lines denotes the fundamental domain of the torus.
There are $\a$ vertical ``wires" (directed upwards), forming $\a$ simple curves in the torus with the same homology class, and $\b \c$ horizontal wires (directed to the right), which form $\b$ simple curves on the torus with another homology class: the top $\b \d$ right ends of horizontal wires cross the top edge and come out on the other side.
We set $q_{ij}$ at the crossing of the $i$-th vertical and the $j$-th horizontal wires, then $Q$ corresponds to the configuration of the $q_{ij}$ on $G$.
When we see $G$ from the inside of the torus, the roles of the
vertical and horizontal wires are interchanged;
there are $\b$ vertical wires forming $\b$ simple curves, and
$\a \c$ horizontal wires forming $\a$ simple curves.
The top $\a \d^{-1}$ ends of horizontal wires cross the top edge
and come out on the other side.
The array $\tilde Q$ corresponds to this ``from-inside" configuration,
where $\tilde q_{ij}$ is placed at
the crossing of the $i$-th vertical and the $j$-th horizontal wires.
See Figure~\ref{fig:3232} for the network in the case of Example~\ref{ex:3232},
where the lines with the same color in the two pictures are identical
simple curves in $G$.
\begin{figure}[h]
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\put(11,81){\scriptsize $1$}
\put(19,81){\scriptsize $2$}
\put(27,81){\scriptsize $\cdots$}
\put(35,81){\scriptsize $\a$}
\put(11,25){\scriptsize $1$}
\put(19,25){\scriptsize $2$}
\put(27,25){\scriptsize $\cdots$}
\put(35,25){\scriptsize $\a$}
\put(43,49){\scriptsize $1$}
\put(51,49){\scriptsize $2$}
\put(59,49){\scriptsize $\cdots$}
\put(67,49){\scriptsize $\a$}
\put(5,78){\scriptsize $1$}
\put(5,70){\scriptsize $2$}
\put(5,62){\scriptsize $\vdots$}
\put(4,54){\scriptsize $\b \d$}
\put(5,46){\scriptsize $\b \d+1$}
\put(5,38){\scriptsize $\vdots$}
\put(4,30){\scriptsize $\b \c$}
\put(5,22){\scriptsize $1$}
\put(5,14){\scriptsize $2$}
\put(5,6){\scriptsize $\vdots$}
\put(4,-2){\scriptsize $\b \d$}
\put(37,78){\scriptsize $1+\b(\c-\d)$}
\put(37,70){\scriptsize $2+\b(\c-\d) $}
\put(37,62){\scriptsize $\vdots$}
\put(36,54){\scriptsize $\b \c$}
\put(37,46){\scriptsize $1$}
\put(37,38){\scriptsize $\vdots$}
\put(36,30){\scriptsize $\b (\c-\d)$}
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\multiput(5,80)(32,0){3}{\line(0,-1){88}}
}
\end{picture}
\caption{Toric network}
\label{fig:networkG}
\end{figure}
\begin{figure}[h]
\unitlength=0.9mm
\begin{picture}(100,85)(0,5)
\put(-8,80){$Q:$}
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\multiput(11,81)(0,-48){2}{\scriptsize $1$}
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\put(29,70){\scriptsize $4$}
\put(29,62){\scriptsize $5$}
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\put(50,80){$\tilde Q:$}
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\put(63,22){\scriptsize $8$}
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}
\end{picture}
\caption{Network for the case of $(3,2,3,2)$}
\label{fig:3232}
\end{figure}
\begin{definition}\label{def:action}
Let $A = \{a_{i,j}\}_{i \in [s], j \in [r]}$ be an $r \times s$ array.
Let $\aa_i = (a_{i,j})_{j \in [r]}$,
and write $\aa_i^T$ for the transpose of $\aa_i$.
For $1 \leq \ell \leq s-1$, define the array $s_\ell(A)$ by
\begin{align}\label{eq:s-ell}
s_{\ell}(A) = \left(\aa_1^T, \ldots, \aa_{\ell-1}^T, (\aa'_{\ell+1})^T,
(\aa'_{\ell})^T,\aa_{\ell+2}^T, \ldots, \aa_s^T \right)
\end{align}
if $A = (\aa_1^T, \ldots, \aa_{\ell-1}^T, \aa_{\ell}^T, \aa_{\ell+1}^T, \aa_{\ell+2}^T, \ldots, \aa_s^T)$,
and $R(\aa_\ell,\aa_{\ell+1}) = (\aa'_{\ell+1},\a'_{\ell})$.
Also define an array $\pi(A)$ by the formula
\begin{align}\label{eq:pi}
\pi(A)_{i,j} = a_{i,j-1}.
\end{align}
Here we consider the first index modulo $r$ and the second index modulo $s$.
\end{definition}
Let $\mM \simeq \C^{\a \b \c}$ be the phase space of our dynamical system
and regard $(q_{a,b,c})_{a \in [\a], b \in [\b], c \in [\c]}$ as
coordinates on $\mM$. Applying the above definition to the array $Q$, we obtain operators $s_{1}, \ldots, s_{\a-1}$ and $\tilde \pi$ acting on $\mM$.
Similarly, applying this definition to the array $\tilde Q$, we obtain operators $\tilde s_{1}, \ldots, \tilde s_{\b-1}$ and $\pi$ acting on $\mM$. We emphasize that \eqref{eq:pi} for $Q$ gives the operation $\tilde \pi$, while $\eqref{eq:pi}$ for $\tilde Q$ gives the operation $\pi$.
The following result generalizes a result of Kajiwara, Noumi and Yamada \cite{KNY}.
\begin{thm}\label{thm:dynamics}
The operators $s_{\ell}$ and $\pi$ generate an action of an extended affine symmetric group $W = (\Z/\a\c\Z) \ltimes \widehat{\mathfrak S}_{\a}$ on $\mM$. Here $\Z/\a\c\Z$ is the cyclic group of order $\a \c$ and $\widehat{\mathfrak S}_{\a}$ is the affine symmetric group (the Coxeter group) of type $\tilde A_{\a-1}$. Similarly, the operators $\tilde s_{\ell}$ and $\pi$ form an action of an extended affine symmetric group $\tilde W = (\Z/\b\c\Z) \ltimes \widehat{\mathfrak S}_{\b}$ on $\mM$, where $(\Z/\b\c\Z)$ is the cyclic group of order $\b \c$. Furthermore, these two actions commute.
Precisely, on $\mM$ the following relations hold.
\begin{align*}
& s_{\ell} s_{\ell+1} s_{\ell}
= s_{\ell+1} s_{\ell} s_{\ell+1},
\qquad
s_{j} s_{\ell} = s_{\ell} s_{j}
\quad (|j-\ell| > 1), \qquad s_\ell^2 = 1,
\\
& \pi s_{\ell+1} = s_{\ell} \pi, \qquad
\pi^{\a \c} = 1,
\end{align*}
\begin{align*}
&\tilde s_{\ell} \tilde s_{\ell+1} \tilde s_{\ell}
=\tilde s_{\ell+1} \tilde s_{\ell} \tilde s_{\ell+1},
\qquad
\tilde s_{j} \tilde s_{\ell} = \tilde s_{\ell} \tilde s_{j}
\quad (|j-\ell| > 1), \qquad \tilde s_\ell^2 = 1,
\\
& \tilde \pi \tilde s_{\ell+1} = \tilde s_{\ell} \tilde \pi, \qquad
{\tilde \pi}^{\b \c} = 1,
\end{align*}
\begin{align*}
& s_{j} \tilde s_{\ell} = \tilde s_{\ell} s_{j},
\qquad
\tilde \pi s_{\ell} = s_{\ell} \tilde \pi,
\qquad
\pi \tilde s_{\ell} = \tilde s_{\ell} \pi,
\qquad
\tilde \pi \pi = \pi \tilde \pi.
\end{align*}
In the above formulae, the index of $s_\ell$ is taken modulo $\a$, and $s_0$ is defined by the equation $ \pi s_{1} = s_{0} \pi$. Similarly, the index of $\tilde s_\ell$ is taken modulo $\b$, and $\tilde s_0$ is defined by the equation $\tilde \pi \tilde s_{1} = \tilde s_{0} \tilde \pi$.
\end{thm}
The proof of Theorem \ref{thm:dynamics} is delayed to \S \ref{sec:dynamics_proof}.
\begin{example}
In Example~\ref{ex:2221} we have
$$ s_1(Q)_{1,1,1}= q_{2,1,1} \frac{q_{1,1,2}q_{1,2,2}q_{1,1,1} + q_{2,2,1}q_{1,2,2}q_{1,1,1} +q_{2,2,1}q_{2,1,2}q_{1,1,1} +q_{2,2,1}q_{2,1,2}q_{2,2,2}}
{q_{1,2,1}q_{1,1,2}q_{1,2,2} + q_{2,1,1}q_{1,1,2}q_{1,2,2} +q_{2,1,1}q_{2,2,1}q_{1,2,2} +
q_{2,1,1}q_{2,2,1}q_{2,1,2}},$$
$$\tilde s_1(\tilde Q)_{1,1,1}= q_{1,2,1} \frac{q_{1,1,1}q_{2,1,2}q_{1,1,2}+q_{1,1,1}q_{2,1,2}q_{2,2,1} + q_{1,1,1}q_{1,2,2}q_{2,2,1} +
q_{2,2,2}q_{1,2,2}q_{2,2,1}}
{q_{2,1,2}q_{1,1,2}q_{2,1,1}+q_{2,1,2}q_{1,1,2}q_{1,2,1} + q_{2,1,2}q_{2,2,1}q_{1,2,1} +
q_{1,2,2}q_{2,2,1}q_{1,2,1}},$$
$$ \tilde \pi(Q)_{1,1,1} = q_{1,2,1}, \qquad \pi(\tilde Q)_{1,1,1} = q_{2,1,2}.$$
We leave it for the reader to verify that $s_1$ and $\pi$ commute with $\tilde s_1$ and $\tilde \pi$, as an easy computational exercise.
\end{example}
We can present the extended affine symmetric groups $W$ and $\tilde W$ as follows. The finite symmetric group $\mS_{\a}$ acts naturally on the lattice $\Z^{\a}$, fixing the subgroup generated by the vector $(1,1,\ldots,1)$. Thus we have an action of $\mS_\a$ on the quotient $\Z^{\a}/\Z(\c,\c,\ldots,\c)$. Then we have that $W = \mS_{\a} \ltimes (\Z^{\a}/\Z(\c,\c,\ldots,\c))$. The commutative normal subgroup $\Z^{\a}/\Z(\c,\c,\ldots,\c)$ is generated by $e_u$ for $1 \leq u \leq \a$:
\begin{align}\label{eq:Za-action}
e_u = (s_u \cdots s_{\a-1})(s_{u-1} \cdots s_{\a-2})
\cdots (s_1 \cdots s_{\a-u}) {\pi}^u.
\end{align}
The element $e_u$ is identified with the vector $\be_u = (1,\ldots,1,0,\ldots,0) \in \Z^a$ with $u$ $1$-s. Note that $e_\a = \tilde \pi^\a$ satisfies $e_\a^\c = 1$ agreeing with the fact that $(\c,\c,\ldots,\c) = 0$ in $\Z^{\a}/\Z(\c,\c,\ldots,\c)$.
Similarly, we have $\tilde W = \mS_\b\ltimes (\Z^{\b}/\Z(\c,\c,\ldots,\c))$,
where the commutative normal subgroup $\Z^{\b}/\Z(\c,\c,\ldots,\c)$ is generated by
$\tilde e_u$ for $1 \leq u \leq \b$:
\begin{align}\label{eq:Zm-action}
\tilde e_u = (\tilde s_u \cdots \tilde s_{\b-1})
(\tilde s_{u-1} \cdots \tilde s_{\b-2})
\cdots (\tilde s_1 \cdots \tilde s_{\b-u}) {\tilde \pi}^u.
\end{align}
Now, the operators $e_u$ and $\tilde e_u$ all commute, and thus give an action of $\Z^{\a}/\Z(\c,\c,\ldots,\c) \times \Z^{\b}/\Z(\c,\c,\ldots,\c)$ on $\mM$! We will think of this as a discrete dynamical system with $\a + \b$ different time evolutions. We shall find a complete set of integrals of motion for this system, and study the initial value problem.
\subsection{Change of indexing}
Instead of the quadruple $(\a,\b,\c,\d)$, we shall also index our systems with triples $(n,m,k)$, given by
$$n=\b \c, \qquad m=\a, \qquad k = \b \d.$$
We shall also set
$$
N := \gcd(n,k) = \b, \qquad n^\prime := n/N = \c, \qquad k^\prime := k / N = \d, \qquad M:= \gcd(n,m+k).$$
The quadruple $(\a,\b,\c,\d)$ can be recovered via
$$\a=m, \qquad \b = \gcd(n,k), \qquad \c=n/\gcd(n,k), \qquad \d = k/\gcd(n,k).$$The involution $Q \longmapsto \tilde Q$ is associated with the following changes of indices:
$$(\a,\b,\c,\d) \longmapsto (\b,\a,\c,\d^{-1}),$$
$$(n,m,k) \longmapsto (m n^\prime,N, \bar k^\prime m),$$
where $\bar k' := (k^\prime)^{-1}$ is taken modulo $n^\prime$.
Through \eqref{eq:q-barq} or \eqref{eq:q-tildeq}
we identify $q = (q_{a,b,c}) \in \mM$ with
$Q = (q_{i,j}) \in \mathrm{Mat}_{n,m}(\C)$ or
$\tilde{Q} = (\tilde q_{i,j}) \in \mathrm{Mat}_{m n^\prime,N}(\C)$.
Correspondingly, the network $G$ has $m$ vertical wires (directed upwards)
which form $m$ simple curves on the torus with the same homology class, and
$n$ horizontal wires (directed to the right) which form $N$
simple curves on the torus with another homology class.
\subsection{Main results}\label{subsec:main}
Let $\mathcal{M} \simeq \C^{\a \b \c} = \C^{mn}$ be the phase space
where the ring $\mathcal{O}(\mM)$ of regular functions on $\mM$ is
generated by $q_{i,j} ~(i \in [m], j \in [n])$. In \S\ref{sec:Lax}, we shall define a map $\psi: \mathcal{M} \to \C[x,y]$. Every coefficient of $f(x,y) = \psi(q)$ is a regular function on $\mM$. We then have (see Corollary \ref{cor:L-WW}) the following result.
\begin{statement}
The actions of the affine symmetric groups $W$ and $\tilde W$ on $\mathcal{M}$ preserve
each fiber $\psi^{-1}(f)$ for $f \in \psi(\mM)$. In particular, the coefficients of $f(x,y)$ are integrals of motion of the commuting $\Z^m$ and $\Z^N$ actions.
\end{statement}
In \S \ref{sec:spectral}, we give a combinatorial description of every coefficient of $f$, as generating functions of path families on a network on the torus.
Now fix a generic $f \in \psi(\mM)$ such that the affine plane curve
$\{(x,y) \mid f(x,y)=0\} \subset \C^2$ is smooth (except for $(0,0)$),
and let $C_f$ be the smooth completion of the affine curve.
We shall call $C_f$ the {\it spectral curve}. In \S\ref{sec:Lax}, we define distinguished {\em special points}, $P$, $A_u ~(u \in [m])$ and $O_u~(u \in [N])$ on $C_f$. We apply the results and techniques of van Moerbeke and Mumford \cite{vMM} to establish the following; see Theorem \ref{thm:phi}.
\begin{statement}
Fix a generic $f \in \psi(\mM)$. There is an injection
$$
\phi : \psi^{-1}(f) \hookrightarrow \Pic^{g}(C_f) \times \mathcal{S}_f \times R_O \times R_A,
$$
where
\begin{enumerate}
\item
$g$ is the genus of $C_f$, and $\Pic^{g}(C_f)$ is the Picard group of $C_f$,
of degree $g$;
\item
$\mathcal{S}_f \simeq (\C^\ast)^{M-1} \subset (\C^\ast)^M$;
\item
$R_O$ is a finite set of cardinality $N!$, identified with the $N!$ orderings of $\{O_1,O_2,\ldots,O_N\}$; and
\item
$R_A$ is a finite set of cardinality $m!$, identified with the $m!$ orderings of $\{A_1,A_2,\ldots,A_m\}$.
\end{enumerate}
\end{statement}
In fact, our Theorem \ref{thm:phi} identifies the image of $\phi$.
Define the divisors $\mathcal{A}_u := uP -\sum_{i=1}^u A_i$ and $\mathcal{O}_u := uP -\sum_{j=N-u+1}^N O_j$.
The commuting time evolutions $e_u$ \eqref{eq:Za-action}
and $\tilde e_u$ \eqref{eq:Zm-action} can be described as follows
(Theorem \ref{thm:finite} and Theorem \ref{thm:commuting-actions}).
\begin{statement}
Suppose $\phi(q) = ([\mathcal{D}], (c_1,\ldots,c_M),O,A)$. Then we have
\begin{align*}
&\phi(e_u (q)) = ([\mathcal{D} - \mathcal{A}_u],
(c_{u+1},\ldots,c_M,c_1,\ldots,c_u),O,A) \quad \text{for $u=1,\ldots,m$},
\\
&\phi(\tilde e_u (q)) = ([\mathcal{D} + \mathcal{O}_u],
(c_{M-u+1},\ldots,c_M,c_1,\ldots,c_{M-u}),O,A) \quad \text{for $u=1,\ldots,N$}.
\end{align*}
Furthermore, the finite symmetric subgroups $\mS_N \subset \tilde W$ and $\mS_m \subset W$ act naturally on $R_O$ and $R_A$ respectively and do not affect the rest of the spectral data.
\end{statement}
In other words, the time evolutions $e_u$ and $\tilde e_u$ are linearized on $\Pic^{g}(C_f)$. Our approach to Theorem \ref{thm:commuting-actions} is similar to that of Iwao \cite{Iwao07}.
When $N = 1$, we give in Theorem \ref{thm:N=1} an explicit formula for the inverse to the map $\phi$, which also explicitly solves the initial value problem for our dynamics. For $t \in \Z^m$, let $(q^t_{j,i})$ denote the point in the phase space after time evolution in the direction $t$.
\begin{statement}
When $N=1$, we have a formula
$$
q^t_{j,i}
=
C \,
\frac{\theta_{i}^{t+\be_{j}}(P) \, \theta_{i-1}^{t+\be_{j-1}}(P)}
{\theta_{i}^{t+\be_{j-1}}(P) \, \theta_{i-1}^{t+\be_{j}}(P)},
$$
where $\theta_i^t(P)$ is a particular value of the Riemann theta function, and $C$ is a constant depending only on $(c_1,\ldots,c_M)$, $O$, and $A$.
\end{statement}
\subsection{Examples}
\label{subsec:example}
\subsubsection{Discrete Toda lattice}
Let $(\a, \b, \c, \d)=(2,1,n,n-1)$ (i.e. $(n,m,k) = (n,2,n-1)$).
We set $q_{i,j} = q_{a,1,c}$ for $i=a, ~j=c$, and
regard $q := (q_{i,j})_{i \in [2], j \in [n]}$ as a coordinate
of $\mM \simeq \C^{2n}$.
The resulting $\Z^2$-action on $\mM$
is generated by ${e}_1$ and ${e}_2$,
where $e_2$ simply acts on $\mM$ as
$ e_2 (q_{i,j}) = q_{i,j+1}$.
As for $e_1$, when we define $q^t := e_1^t(q)$ for $q \in \mM$,
the action of $e_1$ is rewritten as a system of
difference equations:
\begin{align}\label{eq:d-Toda}
\begin{cases}
q_{1,j}^{t+1} q_{2,j}^{t+1} = q_{2,j}^t q_{1,j+1}^t,
\\
q_{1,j+1}^{t+1} + q_{2,j}^{t+1} = q_{2,j+1}^t + q_{1,j+1}^t.
\end{cases}
\end{align}
This is the discretization of $n$-periodic Toda lattice equation
studied in \cite{HTI}.
Actually, we recover the original Toda lattice equation
$\frac{d^2}{dt^2}x_j = \mathrm{e}^{x_{j+1}-x_j} - \mathrm{e}^{x_{j}-x_{j-1}}$
by setting $q_{1,j}^t = 1 + \delta \frac{d}{dt}x_j$ and
$q_{2,j}^t = \delta^2 \mathrm{e}^{x_{j+1}-x_j}$,
and taking the limit $\delta \to 0$.
Here we set $q_{\ast,j}^t = q_{\ast,j}(\delta t)$ for $\ast = 1,2$
and $x_j = x_j(t)$.
A simple generalization of the discrete Toda lattice is
the case of $(\a, \b, \c, \d)=(m,1,n,n-1)$.
Its initial value problem was studied by Iwao \cite{Iwao07,Iwao10}
by applying \cite{vMM}.
He also studied the similar problem in the case of
$(\a, \b, \c, \d) = (m,1,n,0)$ in \cite{Iwao9}.
\subsubsection{Tropicalization and tableaux}\label{sec:tab}
Let $T_1$ and $T_2$ be two semistandard tableaux of rectangular shapes. Define $T_1 \otimes T_2$ to be the concatenation of the two into a skew tableaux by placing $T_2$ North-East of
$T_1$. Let $\jdt(T_1 \otimes T_2)$ denote the straight shape tableau obtained by performing jeu-de-taquin on $T_1 \otimes T_2$. The following result is well-known.
\begin{lem}
There is a unique pair of rectangular semistandard tableaux $T_1'$ and $T_2'$ such that $T_i'$ has the same shape as $T_i$ (for $i = 1,2$), and
$\jdt(T_1 \otimes T_2) = \jdt(T_2' \otimes T_1')$.
\end{lem}
\begin{example}
Suppose
$$
T_1 = \tableau[sY]{1&2\\3&3} \qquad \text{and} \qquad T_2 = \tableau[sY]{1&2&3}.$$
Then one has
$$
T_2' = \tableau[sY]{1&3&3}\qquad \text{and} \qquad T_1' = \tableau[sY]{1&2\\2&3}.$$
\end{example}
The transformation $R(T_1,T_2)=(T_2',T_1')$ is called the {\it {combinatorial $R$-matrix}}. It appears in the theory of crystal graphs as the isomorphism map between tensor products of Kirillov-Reshetikhin
crystals. The following property is well-known.
\begin{thm}
The combinatorial $R$-matrix is an involution. Furthermore, it satisfies the braid relation:
$$(R \otimes Id)(Id \otimes R)(R \otimes Id) = (Id \otimes R)(R \otimes Id)(Id \otimes R).$$
\end{thm}
Now, let $\{q_{a,b,c}\}_{a \in [\a], b \in [\b], c \in [\c]}$ be an array of nonnegative integers. Define $Q = (q_{i,j})$ and $\tilde Q = (\tilde q_{i,j})$ as before:
$$q_{i,j} = q_{a,b,c} \text { for } i=a, \; j=b+\b(c-1)$$ and $$\tilde q_{ij} = q_{a,b,c} \text { for } i=\b+1-b, \; j=\a \d^{-1} c - a +1.$$
We create single-row tableaux from $Q$ and $\tilde Q$ as follows.
For each $i \in [\a]$, let $T_i$ be the single-row tableau with $q_{i,j}$ $j$-s. For each $i \in [\b]$, let $\tilde T_i$ be the single-row tableau with $\tilde q_{i,j}$ $j$-s. We view $T = T_1 \otimes \dotsc \otimes T_\a$ and $\tilde T = \tilde T_1 \otimes \dotsc \otimes \tilde T_\b$ as tropical analogues of $Q$ and $\tilde Q$.
Let us define an action of $W$ on $q_{a,b,c}$ as follows. For $T = T_1 \otimes \dotsc \otimes T_{\a}$ and $1 \leq \ell \leq \a-1$ let
$$s_{\ell}(T) = T_1 \otimes \dotsc \otimes T_{\ell+1}' \otimes T_{\ell}' \otimes \dotsc \otimes T_{\a}.$$
That is, we apply the combinatorial $R$-matrix to the $\ell$-th and $(\ell+1)$-st factors. In addition, let $\tilde \pi$ act on $T$ by applying Schutzenberger's {\it {promotion}} operator \cite{Sch}
to each factor $T_i$. Similarly, we define $\tilde s_\ell$ and $\pi$ acting on $\tilde T$.
\begin{thm}
The operators $s_{\ell}$ and $\pi$ form an action of the extended affine symmetric group $W$ on $q_{a,b,c}$. Similarly,
operators $\tilde s_{\ell}$ and $\tilde \pi$ form an action of the extended affine symmetric group $\tilde W$ on $q_{a,b,c}$. These two actions commute.
These two operations are the tropicalizations of the rational actions of Theorem \ref{thm:dynamics}.
\end{thm}
Here tropicalization refers to the formal operation of substitution
$$\C \mapsto \Z, \qquad \times \mapsto +, \qquad \div \mapsto -, \qquad + \mapsto \min.$$
Under this substitution, the geometric $R$-matrix becomes a piecewise-linear involution of $\Z^n \times \Z^n$, which is the combinatorial $R$-matrix.
In other words, the $W$ dynamics we are considering in this paper is a birational lift of the dynamics of repeated application of the combinatorial $R$-matrix on a sequence of $\a$ single-row tableaux, arranged in a circle.
\begin{example}
Let $(\a,\b,\c,\d)=(2,2,2,1)$ as in Example \ref{ex:2221}. Take $T = 22222344 \otimes 1234$. Then $\tilde T = 122222344 \otimes 134$. We have
$$ s_1(T) = 2234 \otimes 12222344, s_1(\tilde T) = 111122334 \otimes 134;$$
$$\tilde \pi(T) = 11111233 \otimes 1234, \tilde \pi(\tilde T) = 123 \otimes 122222344;$$
$$\tilde s_1(T) = 11112334 \otimes 1233, \tilde s_1(\tilde T) = 124 \otimes 122223344;$$
$$\pi (T) = 1234 \otimes 22222344, \pi(\tilde T) = 111112334 \otimes 234.$$
\end{example}
\begin{remark}
The corresponding commuting crystal actions in the case $\c=1$ were considered by Lascoux in \cite{Las} and by Berenstein-Kazhdan in \cite{BK}.
\end{remark}
\subsubsection{Box-ball systems}
The box-ball system is an integrable cellular automaton introduced by Takahashi
and Satsuma \cite{TS}. It is described by an algorithm to move finitely many
balls in an infinite number of boxes aligned on a line,
where a consecutive array of occupied boxes is regarded as a {\it soliton}.
This system is related to both of the previous two examples;
the global movements of solitons are equivalent to
the tropicalization of the discrete Toda lattice \eqref{eq:d-Toda}.
The symmetry of the system is explained by the crystal base theory,
where the dynamics of bolls is induced by the action of the combinatorial $R$-matrix.
See \cite{IKT} for a comprehensive review of
the combinatorial and tropical aspects of the box-ball system.
\section{Lax matrix and spectral curve}
\label{sec:Lax}
\subsection{Lax matrix}
Fix integers $n,m,k$ such that $n \geq 2$, $m \geq 1$ and
$1 \leq k \leq n$. From now on we shall mainly use $q := (q_{ij})_{i \in [m], j \in [n]}$ as a coordinate
of the phase space $\mM$.
We identify $q \in \mM$ with an $m$-tuple of $n$ by $n$ matrices
$Q:=(Q_i(x))_{i=1,\ldots,m}$ with a spectral parameter $x$,
where
\begin{align}\label{eq:Q}
&Q_i(x) := \left(\begin{array}{cccc}
q_{i1} & 0 & 0 &x\\
1&q_{i2} &0 &0 \\
0&\ddots&\ddots&0 \\
0&0&1&q_{in}
\end{array} \right).
\end{align}
Let $\mL$ be the set of $n$ by $\infty$ scalar matrices
$A := (a_{ij})_{1 \leq i \leq n, \,j \in \Z}$ satisfying the following conditions:
\begin{align}
a_{ij} =
\begin{cases}
1 & j-i = -m-k \\
a_{ij} \in \C & -m-k+1 \leq j-i \leq -k \\
0 & \text{otherwise}.
\end{cases}
\end{align}
In particular, $A$ has finitely many nonzero entries. For $A \in \mL$, we define an $n$ by $n$ matrix
$L(A;x) = (l(x)_{ij})_{1 \leq i,j \leq n}$ by
\begin{align}\label{eq:L-A}
l(x)_{ij} = \sum_{\ell \in \Z} a_{i,j- \ell n} \, x^\ell.
\end{align}
We may identify $\mL$ with $\C^{mn}$, and identify $A \in \mL$ with $L(A;x)$.
Now define a map $\alpha : \mathcal{M} \to \mL$ by
$$
\alpha: Q = (Q_1,Q_2,\ldots,Q_m) \longmapsto L(x) := Q_1(x) Q_2(x) \cdots Q_m(x)P(x)^k,
$$
where
\begin{align}\label{eq:P}
P(x) := \left(\begin{array}{cccc}
0 & 0 & 0 &x\\
1&0 &0 &0 \\
0&\ddots & \ddots&0 \\
0&0&1&0
\end{array} \right).
\end{align}
We call $L(x)$ the \emph{Lax matrix}. Our approach to the study of $L(x)$ is close to that of van Moerbeke and Mumford \cite{vMM}.
We give a combinatorial description of the Lax matrix by using {\em highway paths} on the network $G$.
We introduce the ``$x$-line'' as illustrated in Figure \ref{fig:toda1},
where the top $k$ right ends of horizontal wires cross the $x$-line (the top edge) and come out on the other side.
There are $n$ sources labeled $1,2,\ldots, n$ on the left and $n$ sinks labeled $1,2,\ldots,n$ on the right. Each sink $j$ is connected to the source $j+k \mod n$ by a wire, as illustrated in the figure. There are $mn$ intersection points between the vertical wires and the horizontal wires, which we call the {\it $(i,j)$-crossroads}, where $i = 1,2,\ldots,m$ indexes the vertical wire, and $j = 1,2,\ldots,n$ indexes the horizontal wire.
Let us denote by $G'$ the \emph{cylindrical network} obtained by gluing only the upper and lower edges of Figure \ref{fig:toda1}. In the torus network $G$, source $i$ and sink $i$ are the same point. In the cylindrical network $G'$, source $i$ and sink $i$ are distinct points.
\begin{figure}[h!]
\begin{center}
\scalebox{0.8}{\input{toda1.pstex_t}}
\end{center}
\caption{The toric network $G$, with a coil and a snake path shown.}
\label{fig:toda1}
\end{figure}
Let us introduce the notion of highway paths following \cite{LP}.
A {\it highway path} $p$ is a directed path in the network $G$ (or $G'$) with the following property: at any of the crossroads, if the path is traveling upwards, it must turn right. We shall only consider highway paths that start at one of the sources $1,2,\ldots, n$ and end at one of the sinks $1,2,\ldots,n$.
The weight $\wt(p)$ of a highway path $p$ is defined as follows.
Every time $p$ passes the $x$-line it picks up the weight $x$. Every time $p$ goes through the $(i,j)$-crossroad, it picks up the weight $q_{ij}$ or $1$, according to Figure~\ref{fig:highway}. The weight $\wt(p)$ is the product of all these weights. The condition that a highway path must turn right when travelling up into a crossroad is indicated in the Figure~\ref{fig:highway}: we can think of such a turn as giving weight 0.
Finally, a highway path $p$ may use no edges. In this case, we consider the path $p$ to start at some source $i$, and end at the sink $i$. We declare such paths to be {\it abrupt}, and have weight $\wt(p) = -y$.
\begin{figure}[h!]
\unitlength=0.9mm
\begin{picture}(80,30)(20,0)
\multiput(0,15)(30,0){4}{\vector(1,0){20}}
\multiput(10,5)(30,0){4}{\vector(0,1){20}}
\thicklines
\linethickness{0.45mm}
\put(0,15){\line(1,0){10}}
\put(10,15){\vector(0,1){10}}
\put(30,15){\vector(1,0){20}}
\put(70,5){\line(0,1){10}}
\put(70,15){\vector(1,0){10}}
\put(100,5){\vector(0,1){20}}
\put(-20,-2){weights:}
\put(9,-2){$1$}
\put(39,-2){$q_{ij}$}
\put(69,-2){$1$}
\put(99,-2){$0$}
\end{picture}
\caption{Highway paths}
\label{fig:highway}
\end{figure}
The following lemma gives a highway-path description of the Lax matrix,
which is a variant of the results of \cite{LP}.
It follows directly from the definitions.
\begin{lem}\label{lem:entry}
Let $L(x) = \alpha(q)$ for $q \in \mM$.
For $1 \leq i,j \leq n$, we have
$$
\mbox{$(i,j)$-th entry of $L(x)-y$} = \sum_p \wt(p),
$$
where the summation is over highway paths in $G'$ from source $i$ to sink $j$.
\end{lem}
\subsection{Spectral curve and Newton polygon}\label{sec:spectral}
Define a map $\psi : \mM \to \C[x,y]$
as the composition of $\alpha: \mM \to \mL$ and the map $\beta: \mL \to \C[x,y]$,
$$
Q = (Q_1,Q_2,\ldots,Q_m) \stackrel{\alpha}{\longmapsto} L(x) = Q_1(x) Q_2(x) \cdots Q_m(x)P(x)^k
\stackrel{\beta}{\longmapsto} \det(L(x) - y).
$$
Consequently, for $Q = (Q_1,\ldots,Q_m) \in \mM$
we have an affine plane curve $C'_{\psi(Q)}$ in $\C^2$, given by
the zeros of $\psi(Q)$. We call this curve the \emph{spectral curve}.
Each term of $\psi(Q)$ corresponds to
the weight of specific highway paths as follows.
We say that a pair of paths is {\it noncrossing} if no edge is used twice,
and that a family of paths is noncrossing if every pair of paths is
noncrossing.
Suppose $\p = \{p_1,p_2,\ldots,p_n\}$ is an unordered noncrossing family of $n$ paths in $G'$ using all the sources and all the sinks.
The non-abrupt paths in $\p$ induce a bijection of a subset $S \subset [n]$ with itself. We let $\sign(\p)$ denote the sign of this permutation.
The following theorem is a reformulation \cite{MIT}.
In our language, the proof is very similar to \cite[Theorem 3.5]{TP}.
\begin{thm} \label{thm:mit}
We have
$$
f(x,y)=\det(L(x)-y) = \sum_{\p = \{p_1,p_2,\ldots,p_n\}} {\sign(\p)} \wt(p_1) \wt(p_2) \cdots \wt(p_n),
$$
where the summation is over noncrossing (unordered) families of $n$ paths in $G'$ using all the sources and all the sinks.
In other words, the coefficient $x^ay^b$ in $f(x,y)=\det(L(x)-y)$ counts (with weights) families of $n$ paths that
\begin{itemize}
\item do not cross each other;
\item cross the $x$-line exactly $a$ times;
\item contain exactly $b$ abrupt paths and $n-b$ non-abrupt paths.
\end{itemize}
The overall resulting sign of the monomial $x^ay^b$ is $(-1)^{(n-b-1)a+b}$.
\end{thm}
For $f(x,y) = \sum_{i,j} a_{i,j} x^i y^j \in \C[x,y]$,
we write $N(f) \subset \R^2$ for the Newton polygon of $f$.
This is defined to be the convex hull of the points $\{(i,j) \mid a_{ij} \neq 0\}$.
It is important for us to identify the {\it lower hull} and {\it upper hull} of $N(f)$. The former (resp. latter) is the set of edges of $N(f)$ such that the points directly below (resp. above) these edges do not belong to $N(f)$. We exclude vertical or horizontal edges from the lower and upper hull.
\begin{prop} \label{prop:hull}
For generic $Q = (Q_1,\ldots,Q_m) \in \mM$, the Newton polygon $N(\psi(Q))$ is the triangle with
vertices $(0,n)$, $(k,0)$ and $(k+m,0)$.
In $N(\psi(Q))$, the lower hull (resp. upper hull)
consists of one edge with vertices $(k,0)$ and $(0,n)$
(resp. $(k+m,0)$ to $(0,n)$).
\end{prop}
See \S~\ref{subsec:prop:hull} for the proof.
\subsection{Special points on the spectral curve}\label{subsec:special-pts}
For $f(x,y) \in \C[x,y]$ an irreducible polynomial, let $C'_f= \{(x,y) \mid f(x,y) = 0\} \subset \C^2$ be the corresponding plane curve, and let
$\overline{C'_f} \subset \P^2(\C)$ denote its closure. Let $C_f$ be the normalization of $\overline{C'_f}$, with a map $C_f \to \overline{C'_f}$ that is a resolution of singularities.
We declare points on $C_f$ to be {\it special} if either (1) either $x$ or $y$ is 0, or (2) the point does not lie over $C'_f$ (that is, $x$ or $y$ is $\infty$).
For $f(x,y) \in \psi(\mathcal{M})$,
define a polynomial $g_f(x,y)$ (resp. $h_f(x,y)$) by
$f(x,y) = g_f(x,y) + {\rm other~ terms}$
(resp. $f(x,y) = h_f(x,y) + {\rm other~ terms}$),
where $g_f(x,y)$ (resp. $h_f(x,y)$) consists of
those monomials lying on the lower hull (resp. the upper hull) of $N(f)$.
For this $f(x,y)$ we also define
\begin{align}\label{eq:fc}
f_c:= \sum_{(j,i) \in L_c} f_{i,j},
\end{align}
where
$$
L_c = \{(j,i)
~|~ (m+k) i + n j = n (m+k)-M \}.
$$
We define $\sigma_r \in \mathcal{O}(\mathcal{M})$ by
\begin{align}\label{def:sigma_r}
\sigma_r(q) = \prod_{i=1}^m \prod_{j=0}^{n'-1} q_{i,r+jN},
\end{align}
for $1 \leq r \leq N$, and
define $\epsilon_r \in \mathcal{O}(\mathcal{M})$ by
\begin{align}\label{def:epsilon_r}
\epsilon_r(q):= \prod_{j=1}^n q_{rj}
\end{align}
for $1 \leq r \leq m$.
\begin{remark}
Despite the seeming dissimilarity, $\sigma_r(q)$ and $\epsilon_r(q)$ have the same nature. Indeed, visualize variables $q_{ij}$ as associated to crossings of two families of parallel wires on a torus, as it was done in Section \ref{sec:netw}.
Then $\sigma_r(q)$ is the product of parameters on the $r$-th horizontal wire (out of $N$), while $\epsilon_r(q)$ is the product of parameters on the $r$-th vertical wire (out of $m$). In particular, the symmetry between $Q$ and $\tilde Q$
from Section \ref{sec:netw} switches the $\sigma_r(q)$ and the $\epsilon_r(q)$ into each other.
\end{remark}
\begin{lem}\label{lem:Q}
For $f(x,y) \in \psi(\mathcal{M})$, we have
\begin{align}
&g_f(x,y) = \prod_{r=1}^N ((-y)^{n'} + \sigma_r x^{k'}),
\label{eq:lower-f}
\\
\label{eq:upper-f}
&h_f(x,y) = ((-y)^{\frac{n}{M}}+x^{\frac{k+m}{M}})^M.
\end{align}
\end{lem}
See \S~\ref{subsec:lem:Q} for the proof.
In the following, for $f(x,y) = \psi(q), ~q \in \mM$ we study the special points of $C_f$
related to polynomials $g_f(x,y)$, $h_f(x,y)$ or $f(x,0)$.
We have $f(x,0) = \det P(x)^k \prod_{i=1}^m \det Q_i(x)$,
where $\det Q_i(x) = \prod_{j=1}^n q_{ij} + (-1)^{n-1} x = \epsilon_i(q) + (-1)^{n-1} x$. Thus the roots of $f(x,0)=0$ are exactly the $(-1)^{n} \epsilon_r(q)$, where $\epsilon_r(q)$ is defined by \eqref{def:epsilon_r}. It is also clear that $\epsilon_1,\ldots,\epsilon_m$ depend only on $f(x,y)$.
The following lemma is obtained immediately.
\begin{lem}\label{lem:A}
Suppose that $\epsilon_1,\ldots,\epsilon_m$ are distinct and nonzero. Then there are $m$ special points $A_i = ((-1)^{n} \epsilon_i(q),0)$ on $C_f$ with $y = 0$ and $x$ is nonzero.
\end{lem}
The point $(0,0)$ on $C'_f$ is usually singular, whenever $k \geq 2$.
\begin{lem}\label{lem:O}
Suppose $\sigma_1,\ldots,\sigma_N$ are distinct and nonzero. Then there are $N$ points of $C_f$ lying over $(0,0) \in C'_f$.
\end{lem}
\begin{proof}
By \eqref{eq:lower-f}, the meromorphic function $y^{n'}/x^{k'}$ takes the $N$ distinct values
$(-1)^{n'+1}\sigma_r$ for $r = 1,2,\ldots,N$ as $(x,y) \to (0,0)$.
So there are at least $N$ points on $C_f$.
On the other hand, looking at $N(f)$ we see that analytically near $(0,0)$, the polynomial $f$ can factor into at most $N$ pieces.
\end{proof}
Let $O_1,O_2,\ldots, O_N$ denote the the special points of Lemma \ref{lem:O}.
Near $O_r$ there is a local coordinate $u$ such that \begin{equation*}
(x,y) \sim (u^{n'},-(-\sigma_r)^{1/n'}u^{k'}).
\end{equation*}
It turns out that $C_f' \subset \P^2(\C)$ has only one point at $\infty$.
Due to the polynomial $h_f(x,y)$, in homogeneous coordinates, this point is
$$
P' = \begin{cases} [1:0:0] &\mbox{if $n > k+m$}, \\
[1:1:0] & \mbox{if $n = k+m$},\\
[0:1:0] & \mbox{if $n < k+m$}.
\end{cases}
$$
To compute this, we first homogenize $f(x,y)$ to get $F(x,y,z) = z^d f(x/z,y/z)$where $d := \max(n,m+k)$. Then we solve $F(x,y,0) = 0$.
Recall that $f_c$ was defined in \eqref{eq:fc}.
\begin{lem} \label{lem:inf}
Suppose that $f_c \neq 0$. Then there is a unique point $P \in C_f$ lying over $P'$.
\end{lem}
See \S~\ref{subsec:lem:inf} for the proof.
\subsection{A good condition for the spectral curve}
Let $V_{n,m,k}$ be the subspace of $\C[x,y]$ given by
\begin{align}\label{eq:Vnmk}
V_{n,m,k}
=
\left\{((-y)^{\frac{n}{M}}+x^{\frac{k+m}{M}})^M
+ \sum_{i=0}^{n-1} y^{i} f_i(x) ~\Big|~
f_i(x) =
\sum_{(j,i) \in L_{n,m,k}}
f_{i,j} x^j \in \C[x] \right\},
\end{align}
where we write $L_{n,m,k}$ for the set of lattice points in the
convex hull of $\{(k,0),(0,n),(k+m,0) \}$ but not on the
upper hull. By Proposition \ref{prop:hull} and Lemma \ref{lem:Q},
we have that $\psi(\mM) \subset V_{n,m,k}$.
\begin{definition}\label{def:P:curve}
Define the subset $\mathcal{V} \subset V_{n,m,k}$ as the set of
$f(x,y) \in V_{n,m,k}$, satisfying the conditions:
\begin{enumerate}
\item
$f(x,y)$ is irreducible;
\item
$C'_f$ is smooth;
\item
$f_c \neq 0$;
\end{enumerate}
and such that the special points on $C_f$ consist exactly of:
\begin{enumerate}
\item[(4)]
$m$ distinct points $A_1,A_2,\ldots,A_m$ where $A_r := ((-1)^{n} \epsilon_i,0)$ and $\epsilon_i \neq 0$;
\item[(5)]
$N$ distinct points $O_1,O_2,\ldots, O_N$ lying over $(0,0)$, where near $O_r$
there is a local coordinate $u$ such that
\begin{align}\label{eq:o_r}
(x,y) \sim (u^{n'},-(-\sigma_r)^{1/n'}u^{k'})
\end{align}
and $\sigma_r \neq 0$;
\item[(6)]
a single point $P$ lying over the line at infinity $\P^2(\C) \setminus \C^2$.
\end{enumerate}
\end{definition}
For $f \in \mathcal{V}$, the genus $g$ of $C_f$ is given by
\begin{align} \label{eq:genus}
g = \frac{1}{2}\left( (n-1)m - M -N + 2 \right).
\end{align}
Indeed, it follows from Pick's formula and Proposition \ref{prop:hull} that the number of interior lattice points of $N(f)$ is equal to the right hand side. This formula for the genus then follows from \cite[Corollary on p.6]{Kho}. Alternatively, the genus can be computed using the Riemann-Hurwitz formula, as in \cite{vMM}.
\begin{prop}\label{P:curve}
For $f \in \mathcal{V}$, we have that $\beta^{-1}(f) \neq \emptyset$.
Moreover, the set $\oV := \mathcal{V} \cap \psi(\mathcal{M})$
is a Zariski-dense subset of $\psi(\mathcal{M})$.
\end{prop}
We give the proof in \S~\ref{subsec:P:curve}.
Informally, the last statement of Proposition \ref{P:curve} states that most curves in $\psi(\mathcal{M})$ satisfy the ``niceness" conditions listed in Definition \ref{def:P:curve}.
From Definition~\ref{def:P:curve}, it follows that for $f \in \mathcal{V}$, the meromorphic functions
$x$ and $y$ on $C_f$ satisfy
\begin{align}
(x) = n^\prime \sum_{i=1}^N O_i - n P,
\qquad
(y) = k^\prime \sum_{i=1}^N O_i + \sum_{i=1}^m A_i -(m+k) P.
\end{align}
\begin{remark}
The condition that $A_1,\ldots,A_m$ (resp. $O_1,O_2,\ldots, O_N$) are distinct imply that the quantities $\epsilon_1,\ldots,\epsilon_m$ (resp. $\sigma_1,\ldots,\sigma_N$) are distinct. Many of our main results still apply after a modification even when this condition does not hold.
\end{remark}
\section{Proofs from Sections \ref{sec:netw} and \ref{sec:Lax}}
\label{sec:proofLax}
\subsection{Proof of Proposition \ref{prop:hull}}
\label{subsec:prop:hull}
We use the interpretation of $f(x,y)$ given by Theorem \ref{thm:mit}. Let $\p = (p_1,\ldots,p_n)$ be a family of noncrossing highway paths in $G'$, as in Theorem \ref{thm:mit}, and let $\wt(\p) = \wt(p_1) \cdots \wt(p_n)$.
There are $k+m$ opportunities for a highway path in $\p$ to pick up the weight $x$: from each of the $m$ vertical wires and from the $k$ horizontal wires crossing $x$-line. For a fixed power $y^b$ (where $b = 0,1,\ldots,n$), or equivalently a fixed number of abrupt paths, we will bound the maximal and the minimal possible value of the exponent $a$ of $x$.
\begin{figure}[h!]
\begin{center}
\scalebox{0.8}{\input{toda2.pstex_t}}
\end{center}
\caption{The red path crosses $x$-line less than the green one.}
\label{fig:toda2}
\end{figure}
The first key observation is that we can consider $\p$ to be a family of noncrossing closed cycles on the toric network $G$. An abrupt path $p$ is simply the ``cycle" that starts and ends at the vertex $i$ ($=$ source $i$ and sink $i$ identified) in $G$ and does not move.
Lift such a highway cycle $C$ to the universal cover, as shown in Figure \ref{fig:toda2}. We obtain a path that starts and ends at two vertices labeled with the same integer $i \in \{1,2,\ldots,n\}$, but $\ell$ periods (of $m$ vertical lines each) to the right of the original source. More precisely, if $C$ is obtained by gluing paths $p_{i_1},\ldots,p_{i_\ell}$ in $G'$, then $C$ has length $\ell$. The $x$-line now has a staircase-like shape as shown in Figure \ref{fig:toda2}. We claim that if $C$ has length $\ell$, then it crosses the $x$-line at least $\ell k/n$ times and at most $\ell (k+m)/n$ times.
To see this, observe that if we lift the ending vertex several periods (consisting of $n$ horizontal wires) up, we preserve the length and we increase the
number of crossings with the $x$-line. Similarly, if we lower the ending vertex several periods down, we preserve the length and we decrease the number of crossings of $x$-line. Thus, the smallest and the largest ratios of the quantity (crosses of $x$-line/length) is achieved for the lowest and the highest possible highway paths, shown in Figure \ref{fig:toda2} in purple and green.
The former is the horizontal path, while the latter is an alternating right-up staircase path. For these paths, the ratios are exactly $k/n$ and $(k+m)/n$.
\subsection{Proof of Lemma~\ref{lem:Q}}
\label{subsec:lem:Q}
One of the consequences of the proof of Proposition \ref{prop:hull} is that the monomials for which the lower bound $k/n$ of the ratio is reached are
the ones coming from
horizontal highway paths on the universal cover. Let us call each such closed cycle a {\it {coil}}. For example, the purple line in Figure \ref{fig:toda2} represents a coil passing through source $i$, as well as passing through the $n'$ other vertices between $1$ and $n$ that have residue $i$
modulo $N = \gcd(k,n)$. Thus the terms of $g(x,y)$ are formed in the following way: for each of the $N$ coils we decide whether to include it
into our family of paths, or to make all paths starting at its sources abrupt. The second choice corresponds to the contribution $(-y)^{n'}$. The first choice gives
$(\prod_{i=1}^m \prod_{j=1}^{n'} q_{i,r+jN}) x^{k'}$, which is the weight of that coil. Thus, the $r$-th coil contributes
the factor of $\left((\prod_{i=1}^m \prod_{j=1}^{n'} q_{i,r+jN}) x^{k'} + (-y)^{n'}\right)$, and \eqref{eq:lower-f} follows.
Similarly, the monomials for which the upper bound $(k+m)/n$ of the ratio is reached are
the ones coming from right-up staircase paths on the universal cover. Let us call each such closed cycle a {\it {snake path}}. For example, the
green line in Figure \ref{fig:toda2} represents a snake path passing through source $i$, as well as through all the $n/M$ other vertices between $1$ and $n$ that have residue $i$
modulo $M = \gcd(m+k,n)$. Thus the part contributing to the upper hull of $f(x,y)$ is a product of factors $x^{(m+k)/M}+(-y)^{n/M}$, where
the term $(-y)^{n/M}$ corresponds to choosing to have abrupt paths, while $x^{(m+k)/M}$ corresponds to choosing to have the snake path.
Thus we obtain \eqref{eq:upper-f}.
\subsection{Proof of Lemma~\ref{lem:inf}}
\label{subsec:lem:inf}
We analyze the singularity at $P' \in \P^2(\C)$ on the chart
$y \neq 0$, that is, $\{[x:1:z] ~|~ x,z \in \C\}$.
Let $\tilde{C}_{\overline{f}}$ be the affine curve given by $\overline{f}(x,z) := F(x,1,z) = 0$.
(i) $n < k+m$: using \eqref{eq:upper-f}
we can write $\overline{f}(x,z)$ as
$$
\overline{f}(x,z)
= ((-1)^{\frac{n}{M}}z^{\frac{k+m-n}{M}}
+x^{\frac{k+m}{M}})^M
+ \sum_{j=1}^{\max(m,m+k-n)} z^j \overline{f}_j(x)
$$
where $\deg_x \overline{f}_j(x) \geq 1$.
Then, when $m+k-n = 1$, $\tilde{C}_{\overline{f}}$ is smooth at $(0,0)$ since
$\partial \overline{f}/\partial z|_{(0,0)} = (-1)^{n/M} \neq 0$.
When $m+k-n \geq 2$, the point $(0,0)$ is singular.
Define $K := \C(x)[z]/(\overline{f}(x,z))$.
We consider all valuations $v: K \twoheadrightarrow \Z$ on $K$ satisfying that $v(x) > 0$ and $v(z) > 0$.
We shall show that such valuations are unique,
from which the uniqueness of $P$ follows.
Define $\overline{h}_f(x,z):=((-1)^{\frac{n}{M}}z^{\frac{k+m-n}{M}}
+x^{\frac{k+m}{M}})^M$ which is the part of
$\overline{f}(x,z)$ consisting of monomials
lying on the lower hull of $N(\overline{f})$,
corresponding to the upper hull of $N(f)$.
Then we see that
$v(\overline{f}(x,z)) = v(\overline{h}_f(x,z)) =
M \cdot v(x^{\frac{k+m}{M}}
+ (-1)^{\frac{n}{M}}z^{\frac{k+m-n}{M}})$.
To be consistent with $v(\overline{f}(x,z)) = v(0)= \infty$,
the only choice we have is
$v(x) = \frac{k+m-n}{M}$ and $v(z) = \frac{k+m}{M}$.
(ii) $n = k+m$:
we can write $\overline{f}(x,z)$ as
$$
\overline{f}(x,z)
= (x-1)^n + \sum_{k=1}^m z^k \overline{f}_k(x-1)
$$
where $\overline{f}_k(x-1) \in \C[x-1]$.
Thus $P'$ is smooth point on $C_{\overline{f}}$ if $\overline{f}_1(0) \neq 0$.
Looking at the Newton polygon $N(f)$ and \eqref{eq:fc}, it follows that
$\overline{f}_1(0) = \sum_{i=0}^{n-1} f_{i,n-1-i} = f_c \neq 0$.
(iii) $n >k+m$: this case is nearly the same as the case $n < k+m$.
\subsection{Proof of Proposition \ref{P:curve}}\label{subsec:P:curve}
The first statement follows from Theorem~\ref{thm:eta} in the next section.
In the following, we prove the second statement.
We first show that $\mathcal{V}$ contains a Zariski-dense and open subset of $V_{n,m,k}$. A Zariski-open subset of $V_{n,m,k}$ consists of $f(x,y)$ with Newton polygon $N(f)$ given by Proposition \ref{prop:hull}. This polygon is not a non-trivial Minkowski sum of two other polygons, so $f(x,y)$ is irreducible. Similarly the conditions that $C'_f$ is smooth and $f_c \neq 0$ are Zariski-open conditions on $V_{n,m,k}$. By Lemma \ref{lem:inf}, $f_c \neq 0$ implies that (6) in Definition \ref{def:P:curve} holds. The calculations in the proofs of Lemmas \ref{lem:A} and \ref{lem:O} then imply that $\mathcal{V}$ contains a Zariski-open and dense subset of $V_{n,m,k}$.
It thus suffices to show that $\psi(\mM)$ contains a Zariski-dense subset of $V_{n,m,k}$. We shall use the following result.
\begin{lem}\label{L:mMmL}
The set $\alpha(\mM)$ is Zariski-dense in $\mL$.
\end{lem}
\begin{proof}
This follows from \cite[Proof of Theorem 4.1]{LPgeom} where it is shown that the map $\alpha:\mM \to \mL$ is generically a $m!$ to $1$ map between two spaces of dimension $mn$.
\end{proof}
By the first statement of Proposition \ref{P:curve}, we have $\mathcal{V} \subset \beta(\mL)$. By Lemma \ref{L:mMmL}, $\psi(\mM) = \beta(\alpha(\mM))$ contains a Zariski-dense subset of $\mathcal{V}$. It follows that $\psi(\mM)$ is Zariski-dense in $V_{n,m,k}$. This completes the proof of the second statement of Proposition \ref{P:curve}.
\begin{remark}
The nonconstant coefficients $f_{i,j}$ of $f(x,y)$ can be pulled back to functions on $\mM$ or $\mL$. A consequence of our proof is that $\psi(\mM)$ is Zariski-dense in $V_{n,m,k}$, and therefore the functions $f_{i,j}$ are algebraically independent on $\mM$ or $\mL$. It would be interesting to obtain a direct proof of this.
\end{remark}
\subsection{Proof of Theorem \ref{thm:dynamics}}
\label{sec:dynamics_proof}
We employ the technique first introduced in \cite{LP}.
\begin{figure}[h!]
\begin{center}
\input{wire11.pstex_t}
\end{center}
\caption{Crossing merging and crossing removal moves with vertex weights shown.}
\label{fig:wire11}
\end{figure}
\begin{figure}[h!]
\begin{center}
\scalebox{.7}{\input{wire8.pstex_t}}
\end{center}
\caption{Yang-Baxter move with transformation of vertex weights shown.}
\label{fig:wire8}
\end{figure}
Specifically, we realize the geometric $R$-matrix transformation, dubbed the {\it {whirl move}} in \cite{LP}, as a sequence of local transformations on our toric network.
The transformations we shall employ are of three kinds: crossing merging/unmerging, crossing creation/removal, shown in Figure \ref{fig:wire11}, and Yang-Baxter move shown in Figure \ref{fig:wire8}.
The whirl move $R$ occurs between two parallel wires adjacent to each other and wrapping around a local part of the surface that is a cylinder. The way it is realized as a sequence of local moves is illustrated in
Figure \ref{fig:wire20}. First a crossing is created with weight $0$, and split into two crossings of weight $p$ and $-p$. One of them is pushed through the wires crossing our two distinguished wires,
until it comes out on the other side. As it is proven in \cite[Theorem 6.2]{LP}, there is at most one non-zero value of $p$ for which the end result is again a pair of crossings of weights $p$ and $-p$,
and thus those two can be canceled out. The resulting action on the weights along two parallel wires is exactly the whirl move $R$, and does not depend on where the original auxiliary crossing was
created.
\begin{figure}[h!]
\begin{center}
\input{wire20a.pstex_t}
\end{center}
\caption{For a unique choice of the weight $p$, the weight that comes out on the other side after passing through all horizontal wires is also $p$; the resulting transformation of $x$ and $y$ is exactly the whirl move.}
\label{fig:wire20}
\end{figure}
The value of $p$ does depend on the location $j$ where the new crossings are created, and is given by
$$p = \frac{\prod y_{i} - \prod x_{i}}
{E(\boldsymbol{x}^{(j)},\boldsymbol{y}^{(j)})}.$$
Here the variables $x_i$ and $y_i$ are weights along the two wires as
in Figure \ref{fig:wire20}, and $E(\boldsymbol{x}^{(j)},\boldsymbol{y}^{(j)})$
is the energy of the cyclically shifted vectors $\boldsymbol{x}^{(j)} = (x_{j+1},x_{j+2},\ldots,x_{j})$ and
$\boldsymbol{y}^{(j)} = (y_{j+1},y_{j+2},\ldots,y_{j})$
as introduced in \S~\ref{sec:Rmatrix}.
Now, the fact that $R$ satisfies the braid move, that is, $s_{\ell} s_{\ell+1} s_{\ell} = s_{\ell+1} s_{\ell} s_{\ell+1}$ and $\tilde s_{\ell} \tilde s_{\ell+1} \tilde s_{\ell}
=\tilde s_{\ell+1} \tilde s_{\ell} \tilde s_{\ell+1}$ is shown in \cite[Theorem 6.6]{LP}. Indeed, these relations happen at a local part of the surface (in our case, torus) that looks like a cylinder.
\begin{figure}[h!]
\begin{center}
\scalebox{0.6}{\input{wire25a.pstex_t}}
\end{center}
\caption{}
\label{fig:wire25}
\end{figure}
On the other hand, the commutativity of the $s_{\ell}$ and the $\tilde s_{\ell}$ does not follow from \cite[Theorem 12.2]{LP}. This is because in \cite{LP} we only considered the case when the pairs of parallel wires
intersect once. In our case on the torus however, it is common to have horizontal and vertical wires intersect more than once: this happens for any $\d \not = 1$. The proof in such a situation is essentially the same.
Indeed, if we have two pairs of parallel wires crossing as in Figure \ref{fig:wire25}, but possibly more than once, we can realize each of the two corresponding $R$-moves by a sequence of local moves as above.
It is a local check that performing one of the two sequences does not change
the value of
$p = \frac{\prod y_i - \prod x_i}{E(\boldsymbol{x}^{(j)},\boldsymbol{y}^{(j)})}$
one needs to perform the other, because each of the
$\prod x_i$, $\prod y_i$ and $E(\boldsymbol{x}^{(j)},\boldsymbol{y}^{(j)})$ is not changed.
It is also a local check that the two sequences commute once the parameters $p$ for each are chosen, this is \cite[Proposition 3.4]{LP}.
Thus, commutativity follows.
\section{Eigenvector map}
\label{sec:eigenvector}
In this section, we fix $f \in \mathcal{V}$ (see Definition~\ref{def:P:curve})
and consider the corresponding smooth curve $C_f$.
For $j \in \Z$,
we set $O_j := O_r$
if $j \equiv r$ mod $N$.
Similarly, for $j \in \Z$,
we set $A_j := A_r$
if $j \equiv r$ mod $m$.
\subsection{Generalities}
A divisor $D = \sum_i n_i P_i$ on an algebraic curve $C$
is a finite formal integer linear combination of points $P_i$ on $C$.
We write $D \geq 0$ if $n_i \geq 0$, and say that $D$ is {\it positive} in this case.
The degree of $D$ is given by $\deg(D) = \sum_i n_i$.
Given $h$ a meromorphic function on $C$, we let
$(h) = (h)_0 - (h)_\infty$ be the divisor of $h$. Here, $(h)_0$ denotes the divisor of zeroes, and $(h)_\infty$ denotes the divisor of poles.
Two divisors $D_1, D_2$ are linearly equivalent, $D_1 \sim D_2$,
if there exists a meromorphic function $h$ such that $(h) = D_1-D_2$.
We write $[D]$ for the equivalence class of $D$ with respect to linear equivalence.
The Picard group $\Pic(C)$ is the abelian group of divisors on $C$ modulo linear equivalence.
For $j \in \Z$, we write $\Pic^j(C)$ for the part of the Picard group of $C$
that has degree $j$, that is,
$\Pic^j(C) := \{D : \text{ a divisor on $C$} ~|~ \deg(D) = j \}/\sim$.
To each divisor $D$ we associate a space of meromorphic functions
$$\L(D) = \{f \mid (f) + D \geq 0\}.
$$
If $D = \sum_i n_i P_i \geq 0$, then in words, $\L(D)$ consists of meromorphic functions which are allowed to have poles of order $n_i$ at $P_i$. We have $\dim(\L(D_1)) = \dim(\L(D_2))$ if $D_1$ and $D_2$ are linearly equivalent.
\subsection{Positive divisors of a Lax matrix}\label{subsec:eigenmap}
A divisor $D \in \Pic^g(C_f)$ of degree $g$ is called {\it general} if $\dim \mathcal{L}(D)=1$.
A divisor $D \in \Pic(C_f)$ is called {\it regular} with respect to the points $P$ and $O_j$ if $D$ is general and
if $\dim \mathcal{L}(D+ kP-\sum_{j=n-k}^n O_j)=0$ for $k > 0$. For brevity, we will sometimes just say that a divisor is regular when it is regular with respect to $P$ and the $O_j$.
Let us now fix $L(x) \in \beta^{-1}(f)$. Let
$$
\Delta_{i,j} := (-1)^{i+j} |L(x)-y|_{i,j}
$$
denote the (signed) $(i,j)$-th minor of the matrix $L(x) - y$.
\begin{thm}[cf. \cite{vMM}]\label{thm:double}
There exists a positive divisor $R$ of degree $2g+2$ supported on $S_{P,O} := \{P,O_1,\ldots,O_N\}$, and uniquely defined positive general divisors $D_1,D_2,\ldots,D_n,\bar D_1,\ldots,\bar D_n$ of degree $g$ such that for all $(i,j) \in [n]^2$, we have
$$
(\Delta_{i,j}) = D_j + \bar D_i + (j-i -1 )P + \sum_{r=j+1}^{i -1} O_r - R.
$$
In addition, $D_1,\ldots,D_n$ have pairwise no common points, and $\bar D_1,\ldots,\bar D_n$ have pairwise no common points.
\end{thm}
See \S~\ref{pf:thm:double} for the proof.
Note that $\sum_{r=j}^i O_r$ is to be interpreted in a signed way:
$$
\sum_{r=j}^i O_r := \sum_{r=j}^\infty O_r - \sum_{r=i+1}^\infty O_r
=
\begin{cases}
\sum_{r=j}^i O_r & i \geq j,
\\
0 & i = j-1,
\\
-\sum_{r=i+1}^{j-1} O_r & i \leq j-2.
\end{cases}
$$
For $L(x) \in \beta^{-1}(f)$, define $g_i:=\Delta_{n,i}$.
Since $L(x)-y$ is singular
along $C_f$, the vector $g = (g_1,g_2,\ldots,g_n)^\bot$ (thought of as a vector with entries that are rational functions on $C_f$) is an eigenvector of $L(x)-y$.
We define $g_i$ for $i \in \Z$ by $g_{i+n} = x^{-1}g_i$. We also define $h_i := g_i/g_n$ for $i \in \Z$. Thus $h_n = 1$ and $h_{i+n} = x^{-1}h_i$. The vector $(h_1,h_2,\ldots,h_n)^\bot$ is also an eigenvector of $L(x)-y$.
\begin{definition}\label{def:D}
For $L(x) \in \beta^{-1}(f)$, define the divisor $\D = \D(L(x))$ on $C_f$ to be the minimum positive divisor satisfying
$$
(h_i) + \D \geq \sum_{j=i+1}^n O_j - (n-i) P,
$$
that is, $h_i \in \mL(\D + (n-i) P - \sum_{j=i+1}^n O_j)$,
for $i=1,\ldots,n-1$.
\end{definition}
It follows from Theorem \ref{thm:double} that $\D = D_n$ from the theorem and that this divisor is uniquely determined by Definition \ref{def:D}. In particular, $\D$ is a positive regular divisor of degree $g$ with respect to the points
$P$ and $O_j$, and we have
\begin{align}\label{eq:hdiv}
(h_i) = D_i - \D - (n-i)P + \sum_{j=i+1}^n O_j.
\end{align}
\subsection{The eigenvector map}\label{subsec:eigen}
Let $R_A$ (resp. $R_O$) denote the set of orderings of the $m$ points $A_1,\ldots,A_m$ (resp. $O_1,\ldots,O_N$):
$$
R_A := \{\nu (A_1,\ldots,A_m) ~|~ \nu \in \mathfrak{S}_m \},
\qquad
R_O := \{\tilde \nu(O_1,\ldots,O_N) ~|~ \tilde\nu \in \mathfrak{S}_N \}.
$$
By our assumption that $f \in \mathcal{V}$, the points $A_r$ (resp. $O_r$) are distinct, so $R_A$ has cardinality $m!$ and $R_O$ has cardinality $N!$.
We write $\nu_r ~(r=1,\ldots,m-1)$ (resp. $\tilde \nu_r ~(r=1,\ldots,N-1)$)
for the generator of $\mathfrak{S}_m$ (resp. $\mathfrak{S}_N$),
which permutes the $r$-th and the $(r+1)$-st entry of
$A \in R_A$ (resp. $O \in R_O$).
By our assumption that $f \in \mathcal{V}$, we have $f_c \neq 0$. Define $\mathcal{S}_f \subset (\C^\ast)^M$ by
$$
\mathcal{S}_f
=
\left\{(x_1,\ldots,x_M) \in (\C^\ast)^M ~|~
\prod_{\ell=1}^M x_\ell = f_c \right\}.
$$
For $\ell=1, \ldots, M$, define $c_\ell$ to be the coefficient of the maximal power of $x$ in the $(\ell, \ell+1)$ entry of $L(x)^{n/M}$. (Here, maximal power is the maximal power for a generic $L(x) \in \mL$.) An explicit formula of
$c_\ell$ for $L(x) \in \alpha(\mM)$ is given in Lemma \ref{lem:c-snake}.
We now define the {\it eigenvector map} $\eta_f$ for $f \in \mathcal{V}$ by
\begin{align}
\label{eq:def-eta}
\eta_f : \beta^{-1}(f) &\longrightarrow \Pic^g(C_f) \times \mathcal{S}_f \times R_O\\
L(x) &\longmapsto ([\mathcal{D}],(c_1,\ldots,c_M),O),
\end{align}
where the ordering $O = (O_1,\ldots,O_N)$ is uniquely determined from $(\Delta_{i,j})$ by Theorem \ref{thm:double}. For $f \in \oV$ we define the map $\phi_f$ by
\begin{align}
\label{eq:def-phi}
\phi_f : \psi^{-1}(f) &\longrightarrow \Pic^g(C_f) \times \mathcal{S}_f \times R_O \times R_A \\
~ Q = (Q_1,\ldots,Q_m) &\longmapsto ([\mathcal{D}],(c_1,\ldots,c_M),O, A),
\end{align}
where $A = (((-1)^{n}\epsilon_1,0),\ldots,((-1)^{n}\epsilon_m,0))$, and $\epsilon_r(q)$ is given by \eqref{def:epsilon_r}. That $(c_1,\ldots,c_M)$ lies in $\mathcal{S}_f $ is the content of Lemma \ref{lem:c-f} below.
We will omit the subscripts of $\eta_f$ and $\phi_f$ and just write
$\eta$ and $\phi$ when no ambiguity will arise.
The following theorem should be compared to Theorem 1 in \cite{vMM}.
\begin{thm}\label{thm:eta}
We have a one-to-one correspondence between $L(x) \in \beta^{-1}(\mathcal{V})$
and the following data:
\begin{enumerate}
\item[(a)]
$f \in \mathcal{V}$,
\item[(b)]
$([\D], c, O) \in \Pic^g(C_f) \times \mathcal{S}_f \times R_O$ where
$\D$ is a positive divisor on $C_f$ of degree $g$,
and regular with respect to $P$ and the $O_j$.
\end{enumerate}
\end{thm}
See \S \ref{subsec:thm:eta} for the proof.
It is shown in \cite[Proof of Theorem 4.1]{LPgeom} that the map $\alpha:\mM \to \mL$ is generically $m!$ to $1$. Let $\mathring{\mM}$ denote the open subset of $\mM$ where (a) the map $\alpha$ is $m!$ to $1$ (that is, $\mathring{\mM} = \alpha^{-1}(\alpha(\mathring{\mM})) \to \alpha(\mathring{\mM})$ is a $m!$ to $1$ map), and (b) the finite symmetric subgroup $\mS_m$ acting via the $R$-matrix is well-defined on $\mathring{\mM}$.
\begin{thm}\label{thm:phi}
Fix $f \in \oV$.
We have an injection from $\psi^{-1}(f) \cap \mathring{\mM}$
to the collection of data $([\D], c, O, A) \in \Pic^g(C_f) \times \mathcal{S}_f \times R_O \times R_A$ such that
\begin{enumerate}
\item[(a)]
$\D$ is a positive divisor on $C_f$ of degree $g$.
It is regular with respect to $P$ and $O_j$-s.
\item[(b)]
$c = (c_1,\ldots,c_M) \in \mathcal{S}_f$.
\item[(c)]
$O \in R_O$.
\item[(d)]
$A \in R_A$.
\end{enumerate}
\end{thm}
See \S \ref{subsec:thm:phi} for the proof.
We will usually just write $\psi^{-1}(f)$ instead of $\psi^{-1}(f) \cap \mathring{\mM}$ when no confusion arises (for example, when discussing the action of the $R$-matrix on the spectral data).
Let us perform a quick dimension count. The dimension of $\mM$ or $\mL$ is equal to $mn$. The dimension of $\Pic^g(C_f) \times \mathcal{S}_f$ is equal to $g+M-1$.
The number of lattice points in the convex full of $\{(k,0),(0,n),(k+m,0) \}$ is equal to $g+N+m+M$. Thus the dimension of $V_{n,m,k}$ \eqref{eq:Vnmk}
is equal to $g+N+m-1$. Using \eqref{eq:genus} we obtain $\dim(\mM) = \dim(\Pic^g(C_f) \times \mathcal{S}_f) + \dim(V_{n,m,k})$, consistent with Theorems \ref{thm:eta} and \ref{thm:phi}.
\section{Proofs from Section \ref{sec:eigenvector}}
\label{sec:proofeigenvector}
The first three subsections are devoted to introduce key lemmas for Theorem~\ref{thm:double} and \ref{thm:eta}.
\subsection{Special zeroes and poles of the eigenvector}
\begin{prop} \label{prop:Q}
The eigenvector $g = (g_i)_i^T$ of $L(x) \in \beta^{-1}(f)$ satisfies the following.
\begin{enumerate}
\item[(i)] For any $j \in \Z$,
the rational function $g_j/g_{j+1}$ on $C_f$ has a zero of order one at $O_{j+1}$.
\item[(ii)]
For any $j \in \Z$,
the rational function $g_j/g_{j+1}$ on $C_f$ has a pole of order one at $P$.
\end{enumerate}
\end{prop}
To prove this proposition, we shall need the following generalization of Theorem \ref{thm:mit}, whose proof is essentially the same as that of Theorem \ref{thm:mit}. Suppose $\p = \{p_1,p_2,\ldots,p_{n-1}\}$ is an unordered noncrossing family of $n$ paths in $G'$ using all the sources except $i$, and all the sinks except $j$. Identifying $[n] \setminus i \simeq [n-1] \simeq [n] \setminus j$ via the order-preserving bijection, we have that the non-abrupt paths in $\p$ induce a bijection of a subset $S \subset [n-1]$ with itself. We let $\sign(\p)$ denote the sign of this permutation.
Let $|(L(x)-y)|_{i,j}$ denote the maximal minor of $(L(x)-y)$ complementary to the $(i,j)$-th entry.
\begin{thm} \label{thm:minor}
We have
$$
|(L(x)-y)|_{i,j}= \sum_{\p = \{p_1,p_2,\ldots,p_{n-1}\}} \sign(\p) \wt(p_1) \wt(p_2) \cdots \wt(p_{n-1}),
$$
where the summation is over noncrossing (unordered) families of $n-1$ paths in $G'$ using all the sources except $i$, and all the sinks except $j$. In other words, the coefficient of $x^a y^b$ in $|(L(x)-y)|_{i,j}$ counts (with weights) families of highway paths that
\begin{itemize}
\item start at all sources but $i$ and end at all sinks but $j$;
\item do not cross each other;
\item cross the $x$-line exactly $a$ times;
\item contain exactly $b$ abrupt paths and $n-b-1$ non-abrupt paths.
\end{itemize}
\end{thm}
\begin{example}
Let $(n,m,k)=(3,3,1)$. The Lax matrix is given as follows: $L(x)=$
$$
\left(\begin{array}{ccc}
q_{1,1}x+q_{2,3}x+q_{3,2}x & q_{1,1}q_{2,1}x+ q_{1,1}q_{3,3}x+q_{2,3}q_{3,3}x & q_{1,1}q_{2,1}q_{3,1}x+x^2\\
q_{1,2}q_{2,2}q_{3,2}+x & q_{1,2}x+q_{2,1}x+q_{3,3}x & q_{1,2}q_{2,2}x+q_{1,2}q_{3,1}x+q_{2,1}q_{3,1}x\\
q_{1,3}q_{2,3}+q_{1,3}q_{3,2}+q_{2,2}q_{3,2} & q_{1,3}q_{2,3}q_{3,3}+x & q_{1,3}x +q_{2,2}x+q_{3,1}x
\end{array} \right)
$$
Consider the minor $|(L(x)-y)|_{1,2}=$ with rows $2,3$ and columns $1,3$. Here are some terms that appear in it:
$$|(L(x)-y)|_{1,2}= q_{2,2}x^2-q_{1,3}q_{2,3}q_{1,2}q_{2,2}x-xy - \cdots.$$
The term $q_{2,2}x^2$ for example is formed by two paths: one starting at source $3$, turning at $q_{1,3}$, turning at $q_{1,2}$, going through at $q_{2,2}$, turning at $q_{3,2}$, turning at $q_{3,1}$ and thus finishing in
sink $3$; the other a staircase path starting at source $2$, turning at each of $q_{1,2}$, $q_{1,1}$, $q_{2,1}$, $q_{2,3}$, $q_{3,3}$, $q_{3,2}$ and finishing at sink $1$. The first path contributes weight $q_{2,2}x$,
while the second contributes weight $x$. Note that source $1$ and sink $2$ remain unused, as they should according to the theorem. Also note the the paths give bijection $2 \mapsto 1$ and $3 \mapsto 3$ between the
used sources and sinks. The induced permutation is the identity permutation, and that is why we have sign $+$ in front of this term.
The term $-xy$ corresponds to one abrupt path of weight $-y$ from source $3$ to sink $3$, and one staircase path, the same as for the previous term. The term $-q_{1,3}q_{2,3}q_{1,2}q_{2,2}x$ corresponds to two paths
that induce the map $2 \mapsto 3$ and $3 \mapsto 1$ on sources and sinks, of weights $q_{1,2}q_{2,2}x$ and $q_{1,3}q_{2,3}$ respectively. The minus sign arises since the induced permutation in $S_2$ is a transposition.
\end{example}
Let us prove Proposition~\ref{prop:Q}.
(i) If $N = 1$, the result is easy: $g_j/g_{j+n}$ vanishes to order $n$ at $O_1$. We also have a shift automorphism $s: \mL \to \mL$ (see the proof of Proposition \ref{prop:Q}(ii)) which sends $O_1$ to $O_1$ and pulls $g_j/g_{j+1}$ back to $g_{j+1}/g_{j+2}$. We will thus assume $N >1$.
For each $i$, define
$$v^{(i)} := \left((-1)^1 |L(x)-y|_{i,1}, \ldots, (-1)^n |L(x)-y|_{i,n}\right)^T.$$
We shall think of $v^{(i)}$ as a vector whose entries lie in the coordinate ring of the affine plane curve $\tilde C_f$. Like the vector $g = (g_1,g_2,\ldots,g_n)^T$, the vectors $v^{(i)}$ are (nonzero) eigenvectors of the matrix $L(x)$. The matrix $L(x)$ is singular along the curve $\tilde C_f$, but generically it has rank $n-1$. Thus for any $i$, the vectors $g$ and $v^{(i)}$ are multiples of each other. More precisely, $g/v^{(i)}$ is a rational function on $C_f$. To show that $g_j/g_{j+1}$ has a zero at $O_{j+1}$, we shall calculate using a convenient choice of $v^{(i)}$.
Choose $v = v^{(j)}$. Then $v_j =|L(x)-y|_{j,j}$ and $v_{j+1} =|L(x)-y|_{j,j+1}$.
Thus, $v_j$ counts the families of paths that start at all sources but $j$ and end at all sinks but $j$,
as in Theorem \ref{thm:minor}.
Let us make the substitution $(x,y) \sim (u^{n'},-(-\sigma_j)^{1/n'}u^{k'})$ inside $v_j$, and let $v'_j(x,y)$ denote the terms of in $v_j(x,y)$ that give the lowest degree in $u$ after the substitution. Call this degree $d$. The terms in $v_j(x,y)$ are obtained by either taking abrupt paths, or by taking coils. In the proof of Lemma~\ref{lem:Q} we defined the $N$ coils in the network $G$, which we denote $C_1,C_2,\ldots,C_{N}$. To obtain a path family $\p$ contributing to $v'_j$, instead of $C_j$, we include the $n'-1$ abrupt paths which use the vertices $j'$ where $j \neq j'$ but $j \equiv j' \mod N$. For each of the coils $C_t$ where $t \neq j$, we can either include the coil (that is, include the $n'$ paths in $G'$) in $\p$, or we use the corresponding $n'$ abrupt paths instead. In particular, considering $C_{j+1}$ we see that $v'_j$ has a factor of $(\sigma_{j+1}x^{k'} + (-y)^{n'})$. This factor vanishes under the substitution $(x,y) \sim (u^{n'},-(-\sigma_{j+1})^{1/n'}u^{k'})$, thus creating a zero at the point $O_{j+1}$ of order at least $d+1$.
For $v_{j+1}$, we count families of paths that start at all sources but $j$ and end at all sinks but $j+1$. Let $v'_{j+1}(x,y)$ denote the terms of lowest degree in $v_{j+1}(x,y)$. This lowest degree is again equal to $d$ as well (we caution that if $j = n$, then we take $v_{j+1}:=x^{-1} v_1$). The calculation of $v'_{j+1}(x,y)$ is similar to that of $v_j(x,y)$, except that instead of a single ``incomplete'' coil with index $j$ modulo $N$, we have two incomplete coils $C_j$ and $C_{j+1}$ with indices $j$ and $j+1$ modulo $N$. We obtain
$$
v'_{j+1}(x,y) = a(q) x^\alpha y^\beta \prod_{t \neq j,j+1} (\sigma_{t}x^{k'} + (-y)^{n'})
$$
where $a(q)$ is some nonzero polynomial in $q_{ab}$-s, and $\alpha, \beta$ are nonnegative integers, and the product is over $t \in \{1,2,\ldots N\}$ not equal to $j$ or $j+1$ modulo $N$. As a result, we see that $v_{j+1}$ vanishes at $O_{j+1}$ of order exactly $d$.
Thus $g_j/g_{j+1}$ has a zero at $O_{j+1}$. The fact that $g_j/g_{j+1}$ vanishes to exactly order $1$ follows since we know that $g_j/g_{j+n}$ vanishes to the order of $n'$ at each $O_i$.
(ii) For $L(x) \in \beta^{-1}(f)$, we have
rational functions $g_i/g_{i+1}$ on $C_f$.
To emphasize the dependence of $g_i/g_{i+1}$ on $L$,
we write $g_i/g_{i+1}(L)$.
Let $\varsigma^\ast : \mL \to \mL$ be the shift map given by $L(x) \mapsto P(x) L(x) P(x)^{-1}$. (We will introduce the shift operator $\varsigma$ on $\mM$ in \S~\ref{sec:shift}.) Then $\beta(\varsigma^\ast(L(x))) = \beta(L(x))$ is invariant under $r$.
Let $Y$ be a Zariski dense subset of $\beta^{-1}(\oV)$
such that $\varsigma^\ast(Y) = Y$.
Then there exists a nonnegative integer $a_i$ given by
$$
a_i = \max\{ \mathrm{ord}_P(g_i/g_{i+1}(L))_\infty ~|~ L \in Y \},
$$
where $\mathrm{ord}_P(g_i/g_{i+1}(L))_\infty$ denotes the
order of the pole of $g_i/g_{i+1}(L)$ at $P$.
Since $g_i/g_{i+1}(\varsigma^\ast(L)) = g_{i-1}/g_{i}(L)$ as rational function on
$C_{\beta(L)}$,
we conclude that for all $i$, we have
$\mathrm{ord}_P(g_i/g_{i+1}(L))_\infty = a := \min_i\{a_i\}$.
But we know that $\mathrm{ord}_P(g_{i}/g_{i+n}(L))_\infty=n$
for all $i$ and $L \in \mL$. Thus we must have $a=1$.
Since this value does not depend on the choice of $Y$, the claim follows.
\subsection{A holomorphic differential}
Let $\zeta$ be the differential form on the curve $C_f$ given by
$$
\zeta = \frac{x^{k-1} dy}{\frac{\partial f}{\partial x}}.
$$
\begin{lem}\label{lem:zeta}
The divisor of the differential form $\zeta$ is supported on $S_{P,O} = \{P,O_1,\ldots,O_N\}$:
$$
(\zeta) = (k'-1) \sum_{i=1}^N O_i + \left((n-1)m-k-M \right) P.
$$
When $g \geq 1$, it is holomorphic.
\end{lem}
\begin{proof}
By using local expansions of $f$ around $O_i$ and $P$, we get
\begin{align*}
&\left(\frac{\partial f}{\partial x}\right)
= (n'-k'n) \sum_{i=1}^N O_i + n(k+m-1) P,
\\
&(dy) = (k'-1) \sum_{i=1}^N O_i - (m+k+M)P.
\end{align*}
From these and $(x)$ in \S 4.3, we obtain $(\zeta)$.
Now we prove that if $g \geq 1$ then $(n-1)m-k-M \geq 0$.
Let $p$ be the number of integer points inside a parallelogram in $\R^2$,
whose vertices are $(0,n)$, $(k,0)$, $(k+m,0)$ and $(m,n)$.
We have $p = 2g + M-1$, since the parallelogram is composed of
the Newton polygon $N(f)$ and its copy, sharing the upper hull of $N(f)$.
Then the claim is equivalent to that if $g \geq 1$, then $p + N-1 \geq M+k$.
When $k=n$, $N=n$ follows. Then we have $p+ N-1 - (M+k) = p - (M+1)$.
It reduces to $2g-2$, which is non-negative when $g \geq 1$.
When $m+k=n$, $M=n$ follows. Then we have $g = \frac{1}{2}(n(m-1)-m-N+1)$
which is negative if $m = 1$. So we assume $m \geq 2$.
We have $p+N-1=m(n-1)$ and $M+k = 2n-m$.
Thus, we obtain $p+ N-1 - (M+k) \geq 0$ when $m \geq 2$.
When $n > k$,
we prove the claim by choosing $M+k$ different points from $p$ points
in the parallelogram.
When $m+k < n$ (resp. $m+k > n$), we can choose $M-1$ points on the upper hull
of $N(f)$, $k$ points; $(i,n-i)$ for $i=1,\ldots,k$ inside the upper (resp. lower)
triangle, and one point inside the lower (resp. upper) triangle.
Finally the claim follows.
\end{proof}
\subsection{About $(c_1,\ldots,c_M) \in \mathcal{S}_f$}
Recall that $c_\ell$ is the coefficient of the maximal power of $x$ in the $(\ell, \ell+1)$ entry of $L(x)^{n/M}$ defined in \S \ref{subsec:eigen}.
\begin{lem}\label{lem:c-snake}
When $L(x) = \alpha(q=(q_{ij}))$, we have
$$c_{\ell} = \sum_{i+j \equiv \ell+1 \mod M} q_{ij}.$$
\end{lem}
\begin{proof}
By Lemma \ref{L:mMmL}, it suffices to prove the lemma for $f(x,y) \in \psi(\mM)$.
We apply Lemma \ref{lem:entry} to interpret the $c_{\ell}$ as counting almost snake paths in $G$, that is, paths that turn at all steps but one. The almost snake paths $p$ that are counted start at vertex $\ell$ and end at vertex $\ell+1$, and when drawn in the cylindric network $G'$, they are disjoint unions of $n/M$ paths. In other words, $p$ wraps around the picture Figure \ref{fig:toda1} horizontally $n/M$ times. Such paths $p$ are completely determined by the single vertex that the path goes through. With $\ell$ fixed, the weights of these vertices are exactly the $q_{i,j}$ with $i+j \equiv \ell+1 \mod M$.
\end{proof}
Recall that we defined $f_c$ in \eqref{eq:fc}.
\begin{lem}\label{lem:c-f}
When $L(x) \in \beta^{-1}(f)$, we have
\begin{align}\label{eq:prodc-f}
\prod_{\ell=1}^M c_\ell= f_c = \pm \sum_{(j,i) \in L_c} f_{i,j}.
\end{align}
Thus the product of all $c_\ell$ is constant on $\psi^{-1}(f)$.
\end{lem}
\begin{proof}
We first show that
\begin{equation}
\label{eq:deg}
{\text{degree of $f_{i,j}$ in the $q_{r,s}$}} = (m+k) n - ((m+k) i + n j).
\end{equation}
Indeed, lift a family $\p$ of paths that contributes to $f_{i,j}$ (as in Theorem \ref{thm:mit}) to the universal cover, as in Figure \ref{fig:toda2}. By Theorem \ref{thm:mit}, the family $\p$ includes exactly $i$ abrupt paths and $n-i$ non-abrupt paths. Let us suppose that the sources used by the $n-i$ non-abrupt paths are $(1/2, z_1), \ldots, (1/2, z_{n-i})$ and the sinks are $(m+1/2, w_1), \ldots, (m+1/2, w_{n-i})$, where $1 \leq z_r \leq n$ and $1 \leq w_r \leq n+m$. To agree with our convention in Figure \ref{fig:toda1} that source and sink labels increase as we go down, we will take the $y$-coordinate to increase as we go down in Figure \ref{fig:toda2}; but otherwise, the coordinates we are using are the usual Cartesian coordinates.
For each $r$, let $w'_r \in [1-k,n-k]$ be chosen so that $w'_r \equiv w_r \mod n$.
From Theorem \ref{thm:mit} it also follows that $$\sum_{r=1}^{n-i} (w_{r} - w'_{r})/n = i,$$ as this is the number of times the $x$-line would be crossed by the paths.
Also, we know that $\{w_1, \ldots, w_{n-i}\} = \{z_1-k, \ldots, z_{n-i}-k\} \mod n$ and thus $\{w'_1, \ldots, w'_{n-i}\} = \{z_1-k, \ldots, z_{n-i}-k\}$. This is because our collection of non-abrupt paths must form a collection of cycles in $G$.
The number of the $q_{r,s}$ a path from $(1/2, z_r)$ to $(m+1/2, w_s)$ picks up is equal to $z_r+m-w_s$, and summing over all the non-abrupt paths in $\p$ we get
\begin{align*}
{\text{degree of $f_{i,j}$ in the $q_{r,s}$}} &= m(n-j)+\sum_{r=1}^{n-i} z_r -\sum_{s=1}^{n-i} w_s \\
&= m(n-j)+\sum_{r=1}^{n-i} z_r -\sum_{s=1}^{n-i} w'_s + jn \\
&=m(n-i) + (n-i)k -jn = (m+k) n - ((m+k) i + n j),
\end{align*}
as claimed. This degree is equal to $M$ when $(j,i) \in L_c$.
Now we prove the lemma. Let us partition all the edges in the network $G$ into $M$ snake paths, $H_1, \ldots, H_M$. Suppose $p$ is a closed path in $G$. Then we can partition $p$ into {\it {snake intervals}} -- maximal contiguous segments of $p$ that follow one of the $H_\ell$. Such snake intervals are connected by a {\it horizontal step} going through a vertex (and picking up its weight), and moving from a snake path $H_\ell$ to the next snake path $H_{\ell+1}$. It is easy to see that the vertices which separate $H_{\ell}$ from $H_{\ell +1}$ are exactly the
ones labelled with $q_{ij}$ where $i+j \equiv \ell+1 \mod M$.
In order for $p$ to be closed, it must consist of $c$ horizontal steps, where $c$ is a multiple of $M$, since $p$ must come back to the snake path it started at. When $(j,i) \in L_c$, \eqref{eq:deg} shows that all families of toric paths that contribute to $f_{i,j}$ consist of just a single
closed path in $G$ that picks up exactly one of the $q_{ij}$ from each of the sets $T_\ell = \{q_{ij} \mid i+j \equiv \ell+1 \mod M\}$, for $\ell = 1, \ldots, M$. Using Lemma \ref{lem:c-snake}, we see that each such term appears exactly once in the product $\prod_{\ell=1}^M c_\ell$.
We need to show that this is not only an injection, but a bijection. For each term contributing to
$\prod_{\ell=1}^M c_\ell$, it suffices to find a closed path in $G$ that has this weight.
By \eqref{eq:deg}, such a closed path would necessarily contribute to $f_{i,j}$ for $(j,i) \in L_c$.
For each $q_{r,s}$ appearing in our chosen term of
$\prod_{\ell=1}^M c_\ell$, we draw a horizontal step through the vertex labelled $q_{r,s}$ in the network $G$. From the endpoint of each such horizontal step (say from $H_\ell$ to $H_{\ell+1}$), attach a snake interval in $H_{\ell+1}$ leading to the start point of the horizontal step which goes from $H_{\ell+1}$ to $H_{\ell+2}$. Some of these snake intervals may be empty. When we glue everything together, we get the desired closed path.
\end{proof}
\subsection{Sketch proof of Theorem \ref{thm:double}}\label{pf:thm:double}
The proof is analogous that of \cite{vMM}. We explain the differences.
When $n-k \leq m \leq 2n-k$,
the triple of integers $(N, M, M^\prime)$ in van Moerbeke and Mumford's work \cite{vMM} corresponds to the parameters
$(n,n-k,m+k-n)$ in our case.
One first shows that there is a positive regular divisor of degree $g$, denoted $\D$, satisfying Definition \ref{def:D}. The correspondence $L(x) \mapsto \D$ is the main construction in \cite[Theorem 1]{vMM}. Their results do not formally apply since our Lax matrix is not {\it regular} in their terminology. Nevertheless, the properties of $L(x) \mapsto \D$ is proved in the same manner as \cite[Lemmas 3 and 4]{vMM}, where Proposition~\ref{prop:Q} takes the place of \cite[Lemma 2]{vMM}.
The divisors $D_i$ and $\bar D_j$ of Theorem \ref{thm:double} are obtained by shifting and transposing the Lax matrix. \cite[Proposition 1]{vMM} then says, in our terminology,
$$(\Delta_{i,j} \zeta) = D_j + D'_i + (j-i -1)P + \sum_{r=j+1}^{i-1} O_r$$
where $\zeta$ is the differential of Lemma \ref{lem:zeta}.
We now take $R = (\zeta)$ to obtain Theorem \ref{thm:double}. The statement about common points is on \cite[p.117]{vMM}.
\subsection{Proof of Theorem \ref{thm:eta}}\label{subsec:thm:eta}
We shall also use the following lemma which is proved in a similar way to \cite[Lemma 5]{vMM}.
\begin{lem} \label{lem:D}
Suppose $\D$ is a positive divisor on $C_f$ of degree $g$.
which is regular with respect to the points $P$ and $O_j$. We have
\begin{enumerate}
\item [(i)] $\dim \mathcal{L}(\mathcal{D}+ (n-i)P-\sum_{j=i+1}^n O_j)=1$
for $i \in \Z$.
\item[(ii)]
Suppose for each $i \in \Z$, we have fixed a nonzero element $h_i \in \mathcal{L}(\mathcal{D}+ (n-i)P-\sum_{j=i+1}^n O_j)$.
Then for $i_1 \leq i_2$, and any $h \in \mathcal{L}(\mathcal{D}+ (n-i_1) P-\sum_{j=i_2+1}^n O_j)$, there are unique scalars $b_{i_1},\ldots,b_{i_2}$
such that
$$
h = \sum_{i=i_1}^{i_2} b_i h_i,
$$
with $b_{i_1}, b_{i_2} \neq 0$.
\end{enumerate}
\end{lem}
Let us prove the theorem.
Due to Theorem~\ref{thm:double}, Lemma~\ref{lem:c-f} and the assumption
on $\mathcal{V}$, for $L(x) \in \beta^{-1}(\mathcal{V})$
we have $f := \beta(L(x)) \in \mathcal{V}$,
and $\eta_f(L(x)) = ([\D],c,O)$ satisfying (b).
In the following we construct the inverse of $\eta_f$ \eqref{eq:def-eta}
for $f \in \mathcal{V}$.
Around $P \in C_f$ we have the local expansion
$(x,y) = (x_0 u^{-n}, y_0 u^{-m-k})$ as $u \to 0$.
We take $c=(c_1,\ldots,c_M) \in \mathcal{S}_f$,
and recursively define $d_i ~(i \in \Z)$ by
$$
d_n = 1, \qquad d_i = c_\ell\, d_{i+1} \quad \text{ for } i \equiv \ell \mod M.
$$
Let $h_i \in \mathcal{L}(\mathcal{D}+ (n-i)P-\sum_{j=i+1}^n O_j)$
have an expansion around $P$ as
$h_i = d_i u^{-n+i} + \cdots$.
Since
$y \, h_i \in \mathcal{L}(\mathcal{D}+ (n+m+k-i)
P-\sum_{j=-k+i+1}^n O_j)$,
using Lemma \ref{lem:D}, we have unique scalars $b_{i,j}$ satisfying
$$
y \, h_i = \sum_{j=n-m-k}^{n-k} x \, b_{i,i+j} h_{i+j},
\qquad b_{i,i-m-k}, b_{i,i-k} \neq 0.
$$
By expanding it around $P$, we get
$b_{i,i-m-k} = y_0 d_i / d_{i-m-k} = y_0 / f_c^{\frac{m+k}{M}}$
independent of $i$.
Define an infinite matrix $A = (a_{ij})_{i \in [n], j \in \Z}$ by
$$
a_{i,i+j} =
\begin{cases}
b_{i,i+j} \displaystyle{\frac{f_c^{\frac{m+k}{M}}}{y_0}}
& \text{for } -m-k \leq j \leq -k, \\
0 & \text{otherwise}.
\end{cases}
$$
By using \eqref{eq:L-A} for this $A$,
we obtain $L(A;x) \in \beta^{-1}(f)$.
\subsection{Proof of Theorem \ref{thm:phi}}\label{subsec:thm:phi}
Fix $L(x) \in \beta^{-1}(f)$, and assume that $\alpha^{-1}(L)$ contains $Q \in \mathring{\mM}$. By Lemma \ref{lem:R-matrix}, the action of the finite symmetric group $\mS_m$ via the $R$-matrix action of \S \ref{sec:dynamics} preserves the Lax matrix $L(x)$. (See \S \ref{sec:actions} for more discussion of this action.) Thus $\mS_m \cdot Q \subseteq \alpha^{-1}(L)$.
By \eqref{eq:q1q2}, the $\mS_m$ action induces the permutation action on the $m$ quantities $\epsilon_1(q),\ldots,\epsilon_m(q)$, which corresponds exactly to the permutation action on $R_A$. Also, for $f \in\oV$, these $m$ quantities $\epsilon_1(q),\ldots,\epsilon_m(q)$ are distinct. So $|\mS_m \cdot Q| = m! =|\alpha^{-1}(L)|!$ implying that $\mS_m \cdot Q = \alpha^{-1}(L)$. Thus the map $\alpha^{-1}(L) \to (L,R_A)$ is an injection, and the claim follows.
\iffalse
\remind{This argument is perhaps a bit too sketchy at the moment.}
We sketch an argument that $|\alpha^{-1}(L)| \leq m!$ whenever $\epsilon_1(q),\ldots,\epsilon_m(q)$ are non-zero. Consider a factorization $L = QL'$ where $L'$ is a product of $m-1$ factors $Q_1,Q_2,\ldots,Q_{m-1}$. By induction we may assume that $L'$ has at most $m-1$ factorizations, and so it is enough to show that the factorization problem $L = QL'$ has at most $m$ solutions. The matrix $L$ has interesting entries on $m$ consecutive diagonals, the matrix $Q$ has interesting entries on one diagonal, and the matrix $L'$ has interesting entries on $m-1$ consecutive diagonals. Expanding the equation $L = QL'$ we get $mn$ equations in the entries of $Q$ and $L'$, treating the entries of $L$ as fixed. We first make the observation that the condition that $\epsilon_1(q),\ldots,\epsilon_m(q)$ are non-zero imply that all the $q_{i,j}$ are non-zero, which in turn imply (via these $mn$ equations) that all entries of $Q = (q_1,q_2,\ldots,q_n)$ and $L'$ are determined if we know just one of the $q_i$, say $q_1$. So it is enough to show that there are at most $m$ possible values of $q_1$. Considering the structure of the equations, we can repeatedly substitute them into others until we obtain a single equation in $q_1$. The key observation is that this equation has degree $\leq m$, and thus there are at most $m$ possible values of $q_1$. This completes the proof.
\fi
\section{Actions on $\mathcal{M}$}
\label{sec:actions}
We study the family of commuting extended affine symmetric group actions on $\mathcal{M}$,
introduced in \S~\ref{sec:dynamics}. We also study another family of actions on $\mM$, which we call the {\it snake path actions}.
The main result in this section is Theorem~\ref{thm:commuting-actions}.
Throughout \S \ref{sec:actions}--\ref{sec:fay}, we fix $f \in \oV$.
For simplicity, we write $Q_i := Q_i(x)$ in the rest of this section.
\subsection{The behaviour of spectral data under the extended affine symmetric group actions}\label{sec:WWonM}
Recall the definitions of energy and $R$-matrix from \S \ref{sec:Rmatrix}. Let $A(x)$ and $B(x)$ be two $n$ by $n$ matrices of the shape \eqref{eq:Q}, with diagonal entries $\aa = (a_1,\ldots,a_n)$ and
$\bb = (b_1,\ldots,b_n)$. Define the {\it {energy}} by $E(A,B): = E(\aa,\bb)$ and the {\it $R$-matrix} to be the transformation
$(A(x),B(x)) \mapsto (B'(x),A'(x))$, where $B'(x)$ and $A'(x)$ have diagonal entries $\bb'$ and $\aa'$ respectively, and $R(\aa,\bb) = (\bb',\aa')$.
The following is proved for example in \cite[Corollary 6.4]{LP}.
\begin{lem}\label{lem:R-matrix}
Suppose $R(A(x),B(x)) = (B'(x),A'(x))$. Then we have $A(x)B(x)=B'(x)A'(x)$.
\end{lem}
Indeed, Lemma \ref{lem:R-matrix} and the condition that $R$ is non-trivial uniquely determine the $R$-matrix as a birational transformation.
Now we reformulate the $W \times \tilde W$ actions introduced in \S~\ref{sec:dynamics}, in our current terminology.
The action of the extended affine symmetric group $W$ is generated by
$s_i ~(1 \leq i \leq m-1)$ and $\pi$, where for $1 \leq i \leq m-1$, $s_i$ acts by
$$
s_i: (Q_1,Q_2,\ldots,Q_m) \longmapsto
(Q_1,\ldots,Q'_{i+1},Q'_i,\ldots,Q_m)
$$
where $(Q'_{i+1},Q'_i)$ is the $R$-matrix image of $(Q_i,Q_{i+1})$.
The operator $\pi$ acts as
$$
\pi: (Q_1,Q_2,\ldots,Q_m) \longmapsto (Q_2,Q_3,\ldots,Q_m,P(x)^kQ_1P(x)^{-k}).
$$
The operator $e_u$ acts by moving, via the $R$-matrix, the last $u$ terms of
$$
(Q_{u+1},\ldots,Q_m,P(x)^{k}Q_1P(x)^{-k},\ldots,P(x)^kQ_uP(x)^{-k})
$$
to the first $u$ positions, keeping them in order.
Recall from \S\ref{sec:dynamics} that we also have $\tilde Q = (\tilde Q_1,\ldots,\tilde Q_N) = (\tilde q_{i,j})$ coordinates on $\mM$,
where $\tilde Q_i := \tilde Q_i(x)$ is an $mn'$ by $mn'$ matrix:
$$
\tilde Q_i(x) = \left(\begin{array}{cccc}
\tilde q_{i,1} & 0 & 0 &x\\
1&\tilde q_{i,2} &0 &0 \\
0&\ddots&\ddots&0 \\
0&0&1&\tilde q_{i,mn'}
\end{array} \right).
$$
Similarly,
the action of $\tW$ generated by $\tilde s_r ~(1 \leq r \leq N-1)$
and $\tilde \pi$ on $\tilde Q = (\tilde Q_1,\ldots,\tilde Q_N) \in \mM$
is described as follows:
for $1 \leq r \leq N-1$, $\tilde s_r$ acts by
$$
\tilde s_r: (\tilde Q_1,\ldots, \tilde Q_N) \longmapsto (\tilde Q_1,\ldots,\tilde Q'_{r+1},\tilde Q'_r,\ldots,\tilde Q_N)
$$
where $(\tilde Q'_r,\tilde Q'_{r+1})$ is the $R$-matrix image of $(\tilde Q_{r+1},\tilde Q_r)$.
The operator $\tilde \pi$ acts as
$$
\tilde \pi: (\tilde Q_1,\tilde Q_2,\ldots,\tilde Q_N) \longmapsto
(\tilde Q_2,\tilde Q_3,\ldots,\tilde Q_N,\tilde P(x)^{m \bar k'}\tilde Q_1
\tilde P(x)^{-m \bar k'}),
$$
where $\tilde P(x)$ is the $mn'$ by $mn'$ version of $P(x)$ \eqref{eq:P}.
Recall that we denote by $\mS_m \subset W$ (resp. $\mS_N \subset \tilde W$) the finite symmetric group generated by $s_1,\ldots,s_{m-1}$ (resp. $\tilde s_1,\ldots,\tilde s_{N-1}$). Then $W$ is generated by $\mS_m$ and $\pi$, and $\tilde W$ is generated by $ \mS_N$ and $\tilde \pi$.
Recall that in \S~\ref{subsec:main}
we define divisors $\mathcal{O}_u$ and $\mathcal{A}_u$ on $C_f$ as follows:
\begin{align}
\label{def:O}
&\mathcal{O}_{u} = uP-\sum_{i=N-u+1}^N O_i,
\qquad \mathcal{A}_u = uP-\sum_{i=1}^u A_i,
\end{align}
for $u \in \Z_{\geq 0}$.
Let $\tau$ acts on $\mathcal{S}_f$ by $(c_1,\ldots,c_M) \mapsto (c_M, c_1,\ldots,c_{M-1})$.
\begin{thm}\label{thm:finite}
\begin{enumerate}
\item[(i)]
The actions of $\mS_m$ and $\pi$ on $\psi^{-1}(f)$ induce the following
transformations on $\Pic^{g}(C_f) \times \mathcal{S}_f \times R_O \times R_A$: for $r=1, \ldots, m-1$,
$s_r$ induces
\begin{align}\label{eq:vertical-s}
([\mathcal{D}],c,O,A)
\mapsto ([\mathcal{D}],c,O,\nu_{r}(A)),
\end{align}
and $\pi$ induces
\begin{align}\label{eq:vertical-pi}
([\mathcal{D}],c,O,A) \mapsto
([\mathcal{D} - \mathcal{A}_1],\tau^{-1}(c),O, \nu_{m-1} \nu_{m-2} \cdots \nu_{1}(A)).
\end{align}
\item[(ii)]
The actions of $\mS_{N}$ and $\tilde\pi$ on $\psi^{-1}(f)$ induce
the following transformations
on $\Pic^{g}(C_f) \times \mathcal{S}_f \times R_O \times R_A$:
for $r=1, \ldots, N-1$, $\tilde s_r$ induces
\begin{align}\label{eq:horizontal-s}
([\mathcal{D}],c,O,A)
\mapsto ([\mathcal{D}],c,\tilde \nu_{N-r}(O),A),
\end{align}
and $\tilde \pi$ induces
\begin{align}\label{eq:horizontal-pi}
([\mathcal{D}],c,O,A) \mapsto
([\mathcal{D} + \mathcal{O}_1],\tau(c), \tilde \nu_1 \cdots \tilde \nu_{N-1}(O),A).
\end{align}
\end{enumerate}
\end{thm}
Theorems \ref{thm:finite} will be proved in \S\ref{sec:prooffinite}.
The following theorem states that the commuting $\Z^m$ and $\Z^N$ actions on $\Pic^g(C_f)$ are linearized by the spectral map $\phi$.
\begin{thm}\label{thm:commuting-actions}
The following diagrams are commutative:
\begin{align}\label{eq:comm-e}
\xymatrix{
\psi^{-1}(f) \ar[d]_{e_u} \ar[r]^{\phi \qquad \qquad}
& \Pic^{g}(C_f) \times \mathcal{S}_f \times R_O \times R_A\
\ar[d]^{(-[\mathcal{A}_u],\,\tau^{-u}, \,id,\,id)}
\\
\psi^{-1}(f) ~~ \ar[r]_{\phi \qquad \qquad}
& \Pic^{g}(C_f) \times \mathcal{S}_f \times R_O \times R_A, \\
}
\end{align}
for $u=1,\ldots,m$, and
\begin{align}\label{eq:comm-ve}
\xymatrix{
\psi^{-1}(f) \ar[d]_{\tilde e_u} \ar[r]^{\phi \qquad \qquad}
& \Pic^{g}(C_f) \times \mathcal{S}_f \times R_O \times R_A
\ar[d]^{(+[\mathcal{O}_u],\,\tau^{u},\,id,\,id)}
\\
\psi^{-1}(f) ~~ \ar[r]_{\phi \qquad \qquad}
& \Pic^{g}(C_f) \times \mathcal{S}_f \times R_O \times R_A, \\
}
\end{align}
for $u=1,\ldots,N$.
Here $-[\mathcal{A}_{u}]$ and $+[\mathcal{O}_u]$ respectively
act on $\Pic^g(C_f)$ as
$[\mathcal{D}] \mapsto [\mathcal{D} - \mathcal{A}_{u}]$ and
$[\mathcal{D}] \mapsto [\mathcal{D} + \mathcal{O}_{u}]$.
\end{thm}
\begin{proof}
We shall show \eqref{eq:comm-e}.
The proof of \eqref{eq:comm-ve} is done in the similar way by exchanging
the rules of the special points $\{A_i\}_{i \in [m]}$ for that of
$\{O_j\}_{j \in [N]}$.
From \eqref{eq:vertical-s} and \eqref{eq:vertical-pi}, it follows that
a non-trivial part is to show $e_u$ \eqref{eq:Za-action} induces the action on
$\Pic^{g}(C_f) \times R_A$:
$$
([\mathcal{D}],A) \mapsto
([\mathcal{D} - \mathcal{A}_u],A).
$$
Note that for $1 \leq i < j \leq m$,
$\nu_{j-1} \nu_{j-2} \cdots \nu_i \in \mathfrak{S}_m$ acts on $R_A$ as
\begin{align}\label{eq:nuA}
(A_1,\ldots,A_m)
\mapsto (A_1,\ldots,A_{i-1},A_{i+1},\ldots,A_j,A_i,A_{j+1},\ldots,A_m).
\end{align}
Thus, \eqref{eq:vertical-pi} indicates that $\pi$ induces
$([\mathcal{D}],(A_1,\ldots,A_m)) \mapsto
([\mathcal{D}-\mathcal A_1],(A_2,\ldots,A_m,A_1))$, and $\pi^u$ induces
$$
([\mathcal{D}],(A_1,\ldots,A_m)) \mapsto
([\mathcal{D}-\mathcal A_u],(A_{u+1},\ldots,A_m,A_1,A_2,\ldots,A_m)).
$$
Further, \eqref{eq:vertical-s} denotes that $s_r$ does not change a point
in $\Pic^g(C_f)$ and acts on $R_A$ as a permutation $\nu_r \in \mathfrak{S}_m$.
The rest part $(s_u \cdots s_{m-1})(s_{u-1} \cdots s_{m-2})
\cdots (s_1 \cdots s_{m-u})$ of $e_u$ changes
$(A_{u+1},\ldots,A_m,A_1,A_2,\ldots,A_m) \in R_A$ following
(the inverse of) \eqref{eq:nuA} as
\begin{align*}
&(A_{u+1},\ldots,A_m,A_1,A_2,\ldots,A_m)
\stackrel{\nu_1 \cdots \nu_{m-u}}{\mapsto}
(A_1,A_{u+1},\ldots,A_m,A_2,\ldots,A_m)
\\
&\stackrel{\nu_2 \cdots \nu_{m-u+1}}{\mapsto}
\cdots
\stackrel{\nu_{u-1} \cdots \nu_{m-2}}{\mapsto}
(A_1,\ldots,A_{u-1},A_{u+1},\ldots,A_m,A_u)
\stackrel{\nu_u \cdots \nu_{m-1}}{\mapsto}
(A_1,\ldots,A_m). \qedhere
\end{align*}
\end{proof}
We remark that $e_u$ and $\tilde e_u$ act on the sets $R_A$ and $R_O$ as the identity transformations.
\subsection{The shift operator}\label{sec:shift}
We define a shift operator $\varsigma$ acting on $\mM$ as
\begin{align}
\varsigma: (Q_i)_{1 \leq i \leq m} \mapsto (P(x) Q_i P(x)^{-1})_{1 \leq i \leq m}.
\end{align}
It is easy to see that it induces an action $\varsigma^\ast: \mL \to \mL$ given by
$\varsigma^\ast : L(x) \mapsto P L(x) P^{-1}$. The following result generalizes \cite[Proposition~2.6]{Iwao07}.
\begin{prop}\label{prop:shift}
The following diagram is commutative:
\begin{align}\label{eq:comm-s}
\xymatrix{
\psi^{-1}(f) \ar[d]_{\varsigma} \ar[r]^{\phi \qquad \qquad }
& \Pic^{g}(C_f) \times \mathcal{S}_f \times R_O \times R_A~~\
\ar[d]^{(+[P-O_N],\,\tau,\,\tilde \nu_1 \cdots \tilde \nu_{N-1}, \,{id})}
\\
\psi^{-1}(f) ~~ \ar[r]_{\phi \qquad \qquad}
& \Pic^{g}(C_f) \times \mathcal{S}_f \times R_O \times R_A, \\
}
\end{align}
where $+[P-O_N]$ acts on $\Pic^g(C_f)$ as
$[\mathcal{D}] \mapsto [\mathcal{D} + P - O_N]$.
\end{prop}
\begin{proof}
Suppose $Q \in \psi^{-1}(f)$ and $\phi (Q) = ([\mathcal{D}],c,O,A)$.
Then
$\phi \, \circ \, \varsigma (Q) = ([\D'],\tau(c),\tilde \nu_1 \cdots \tilde \nu_{N-1}(O),A)$,
where $\D'$ is some positive divisor of degree $g$.
We set $(O'_1,\ldots,O'_N) := \tilde \nu_1 \cdots \tilde \nu_{N-1}(O_1,\ldots,O_N) = (O_N,O_1,\ldots,O_{N-1})$.
Recall that $g=(g_1,g_2,\ldots,g_n)^T$ denotes the eigenvector of $\alpha(Q)$.
An eigenvector of $\alpha \circ \varsigma(Q)$ is $(g'_1,g'_2,\ldots,g'_n)^T:=
(x g_n, g_1,g_2,\ldots,g_{n-1})^T$.
Define $h_i' := g_i'/g_n' = g_{i-1}/g_{n-1} = h_{i-1}/h_{n-1}$; these ratios do not depend on which eigenvector of $\alpha \circ \varsigma(q)$ we chose. Then by \eqref{eq:hdiv}
\begin{align*}
(h'_i) = (h_{i-1})-(h_{n-1}) &= (D_{i-1} - \D - (n-i+1)P + \sum_{j=i}^n O_j) - (D_{n-1} - \D - P + O_n) \\&= D_{i-1} - D_{n-1} - (n-i)P + \sum_{j=i+1}^{n}O'_j.
\end{align*}
By the uniqueness of $\D'$ it follows that the divisor $\D'$ (resp. $D'_i$) is equal to $D_{n-1}$ (resp. $D_{i-1}$). Furthermore, $\D'-\D \sim P - O_n$, as required.
\end{proof}
\subsection{$W \times \tilde W$ actions on Lax-matrix}
For $r=1, \ldots, N-1$, $i=0, \ldots, n'-1$ and $s =0,\ldots,m-1$,
using the energy in \S~\ref{sec:Rmatrix} define
$$E^{(s)}_{r+ki}: = E(\tilde P(x)^{-mi-s} \tilde Q_r \tilde P(x)^{mi+s}, \tilde P(x)^{-mi-s} \tilde Q_{r+1} \tilde P(x)^{mi+s}).$$
Define an $n$ by $n$ matrix $B_{s,r}$ by:
$$
B_{s,r} = \mathbb{I}_n
+ \sum_{i=0}^{n'-1} \kappa_{r+ki}^{(s)} E_{r+ki,r+ki+1},
\quad \kappa_{r+ki}^{(s)} = \frac{\sigma_{r+1}(q)-\sigma_r(q)}{E^{(s)}_{r+ki}},
$$
where $(E_{a,b})_{c,d} = \delta_{a,c} \delta_{b,d}$
for $a,b,c,d \in \Z / n \Z$, and $\mathbb{I}_n $ denotes the identity matrix.
\begin{lem}\label{lem:vertical-L}
\begin{enumerate}
\item[(i)]
The action of $s_r ~(r=1,\ldots,m-1)$ on $\mM$ induces
the identity map on $\mL$,
and the action of $\pi$ on $\mM$ induces
the adjoint transformation
$$
{\pi}^\ast : L(x) \mapsto Q_1^{-1} L(x) Q_1
$$
on $\mL$.
\item[(ii)]
The action of $\tilde s_r$ and $\tilde \pi$ on $\mM$ induces
adjoint transformations on $\mL$, given by
$$
\tilde{s}_r^\ast : L(x) \mapsto (B_{0,r})^{-1} L(x) B_{0,r},
$$
for $r=0, \ldots, N-1$, and
$$
{\tilde \pi}^\ast : L(x) \mapsto P(x) L(x) P(x)^{-1}.
$$
\end{enumerate}
\end{lem}
\begin{proof}
(i) Due to Lemma~\ref{lem:R-matrix}, we have $\alpha \circ s_r(Q) = \alpha(Q)$ for $1 \leq r \leq m-1$.
The induced action of $\pi$ is
\begin{align*}
\pi^\ast (Q_1 \cdots Q_m P(x)^k)
&= Q_2 \cdots Q_m P(x)^k Q_1,
\end{align*}
and the claim follows.
\\
(ii) By \cite[Theorem 6.2]{LP},
the action of $\tilde s_r$ on $\mM$ is given by
\begin{align}\label{eq:actionB}
(Q_s)_{s=1,\ldots,m} \mapsto
((B_{s,r})^{-1} Q_s \,B_{s+1,r})_{s=1,\ldots,m}.
\end{align}
By definition $E^{(s)}_{r+ki}$ satisfies
$E^{(s+m)}_{r+ki} = E^{(s)}_{r+k(i-1)}$,
and we have $B_{s+m,r} = P(x)^k B_{s,r} P(x)^{-k}$.
The claim follows.
The second part is easy to see, as the transformation $\tilde \pi$ just cycles
the indices inside the $Q_i$ (Definition \ref{def:action}).
\end{proof}
Since similar matrices have the same
characteristic polynomial, we obtain the following:
\begin{cor}\label{cor:L-WW}
For each $q \in \mM$, $\psi(q)$ is invariant under the action of
$W \times \tilde W$. In particular, the coefficients of $f(x,y) = \psi(q)$ are conserved quantities for $W \times \tilde W$.
\end{cor}
\begin{lem}\label{lem:pi-c}
Each $c \in \mathcal{S}_f$ is invariant under actions of $s_i$ and $\tilde s_i$.
The cyclic shift $\pi$
acts on $\mathcal{S}_f$ via
$$\tau^{-1}: (c_1, \ldots, c_M) \mapsto (c_2, \ldots, c_M, c_1),$$
and $\tilde \pi$ acts on $\mathcal{S}_f$ via
$$\tau: (c_1, \ldots, c_M) \mapsto (c_M, c_1, \ldots, c_{M-1}).$$
\end{lem}
\begin{proof}
From the definition of the cyclic group action \eqref{eq:pi},
it follows that $\tilde \pi(Q)_{ij} = q_{i,j-1}$ and $\pi(Q)_{ij} = q_{i+1,j}$. Thus, using Lemma~\ref{lem:c-snake}, we see that
$\tilde \pi$ induces
$$
c_\ell = \sum_{i+j \equiv \ell+1 \mod M} q_{i,j}
~\mapsto \sum_{i+j \equiv \ell+1 \mod M} q_{i,j-1}
= \sum_{i+j \equiv \ell \mod M} q_{i,j} = c_{\ell -1},
$$
and that $\pi$ induces $c_{\ell} \mapsto c_{\ell+1}$ similarly.
\end{proof}
\subsection{Eigenvector change under conjugation}
For a vector $v = (v_1,v_2,\ldots,v_n)$ of rational functions on $C_f$, define
$$
D(v) = \text{(common zeroes of $v_i$)} - \text{(common poles of $v_i$)} + R + O_n
$$
where $R$ is the divisor supported on $S_{P,O} = \{P,O_1,\ldots,O_N\}$,
which appears in Theorem \ref{thm:double}. (See Lemma~\ref{lem:zeta}
for the explicit formula of $R$.)
The following result is immediate.
\begin{lem}\label{L:scale}
Let $f$ be a rational function on $C$. Then $D((fv_1,\ldots,fv_n))$ is linearly equivalent to $D((v_1,\ldots,v_n))$.
\end{lem}
\begin{lem}\label{L:D}
The positive general divisor $\D = \D(L(x))$ of degree $g$ associated to $L(x)$ is equal to $D((\Delta_{1,n},\ldots,\Delta_{n,n}))$.
\end{lem}
\begin{proof}
By Theorem \ref{thm:double}, $\D = D_n$ belongs to the common zeroes. Also by Theorem \ref{thm:double}, $\bar D_1,\ldots,\bar D_n$ have no common points, so they do not contribute to $D((\Delta_{1,n},\ldots,\Delta_{n,n}))$.
\end{proof}
\begin{lem}\label{L:Qv}
Suppose $M = M(x,y)$ is a $n \times n$ matrix whose entries are polynomials in $\C[x,y]$, thought of as rational functions on $C_f$ with poles supported at $P$. Let $D(M \cdot v) - D(v) = D_+ - D_-$, where $D_+$ and $D_-$ are
positive divisors.
Then restricted to $ C_f \setminus P$, we have that $(\det M)_0 - D_+$ is a positive divisor. Also, $D_-$ is supported at $P$.
\end{lem}
\begin{proof}
Let $p \in C_f \setminus P$. Then the entries of $M$ are regular at $p$. We shall show that the multiplicity of $p$ in $D_+$ is less than the multiplicity of $p$ as a zero in $\det M$. Since $v'_i = \sum_{j} M_{ij}(x,y) v_j$, and $M_{ij}$ is regular at $p$, it is clear that $\mult_p D(Mv) \geq \mult_p D(v)$, where $\mult_p$ denotes the multiplicity of a divisor at a point $p$. We also have $$v_i = \sum_{j} (M^{-1})_{ij}(x,y) v'_j = \dfrac{1}{(\det M)(x,y)}\sum_j \pm |M|_{i,j}(x,y) v'_j.$$
Since $|M|_{i,j}(x,y)$ is regular at $p$, we have $\mult_p D(v) \geq \mult_p(Mv) - \mult_p(\det M)$. Both claims follow.
\end{proof}
Suppose $L'(x) = Q_1^{-1}L(x)Q_1$ and $\D' = \D(L'(x))$. We shall compute $\D' - \D$ up to linear equivalence. Note that $v = (\Delta_{1,n},\ldots,\Delta_{n,n})^T$ is an eigenvector of $L(x)^T-y$, and $L'(x)^T = Q_1^TL(x)^T(Q_1^{-1})^T$, so an eigenvector of $L'(x)^T-y$ is equal to $v' = Q_1^Tv$, where
$$
Q_1^T= Q_1^T(x) =
\left(\begin{array}{cccc}
q_{1,1} & 1 & 0 &0\\
0&q_{1,2} &1 &0 \\
0&\ddots&\ddots&1 \\
x&0&0&q_{1,n}
\end{array} \right).
$$
Extend the vector $v$ into an infinite vector $\tv$ by $v_{i+n} = xv_i$. As in \S\ref{sec:Lax}, let $A = (a_{i,j})_{i \in [n], j \in \Z}$ be the infinite ``unfolded" version of $L^T(x)$, satisfying $(L^T)_{i,j}(x) = \sum_\ell {a}_{i,j+\ell n} x^\ell$. (Note that $A$ is a matrix of scalars, but $\tv = (\ldots,v_{-1},v_0,v_1,\ldots)^T$ is a matrix of functions.) Then we have
\begin{equation}\label{E:unfolded}
A \cdot \tv = y \tv.
\end{equation}
Define $w = (v_1,v_2,\ldots,v_{k+m})^T$ and $w' = (v'_1,v'_2,\ldots,v'_{k+m})^T$. We claim that there exists a $(k+m) \times (k+m)$ matrix $M = M(y)$ such that $w' = M(y) \cdot w$. Since $v' = Q_1^T v$, we have $v'_i = q_i v_i + v_{i+1}$ (where $q_i$ is extended periodically if $i \geq n$). We now write $v_{k+m+1}$ in terms of $v_1,\ldots,v_{k+m}$ using \eqref{E:unfolded}. The matrix $A$ is supported on the $m+1$ diagonals $k,k+1,\ldots,k+m$. The matrix $A - y$ is supported on the $(k+m+1)$ diagonals $0,1,\ldots,k+m$. Thus \eqref{E:unfolded} gives
\begin{equation}\label{E:km1}
a_{1,k+1}v_{k+1} + \cdots + a_{1,k+m+1} v_{k+m+1} = y v_{1}.
\end{equation}
Also note that $a_{1,k+m+1} = 1$. So
$$
M(y) = \left(\begin{array}{ccccccc}
q_{1,1} & 1 & 0 & \cdots & \cdots&0&0\\
0&q_{1,2} &1 &0&0&0&0 \\
\vdots & & \ddots & \ddots & \\
0&\cdots && q_{1,k+1} & 1\\
\vdots & && & \ddots & \ddots \\
0 &\cdots && & &q_{k+m-1}&1 \\
y&0&\cdots&-a_{1,k+1}& \cdots & -a_{1,k+m-1} &q_{1,k+m}-a_{1,k+m}
\end{array} \right).
$$
\begin{lem}\label{lem:time}
With the above conventions, we have $$\D' \sim \D +A_1 - P.$$
\end{lem}
\begin{proof}
We shall show that $D(v') - D(v) = A_1 - P$. This suffices by Lemmas \ref{L:scale} and \ref{L:D}.
From Lemma \ref{L:Qv}, we have $D(v') - D(v) = D_+ - D_-$ where $D_{\pm}$ are positive divisors, with $D_+$ when restricted to $C_f \setminus P$ supported on $\{(\det Q^T)(x) = 0\}$, and $D_-$ supported at $P$. Since $D(v')$ is a positive divisor of degree $g$
by the construction of $L'(x)^T$, we have that $D_+$ and $D_-$ have the same degree.
We now calculate $D_-$. By Theorem \ref{thm:double}, we have that $\mult_P(v_i) = C - i$ for some integer $C$ (we use the convention that negative multiplicity is a pole), and this formula still holds for the infinite vector $(\ldots,v_{-1},v_0,v_1,\ldots)$. Since $v'_i = q_i v_i + v_{i+1}$, we have $\mult_P(v'_i) = C - (i+1)$. Thus $D_- = P$. It follows that $D_+$ is a single point in $C_f \setminus P$.
We shall show that $D_+$ must be supported on $A_1$.
By Theorem \ref{thm:double},
the common zeros of $v_i$ and $v_j$ for any $i \neq j$ are only $D_n$ except for
the points in $S_{P,O}$. Thus we have $D(w) = D_n + R'$ where $R'$ is
supported on $S_{P,O}$.
But $\{(\det Q^T)(x) = 0\}$ does not intersect $S_{P,O}$. Using Lemma \ref{L:Qv} now applied to $w' = M(y)w$, we see that $D_+$ must be supported on $\{\det M(y) = 0\}$.
By Lemma \ref{L:A} below, we conclude that $D_+$ is a multiple of $A_1$. Since $D(v') - D(v)$ is a divisor of degree 0, we must have
$$
D(v') - D(v) = A_1 - P,
$$
as required.
\end{proof}
\begin{lem}\label{L:A}
The intersection of $(\det Q^T)(x) = 0$ and $(\det M(y)) = 0$ is the single point $A_1$.
\end{lem}
\begin{proof}
We check that $\det M(y) = \pm y$.
It is clear that
$$
\det M(y) = (-1)^{m+1} y + q_{1,1} \cdots q_{1,k}
\det \begin{pmatrix}
q_{1,k+1} & 1 \\
& q_{1,k+2} & 1 \\
& & \ddots & \quad \ddots \\
& & & q_{1,k+m-1} & 1\\
-\tl_{1,k+1} & -\tl_{1,k+2} & \cdots & -\tl_{1,k+m-1} & q_{1,k+m} - \tl_{1,k+m}
\end{pmatrix}.
$$
Write $u_i~(i=1,\ldots,m)$ for the $m$ columns of the above $m$ by $m$ matrix. We show that they are linearly dependent as
$$
u_1 - q_{1,k+1} \big(u_2 - q_{1,k+2} (u_3 - \cdots (u_{m-1} - q_{1,k+m-1} u_m) \cdots ) \big) = 0,
$$
which reduces to
\begin{align}\label{eq:detM=0}
&-\tl_{1,k+1} + q_{k+1} \tl_{1,k+2} - q_{1,k+1} q_{1,k+2} \tl_{1,k+3} +
\cdots \\
&+(-1)^{m} q_{k+1}q_{k+2} \cdots q_{k+m-1} \tl_{1,k+m}
+ (-1)^{m+1} q_{k+1}q_{k+2} \cdots q_{k+m} = 0. \nonumber
\end{align}
By the definition of $A$, we have
$$
\tl_{1,j} = \begin{cases}
l_{j,1} & j=k+1,\ldots,n, \\
l_{j-n,1} & j=n+1,\ldots,n+k, \\
\end{cases}
$$
where $L = (l_{ij})_{i \in [n], j \in \Z}$ is the infinite unfolded version of $L(x)$.
On the network $G'$, $l_{k+i,1}$ is the weight generating function of paths
from the source $k+i$ to the sink $k+1$.
Note that each weight in $l_{k+i,1}$ is of degree $m+1-i$ in $q$-s.
We divide $l_{k+i,1}$ into two parts:
$$
l_{k+i,1} = l_{k+i,1}' + l_{k+i,1}'',
$$
where $l_{k+i,1}'$ is the sum of the weights {\em with} $q_{1,k+i}$,
and $l_{k+i,1}'$ is the sum of the weights {\em without} $q_{1,k+i}$.
We claim that there is one-to-one correspondence between
the paths contributing to $l_{k+i,1}'$
and the paths contributing to $l_{k+i+1,1}''$.
Precisely, we have $q_{1,k+i} l_{k+i+1,1}'' = l_{k+i,1}'$.
Combining with $l_{k+1,1}'' = 0$ and $l_{k+m,1}' = q_{1,k+m}$,
we obtain \eqref{eq:detM=0}.
\end{proof}
\subsection{Proof of Theorem \ref{thm:finite}}\label{sec:prooffinite}
(i) Due to the definition of $s_r$, Lemma~\ref{lem:vertical-L}(i) and
Lemma~\ref{lem:pi-c},
for $Q=(Q_1,\ldots,Q_m) \in \psi^{-1}(f)$ with
$\phi (Q) = ([\mathcal{D}],c,O,(A_1,\ldots,A_m))$, we have
$$
\phi \circ s_r(Q_1,\ldots,Q_m) = ([\mathcal{D}],c,O,(A_1,\ldots,A_{r-1},
A_{r}(Q'),A_{r+1}(Q'),A_{r+2},\ldots,A_m)),
$$
where $Q' = s_r(Q)$. From \eqref{eq:q1q2} we see that
$\epsilon_{r}(q') = \epsilon_{r+1}(q)$ and $\epsilon_{r+1}(q') = \epsilon_{r}(q)$,
and \eqref{eq:vertical-s} follows.
It is easy to see that for $L(x) = \alpha(Q)$ we have
$\eta \circ \pi(L(x)) = ([\D'], \tau^{-1}(c),O')$,
where $\D'$ is some divisor. By Lemma \ref{lem:time}, we have $[\D] = [\D' + A_1 - P]$. The claim follows.
\\
(ii) First we prove \eqref{eq:horizontal-s}.
Let $g = (g_1,\ldots,g_n)^t$ be the eigenvector of $L(x) \in \psi^{-1}(f)$,
and set $(g_1',\ldots,g_n')^t := (B_{0,r})^{-1} g$.
Define $h_i' := g_i' / g_n'$ for $i=1,\ldots,n-1$.
Then we have
$$g_j'
= \begin{cases}
g_{r+ki} - \kappa^{(0)}_{r+ki} g_{r+ki+1} & j \equiv r+ki \mod n
\\
g_j & \text{otherwise}.
\end{cases}
$$
We must show that $(h_j')_{\infty} = (h_j)_{\infty}$
for $r=1,\ldots,N-1$ and $j=1,\ldots,n-1$. It is enough to consider the case of $r=1$ and $j=1$.
Due to Theorem \ref{thm:double} and \eqref{eq:hdiv}, there are positive divisors
$\mathcal{D}, \mathcal{D}'$
of degree $g$ such that
$(h_1)_\infty = (n-1) P + \mathcal{D}$,
$(h_1')_\infty = (n-1) P + \mathcal{D}'$.
Since $h_1' = h_1 - \kappa^{(0)}_{1} h_2$ and $(h_1)_\infty > (\kappa^{(0)}_{1} h_2)_\infty = (h_2)_\infty$,
we get $(h_1')_\infty \leq (h_1)_\infty$.
Thus it follows that $\mathcal{D}' = \mathcal{D}$.
From \eqref{eq:actionB} and Lemma~\ref{lem:pi-c}
it follows that both $A \in R_A$ and $c \in \C^M$ are not changed by
$\tilde s_r$.
From the facts that $\sigma_{N+1-r}(q)$ is the product of all
diagonal elements of $\tilde Q_r$, and that
the $R$-matrix action changes $(\tilde Q_r, \tilde Q_{r+1})$
to $(\tilde Q_{r+1}',\tilde Q_r')$,
it follows that $\tilde s_r$ exchanges the $(N-r)$-th and the $(N-r+1)$-th
elements of $O \in R_O$.
Then we get \eqref{eq:horizontal-s}.
Since $\tilde{\pi}$ induces the shift $\varsigma^\ast$ of $L(x)$,
\eqref{eq:horizontal-pi} follows from Proposition~\ref{prop:shift}.
\subsection{Snake path actions}
Recall that $M=\gcd(n,k+m)$.
The {\em snake path actions} are torus actions on $\mM$, considered in \cite{LP}.
Recall from \S~\ref{subsec:lem:Q}
that a snake path is a closed path on $G$ that turns at every vertex.
Thus it alternates between going up or going right.
For $1 \leq s \leq M$ and $t \in \C^\ast$,
the action $T_s := T_s(t)$ on $\mathcal{M}$ is given by
\begin{align}
\label{eq:snake-q}
T_s(q)_{ij} =
\begin{cases}
t \, q_{ij} & \text{ if } j \equiv s-i+1 \mod M\\
\displaystyle{\frac{1}{t}}\, q_{ij} & \text{ if } j \equiv s-i \mod M\\
q_{ij} & \text{ otherwise}.
\end{cases}
\end{align}
Informally, $T_s(t)$ multiplies all left turns on a snake path by $t$, and all right turns by $1/t$.
\begin{lem}\label{lem:snake-h}
The induced action $T^\ast_s ~(1 \leq s \leq M)$ on $\mL$ is given by
$$
T^\ast_s: ~ L(x) \mapsto D_s(t) L(x) D_s(t)^{-1},
$$
where
$$
D_s(t) = \mathrm{diag}(d_i)_{1 \leq i \leq n},
\quad
d_i = \begin{cases}
t & \text{ if $i \equiv s \mod M$} \\
1 & \text{ otherwise }.
\end{cases}
$$
\end{lem}
\begin{proof}
For simplicity we write $D_s := D_s(t)$ and $P := P(x)$.
We rewrite $D_s L(x) D_s^{-1}$ as
\begin{align}\label{eq:DLD}
D_s L(x) D_s^{-1}
&=
\left( \prod_{i=1}^{m} (P^{-i+1} D_s P^{i-1}) Q_i (P^{-i} D_s P^{i})^{-1}
\right)
\cdot (P^{-m} D_s P^{m}) P^k D_s^{-1}.
\end{align}
This is equal to $\alpha(T_s(Q))$ since $(P^{-m} D_s P^{m}) \cdot P^k D_s^{-1} = P^k$.
\end{proof}
\begin{prop}\label{prop:snake-dj}
We have the following commutative diagram:
$$
\xymatrix{
\mM ~~\ar[d]_{{e}_u} \ar[r]^{T_s}
& ~~\mM \ar[d]^{{e}_u} \\
\mM ~~~ \ar[r]_{\varsigma^{-u} \circ T_s \circ \varsigma^{u}} & ~~~\mM .\\
}
$$
\end{prop}
\begin{proof}
It is enough to prove
$((\varsigma^\ast)^{-u} \circ T^\ast_s \circ (\varsigma^\ast)^{u}) \circ e^\ast_u
= e^\ast_u \circ T^\ast_s$ acting on $\mL$.
For $L(x) = Q_1 \cdots Q_m P^k \in \mathcal{L}$, we have
\begin{align*}
L(x) &\stackrel{e^\ast _u}{\longmapsto}
Q_{u+1}\cdots Q_m P^k Q_1 \cdots Q_u
\\
&\stackrel{(\varsigma^\ast)^{-u} \circ \tilde{T}_s \circ (\varsigma^\ast)^{u}}{\longmapsto}
(P^{-u} D_s P^u)Q_{u+1}\cdots Q_m P^k Q_1 \cdots Q_u(P^{-u} D_s P^u)^{-1}
\\
&= \prod_{i=u+1}^{m} (P^{-i+1} D_s P^{i-1}) Q_i (P^{-i} D_s P^{i})^{-1}
\cdot (P^{-m} D_s P^{m}) P^k D_s^{-1}
\\
& \qquad \qquad
\cdot
\prod_{i=1}^{u} (P^{-i+1} D_s P^{i-1}) Q_i (P^{-i} D_s P^{i})^{-1}
\\
&= Q_{u+1}'\cdots Q_m' P^k Q_1' \cdots Q_u',
\end{align*}
where $Q_i'= (P^{-i+1} D_s P^{i-1}) Q_i (P^{-i} D_s P^{i})^{-1}$.
On the other hand we have
\begin{align*}
L(x) &\stackrel{T^\ast_s}{\longmapsto}
D_s L(x) D_s^{-1}
\stackrel{\eqref{eq:DLD}}{=} Q_1'\cdots Q_m' P^k
\\
&\stackrel{e^\ast_u}{\longmapsto}
Q_{u+1}' \cdots Q_m' P^k Q_1' \cdots Q_u'.
\end{align*}
Thus the claim is obtained.
\end{proof}
From Lemma~\ref{lem:c-snake} and \ref{lem:snake-h} we obtain
\begin{prop}\label{prop:snake}
The following diagram is commutative:
\begin{align}\label{eq:comm-T}
\xymatrix{
\psi^{-1}(f) \ar[d]_{T_s} \ar[r]^{\phi \qquad \qquad}
& \Pic^{g}(C_f) \times \mathcal{S}_f \times R_O \times R_A~~\ \ar[d]^{(id,\,t_s,\,id, \,id)}
\\
\psi^{-1}(f) ~~ \ar[r]_{\phi \qquad \qquad}
& \Pic^{g}(C_f) \times \mathcal{S}_f \times R_O \times R_A, \\
}
\end{align}
where $t_s := t_s(t)$ acts on $\mathcal{S}_f$ by
\begin{align}
\label{eq:snake-c}
t_s(c_{\ell}) =
\begin{cases}
t \, c_{\ell} & \text{ if } s=\ell \\
\displaystyle{\frac{1}{t}}\, c_{\ell} & \text{ if } s=\ell +1\\
c_{\ell} & \text{ otherwise}.
\end{cases}
\end{align}
\end{prop}
Thus the snake path actions $T_s(t)$ act transitively on $\mathcal{S}_f$.
\section{Theta function solution to initial value problem}
\label{sec:theta}
\subsection{Riemann theta function}
Fix $f \in \oV$.
We fix a universal cover of $C_f$ and $P_0 \in C_f$,
and define the Abel-Jacobi map $\iota$ by
$$
\iota: C_f \to \C^g; ~ X \mapsto \left(\int_{P_0}^{X} \omega_1,\ldots,\int_{P_0}^{X} \omega_g \right),
$$
where $\omega_1,\ldots,\omega_g$ is a basis of holomorphic differentials on $C_f$.
We also write $\iota$ for the induced map $\mathrm{Div}^0(C_f) \to \C^g$.
Let $\Omega$ be the period matrix of $C_f$, and write
$\Theta(z ) := \Theta(z;\Omega)$ for the Riemann theta function:
$$
\Theta(z;\Omega)
=
\sum_{{\bf m}\in \Z^g}
\exp \left(\pi \sqrt{-1}
{\bf m} \cdot (\Omega {\bf m}+ 2 z)\right),
\quad z \in \C^g.
$$
It is known to satisfy the quasi-periodicity:
$$
\Theta(z + {\bf m} + {\bf n} \Omega)
= \exp \left(- \pi \sqrt{-1}\, {\bf n} \cdot (\Omega {\bf n} + 2z) \right)
\Theta(z),
$$
for ${\bf m}, {\bf n} \in \Z^g$.
Let $\mathcal{D}_0 \in \mathrm{Div}^{g-1}(C_f)$ be the Riemann characteristic,
that is, $2 \mathcal{D}_0$ is linearly equivalent to the
canonical divisor $K_{C_f}$.
Then the well known properties of the Riemann theta function gives
\begin{lem}
For any point $Y \in C_f$, the function
$$
X \mapsto \Theta(\iota(X-Y))
$$
is a section of the line bundle $\OO(\D_0+Y)$ which has zeroes exactly at $\D_0$ and at $Y$, and no poles. For any positive divisor $\D$ of degree $g$, the function
$$
X \mapsto \Theta(\iota(X - \D + \D_0))
$$
is a section of the line bundle $\OO(\D)$ which has zeroes exactly at $\D$ and no poles.
\end{lem}
\subsection{The inverse of $\phi$}
For a positive divisor $\D$ of degree $g$, define the function $\psi_i ~(i \in \Z)$ on $C_f$ by
$$
\psi_i(X)
= \frac{\Theta(\beta - \iota(\OO_{n-i}) + \iota(X - O_n))
\prod_{\ell=i+1}^n \Theta(\iota(X - O_\ell))}{\Theta(\beta + \iota(X-O_n))
\Theta(\iota(X-P))^{n-i}},
$$
where $\beta = \iota(\D_0 + O_n - \D)$.
\begin{lem}\label{lem:psi-h}
$\psi_i(X)$ is a meromorphic function on $C_f$.
When $\D$ is the positive divisor defined by Definition~\ref{def:D}, we have
\begin{align}\label{eq:h-div}
(\psi_i) = (h_i)
=
\mathcal{D}_i + \sum_{j=i+1}^n O_j
- \mathcal{D} - (n-i) P.
\end{align}
Thus $\psi_i(X)$ is equal to $h_i(X)$ up to a scalar.
\end{lem}
\begin{proof}
To check that $\psi_i(X)$ is a meromorphic function on $C_f$ we use the functional equation for $\Theta$, and note that the multiset of points (with signs and multiplicities) that appear in the numerator is equal to the same for the denominator.
This multiset in additive notation is $\beta - (n-i)P + (n-i+1)X - O_n$.
For the second statement, we note that $\psi_i(X)$ has zeroes at $O_{i+1},O_{i+2},\ldots,O_{n}$, a pole of order $n-i$ at $P$, and also poles at $\D$. By \eqref{eq:hdiv}, we have $(\psi_i) = (h_i)$.
\end{proof}
Now we study the inverse of $\phi$.
We focus on the $\Z^m$ action.
For $t=(t_1,t_2,\ldots,t_m) \in \Z^m$,
let $q^t := (q_{ji}^t)_{ji} \in \psi^{-1}(f)$ be a configuration at time $t$.
For $L^t(x) := \alpha(q^t)$, let $g^t = (g^t_1,\ldots, g^t_n)^\bot$ be
its eigenvector and set $h^t_i := g^t_i/g^t_n$.
Let $\phi(q^t)$ be
$$
(\D^t, (c^t_1,\cdots,c^t_M),(A_1,\ldots,A_m),(O_1,\ldots,O_N)),
$$
where $\D^t = \D - \sum_{j=1}^m t_j \mathcal{A}_j$
for some positive divisor $\D$
of degree $g$.
Define
$\be_j := (\underbrace{1,\ldots,1}_{j},\underbrace{0,\ldots,0}_{m-j})$
for $j=0,\ldots,m-1$, and recursively set
$\be_{j+m} = (1,\ldots,1) + \be_j$.
Define functions $\theta^t_i$ and $\Psi_{i}^{t}$ on $C_f$ by
$$
\theta^{t}_{i}(X) = \Theta(\beta^t + \iota(X-O_n
- \OO_{n-i})),
$$
where $\beta^t = \iota(\D_0 + O_n - \D^t)$, and by
$$
\Psi_{i}^{t+\be_j}(X)
=
\frac{\theta_{i-N}^{t+\be_{j-1}}(X) \,\theta_{i}^{t+\be_{j}}(X)}
{\theta_{i}^{t+\be_{j-1}}(X) \, \theta_{i-N}^{t+\be_{j}}(X)}.
$$
By Lemma \ref{lem:psi-h}, there is a constant $b_i^{t+\be_j}$ depending on $C_f$, such that
we have
$$
\Psi_{i}^{t+\be_j}(X)
=
b_i^{t+\be_j}
\frac{h_{i-N}^{t+\be_{j-1}}(X) \, h_{i}^{t+\be_{j}}(X)}
{h_{i}^{t+\be_{j-1}}(X) \, h_{i-N}^{t+\be_{j}}(X)}.
$$
\begin{lem}(Cf. Lemma~3.1 \cite{Iwao10})\label{lem:Psi-OP}
We have
\begin{align}
\label{eq:Psi-O}
&\Psi_{i-1}^{t+\be_j}(P) = \Psi_{i}^{t+\be_j}(O_i)
= b_i^{t+\be_j} \frac{q_{j,i-N}^t}{q_{j,i}^t},
\\
\label{eq:Psi-P}
&\Psi_{i}^{t+\be_j}(P)
=
b_i^{t+\be_j}
\frac{d_{i-N-1}^{t+\be_{j}} \, d_{i}^{t+\be_{j}}}
{d_{i-1}^{t+\be_{j}} \, d_{i-N}^{t+\be_{j}}},
\end{align}
where $d_{i}^{t}$ is a coefficient of the leading term of
$h_{i}^{t}(X)$ around $X=P$.
\end{lem}
\begin{proof}
The first equality of \eqref{eq:Psi-O} follows from
$\theta^t_i(O_i) = \theta^t_{i-1}(P)$ and $O_i = O_{i-N}$.
By Lemma~\ref{lem:vertical-L} we have
$g^{t+\be_{j-1}} = Q_j^t \, g^{t+\be_{j}}$ which gives
$$
\Psi_{i}^{t+\be_j}(X)
=
b_i^{t+\be_j}
\frac{(h_{i-N-1}^{t+\be_{j}}(X) + q_{j,i-N} \, h_{i-N}^{t+\be_{j}}(X))
h_{i}^{t+\be_{j}}(X)}
{(h_{i-1}^{t+\be_{j}}(X) + q_{j,i} \, h_{i}^{t+\be_{j}}(X))
h_{i-N}^{t+\be_{j}}(X)}.
$$
Then, from \eqref{eq:h-div} the second equalities of \eqref{eq:Psi-O}
and \eqref{eq:Psi-P} follow.
\end{proof}
\begin{lem} We have
\begin{align}\label{eq:sneke-d}
\frac{d_{i}^{t+\be_{j}}}{d_{i+1}^{t+\be_{j}}}
=
c^t_{i+j}, \qquad i+j \equiv \ell \mod M,
\end{align}
where we extend $c_i^t$ to $i \in \Z$ by setting $c_i^t := c_\ell^t$
if $i \equiv \ell \mod M$ for $1 \leq \ell \leq M$.
\end{lem}
\begin{proof}
It is enough to show the case of $j=0$, $1 \leq i \leq M$.
Then the other cases follow from Lemma~\ref{lem:pi-c}.
We will show that the coefficient of the leading term in
$\frac{g_i}{g_{i+1}}$ at $X=P$ is equal to $c_i$.
To obtain the ratio $\frac{g_i}{g_{i+1}}$ we may choose to compute
maximal minors of $L(x)-y$ with respect to any row. Let us choose
row $i+1$. Then by Theorem \ref{thm:minor}, we are counting families
of paths that do not start at source $i+1$ and do not end at sink $i$
(for $g_i$) or sink $i+1$ (for $g_{i+1}$).
Since we are computing at $X=P$, the only terms that contribute are
the ones lying on the upper hull of the Newton polygon, that is, the
edge with slope $-n/(m+k)$.
In terms of path families this means that we only take paths that are
as close to a snake path as possible. In the case of $g_{i+1}$ this
means that we take a subset of closed snake paths. Each of them picks
up no weight, and thus overall we just get a constant.
In the case of $g_{i}$ one of the closed staircase paths skips one
term, and thus we get a long path starting at source $i$, winding
around the torus and ending at sink $i+1$.
Such path essentially by definition picks up weight $c_i$.
The other snake paths may or may not appear, thus creating only a
constant factor in front.
\end{proof}
Note that \eqref{eq:sneke-d} is compatible with the snake path
actions in Lemma~\ref{lem:snake-h}.
\begin{thm}\label{thm:N=1}
When $N=1$, the inverse map of $\phi$ is given by
$$
q_{j,i}^t
=
f_c^{-\frac{1}{M}} \cdot a_j \cdot c_{i+j-1}^t \cdot
\frac{\theta_{i}^{t+\be_{j}}(P) \, \theta_{i-1}^{t+\be_{j-1}}(P)}
{\theta_{i}^{t+\be_{j-1}}(P) \, \theta_{i-1}^{t+\be_{j}}(P)},
$$
where
\begin{align}\label{eq:q-a}
a_j = x(A_j)^{\frac{1}{n}} \cdot
\mathrm{exp}\left[\frac{2 \pi \sqrt{-1}}{n} \, b \cdot \iota(P-A_j)\right],
\end{align}
and
$b \in \Z^g$ is determined by
$a,b \in \Z^g$ such that $a + b \Omega := \iota(\OO_n)$.
\end{thm}
\begin{proof}
Let $p_{j,i}^t = q_{j,i}^t q_{j,i-1}^t \cdots q_{j,i-N+1}^t$.
By Lemma~\ref{lem:Psi-OP} we get the equations
$$
\alpha^t_j := \frac{p_{j,i}^t}{\Psi_{i}^{t+\be_{j}}(P)}
\frac{d_{i}^{t+\be_{j}}}{d_{i-N}^{t+\be_{j}}}
=
\frac{p_{j,i-1}^t}{\Psi_{i-1}^{t+\be_{j}}(P)}
\frac{d_{i-1}^{t+\be_{j}}}{d_{i-N-1}^{t+\be_{j}}}
=
\cdots.
$$
We show that $\alpha^t_j$ does not depend on $t$.
Consider the product
$$
x(A_j) = p_{j,i}^t p_{j,i+N}^t \cdots p_{j,i+(n'-1)N}^t,
$$
which is equal to
$$
(\alpha^t_j)^{n^\prime}
\frac{d_{i-N}^{t+\be_{j}}}{d_{i+n-N}^{t+\be_{j}}}\,
\frac{\theta_{i-N}^{t+\be_{j-1}}(P) \,\theta_{i+n-N}^{t+\be_{j}}(P)}
{\theta_{i+n-N}^{t+\be_{j-1}}(P) \,\theta_{i-N}^{t+\be_{j}}(P)}.
$$
Since $\OO_n$ is equivalent to $0$ in $\Pic^0(C_f)$,
there exists $a,b \in \Z^g$ such that
$a + b \Omega = \iota(\OO_n)$.
Due to the quasi-periodicity of $\Theta(z)$, we have
\begin{align}\label{theta-quasi1}
\theta_{i+n}^t(P)
= \exp \left[-2 \pi \sqrt{-1} b \cdot
\left(\beta^t - \iota(\OO_{n-i}-\OO_1) + b \Omega / 2 \right)
\right]\cdot \theta_i^t(P).
\end{align}
By using \eqref{eq:sneke-d} and \eqref{theta-quasi1} we obtain
$\alpha^t_j$ independent of $t$ as
$$
\alpha_j^t
=
f_c^{-\frac{N}{M}}
a_j^N.
$$
Hence we get
\begin{align}\label{eq:p-theta}
p_{j,i}^t
=
f_c^{-\frac{N}{M}} a_j^N
\cdot \prod_{\ell=1}^N c_{i+j-\ell}^t \cdot
\frac{\theta_{i}^{t+\be_{j}}(P) \, \theta_{i-N}^{t+\be_{j-1}}(P)}
{\theta_{i}^{t+\be_{j-1}}(P) \, \theta_{i-N}^{t+\be_{j}}(P)}.
\end{align}
When $N=1$, it is nothing but the claim.
\end{proof}
\subsection{Conditional solution for $N>1$}
When $N > 1$, by factorizing \eqref{eq:p-theta} we obtain
\begin{align}\label{eq:q-gamma}
q_{j,i}^t
=
f_c^{-\frac{1}{M}} \cdot \gamma_{j,i} \cdot a_j \cdot
c_{i+j-1}^t
\frac{\theta_{i}^{t+\be_{j}}(P) \, \theta_{i-1}^{t+\be_{j-1}}(P)}
{\theta_{i}^{t+\be_{j-1}}(P) \, \theta_{i-1}^{t+\be_{j}}(P)}.
\end{align}
Here $\gamma_{j,i}$ satisfies
\begin{align}\label{gamma-period}
\prod_{\ell=1}^N \gamma_{j,i+\ell} = 1,
\end{align}
which denotes that $\gamma_{j,i} = \gamma_{j,i+N}$.
\begin{lem}
For $i$ such that $i \equiv r \mod N$ we have
\begin{align}\label{gamma-m}
\prod_{j=1}^m \gamma_{j,i}
=
\sigma_r^\frac{1}{n'} \cdot \prod_{j=1}^m a_j^{-1} \cdot
\mathrm{exp}\left(2 \pi \sqrt{-1} (b' - b \, \frac{k'}{n'})
\cdot \iota(P-O_r)\right).
\end{align}
Here $b'$ is given by $a', b' \in \Z^g$ such that
$a' + b' \Omega := \iota(\mathcal{A}_m + \OO_k)$.
\end{lem}
\begin{proof}
From \eqref{def:sigma_r} and \eqref{eq:q-gamma} it follows that
\begin{align}\label{sigma-gamma}
\sigma_r = \left(\prod_{j=1}^m \gamma_{j,i} \,a_j \,f_c^{-\frac{1}{M}}\right)^{n'}
\cdot \prod_{j=1}^m \prod_{p=0}^{n'-1} c_{j+i+pN-1}^t \cdot
\prod_{j=0}^{n'-1}
\frac{\theta^t_{i+jN-1}(P) \, \theta_{i+jN}^{t+\be_{m}}(P)}
{\theta_{i+jN-1}^{t+\be_{m}}(P) \, \theta^t_{i+jN}(P)},
\end{align}
for $r=1,\ldots,N$ and $i \equiv r \mod N$.
First we show that the second factor of \eqref{sigma-gamma} is equal to
$f_c^{\frac{m n'}{M}}.$
It is enough to prove in the case of $i=1$, where the second factor is
$
\prod_{p=0}^{n'-1}(c_{1+pN} \cdots c_{m+pN}).
$
When $n'=k'$, $M$ is a divisor of $m$ and we have
$c_{1+pN} \cdots c_{m+pN} = f_c^{m/M}$.
When $n' \neq k'$, define an automorphism $\nu$ of
$\mathcal{N}' := \{0,\ldots,n'-1\}$ by
$\nu : p \mapsto p+n'-k' \mod n'$.
Since $n'$ and $k'$ are coprime, a set
$\{\nu^\ell(1) ~|~ \ell \in \mathcal{N}' \}$ coincides with
$\mathcal{N}'$.
Thus, from $M=\gcd(n,k+m)$ and $m+pN \equiv \nu(p) N \mod M$,
it follows that
the product $\prod_{p=0}^{n'-1}(c_{1+pN} \cdots c_{m+pN})$
is reordered to be $\prod_{j=1}^{mn'} c_j$.
Further, since $M$ is a divisor of $mn'$, we obtain the claim.
Next, we can show that the third factor of \eqref{sigma-gamma} is equal to
$
\mathrm{exp}\left(2 \pi \sqrt{-1} (-b'n' + b k') \cdot \iota(P-O_r)\right),
$
by using \eqref{theta-quasi1} and
$$
\theta_{i}^{t+\be_m}(P) = \exp \left[-2 \pi \sqrt{-1} b' \cdot \left(\beta^t
- \iota(\OO_{n-i+k} - \OO_1) + b' \Omega / 2 \right) \right]\cdot \theta_{i-k}^t(P).
$$
Here $b'$ is given by $a', b' \in \Z^g$ such that
$a' + b' \Omega := \iota(\mathcal{A}_m + \OO_k)$.
Finally from \eqref{sigma-gamma} we obtain
$$
\left(\prod_{j=1}^m \gamma_{j,i}\right)^{n'}
=
\sigma_r \cdot \prod_{j=1}^m a_j^{-n'} \cdot
\mathrm{exp}\left(2 \pi \sqrt{-1} (n'b' - b k') \cdot
\iota(P-O_r)\right),
$$
and the claim follows.
\end{proof}
Unfortunately, we have not been able to obtain $\gamma_{j,i}$ from
\eqref{gamma-period} and \eqref{gamma-m}.
Nevertheless, if we assume that $\gamma_{j,i}$ is constant on $\Pic^g(C_f)$, then we obtain the following conditional result.
\begin{prop}\label{prop:N>1}
Suppose that $\gamma_{j,i}$ is a constant function on $\Pic^g(C_f)$.
Then we have
\begin{align}\label{eq:gamma}
\gamma_{j,i}
=
\sigma_r^\frac{1}{n'm} \cdot \prod_{j=1}^m a_j^{-\frac{1}{m}}
\cdot
\mathrm{exp}\left(\frac{2 \pi \sqrt{-1}}{n'm} (n'b' - k' b)
\cdot \iota(P-O_r)\right).
\end{align}
In particular, the inverse of $\phi$ is given by
$$
q_{j,i}^t
=
Q \cdot a_j \cdot o_i \cdot c_{i+j-1}^t \,
\frac{\theta_{i}^{t+\be_{j}}(P) \, \theta_{i-1}^{t+\be_{j-1}}(P)}
{\theta_{i}^{t+\be_{j-1}}(P) \, \theta_{i-1}^{t+\be_{j}}(P)},
$$
where $a_j$ is defined by \eqref{eq:q-a}, and
\begin{align*}
&Q = f_c^{-\frac{1}{M}} \cdot \prod_{j=1}^m x(A_j)^{-\frac{1}{nm}},
\\
&o_i = \left((-1)^{n'+1} \frac{y(O_r)^{n'}}{x(O_r)^{k'}}
\right)^{\frac{1}{n'm}}
\cdot
\mathrm{exp}\left[\frac{2 \pi \sqrt{-1}}{nm}
\left((n b'- kb) \cdot \iota(P-O_i) - b \cdot \iota(\mathcal{A}_m) \right)
\right],
\\
&\hspace*{12cm} i \equiv r \mod N.
\end{align*}
\end{prop}
\begin{proof}
To have \eqref{eq:q-gamma} compatible with the snake path action
\eqref{eq:snake-q} and \eqref{eq:snake-c},
$\gamma_{i,j}$ has to be constant on $\mathcal{S}_f$.
So $\gamma_{j,i}$ is regarded as a function on $R_A \times R_O$.
If $\gamma_{j,i}$ is not constant on $R_A$, \eqref{gamma-m} implies
that $\gamma_{j,i}$ depends on $a_j^{-1}$, but this contradicts
\eqref{gamma-period}.
Thus $\gamma_{j,i}$ is constant on $R_A$, and we have \eqref{eq:gamma}
which fulfills \eqref{gamma-period}.
By substituting \eqref{eq:gamma} in \eqref{eq:q-gamma} and
using \eqref{eq:o_r}, we obtain the final claim.
\end{proof}
\section{Octahedron recurrence}
\label{sec:fay}
We shall prove that the function $\theta_i^t(P)$
satisfies a family of octahedron recurrences \cite{Spe},
as a specialization of Fay's trisecant identity.
We follow the definitions in \S \ref{sec:theta}.
In the following we assume $N = \gcd(n,k) = 1$ and
write $O$ for the unique special point $O_1$ over $(0,0) \in \tilde{C}_f$.
\subsection{Fay's trisecant identity}
We introduce Fay's trisecant identity in our setting.
For $\alpha, \,\beta \in \R^g$,
we define a generalization of the Riemann theta function:
\begin{align}\label{def:gen-theta}
\Theta[\alpha,\beta](z)
:=
\exp \left(\pi \sqrt{-1} \beta \cdot (\beta \Omega + 2 z + 2 \alpha
\right)
\cdot \Theta(z + \Omega \beta + \alpha).
\end{align}
We call $[\alpha,\beta]$ {\it a half period} when
$\alpha, \, \beta \in (\Z/2)^g$. Furthermore, a half period $[\alpha,\beta]$
is {\it odd} when
$\Theta[\alpha,\beta](z)$ is an odd function of $z$,
that is, $\Theta[\alpha,\beta](-z) = - \Theta[\alpha,\beta](z)$.
We note that the Riemann theta function itself is an even function:
$\Theta(z) = \Theta(-z)$.
It is easy to check that a half period $[\alpha,\beta]$ is odd
if and only if $4 \alpha \cdot \beta \equiv 1 \mod 2$.
\begin{thm}\cite[\S II]{Fay73}
\label{thm:Fay}
For four points $P_1,P_2,P_3,P_4$ on the
universal cover of $C_f$, $z\in \C^g$, and
an odd half period $[\alpha,\beta]$, the formula
\begin{align*}
&\Theta(z+\iota(P_3-P_1)) \, \Theta(z+\iota(P_4-P_2)) \,
\Theta[\alpha,\beta](\iota(P_2-P_3)) \,\Theta[\alpha,\beta](\iota(P_4-P_1))\\
&+\Theta(z+\iota(P_3-P_2)) \, \Theta(z+\iota(P_4-P_1)) \,
\Theta[\alpha,\beta](\iota(P_1-P_3)) \, \Theta[\alpha,\beta](\iota(P_2-P_4))\\
&=
\Theta(z+\iota(P_3+P_4-P_1-P_2)) \, \Theta({\bf z}) \,
\Theta[\alpha,\beta](\iota(P_4-P_3)) \, \Theta[\alpha,\beta](\iota(P_2-P_1)).
\end{align*}
holds.
\end{thm}
\subsection{$m=2$ case}
When $m=2$, the vertical actions $e_u ~(u=1,2)$ are written as
difference equations expressed as
\begin{align}
\label{eq:evolm=2-1}
&q_{2,i}^t q_{1,i-k}^t = q_{1,i}^{t+\be_1} q_{2,i}^{t+\be_1},
\\
\label{eq:evolm=2-2}
&q_{2,i+1}^t + q_{1,i-k}^t = q_{1,i+1}^{t+\be_1} + q_{2,i}^{t+\be_1},
\\
\label{eq:evolm=2-3}
&q_{j,i-k}^t = q_{j,i}^{t+\be_2} \qquad (j=1,2).
\end{align}
For simplicity we write $\theta_i^t$ for
$\theta_i^t(P) = \Theta(\beta^t + \iota((n-i-1) (O-P)))$.
By construction the theta function solution of $q_{j,i}^t$
(Theorem \ref{thm:N=1}),
\begin{align}\label{eq:theta-sol}
q_{j,i}^t
=
f_c^{-\frac{1}{M}} \cdot a_j \cdot c_{i+j-1}^t \cdot
\frac{\theta_{i}^{t+\be_{j}} \, \theta_{i-1}^{t+\be_{j-1}}}
{\theta_{i}^{t+\be_{j-1}} \, \theta_{i-1}^{t+\be_{j}}},
\end{align}
satisfies \eqref{eq:evolm=2-1}--\eqref{eq:evolm=2-3}.
\begin{thm}\label{thm:octahedron}
For any $t \in \Z^2$ and $i \in \Z$, the $\theta_i^t$ satisfy
an octahedron recurrence relation,
\begin{align}\label{eq:m=2-oct}
a_2 \,\theta_{i+1}^{t+\be_2} \theta_i^{t+2 \be_1}
- a_1 \,\theta_{i+1}^{t+2 \be_1} \theta_i^{t+\be_2}
=
c \,\theta_{i}^{t+\be_1 + \be_2} \theta_{i+1}^{t+\be_1}.
\end{align}
Here $c$ is a constant given by
$$
c = a_2 \, \frac{\Theta(p_0+\iota(A_1-A_2)) \Theta(p_0+\iota(O-P))}
{\Theta(p_0+\iota(O-A_2)) \Theta(p_0+\iota(A_1-P))},
$$
where $p_0$ is a zero of the Riemann theta function: $\Theta(p_0) = 0$.
\end{thm}
\begin{proof}
By setting $(P_1,P_2,P_3,P_4) = (A_2,O,P,A_1)$ and using \eqref{def:gen-theta},
we obtain:
\begin{align}\label{Fay-basic}
\begin{split}
&T_1 \, \Theta(z+\iota(P-A_2)) \, \Theta(z+\iota(A_1-O))
+ T_2 \, \Theta(z+\iota(P-O)) \, \Theta(z+\iota(A_1-A_2))\,
\\
&\qquad = T_3 \, \Theta(z+\iota(P+A_1-A_2-O)) \, \Theta(z) \,
\end{split}
\end{align}
where $p := \Omega \beta + \alpha$, and
\begin{align}\label{eq:T}
\begin{split}
&T_1 := \Theta(p+\iota(O-P)) \, \Theta(p+\iota(A_1-A_2)),
\\
&T_2 := \mathrm{e}^{4 \pi \sqrt{-1} \beta \cdot \iota(A_2-A_1)} \,
\Theta(p+\iota(A_2-P)) \, \Theta(p+\iota(O-A_1)),
\\
&T_3 := \Theta(p+\iota(A_1-P)) \, \Theta(p+\iota(O-A_2)).
\end{split}
\end{align}
By setting $z = \beta^t + \iota((n-i-1)(O-P) + 2 \mathcal{A}_1)$,
\eqref{Fay-basic} turns out to be
\begin{align}\label{eq:oct}
T_1 \, \theta_i^{\be_1+\be_2} \, \theta_{i+1}^{\be_1}
+ T_2 \, \theta_{i+1}^{2\be_1} \, \theta_{i}^{\be_2}
=
T_3 \, \theta_{i+1}^{\be_2} \, \theta_{i}^{2 \be_1}.
\end{align}
\begin{lem}\label{lem:T23}
We have
$$
\frac{T_2}{T_3} = \frac{a_1}{a_2}.
$$
\end{lem}
\begin{proof}
For $q^0 \in \psi^{-1}(f)$,
take $t_0 \in \Z^m$ such that $\beta^{t_0} \in \C^g$ is a zero of
the Riemann theta function,
which is always possible by choosing $q^0$ appropriately.
We write $-p_0$ for such $\beta^{t_0}$.
Then we have $\theta^{t_0}_{n-1} = \Theta(-p_0) = 0$, and
$q_{1,n}^{t_0} = q_{2,n}^{t_0-\be_1} = 0$.
From \eqref{eq:evolm=2-1} and \eqref{eq:evolm=2-2},
we obtain
\begin{align}\label{eq:rel-at-t0}
q_{2,n-1}^{t_0-\be_1} q_{1,n-1-k}^{t_0-\be_1}
= q_{1,n-1}^{t_0} q_{2,n-1}^{t_0},
\qquad
q_{1,n-k-1}^{t_0-\be_1} = q_{2,n-1}^{t_0}.
\end{align}
On the other hand,
when $z = p_0 + \iota(A_2-P)$, \eqref{Fay-basic} becomes
$$
T_2 \, \Theta(p_0+\iota(A_2-O)) \, \Theta(p_0+\iota(A_1-P))
= T_3 \, \Theta(p_0+\iota(A_1-O)) \, \Theta(p_0+\iota(A_2-P)).
$$
It is rewritten as (using that $\Theta(z)$ is even function)
$$
\frac{T_2}{T_3}
=
\frac{\theta^{t_0 + \be_1}_{n-2} \, \theta^{t_0 + \be_2-\be_1}_{n-1}}
{\theta^{t_0 + \be_2-\be_1}_{n-2} \, \theta^{t_0 + \be_1}_{n-1}}
=
\frac{a_1}{a_2} \cdot \frac{q_{2,n-1}^{t_0-\be_1}}{q_{1,n-1}^{t_0}},
$$
where we use \eqref{eq:theta-sol} to get the last equality.
It follows from \eqref{eq:rel-at-t0} that
$q_{2,n-1}^{t_0-\be_1}/q_{1,n-1}^{t_0} = 1$, and we obtain the claim.
\end{proof}
We continue the proof of the theorem.
By setting
$z = p_0 + \iota(O-P)$ in \eqref{Fay-basic}, we obtain
$$
T_1 \, \Theta(p_0+\iota(O-A_2)) \, \Theta(p_0+\iota(A_1-P))
= T_3 \, \Theta(p_0+\iota(A_1-A_2)) \, \Theta(p_0+\iota(O-P)).
$$
Using this and the above lemma, \eqref{eq:oct} is shown to be
\begin{equation*}
c \, \theta_i^{\be_1+\be_2} \, \theta_{i+1}^{\be_1}
+ a_1 \, \theta_{i+1}^{2\be_1} \, \theta_{i}^{\be_2}
=
a_2 \, \theta_{i+1}^{\be_2} \, \theta_{i}^{2 \be_1}. \qedhere
\end{equation*}
\end{proof}
Conversely, we have the following.
\begin{prop}\label{prop:thetaq}
Suppose that $\theta_i^t$ satisfy \eqref{eq:m=2-oct}. Then $q_{j,i}^t$ defined by \eqref{eq:theta-sol}
satisfies \eqref{eq:evolm=2-1}--\eqref{eq:evolm=2-3}.
\end{prop}
\begin{proof}
It is very easy to see that \eqref{eq:theta-sol} satisfies
\eqref{eq:evolm=2-1} and \eqref{eq:evolm=2-3}.
We check \eqref{eq:evolm=2-2}.
We consider a ratio
$$
f_i^t
:= \frac{q_{2,i+1}^t -q_{1,i+1}^{t+\be_1}}
{q_{2,i}^{t+\be_1} - q_{1,i}^{t+\be_2}}.
$$
By substituting \eqref{eq:theta-sol} in $f_i^t$ we obtain
\begin{align}
f_i^t
=
\frac{
a_2 c_{i+2} \frac{\theta_{i+1}^{\be_2} \, \theta_i^{\be_1}}
{\theta_{i+1}^{\be_1} \, \theta_i^{\be_2}}
- a_1 c_{i+1}^{\be_1} \frac{\theta_{i+1}^{2 \be_1} \, \theta_{i}^{\be_1}}
{\theta_{i+1}^{\be_1} \, \theta_{i}^{2 \be_1}}
}
{
a_2 c_{i+1}^{\be_1} \frac{\theta_{i}^{\be_1+\be_2} \, \theta_{i-1}^{2 \be_1}}
{\theta_{i}^{2 \be_1} \, \theta_{i-1}^{\be_1+\be_2}}
- a_1 c_{i}^{\be_2} \frac{\theta_{i}^{\be_1+\be_2} \, \theta_{i-1}^{\be_2}}
{\theta_{i}^{\be_2} \, \theta_{i-1}^{\be_1+\be_2}}
}
=
\frac{
\theta_i^{\be_1} \, \theta_{i-1}^{\be_1+\be_2}
\left( a_2 \theta_{i+1}^{\be_2} \, \theta_{i}^{2 \be_1}
- a_1 \theta_{i+1}^{2 \be_1} \, \theta_{i}^{\be_2} \right)
}
{
\theta_i^{\be_1 + \be_2} \, \theta_{i+1}^{\be_1}
\left( a_2 \theta_{i}^{\be_2} \, \theta_{i-1}^{2 \be_1}
- a_1 \theta_{i}^{2 \be_1} \, \theta_{i-1}^{\be_2} \right)
},
\end{align}
where we omit the superscripts $t$ for simplicity.
At the second equality we have canceled all the $c_i$
using $c_i^{\be_u} = c_{i+u}$.
Due to \eqref{eq:m=2-oct}, we obtain $f_i^t = 1$.
Thus, using \eqref{eq:evolm=2-3}, we obtain \eqref{eq:evolm=2-2} .
\end{proof}
\subsection{General $m$ case}
The vertical actions $e_u ~(u=1,\ldots,m)$ are expressed as
matrix equations:
\begin{align}
\label{Q-m-1}
Q_1^{t+\be_{u}}\cdots Q_m^{t+\be_{u}}
=
Q_{u+1}^t \cdots Q_m^t P(x)^k Q_1^t \cdots Q_{u}^t P(x)^{-k}.
\end{align}
Among the family of difference equations we will use the following
ones later:
\begin{align}
\label{eq:evol1}
&\prod_{j=1}^m q_{j,i}^{t+\be_{u}}
=
\prod_{j=u+1}^m q_{j,i}^t \cdot \prod_{j=1}^u q_{j,i-k}^t,
\\
\label{eq:evol2}
&\sum_{j=1}^m q_{1,i}^{t+\be_u} \cdots q_{j-1,i}^{t+\be_u} \,
q_{j+1,i-1}^{t+\be_u} \cdots q_{m,i-1}^{t+\be_u}
=
\sum_{j=1}^m q_{s(1,i)}^{t} \cdots q_{s(j-1,i)}^{t} \,
q_{s(j+1,i-1)}^{t} \cdots q_{s(m,i-1)}^{t},
\end{align}
for $u=1,\cdots,m$. Here we define
$$
s(j,i) = \begin{cases}
(j+u,i) & j=1,\ldots m-u, \\
(j+u-m,i-k) & j=m-u+1,\ldots,m.
\end{cases}
$$
\begin{thm}\label{thm:octahedron-general}
For any $t \in \Z^m$, $i \in \Z$ and $1 \leq p < r \leq m$,
the $\theta_i^t$ satisfy
an octahedron recurrence relation,
\begin{align}\label{eq:m-oct}
\delta_{p,r} \,\theta_{i+1}^{t+\be_{p-1}+\be_{r-1}}
\theta_{i}^{t+\be_{p}+\be_{r}}
+ a_p \,\theta_{i+1}^{t+\be_{p}+\be_{r-1}} \theta_i^{t+\be_{p-1}+\be_{r}}
=
a_r \,\theta_{i+1}^{t+\be_{p-1}+\be_{r}} \theta_i^{t+\be_{p}+\be_{r-1}}.
\end{align}
Here $\delta_{p.r}$ is a constant given by
$$
\delta_{p,r} = a_r \, \frac{\Theta(p_0+\iota(A_p-A_r)) \Theta(p_0+\iota(O-P))}
{\Theta(p_0+\iota(O-A_r)) \Theta(p_0+\iota(A_p-P))},
$$
and $p_0$ is a zero of the Riemann theta function: $\Theta(p_0) = 0$.
\end{thm}
\begin{proof}
The proof is similar to the $m=2$ case. We explain the outline.
In the case $(P_1,P_2,P_3,P_4) = (A_r,O,P,A_p)$ of Theorem~\ref{thm:Fay},
we obtain
\begin{align}\label{Fay-basic-m}
\begin{split}
&T_1 \, \Theta(z+\iota(P-A_r)) \, \Theta(z+\iota(A_p-O))
+ T_2 \, \Theta(z+\iota(P-O)) \, \Theta(z+\iota(A_p-A_r))\,
\\
&\qquad = T_3 \, \Theta(z+\iota(P+A_p-A_r-O)) \, \Theta(z) \,
\end{split}
\end{align}
where $T_1$, $T_2$ and $T_3$ are given by \eqref{eq:T}, but
we replace $A_1$ (resp. $A_2$) with $A_p$ (resp. $A_r$).
By setting
$z = \beta^t + \iota((n-i-1)(O-P) + \mathcal{A}_p + \mathcal{A}_{r-1})$ at
\eqref{Fay-basic-m}, we get
$$
T_1 \,\theta_{i+1}^{t+\be_{p-1}+\be_{r-1}}
\theta_{i}^{t+\be_{p}+\be_{r}}
+ T_2 \,\theta_{i+1}^{t+\be_{p}+\be_{r-1}} \theta_i^{t+\be_{p-1}+\be_{r}}
=
T_3 \,\theta_{i+1}^{t+\be_{p-1}+\be_{r}} \theta_i^{t+\be_{p}+\be_{r-1}}.
$$
We take $t_0 \in \Z^m$ and define $p_0 := - \beta^{t_0} \in \C^g$
in the same manner as in the proof of Lemma~\ref{lem:T23}.
Then $q_{u,n}^{t_0-\be_{u-1}} = 0$ holds for $u=1,\ldots,m$.
From \eqref{eq:evol1} (resp. \eqref{eq:evol2})
of $i=n-1$ (resp. $i=n$) and $u=p-1$ or $r-1$, it follows that
$$
\frac{q_{p,n-1}^{t_0-\be_{p-1}}}{q_{1,n-1}^{t_0}}
= \frac{q_{r,n-1}^{t_0-\be_{r-1}}}{q_{1,n-1}^{t_0}} = 1.
$$
On the other hand,
by setting $z=p_0 + \iota(A_r-P)$ at \eqref{Fay-basic-m}, we obtain
$$
\frac{T_2}{T_3}
=
\frac{\theta^{t_0 + \be_{p} - \be_{p-1}}_{n-2} \, \theta^{t_0 + \be_{r}-\be_{r-1}}_{n-1}}
{\theta^{t_0 + \be_{r}-\be_{r-1}}_{n-2} \, \theta^{t_0 + \be_{p}-\be_{p-1}}_{n-1}}
=
\frac{q_{r,n-1}^{t_0-\be_{r-1}}}{q_{p,n-1}^{t_0-\be_{p-1}}} \cdot \frac{a_p}{a_r}.
$$
From the above two relations, $T_2/T_3 = a_p/a_r$ follows.
Finally, by setting $z=p_0+\iota(O-P)$ at \eqref{Fay-basic-m}
we obtain the formula of $\delta_{p,r}$.
\end{proof}
The following conjecture extends Proposition \ref{prop:thetaq}.
\begin{conjecture}
Suppose $\theta_i^t$ satisfy \eqref{eq:m-oct}. Then $q_{j,i}^t$ defined via \eqref{eq:theta-sol}
satisfy all the difference equations \eqref{Q-m-1}.
\end{conjecture}
We have checked the conjecture for $m \leq 3$.
\section{The transposed network}
\label{sec:transpose}
\subsection{Transposed Lax matrix and spectral curve}
Corresponding to $\tilde Q$ of \eqref{eq:q-tildeq},
we also identify $q \in \mM$ with an $N$-tuple of $n'm \times n'm$ matrices
$\tilde Q := (\tilde Q_i(x))_{i \in [N]}$ as \S\ref{sec:WWonM}.
The matrices $\tilde Q_i(x)$ are given in terms of the $q_{ji}$ by
\begin{align*}
&\tilde Q_{N+1-i}(x) :=
\tilde P(x) + \mathrm{diag}[q_{m,i},\ldots,q_{1,i},q_{m,i+k},\ldots,q_{1,i+k},\ldots,
q_{m,i-k},\ldots,q_{1,i-k}],
\end{align*}
where $\tilde P(x)$ is the $mn' \times mn'$ matrix:
$$
\tilde P(x) := \left(\begin{array}{cccc}
0 & 0 & 0 &x\\
1&0 &0 &0 \\
0&\ddots & \ddots&0 \\
0&0&1&0
\end{array} \right).
$$
We define $\tilde L(x) := \tilde Q_1(x) \cdots \tilde Q_N(x)
\tilde P(x)^{\bar k^\prime m}$, which is another Lax matrix.
Also define a map $\tilde \psi : \mM \to \C[x,y]$ given as a composition,
$$
\tilde Q \mapsto \tilde L(x) \mapsto \det(\tilde L(x) - y).
$$
Consequently, for $q \in \mM$
we have two affine plane curves $\tilde C_{\psi(q)}$ and
$\tilde C_{\tilde \psi(q)}$in $\C^2$, given by
the zeros of $\psi(q)$ and $\tilde \psi(q)$ respectively.
The proof of Proposition \ref{prop:transposehull} is similar to that of Proposition \ref{prop:hull}.
\begin{prop} \label{prop:transposehull}
The Newton polygon $N(\tilde \psi(q))$ is the triangle with
vertices $(0,m n')$, $(m \bar k',0)$ and $(m \bar k'+N,0)$,
where the lower hull (resp. upper hull)
consists of one edge with vertices $(m \bar k',0)$ and $(0,m n')$
(resp. $(m \bar k'+N,0)$ to $(0,m n')$).
\end{prop}
\begin{lem}
The affine transformation
\begin{align}\label{eq:aff-trans}
\begin{pmatrix}
i \\ j
\end{pmatrix}
\mapsto
\begin{pmatrix} \bar k'(k + m) \\ -n' k \end{pmatrix}
+
\begin{pmatrix} -\bar k^\prime &
(1- k^\prime \bar k^\prime)/n' \\
n' & k^\prime \end{pmatrix}
\begin{pmatrix} i \\ j \end{pmatrix}
\end{align}
sends integer points of $N(\psi(q))$ into integer points
of $N(\tilde \psi(q))$.
\end{lem}
\begin{proof}
It is easy to see that it sends the vertices correctly. By the
definition of
$\bar k'$ we know $(1-k' \bar k')/n'$ is an integer. Thus
this transformation sends integer points to integer points.
So does the inverse, as the determinant of the matrix involved is $-1$.
\end{proof}
\begin{example}
Let $(n,m,k) = (6,3,4)$.
The two Newton polygons $N(\psi(q))$ and $N(\tilde \psi(q))$ are illustrated in
Figure \ref{fig:mnk1}. Here $i$ labels the horizontal axis and $j$
labels the vertical axis.
\begin{figure}[ht]
\begin{center}
\vspace{-.1in}
\input{mnk1.pstex_t}
\vspace{-.1in}
\end{center}
\caption{}
\label{fig:mnk1}
\end{figure}
The dots of the same color show integer points inside the Newton polygon
that get sent to each other by the transformation \eqref{eq:aff-trans}. The formula
for the transformation in this case is
$$
\left(\begin{array}{c}
i \\
j
\end{array} \right)
\mapsto
\left(\begin{array}{c}
14 \\
-12
\end{array} \right)
+
\left(\begin{array}{cc}
-2 & -1 \\
3 & 2
\end{array} \right)
\left(\begin{array}{c}
i \\
j
\end{array} \right).
$$
\end{example}
\begin{prop}\label{prop:two-polys}
For $q \in \mM$, the polynomials $\psi(q)$ and $\tilde \psi(q)$
coincide up to the monomial transformation induced by \eqref{eq:aff-trans}.
The signs of the new monomials are derived from the rule given in Theorem \ref{thm:mit}: the sign of $x^a y^b$ is $(-1)^{(mn'-b-1)a+b}$.
\end{prop}
See \S~\ref{proof:two-polys} for the proof.
\subsection{Special points on the transposed curve}
We write $\tilde f(x,y)$ for the polynomial obtained from
the fixed polynomial $f(x,y)$ in \S\ref{subsec:special-pts} via
the affine transformation \eqref{eq:aff-trans}.
Let $C_{\tilde{f}}$ be the smooth compactification of the affine plane curve
$\tilde{C}_{\tilde f}$ given by $\tilde f(x,y) = 0$.
As for $C_{f}$ (Lemma~\ref{lem:inf}),
$C_{\tilde f}$ has a unique point $\tilde P$ lying over $\infty$.
Due to this fact and Proposition~\ref{prop:two-polys}, the two curves $C_{f}$ and $C_{\tilde{f}}$ are isomorphic.
Let $\tau : C_{\tilde f} \to C_f$ be the birational isomorphism, which is given by
$$
(x,y) \mapsto ( y^{n'} x^{k'}, x^\frac{1-k' \bar k'}{n'} y^{k'} )
$$
when $(x,y) \in C_{\tilde f} \cap (\C^\ast)^2$.
Besides $\tilde P$, on $C_{\tilde{f}}$ we have special points
$\tilde O_r = ((-1)^{m n'} \sigma_r,0)$ for $r \in [N]$, and $\tilde A_i$ for $i \in [m]$,
where near $\tilde A_i$ there is a local coordinate $u$ such that
$$
(x,y) \sim (u^{n'}, -(- \epsilon_i)^{\frac{1}{n'}} u^{\bar k'}).
$$
We see that $\tau (\tilde P) = P$, $\tau (\tilde A_i) = A_i$ and
$\tau (\tilde O_r) = O_r$.
\subsection{Proof of Proposition~\ref{prop:two-polys}}
\label{proof:two-polys}
The terms contributing to a particular coefficient of the spectral
curve are described by Theorem \ref{thm:mit}. We shall exhibit a bijection between
the terms of the coefficient associated with the lattice point $(i,j)$
inside the Newton polygon $N(\psi(q))$ and the terms contributing to the
coefficient associated with the image of that point in $N(\tilde \psi(q))$
under the affine transformation \eqref{eq:aff-trans}.
An \emph{underway path} in the network $G$ is the mirror symmetric version of a highway path. Thus the weights of an underway path are given by Figure \ref{fig:highway} with $0$ and $q_{ij}$ swapped. The crucial observation is that families of highway paths on the
network associated with $\tilde Q$ are families of underway paths
on the original network,
parsed in the opposite direction. This is because by the construction
of the transpose map between $Q$ and $\tilde Q$, the corresponding
toric networks are the same but are viewed
from opposite sides of the torus (inside vs outside).
Now, assume we have a closed family of highway paths contributing to
one of the coefficients of $\psi(q)$.
Simply complement all edges that ended up on our family of closed
highway paths inside the set of all edges of the network. We claim
that the result can be parsed as the desired
family of closed underway paths.
Indeed, the original family can be viewed as a number of horizontal
``steps through'' picking up a weight $q_{ij}$ at some node,
connected by intervals of staircase paths that
do not pick up any weight. We can interpret our procedure as
complementing used intervals of staircase paths, that is, making them not
used, and vice versa. As a result, locally around
each weight $q_{ij}$ that was picked up the new path will look
like what is shown in Figure \ref{fig:mnk2}. Thus, it will be an underway path, and it will
pick up exactly such $q_{ij}$. In other words, the
weight of the original highway family is the same as the weight of
this new underway family.
\begin{figure}[ht]
\begin{center}
\vspace{-.1in}
\input{mnk2.pstex_t}
\vspace{-.1in}
\end{center}
\caption{}
\label{fig:mnk2}
\end{figure}
It is also easy to see that the new underway family is closed.
This completes the proof.
\begin{example}
Let $(\a,\b,\c,\d)= (3,2,3,2)$ as in Example \ref{ex:3232}. Figure
\ref{fig:mnk3} shows an example of a family of paths contributing to
the purple term of the spectral curve
as marked in Figure \ref{fig:mnk1}. The first step takes the
complement of edges of this family inside the set of all edges of this
network.
\begin{figure}[ht]
\begin{center}
\vspace{-.1in}
\input{mnk3.pstex_t}
\vspace{-.1in}
\end{center}
\caption{}
\label{fig:mnk3}
\end{figure}
The second step does not change the network or the paths, it just
changes the point of view and reverses paths' directions.
\end{example}
\section{Relation to the dimer model}
\label{sec:cluster}
In this section, we give the explicit relation between
the $R$-matrix dynamics on our toric network and
cluster transformations on the honeycomb dimer on a torus.
See \cite{GonchaKenyon13} for background on the dimer model.
\subsection{Cluster transformations on the honeycomb dimer}
\label{subsec:cluster-trans}
The calculation in this section is the dimer analogue of the highway network computation of the geometric $R$-matrix (cf. \cite[Theorem 6.2]{LP}), which was explained earlier in \S \ref{sec:dynamics_proof} (Figure \ref{fig:wire20}).
Fix a positive integer $L$, and consider a honeycomb bipartite graph on a cylinder as in
Figure~\ref{fig:honeycone-dimer},
where $\alpha_i, \beta_i, \gamma_i ~(i \in \Z/L \Z)$ are the weights of the faces in three cyclically consecutive rows.
We write $\alpha := (\alpha_i, \beta_i, \gamma_i)_{i \in \Z / L \Z}$.
\begin{figure}[h]
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\multiput(0,50)(20,0){6}{\line(0,-1){10}}
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\multiput(10,22)(20,0){6}{\circle*{1.5}}
\multiput(0,14)(20,0){6}{\circle{1.5}}
\put(-3,63){$\cdots$}
\put(15,63){$\gamma_{L-1}$}
\put(37,63){$\gamma_L$}
\put(58,63){$\gamma_1$}
\put(78,63){$\gamma_2$}
\put(98,63){$\gamma_3$}
\put(5,45){$\alpha_{L-1}$}
\put(27,45){$\alpha_L$}
\put(48,45){$\alpha_1$}
\put(68,45){$\alpha_2$}
\put(88,45){$\alpha_3$}
\put(107,45){$\cdots$}
\put(-3,27){$\cdots$}
\put(17,27){$\beta_L$}
\put(38,27){$\beta_1$}
\put(58,27){$\beta_2$}
\put(78,27){$\beta_3$}
\put(98,27){$\beta_4$}
\end{picture}
\caption{Honeycomb dimer on a cylinder}
\label{fig:honeycone-dimer}
\end{figure}
For the weights $\alpha$ of the honeycomb,
we define a transformation $R_\alpha$ of $\alpha$ in a following way:
first we split the $\alpha_1$-face into two, by inserting a digon
of weight $-1$ as the top of Figure~\ref{fig:dimer-mutation}. We thank R. Kenyon for explaining this operation to us.
Set the weights of the new two faces to be
$-c$ and $\frac{\alpha_1}{c}$, where $c$ is a nonzero parameter which
will be determined.
Let $D$ be the quiver dual to the bipartite graph, drawn in blue in the figure.
The weights of faces are to be regarded as {\em coefficient variables}
associated to each vertex of $D$.
With $1,\ldots,L,a$ and $b$,
we assign the vertices of $D$ in the middle row, as depicted.
Next, we apply the cluster mutations $\mu_1$, $\mu_2, \ldots, \mu_L$ to $(D,\alpha)$
in order, recursively defining $\omega_i ~(i=1,\ldots,L)$ by
$$
\omega_1 := \frac{\alpha_1}{c}, \qquad \omega_i := \alpha_i(1+\omega_{i-1}).
$$
The condition that the digon's weight is again $-1$ after the
$L$ mutations gives an equation for $c$ as $-c (1+\omega_L) = -1$.
By solving it, $c$ is determined to be
$$
c = \frac{1-\prod_{s=1}^L \alpha_s}
{\sum_{t=0}^{L-1} \prod_{s=1}^t \alpha_{1-s}},
$$
and $\omega_i := \omega_i(\alpha_1,\ldots,\alpha_L)$ is obtained as
\begin{align}\label{eq:omega}
\omega_i(\alpha_1,\ldots,\alpha_L)
= \frac{\alpha_i \sum_{t=0}^{L-1} \prod_{s=1}^t \alpha_{i-s}}
{1- \prod_{s=1}^L \alpha_s}.
\end{align}
\begin{figure}
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\qbezier(45,62)(57,42.5)(45,23)
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\put(42,35){\small $-1$}
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\put(73,35){$\alpha_2$}
\put(4,65){\small $\gamma_{L-1}$}
\put(32,65){\small $\gamma_L$}
\put(63,65){\small $\gamma_1$}
\put(90,65){\small $\gamma_2$}
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\put(76,-37){\tiny{$\alpha_2(1+\frac{\alpha_1}{c})$}}
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\multiput(30,-58)(31,0){2}{\vector(1,4){5.8}}
}
\put(44,-71){$\downarrow ~ \mu_{L} \cdots \mu_2$}
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\multiput(0,-100)(30,0){4}{\line(0,-1){15}}
\multiput(15,-127)(30,0){3}{\line(0,-1){10}}
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\multiput(30,-100)(30,0){3}{\line(-5,4){15}}
\multiput(15,-127)(30,0){3}{\line(5,4){15}}
\multiput(15,-127)(30,0){3}{\line(-5,4){15}}
\multiput(15,-88)(30,0){3}{\circle*{2}}
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\multiput(0,-115)(30,0){4}{\circle*{2}}
\multiput(15,-127)(30,0){3}{\circle{2}}
\qbezier(45,-88)(33,-107.5)(45,-127)
\qbezier(45,-88)(57,-107.5)(45,-127)
\put(10,-119){$\frac{1+\omega_L}{\omega_{L-1}}$}
\put(33,-115){$\frac{1}{\omega_L}$}
\put(37,-102){\tiny $-c(1+\omega_L)$}
\put(45,-113){\tiny $-(1+\omega_1)$}
\put(71,-119){$\frac{1+\omega_2}{\omega_1}$}
\put(-17,-85){\tiny $\gamma_{L-1}(1+\frac{1}{\omega_{L-1}})^{-1}$}
\put(20,-85){\tiny $\gamma_L (1+\frac{1}{\omega_{L}})^{-1}$}
\put(51,-85){\tiny $\gamma_1 (1+\frac{1}{\omega_{1}})^{-1}$}
\put(80,-85){\tiny $\gamma_2 (1+\frac{1}{\omega_{2}})^{-1}$}
\put(-17,-132){\tiny $\beta_{L}(1+\frac{1}{\omega_{L-1}})^{-1}$}
\put(20,-132){\tiny $\beta_1 (1+\frac{1}{\omega_{L}})^{-1}$}
\put(51,-132){\tiny $\beta_2 (1+\frac{1}{\omega_{1}})^{-1}$}
\put(80,-132){\tiny $\beta_3 (1+\frac{1}{\omega_{2}})^{-1}$}
{\color{blue}
\multiput(0,-81)(30,0){4}{\circle*{1.5}}
\multiput(15,-108)(30,0){3}{\circle*{1.5}}
\multiput(36,-108)(18,0){2}{\circle*{1.5}}
\multiput(0,-135)(30,0){4}{\circle*{1.5}}
\put(32,-106){\tiny $L$}
\put(44,-106){\tiny $a$}
\put(56,-106){\tiny $b$}
\put(74,-105){\tiny $1$}
\put(14,-105){\tiny $L-1$}
\multiput(28,-81)(30,0){3}{\vector(-1,0){26}}
\multiput(28,-135)(30,0){3}{\vector(-1,0){26}}
\put(37.5,-108){\vector(1,0){6}}
\put(34.5,-108){\vector(-1,0){18}}
\put(73.5,-108){\vector(-1,0){18}}
\put(46.5,-108){\vector(1,0){6}}
\put(13,-108){\vector(-1,0){26}}
\put(103,-108){\vector(-1,0){26}}
\multiput(1.5,-83)(60,0){2}{\vector(1,-2){12}}
\multiput(16,-106.5)(60,0){2}{\vector(1,2){12}}
\multiput(16,-109.5)(60,0){2}{\vector(1,-2){12}}
\put(1.5,-133){\vector(1,2){12}}
\put(30,-83){\vector(1,-4){5.8}}
\put(54,-106){\vector(1,4){5.8}}
\put(54,-109.5){\vector(1,-4){5.8}}
\put(30,-133){\vector(1,4){5.8}}
\put(62,-133){\vector(1,2){12}}
}
\end{picture}
\caption{Honeycomb dimer model on a cylinder.}
\label{fig:dimer-mutation}
\end{figure}
\begin{definition}\label{prop:Rcluster}
Let $R_{\alpha}$ be the transformation of $\alpha$ given by
\begin{align}\label{eq:Rtrans}
(\alpha_i,\beta_i,\gamma_i) \mapsto
\left( \omega_{i-1}^{-1} (1+\omega_i),~
\beta_i (1+\omega_{i-1}^{-1})^{-1},~
\gamma_i (1+\omega_{i}^{-1})^{-1} \right),
\end{align}
which is induced by
the sequence of mutations $\mu_L \mu_{L-1} \cdots \mu_1$,
\end{definition}
Due to the expression of $\omega_i$,
we see that
$R_\alpha$ does not depend on which $\alpha_j$-face we split
at the first stage.
\begin{remark}
Note that our derivation of $R_\alpha$ is not entirely a cluster algebra computation since we began with the ``digon insertion" operation, which does not have a clear cluster algebra interpretation. In an upcoming work \cite{ILP}, we plan to further clarify the cluster nature of the geometric $R$-matrix.
\end{remark}
\subsection{Relation with the $(n,m,k)$-network}
We consider the honeycomb bipartite graph on a torus as in Figure~\ref{fig:Toda-dimer}.
We assign each face (resp. edge) with a weight $x_{ji}$
(resp. $q_{ji}$ or $1$),
satisfying the periodicity conditions $x_{j,i+n} = x_{j,i}$ and $x_{m+j,i} = x_{j,i-k}$
(resp. $q_{j,i+n} = q_{j,i}$ and $q_{m+j,i} = q_{j,i-k}$).
In the figure we omit weights that are equal to $1$.
The $x_{ji}$ and $q_{ji}$ are related by
\begin{align}\label{eq:x-q}
x_{j,i} = \frac{q_{j,i+1}}{q_{j+1,i}} ~(j=1,\ldots,m-1),
\qquad
x_{m,i} = \frac{q_{m,i+1}}{q_{1,i-k}},
\end{align}
hence the product of all the $x_{ji}$ is equal to $1$.
\begin{figure}
\unitlength=0.9mm
\begin{picture}(100,100)(5,0)
\multiput(0,86)(20,0){6}{\line(0,-1){10}}
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\multiput(0,50)(20,0){6}{\line(0,-1){10}}
\multiput(10,32)(20,0){6}{\line(0,-1){10}}
\multiput(0,14)(20,0){6}{\line(0,-1){10}}
\multiput(0,76)(20,0){6}{\line(5,-4){10}}
\multiput(10,68)(20,0){5}{\line(5,4){10}}
\multiput(10,58)(20,0){5}{\line(5,-4){10}}
\multiput(0,50)(20,0){6}{\line(5,4){10}}
\multiput(0,40)(20,0){6}{\line(5,-4){10}}
\multiput(10,32)(20,0){5}{\line(5,4){10}}
\multiput(10,22)(20,0){5}{\line(5,-4){10}}
\multiput(0,14)(20,0){6}{\line(5,4){10}}
\multiput(0,86)(20,0){6}{\circle{1.5}}
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\multiput(10,68)(20,0){6}{\circle{1.5}}
\multiput(10,58)(20,0){6}{\circle*{1.5}}
\multiput(0,50)(20,0){6}{\circle{1.5}}
\multiput(0,40)(20,0){6}{\circle*{1.5}}
\multiput(10,32)(20,0){6}{\circle{1.5}}
\multiput(10,22)(20,0){6}{\circle*{1.5}}
\multiput(0,14)(20,0){6}{\circle{1.5}}
\multiput(0,4)(20,0){6}{\circle*{1.5}}
\put(5,81){$x_{3,n-2}$}
\put(25,81){$x_{3,n-1}$}
\put(47,81){$x_{3,n}$}
\put(67,81){$x_{3,1}$}
\put(87,81){$x_{3,2}$}
\put(107,81){$\cdots$}
\put(-3,63){$\cdots$}
\put(15,63){$x_{2,n-1}$}
\put(37,63){$x_{2,n}$}
\put(57,63){$x_{2,1}$}
\put(77,63){$x_{2,2}$}
\put(97,63){$x_{2,3}$}
\put(5,45){$x_{1,n-1}$}
\put(27,45){$x_{1,n}$}
\put(47,45){$x_{1,1}$}
\put(67,45){$x_{1,2}$}
\put(87,45){$x_{1,3}$}
\put(107,45){$\cdots$}
\put(-3,27){$\cdots$}
\put(17,27){$x_{m,k}$}
\put(35,27){$x_{m,k+1}$}
\put(55,27){$x_{m,k+2}$}
\put(75,27){$x_{m,k+3}$}
\put(95,27){$x_{m,k+4}$}
\put(4,9){$x_{m-1,k}$}
\put(22,9){$x_{m-1,k+1}$}
\put(42,9){$x_{m-1,k+2}$}
\put(62,9){$x_{m-1,k+3}$}
\put(82,9){$x_{m-1,k+4}$}
\put(107,9){$\cdots$}
\put(12,71){$q_{3,n-1}$}
\put(32,71){$q_{3,n}$}
\put(52,71){$q_{3,1}$}
\put(72,71){$q_{3,2}$}
\put(92,71){$q_{3,3}$}
\put(3,53){$q_{2,n-1}$}
\put(22,53){$q_{2,n}$}
\put(42,53){$q_{2,1}$}
\put(62,53){$q_{2,2}$}
\put(82,53){$q_{2,3}$}
\put(12,35){$q_{1,n}$}
\put(32,35){$q_{1,1}$}
\put(52,35){$q_{1,2}$}
\put(72,35){$q_{1,3}$}
\put(92,35){$q_{1,4}$}
\put(3,17){$q_{m,k}$}
\put(22,17){$q_{m,k+1}$}
\put(42,17){$q_{m,k+2}$}
\put(62,17){$q_{m,k+3}$}
\put(82,17){$q_{m,k+4}$}
\end{picture}
\caption{$(n,m,k)$-dimer}
\label{fig:Toda-dimer}
\end{figure}
Let $\mathcal{X} \simeq \C^{mn}$ be a space of the weights of faces whose
coordinates are given by
$\underline{x} := (x_{ji})_{j \in \Z/m \Z,~ i \in \Z/n\Z}$.
Let $\rho$ be an embedding map from $\C(\mathcal{X})$ to $\C(\mathcal{M})$
given by \eqref{eq:x-q}.
We define three types of actions $\bar R_j$, $\tilde R_i$ and
$\hat R_i$ on $\mathcal{X}$, corresponding to the three directions that the honeycomb bipartite graph can be arranged into rows.
For $j =1,\ldots,m$, define
$\bar x(j) := (x_{j,i}, x_{j-1,i}, x_{j+1,i})_{i \in \Z/ n \Z}$.
Let $\bar R_j$ be the action on $\mathcal{X}$ induced by $R_{\bar x(j)}$
\eqref{eq:Rtrans}
with $L=n$. More precisely, $R_j(\underline{x}) = \underline{x}^\prime$ is given by
$$
x_{li}^\prime =
\begin{cases}
\omega_{i-1}^{-1}(1+\omega_i) & l = j,
\\
x_{j-1,i} (1+\omega_{i-1}^{-1})^{-1} & l=j-1,
\\
x_{j+1,i} (1+\omega_{i}^{-1})^{-1} & l=j+1,
\\
x_{li} & \text{otherwise},
\end{cases}
$$
with $\omega_i := \omega_i(x_{j,1},\ldots,x_{j,n})$ \eqref{eq:omega}.
In a similar way,
for $i=1,\ldots,N$, define
$\tilde x(i) := (x_{m-j,N-i}, x_{m-j,N-i-1}, x_{m-j,N-i+1})_{j \in \Z/ m n'\Z}$
and let $\tilde R_i$ be the action on $\mathcal{X}$ given by $R_{\tilde x(i)}$
\eqref{eq:Rtrans}
with $L=mn'$.
For $i =1,\ldots,M$, define
$\hat x(i) := (x_{j,i+1-j}, x_{j,i+2-j}, x_{j,i-j})_{j \in \Z/ \frac{mn}{M} \Z}$, and let $\hat R_i$ be the action on $\mathcal{X}$
given by $R_{\hat x(i)}$ \eqref{eq:Rtrans}
with $L=\frac{mn}{M}$.
Recall that we also have three types of actions on $\mathcal{M}$ given by
the elements $s_j ~(j=0,1,\ldots,m-1)$ of
the extended symmetric group $W$,
the elements $\tilde s_i ~(i=0,1,\ldots,N-1)$ of
the extended symmetric group $\tilde W$,
and the snake path action $T_s ~(s=1,\ldots,M)$.
\begin{prop}\
\begin{enumerate}
\item[(i)]
The actions $s_j$ and $\tilde s_i$ are compatible with
the actions $\bar R_j$ and $\tilde R_j$ respectively:
for $\underline{x} \in \mathcal{X}$, we have
\begin{align}
\label{R-s-1}
&\rho \circ \bar R_j (\underline{x})
= s_j^\ast \circ \rho (\underline{x}),
\quad j \in \Z / m \Z,
\\
\label{R-s-2}
&\rho \circ \tilde R_i (\underline{x})
= \tilde s_i^\ast \circ \rho (\underline{x}),
\quad i \in \Z / N \Z.
\end{align}
\item[(ii)] For $\underline{x} \in \mathcal{X}$,
the action $\hat R_i$ satisfies
\begin{align}
\label{R-T}
&\rho \circ \hat R_i (\underline{x})= \rho (\underline{x}),
\quad i=1,\ldots,M.
\end{align}
\item[(iii)]
For $\underline{x} \in \mathcal{X}$, the snake path action $T_i^\ast$ satisfies
$$T_i^\ast \circ \rho(\underline{x}) = \rho(\underline{x}), \quad i=1,\ldots,M.$$
\end{enumerate}
\end{prop}
\begin{proof}
(i) To show \eqref{R-s-1}, it is enough to prove the $j=1$ case.
We write $E_i$ for the energy $E(P^{-i} Q_1 P^i, P^{-i} Q_2 P^i)$.
The operator $s_1$ acts on $\mathcal{M}$ as
$s_1(q) = q^\prime$:
\begin{align}
q_{1,i}^\prime
=
\displaystyle{q_{2,i} \,\frac{E_{i}}{E_{i-1}},}
\qquad
q_{2,i}^\prime
=
\displaystyle{q_{1,i} \, \frac{E_{i-1}}{E_{j,i}}},
\end{align}
and the other $q_{ji}$ do not change.
By definition, $R_1(\underline{x}) = \underline{x}^\prime$ is obtained as
$$
x_{ji}^\prime
=
\begin{cases}
\omega_{i-1}^{-1} (1+\omega_i) & j=1,
\\
x_{m,i+k} (1+\omega_{i-1}^{-1})^{-1} & j=m,
\\
x_{2,i} (1+\omega_{i}^{-1})^{-1} & j=2,
\\
x_{ji} & \text{otherwise}.
\end{cases}
$$
where $\omega_i := \omega_i(x_{1,1},\ldots,x_{1,n})$.
On the other hand, by direct computation we obtain
$$
\rho(\omega_i) = \frac{q_{1,i+1} E_{i}}
{\prod_{s=1}^n q_{2,s} - \prod_{s=1}^n q_{1,s}},
\qquad
\rho(1+\omega_i) = \frac{q_{2,i+1} E_{i+1}}
{\prod_{s=1}^n q_{2,s} - \prod_{s=1}^n q_{1,s}}.
$$
Thus we get
\begin{align*}
&\rho \left(\omega_{i-1}^{-1} (1+\omega_i)\right)
= \frac{q_{2,i+1} E_{i+1}}{q_{1,i} E_{i-1}}
= s_1^\ast \left(\frac{q_{1,i+1}}{q_{2,i}}\right)
= s_1^\ast \circ \rho(x_{1,i}),
\\
&\rho \left(x_{m,i+k} (1+\omega_{i-1}^{-1})^{-1}\right)
= \frac{q_{m,i+k+1} E_{i-1}}{q_{2,i} E_{i}}
= s_1^\ast \left(\frac{q_{m,i+k+1}}{q_{1,i}}\right)
= s_1^\ast \circ \rho(x_{m,i+k}),
\\
&\rho \left(x_{2,i} (1+\omega_{i}^{-1})^{-1}\right)
= \frac{q_{1,i+1} E_{i}}{q_{3,i} E_{i+1}}
= s_1^\ast \left(\frac{q_{2,i+1}}{q_{3,i}}\right)
= s_1^\ast \circ \rho(x_{2,i}),
\end{align*}
and \eqref{R-s-1} follows.
We can prove \eqref{R-s-2} in a similar manner,
by replacing $Q_i$ with $\tilde Q_i$, $P$ with $\tilde P$, and so on.
\noindent
(ii) Again, it is enough to prove the case of $i=1$.
For simplicity, we write $L$ for $\frac{mn}{M}$, and set
$x_j := x_{j,2-j}$, $q_j := q_{j,3-j}$ for $j \in \Z /L \Z$.
From \eqref{eq:x-q}, it follows that
\begin{align}\label{eq:x-s-rho}
\rho(x_j) = \frac{q_j}{q_{j+1}}, \qquad
\rho\left(\prod_{j=1}^L x_j\right) = 1.
\end{align}
Define $\omega_j = \omega_j(x_1,\ldots,x_L)$ for $j=1,\ldots,L$.
Using the definition of $\omega_j$ and \eqref{eq:x-s-rho},
we obtain
\begin{align}
\rho\left(\frac{1 + \omega_j}{\omega_{j-1}}\right)
= \frac{q_j}{q_{j+1}},
\qquad
\rho\left(\frac{\omega_j}{1 + \omega_j}\right) = 1,
\end{align}
by the following calculation:
\begin{align*}
\frac{1 + \omega_j}{\omega_{j-1}}
&=
\frac{1 - \prod_{s=1}^L x_s + x_i \sum_{t=0}^{L-1} \prod_{s=1}^t x_{i-s}}
{x_{i-1} \sum_{t=0}^{L-1} \prod_{s=1}^t x_{i-1-s}}
\\
&\stackrel{\rho}{\longmapsto}
\frac{1 - 1 + \sum_{t=0}^{n-1} \frac{q_{i-t}}{q_{i+1}}}
{\sum_{t=1}^{n} \frac{q_{i-t}}{q_{i}}}
= \frac{q_i}{q_{i+1}},
\\
\frac{\omega_{j}}{1 + \omega_j}
&=
\frac{x_{i} \sum_{t=0}^{L-1} \prod_{s=1}^t x_{i-s}}
{1 - \prod_{s=1}^L x_s + x_i \sum_{t=0}^{L-1} \prod_{s=1}^t x_{i-s}}
\stackrel{\rho}{\longmapsto} 1.
\end{align*}
Thus we have
$\rho \circ R_{\hat x(1)}(\hat x(1))
= \rho(\hat x(1))$,
and \eqref{R-T} follows.
\noindent
(iii)
The snake path action $T_s$
changes $q_{j,i}$ and $q_{j^\prime,i^\prime}$ in the same way
if $i+j \equiv i^\prime + j^\prime \mod M$.
This condition is satisfied by $q_{j,i+1}$ and $q_{j+1,i}$,
then the change is cancelled in $\rho(x_{j,i})$.
Thus we see that $T_s \circ \rho(x_{j,i}) = \rho(x_{j,i})$.
\end{proof}
|
{
"timestamp": "2016-06-22T02:09:17",
"yymm": "1504",
"arxiv_id": "1504.03448",
"language": "en",
"url": "https://arxiv.org/abs/1504.03448"
}
|
\section{Introduction}
The compound HoPdBi is member of the large family of half-Heuslers that crystallize in the non-centrosymmetric space group $F\overline{4}3m$~\cite{Haase2002}. Ternary Heuslers with composition 2:1:1 and half Heuslers with composition 1:1:1 attract ample attention because of their flexible electronic structure, which gives rise to applications as multifunctional materials in, for example, the fields of spintronics and thermoelectricity~\cite{Graf2011}. Recently, a new incentive to investigate half-Heusler compounds was provided by first principles calculations~\cite{Lin2010,Chadov2010,Feng2010}. By employing the hybridization strength and the magnitude of the spin-orbit coupling is was established that "heavy-element" half-Heusler compounds may show a non-trivial band inversion, similar to the prototypical topological insulator HgTe~\cite{Bernevig2006,Hasan&Kane2010,Qi&Zhang2010}. Topological half-Heuslers are zero-gap semiconductors, where the topological surface states can be created by applying strain that opens the gap. The emergence of magnetic order and superconductivity at low temperatures in half-Heuslers provides a new research direction to study the interplay of topological states, superconductivity and magnetic order~\cite{Pan2013}.
Recently, much research has been devoted to the band-inverted half-Heusler platinum and palladium bismuthides, notably because some of these superconduct at low temperatures~\cite{Yan&deVisser2014}: LaPtBi ($T_c = 0.9$~K~\cite{Goll2008}), YPtBi ($T_c = 0.77$~K~\cite{Butch2011,Bay2012b}), LuPtBi ($T_c = 1.0$~K~\cite{Tafti2013}) and LuPdBi ($T_c = 1.7$~K~\cite{Xu2014,Pavlosiuk2015}). The $s$-$p$ inverted band order ($\Gamma _8 > \Gamma _6$) makes these compounds candidates for topological superconductivity~\cite{Kitaev2009,Schnyder2009}. Topological superconductors are predicted to be unconventional superconductors, with an admixture of even- and odd-parity Cooper pair states in the bulk, and protected Majorana states at the surface~\cite{Hasan&Kane2010,Qi&Zhang2010}. Since the crystal structure lacks inversion symmetry these superconductors might also be classified as non-centrosymmetric. Here theory predicts a mixed even- and odd-parity Cooper pair state~\cite{Frigeri2004} as well.
Recently, we reported the observation of superconductivity ($T_c = 1.22$~K) and antiferromagnetic order ($T_N = 1.06$~K) in the half-Heusler ErPdBi. In addition, electronic structure calculations predicted a sizeable band inversion, which makes ErPdBi a first candidate for the investigation of the interplay of topological states, superconductivity and magnetic order. Another candidate proposed is CePdBi ($T_c = 1.3$~K, $T_N = 2.0$~K), however, here superconductivity develops in a small sample volume and is associated with a disordered phase~\cite{Goraus2013}. The coexistence of superconductivity and magnetic order in ErPdBi provided a strong motivation to search for similar phenomena in other REPdBi compounds (RE = Rare Earth), such as HoPdBi~\cite{Haase2002,Gofryk2005}. HoPdBi was reported to be an antiferromagnet with a N\'{e}el temperature $T_N = 2.2$~K~\cite{Gofryk2005}. Its magnetic susceptibility follows a Curie-Weiss law with an effective moment $\mu_{\rm {eff}}$ = 10.6 $\mu_B$, in good agreement with the theoretical value for free RE$^{3+}$ ions with Russell-Saunders coupling ($J=8$). The weak antiferromagnetic coupling is furthermore inferred from the low value of the Curie-Weiss constant $\Theta _P = - 6.4$~K. Here we report results of transport, magnetic and thermal measurements on single crystalline HoPdBi. Our flux-grown crystals show antiferromagnetic order at $T_N = 2.0$~K and superconductivity at $ T_c \sim 1.9$~K. However, the transition to bulk superconductivity, inferred from the full diamagnetic screening signal (100 \% volume fraction), sets in near $ T_c ^{bulk} \sim 0.75$~K. We report the magnetic and superconducting phase diagram in the field-temperature plane, as well as electronic structure calculations that reveal a non-trivial band inversion of 0.25~eV at the $\Gamma$-point. In the course of our investigations, Nakajima \textit{et al}.~\cite{Nakajima2015} reported that superconductivity occurs in almost the entire antiferromagnetic REPdBi series, with $T_c \sim 1$~K for HoPdBi.
\section{Methods}
Several single crystalline batches of HoPdBi were prepared using Bi flux. The high purity elements Ho, Pd and Bi (purity 3N5, 4N and 5N, respectively) served as starting material. First an ingot of HoPdBi was prepared by arc-melting and placed in an alumina crucible with excess Bi flux. The crucible was sealed in a quartz tube under a high-purity argon atmosphere (0.3 bar). Next the tube with crucible was heated in a furnace to 1150 $^{\circ}$C and kept at this temperature for 36 h. Then the tube was slowly cooled to 500 $^{\circ}$C at a rate of 3 $^{\circ}$C per hour to form the crystals. The single-crystalline nature of the crystals was checked by Laue backscattering. After cutting the crystals in suitable shapes for the various experiments, the surfaces were cleaned by polishing. Powder X-ray diffraction confirmed the F$\bar{4}$3m space group and the extracted lattice parameter, $a$ = {6.605} {\AA}, is in good agreement with the literature~\cite{Haase2002}. Several tiny extra diffraction lines point to the presence of a small amount of an impurity phase ($< 5$~\%). These lines could not be matched to potential inclusions or impurity phases, such as Bi, PdBi, $\alpha$-Bi$_2$Pd and $\beta$-Bi$_2$Pd. The results presented here are taken on samples that all come from the same single-crystalline batch. The measured samples are labeled \#2-\#4, and show very similar magnetic and superconducting properties.
Magnetic and transport measurements were carried out in a Physical Property Measurements System (PPMS, Quantum Design) in the temperature range 1.8-300~K. The magnetization and susceptibility were measured utilizing a Vibrating Sample Magnetometer, while the resistance and Hall effect were measured using a standard four-point low-frequency ac-technique with an excitation current of 1 mA. The specific heat was measured down to 2.0 K on a crystal with a mass of 10 mg in the PPMS as well. Low temperature, $T = 0.24 - 10$~K, resistance and ac-susceptibility measurements were made in a $^3$He refrigerator (Heliox, Oxford Instruments) using a low-frequency lock-in technique with low excitation currents ($I \leq 200$~$\mu$A). The coefficient of linear thermal expansion, $\alpha$ = L$^{-1}$(dL/dT), with $L$ the sample length, was measured using a three-terminal parallel-plate capacitance method using a home-built sensitive dilatometer based on the design reported in Ref.~\cite{Schmiedeshoff2006}. Magnetoresistance data were taken in a dilution refrigerator (Kelvinox, Oxford Instruments) for $T = 0.03 - 1$~K and high magnetic fields up to 17 T. Additional low-temperature dc-magnetization and ac-susceptibility measurements were made using a SQUID magnetometer, equipped with a miniature dilution refrigerator, developed at the N\'{e}el Institute.
\section{Results and analysis}
\subsection{Superconductivity}
\begin{figure}
\begin{center}
\includegraphics[height=6cm]{fig1.eps}
\caption{Resistivity and carrier concentration (inset) as a function of temperature of HoPdBi sample~\#2.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[height=6cm]{fig2.eps}
\caption{Resistivity as a function of temperature for HoPdBi sample~\#2 measured in fixed magnetic fields: from right to left 0~T to 1.1~T in steps of 0.1~T, 1.2~T to 2~T in steps of 0.2~T, and 2.25~T, 2.5~T and 3~T. Lower inset: Derivative d$\rho$/d$T$ \textit{versus} $T$ for $B = 1.0$~T; arrows indicate $T_c$ and $T_N$. Upper inset: $\rho (T)$ \textit{versus} $T$ in zero field for sample~\#3.}
\end{center}
\end{figure}
In fig.~1 we show the resistivity $\rho(T)$ of a single crystal of HoPdBi (sample \#2). The resistivity values are rather large and $\rho(T)$ has a broad maximum around 80 K. The charge carrier concentration $n_h$ extracted from the low field ($B < 2$~T) Hall effect measurements is traced in the inset of fig.~1. The carriers are holes and at the lowest temperature ($T=2.0$~K) $n_h = 1.3\times 10^{19}$~cm$^{-3}$. These transport parameters reveal semimetallic-like behaviour. At low temperatures, the resistivity drops to zero, which signals the superconducting transition.
The superconducting transition measured by resistivity in zero and applied magnetic fields is reported in detail in fig.~2. The superconducting transition at $B=0$ is fairly broad, with an onset temperature of 2.3~K and zero resistance at 1.6~K. By tracing the maximum in the derivative d$\rho$/d$T$ we obtain $T_c$ = 1.86 K. The broad transition is most likely related to the simultaneous development of antiferromagnetic order with $T_N = 2.0$~K (see below). A similar rounded $\rho(T)$ near the onset for superconductivity was observed for ErPdBi, where $T_c \simeq T_N$ as well~\cite{Pan2013}. We have determined the upper critical field $B_{c2}$ by taking $T_c (B)$ as the maximum in d$\rho$/d$T$ obtained in constant magnetic fields. See the inset of fig.~2 for the data at $B = 1.0$~T. Here the structure at 2.0 K signals the antiferromagnetic transition. The temperature variation $B_{c2}(T)$ is reported in the phase diagram, fig.~7. In the upper inset of fig.~2 we show the zero-field data for sample \#3. The resistivity data for this sample (data in field are not shown) are in good agreement with the results obtained for sample \#2.
\begin{figure}
\begin{center}
\includegraphics[height=6cm]{fig3.eps}
\caption{Real part of the ac-susceptibility of HoPdBi sample \#3 for driving fields of 0.0025, 0.005, 0.025, 0.05, 0.10 and 0.25~mT, from bottom to top. Arrows indicate $T_N$, $T_c$ and $T_c ^{bulk}$. The inset shows the corresponding imaginary part of $\chi_{ac}$. }
\end{center}
\end{figure}
In fig.~3 we show ac-susceptibility ($\chi _{ac}$) data taken on sample \#3 for different driving fields $B_{ac}$ ranging from $0.0025$ to $0.25$~mT. Upon cooling $\chi _{ac}(T)$ first increases and has a pronounced maximum at 2.0~K, which locates the antiferromagnetic phase transition temperature. Upon further cooling a large superconducting signal appears below 0.75~K. For the smallest driving fields $B_{ac} \leq 0.05$~mT the screening fraction associated with this diamagnetic signal is $\sim$~90~\%, which provides strong support for bulk superconductivity. Upon increasing the driving field to 0.25~mT the magnitude of the diamagnetic signal is rapidly depressed, which indicates the lower critical field $B_{c1}$ is very small. However, a close inspection of the data shows that a weaker diamagnetic signal develops near 1.8 K, which is close to midpoint of the superconducting transition at 1.9 K of sample \#3 (see upper inset fig.~2). The amplitude of the diamagnetic signal varies with $B_{ac}$. This clearly indicates a smaller volume fraction ($\sim 20$~\%) of the sample becomes superconducting at 1.8 K. Here we have neglected the demagnetization factor that we estimate to be $N = 0.07 \pm 0.02$ for our needle-shaped sample \#3. We remark that after correcting for demagnetization effects the total superconducting volume fraction is close to $\sim 100$~\%. These observations are corroborated by the imaginary part of $\chi_{ac}(T)$ shown in the inset of fig.~3. The transition to the bulk superconducting state reveals the presence of a remaining sample inhomogeneity, as indicated by the double peak structure in $\chi _{ac} ^{\prime \prime}$ for small values $B_{ac} \leq 0.005$~mT.
\subsection{Antiferromagnetic order}
Magnetic susceptibility measurements carried out in a field of 0.1~T confirm local moment behaviour of the Ho moments. For $T > 50$~K Curie-Weiss behaviour is observed with an effective moment $\mu_{eff} = 10.3~\mu_B$ and a negative Curie-Weiss constant $\Theta_P = -5.7$~K indicating the presence of antiferromagnetic interactions. These values are close to the ones reported in Ref.~\cite{Gofryk2005}. In fig.~4 we show the dc-susceptibility around the antiferromagnetic transition measured in fields up to 3~T down to very low temperatures (0.1 K). The N\'{e}el temperature is progressively shifted with field and can no longer be discerned at the largest field. The variation $T_N (B)$ is traced in the phase diagram (fig.~7). In the right panel we show the magnetization measured at $T= 0.075$~K, deep in the antiferromagnetic phase, and at 2~K and 5~K, measured along the [100] axis. At the lowest temperature the magnetization increases rapidly until it reaches a value of $\sim 6~ \mu _B$ at 3~T after which the increase is more gradual. In a field of 8~T the magnetization attains a value of $7.2~\mu _B$ which is still considerably below the saturation value $m_s = gJ = 10 ~\mu_B$ (here $g = 5/4$ is the Land\'{e} factor). An explanation for this is offered by the sizeable magnetic anisotropy associated with the Ho ions (see section 4). The presence of magnetocrystalline anisotropy can also be inferred from the very weak signature of a metamagnetic-like transition observed in $M(B)$ below $T_N$.
\begin{figure}
\begin{center}
\includegraphics[height=6cm]{fig4.eps}
\caption{Left panel: Magnetic susceptibility of HoPdBi sample \#3 as a function of temperature measured in magnetic fields up to 3~T as indicated. Right panel: Magnetization in fields up to 8~T at temperatures of 0.075, 2.0 and 5.0 K.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[height=6cm]{fig5.eps}
\caption{Left panel: Coefficient of linear thermal expansion as a function of temperature of HoPdBi (sample \#4) measured in zero and applied fields as indicated. Right panel: Specific heat (sample \#2) in a plot of $c$ versus $T$ up to 50~K. The solid line indicates the lattice contribution given by the Debye function with $\theta _D = 193$~K.}
\end{center}
\end{figure}
In fig.~5 we show the coefficient of linear thermal expansion $\alpha (T)$ around the antiferromagnetic transition. Upon cooling a large positive step $\Delta \alpha$ is observed, where the midpoint nicely agrees with $T_N$ determined by the magnetic susceptibility. The superconducting transition is not observed in the thermal expansion because of the relatively small entropy involved (see the discussion section). In the right panel we show the specific heat measured in the temperature range 2-50 K. The increase at low temperatures upon approaching 2 K is due to the antiferromagnetic transition. When we compare $c(T)$ with the lattice specific heat, estimated by a Debye function with a Debye temperature $\theta _D = 193$~K, a broad magnetic contribution centered around 15~K becomes noticeable. This magnetic contribution with an entropy of $ \sim 1.7 \times R$, where $R=8.31$ J/molK is the gas constant, is most likely related to the partly lifting of the 17-fold degenerated crystalline electric field ground state in the cubic symmetry. We remark that the magnetic susceptibility starts to deviate from the Curie-Weiss behaviour below 50~K as well. By combining the thermal expansion and specific heat data we can make an estimate of the pressure variation of $T_N$ using the Ehrenfest relation d$T_N / $ d$p = V_m \Delta \beta / (\Delta c/T_N)$. Here the coefficient of volume expansion $\beta = 3 \alpha$ and $V_m = 4.3 \times 10^{-5}$~m$^3$/mol is the molar volume. Using the experimental values $\Delta \alpha = 1.2 \times 10^{-4}$~K$^{-1}$ and $\Delta c/T_N \sim$ 5~J/molK$^2$ we obtain d$T_N / $ d$p \sim 0.3$~K/kbar. In an applied field, the longitudinal thermal expansion measured along the direction of the magnetic field, $\alpha _\parallel$, develops a pronounced minimum. The negative $\alpha _\parallel$ indicates a large magnetocrystalline anisotropy, with an expansion along the field and a contraction perpendicular to the field, upon cooling below $T_N$. Notice, in magnetic field $ \beta = \alpha_\parallel + 2 \alpha _\perp$. The variation $T_N (B)$ extracted from the thermal expansion is traced in the phase diagram fig.~7.
\subsection{Quantum oscillations}
The magnetoresistance was measured in magnetic fields up to 17~T at low temperatures in the dilution refrigerator. A typical trace taken at $T= 0.05$~K is shown in fig.~6. After the suppression of superconductivity, the kink at $B_M = 3.6$~T locates the antiferromagnetic phase boundary. In fields exceeding 8~T we observed Shubnikov-de Haas oscillations. Since the oscillatory component is rather small ($\Delta \rho / \rho \sim 0.1 \%$) and superimposed on a non-monotonic background, it is difficult to extract the precise magnitude as a function of $1/B$. In the lower inset of fig.~6 we trace the resulting oscillations in $\Delta \rho$ as a function of $1/B$. A fast Fourier transform (upper inset) shows the frequency is $75 \pm 5$~T. For a circular extremal cross section $A_k (E_F) = \pi k_F^2$ we calculate with help of the Onsager relation a Fermi wave vector $k_F = 4.8\times 10^8$~m$^{-1}$. Assuming a spherical Fermi surface pocket, the corresponding 3D carrier density $n=k_F^3/3\pi^2 = 3.7\times 10^{18}$~m$^3$ which is a factor $\sim 3.5$ smaller than the value deduced from the Hall effect measurements. Additional measurements at $T= 0.3$ and 0.6~K show the thermal damping is very weak, which points to an effective mass much smaller than the free electron mass, $m_{eff} < 0.5 m_e$, as reported for other topological semimetals~\cite{Butch2011,Goll2002}.
\begin{figure}
\begin{center}
\includegraphics[height=6cm]{fig6.eps}
\caption{Resistance of HoPdBi sample \#2 as a function of the magnetic field at $T=0.05$~K. The arrow indicates the antiferromagnetic phase boundary at $B_M$. Lower inset: $\Delta \rho$ as a function of $1/B$ showing the Shubnikov - de Haas effect at 0.05, 0.3 and 0.6 K. Upper inset: fast Fourier transform of the data with frequency $F= 75 \pm 5$~T. }
\end{center}
\end{figure}
\subsection{Phase diagram}
In fig.~7 we present the magnetic and superconducting phase diagram of HoPdBi. The upper critical field $B_{c2}$ shows a smooth variation with temperature and extrapolates to the value of 1.1~T at $T=0$. This tells us $B_{c2}(T)$ is associated with the superconducting transition observed at 1.9~K in zero field. When we compare the $B_{c2}$-data to the Werthamer-Helfand-Hohenberg (WHH) curve for a weak-coupling spin-singlet orbital-limited superconductor, some departure appears below $\sim 0.7$~K. We remark this is close to the temperature below which the diamagnetic screening signal shows the sudden increase towards bulk superconductivity (see fig.~2). In the WHH model the clean-limit zero-temperature orbital critical field can be calculated from the relation $B_{c2}^{orb}(0) = 0.72 \times T_c |$d$B_{c2}/$d$T|_{T_c}$ and attains a value of 0.9 T (in the dirty limit the coefficient 0.72 is replaced by 0.69). From the experimental value $B_{c2}(0)=1.1$~T we calculate, utilizing the relation $B_{c2} = \Phi _0 / 2 \pi \xi ^2$, where $\Phi _0$ is the flux quantum, a superconducting coherence length $\xi = 17$~nm. A departure from the WHH model has also been observed for ErPdBi~\cite{Pan2013} and notably for YPtBi~\cite{Butch2011,Bay2012b}, where is was taken as a strong indication of an odd-parity component in the superconducting order parameter. We remark, the superconducting phase is completely embedded in the antiferromagnetic phase.
The antiferromagnetic phase boundary has been determined by resistivity, magnetization and thermal expansion measurements, see fig.~7. After an initial ($B < 1$~T) steep increase with field, $T_N$ smoothly decreases, and the antiferromagnetic phase transition is suppressed at $B_M (0)= 3.6$~T. Values for $T_N$ extracted from the magnetization and thermal expansion are somewhat lower than the values obtained by transport, but all three experiments track the same phase boundary. These differences might be partly attributed to different orientations of the crystals in the magnetic field in connection to the large magnetic anisotropy. For the magnetization experiments the magnetic field was applied along [100], while for the other experiments the field was applied along an arbitrary crystal direction. In the figure we compare the phase boundary with the phenomenological order parameter function $B_M (T) = B_M (0)(1-(T/T_N)^{\alpha})^{\beta}$ with $T_N = 2.0$~K, $B_M (0) =3.6$~T, $\alpha = 2$ and $\beta = 0.34$. The latter value is close to the theoretical value $\beta = 0.38$ expected for a 3D Heisenberg antiferromagnet~\cite{Domb1996}.
\begin{figure}
\begin{center}
\includegraphics[height=6cm]{fig7.eps}
\caption{Magnetic and superconducting phase diagram of HoPdBi.
Data points for the superconducting (SC) transition are taken on sample \#2 where $T_c (B)$ is defined by the temperature of the maximum in d$\rho /$d$T$ at fixed fields (blue circles) or the maximum in d$\rho /$d$B$ at fixed temperatures (blue triangles).
The blue solid line is a comparison of the $B_{c2}(T)$ data with the WHH model (see text). The blue dashed line represents a linear fit to the data. Data points for the antiferromagnetic (AFM) transition are taken from the resistivity (sample \#2: red circles and triangles from d$\rho /$d$T$ and d$\rho /$d$B$, respectively), the magnetization (sample \#3: open stars) and thermal expansion (sample \#4: open circles). The red solid line represents a phenomenological order parameter fit with $T_N = 2.0$~K and $B_M = 3.6$~T at $T=0$ (see text)}.
\end{center}
\end{figure}
\subsection{Electronic structure}
\begin{figure}
\begin{center}
\includegraphics[height=7cm]{fig8.eps}
\caption{Calculated bulk band structure of the half-Heusler HoPdBi. The projection of the Pd-$s$ states is highlighted by the size of the filled red circles. Bands $\Gamma _{6,7,8}$ are labeled according to the symmetry of their wave functions. The Fermi energy, $E_F$, is shifted to zero (horizontal dashed line). The measured sample is slightly $p$-doped, which will cause $E_F$ to lie marginally below zero.}
\end{center}
\end{figure}
To understand the electronic properties, we have performed \textit{ab initio} band structure calculations~\cite{Kresse&Furthmuller1996} based on the density-functional theory. The core electrons are represented by the projector-augmented-wave potential. We keep only three valence electrons of Ho (equivalent to La) and freeze all other $f$ electrons as core electrons, to simplify the understanding of the topology. The calculated bulk band structure of HoPdBi exhibits a zero-gap, and is quite similar to that of ErPdBi~\cite{Pan2013} and other half-Heusler topological semimetals ~\cite{Lin2010,Chadov2010}. The $\Gamma_6$ band (contributed by Pd-$s$ states) is below the $\Gamma_8$ bands, showing the hall mark of band inversion. Therefore, we can clearly conclude that HoPdBi is also a topological semimetal. The effect of magnetization of Ho-$f$ states is expected to possibly split the spin degeneracy slightly, but to preserve the inverted band order because the band inversion strength (the energy difference between $\Gamma_8$ and $\Gamma_6$ bands) is as large as $0.25$~eV. The measured sample is slightly $p$-doped, which will cause $E_F$ to lie marginally below zero.
\section{Discussion}
Our systematic transport, magnetic and thermal properties study demonstrates that HoPdBi combines local moment magnetism and superconductivity and thus behaves very similar to the related half-Heusler compound ErPdBi~\cite{Pan2013}. Superconductivity occurs at $T_c = 1.9$~K as determined by the resistive transition and the onset in $\chi _{ac}$, however, the transition to a full diamagnetic screening fraction sets in at a lower temperature, near 0.75~K. These results are in line with the recent work by Nakajima \textit{et al}.~\cite{Nakajima2015}, but nevertheless there are some noticeable differences: the midpoint of the resistive transition is at $\sim 1.0$~K, and the onset of the diamagnetic signal is at $\sim 0.9~$~K ~\cite{Nakajima2015}. Moreover, the transition measured by $\chi _{ac}$ ~\cite{Nakajima2015} is very broad and extends to the lowest temperature measured ($\sim 0.1$~K). A comparison of both studies suggests the superconducting phase transition found below $\sim 0.9~$~K by Nakajima \textit{et al}.~\cite{Nakajima2015} is to be associated with the transition to the bulk superconducting state at 0.75 K as reported in fig.~2. The presence of a small volume fraction of our sample that superconducts at 1.9~K remains a puzzling aspect. A possible explanation is the presence of an impurity phase. Among the binary Bi-Pd alloys $\alpha$-Bi$_2$Pd is reported to superconduct at 1.7~K~\cite{Matthias1963}. However, the few extra tiny lines in the diffraction pattern could not be matched to the structure of $\alpha$-Bi$_2$Pd. Moreover, $B_{c2}(0)$ of $\alpha$-Bi$_2$Pd is small~\cite{Nakajima2015}, which is at variance with the measured value $B_{c2}(0) = 1.1$~T.
A second, appealing explanation for superconductivity at 1.9~K is the occurrence of surface superconductivity associated with the topological surface states. Surface superconductivity at temperatures higher than the bulk $T_c$ was recently proposed by Kapitulnik \textit{et al.}~\cite{Kapitulnik2014,Kapitulnik2015} for the topological half-Heuslers YPtBi and LuPtBi. Experimental evidence for this was provided by (surface sensitive) Scanning Tunneling Spectroscopy measurements. Surface superconductivity may also offer an explanation for the "unknown impurity phase" with $T_c = 1.7$~K, $i.e.$ above the bulk $T_c$ of 1.2~K, reported for ErPdBi~\cite{Pan2013}. Interestingly, Nakajima \textit{et al}.~\cite{Nakajima2015} claim that surface superconductivity with $T_c = 1.6$~K can be induced by heat treatments at 200~$^{\circ}$C for all REPdBi. These results may open a new, fascinating research direction in the field of topological superconductivity. Future experiments have to be directed to disentangle surface and bulk superconductivity in order to put these ideas on a firmer footing.
Local-moment antiferromagnetic order has been detected for most members of the REPdBi series~\cite{Riedemann1996,Gofryk2005,Pan2013,Nakajima2015}, with $T_N$ following the De Gennes scaling: $T_N \propto (g_J -1)^2 J(J+1)$ with $g_J$ the Land\'{e} factor~(see \textit{e.g.} \cite{Jensen&Mackintosh1991}). The Ne\'{e}l temperature $T_N = 2.0$~K of our crystals is in between the values 2.2~K and 1.9 K reported in Refs.~ \cite{Gofryk2005} and~\cite{Nakajima2015}, respectively. The phase diagram (fig.~7) shows that antiferromagnetic order entirely encloses the superconducting phase. The low-temperature $\chi _{ac}$ and dc-magnetization measurements indicate that superconductivity and antiferromagnetic order coexist. In order to provide solid proof for this muon spin rotation/relaxation ($\mu$SR) and/or nuclear magnetic resonance experiments (NMR) would be very helpful. Alternatively, both ordering phenomena could possibly occupy different sample volumes. From recent neutron diffraction experiments~\cite{Nakajima2015} it was concluded that HoPdBi has a type-II antiferromagnetic structure: ferromagnetic planes couple antiferromagnetically with a propagation vector along the cube diagonal, $Q = (0.5, 0.5, 0.5)$. The magnetization curves (fig.~4) and the thermal expansion in field (fig.~5) reveal the magnetocrystalline anisotropy is rather strong. This offers the following scenario for the magnetization curves. When the field is increased along the [100] axis the spins progressively reorient towards the field, while still pointing along the [111] direction. Near the antiferromagnetic phase boundary the component of the spin along the [100] direction is $m_s / \sqrt 3$ = 5.8~ $\mu_B$ which is close to the experimental value. For fields above 3~T the spins are being pulled against the anisotropy field, which explains the reduced value with respect to $m_s = 10 ~\mu _B$. Interestingly, theoretical work based on symmetry classifications predicts the type II antiferromagnetic structure in these half-Heuslers produces a new ground state: an antiferromagnetic topological insulator~\cite{Mong2010}. Recently, it was proposed that a first realization of this novel electronic state is found in the half-Heusler antiferromagnet GdPtBi ($T_N=9$~K)~\cite{Muller2014}.
The large contribution of antiferromagnetic order to the thermal properties (fig.~5) hampers the detection of superconductivity in the thermal expansion and specific heat. An estimate for the step-size $\Delta c$ in the specific heat at $T_c$ can be calculated from the standard BCS relation $(\Delta c /T_c )/\gamma = 1.43$, where $\gamma$ is the Sommerfeld coefficient. It appears that $\gamma$ cannot reliably be deduced from the experimental data (see fig.~5) because of the presence of the large magnetic contribution centered around 15~K. However, an estimate for $\gamma$ can be obtained from the Shubnikov - de Haas data using the relation $\gamma = \pi^2 k_B^2 g(E_F)/3$, where $g(E_F)=m^* k_F / \hbar^2 \pi^2$ is the density of states. With the parameters calculated in section 3.3 we obtain $\gamma < 0.1$~mJ/molK$^2$ and $\Delta c < 0.1$~mJ/K at $T_c$, which is very small indeed. An estimate for the step size in the thermal expansion coefficient, $\Delta \alpha$, at $T_c$ can be obtained from the Ehrenfest relation d$T_c /$d$p = V_m 3 \alpha / (\Delta c/T_c)$. Assuming a typical value for d$T_c / $ d$p$ = 0.044 K/GPa~\cite{Bay2012b} we calculate $\Delta \alpha$ at $T_c$ is smaller than $4.5 \times 10^{-11}$~K$^{-1}$, which is beyond the resolution of the dilatometer.
The occurrence of superconductivity in a Ho-based compound is uncommon. Rare examples are found among the Chevrel phases~\cite{Shelton1983} and the heavy-rare earth cuprates~\cite{Hor1987} and pnictides~\cite{Yang2009}. Coexistence of superconductivity and antiferromagnetic order is found in the borocarbide HoNi$_2$B$_2$C~\cite{Cava1994}, but here $T_c = 10.5$~K and exceeds $T_N = 6.0$~K. An important new aspect in the REPdBi series is the non-centrosymmetric crystal structure and the topological nature of the electronic structure. This provides a new opportunity to investigate unconventional superconductivity, due to mixed even and odd parity Cooper pair states, and its interplay with long-range magnetic order.
\section{Summary}
We have investigated the transport, magnetic and thermal properties of the half-Heusler antiferromagnet HoPdBi. Electrical resistivity and ac-susceptibility data taken on flux-grown single crystals show superconductivity occurs below 1.9~K. We demonstrate HoPdBi is a bulk superconductor. However, the transition to bulk superconductivity, as indicated by a diamagnetic screening fraction close to 100 \%, sets in at a lower temperature of 0.75 K. The N\'{e}el temperature $T_N$ is $2.0$~K as determined by thermal expansion and dc-magnetization measurements. The superconducting and magnetic phase diagram in the $B-T$ plane has been determined: superconductivity is confined to the antiferromagnetic phase. Electronic structure calculations show HoPdBi is a topological semimetal with a band inversion of 0.25~eV at the $\Gamma$-point. This is in-line with the semimetallic behaviour observed in the electrical resistivity and the low carrier concentration $n_h = 3.7 \times 10^{18}$~cm$^{-3}$ extracted from the Shubnikov-de Haas effect. We conclude, HoPdBi belongs to the half-Heusler REPdBi series with a topological band structure and presents a new laboratory tool to study the interplay of antiferromagnetic order, superconductivity and topological quantum states~\cite{Pan2013,Nakajima2015}.
\ack{This work is part of the research programme on Topological Insulators of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientific Research (NWO). B.Y. acknowledges financial support from a European Research Council (ERC) Advanced Grant (291472).}
\section*{References}
|
{
"timestamp": "2015-04-14T02:14:36",
"yymm": "1504",
"arxiv_id": "1504.03181",
"language": "en",
"url": "https://arxiv.org/abs/1504.03181"
}
|
\section{Experimental}
Figure~\ref{Figure1} shows a schematic of a DNA-functionalised GUV adhering to a DNA-functionalised SBL. GUVs and SBLs are prepared and separately functionalised with cholesterol labeled DNA constructs. The hydrophobic cholesterol inserts into the lipid bilayer allowing DNA tethers to freely diffuse. Connected to the cholesterol anchor there is a section of length $L=9.8$ nm (29 base pairs\cite{Smith_Science_1996}) of double-strands DNA (dsDNA), terminating in a single-stranded DNA (ssDNA) \emph{sticky end}, which mediates the attractive interactions. To further facilitate the pivoting motion of the DNA tethers, 4 unpaired adenine bases are left between the cholesterol anchor and the dsDNA spacer. One unpaired adenine is left between the dsDNA spacer and the sticky end. Two mutually complementary sticky ends are used: $a$ and $a'$, of 7 bases each (see Fig.~\ref{Figure1}). GUVs and SBLs are functionalised with equal molar fractions of both $a$ and $a'$. Tethers can therefore form intra-membrane \emph{loops} and inter-membrane \emph{bridges}. The latter are responsible for the observed adhesion and are confined within the contact area between the GUVs and the SBLs. We use FRET spectroscopy to estimate the fraction of formed DNA bonds. To enable these measurements the termini of the sticky ends $a$ and $a'$ are functionalised with a Cy3 (indocarbocyanine) and a Cy5 (indodicarbocyanine) molecules respectively, spaced by an unpaired adenine base. Note that the sticky ends used here are shorter than those adopted in our previous study\cite{Parolini_NatComms_2015} (7 base-pairs instead of 9), this choice was made to lower the melting temperature of the DNA to within an experimentally accessible temperature, and reduce the overall strength of the DNA-mediated adhesion, making it easier to measure by flickering spectroscopy.
\subsection{Materials and sample preparation}
\subsubsection*{GUVs electroformation~~}1,2-Dioleoyl-sn-glycero-3-phosphocholine (DOPC, Avanti Polar Lipids) GUVs are prepared by standard electroformation in 300 mM sucrose (Sigma Aldrich) solution in double-distilled water.\cite{Angelova_FD_1989,Angelova_PCPS_1992} Full details can be found in ref.\cite{Parolini_NatComms_2015}
\subsubsection{Supported bilayers}DOPC supported bilayers (SBLs) are formed by rupture of Small Unilamellar Vesicles (SUVs $\sim$ 100 nm) on the hydrophilised glass bottom of sample chambers.\cite{Cremer_JPCB_1999,Brian_PNAS_1984}
SUVs are prepared by standard extrusion. Briefly, 200 $\mu$l of 25 mg/ml DOPC solution in chloroform are left to dry in a glass vial, then hydrated by adding 500 $\mu$l of 300 mM sucrose solution and mixed by vortexing for at least 5 minutes. The solutions are then transferred in plastic vials and treated with 5 cycles of rapid freezing/unfreezing by alternatively immersing the vial in baths of liquid nitrogen and warm water.\cite{Beales_JPCA_2007} Extrusion is carried out using a hand-driven mini-extruder (Avanti Polar Lipids) with a polycarbonate track-etched membrane (100 nm pores, Whatman). To facilitate rupture of the SUVs and bilayer formation, the extruded solution is diluted in a 1:9 ratio in iso-osmolar solution containing TE buffer (10 mM tris(hydroxymethyl) aminomethane, 1~mM ethylenediaminetetra acetic acid, Sigma Aldrich), 5~mM MgCl$_{2}$ and 272~mM glucose (Sigma Aldrich).\\
Sample chambers are obtained by applying adhesive silicone-rubber multi-well plates (6.5 mm $\times$ 6.5 mm $\times$ 3.2 mm, Flexwell, Grace Biolabs) on glass coverslips (26$\times$ 60 mm, No.1, Menzel-Gls\"{a}er), cleaned following a previously reported protocol.\cite{Di-Michele_NatComm_2013} To form SBLs, the glass bottom of the cells is hydrophilised by plasma cleaning on a plasmochemical reactor (Femto, Diener electronic, Germany), operated at frequency of 40 kHz, pressure of 30 Pa, and power input of 100 W for 5 minutes. Within 5 minutes from plasma cleaning, each cell is filled with $100$ $\mu$l of diluted SUV solution and incubated for at least 30 minutes at room temperature to allow bilayer formation. To remove excess lipid and magnesium, the cells are repeatedly rinsed with the experimental solution (TE buffer, 100 mM NaCl, 87 mM glucose). Care is taken to always keep the bilayers covered in buffer to avoid exposure to air.\\
For experiments not involving FRET spectroscopy GUVs and SBLs are stained with 0.8-1.2\% molar fraction of Texas Red 1,2-Dihexadecanoyl-sn-Glycero-3-Phosphoethanolamine, Triethylammonium Salt (Texas Red DHPE, Life Technologies).
\subsubsection*{DNA preparation~~}
The DNA tethers are pre-assembled from two ssDNA strands, one of them (\emph{i}) functionalized with a cholesterol molecule and the second (\emph{ii}) carrying the sticky end:\\
\begin{itemize}
\item[\emph{i}] $5'$--{\bf CGT GCG CTG GCG TCT GAA AGT CGA TTG CG} {\it AAAA} [Cholesterol TEG] --$3'$
\item[\emph{ii}] $5'$--{\bf GC GAA TCG ACT TTC AGA CGC CAG CGC ACG} {\it A} [Sticky End] {\it A} Cy3/Cy5 --$3'$.
\end{itemize}
The bold letters indicate the segments forming the dsDNA spacer, the italic letters the inert flexible spacers. DNA is purchased lyophilized from Integrated DNA Technologies, reconstituted in TE buffer, aliquoted, and stored at -20 $^\circ$C. For assembling the constructs, we dilute equal amounts of the two single strands, \emph{i} and \emph{ii}, to 1.6 $\mu$M in TE buffer containing 100 mM NaCl. Hybridization is carried out by ramping down the temperature from 90$^\circ$C to 20$^\circ$C at a rate of -0.2$^\circ$C min$^{-1}$ on a PCR machine (Eppendorf Mastercycler).\cite{Parolini_NatComms_2015}
\subsubsection*{Membrane functionalisation.~~}
Fuctionalisation of the supported bilayers is carried out by injecting 90 $\mu$L of iso-osmolar experimental solution (TE buffer, 87 mM glucose, 100 mM NaCl) containing
overall $X$ moles of DNA constructs into each of the silicone-rubber cells, with equal molarity of $a$ and $a'$ strands. Similarly, GUVs are functionalised by diluting 10 $\mu$L of electroformed vesicle solution in 90 $\mu$L of iso-osmolar experimental solution containing $X$ moles of DNA constructs. GUVs and SLBs are incubated for at least 1 hour at room temperature to allow grafting. After incubation, 10 $\mu$l of the the liposome solution are injected into the sample chambers, which are immediately closed with a second clean coverslip and sealed using rapid epoxy glue (Araldite). Care is taken to prevent the formation of air bubbles. Sedimentation of the GUVs results in the formation of an adhesion patch between GUVs and supported bilayer. In a typical sample a fraction of the GUVs is found to form clusters. We limit our analysis to isolated GUVs.\\
We tested samples at different DNA concentrations, obtained by setting $X$ to $X_\mathrm{low}=0.09,$ $X_\mathrm{med}=0.9$ and $X_\mathrm{high}=2.7$~pmoles. We have previously quantified the surface coverage of GUVs functionalised with $X_\mathrm{med}=0.9$ pmoles in $\rho_\mathrm{DNA}^\mathrm{med}=390\pm90$ $\mu$m$^{-2}$.\cite{Parolini_NatComms_2015} Here we proportionally assume $\rho_\mathrm{DNA}^\mathrm{low}=39\pm9$ $\mu$m$^{-2}$, $\rho_\mathrm{DNA}^\mathrm{high}=1200\pm300$ $\mu$m$^{-2}$. Having estimated that the overall surface of the SBL is approximately equal to the surface of the GUVs used for each sample, we assume equal DNA coverages for the SBL. This assumption is confirmed by fluorescence emission measurements: at sufficiently high temperature, when no DNA bridges are formed between GUVs and SBL, DNA is uniformly distributed on both interfaces. In this regime the fluorescence emission from DNA located within the contact area between a GUV and the SBL approximately equals twice the intensity measured on the free SBL ($2.1\pm 0.1$), confirming that GUVs and SBL have, within experimental errors, the same DNA coverage.\\
\subsection{Imaging and image analysis}
\subsubsection*{Imaging and temperature cycling.~~}
The samples are imaged on a Leica TCS SP5 laser-scanning confocal microscope. Texas Red DHPE is excited with a He-Ne laser (594 nm). For FRET spectroscopy measurements, Cy3 is excited with an Ar-ion laser line (514 nm). The temperature of the sample is regulated with a home-made Peltier device controlled by a PID (proportional-integral-derivative) controller, featuring a copper plate to which the sample chamber is kept in thermal contact. Two thermocouples are used as temperature sensors. The first sensor, kept in contact with the copper plate, serves as a feedback probe for the PID controller. The second thermocouple is inserted in a dummy experimental chamber, filled with water, and used to precisely probe the temperature of the sample. For all the temperature-dependent experiments imaging is carried out using a Leica HCX PL APO CS 40$\times$ 0.85 NA dry objective, to prevent heat dissipation.\\
The temperature-dependent morphology of adhering GUVs is captured via confocal z-stacks and reconstructed using a custom script written in Matlab. Briefly: Each z-stack contains a single adhering GUV. The plane of the SBL/adhesion patch is identified as the one with maximum average intensity, then the adhesion patch is reconstructed by thresholding, following the application of a bandpass filter to flatten the background and remove pixel-level noise. From the area $A_\mathrm{p}$ of the adhesion patch we extract the patch-radius as $R_\mathrm{p}=\sqrt{A_\mathrm{p}/\pi}$. The portion of the z-stack above the SBL is then scanned, and in each plane the contour of the vesicle, suitably highlighted by filtering and thresholding, is fitted with a circle. The slice featuring the largest circle is identified as the equatorial plane, determining the vesicle radius $R$. The contact angle is derived as $\theta=\sin^{-1}{\left(R_\mathrm{p}/R\right)}$ (see Fig.~\ref{Figure1}).\\
For flickering experiments, movies are recorded across the equatorial plane. Details of the flickering analysis are reported in the next section. To improve the quality of the signal, imaging for morphological characterisation and flickering analysis is carried out on samples stained with Texas Red DHPE.\\
FRET spectroscopy measurements are carried out by performing a spectral emission scan ($\lambda$-scan) of the contact area between an adhering GUV and the surrounding free SBL. While exciting the donor (Cy3), the emission of donor and acceptor is reconstructed by scanning the acquisition window from 530 to 785 nm, with a 6.75 nm binning. Similarly to the case of z-stacks, the adhesion patch is identified in each image by filtering and thresholding. For FRET imaging, non-fluorescent lipids are used in order to prevent undesired energy transfer between the bilayer and the DNA tethers.\\
\subsubsection{Flickering analysis}
\begin{figure}[ht!]
\centering
\includegraphics[width=7cm]{./Figure2.eps}
\caption{Intermediate steps of the contour detection algorithm. \textbf{a}, The position of the centra and the mean radius $R$ of the vesicle define an annular region in the frame;
\textbf{b}, the annular region $C(r,\varphi)$ extrapolated from the frame is used to compute an averaged template $\langle C(r) \rangle$; a 2D correlation $Q(r,\phi)$ of the two is computed and fitted to find the membrane position with sub-pixel precision, as described in the text (steps 1-6). The resulting contour is reported as a black line; \textbf{c}, colormap representation of the membrane fluctuations relative to the average radius. The colorbar reports the relative fluctuation as percentage in the range $-5\%$, $+5\%$ }
\label{FigureFlickering}
\end{figure}
Typically, in case of phase-contrast imaging, the contour of fluctuating GUVs is reconstructed by finding the inflection point in the radial intensity profile, as in reference.\cite{Pecreaux_EPJE_2004} In case of fluorescence imaging, the maximum of the intensity profile is commonly used to mark the membrane position.
Here, we designed a Matlab algorithm to reconstruct the contour of GUVs from their equatorial cross section with sub-pixel precision, independently on the imaging mode employed.
The algorithm proceeds following these steps:
\begin{enumerate}
\item The position of the centre, and a rough-guess value for the radius $R$ of the GUV are manually selected by the user on the first frame of the video (see Fig.~\ref{FigureFlickering}a).
\item The image of the radial profile $C(r,\varphi)$ is selected within an annular region of user-defined width that contains the membrane. The annular region is then mapped onto a rectangular stripe using a cubic interpolation (see Fig.~\ref{FigureFlickering}b, top).
\item A template $\langle C(r) \rangle$ of the radial intensity profile is calculated from the annular region, either by manual selection, or by automatically averaging along $\varphi$ (see Fig.~\ref{FigureFlickering}b, top-right).
\item The algorithm computes the 2D correlation function $Q(r,\varphi)$ of $C(r,\varphi)$ with the template $\langle C(r) \rangle$. For each value of the polar angle $\varphi$, $Q(r,\varphi)$ is maximum at the radial coordinate $r$ where the template best correlates with the membrane profile (see Fig.~\ref{FigureFlickering}b, bottom).
\item A rough measurement of the contour position $r(\varphi)$ is obtained by fitting the peak of the correlation image with a parabola around its maximum, for every value of $\varphi$. The algorithm cycles three times from point 2 to 5 , each time refining the estimate of the vesicle's centre and mean radius using the obtained contour.
\item The rough contour estimate is used as the initial point of a global minimisation of the energy-like functional
\begin{equation}
\epsilon (r, \varphi) = \sum_{\varphi} \left[ \left[ 1 - Q \left( r(\varphi),\varphi \right) \right] + \alpha \frac{\partial^2 r(\varphi) }{\partial \varphi^2} \right]
\end{equation}
here, the membrane position corresponds to the minimum of $1-Q$, corrected by a term that gives low weight to high curvature features, commonly due to random noise in the image. This retrieves the measurement of the contour position $r(\varphi)$. The minimisation makes use of the Nelder-Mead algorithm\footnote{Matlab's \textit{fminsearch} function}.
\end{enumerate}
The procedure is repeated for every frame in a video of the fluctuating membrane, each time using the center and radius values of the previous frame as the starting point of the algorithm. An example of the temporal evolution of the membrane radial profile $r(\varphi)$ is shown in Fig.~\ref{FigureFlickering}c.
The spectrum of the thermal fluctuations of the contour is calculated using Matlab Fast Fourier Transform algorithm, using the formula
$$ \langle |h^{2} (q_x) | \rangle = \frac{ \langle \left| FFT \left ( R(\varphi,t) \right ) \right|^{2} \rangle_{t}}{M^{2}}$$
where M is the number points used to map the contour, and then averaged over the ensemble of frames.
\subsubsection{Control experiments}
Control experiments are performed to measure temperature-dependent membrane tension of non-adhering GUVs. Plain, non-functionalised, GUVs are imaged while laying on a glass substrate passivated with bovine serum albumin (BSA, Sigma Aldrich). Part of the images is carried out in bright-field microscopy, using a Nikon Eclipse Ti-E inverted microscope, a Nikon PLAN APO 40$\times$ 0.95 N.A. dry objective and a IIDC Point Grey Research Grasshopper-3 camera. For control experiments GUVs are diluted in a 1:9 ratio in iso-osmolar glucose solution to enable sedimentation.
\section{Theoretical model}
We present a quantitative model describing the DNA-mediated adhesion of GUVs on flat supported bilayers. The model is adapted from reference,\cite{Parolini_NatComms_2015} where we treated the case of two identical adhering GUVs. Let us consider the interaction between an infinite SBL and a GUV adhering to it
\begin{equation}\label{U1}
U=U_\mathrm{membrane}+U_\mathrm{DNA}+U_0,
\end{equation}
where $U_\mathrm{membrane}$ accounts for the mechanical deformation of the GUV, $U_\mathrm{DNA}$ encodes for DNA-mediated adhesion, and $U_0$ is the reference energy, calculated for isolated GUV and SBL.
\subsection{Membrane deformation}
In Eq.~\ref{U1}, $U_\mathrm{membrane}$ summarises three contributions: stretching energy, bending energy, and the entropic cost of suppressing thermal fluctuations of the contact area between {}the GUV and the SBL.\cite{Ramachandran_Langmuir_2010} In the limit of \emph{strong adhesion} the stretching energy dominates over the other two contributions, which we can neglect.\cite{Ramachandran_Langmuir_2010} As discussed later, for the case of DNA-mediated interactions, this condition is generally verified at low enough temperature. We can rewrite
\begin{equation}\label{U2}
U_\mathrm{membrane}(\theta)=U_\mathrm{stretching}(\theta)=K_\mathrm{a}\frac{\left[A(\theta)-\tilde{A}\right]^2}{2\tilde{A}},
\end{equation}
where $K_\mathrm{a}$ is the stretching modulus of the membrane, $A(\theta)$ is the overall (stretched) area of the GUV, and $\tilde{A}$ is the reference, un-stretched, area. In the limit of strong adhesion the GUV will take the shape of a truncated sphere with contact angle $\theta$ (Fig.~\ref{Figure1}), which we take as the independent variable of our model.\cite{Ramachandran_Langmuir_2010} Within the assumption of constant inner volume $V=4\pi R_0^3/3$, where $R_0$ is a reference radius, the total area $A$ of the GUV and the adhesion patch area $A_\mathrm{p}$ can be expressed as a function of the contact angle $\theta$(see Appendix, section \ref{App_geometry}). The un-stretched vesicle area $\tilde{A}$ exhibits a strong temperature-dependence
\begin{equation}\label{eqn:unstretched}
\tilde{A}=A_0\left[1+\alpha(T-T_0)\right],
\end{equation}
where $\alpha$ is the area thermal expansion coefficient\cite{Evans_JPC_1987} and $T_0$ is the \emph{neutral temperature} of the GUVs, at which its reduced volume equals unity and $A=4\pi R_0^2$ (see Appendix, section \ref{App_geometry}).
\subsection{DNA-mediated adhesion}
We now focus on the DNA-mediated contribution to the interaction energy in Eq.~\ref{U1}: $U_\mathrm{DNA}$.\\
Given that the persistence length of dsDNA is $\sim50$~nm~$\gg$~$L=9.8$~nm,\cite{Smith_Science_1996} we can model the dsDNA spacers as rigid rods that, thanks to the fluidity of the membrane, can freely diffuse on the surface of the bilayers.\cite{Parolini_NatComms_2015,Angioletti-Uberti_PRL_2014} Free pivoting motion is guaranteed by the flexibility of the joint between the cholesterol anchor and the dsDNA spacer (Fig.~\ref{Figure1}). As demonstrated in ref.\cite{Parolini_NatComms_2015}, we can regard the sticky ends as point-like reactive sites, neglecting their physical dimensions. Moreover we can safely assume that the distance between the adhering membranes within the contact area is equal to $L$.\cite{Parolini_NatComms_2015} Finally, we neglect excluded volume interactions between unbound DNA tethers.\cite{Bracha_PNAS_2013,Parolini_NatComms_2015} The last two assumptions guarantee a uniform distribution of unbound DNA tethers and loops over the GUV and SBL surfaces.\\
The free energy change associated to the formation of a single bridge (b) or a loop ($\ell$) within the GUV is
\begin{equation}\label{eqn:singleenergy}
\Delta G_{\mathrm{b}/\ell}=\Delta G^0 - T \Delta S^\mathrm{conf}_{\mathrm{b}/\ell},
\end{equation}
where $\Delta G^0=\Delta H^0 - T \Delta S^0$ is the hybridisation free energy of untethered sticky ends, which can be calculated from the nearest-neighbour thermodynamic model,\cite{SantaLucia_PNAS_1998} eventually corrected to account for neighbouring non-hybridised bases.\cite{Di-Michele_JACS_2014} In Eq.~\ref{eqn:singleenergy}, the term $-T\Delta S^\mathrm{conf}_{\mathrm{b}/\ell}$ accounts for the confinement entropic loss taking place when tethered sticky ends hybridise,\cite{Angioletti-Uberti_NMat_2012,Mognetti_SoftMatter_2012,Varilly_JChemPhys_2012} and can be estimated as (see Appendix, section \ref{App_FreeEnergy})
\begin{equation}\label{DeltaG}
\Delta G(\theta) =\Delta G^0 - k_\mathrm{B}T\log\left[\frac{1}{\rho_0 L A(\theta)}\right],
\end{equation}
where $\rho_0=1$M is a reference concentration, and we highlighted the coupling with the geometry of the GUV via the $\theta$-dependence. Note that, contrary to the case of two adhering GUVs,\cite{Parolini_NatComms_2015} here $\Delta S^\mathrm{conf}_\mathrm{b}=\Delta S^\mathrm{conf}_\ell$. The local roughness of the membranes could also influence $\Delta G$\cite{Hu_PNAS_2013} -- this effects will be studied elsewhere.\\
We indicate with $N$ the total number of $a$ and $a'$ tethers on the GUVs, and model the SBL as an infinite reservoir of tethers. A combinatorial calculation detailed in the Appendix (sections \ref{App_SBL} and \ref{App_Comb}) allows the derivation of the overall hybridisation energy for the system of linkers
\begin{equation}\label{UHyb}
U_\mathrm{hyb}=N k_\mathrm{B} T \left[x_\ell + 2 \log \left(1-x_\ell-x_\mathrm{b}\right)-2\frac{\bar{N}_\mathrm{f}}{N}\right],
\end{equation}
In Eq.~\ref{UHyb}, $x_\mathrm{b}$ and $x_\ell$ are the fraction of tethers involved in bridges/loops, given by
\begin{equation}\label{BridgesImpl}
\frac{x_\mathrm{b}}{\left(1-x_\ell-x_\mathrm{b}\right)}=q_\mathrm{b},
\end{equation}
\begin{equation}\label{LoopsImpl}
\frac{x_\ell}{\left(1-x_\ell-x_\mathrm{b}\right)^2}=q_\ell
\end{equation}
where
\begin{eqnarray}
q_\ell&=&\exp\left[-\beta \Delta G^*\right], \label{QLoops}\\
q_\mathrm{b}&=&\frac{\bar{N}_\mathrm{f}}{N}\exp\left[-\beta \Delta G^*\right].\label{QBridges}
\end{eqnarray}
In Eqs.~\ref{QLoops} and \ref{QBridges} the hybridisation free energy is re-defined as $\Delta G^* = \Delta G - k_\mathrm{B}T \log N$. The combinatorial contribution $-k_\mathrm{B}T \log N$, typically estimated in $\sim-10 k_\mathrm{B}T$, has a stabilising effect.\\
The quantity $\bar{N}_\mathrm{f}$ appearing in Eq.~\ref{UHyb} indicates the average number of unbound DNA tethers anchored to the SBL available within the adhesion patch. The concentration $c_\mathrm{f}$ of unbound tetheres of each type ($a$ or $a'$), such that $\bar{N}_\mathrm{f}(\theta)=c_\mathrm{f}A_\mathrm{p}(\theta)$, is given by
\begin{equation}\label{eqn:freeSBL}
c_0-c_\mathrm{f}=c_\mathrm{f}^2\frac{\exp[-\beta \Delta G_0]}{\rho_0 L},
\end{equation}
where $c_0=\rho_\mathrm{DNA}/2$ is the total concentration of $a$ and $a'$ tethers. The full derivation of Eq.~\ref{eqn:freeSBL} is provided in the Appendix (section \ref{App_Comb}). Note that the concentration of loops on the SBL is $c_\ell=c_0-c_\mathrm{f}$.\\
Equation \ref{UHyb} generalises the expression found in ref.\cite{Parolini_NatComms_2015} to the case in which vesicles are in contact with an infinite reservoir of thethers.
The overall DNA contribution to the interaction energy in Eq.~\ref{U1} is
\begin{equation}\label{UDNA}
U_\mathrm{DNA}=U_\mathrm{hyb}-2N k_\mathrm{B} T \log\left(\frac{A}{A_0}\right),
\end{equation}
where the term on the right-hand side accounts for the change in overall confinement entropy following the area-change of the GUV.\cite{Parolini_NatComms_2015}\\
\subsection{Overall interaction energy.~~}
By combining Eqs.~\ref{U2}, \ref{UHyb}, and \ref{UDNA} into Eqs.~ \ref{U1}, we obtain an analytical expression for the interaction energy of the GUV+SBL system
as a function of the only independent variable $\theta$
\begin{multline}
U(\theta)-U_0=K_\mathrm{a}\frac{\left[A(\theta)-\tilde{A}\right]^2}{2\tilde{A}}+ N k_\mathrm{B} T \Bigl[x_\ell(\theta) + \\
2 \log \left[1-x_\ell(\theta)-x_\mathrm{b}(\theta)\right]-2\frac{\bar{N}_\mathrm{f}(\theta)}{N} - 2 \log\left(\frac{A(\theta)}{A_0}\right) \Bigr].
\end{multline}
Note that $U$ depends on $\theta$ though the adhesion area $A_\mathrm{p}$ (see Eq.~\ref{Nbar}) and the total area $A$ of the GUV. On the other hand, the reference energy $U_0$ is $\theta$-independent (see derivation in the Appendix, section \ref{App_Ref}). The interaction energy can be minimised to calculate all the morphological observables (e.g. the contact angle $\theta$, area of the adhesion patch $A_\mathrm{p}$, area of the spherical section of the GUV $A$), as well as the fraction of formed bridges and loops ($x_{\mathrm{b}/\ell}$).
\begin{figure}[ht!]
\centering
\includegraphics[width=8cm]{./Figure3.eps}
\caption{From left to right: 3D reconstruction from confocal z-stack, and confocal cross section of an adhering GUV (\textbf{a}) and a non-adhering GUV on a glass substrate (\textbf{b}). For 3D rendering, confocal images have been acquired with a Leica HCX PL APO CS 63$\times$ 1.4 NA oil immersion objective for better resolution, and deconvolved using the experimental point-spread function. Scale bars: $10$ $\mu$m.}
\label{FigureStacks}
\end{figure}
\subsection{Model parameters and error propagation}
The model features seven input parameters: the thermal expansion coefficient $\alpha$, the stretching modulus $K_\mathrm{a}$, the length of the dsDNA tether $L$, the hybridisation enthalpy $\Delta H^0$ and entropy $\Delta S^0$ of the sticky ends, the DNA coating density $\rho_\mathrm{DNA}$ (used to calculate the overall number of strands per GUV -- $2N$), the neutral temperature $T_0$ and radius $R_0$.\\ The stretching modulus is estimated from literature data as $K_\mathrm{a} = 240\pm90$ mN~m$^{-1}$, with the error bar covering the entire range of reported values.\cite{Rawicz_BJ_2000,Rawicz_BJ_2008,Fa_BBA_2007,Pan_BJ_2008} The thermal expansion coefficient has been experimentally estimated as $\alpha=1.3\pm0.7$ K$^{-1}$.\cite{Parolini_NatComms_2015} The hybridisation enthalpy and entropy are estimated according to the nearest neighbours thermodynamic rules\cite{SantaLucia_PNAS_1998} as $\Delta H^0=-54\pm5$ kcal~mol$^{-1}$ and $\Delta S^0=-154\pm13$ cal~mol$^{-1}$~K$^{-1}$. It is not clear whether the stabilising effect of the non-hybridised dangling bases neighbouring the sticky ends, in the present case the adenine bases present at both sides of the sequences (see Fig.~\ref{Figure1}), should be taken into account. It is indeed possible that their attractive contribution is compensated or overwhelmed by the repulsive effect of the long inert DNA connected to the sticky ends.\cite{Di-Michele_JACS_2014} The errorbars in $\Delta H^0$ and $\Delta S^0$ are included to cover both these scenarios. The length $L=9.8$~nm of the dsDNA spacers can be precisely estimated from the contour length of dsDNA ($0.388$ nm per base-pair\cite{Smith_Science_1996}). The DNA coating density is estimated for different samples as explained in the experimental section. The neutral temperature $T_0$ changes widely from vesicle to vesicle due to the polydispersity of electroformed samples. We generally use $T_0$ as a fitting parameter. For a known $T_0$, the neutral radius $R_0$ is experimentally determined as $R_0=R|_{T=T_0}$. If $T_0$ falls outside the experimentally accessible temperature range, we extrapolate
\begin{equation}
R_0\approx \sqrt{\frac{A(T_1)}{\pi \left[1+\alpha\left(T_1-T_0\right)\right]}},
\end{equation}
where $T_1$ is the minimum experimentally accessible temperature.\\
Errors in the input parameters are numerically propagated to the theoretical predictions.\cite{Parolini_NatComms_2015} Briefly, we sample the results of the model using random values of the input parameters $X\pm \Delta X$ extracted from a Gaussian distribution with mean equal to $X$ and standard deviation equal to $\Delta X$. From a sample of size $\ge10000$, we estimate the theoretical prediction of each observable as the median of the sampled distribution. Errobars cover the interval between the $16^\mathrm{th}$ and the $84^\mathrm{th}$ percentile.
\begin{table}[ht!]
\small
\caption{\textbf{Input parameters of the model}}
\label{Table}
\begin{tabular*}{0.5\textwidth}{@{\extracolsep{\fill}}ccc}
\hline
& $\alpha = 1.3\pm0.7$ K$^{-1}$ & \\
& $K_\mathrm{a} = 240\pm90$ mN~m$^{-1}$ &\\
& $L=9.8$ nm &\\
& $\Delta H^0=-54\pm5$ kcal~mol$^{-1}$ &\\
& $\Delta S^0=-154\pm13$ cal~mol$^{-1}$~K$^{-1}$& \\
\hline
\end{tabular*}
\end{table}
\begin{figure}[ht!]
\centering
\includegraphics[width=8.8cm]{./Figure4.eps}
\caption{Experimental and theoretical temperature-dependent vesicle adhesion for samples with intermediate DNA coating density $\rho_\mathrm{DNA}^\mathrm{med}=390\pm90$ $\mu$m$^{-2}$ (see experimental section). In the top row we demonstrate the temperature-dependence of the contact angle $\theta$ (\textbf{a}), adhesion-patch radius $R_\mathrm{p}$ (\textbf{b}), and vesicle radius $R$ (\textbf{c}) for a typical adhering GUV. Points indicate experimental data, solid lines mark theoretical predictions, with errorbars visualised as grey-shaded regions. Fitting parameter $T_0=-40^\circ$C. In the bottom row we summarise the results for 5 vesicles. Points represent the relative deviation of experimental data from theoretical predictions $(X^\mathrm{exp}-X^\mathrm{th})/X^\mathrm{th}$, with $X=\theta$ (\textbf{d}), $R_\mathrm{p}$ (\textbf{e}), and $R$ (\textbf{f}). Grey-shaded regions mark the uncertainty interval of the theory. Model parameters are reported in Table \ref{Table}.}
\label{FigureShapeHigh}
\end{figure}
\section{Results and discussion}
\subsection{Qualitative observations}
In Fig.~\ref{FigureStacks} we can visually compare confocal images of a DNA-functionalised GUV adhering to a SBL (a) and a non-adhering GUV on a passivated glass surface (b). Both GUVs and SBL are stained with fluorescent lipids.\\
It is clear from both the 3D reconstruction and the vertical cross section that the adhering GUV takes the shape of a truncated sphere, with a flat and circular contact region. This evidence confirms the assumption that, at low enough temperature, DNA-mediated adhesion is strong enough to guarantee the dominance of stretching over bending and other contributions to the deformation energy. The fluorescence intensity measured within the adhesion patches is almost exactly equal to twice the value measured on the SBL outside the adhesion region (1.95 times for the vesicle shown in Fig.~\ref{FigureStacks}a). This evidence confirms the presence of two lipid bilayers in close contact within the adhesion area and excludes the possibility of DNA-mediated fusion of the two membranes.\cite{Stengel_JCPB_2008}\\ The non-adhering GUV displayed in Fig.~\ref{FigureStacks}b does not exhibit a flat adhesion patch, as clear from the vertical cross-section. Note that for both the adhering and the non-adhering GUVs, the bottom part of the stacks appears brighter due to the z-dependent response of the instrument.\\
\begin{figure}[ht!]
\centering
\includegraphics[width=8.8cm]{./Figure5.eps}
\caption{Experimental and theoretical temperature-dependent vesicle adhesion for samples with low DNA coating density $\rho_\mathrm{DNA}^\mathrm{low}=39\pm9$ $\mu$m$^{-2}$ (see experimental section). In the top row we demonstrate the temperature-dependence of the contact angle $\theta$ (\textbf{a}), adhesion-patch radius $R_\mathrm{p}$ (\textbf{b}), and vesicle radius $R$ (\textbf{c}) for a typical adhering GUV. Points indicate experimental data, solid lines mark theoretical predictions, with errorbars visualised as grey-shaded regions. Fitting parameter $T_0=-20^\circ$C. In the bottom row we summarise the results for 5 vesicles. Points represent the relative deviation of experimental data from theoretical predictions $(X^\mathrm{exp}-X^\mathrm{th})/X^\mathrm{th}$ for $X=\theta$ (\textbf{d}), $R_\mathrm{p}$ (\textbf{e}), and $R$ (\textbf{f}). Grey-shaded regions mark the uncertainty interval of the theory. Model parameters are reported in Table \ref{Table}.}
\label{FigureShapeLow}
\end{figure}
\subsection{Temperature-dependence of the geometrical observables}
In this section we discuss the temperature-dependence of the morphology of adhering GUVs. In Fig.~\ref{FigureShapeHigh}a-c we show the experimentally determined contact angle $\theta$, adhesion-patch radius $R_\mathrm{p}$, and vesicle radius $R$ for a typical adhering GUV is a sample with DNA concentration equal to $\rho_\mathrm{DNA}^\mathrm{med}=390\pm90$ $\mu$m$^{-2}$. The contact angle displays a non-monotonic behaviour as a function of temperature, with a positive slope at low $T$, followed by a sudden decay for $T$ higher than $\sim30^\circ$C. The adhesion radius follows the same trend, as does the vesicle radius, which however displays much smaller relative variations. The solid lines in Fig.~\ref{FigureShapeHigh}a-c represent theoretical predictions calculated using the input parameters in Table.~\ref{Table}, and using the neutral temperature $T_0$ as a fitting parameter. Grey-shaded regions indicate propagated uncertainty in the theoretical predictions. The agreement between theory and experiments is quantitative at low temperatures. At high $T$, the theory fails to predict the drop in contact angle observed in experiments. This behaviour is expected since our theoretical description is valid in the limit of strong adhesion, where the attractive forces are sufficient to suppress bending contributions in the interaction energy. At high temperature the DNA, which in the present experiment features relatively short sticky ends, starts to melt, causing the loosening of the adhesive forces and a change in the GUV shape, detected as a shrinkage of the adhesion area. The temperature-dependence of the fraction of DNA bonds is quantified and discussed in the following sections.\\ In Fig.~\ref{FigureShapeHigh}d-f we show the relative deviations of the experimentally-determined morphological observables from the theoretical predictions, defined as $(X^\mathrm{exp}-X^\mathrm{th})/X^\mathrm{th}$, for $X=\theta$, $A_\mathrm{p}$, and $A$. The data, collected from 5 vesicles are consistent: the experimental data fall within theoretical errorbars at low $T$, deviating at higher temperature due to the failure of the strong adhesion assumption.\\
In Fig.~\ref{FigureShapeLow} we show experimental, and theoretically predicted morphological observables for the case of low DNA concentration, $\rho_\mathrm{DNA}^\mathrm{low}=39\pm9$ $\mu$m$^{-2}$. Similarly to the case of higher DNA concentration, experimental data are backed by theoretical predictions. In this case, however, the high-temperature deviation of the experiments from the theoretical predictions appears to be less evident, and shifted towards higher temperatures. This indicates the presence of an (however small) adhesive force hindering the partial detachment of the GUVs. We ascribe this behaviour to the effect of non-specific membrane-membrane adhesion, e.g. dispersion attraction, that for higher DNA coverage is suppressed by steric repulsion.\\
\begin{figure*}[ht!]
\centering
\includegraphics[width=14cm]{./Figure6.eps}
\caption{In-situ FRET spectroscopy characterisation of the temperature-dependence of the fraction of DNA bonds. \textbf{a}, Confocal image of the adhesion patch of a GUV. At the bottom we show the same image segmented with our software to separate the adhesion patch (blue) from the free supported bilayer (green). Scale bar: 15~$\mu$m. \textbf{b}, Fluorescence emission spectra measured within the adhesion patch for a typical vesicle (blue area in panel \textbf{a}). Colours from blue to red mark low to high temperatures in the interval $14.5\le T \le 62.9^\circ$C. The curves are normalised to the emission peak of Cy3 ($568$nm). In panel \textbf{c} we show the emission spectra measured on the free SBL (green area in panel \textbf{a}). \textbf{d}, Relative intensity of the Cy5 emission peak ($665$nm) measured within (blue circles) and outside (green lozenges) the adhesion patch. The curves are relative to 3 different vesicles. The amplitude of the Cy5 and Cy3 emission peaks is determined through a Gaussian fit of the 5 data points closest to the maximum. \textbf{e}, Experimental (symbols) and theoretically predicted (solid lines) fraction of formed DNA bonds within (blue circles) and outside (green lozenges) the adhesion patch. Experimental data are extracted from the curves in panel \textbf{d} as described in the text. Theoretical curves are calculated using the parameters in Table~\ref{Table}, DNA-coating density $\rho^\mathrm{high}_\mathrm{DNA}=1200\pm300$ $\mu$m$^{-2}$, $T_0=-20$, and $R_\mathrm{0}=10$ $\mu$m. Note that the value of $T_0$ does not significantly affect these quantities. Blue and Green shaded regions indicate propagated uncertainties of the blue and green solid lines. Cyan shaded region marks their overlap.}
\label{FigureFRET}
\end{figure*}
\subsection{DNA-melting}
We investigate the temperature-dependence of the fraction of formed DNA bonds via in-situ FRET measurements. Cyanine fluorophores, Cy3 (donor) and Cy5 (acceptor), are connected to the $3'$ termini of $a$ and $a'$ sticky ends, as sketched in Fig.~\ref{Figure1}. FRET efficiency is described by
\begin{equation}
E=\frac{1}{1+\left(\frac{d}{R_0}\right)^6},
\end{equation}
where $d$ is the distance between the fluorohpores and $R_0$ is the Forster radius, equal to $5.4$~nm for the case of Cy3-Cy5.\cite{Yuan_NAR_2007} When sticky ends are bound to form a loop or a bridge, the distance between Cy3 and Cy5 is approximately equal to the length of the hybridised sticky ends -- 2.7~nm, therefore we can assume a very high energy transfer efficiency for bound linkers. The average distance between unbound linkers is sufficiently high to guarantee a comparatively very low transfer efficiency between unpaired tethers. Note that for FRET experiments the lipid membrane is not stained with Texas Red to avoid spurious signal (energy transfer between Texas Red and Cy5). In Fig.~\ref{FigureFRET}a we show the confocal image of an adhesion patch (top), segmented to separate the actual adhesion area from the surrounding free SBL. This enables an efficient characterisation of FRET efficiency in-situ. The emission spectra, collected within the patch and on the SBL while exciting the donor at 514~nm, are shown in Fig.~\ref{FigureFRET}c and d respectively. Colours from blue to red indicate low to high temperature. Spectra are normalised to the emission peak of Cy3, correctly found at $\sim568$~nm. Note that for this experiment we used high DNA concentration $\rho_\mathrm{DNA}^\mathrm{high}$ to strengthen the signal that otherwise would be too weak for a wavelength scan. As expected, the emission of the acceptor, peaked at $\sim665$~nm, visibly drops at high temperature. We quantify this effect in Fig.~\ref{FigureFRET}d, where we plot the normalised acceptor emission intensity $\iota=I_\mathrm{Cy5}/\left(I_\mathrm{Cy5}+I_\mathrm{Cy3}\right)$,\cite{McCann_BJ_2010} where the $I_\mathrm{Cy5/Cy3}$ are the peak-intensities estimated through a local Gaussian fit. Although qualitatively similar, the $\iota$ curves measured within and outside the adhesion patch exhibit some differences. For the case of free SBL, the emission ratio remains constant ($\sim 0.4$) or slightly decreases upon heating, up to $\sim 40^\circ$C, then it gradually drops down to $\sim 0.15$. This decay is ascribed to the melting transition of DNA loops formed within the SBL. The FRET signal measured within the patch is higher at low temperatures ($\sim 0.6$). This effect is probably due to the higher DNA density found within the patch at low temperature, which increases the probability of energy transfer between unpaired strands. Indeed we find that the overall fluorescence intensity measured within the patch at $T<20^\circ$C is between 6 and 12 times higher than the intensity measured on the SBL. The FRET signal measured within the patch is also found to increase upon heating, before suddenly decreasing at $T\sim45^\circ$C. The increase in FRET efficiency cannot be explained by an increase in DNA density within the patch, since the local DNA density decreases as the adhesion area becomes larger upon heating. A possible explanation of this behaviour could be radiative cross-excitation between the two fluorophores, that becomes more efficient as the adhesion patch gets less crowded upon heating. At high temperatures the FRET signal measured within the patch plateaus at $\sim 0.15$, in line with what we measure on the SBL. This confirms that at high enough temperature, when no bonds are formed, the DNA concentration is uniform across all the surfaces.\\
The curves $\iota(T)$ can be used to semi-quantitatively estimate the temperature dependence of the overall fraction of DNA bonds. We fit the low temperature plateaus ($T<35^\circ$C) in Fig.~\ref{FigureFRET}d with linear baselines $B(T)$ and assume that $\iota(T)$ plateaus to a constant value for $T>57^\circ$C. The fraction of formed DNA bonds is thereby estimated as
\begin{equation}
\phi(T)=\frac{\iota(T)-\langle \iota_{T>57^\circ\mbox{C}}\rangle}{B(T)-\langle \iota_{T>57^\circ\mbox{C}}\rangle}.
\end{equation}
A better estimate of $\phi(T)$ could be obtained by measuring $\iota(T)$ up to higher temperatures, and fitting the high-temperature plateau with a second linear baseline. However, temperatures higher than $65^\circ$C cannot be safely probed due to the risk of destabilisation of the dsDNA spacers. The experimental $\phi(T)$ data in Fig.~\ref{FigureFRET}e indicate that the DNA melting transition is relatively broad, spanning more than $30^\circ$C.\cite{Meulen_JACS_2013} Moreover, the melting seems to occur at a higher temperature (by about $5^\circ$C) within the adhesion patch.\\
The experimentally estimated fraction of DNA bonds can be compared with theoretical predictions. On the free SBL, only loops can form, and therefore $\phi(T)$ should be compared to the fraction of loops $x_\ell^\mathrm{SBL}$, calculated accordingly to Eqs.~\ref{eqn:freeSBL}, \ref{fractionofloopsSBL}-\ref{qloopsSBL}. Within the adhesion patch we count contributions from bridges, loops formed on the GUV, and loops formed on the SBL. By assuming evenly distributed loops, the overall number of DNA bonds found within the patch is
\begin{equation}
N_\mathrm{bound}=N\left(\frac{A_\mathrm{p}}{A}x_\ell+x_\mathrm{b}\right)+c_0 x_\ell^\mathrm{SBL} A_\mathrm{p},
\end{equation}
where $x_{\mathrm{b}/\ell}$ are the fractions of loops and bridges on the GUV, given by Eqs.~\ref{Bridges} and \ref{Loops}. The overall number of DNA tethers, bound and unbound, found within the patch is
\begin{equation}
N_\mathrm{tot}=N\left[\frac{A_\mathrm{p}}{A}\left(1-x_\mathrm{b}\right)+x_\mathrm{b}\right]+c_0A_\mathrm{p},
\end{equation}
where we assume that also unbound DNA is evenly distributed across the surfaces.
The theoretically predicted fraction of bonds within the patch is thus $N_\mathrm{bound}/N_\mathrm{tot}$. In Fig.~\ref{FigureFRET}e we compare theoretical $\phi(T)$ with experimental estimates. Since the choice of the neutral temperature $T_0$ and radius $R_0$ do not noticeably affect the melting curves, theoretical predictions are calculated using a fixed $T_0=-20^\circ$C and $R_0=10$ $\mu$m, with no fitting parameters. Our model captures the width of the DNA transition as well as the difference in melting temperature between the patch and the free SBL. However the theory underestimates the average melting point by $5-10^\circ$C. This deviation is, at least partially, ascribable to the attractive effect of Cy3 and Cy5 molecules on the stability of DNA. For duplexes labeled with either of the molecules, the stabilisation has been quantified in a positive melting-temperature shift of $1.4-1.5^\circ$C.\cite{Moreira_BBRC_2005} The presence of both molecules is expected to cause a greater shift. Another explanation could be an underestimation of the DNA concentration $\rho_\mathrm{DNA}^\mathrm{high}$. As discussed below, this hypothesis is consistent with tension measurements. The impossibility of probing the high-temperature baselines could also play a role.\\
\begin{figure}[ht!]
\centering
\includegraphics[width=8.8cm]{./Figure7.eps}
\caption{Experimental and theoretical temperature-dependent membrane tension. \textbf{a}, Membrane tension $\sigma$ measured from filckering experiments. Blue circles indicate adhering GUVs with low DNA coverage ($\rho_\mathrm{DNA}^\mathrm{low}=39\pm9$ $\mu$m$^{-2}$). Red lozengges indicate non-adhering GUVs on a BSA-coated glass surface. Grey-shaded regions mark regimes in which membrane tension is currently inaccessible to our technique. The blue-shaded band indicates the theoretical prediction for $\sigma$ calculated with the fitting parameter $-60\le T_0 \le 0^\circ$C and low DNA coverage. Blue lines mark the corresponding errorbars. The green-shaded band indicate the theoretical prediction for $\sigma$ for higher DNA coverage ($\rho_\mathrm{DNA}^\mathrm{med}=390\pm90$ $\mu$m$^{-2}$). Green lines mark the corresponding errorbars. Model parameters are reported in Table \ref{Table}.\cite{Parolini_NatComms_2015} \textbf{b},~Fluctuation-amplitude spectra for adhering GUVs with low DNA coverage and various temperatures. Symbols indicate experimental data and solid lines indicate fits according to Eq.~\ref{eqn:fitspectrum}. From top to bottom the fitted values of the membrane tension are $\sigma=4.1\pm0.5$, $2.42\pm0.09$, $1.88\pm0.09\times10^{-7}$~Nm.}
\label{FigureTension}
\end{figure}
\subsection{Membrane tension}
The temperature dependence of the membrane tension is measured by flickering analysis of the equatorial cross sections of GUVs. The tension is extracted by fitting the power spectra of the thermal fluctuations, determined as explained in the experimental section, with the function
\begin{equation}\label{eqn:fitspectrum}
\langle |h^2(q_x)| \rangle = \frac{k_\mathrm{B}T}{2 \mathcal{L} \sigma} \left[\frac{1}{q_x}-\frac{1}{\sqrt{q_x^2+\frac{\sigma}{\kappa}}}\right],
\end{equation}
where $\mathcal{L}$ is the contour length of the equatorial cross-section of the GUV, $\sigma$ is the membrane tension, $\kappa$ the bending modulus, and $q_x$ the wave vector evaluated along the contour.
Equation \ref{eqn:fitspectrum} is derived from the original work of Helfrich,\cite{Helfrich} describing the fluctuations of an infinite 2D membrane, and corrected to account for the fact that, by imaging an equatorial cross-sections, only modes propagating along the horizontal direction should be considered.\cite{Pecreaux_EPJE_2004} Of the discrete set of wave vectors $q_x(n)=2 \pi n / \mathcal{L}$, modes with $n<6$ are excluded from the analysis. Mode $n=0$ and $n=1$ correspond to size changes and translations of the GUV. Modes with $n>2$ describe thermal fluctuations. However, Eq.~\ref{eqn:fitspectrum} is derived for a planar membrane, and should not be used to describe modes with $n<6$, which are influenced by the spherical geometry of the GUV.\cite{Pecreaux_EPJE_2004} For our analysis we fit the spectra for modes $6\le n \le 16$. At higher $q$ we approach the resolution limits of the current method.\\
In Fig.~\ref{FigureTension}a blue circles mark the tension measured as a function of temperature for adhering GUVs. In Fig.~\ref{FigureTension}b we show examples of power spectra fitted by Eq.~\ref{eqn:fitspectrum}. The tension typically lies in the interval $2\times 10^{-7}-2\times10^{-6}$ N~m$^{-1}$, with clear variations between different GUVs. In the tested range, the tension consistently displays a weak dependence on temperature changes.\\
For comparison, the membrane tension is measured on non-adhering GUVs supported by a passivated glass substrate. The values of $\sigma$ measured for non-adhering GUVs (red lozenges in Fig.~\ref{FigureTension}a) are significantly lower than those measured on adhering vesicles, falling within the range $10^{-9}-10^{-6}$ N~m$^{-1}$, and being clustered around $10^{-8}$ N~m$^{-1}$. Furthermore, membrane tension of non-adhering GUVs typically displays a strong decrease upon increasing temperature. The large variability observed in the tension of adhering and, in particular, of free GUVs, is ascribed to the polydispersity of electroformed samples, which produces vesicle populations with very different excess areas ($T_0$).\\
With the present technique we cannot access the tension of vesicles adhering to SBL for the case of higher DNA concentrations. Indeed, for values of $\sigma$ in the grey-shaded region on Fig.~\ref{FigureTension}a, the relevant portions of the fluctuation power spectra are masked by experimental noise deriving from the finite resolution of the contour-tracking procedure.\\
The membrane tension can be evaluated within the framework of our model. At equilibrium, the derivative of the interaction energy in Eq.~\ref{U1}, taken with respect to the GUV area, is
\begin{equation}\label{eqn:tensionDNA}
\frac{\partial U}{\partial A}=\sigma+\frac{\partial U_\mathrm{DNA}}{\partial A}=0,
\end{equation}
where we used
\begin{equation}
\sigma = \frac{\partial U_\mathrm{stretching}}{\partial A} = K_\mathrm{a} \frac{A-\tilde{A}}{\tilde{A}}.
\end{equation}
Equation \ref{eqn:tensionDNA} suggests that by measuring $\sigma$ we can directly probe the DNA-mediated forces. The blue-shaded region in Fig.~\ref{FigureTension}a marks the model predictions for $\sigma$, calculated using the parameters Table~\ref{Table}, $\rho_\mathrm{DNA}^\mathrm{low}=39\pm9$ $\mu$m$^{-2}$, and values of the neutral temperature covering the experimentally observed range ($-60\le T_0 \le 0^\circ$C). The size of the vesicles does not impact the predictions of $\sigma$, therefore we fix $R_\mathrm{0}=10$ $\mu$m. Solid blue lines mark the uncertainty interval propagating from the errorbars of the model parameters (Table~\ref{Table}). With no fitting parameters, we observe a semi-quantitative agreement between theory and experiments. In particular, the theory predicts the weak temperature-dependence of $\sigma$ observed in the experiments. The slight underestimation of the theoretical predictions as compared to the measured tension could be explained by an underestimation of the DNA-coating density. This evidence is consistent with the underestimation of the DNA melting temperature discussed above.\\
The green-dashed region in Fig.~\ref{FigureTension}a indicates the theoretical prediction calculated using $\rho_\mathrm{DNA}^\mathrm{med}=390\pm90$ $\mu$m$^{-2}$. The predicted tension falls within the non-accessible region.\\
\section{Conclusion}
In this article we experimentally investigate temperature-dependent adhesion of Giant-Unilamellar-Vesicles on supported lipid bilayers mediated by mobile DNA linkers.
The simple geometry of the problem allows for an accurate characterisation of the morphology of adhering GUVs and the temperature dependent fraction of bound DNA tethers by means of
confocal microscopy. For the first time to our knowledge, we quantify the temperature-dependent membrane tension induced by DNA bonds by analysing the thermal fluctuations of the GUVs imaged across their equatorial plane.\\
The experimental results are compared to theoretical predictions from our recently developed model,\cite{Parolini_NatComms_2015} which we here extend to the case of vesicle-plane adhesion. The model takes into account both the elastic deformation of the GUV and the statistical-mechanical details of the DNA-mediated interactions.\\
For sufficiently high DNA coverage, the adhesion contact angle exhibits a re-entrant temperature dependence. Upon heating from low temperature the contact angle increases, reaching a maximum at $T\simeq30-40^\circ$C. Upon further temperature increase, the contact angle drops. The re-entrance is less pronounced or absent for lower DNA coverage.
With a single fitting parameter, the model is capable of quantitatively predicting the low temperature regime and ascribes the increase in contact angle to the interplay between the temeperature-dependent excess area of the GUV and the entropic coupling between the hybridisation free-energy of the mobile tethers and the adhesion area. The theory is developed in the limit of strong adhesion, therefore it fails to predict the re-entrant behaviour of the adhesion area, caused by the weakening of the DNA bonds. The less-pronounced re-entrance observed for low DNA concentrations is ascribable to non-specific adhesive interactions that kick-in at high temperature, and are suppressed by steric repulsion for samples with high DNA coverage.\\
The melting of DNA bonds is investigated in-situ by FRET measurements. We observe a broad melting transition and find that bonds formed within the GUV-plane adhesion patch are more stable than in-plane bonds formed on free bilayers. With no fitting parameters our model can semi-quantitatively reproduce these features, although an underestimation of the melting temperature is observed.\\
Membrane tension measurements performed on adhering GUVs demonstrate a weak temperature dependence. In a similar range of temperatures, non-adhering GUVs exhibit significantly lower tension, rapidly decreasing upon heating. The differences in magnitude and trend demonstrates the role played by DNA in mediating membrane adhesion.
Experimental results are in semi-quantitative agreement with theoretical prediction, which further demonstrates the accuracy of the model used to describe hybridisation free-energy of tethered mobile linkers, and in particular the translational-entropic contributions that couples it to the adhesion area.\\
Our experimental observations and the agreement with the theoretical predictions help to clarify the complex mechanisms controlling adhesion of soft units mediated by multiple linkers. Besides the fundamental interest for the still poorly understood physics of multivalent interactions, our findings can help the design of functional, responsive, tissue-like materials with promised applications in biosensing, encapsulation-release mechanisms, and filtration. Finally, the conclusions drawn for our model system can be adapted to the quantitative description of cell adhesion and spreading on solid substrates, with possible biomedical implications in prosthetics and scaffoldings for tissue regeneration.\cite{Dalva_NatureRev_2007,Parsons_NatureRev_2010,Alberts,Bao_NMat_2003,Discher_Science_2005}
\subsection*{Acknowledgements}
The authors acknowledge financial support from the EPSRC Programme Grant CAPITALS number EP/J017566/1 (PC, LP, LDM), the Ernest Oppenheimer Fund and Emmanuel College Cambridge (LDM), Universit\'{e} libre de Bruxelles (BMM), and Grant-in-Aid for JSPS Fellows Grant (No. 25-1270) (SFS), Project SPINNER 2013, Regione Emilia-Romagna (It), European Social Fund (DO). In compliance with our funders' requirements all the data underlying this article are available trough the corresponding authors.\\
\begin{appendix}
\section{Details on model derivation}
\subsection{Geometrical expressions}\label{App_geometry}
In the limit of strong adhesion the GUV will take the shape of a truncated sphere with contact angle $\theta$ (Fig.~\ref{Figure1}), which we take as the independent variable of our model.\cite{Ramachandran_Langmuir_2010} In this simple geometry, the contact (patch) area, total area, and volume of the GUV are respectively
\begin{eqnarray}
&A_\mathrm{p}=\pi R^2 \sin^2 {\theta} \label{eqn:Ap} \\
&A=\pi R^2 \left(1+\cos {\theta}\right)\left(3-\cos {\theta}\right) \label{eqn:Area} \\
&V=\frac{\pi R^3}{3}\left(1+\cos {\theta}\right)^2\left(2-\cos {\theta} \right) \label{eqn:V}.
\end{eqnarray}
In the limit of water-impermeable membranes, the internal volume of the GUVs can be taken as a constant
\begin{equation}\label{ConstantVolume}
V=\frac{4}{3}\pi R_0^3,
\end{equation}
where we introduce a reference radius $R_0$. By using Eqs.~\ref{eqn:V} and \ref{ConstantVolume} we obtain
\begin{equation}\label{eqn:RofTheta}
R=R_0\left[\frac{4}{\left(1+\cos {\theta}\right)^2\left(2-\cos {\theta}\right)}\right]^{1/3},
\end{equation}
which can be inserted into Eqs. \ref{eqn:Ap} and \ref{eqn:Area} to make the $\theta$-dependence of $A_\mathrm{p}$ and $A$ explicit.\\
Let us now recall the definition of reduced volume of a vesicle\cite{Tordeux_PRE_2002,Ramachandran_Langmuir_2010}
\begin{equation}\label{eqn:reducedvolume}
v=\frac{\frac{3V}{4\pi}}{\left(\frac{\tilde{A}}{4\pi}\right)^{3/2}}.
\end{equation}
By combining Eq.~\ref{eqn:reducedvolume} with expression for the temperature-dependent unstretched area in Eq.~\ref{eqn:unstretched}, we obtain
\begin{equation}
v=\left[1+\alpha(T-T_0)\right]^{-3/2}.
\end{equation}
The reference temperature $T_0$ is therefore defined as the temperature at which a GUV has reduced volume equal to 1. For $T<T_0$, when $v>1$, an isolated vesicle resembles a turgid sphere, with non-zero membrane tension whereas for $T>T_0$, $v<1$, it assumes the features of a ``floppy" balloon, with excess area. At $T=T_0$ an isolated vesicle is a perfect sphere with zero-membrane tension and radius equal to the reference radius $R_0$.\\
By combining Eqs.~\ref{eqn:Area}, \ref{eqn:RofTheta} and \ref{eqn:unstretched} we obtain an explicit, $\theta$-dependent expression for the stretching energy in Eq.~\ref{U2}. Note that Eq.~\ref{eqn:RofTheta} has been derived under a constant-volume assumption (Eq.~\ref{ConstantVolume}). Alternatively, an equivalent relation can be derived for water permeable -- solute impermeable -- GUVs, in which the volume is set by the balance between the osmotic pressure drop across the membrane and the Laplace pressure.\cite{Ramachandran_Langmuir_2010} In relevant experimental conditions the two assumptions lead to very similar results.\cite{Parolini_NatComms_2015}\\
\subsection{Free energy for bridge and loop formation.~~}\label{App_FreeEnergy}
For the case of mobile linkers, the configurational entropic contribution to the bridge/loop formation free energy $\Delta S^\mathrm{conf}_{\mathrm{b}/\ell}$ (Eq.~\ref{eqn:singleenergy}) can be split into a rotational and translational contribution
\begin{equation}\label{eqn:confent_app}
\Delta S_{\mathrm{b}/\ell}^\mathrm{conf}=\Delta S^\mathrm{rot} +\Delta S_{\mathrm{b}/\ell}^\mathrm{trans}.
\end{equation}
The rotational contribution takes the same expression for loop and bridge formation, and encodes for the reduction of configurational entropy following the hybridisation of two rigid tethers with fixed grafting sites\cite{Leunissen_JCP_2011,Parolini_NatComms_2015}
\begin{equation}
\Delta S^\mathrm{rot} =k_\mathrm{B}\log\left[\frac{1}{4 \pi \rho_0 L^3}\right],\label{eqn:rotent}
\end{equation}
where $\rho_0=1$M is a reference concentration.
The translational contribution encodes for the lateral confinement following the binding of two mobile tethers. For the case of loops, upon binding, two tethers initially capable of exploring the entire GUV surface area $A$, are confined to within a region $\sim L^2$ from each other\cite{Parolini_NatComms_2015}
\begin{equation}
\Delta S^\mathrm{trans}_\ell=k_\mathrm{B}\log\left[\frac{4\pi L^2 A}{A^2}\right]=k_\mathrm{B}\log\left[\frac{4\pi L^2}{A}\right]\label{eqn:transentLoops}.
\end{equation}
For the case of bridge formation, we consider a free linker on the SBL, initially located within the contact area $A_\mathrm{p}$, and a second linker on the GUV, which is free to explore the entire surface area $A$. Upon binding, the area available to the pair is reduced to $4\pi L^2 A_\mathrm{p}$, resulting in
\begin{equation}
\Delta S^\mathrm{trans}_\mathrm{b}=k_\mathrm{B}\log\left[\frac{4\pi L^2 A_\mathrm{p}}{A A_\mathrm{p}}\right]=k_\mathrm{B}\log\left[\frac{4\pi L^2}{A}\right]\label{eqn:transentBridges}.
\end{equation}
We notice that, contrary to the case of two adhering GUVs,\cite{Parolini_NatComms_2015} here $\Delta S^\mathrm{trans}_\mathrm{b}=\Delta S^\mathrm{trans}_\ell$. By combining Eqs.~\ref{eqn:confent_app}--\ref{eqn:transentBridges} with Eq.~\ref{eqn:singleenergy}, we obtain the hybridisation free energy of bridge formation on the loops formation on the GUV in Eq.~\ref{DeltaG}.
\subsection{Fraction of loops and free tethers on the SBL.}\label{App_SBL}
We now focus on the description of the tethers anchored to the SBL, and calculate the equilibrium fraction of formed loops in the absence of an adhering GUV. This information is needed for the calculation of the GUV-SBL adhesive interaction as well as for a direct comparison with experimental data.\\
Let us consider a finite portion of the SBL of area $\Sigma$, containing two populations of $N$ linkers with $a$ and $a'$ sticky ends. Following Eq.~\ref{DeltaG}, the free energy for loop ($\ell$) formation on the SBL can be written as
\begin{equation}\label{DeltaGLoopsSBL}
\Delta G_\ell^\mathrm{SBL} =\Delta G^0 - k_\mathrm{B}T\log\left[\frac{1}{\rho_0 L \Sigma}\right].
\end{equation}
By indicating as $N_\ell$ the number of loops within the SBL, and taking into account combinatorics, we can write the partition function of this systems as\cite{Parolini_NatComms_2015}
\begin{equation}
Z=\sum_{N_\ell} {N\choose N_\ell}^2 N_\ell ! \exp \left(-\beta N_\ell \Delta G_\ell^\mathrm{SBL}\right),
\end{equation}
which can be rearranged as
\begin{equation}\label{ZSBL}
Z=\sum_{N_\ell} e^{-S(N_\ell)}.
\end{equation}
We now consider the limit of an infinite SBL with a constant DNA surface density $c_0$, i.e. we take $N,\Sigma \rightarrow \infty$, with $c_0=N/\Sigma=\rho_\mathrm{DNA}/2$. By using the Stirling approximation we obtain
\begin{multline}
S(c_\ell) =\mathrm{const}\cdot \Sigma [-c_\ell \log c_\ell -2 (c_0-c_\ell)\log(c_0-c_\ell)\\
-c_\ell \beta \Delta G^0-c_\ell \log (\rho_0L) - c_\ell +\mathrm{const}],
\end{multline}
where we define the density of loops as $c_\ell=N_\ell/\Sigma$. Within the saddle-point approximation,\cite{Parolini_NatComms_2015} the sum in Eq. \ref{ZSBL} is dominated by the stationary point of $S$
\begin{equation}\label{saddleSBL}
\frac{\partial S}{\partial c_\ell}=0.
\end{equation}
By solving Eq.~\ref{saddleSBL} we obtain the expression in Eq.~\ref{eqn:freeSBL} in the text,
where we introduce the concentration of free tethers on the SBL $c_\mathrm{f}=c_0-c_\ell$. Note that with Eq.~\ref{eqn:freeSBL} we recover a simple mass-balance relation between loops and free tethers on the SBL, which ultimately results in
\begin{equation}\label{fractionofloopsSBL}
c_\ell=c_0\frac{2 q_\mathrm{SBL} + 1 - \sqrt{4 q_\mathrm{SBL}+1}}{2 q_\mathrm{SBL}},
\end{equation}
where
\begin{equation}\label{qloopsSBL}
q_\mathrm{SBL}=\frac{c_0}{\rho_0 L} \exp[-\beta \Delta G_0].
\end{equation}\\
The fraction of loops formed within the bilayer is $x_\ell^\mathrm{SBL}=c_\ell/c_0$.
\subsection{Combinatorial effects.~~}\label{App_Comb}
Given the expressions for the hybridisation free-energy of a single bridge/loop (Eq.~\ref{DeltaG}), a combinatorial approach is required to compute the overall DNA-mediated interaction energy.\\
Following the derivation carried out to describe loop formation on the SBL, we indicate the total number of tethers with $a$ ($a'$) sticky ends on the GUV as $N$, and define $N_\ell$ as the number of those tethers linked in loops. We indicate as $N_{\mathrm{b}i}$, with $i=1,2$, the number of tethers forming bridges with those on the SBL, with the index $i$ referring to $a$ and $a'$ sticky ends. We label as $N_{\mathrm{f}i}$ the number of free tethers on the SBL located within the adhesion patch, where the index $i=1,2$ now refers to $a'$ and $a$ sticky ends. The partition function of the system of linkers can be written as
\begin{equation}\label{eqn:z1}
\begin{split}
z(N_{\mathrm{f}1},N_{\mathrm{f}1},N) \nonumber = \sum_{N_{\mathrm{b}i},N_\ell} \bigl[ \Omega_{N_{\mathrm{f}1},N_{\mathrm{f}2},N}(N_{\mathrm{b}1},N_{\mathrm{b}1},N_\ell)\\
\exp \left[-\beta (N_\mathrm{b1} + N_\mathrm{b2} +N_\ell) \Delta G\right]\bigr],
\end{split}
\end{equation}
where the number of possible configurations for a given $N_\ell$ and $N_{\mathrm{b}i}$ is
\begin{multline}
\Omega_{N_{\mathrm{f}1},N_{\mathrm{f}2},N}(N_{\mathrm{b}1},N_{\mathrm{b}1},N_\ell)= \\ N_\ell!\prod_{i=1,2} {N_{\mathrm{f}i} \choose N_{\mathrm{b}i}}{N \choose N_{\mathrm{b}i}} { {N-N_{\mathrm{b}i}} \choose N_\ell}N_{\mathrm{b}i}!
\end{multline}
To account for strand-concentration fluctuations within the adhesion patch, we need to consider that $N_{\mathrm{f}i}$ is Poisson-distributed around its average value $\bar{N}_{\mathrm{f}}$
\begin{equation}\label{enq:Poisson}
P(N_{\mathrm{f}i},\bar{N}_\mathrm{f})=\exp[-\bar{N}_\mathrm{f}]\frac{\bar{N}_\mathrm{f}^{N_{\mathrm{f}i}}}{N_{\mathrm{f}i} !}.
\end{equation}
Using Eq.~\ref{fractionofloopsSBL}, and recalling that $c_\mathrm{f}=c_0-c_\ell$ is the concentration of free tethers within the SBL, we find
\begin{equation}\label{Nbar}
\bar{N}_{\mathrm{f}}(\theta)=c_\mathrm{f} A_\mathrm{p}(\theta),
\end{equation}
where we highlighted the strong dependence on the contact angle $\theta$.
Using Eqs. \ref{eqn:z1} and \ref{enq:Poisson} we write the full partition function as
\begin{equation}\label{eqn:z2}
Z_{N_{\mathrm{f}1},N_{\mathrm{f}2}}(\bar{N}_\mathrm{f},N)=\sum_{N_\mathrm{f}} P(N_{\mathrm{f}1},\bar{N}_\mathrm{f}) P(N_{\mathrm{f}2},\bar{N}_\mathrm{f}) z(N_{\mathrm{f}1},N_{\mathrm{f}1},N).
\end{equation}
Equation \ref{eqn:z2} can be rearranged as
\begin{equation}\label{eqn:z3}
Z_{N_{\mathrm{f}1},N_{\mathrm{f}2}}(\bar{N}_\mathrm{f},N)=\sum_{N_{\mathrm{f}1},N_{\mathrm{f}2},N_\ell,N_{\mathrm{b}1},N_{\mathrm{b}2}} \exp[-N \mathcal{A}].
\end{equation}
By defining the fractions $x_y=N_\mathrm{y}/N$ ($y=$b1, b2, l, f1, f2), and using the Stirling approximation, $\mathcal{A}$ can be expressed as
\begin{multline}\label{eqn:A}
\mathcal{A}= \beta \Delta G^*x_\ell + x_{\ell}\left(\log x_{\ell} +1\right) + \\
\sum_{i=1,2} \Bigl[ \beta \Delta G^*x_{\mathrm{b}i} + x_{\mathrm{f}i} \left( \log N - \log \bar{N}_\mathrm{f} -1 \right) + \\
x_{\mathrm{b}i} \left(\log x_{\mathrm{b}i} +1\right) +\left(x_{\mathrm{f}i}-x_{\mathrm{b}i}\right)\log \left(x_{\mathrm{f}i}-x_{\mathrm{b}i}\right)\\
+ \left(1-x_{\ell}-x_{\mathrm{b}i}\right) \log \left(1-x_{\ell}-x_{\mathrm{b}i}\right) \Bigr].
\end{multline}
Note that in Eq.~\ref{eqn:A} we re-defined the hybridisation free energy for bridge and loop formation as
\begin{equation}\label{eqnDGstar}
\Delta G^* = \Delta G - k_\mathrm{B}T \log N.
\end{equation}
For typical experimental conditions, the attractive combinatorial term $-k_\mathrm{B}T \log N$ in Eq. \ref{eqnDGstar} can be estimated in $\approx-10 k_\mathrm{B}T$.\cite{Parolini_NatComms_2015} Within the saddle-point approximation, the sum in Eq.~\ref{eqn:z3} is dominated by the stationary point of $\mathcal{A}$
\begin{equation}\label{SaddlePoint}
\frac{\partial \mathcal{A}}{\partial x_y} = 0 \mbox{ with $y$=b1, b2, l, f1, f2}.
\end{equation}
From the saddle-point equations Eq.~\ref{SaddlePoint} we obtain Eqs.~\ref{BridgesImpl} and \ref{LoopsImpl} in the text, where we find $x_{\mathrm{b}1}=x_{\mathrm{b}2}=x_\mathrm{b}$.
By solving Eqs.~\ref{BridgesImpl} and ~\ref{LoopsImpl} we obtain
\begin{equation}
x_\mathrm{b}=\frac{q_\mathrm{b}\left(\sqrt{q_\mathrm{b}^2+2q_\mathrm{b}+4q_\ell +1}- q_\mathrm{b}- 1 \right) }{2q_\ell} \label{Bridges}\\
\end{equation}
\begin{equation}
x_\ell=\frac{ q_\mathrm{b}^2+2q_\mathrm{b}+2q_\ell +1 -\left(q_\mathrm{b}+1\right) \sqrt{q_\mathrm{b}^2+2q_\mathrm{b}+4q_\ell +1}}{2q_\ell}. \label{Loops}
\end{equation}
Note that for simplicity the fraction of bridges and loops are indicated as $x_{\mathrm{b}/\ell}$.
The saddle point equations for $x_\mathrm{f}$ ($x_{\mathrm{f},1}=x_{\mathrm{f},2}$) read $x_\mathrm{f}-x_\mathrm{b}=\bar{N_\mathrm{f}}/N$, which confirms that the density of the free tethers in in the patch region is equal to that of the reservoir, as expected.\\
By inserting the saddle-point solutions for $x_\ell,$ $x_\mathrm{b}$, and $x_\mathrm{f}$ in Eqs.~\ref{eqn:z3} and \ref{eqn:A} we can calculate the free energy $U_\mathrm{hyb}$ (Eq.~\ref{UHyb}).\cite{Angioletti-Uberti_JCP_2013}
\subsection{Reference energy.}\label{App_Ref}
The reference energy $U_0$ in Eq.~\ref{U1} is calculated for isolated GUV and SBL and can be written as
\begin{equation}\label{U0}
U_0=U_{0}^\mathrm{stretching}+U_{0}^\mathrm{DNA}.
\end{equation}
The stretching term is\cite{Parolini_NatComms_2015}
\begin{equation}\label{UST0}
U_0^\mathrm{stretching}=
\begin{cases}
0 \mbox{ if } T \ge T_0\\
K_\mathrm{a}\frac{\left(A_0-\tilde{A}\right)^2}{A_0} \mbox{ if } T< T_0.
\end{cases}
\end{equation}
Note that the stretching contribution is only present for pre-stretched vesicles, i.e. if the reduced volume is $v>1$ (i.e. $T<T_0$).~\cite{Ramachandran_Langmuir_2010}
The DNA contribution is calculated for a GUV of area equal to the unstretched area $\tilde{A}$, in which only loops can form. By following the steps outlined in section \ref{App_Comb} and in ref.\cite{Parolini_NatComms_2015} we calculate the fraction of tethers involved in loops
\begin{equation}
{x}^0_\ell=\frac{ 2 q_\ell +1 - \sqrt{4 q_\ell+1}}{2 q_\ell},
\end{equation}
where $q_\ell$ is given by Eq.~\ref{QLoops}. The DNA part of the reference energy is
\begin{equation}\label{UDNA0}
U_0^\mathrm{DNA}=N k_\mathrm{B} T \left[x^0_\ell + 2 \log \left(1-x^0_\ell\right)-2\frac{\tilde{N}_\mathrm{f}}{N}-2 \log\left(\frac{\tilde{A}}{A_0}\right)\right],
\end{equation}
where $\tilde{N}_\mathrm{f}=c_\mathrm{f}\tilde{A}_\mathrm{p}$ is the number of free tethers present within area $\tilde{A}_\mathrm{p}$ on the SBL. $\tilde{A}_\mathrm{p}$ is the zero-stretching adhesion area, which the GUV-SBL system would form for negligibly small attractive forces when $T>T_0$, as derived in ref.\cite{Parolini_NatComms_2015}\\
Note that $U_0$ does not depend on the contact angle $\theta$ therefore its form does not influence the equilibrium features of the system.
\end{appendix}
|
{
"timestamp": "2015-04-20T02:02:59",
"yymm": "1504",
"arxiv_id": "1504.03172",
"language": "en",
"url": "https://arxiv.org/abs/1504.03172"
}
|
\section{Introduction}
The correlated electron systems like cobaltates~\cite{qian06,tera97,tekada03}, $GdI_{2}$~\cite{tara08} and its doped variant $GdI_{2}H_{x}$ ~\cite{tulika06,fel,ryaz}, $NaTiO_{2}$~\cite{clarke98,pen97,khom05}, $MgV_{2}O_{4}$~\cite{rmn13} etc. have attracted great interest recently as they exhibit a number of remarkable cooperative phenomena such as valence and metal-insulator transition, charge, orbital and spin/magnetic order, excitonic instability and possible non-fermi liquid states~\cite{tara08}. These are layered triangular lattice systems and are characterized by the presence of localized (denoted by $f$-) and itinerant (denoted by $d$-) electrons. The geometrical frustration from the underlying triangular lattice coupled with strong quantum fluctuations give rise to a huge degeneracy at low temperatures resulting in competing ground states close by in energy. Therefore, for these systems one would expect a fairly complex ground state magnetic phase diagram and the presence of soft local modes strongly coupled with the itinerant electrons. It has recently been proposed that these systems may very well be described by different variants of the two-dimensional Falicov-Kimball model (FKM)~\cite{tara08,tulika06} on the triangular lattice.
Originally the FKM was proposed to describe the metal-insulator transition in mixed valence compounds~\cite{fkm69,fkm70}.
Later the FKM was used to study the tendency of formation of charge density wave (CDW)
order as well~{\cite{brandt,schmidt,freer,hassan}. Recently we have studied the ground state and finite temperature properties of the FKM and its different extensions on the triangular lattice~\cite{umesh1,umesh2}. We have reported several interesting results like various charge order, metal-insulator transitions and resolved the issue of spontaneous symmetry breaking (SSB)~\cite{umesh2} in the ground state and explored the metal-insulator transition at finite-temperature~\cite{umesh3,umesh4,umesh5} in the different regime of parameters. In all these studies the spin-degree of freedom was ignored and the interactions between electrons were spin-independent. Recent experimental results show that a charge order generally occurs with an attendant spin/magnetic order in many correlated systems ~\cite{chen,tranq1,tranq2}. In order to describe both the charge and magnetic orders in a unified way we use a generalized FKM Hamiltonian~\cite{lemanski05} that includes spin-dependent local interactions:
\begin{eqnarray}
H=-\,\sum\limits_{\langle ij\rangle\sigma}(t_{ij}+\mu\delta_{ij})d^{\dagger}_{i\sigma}d_{j\sigma}
+\,(U-J)\sum\limits_{i\sigma}f^{\dagger}_{i\sigma}f_{i\sigma}d^{\dagger}_{i\sigma}d_{i\sigma}
\nonumber \\
+\,U\sum_{i\sigma}f^{\dagger}_{i,-\sigma}f_{i,-\sigma}d^{\dagger}_{i\sigma}d_{i\sigma}
+\,U_{f}\sum\limits_{i\sigma}f^{\dagger}_{i\sigma}f_{i\sigma}f^{\dagger}_{i,-\sigma}f_{i,-\sigma}
\nonumber \\
+E_{f}\sum\limits_{i\sigma}f^{\dagger}_{i\sigma}f_{i\sigma}
\end{eqnarray}
\noindent here $\langle ij\rangle$ denotes the nearest neighbor ($NN$) lattice sites. The $d^{\dagger}_{i\sigma}, d_{i\sigma}\,(f^{\dagger}_{i\sigma},f_{i\sigma})$ are, respectively, the creation and annihilation operators for $d$- ($f$-) electrons with spin $\sigma=\{\uparrow,\downarrow\}$ at the site $i$. First term is the band energy of the $d$-electrons and $\mu$ is the chemical potential. The hopping parameter $t_{\langle ij\rangle} = t$ for $NN$ hopping and zero otherwise. The interaction between $d$-electrons is neglected in FKM as usual. The second term is the on-site interaction between $d$ and $f$-electrons of same spin with coupling strength ($U - J$) (where $U$ is the usual spin-independent Coulomb term and $J$ is the exchange interaction; the term follows from Hund's coupling). The third term is the on-site interaction $U$ between $d$- and $f$-electrons of opposite spins. { Here $J$ basically represents the spin dependent local interactions between localized ($f$-) and itinerant ($d$-) electrons that stabilizes parallel over anti-parallel alignment between $f$- and $d$-electrons. Inclusion of the exchange or Hund's coupling term enables us to study the magnetic structure of the $f$-electrons and band magnetism of the $d$-electrons. Fourth term is on-site Coulomb repulsion $U_f$ between opposite $f$-spins while the last term is the spin-independent, dispersionless energy level$E_f$ of the $f$-electrons.
There are some theoretical results available for the spin-dependent FKM on a bipartite lattice ~\cite{lemanski05,farkov02}. A few ground state charge and magnetic configurations exist for certain fixed values of $U$ and $J$. There is hardly any study available for the spin-dependent FKM on non-bipartite lattices. Therefore, in the present work we take up model systems that represent layered materials with triangular lattice (hence geometrically frustrated). Within second order perturbation theory, the spinless FKM with extended interactions can be shown to map to an effective Ising model with antiferrmagnetic interactions in the large U limit~\cite{umesh2}. The AFM coupling on triangular lattice is frustrated and leads to large degeneracies at low temperature. It turns out that this frustration is lifted~\cite{gruber1,gruber2} in the higher order perturbation in $\frac{1}{U}$~\cite{footnote}. Therefore it would be quite interesting to see the role of spin degree of freedom of electrons on the ground state properties on such lattices with different values of parameters $U$ and $J$. We study FKM at different range of interactions $U$ and $J$ for different electronic filling fractions on a triangular lattice.
\section{Methodology}
All the interactions in the Hamiltonian $H$ (Eq.$1$) preserve local occupation and spin of the $f$-electrons, i.e. the $d$-electrons traveling through the lattice change neither occupation numbers nor spins of the $f$-electrons. The local $f$-elctron occupation number $\hat{n}_{fi\sigma}=f_{i\sigma}^{\dagger}f_{i\sigma}$ is conserved as $\big[\hat{n}_{fi\sigma},H\big]=0$ for all $i$ and $\sigma$. This implies that $\omega_{i\sigma}=f_{i\sigma}^{\dagger}f_{i\sigma}$ is a good quantum number taking values only $1$ or $0$ as the site $i$ is occupied or unoccupied by an $f$-electron of spin $\sigma$, respectively. Following this local conservation, $H$ can be rewritten as
\begin{eqnarray}
H=\sum\limits_{\langle ij \rangle \sigma}\, h_{ij}(\{\omega_{\sigma}\})\,d_{i\sigma}^{\dagger}d_{j\sigma}
+\,U_{f}\sum\limits_{i\sigma}{\omega_{i\sigma}\omega_{i,-\sigma}}
\nonumber \\
+\,E_{f}\sum\limits_{i\sigma}\,{\omega_{i\sigma}}
\end{eqnarray}
\noindent where $h_{ij}(\{\omega_{\sigma}\})=\big[-t_{ij}+\{(U-J)\omega_{i\sigma}+U\omega_{i,-\sigma}-\mu\}\delta_{ij}\big]$ and $\{\omega_{\sigma}\}$ is a chosen configuration of $f$-electrons of spin $\sigma$.
The Hamiltonian $H$ in Eq.$2$ shows that the $f$-electrons act as an external charge and spin-dependent potential or annealed disordered background for the non-interacting $d$-electrons. This external potential of $f$-electrons can be ``annealed'' to find the minimum energy of the system. It is clear that there is inter-link between subsystems of $f$- and $d$-electrons. This inter-link is responsible for the long range ordered configurations and different charge and magnetic structures of $f$-electrons in the ground state.
We set the scale of energy with $t_{\langle ij \rangle} = 1$. The value of $\mu$ is chosen such that the filling is ${\frac{(N_{f}~ + ~N_{d})}{4N}}$ (e.g. $N_{f} + N_{d} = N$ is one-fourth case and $N_{f} + N_{d} = 2N$ is half-filled case etc.), where $N_{f} = (N_{f_{\uparrow}}+N_{f_{\downarrow}})$, $N_{d} = (N_{d_{\uparrow}} + N_{d_{\downarrow}})$ and $N$ are the total number of $f-$ electrons, $d-$ electrons and sites respectively. For a lattice of $N$ sites the $H(\{\omega_{\sigma}\})$ (given in Eq.2) is a $2N\times 2N$ matrix for a fixed configuration $\{\omega_{\sigma}\}$. For one particular value of $N_f(= N_{f_{\uparrow}} + N_{f_{\downarrow}})$, we choose values of $N_{f_{\uparrow}}$ and $N_{f_{\downarrow}}$ and their configuration $\{\omega_{\uparrow}\} = \{{\omega_{1\uparrow}, \omega_{2\uparrow},\ldots, \omega_{N\uparrow}}\}$ and $\{\omega_{\downarrow}\} = \{{\omega_{1\downarrow}, \omega_{2\downarrow},\ldots, \omega_{N\downarrow}}\}$. Choosing the parameter $U$ and $J$, the eigenvalues $\lambda_{i\sigma}$($i = 1\ldots N$) of $h(\{\omega_{\sigma}\})$ are calculated using the numerical diagonalization technique on the triangular lattice of finite size $N(=L^{2}, L = 12)$ with periodic boundary conditions (PBC).
The partition function of the system is written as,
\begin{eqnarray}
\it{Z}=\,\sum\limits_{\{\omega_{\sigma}\}}\,Tr\,\left(e^{-\beta H(\{\omega_{\sigma}\})}\right)
\end{eqnarray}
\noindent where the trace is taken over the $d-$electrons, $\beta=1/k_{B}T$. The trace is calculated from the eigenvalues $\lambda_{i\sigma}$ of the matrix $h(\{\omega_{\sigma}\})$ (first term in Eq.2). The partition function can, therefore, be recast in the form,
\begin{eqnarray}
\it{Z}=\,\sum\limits_{\{\omega_{\sigma}\}}\,
\prod\limits_{i}\,\left(e^{-\beta\big[U_{f}\omega_{i\sigma}\omega_{i,-\sigma} + E_{f}\omega_{i\sigma} \big]}\right)\,
\nonumber \\
\prod\limits_{j}\,\left(e^{-\beta\big[\lambda_{j\sigma}(\{\omega_{\sigma}\})-\mu\big]}+1 \right)
\end{eqnarray}
Now, the thermodynamic quantities can be calculated as averages over various configurations $\{\omega_{\sigma}\}$ with
statistical weight $P(\{\omega_{\sigma}\})$ is given by
\begin{eqnarray}
P(\{\omega_{\sigma}\})=\frac{e^{-\beta\,F(\{\omega_{\sigma}\})}}{\it{Z}}
\end{eqnarray}
\noindent where the corresponding free energy is given as,
\begin{eqnarray}
F(\{\omega_{\sigma}\})=\,-\frac{1}{\beta}\,\bigg[ln\,\left(\prod\limits_{i}\,
e^{-\beta\big[U_{f}\omega_{i\sigma}\omega_{i,-\sigma} + E_{f}\omega_{i\sigma} \big]}\right)
\nonumber \\
+\sum\limits_{j}\,ln\,\left(e^{-\beta\big[\lambda_{j\sigma}(\{\omega_{\sigma}\})-\mu\big]}+1 \right) \bigg]
\end{eqnarray}
\noindent The ground state total internal energy $E(\{\omega_{\sigma}\})$ is calculated as,
\begin{eqnarray}
E(\{\omega_{\sigma}\})=\,\lim_{T\rightarrow 0} F(\{\omega_{\sigma}\})
=\sum\limits_{i\sigma}^{N_{d}}\lambda_{i\sigma}(\{\omega_{\sigma}\})
\nonumber \\
+U_{f}\sum\limits_{i\sigma}\omega_{i\sigma}\omega_{i,-\sigma}
+E_{f}\sum\limits_{i\sigma}\omega_{i\sigma}
\end{eqnarray}
Our aim is to find the unique ground state configuration (state with minimum total internal energy $E(\{\omega_{\sigma}\}$)) of $f$-electrons out of exponentially large possible configurations for a chosen $N_{f}$. In order to achieve this goal, we have used classical Monte Carlo simulation algorithm by annealing the static classical variables $\{\omega_{\sigma}\}$ ramping the temperature down from a high value to a very low value. Details of the method can be found in our earlier
papers~\cite{umesh1,umesh2,umesh3,umesh4,umesh5,umesh6}.
\section{Results and discussion}
\begin{figure}[h]
\begin{center}
\includegraphics[trim=0.5mm 0.5mm 0.5mm 0.5mm,clip,width=8.9cm,height=7.0cm]{Fig1.eps}
\caption{(Color online) Variation of magnetic moment of $d$-electrons $m_{d}$ and $f$-electrons $m_{f}$ with number of $d-$electrons $N_d$ for $n_{f}=1$, $U = 5$, $U_{f} = 10$ and for $J = 5$ and $3$.}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[trim=0.4mm 0.4mm 0.4mm 0.1mm,clip,width=8.9cm,height=4.0cm]{Fig2.eps}
\caption{(Color online) (a) Up-spin and (b) down-spin $d$-electron densities are shown on each site for $J = 5$, $U = 5$, $U_{f} = 10$, $n_{f} = 1$ and $N_{d} = 55$. The color coding and radii of the circles indicate the $d$-electron density profile. Triangle-up and triangle-down, filled by black and red colors correspond to the sites occupied by up-spin and down-spin $f$-electrons, respectively.}
\end{center}
\end{figure}
We have studied the variation of magnetic moment of $d$-electrons $\large(m_{d} = \frac{M_{d}}{N} = \frac{\large(N_{d_{\uparrow}}~ - ~N_{d_{\downarrow}}\large)}{N}\large)$ (as $d$-electrons are spread over all $N$ number of sites) and magnetic moment of
$f-$electrons $\large(m_{f} = \frac{M_{f}}{N_f} = \frac{\large(N_{f_{\uparrow}}~ - ~N_{f_{\downarrow}}\large)}{N_f}\large)$
(as $f$-electrons are confined to only $N_f$ number of sites) with number of $d$-electrons $N_{d}$ at a fixed value of $U$, $U_{f}$ and $J$ for $n_{f} = 1~\large(n_{f} = \frac{N_{f}}{N} = \frac{\large(N_{f_{\uparrow}}~ + ~N_{f_{\downarrow}}\large)}{N}\large)$. We have also studied the density of $d$-electrons at each site for the above case. Fig.$1$ shows the variation of magnetic moment of $d$-electrons ($m_{d}$) and $f$-electrons ($m_{f}$) with number of $d$-electrons ($N_d$) for two different values of exchange correlation $J$ i.e. $J = 5$ and $J = 3$ at a fixed value of on site coulomb repulsion $U = 5$ and $U_{f} = 10$. From Fig.$1$ one can note that when $N_{d} = 144$ the ground state is Neel ordered anti-ferromagnetic (AFM) in nature. The reason for this can be understood in the following way. It is clear From Eq.$1$ that for $U = J$, there is no repulsion between $d$- and $f$-electrons of same spins on the same site. The repulsion between $d$- and $f$-electrons of the opposite spins on the same site is $U$. Therefore it is energetically favorable that a $d$-electron of same spin (as that of $f-$) occupies the site. For the FM arrangement of $f$-electrons, all $d$-electrons occupying the sites shall have FM arrangement themselves. Similarly for the AFM arrangement of $f$-electrons, $d$-electrons occupying the sites shall have AFM arrangement. For FM arrangements of $d$- and $f$-electrons, there is no hopping possible for $d$-electrons due to Pauli$^{\prime}$s exclusion principle, but for AFM arrangement of $f$- and $d$-electrons, there will be finite hopping of $d$-electrons between neighboring sites. This hopping reduces the kinetic energy of $d$-electrons and hence total band energy of $d$-electrons. Hence AFM arrangement of spins corresponds to minimum energy. Thus the ground state is AFM. In fact, it remains AFM up to $N_{d} = 70$ for $J = 5$ and up to $N_{d} = 65$ for $J = 3$. The magnetic moment for $f$-electrons starts increasing for $N_{d} < 70$ (for $J = 5$) and for $N_{d} < 65$ (for $J = 3$). Fully FM state is observed at $N_{d} \le 40$ for $J=5$ and at $N_{d} \le 45$ for $J = 3$, as now $d$-electrons find a plenty of sites (where no $d$-electrons are present) to hop to.
By changing the number of $d$-electrons, basically we are varying the doping and the extent of doping caused the phase transition from Neel order AFM to FM via mixture of both state. These type of phase transition also observed using band structure calculation for $GdI_{2}H_{x}$, by doping hydrogen as reported by authors~\cite{tulika06,fel,ryaz}.
For the above mentioned parameter value $U = 5$ and $J = 5$ and $U = 5$ and $J = 3$, we have seen $f$-electrons configuration and $d$-electrons density at each site for different $N_{d}$. Fig.$2$ shows the density of $d$-electrons at each site for $U = 5, J = 5$ for $N_d = 55$ (say). Table $1$ and $2$, respectively summaries the density of $d$-electrons for two cases $U = 5$ and $J = 5$ (Fig.$3$) and $U = 5$ and $J = 3$ (not shown here) with FM and AFM arrangement of $f$-electrons. The density of $d$-electrons at each site strongly depends upon the value of exchange correlation $J$. Let us compare the density of $d$-electrons at sites with up-spin and down-spin $f$-electrons for a fixed value of $N_{d}$ (say $N_{d} = 55$) and for $J = 5$ and $J = 3$ (given in Table $1$ and $2$). We note from Table $1$ that the density of $d$-electrons at sites where $d$- and $f$-electrons have same spins is large for $J = 5$ and less for $J = 3$. This is expected (as seen from Hamiltonian (Eq.$1$)), because for $U = J = 5$, there is no on-site repulsion between $d$- and $f$-electrons of the same spins, but for $U = 5$ and $J = 3$ there is finite repulsion between $d$- and $f$-electrons of the same spins. Also from Table $1$ and $2$, we note that density of $d$-electrons with up-spin ($n_{d_{\uparrow}}$) at sites with FM arrangement of $f$-electrons is lesser than that at sites with AFM arrangement of $f$-electrons. This is so because in later case the $d$-electrons may hop to either empty sites or to sites having down-spin $f$-electrons. This hopping reduces the kinetic energy of $d$-electrons and hence total band energy of $d$-electrons. For the FM case, the $d$-electrons can hop only to empty sites and the reduction in its total band energy is less. Same is true for density of $d$-electrons with down-spin ($n_{d_{\downarrow}}$).
\begin{table*}
\caption{The density of $d$-electrons with FM arrangement of $f$-electrons}
\centering
\begin{tabular}{|*{2}{c|}*{2}{c|}c|}
\hline
\multicolumn{2}{|c|}{Density of $d$-electrons} & \multicolumn{2}{|c|}{Density of $d$-electrons}& $U = 5$ \\
\multicolumn{2}{|c|}{with up spin ($n_{d_{\uparrow}}$)} & \multicolumn{2}{|c|}{with down spin ($n_{d_{\downarrow}}$)}& $N_{d} = 55$ \\
\hline
Sites with $f_{\uparrow}$ & Sites with $f_{\downarrow}$ & Sites with $f_{\uparrow}$ & Sites with $f_{\downarrow}$ & \\
\hline
$0.3$ & 0 & $0.001 - 0.03$ & $0$ & $J = 5$ \\
\hline
$0.27$ & 0 & $0.058 - 0.072$ & $0$ & $J = 3$ \\
\hline
\end{tabular}
\end{table*}
\begin{table*}
\caption{The density of $d$-electrons with AFM arrangement of $f$-electrons}
\centering
\begin{tabular}{|*{2}{c|}*{2}{c|}c|}
\hline
\multicolumn{2}{|c|}{Density of $d$-electrons} & \multicolumn{2}{|c|}{Density of $d$-electrons} & $U = 5$ \\
\multicolumn{2}{|c|}{with up spin ($n_{d_{\uparrow}}$)} & \multicolumn{2}{|c|}{with down spin ($n_{d_{\downarrow}}$)}& $N_{d} = 55$ \\
\hline
Sites with $f_{\uparrow}$ & Sites with $f_{\downarrow}$ & Sites with $f_{\uparrow}$ & Sites with $f_{\downarrow}$ & \\
\hline
$0.37$ & $0.06$ & $0.06$ & $0.41$ & $J = 5$ \\
\hline
$0.32$ & $0.1$ & $0.1$ & $0.3$ & $J = 3$ \\
\hline
\end{tabular}
\end{table*}
We have also studied the ground state magnetic phases for up-spin and down-spin $f$-electrons, magnetic moments of $d$- and $f$-electrons and the density of $d$-electrons on each site for the range of values of parameters $U$, $U_{f}$ and $J$ for two cases (i) $n_{f} + n_{d} = 1$ (one-fourth filled case) and (ii) $n_{f} + n_{d} = 2$ (half filled case). We have chosen large value of $U_{f}$ so that double occupancy of $f$-electrons is avoided.
\subsection{One-fourth filled case ($n_{f} + n_{d} = 1$):}
\begin{figure}[h]
\includegraphics[trim=0.4mm 0.4mm 0.4mm 0.1mm,clip,width=8.9cm,height=4.0cm]{Fig3.eps}
\caption{(Color online) The ground-state magnetic configurations of $f$-electrons for $n_{f} = \frac{1}{2}$, $n_{d} = \frac{1}{2}$, $U = 5$, $U_{f} = 10$ and for various values of $J$. Triangle-up and triangle-down, filled by black and red colors correspond to the sites occupied by up-spin and down-spin $f$-electrons, respectively. Open green circles correspond to the unoccupied sites.}
\end{figure}
\begin{figure}[h]
\includegraphics[trim=0.5mm 0.0mm 0.5mm 0.1mm,clip,width=8.9cm,height=5.5cm]{Fig4.eps}
\caption{(Color online) Variation of magnetic moment of $d$-electrons $m_{d}$ and $f$-electrons $m_{f}$ with exchange correlation $J$ for $n_{f} = \frac{1}{2}$, $n_{d} = \frac{1}{2}$ at (a) $U = 3.1$, $U_{f} = 7$ and (b) $U=5$, $U_{f} = 10$.}
\end{figure}
\begin{figure}[h]
\includegraphics[trim=0.4mm 0.4mm 0.4mm 0.1mm,clip,width=8.9cm,height=4.0cm]{Fig5.eps}
\caption{(Color online) Up-spin $d$-electron densities are shown on each side for $J = 0$ (a) and $J = 3.5$ (b), keeping other parameters same i.e. $U = 5$, $U_{f} = 10$, $n_{f} = \frac{1}{2}$ and $n_{d} = \frac{1}{2}$. The color coding and radii of the circles indicate the $d$-electron density profile. Triangle-up and triangle-down, filled by black and red colors correspond to the sites occupied by up-spin and down-spin $f$-electrons, respectively. Open green circles correspond to the unoccupied sites.}
\end{figure}
In Fig.$3$ the ground state magnetic configurations of up-spin and down-spin $f$-electrons are shown for $U = 5$, $U_{f} = 10$ and for different $J$ values. The ground state configurations are observed to be long range ordered Neel ordered AFM arrangement of spins (Fig.3(a)) or complete FM arrangement (Fig.3(b)) or mixture of both arrangements (Fig.3(c). Here we note that $U=3.1$ is the critical value of the on-site Coulomb correlation below which we do not get finite magnetic moment. Complete AFM phases are observed below $U = 3.1$.
The variation of magnetic moment of $d$-electrons ($m_{d}$) and magnetic moment of $f$-electrons ($m_{f}$) with exchange correlation $J$ is shown in Fig.$4(a)$ and Fig.$4(b)$ for $U=3.1$ and $U_{f} = 7$ and for $U=5$ and $U_{f} = 10$ respectively. We note that in both cases the magnetic moment of $d$- and $f$-electrons increases with increasing $J$. Complete FM phase is observed at a particular value of $J$ (e.g. $J = 0.25$ for $U=5$ and $U_{f} = 10$) and it continues up to some critical value of $J$ (up to $J = 2$ for $U=5$ and $U_{f} = 10$). With further increase in $J$ mixed states are observed and finally AFM state is observed for larger values of $J$.
Figs.$5(a)$ and $5(b)$ show the density of up-spin $d$-electrons at a fixed value of $U = 5$, $U_{f} = 10$ and for $J = 0$ and $J = 3.5$ respectively. When $J = 0$ the interaction between $d$- and $f$-electrons is same irrespective of their spins so the density of $d$-electrons at sites occupied by f-electrons are same, while it is maximum at unoccupied sites. With the increase in $J$ value density of $d$-electrons at sites where $f$-electrons of same spin are present increases and at empty sites it decreases, because as $J$ increases, the interaction $(U-J)$ between $d$- and $f$-electrons of the same spins decreases.
These results can be explained as we have three kinds of competing energies in the system namely the kinetic energy of $d$-electrons and on-site interaction energies, `$U$' between $d$- and $f$-electrons of opposite spins and `$(U - J)$' between $d$- and $f$-electrons of same spins. For $J = 0$, the on-site interaction energies between $d$- and $f$-electrons of opposite spins and $d$- and $f$-electrons of same spins will be the same. Hence the ground state configuration is AFM type as possible hopping of $d$-electrons minimizes the energy of the system. It is clearly shown in the variation of $d$-electron density at each site in Fig.$5(a)$. As a result both the $m_d$ and $m_f$ are zero (as shown in Fig.$4$ and $7$). For finite but small value of $J$, the on-site interaction energy between $d$- and $f$-electrons of same spins will be smaller in comparison to the on-site interaction energy between $d$- and $f$-electrons of opposite spins. So few sites with FM arrangement of spin-up $f$-electrons will be occupied by down-spin $d$-electrons and few by up-spin $d$-electrons (as $m_d$ is very small). With this arrangement there is finite hopping possible for $d$-electrons which reduces its kinetic energy and hence total energy of the system. Therefore the ground state configuration is either mixture of AFM and FM types or completely FM type. As a result both the $m_{d}$ and $m_f$ increase with increasing $J$. Finally for large value of $J$ ($J \sim U$), the ground state is AFM in nature. We have explained that in earlier section.
\subsection{Half-filled case ($n_{f}+n_{d}=2$):}
\begin{figure}[h]
\includegraphics[trim=0.4mm 0.4mm 0.4mm 0.1mm,clip,width=8.9cm,height=4.0cm]{Fig6.eps}
\caption{(Color online) The ground-state magnetic configurations of $f$-electrons for $U = 5$, $U_{f} = 10$ and for various values of $J$ with condition $n_{f} + n_{d} = 2$. Triangle-up and triangle-down, filled by black and red colors correspond to the sites occupied by up-spin and down-spin $f$-electrons, respectively.}
\end{figure}
\begin{figure}[h]
\includegraphics[trim=0.5mm 0.5mm 0.5mm 0.1mm,clip,width=8.9cm,height=5.5cm]{Fig7.eps}
\caption{(Color online) Variation of magnetic moment of $d$-electrons $m_{d}$ and $f$-electrons $m_{f}$ with exchange correlation $J$ for (a) $U = 2$, $U_{f}= 5$ and (b) $U = 5$, $U_{f} = 10$ and for $n_{f} = 1$ and $n_{d} = 1$.}
\end{figure}
\begin{figure}[h]
\includegraphics[trim=0.4mm 0.4mm 0.4mm 0.1mm,clip,width=8.9cm,height=4.0cm]{Fig8.eps}
\caption{ Up-spin $d-$electron densities are shown on each side for $J = 0$ (a) and $J = 0.5$ (b), keeping other parameters same i.e. $U = 5$, $U_{f} = 10$, $n_{f} = 1$ and $n_{d} = 1$. The color coding and radii of the circles indicate the $d$-electron density profile. Triangle-up filled by black color corresponds to the sites occupied by up-spin $f$-electrons.}
\end{figure}
Shown in Fig.$6$ are the ground state, magnetic configurations of up-spin and down-spin $f$-electrons for $U = 5$, $U_{f} = 10$ and for different $J$ values. At $J = 0$, again regular Neel ordered AFM structure is seen. With increasing $J$, mixed phase of FM and Neel type AFM is seen (Fig.6(a)). On futher increasing $J$ the phase becomes fully FM and remains the same up to a value of $J$ nearly equal to $0.75$. Then the mixed phase comes back on increasing $J$ at around $J = 1$ (Fig.6(b)) and finally the ground state becomes Neel ordered AFM again for $J > 1.80$ and continues up to $J = 5$ (Fig.6(c)). Similar FM, AFM and mixed magnetic configurations of up-spin and down-spin $f$-electrons also observed for $U = 2$, $U_{f} = 5$ and for different $J$ values (not shown here).
Corresponding variation of magnetic moment of $d$-electrons ($m_d$) and of $f$-electrons ($m_f$) with exchange correlation $J$ is shown in Fig.$7(a)$ and Fig.$7(b)$ for $U=2$ and $U_{f} = 5$ and for $U=5$ and $U_{f} = 10$ respectively. As we have already explained that the magnetic moments of $d$- and $f$-electrons increase with increasing $J$ in both cases. Increase in magnetic moment of $f$-electrons is observed when one goes from Neel type AFM to complete FM phase through the intermediate mixed phase. Similarly, $m_d$ also increases with $J$ but the increase starts after a certain value of $J$. Here we observe that $m_d$ and $m_f$ increase initially with increasing $J$ but after reaching a maximum the moments drop down at larger $J$ values and finally both $m_d$ and $m_f$ become zero. We have found that for large value of $U$, magnetization decreases sharply with $J$ in comparison to that for small values of $U$.
Figs.$8(a)$ and $8(b)$ show the density of $d$-electrons at a fixed value of $U = 5$, $U_{f} = 10$ and for $J = 0$ and $J = 0.5$ respectively. For $J = 0$ the density of $d$-electrons at all sites is same irrespective of spins of $f$-electrons. As value of $J$ increases, density of $d$-electrons at sites where $f$-electrons of same spins are present increase as compared to at sites where $f$-electrons of opposite spins are present. This is expected because as $J$ increases, the repulsion between $d$- and $f$-electrons of the same spins decreases and hence they prefer to sit on the sites where $f$-electrons of the same spin are present.
In conclusion, the ground state magnetic properties of two dimensional spin-$1/2$ FKM on a triangular lattice for different range of values of parameters like $d$- and $f$-electron fillings, on-site Coulomb correlation $U$ and exchange correlation $J$ are studied. We have found that the magnetic moments of $d$- and $f$-electrons depend strongly on the values of $J$ and on the number of $d$-electrons $N_{d}$. We have seen for one-fourth filling that there are no magnetic moments of $d$- and $f$-electrons for $U$ less than $3.1$. At half-filling we have found that the magnetic moments of $d$- and $f$-electrons decrease sharply with $J$ at larger $U$ in comparison to smaller values of $U$. At both fillings we note that the density of $d$-electrons depends upon the value of exchange correlation $J$. Also, various charge and magnetic ordered phases of the localized $f$-electrons in the ground state have been observed at different values of $J$. The ground state configuration is observed to be long range ordered (either in some form of AFM arrangement of spins, complete FM arrangement or a mixture of both). The magnetic moments of $d$- and $f$-electrons start to increase sharply with increasing $J$ and persist up to a larger value of $J$ for one-fourth filling in comparison to one-half filling case (see Fig.4(b) and Fig.7(b)). There is rarely any calculation available for the spin-dependent FKM on a triangular lattice. Our results may motivate further studies of the magnetic properties of frustrated systems of recent interest like Cobaltates, $GdI_{2}$, $NaTiO_{2}$ and $MgV_{2}O_{4}$.
$Acknowledgments.$ SK acknowledges MHRD, India for a research fellowship. UKY acknowledges CSIR, India for a research fellowship through SRF grant and the UGC, India for Dr. D. S. Kothari Post-doctoral Fellowship through grant No.F.4-2/2006(BSR)/13-762/2012(BSR).
|
{
"timestamp": "2015-04-15T02:05:11",
"yymm": "1504",
"arxiv_id": "1504.03429",
"language": "en",
"url": "https://arxiv.org/abs/1504.03429"
}
|
\section{Orientation of Titan's dunes from radar images}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=15cm]{figure_s5.pdf}
\end{center}
\caption{\textbf{Map of Titan's dune orientation.} Map of radar-measured dune orientation vectors \citep{lorenz09} (CREDITS: NASA/JPL-Caltech/ASI/Space Science Institute, PIA11801), showing the global eastward propagation and the divergence from the equator for latitudes higher than 10$^\circ$.}
\label{figure_s5}
\end{figure}
Fig. \ref{figure_s5} shows the map of dune orientation and propagation, indicated with vectors, over a near-infrared basemap derived from Cassini ISS (Imaging Science Subsystem) images \citep{lorenz09}. The direction of propagation is obtained by looking at dune morphology around obstacles and dune terminations, with for instance some dunes stopping on the west side of obstacles and others diverting and recombining beyond the east side of obstacles. This analysis reveales that all dunes propagate eastward. Concerning the North-South orientation, a statistical analysis indicates that dunes tend to diverge poleward for latitudes higher than around 10$^\circ$ N/S\citep{lucas14}.
Images of Titan's surface obtained by the Cassini's radar SAR (Synthetic Aperture Radar) suffer from errors from a variety of sources, the most prominent being speckle noise. This noise refers to the constructive and destructive interference of scattered energy from roughness elements on a scale smaller than the size of a SAR pixel. The result is a multiplicative noise, which hinders interpretation. An advanced denoising algorithm has been adapted to Cassini SAR data \citep{lucas14b}. It has been applied to the original image (i.e. the image by Cassini Radar, T8 flyby) of Fig. 4a of the letter to significantly reduce the noise.
Fig. 3a and 3b of the article show dune orientations obtained in the same way as Fig. \ref{figure_s5} but with denoised radar images whose dunes segments have been detected \citep{lucas14} and their orientation averaged over areas of 8$^\circ$$\times$8$^\circ$ in longitude-latitude.
\section{Calculation of general circulation winds}
\textbf{Description of the IPSL Titan GCM}\\
For this study, we used a 3-dimensional GCM \citep{lebonnois12,charnay12}. A horizontal resolution of 32 $\times$ 48, corresponding to resolutions of 3.75$^{\circ}$ latitude by 11.25$^{\circ}$ longitude, is used for the simulations. This GCM covers altitudes from the ground (first level at 35 m) to 500~km. The dynamical core is based on the most up-to-date version of the LMDZ \citep{hourdin06}. It is a finite-difference discretization scheme that conserves both potential enstrophy for barotropic nondivergent flows, and total angular momentum for axisymmetric flows. The version used in this study includes gravitational tides \citep{tokano06b}, though the impact in the troposphere does not influence the effects described in this work. We found tidal effects on the pressure similar to previous works \citep{tokano06b}, but tidal winds are much weaker in our model. We use a fully coupled aerosol microphysics calculated in 2D (zonally averaged) \citep{rannou04b}. The present model is dry and does not take into account the methane cycle. The profile of methane is fixed (close to the HASI profile \citep{niemann05}) for the radiative transfer. The latter is based on the McKay radiative code \citep{mckay89} and includes the diurnal cycle. The radiative transfer is called 200 times per Titan day.
For the surface, we use an albedo of 0.3, a rugosity length of 0.005~m, an emissivity of 0.95 and a thermal inertia of 400~J/m$^{2}$/K. In this study, we run the GCM with a flat topography.
To calculate the friction speed $u_{\star}$ from the GCM wind $u$ at an altitude $z$, we use the relation:
\begin{equation}
u_{\ast}=\frac{\kappa}{\ln(z/z_0)} u
\end{equation}
with $\kappa$ = 0.4 the Von-Karman constant and $z_0$ = 0.005 m the rugosity length.
The threshold $u_{\ast t}$=0.04 m/s corresponds to a wind speed of 0.89 m/s at 35 m.
\\
\\
\textbf{GCM wind statistics:}\\
For all calculation implying the GCM wind statistics (e.g. sand fluxes), we use instantaneous GCM winds with 20 outputs per Titan day and combining wind series at all longitudes. The dissimilarities for wind statistics obtained for different Titan years are negligeable.
Fig. \ref{figure_s1} shows the surface wind roses produced by the GCM in the equatorial band.
Surface winds are essentially bimodal. At the equator, they blow from NE to SW in northern winter and from SE to NW in northern summer. For latitudes higher than 5$^\circ$ N/S, the summer winds are eastward. Because of Saturn's eccentricity, the southern summer is shorter and hotter than the northern summer. This implies that northerly winds are less frequent but stronger than southerly winds.
\\
\begin{figure}[!h]
\begin{center}
\includegraphics[width=5.6cm]{figure_s1a.pdf}
\includegraphics[width=5.6cm]{figure_s1b.pdf}
\includegraphics[width=5.6cm]{figure_s1c.pdf}
\end{center}
\caption{\textbf{Wind roses (direction and speed) produced by the GCM}. The roses have been obtained with instantaneous winds at 35 m at 0$^\circ$N (left) 10$^\circ$N (middle) and 20$^\circ$N (right). The frequency is given in percent.}
\label{figure_s1}
\end{figure}
\\
\begin{figure}[!h]
\begin{center}
\includegraphics[width=9.cm]{figure_s2.pdf}
\end{center}
\caption{\textbf{Statistics of GCM winds at the equator.} The different lines correspond to the relative probability of friction wind speed per bin of 0.001 m/s for the GCM winds (in red) and the GCM winds with increased gust (in orange) at the equator. The black line corresponds to the Weibull distribution fitting the GCM wind statistics (coefficient k=2.15 and c=0.0157 m/s) and the blue line the distribution fitting the GCM wind statistics with increased gust (coefficient k=2.15 and c=0.019 m/s).}
\label{figure_s2}
\end{figure}
Fig. \ref{figure_s2} shows the relative probability of the friction speed from the GCM (in red) at the equator and per bin of 0.001 m/s. Wind speed exceeds the threshold friction speed of 0.04 m/s only around 0.06 $\%$ of the time. The GCM wind statistics are well described by a Weibull distribution (in black in the figure), for which the probability of exceeding a friction speed $U$ is $P(>U)=\exp(-(U/c)^k)$, with $c$=0.0157 m/s the scale parameter and $k$=2.15 the shape parameter. In order to represent the higher variability of Titan's winds due to local gusts that are not captured with the low-resolution GCM grid (typical size of the spatial grid: 500 km$\times$170 km), we have also considered the case of an increase of wind speed by 20$\%$ (in orange in the figure). This arbitrary value corresponds to a significant increase of wind and leads to a one order of magnitude stronger sand transport. Because of the weak turbulence in the boundary layer \citep{tokano06}, it is likely to be a quite high value for high value for the representation of missing gusts. Concerning wind statistics, it is equivalent to an increase of the scale parameter of the Weibull distribution by 20$\%$ (in blue in the figure). In these conditions, the wind speed exceeds the threshold of 0.04 m/s around 0.7 $\%$ of the time (i.e. around ten times more than without correction).
\section{Simulation of methane storms}
\textbf{Description of the TRAMS model}\\
The Titan Regional Atmospheric Modeling System (TRAMS) is a coupled, regional-scale dynamics and column microphysics model \citep{barth07,barth10}. The governing equations for the dynamical core are the standard non-hydrostatic Reynolds-averaged primitive equations \citep{rafkin01}. The microphysics package was adapted from Barth et al. (2006)\citep{barth06} and operates independently on each column. Methane cloud particles form through nucleation onto submicron-sized haze particles. Through condensation and coalescence, methane particles can grow up to millimeter sizes. Depending on the temperature of the surrounding environment, the methane clouds form as either ice particle or droplets; melting and freezing of cloud particles is also included. Liquid droplets are treated as a mixture of N2/CH4 following the work of Thompson et al. (1992) \citep{thompson92}.
TRAMS is run here as a 2-D model. The 2-D simulations are run with a horizontal grid spacing of 1 km, and a total horizontal extent of 1000 km. Cyclic horizontal boundary conditions are employed. The vertical grid extends up to about 50 km; vertical layers are more compact near the surface, starting at about 15 m spacing and extend to constant 2 km spacing above 15 km altitude. The atmosphere is initialized horizontally homogeneously using the temperature-pressure profile measured by the HASI instrument on the Huygens probe \citep{fulchignoni05}. An initial horizontal wind is included using the $u$-velocity component from the GCM simulations. For methane, we construct profiles using a fixed amount of convective available potential energy (CAPE); we look at cases with CAPE=250 and CAPE=500 (equivalent to a methane mixing ratio of 5 g/kg, or 10 g/kg, respectively, near the surface). Cloud formation is initially triggered by perturbing the atmosphere with a warm bubble (the air is locally warmed in the first km), which has a maximum atmospheric temperature increase of 2 K. This 2K perturbation is a classical method to trigger deep convection in mesoscale simulations of terrestrial storm \citep{weisman88}. It is also possible to trigger deep convection with initial vertical updrafts (typically 1 m/s) but both methods lead to similar cloud dynamics \citep{hueso06}. In reality, the moist convection would be triggered by updrafts produced by planetary waves, the Hadley cell convergence, the diurnal cycle or the topography. The updraft velocities at largescale trigerring storms could be obtained from a GCM.
\\
\\
\textbf{Dynamics of convective cloud systems}\\
2D simulations of methane storms have been performed with our mesoscale model investigating in more details the effect of wind shear and CAPE on the storm dynamics, morphology and life time \citep{rafkin14}. It revailed behaviors similar to 2D simulations of terrestrial storms \citep{rotunno88}. We therefore expect the dynamics of the various kinds of Titan's storms (i.e. from individual convective cell to mesoscale convective systems) to be very similar to those of terrestrial storms. In particular, in our simulations the propagation and lifetime of Titan's storms is mostly controlled by the gust front. The leading edge of the gust front is able to lift moist air from the surface to trigger a new cell. When this new cell is close enough to the previous cell they merge together, increasing the lifetime of the storm and the strength of the gust front. When the gust front moves too quickly compared to the storm, they become disconnected and the storm dissipates. If a new cell were produced it would not be able to merge with the previous one.
Fig. 1 of the letter in the manuscript reveals this behaviour. A new cell is produced in Fig. 1c at around 20 km of the pre-existing cell and merges with it. Other cells were produced by the leading edge of the gust front before this event. We found that these new cells form at 20-40 km in front of the pre-existing cell before merging with it. In figure 4d, the gust front has moved away from the mean cell. It triggers a new cell but too far (100 km) from the pre-existing cell and they dissipate.
Our idealized 2D simulations of methane storms and gust fronts are representative of large storms or mesoscale convective systems (typically 100-1000 km in latitude), as those observed by Cassini \citep{turtle11b, turtle11a}. Small isolated convective clouds, as the ones simulated in 3D by Hueso and Sanchez-Lavega 2006 \citep{hueso06}, should produce weaker gust fronts than in our simulations and with a more istropic direction. However, multiple small convective clouds should merge into larger storms (see supplementary figure 2 in Hueso and Sanchez-Lavega 2006) associated with stronger gust fronts and with a spanwise dimension similar to the one of these cloud systems (i.e. 100-1000 km).
We also expect that some large storms or mesoscale convective systems on Titan evolve to produce squall lines or bow echos as they generally do on Earth \citep{bluestein85,weisman88}.
On Earth, a mesoscale convective system has a 3D structure, yet it becomes mostly 2D
when the wind shear is essentially 2D \citep{weisman88,parker04, coniglio06, mahoney09}. On Titan, the meridional wind shear is very weak compared to the longitudinal wind shear corresponding to the superrotation. Moreover, the Coriolis force is particularly weak at low latitude on Titan. This implies that equatorial mesocale convective systems should propagate westerly, forming fronts with no deformation by the Coriolis force. We therefore expect Titan's storm dynamics to be essentially 2D. This implies that surface winds produced by Titan's storms are primarily oriented east-west and that gust fronts should be qualitatively well represented by our idealized 2D simulations.
Finally, the direction of propagation of the gust front produced by a mesoscale convective system is essentially controlled by the transport of momentum from the mid-troposphere to the surface by cold pools \citep{mahoney09}. The presence of the superrotation necessarily implies a favoured eastward propagation for gust fronts on Titan.
A good analogy with the generation of eastward surface winds by Titan's storms is the impact of mesoscale convective systems over the Pacific warm pool during the Madden-Julian Oscillation. During such periods, mesoscale convective systems accerelate surface winds eastward in regions of fast high eastward winds \citep{mechem06, houze00, moncrieff97}.
\section{Estimation of storm frequency}
If we consider only the giant storm which was observed on 27 September 2009 \citep{turtle11a}, and if we assume that the same event occurs every equinox, we can evaluate a lower limit for the storm frequency as:
\begin{equation}
f_{min}= \frac{2 A_{\rm storm}}{A_{\rm equator} T_{\rm Titan}}
\end{equation}
$A_{\rm storm}$ is the area covered by the passage of the storm, $A_{\rm equator}$ $\approx$ 4$\times$10$^7$ km$^2$ is the area of the equatorial band (30$^\circ$ S to 30$^\circ$ N)
and $T_{\rm Titan}$ is the length of a Titan year ($\approx$29.5 terrestrial years).
The width of the storm was around 2000 km and it traveled at least 4000 km during probably a few days to produce the observed surface changes \citep{turtle11a}. Thus, thee passage of this storm covered around 20\% of Titan's equatorial band. This gives an average storm frequency of 0.4 per Titan's year or 0.2 storm per equinox. It seems reasonable to consider this value as a lower limit for the storm frequency.
Other large cloud systems have been observed during this equinox \citep{schaller09, griffith09, turtle11a, rodriguez11}. Moreover, Titan has only been observed during a short period of the equinox and not globally. Possible observations of liquid ethane on Titan's surface suggest that moderate and small rainstorms are quite common during the equinoctial season at the equator \citep{dalba12}.
We therefore expect that the mean storm frequency should rather be in the order of one storm per equinox (i.e. 2 storms per Titan year or 0.068 storms per terrestrial year).
\section{Sand flux calculations}
\textbf{Saltation threshold}
The saltation threshold corresponds to the minimal friction speed for which the wind stress is sufficient to lift particle \citep{kok12}. It has been estimated to be around 0.04 m/s for Titan \citep{greeley85,lorenz95,kok12,lorenz13} for an optimum particle diameter of 200-300 $\mu m$ and a sediment density of around 1000 kg/m$^2$.
A simple expression of the threshold friction speed for saltation is \citep{shao00}:
\\
\begin{equation}
u_{\ast t}=A_N \sqrt{\frac{\rho_{\rm sed} - \rho_{\rm air}}{\rho_{\rm air} } g D+ \frac{\gamma}{\rho_{\rm air} D}}
\end{equation}
\\
with $A_N$=0.111 a dimensionless parameter, $\rho_{\rm air}$ the air density, $\rho_{\rm sed}$ the sediment density, $D$ the particle diameter, $g$ Titan's gravity, $\gamma$ a parameter scaling the strength of the interparticle forces. $A_N$ actually depends on the particle friction Reynolds number \citep{iversen82}, but this dependence remains limited and $A_N$ can be considered constant at first approximation \citep{shao00}.
Fig. \ref{figure_s3} shows the threshold friction speed according to particle diameter for $g$ = 1.35 m s$^{-2}$, $\rho_{\rm sed}$=1000 kg/m$^3$, $\rho_{\rm air}$=5.3 kg/m$^3$ and $\gamma$=1.5$\times$10$^{-4}$N/m. For these values, the minimal threshold friction speed is 0.045 m/s for an optimum diameter of 350 $\mu \rm m$.
For pebbles with a diameter of 1-5 cm, the threshold friction speed is 0.18-0.4 m/s. This corresponds to a wind speed at 35 m of 4-9 m/s (see below).
For dust particles with a diameter of 10 $\mu \rm m$, the threshold friction speed is 0.19 m/s. This corresponds to a wind speed at 35 m of around 4 m/s.
General circulation winds never reach such high thresholds. In contrast, gust fronts produced by methane storms may be strong enough to transport centimetric pebbles and dust particles.
Dust particles can also be raised by the impact of saltating sand, thus with a wind speed lower than threshold friction speed.
The release of dust particles could lead to the formation of dust storms that could persist in the lower atmosphere several hours or days after the passage of the methane storm.
\begin{figure}[!h]
\begin{center}
\includegraphics[width=11.5cm]{figure_s3.pdf}
\end{center}
\caption{\textbf{Threshold friction speed for saltation as a function of the particle diameter.}
The saltation speed has been calculated using relation (3) for Titan's conditions \citep{lorenz13}, with a sediment density of 1000 kg/m$^3$ and a parameter $\gamma$=1.6$\times$10$^{-4}$N/m.}
\label{figure_s3}
\end{figure}
\\
\textbf{Sand transport law}\\
For calculating sand transport, we use the law from Kawamura (1951) \citep{kawamura51,white79,lorenz95}:
\begin{equation}
Q=2.6 \left(\frac{\rho_{\rm air}}{g \rho_{\rm sed}}\right) (u_{\ast}-u_{\ast t})(u_{\ast}+u_{\ast t})^2
\end{equation}
with $Q$ the sand flux per unit of width in m$^2$/s, $\rho_{\rm air}$ = 5.3 kg/m$^3$ the air density, $\rho_{\rm sed}$ = 1000 kg/m$^3$ the sediment density, $g$ = 1.35 m s$^{-2}$ Titan's gravity, $u_{\ast}$ the friction speed and $u_{\ast t}$ the threshold friction speed for transport.
To calculate $u_{\ast}$ from $u$, the GCM or meso-scale wind at an altitude $z$, we use the relation (1). Several other sand transport laws have been proposed \citep{kok12}. Their uses instead of the Kawamura's law could quantitively change our results (e.g. mean sand flux values) but not qualitatively (e.g. the reorientation of sand flux by methane storms).
\\
\\
\textbf{Calculation of the storm impact on the sand flux} \\
To calculate the impact of storms on the sand flux, we integrate formula (4) for the sand flux produced by the passage of one storm in our simulation and averaged it over all the domain (1000 km). In our simulations, storms are around 40 km width (in longitude) and travel around 200 km during around one day. Thus, we multiply the previous value by 5 (1000 km/200 km) to get the average impact of one storm. Finally, we divide this value by a period of half a Titan year. This yields to the mean sand flux produced by one storm per equinox (typically 0.15 m$^3$/m/year).
To obtain the total sand flux rose, we added the eastward and westward component of this sand flux, multiplied by the storm frequency per equinox, to the rose obtained with the GCM. In our mesoscale simulations, the eastward sand flux is around 2.7 times higher than the westward sand flux. The direction of surface winds produced by real storms is not likely to be purely west-east as in our 2D simulations. However, the favoured direction is eastward. Thus, the sand flux and the RDD are eastward on average.
To get more realistic figures, we added a small ad hoc angular dispersion, for the sand flux rose produced by storms: $Q\propto e^{-(\frac{\theta}{ 70^\circ})^2) }$ and $Q$ = 0 for $|\theta|$>20$^\circ$ where $\theta$ is the direction.
Fig. \ref{figure_s4} shows to the sand flux roses for different latitudes, averaging the effect of Saturn's eccentricity.
\\
\begin{figure}
\begin{center}
\includegraphics[width=5.2cm]{figure_s4a.pdf}
\includegraphics[width=5.2cm]{figure_s4b.pdf}
\includegraphics[width=5.2cm]{figure_s4c.pdf}
\end{center}
\caption{\textbf{Sand flux roses obtained by combining the GCM winds with winds produced during one typical gust front every equinox.} The arrows correspond to the resultant drift directions. a, b and c correspond to 0$^\circ$, 10$^\circ$ and 20$^\circ$ N latitudes, respectively, with GCM winds (speed increased by 20 $\%$) and a threshold of 0.04 m/s, and averaging the effect of Saturn's eccentricity.}
\label{figure_s4}
\end{figure}
\\
\textbf{Longitudinal dune extension rate}
The extension rate of a longitudinal dune can be expressed as:
\begin{equation}
c=Q_{\rm RDP} \times \frac{W}{S}
\end{equation}
\\
$c$ the extension rate (in m/s), $Q_{\rm RDP}$ is the norm of the mean flux, also called the resultant drift potential (in m$^2$/s), $W$ is the width of the dune (in m) and $S$ is the cross section (in m$^2$).
For a linear dune, $S=W \times H/2$ (i.e. the surface of an isocele triangle) where $H$ is the height of the dune. We have then $c=2Q_{\rm RDP}/H$.
For Titan's dunes, $H\approx$ 100 m. If we consider the impact of one storm every equinox, $Q_{\rm RDP}$ = 0.15 m$^2$/yr and the extension rate is around 3 mm/yr.
\\
\\
\textbf{Sand flux in southern Arabia and in Egypt}
To calculate the sand flux in Rub'al-Kali desert (Fig. 4b of the letter) and in the Great Sand Sea (Fig. \ref{figure_s6}), we use the wind outputs of the ERA-Interim-project, a reanalysis from the European Centre for Medium-Range Weather Forecasts \citep{simmons06}. We analyzed the 10 m wind data for the period between the 1/1/1979 and the 31/12/2012, at 18$^\circ$N and 48$^\circ$E for Rub'al-Kali desert, and at 25.5$^\circ$N and 26.25$^\circ$E for the Great Sand Sea. Mean sand flux roses (see Fig. 4b of the letter and Fig. \ref{figure_s6}) have been produced using these wind data and the formula (4). The mean sand flux direction (i.e. the RDD) is shown with an arrow and is parallel to the longitudinal dune orientation as predicted \citep{courrech14}.
\begin{figure}[!h]
\begin{center}
\includegraphics[width=8.04cm]{figure_s6a.pdf}
\includegraphics[width=8cm]{figure_s6b.pdf}
\end{center}
\caption{\textbf{Longitudinal dunes in Egypt and Algeria.} Map data: Google. (a) Dune field in Egypt (25.5$^\circ$N, 26.25$^\circ$E) with the sand flux roses calculated from winds at 10 m. (b) Dune field in Algeria (29.5$^\circ$ N, 5.5$^\circ$ E). The arrow corresponds to the resultant drift direction calculated with the sand flux rose.}
\label{figure_s6}
\end{figure}
\newpage
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|
{
"timestamp": "2015-04-15T02:03:38",
"yymm": "1504",
"arxiv_id": "1504.03404",
"language": "en",
"url": "https://arxiv.org/abs/1504.03404"
}
|
\section{\label{sec:intro}Introduction}
Any network found in the literature is inevitably just a sampled representative of its real-world analogue under study. For instance, many networks change quickly over time and in most cases merely incomplete data is available on the underlying system. Additionally, network sampling techniques are lately often applied to large networks to allow for their faster and more efficient analysis. Since the findings of the analyses and simulations on such sampled networks are implied for the original ones, it is of key importance to understand the structural differences between the original networks and their sampled variants.
A large number of studies on network sampling focused on the changes in network properties introduced by sampling. Lee~et~al.~\cite{LKJ06} showed that random node and link selection overestimate the scale-free exponent~\cite{BA99} of the degree and betweenness centrality~\cite{Fre77} distributions, while they preserve the degree mixing~\cite{Newman02}. On the other hand, random node selection preserves the degree distribution of different random graphs~\cite{SWM05} and performs better for larger sampled networks~\cite{SCBFGP12}. Furthermore, Leskovec~et~al.~\cite{LF06} showed that the exploration sampling using random walks or forest-fire strategy~\cite{LKF05} outperforms the random selection techniques in preserving the clustering coefficient~\cite{WS98}, different spectral properties~\cite{LF06}, and the in-degree and out-degree distributions. More recently, Ahmed~et~al.~\cite{ANK11} proposed random link selection with additional induction step, which notably improves on the current state-of-the-art. Their results confirm that the proposed technique well captures the degree distributions, shortest paths~\cite{WS98} and also the clustering coefficient of the original networks. Lately, different studies also focus on finding and correcting biases in sampling process, for example observing the changes of user attributes under the sampling of social networks~\cite{PM13}, analyzing the bias of traceroute sampling~\cite{LBCX03} and understanding the changes of degree distribution and hubs inclusion under various sampling techniques~\cite{MB11}. However, despite all those efforts, the changes in network structure introduced by sampling and the effects of network structure on the performance of sampling are still far from understood.
Real-world networks commonly reveal communities (also link-density community~\cite{LWZY07}), described as densely connected clusters of nodes that are loosely connected between~\cite{GN02}. Communities possibly play important roles in different real-world systems, for example in social networks communities represent friendship circles or people with similar interest~\cite{SC11}, while in citation networks communities can help us to reveal relationships between scientific disciplines~\cite{RB08}. Furthermore, community structure has a strong impact on dynamic processes taking place on networks~\cite{ADP06} and thus provides an important insight into structural organization and functional behavior of real-world systems. Consequently, a number of community detection algorithms have been proposed over the last years~\cite{WH04,RB07,RAK07,SB11} (for a review see~\cite{Fortunato10}). Most of these studies focus on classical communities characterized by higher density of edges~\cite{RCCLP04}. However, some recent works demonstrate that real-world networks reveal also other characteristic groups of nodes~\cite{NL07,PSR10} like groups of structurally equivalent nodes denoted modu\-les~\cite{NL07,SB12} (also link-pattern community~\cite{LWZY07} and other~\cite{RW07}), or different mixtures of communities and modules~\cite{SBB13}.
Despite community structure appears to be an intrinsic property of many real-world networks, only a few studies considered the effects between the community structure and network sampling. Salehi et al.~\cite{SRR12} proposed Page-Rank sampling, which improves the performance of sampling of networks with strong community structure. Furthermore, expansion sampling~\cite{MBT10} directly constructs a sample representative of the community structure, while it can also be used to infer communities of the unsampled nodes. Other studies, for example analyzed the evolution of community structure in collaboration networks and showed that the number of communities and their size increase over time~\cite{LLPP14}, while the network sampling has a potential application in testing for signs of preferential attachment in the growth of networks~\cite{perc14}. However, to the best of our knowledge, the question whether sampling destroys the structure of communities and other groups of nodes or are sampled nodes organized in a similar way than nodes in original network remains unanswered.
In this paper, we study the presence of characteristic groups of nodes in different social and information networks and analyze the changes in network group structure introduced by sampling. We consider six sampling techniques including random node and link selection, network exploration and expansion sampling. The results first reveal that nodes in social networks form densely linked community-like groups, while the structure of information networks is better described by modules. However, regardless of the type of the network and consistently across different sampling techniques, the structure of sampled networks exhibits much stronger characterization by community-like groups than the original networks. We therefore conclude that the rich community structure is not necessary a result of for example homophily in social networks.
The rest of the paper is structured as follows. In Section~\ref{sec:sampl}, we introduce different sampling techniques considered in the study, while the adopted node group extraction framework is presented in Section~\ref{sec:nodegroups}. The results of the empirical analysis are reported and formally discussed in Section~\ref{sec:analys}, while Section~\ref{sec:conclusion} summarizes the paper and gives some prominent directions for future research.
\section{\label{sec:sampl}Network sampling}
\begin{figure}[!t]
\centering
\subfigure[RND]{\includegraphics[width=0.3\columnwidth]{RND}\label{subfig:RND}} \quad
\subfigure[RLS]{\includegraphics[width=0.3\columnwidth]{RLS}\label{subfig:RLS}} \quad
\subfigure[RLI]{\includegraphics[width=0.3\columnwidth]{RLI}\label{subfig:RLI}}
\caption{Random selection techniques applied to a small toy network, where the samples are shown with highlighted nodes and links. \subref{subfig:RND}~In random node selection by degree, the nodes are selected with probability proportional to their degrees, while their mutual links are included in the sample. \subref{subfig:RLS}~In random link selection, the sample consists of links selected uniformly at random. \subref{subfig:RLI}~In random link selection with induction, the sample consists of randomly selected links (solid lines) and the links between their endpoints (dashed lines).}
\label{fig:selection}
\end{figure}
Network sampling techniques can be roughly divided into two categories: random selection and network exploration techniques. In the first category, nodes or links are included in the sample uniformly at random or proportional to some particular characteristic like the degree of a node or its PageRank score~\cite{BP98}. In the second category, the sample is constructed by retrieving a neighborhood of a randomly selected seed node using random walks, breadth-first search or another strategy. For the purpose of this study, we consider three techniques from each of the~categories.
\subsection{Random selection}
From the random selection category, we first adopt random node selection by degree~\cite{LF06} (RND). Here, the nodes are selected randomly with probability proportional to their degrees, while all their mutual links are included in the sample (Fig.~\ref{subfig:RND}). Note that RND improves the performance of the basic random node selection~\cite{LF06,BSB14}, where the nodes are selected to the sample uniformly at random. RND fits better spectral network properties~\cite{LF06} and produces the sample with larger weakly connected component~\cite{BSB14}. Moreover, it shows good performance in preserving the clustering coefficient and betweenness centrality distribution of the original networks~\cite{BSB14}. Nevertheless, it can still construct a disconnected sample network, despite a fully connected original network.
Next, we adopt random link selection~\cite{LF06} (RLS), where the sample consists of links selected uniformly at random (Fig.~\ref{subfig:RLS}). RLS overestimates degree and betweenness centrality exponent, underestimate the clustering coefficient and accurately matches the assortativity of the original network~\cite{LKJ06}. The samples created with RLS are sparse and the connectivity of the original network is not preserved, still RLS is likely to capture the path length of the original network~\cite{ANK12}.
Last, we adopt random link selection with induction~\cite{ANK11} (RLI), which improves the performance of RLS. In RLI, the sample consists of randomly selected links as before, while also all additional links between their endpoints (Fig.~\ref{subfig:RLI}). RLI outperforms several other methods in capturing the degree, path length and clustering coefficient distribution. It selects nodes with higher degree than RLS, thus the connectivity of the sample is increased~\cite{ANK11}.
Techniques from random selection category imitate classical statistical sampling approaches, where each individual is selected from population independently from others until desired size of the sample is reached.
\subsection{Network exploration}
\begin{figure}[t]
\centering
\subfigure[BFS]{\includegraphics[width=0.3\columnwidth]{BFS}\label{subfig:BFS}} \quad
\subfigure[FFS]{\includegraphics[width=0.3\columnwidth]{FFS}\label{subfig:FFS}} \quad
\subfigure[EXS]{\includegraphics[width=0.3\columnwidth]{EXS}\label{subfig:EXS}}
\caption{Network exploration techniques applied to a small toy network, where the samples are shown with highlighted nodes and links. \subref{subfig:BFS}~In breadth-first sampling, a seed node is first selected uniformly at random, while its broad neighborhood retrieved from breadth-first search is included in the sample. \subref{subfig:FFS}~In forest-fire sampling, the broad neighborhood of a randomly selected seed node is retrieved from partial breadth-first search, where only a fraction of neighbors is included in the sample. \subref{subfig:EXS}~In expansion sampling, the seed node is selected uniformly at random, while the remaining nodes are selected from the neighborhood of sampled nodes with probability proportional to their contribution to the expansion factor (see text).}
\label{fig:exploration}
\end{figure}
From the network exploration category, we first adopt breadth-first sampling~\cite{LKJ06} (BFS). Here, a seed node is selected uniformly at random, while its broad neighborhood retrieved from the basic breadth-first search is included in the sample (Fig.~\ref{subfig:BFS}). The sample network is thus a connected subgraph of the original network. BFS is biased towards selecting high-degree nodes in the sample~\cite{KMT10}. It captures well the degree distribution of the networks, while it performs worst in inclusion of hubs in the sample quickly in the sampling process~\cite{MB11}. BFS imitates the snowball sampling approach for collecting social data used especially when the data is difficult to reach~\cite{goodman61}. Selected seed participant is asked to report his friends, which are than invited to report their friends. The procedure is repeated until the desired number of people is sampled.
Next, we adopt a modification of BFS denoted forest-fire sampling~\cite{LF06} (FFS). In FFS, the broad neighborhood of a randomly selected seed node is retrieved from partial breadth-first search, where only some neighbors are included in the sample on each step (Fig.~\ref{subfig:FFS}). The number of neighbors is sampled from a geometric distribution with mean $p/(1-p)$, where $p$ is set to $0.7$~\cite{LF06}. FFS matches well spectral properties~\cite{LF06}, while it underestimates the degree distribution and fails to match the path length and clustering coefficient of the original networks~\cite{ANK12}. However, FFS corresponds to a model by which one author collects the papers to cite and include them in the bibliography~\cite{LKF05}. The author starts with one paper, explores its bibliography and selects the papers to cite. The procedure is recursively repeated in selected papers until desired collection of citations is reached.
Last, we adopt expansion sampling~\cite{MBT10} (EXS), where the seed node is again selected uniformly at random, while the neighbors of the sampled nodes are included in the sample with probability proportional to
\begin{equation}
1-\beta^{|N(\{v\})-(N(S) \cup S)|},
\end{equation}
where $v$ is the concerned node, $S$ the current sample and $N(S)$ the neighborhood of nodes in $S$ (Fig.~\ref{subfig:EXS}). Expression $|N(\{v\})-(N(S) \cup S)|$ denotes the expansion factor of node $v$ for sample $S$ and means the number of new neighbors contributed by $v$. The parameter $\beta$ is set to $0.9$~\cite{MBT10}. Note that EXS ensures that the sample consists of nodes from most communities in the original network and that the nodes that are grouped together in the original network, are also grouped together in the sample~\cite{GN02}. EXS imitates the modification of snowball sampling approach mentioned above, where for example we want to gather the data about individuals from different countries. Thus, on each step we include in the sample the individuals, which knows larger number of others from various countries.
\section{\label{sec:nodegroups}Group extraction}
The node group structure of different networks is explored by a group extraction framework~\cite{SBB13,ZLZ11,Weiss} with a brief overview below.
Let the network be represented by an undirected graph $G(V, L)$, where $V$ is the set of nodes and $L$ the set of links. Next, let $S$ be a group of nodes and $T$ a subset of nodes representing its corresponding linking pattern (i.e., the pattern of connections of nodes from $S$ to other nodes~\cite{NL07}), $S,T\subseteq V$. Denote $s=|S|$ and $t=|T|$. The linking pattern $T$ is selected to maximize the number of links between $S$ and $T$, and minimize the number of links between $S$ and $T^C$, while disregarding the links with both endpoints in $S^C$. For details on the group objective function see~\cite{SBB13,SZBB14}.
\begin{figure}[!t]
\centering
\subfigure[Community]{\includegraphics[width=0.3\columnwidth]{community}\label{subfig:comm}}\quad
\subfigure[Module]{\includegraphics[width=0.3\columnwidth]{module}\label{subfig:mod}}
\caption{Toy examples of groups of nodes in networks, where groups $S$ and their corresponding linking patterns $T$ are shown with highlighted and squared nodes, respectively (see text). \subref{subfig:comm}~Communities are densely connected groups of nodes with $S=T$. \subref{subfig:mod}~Modules are possibly disconnected groups of structurally equivalent nodes with $S\cap T=\emptyset$. Groups spanning between communities and modules are denoted mixtures.}
\label{fig:groups}
\end{figure}
The above formalism comprises different types of groups commonly analyzed in the literature (Fig.~\ref{fig:groups}). It consider communities~\cite{GN02} (i.e., link-density community~\cite{LWZY07}), defined as a (connected) group of nodes with more links toward the nodes in the group than to the rest of the network~\cite{RCCLP04}. Communities are characterized by $S=T$. Furthermore, the formalism consider possibly disconnected groups of structurally equivalent nodes denoted modules~\cite{NL07,SB12} (i.e., link-pattern community~\cite{LWZY07}), defined as a (possibly) disconnected group of nodes with more links towards common neighbors than to the rest of the network~\cite{RCCLP04}. Modules have $S \cap T = \emptyset$. Communities and modules represent two extreme cases with all other groups being the mixtures of the two~\cite{SBB13}, $S \cap T \subset S$ and/or $S \cap T \subset T$. The reader may also find it interesting that the core-periphery structure is a mixture with $S\subset T$, while the hub \& spokes structure is a module with $t=1$.
The type of group $S$ can in fact be determined by the Jaccard index~\cite{jaccard1901} of $S$ and its corresponding linking pattern $T$. The group parameter $\tau$~\cite{SBB13}, $\tau \in [0,1]$, is defined as
\begin{equation}
\tau(S,T)=\frac{|S \cap T|}{|S \cup T|}.
\end{equation}
Communities have $\tau = 1$, while modules are indicated by $\tau = 0$. Mixtures correspond to groups with $0 <\tau < 1$. For the rest of the paper, we refer to groups with $\tau\approx 1$ as community-like and groups with $\tau\approx 0$ as module-like.
Groups in networks are revealed by a sequential extraction procedure proposed in~\cite{ZLZ11,SBB13,Weiss}. One first finds the group $S$ and its linking pattern $T$ with random-restart hill climbing~\cite{RN03} that maximizes the objective function. Next, the revealed group $S$ is extracted from the network by removing the links between groups $S$ and $T$, and any node that becomes isolated. The procedure is then repeated on the remaining network until the objective function is larger than the $99$th percentile of the values obtained under the same framework in a corresponding Erd\H{o}s-R{\'e}nyi random graph~\cite{ER59}. All groups reported in the paper are thus statistically significant at $1\%$~level. Note that the above procedure allows for overlapping~\cite{PDFV05}, hierarchical~\cite{RSMOB02}, nested and other classes of groups.
\section{\label{sec:analys}Analysis and discussion}
Section~\ref{subsec:nets} introduces real-world networks considered in the study. Section~\ref{subsec:orig} reports the node group structure of the original networks extracted with the framework described in Section~\ref{sec:nodegroups}. The groups extracted from the sampled networks are analyzed in Section~\ref{subsec:sampled}. For a complete analysis, we also observe the node group structure of a large network with more than a million links in Section~\ref{subsec:large}.
\subsection{\label{subsec:nets}Network data}
\begin{table}[!t]
\scriptsize
\centering
\caption{Social and information networks considered in the study.}
\label{tbl:nets}
\begin{tabular}{clrr}
\hline\noalign{\smallskip}
\multicolumn{1}{c}{Network} & \multicolumn{1}{c}{Description} & \multicolumn{1}{c}{\# Nodes} & \multicolumn{1}{c}{\# Links} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
\textit{Collab} & High Energy Physics collaborations~\cite{LKF05} & $9877$ & $25998$ \\
\textit{PGP} & Pretty Good Privacy web-of-trust~\cite{BPDA04} & $10680$ & $24340$ \\
\noalign{\smallskip}\hline\noalign{\smallskip}
\textit{P2P} & Gnutella peer-to-peer file sharing~\cite{LKF05} & $8717$ & $31525$ \\
\textit{Citation} & High Energy Physics citations~\cite{LKF05} & $27770$ & $352807$ \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
The empirical analysis in the following sections was performed on four real-world social and information networks. Their main characteristics are shown in Table~\ref{tbl:nets}.
The \textit{Collab}~\cite{LKF05} is a social network of scientific collaborations among researchers, who submitted their papers to High Energy Physics -- Theory category on the arXiv in the period from January $1993$ to April $2003$. The nodes represent the authors, while undirected links denote that two authors co-authored at least one paper together.
The \textit{PGP}~\cite{BPDA04} is a social network, which corresponds to the interaction network of users of the Pretty Good Privacy algorithm collected in July $2001$. The nodes represent users, while undirected links indicate relationships between those, who sign each other's public key.
The \textit{P2P}~\cite{LKF05} is an information network, which contains a sequence of snapshots of the Gnu\-te\-lla peer-to-peer file sharing network collected in August $2002$. The nodes represent hosts in the Gnutella network, which are linked by undirected links if there exist connections between them.
The \textit{Citation}~\cite{LKF05} is an information network, again gathered from the High Energy Physics -- Theory category from the arXiv in the period from January $1993$ to April $2003$ and includes the citations among all papers in the dataset. The network consists of nodes, which represent papers, while links denote that one paper cite another.
\subsection{\label{subsec:orig}Group structure of original networks}
We first analyze the properties of groups extracted from the original networks summarized in Table~\ref{tbl:orig}.
The number of groups differs among networks, still the mean group size $s$ (denoted $\left<s\right>$) is comparable across network types. Groups $S$ in social networks consist of around $64$ nodes, while $\left<s\right>$ in information networks exceeds $150$ nodes. The mean linking pattern size $t$ (denoted $\left<t\right>$) of social networks is comparable to $\left<s\right>$. The latter relation $\left<t\right> \approx \left<s\right>$ is expected due to the pronounced community structure commonly found in social networks~\cite{NP03}. On the other hand, $\left<t\right> > \left<s\right>$ is expected for information networks, due to the abundance of module-like groups.
The characteristic group structure of networks is reflected in the group parameter $\tau$. For social networks, its values are around $0.556$, which indicates the presence of communities, modules and mixtures of these. In contrast to social networks, the information networks have $\tau$ closer to $0$ and consist mostly of module-like groups.
To summarize, social networks represent people and interactions between them, like a few authors writing a paper together, therefore we can expect a larger number of community-like groups in these networks. On the other hand, in information networks the homophily is less typical and thus the structure of these networks seem better described by module-like groups.
\begin{table}[!t]
\scriptsize
\centering
\caption{Groups of nodes extracted from social and information networks. We report the number of groups \#, the mean group size $s$, the mean pattern size $t$, the mean group parameter $\tau$, the median group parameter denoted $m_{\tau}$ and the distribution over different types of groups (see text). Notice that social networks consist of smaller groups with larger $\tau$ than information~networks.
}
\label{tbl:orig}
\begin{tabular}{crrrrrrrr}
\hline\noalign{\smallskip}
\multicolumn{1}{c}{Network} & \multicolumn{5}{c}{Group} & \multicolumn{1}{c}{Community} & \multicolumn{1}{c}{Mixture} & \multicolumn{1}{c}{Module}\\
& \multicolumn{1}{c}{\#} & \multicolumn{1}{c}{$\left\langle s \right\rangle$} & \multicolumn{1}{c}{$\left\langle t \right\rangle$} & \multicolumn{1}{c}{$\left\langle \tau \right\rangle$} & \multicolumn{1}{c}{$m_{\tau}$}& \multicolumn{3}{c}{Distribution \%} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
\textit{Collab} & $129$ & $66.9$ & $67.2$ & $0.568$ & $0.554$ & $1.6 \%$ & $96.8\%$ & $1.6\%$ \\
\textit{PGP} & $87$ & $62.2$ & $61.9$ & $0.568$ & $0.536$ & $4.6 \%$ & $94.3\%$ & $1.1\%$ \\
\noalign{\smallskip}\hline\noalign{\smallskip}
\textit{P2P} & $70$ & $154.8$ & $177.0$ & $0.057$ & $0.000$ & $0.0 \%$ & $44.3 \%$ & $55.7 \%$ \\
\textit{Citation} & $284$ & $271.7$ & $280.6$ & $0.186$ & $0.120$ & $0.0 \%$ & $96.8 \%$ & $3.2 \%$ \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
\subsection{\label{subsec:sampled}Group structure of sampled networks}
Sampling techniques outlined in Section~\ref{sec:sampl} enable setting the size of the sampled networks in advance. We consider sample sizes of $15\%$ of nodes from the original networks, that provides for an accurate fit of several network properties~\cite{LF06,BSB14}.
Table~\ref{tbl:samplsoc} and~\ref{tbl:samplinf} present the properties of the node group structure of sampled social and information networks, respectively. Notice that RLS and FFS show different performance than other techniques. The samples obtained with RLS and FFS contain less groups, which consist of no more than $36$ nodes. Additionally, almost all groups in these samples are modules, which reflects in the mean group parameter $\tau$ (denoted $\left<\tau\right>$) approaching $0$ for all networks.
To verify the above findings, we compute externally studentized residuals of the sampled networks that measure the consistency of each sampling technique with the rest. The residuals are calculated for each technique as the difference between the observed value of considered property and its mean divided by the standard deviation. The mean value and standard deviation are computed for all sampling techniques, excluding the observed one (for details see~\cite{SFB14}). Statistically significant inconsistencies between techniques are revealed by two-tailed Student $t-$test~\cite{CW82} at $P-$value of $0.1$, rejecting the null hypothesis that the values of the considered property are consistent across the sampling techniques.
Statistical comparison of sampling techniques for the number of groups and the mean group parameter $\tau$ is shown on Fig.~\ref{fig:stau}. We confirm that the samples obtained with RLS and FFS reveal significantly less groups with significantly smaller $\left<\tau\right>$ than other sampling techniques. Moreover, if we compare the number of links in the sampled networks, RLS and FFS create samples that contain on average $3\%$ of links from the original networks. In contrast, the samples obtained with RND, RLI, BFS and EXS consist of around $16\%$ of links from the original networks. As mentioned before, the sizes of all samples are $15\%$ of the original networks, thus the sampled networks obtained with RLS and FFS are much sparser than others. In addition, the performance of RLS and FFS can also be explained by their definition. Since in RLS we include only randomly selected links in the sample, the variance is very high, while it commonly contains a large number of sparsely linked components, whose structure is best described as module-like. On the other hand, the samples obtained with FFS consist of one connected component with a low average degree of $2.33$. Thus, the sparsely connected nodes also form groups, which are more similar to modules. Due to the above reasons, we exclude RLS and FFS from further analysis. We focus on RND, RLI, BFS, and EXS, whose performance is clearly more comparable.
The selected sampling techniques perform similarly ac\-ross all networks as shown in Table~\ref{tbl:samplsoc} for social and Table~\ref{tbl:samplinf} for information networks. The samples consist of various number of groups, still in most cases less than the original networks. The mean sizes $s$ and $t$ are around $40$, in contrast to groups with $143$ nodes on average in the original networks. Still, $\left<s\right> \approx \left<t\right>$ irrespective of network type and the sampling technique, which implies stronger characterization by community-like groups, as already argued in the case of social networks in Section~\ref{subsec:orig}.
\begin{table}[!t]
\scriptsize
\centering
\caption{Groups of nodes extracted from sampled social networks over $100$ realizations of different sampling techniques (see text). We report the number of groups \# and standard deviation, the mean group size $s$, the mean pattern size $t$, the mean group parameter $\tau$ and standard deviation, the median group parameter denoted $m_{\tau}$ and the distribution over different types of groups. Notice that sampled networks expectedly consist of smaller groups, but with larger $\tau$ than original social~networks (see $\left\langle \tau \right\rangle$ and $m_{\tau}$).}
\label{tbl:samplsoc}
\begin{tabular}{ccrrrrrrrrrr}
\hline\noalign{\smallskip}
\multicolumn{1}{c}{Network} & \multicolumn{1}{c}{Sampling} & \multicolumn{5}{c}{Group} & \multicolumn{1}{c}{Community} & \multicolumn{1}{c}{Mixture} & \multicolumn{1}{c}{Module}\\
& & \multicolumn{1}{c}{\#} & \multicolumn{1}{c}{$\left\langle s \right\rangle$} & \multicolumn{1}{c}{$\left\langle t \right\rangle$} & \multicolumn{1}{c}{$\left\langle \tau \right\rangle$} & \multicolumn{1}{c}{$m_{\tau}$} & \multicolumn{3}{c}{Distribution \%} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
\multirow{7}{*}{\textit{Collab}} & / & \multicolumn{1}{l}{$129.0$} & $66.9$ & $67.2$ & \multicolumn{1}{l}{$0.568$} & $0.554$ & $1.6 \%$ & $96.8 \%$ & $1.6 \%$\\\noalign{\smallskip}\cline{2-10}\noalign{\smallskip}
& RND & $65.4\pm3.7$ & $13.5$ & $13.7$ & $0.851\pm0.030$& $0.989$ & $54.7 \%$ & $41.9 \%$ & $3.4 \%$ \\
& RLS & $1.2\pm0.5$ & $1.5$ & $4.8$ & $0.047\pm0.149$ & $0.048$ & $0.0 \%$ & $8.3 \%$ & $91.7 \%$ \\
& RLI & $74.7\pm4.6$ & $13.7$ & $13.9$ & $0.846\pm0.030$ & $0.979$ & $52.7 \%$ & $43.4 \%$ & $3.9 \%$ \\
& BFS & $104.0\pm6.5$ & $18.2$ & $18.5$ & $0.787\pm0.032$ & $0.861$ & $30.3 \%$ & $66.5 \%$ & $3.2 \%$ \\
& FFS & $4.0\pm1.6$ & $16.8$ & $29.8$ & $0.000\pm0.000$ & $0.000$ & $0.0 \%$ & $0.0 \%$ & $100.0 \%$ \\
& EXS & $87.0\pm5.8$ & $18.4$ & $18.9$ & $0.741\pm0.026$ & $0.791$ & $21.4 \%$ & $76.3 \%$ & $2.3 \%$ \\
\noalign{\smallskip}\hline\noalign{\smallskip}
\multirow{7}{*}{\textit{PGP}} & / & \multicolumn{1}{l}{$\phantom{7}87.0$} & $62.2$ & $61.9$ & \multicolumn{1}{l}{$0.568$} & $0.536$ & $4.6 \%$ & $94.3 \%$ & $1.1 \%$ \\\noalign{\smallskip}\cline{2-10}\noalign{\smallskip}
& RND & $68.2\pm4.5$ & $15.8$ & $16.0$ & $0.891\pm0.024$ & $1.000$ & $67.8 \%$ & $28.7 \%$ & $3.5 \%$ \\
& RLS & $2.8\pm1.0$ & $5.7$ & $7.6$ & $0.304\pm0.233$ & $0.263$ & $21.4 \%$ & $28.6 \%$ & $50.0 \%$ \\
& RLI & $74.3\pm4.3$ & $15.8$ & $16.1$ & $0.883\pm0.024$ & $1.000$ & $65.1 \%$ & $31.1 \%$ & $3.8 \%$ \\
& BFS & $95.4\pm9.2$ & $17.5$ & $17.7$ & $0.784\pm0.025$ & $0.909$ & $39.2 \%$ & $55.6 \%$ & $5.2 \%$ \\
& FFS & $3.6\pm1.3$ & $13.5$ & $32.6$ & $0.000\pm0.000$ & $0.000$ & $0.0 \%$ & $0.0 \%$ & $100.0 \%$ \\
& EXS & $80.9\pm6.5$ & $15.6$ & $15.8$ & $0.779\pm0.028$ & $0.873$ & $34.5 \%$ & $61.2 \%$ & $4.3 \%$ \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
\begin{table}[!t]
\scriptsize
\centering
\caption{Groups of nodes extracted from sampled information networks over $100$ realizations of different sampling techniques (see text). We report the number of groups \# and standard deviation, the mean group size $s$, the mean pattern size $t$, the mean group parameter $\tau$ and standard deviation, the median group parameter denoted $m_{\tau}$ and the distribution over different types of groups. Notice that sampled networks expectedly consist of smaller groups, but with larger $\tau$ than original information~networks (see $\left\langle \tau \right\rangle$ and $m_{\tau}$).}
\label{tbl:samplinf}
\begin{tabular}{ccrrrrrrrrr}
\hline\noalign{\smallskip}
\multicolumn{1}{c}{Network} & \multicolumn{1}{c}{Sampling} & \multicolumn{5}{c}{Group} & \multicolumn{1}{c}{Community} & \multicolumn{1}{c}{Mixture} & \multicolumn{1}{c}{Module} \\
& & \multicolumn{1}{c}{\#} & \multicolumn{1}{c}{$\left\langle s \right\rangle$} & \multicolumn{1}{c}{$\left\langle t \right\rangle$} & \multicolumn{1}{c}{$\left\langle \tau \right\rangle$} & \multicolumn{1}{c}{$m_{\tau}$} & \multicolumn{3}{c}{Distribution \%} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
\multirow{7}{*}{\textit{P2P}} & / & \multicolumn{1}{l}{$\phantom{7}70.0$} & $154.8$ & $177.0$ & \multicolumn{1}{l}{$0.057$} & $0.000$ & $0.0 \%$ & $44.3 \%$ & $55.7 \%$ \\\noalign{\smallskip}\cline{2-10}\noalign{\smallskip}
& RND & $23.3\pm3.9$ & $24.2$ & $24.4$ & $0.163\pm0.049$ & $0.034$ & $4.2 \%$ & $45.8 \%$ & $50.0 \%$ \\
& RLS & $1.6\pm0.9$ & $1.2$ & $3.6$ & $0.000\pm0.008$ & $0.000$ & $0.0 \%$ & $0.0 \%$ & $100.0 \%$ \\
& RLI & $26.2\pm4.4$ & $27.5$ & $28.1$ & $0.161\pm0.039$ & $0.035$ & $3.8 \%$ & $48.8 \%$ & $47.4 \%$ \\
& BFS & $34.1\pm5.5$ & $31.3$ & $27.9$ & $0.131\pm0.042$ & $0.034$ & $2.3 \%$ & $50.7 \%$ & $47.0 \%$ \\
& FFS & $3.6\pm1.4$ & $17.8$ & $28.3$ & $0.000\pm0.000$ & $0.000$ & $0.0 \%$ & $0.0 \%$ & $100.0 \%$ \\
& EXS & $34.0\pm5.9$ & $36.9$ & $37.3$ & $0.125\pm0.030$ & $0.035$ & $2.4 \%$ & $53.8 \%$ & $43.8 \%$ \\
\noalign{\smallskip}\hline\noalign{\smallskip}
\multirow{7}{*}{\textit{Citation}} & / & \multicolumn{1}{l}{$284.0$} & $271.7$ & $280.6$ & \multicolumn{1}{l}{$0.186$} & $0.120$ & $0.0 \%$ & $96.8 \%$ & $3.2 \%$ \\\noalign{\smallskip}\cline{2-10}\noalign{\smallskip}
& RND & $121.4\pm4.9$ & $74.9$ & $78.1$ & $0.405\pm0.016$ & $0.329$ & $0.2 \%$ & $80.9 \%$ & $18.9 \%$ \\
& RLS & $1.5\pm1.2$ & $1.4$ & $15.3$ & $0.014\pm0.073$ & $0.014$ & $0.0 \%$ & $0.0 \%$ & $100.0 \%$ \\
& RLI & $124.8\pm5.5$ & $76.3$ & $79.9$ & $0.415\pm0.014$ & $0.344$ & $0.2 \%$ & $82.6 \%$ & $17.2 \%$ \\
& BFS & $120.4\pm7.1$ & $99.2$ & $100.9$ & $0.359\pm0.047$ & $0.244$ & $0.1 \%$ & $77.5 \%$ & $22.4 \%$ \\
& FFS & $10.6\pm4.2$ & $35.5$ & $30.0$ & $0.000\pm0.000$ & $0.000$ & $0.0 \%$ & $0.0 \%$ & $100.0 \%$ \\
& EXS & $131.2\pm6.0$ & $91.4$ & $95.4$ & $0.388\pm0.019$ & $0.284$ & $0.2 \%$ & $82.0 \%$ & $17.8 \%$ \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
Indeed, the majority of groups found in sampled social networks are community-like, which reflects in the parameter $\tau > 0.7$. In sampled information networks the number of mixtures decreases and communities appear, thus $\tau$ is larger than in the original networks. Fig.~\ref{fig:tau-hists} shows a clear difference in the distribution of $\tau$ between the original and sampled networks. Furthermore, to confirm that differences exist between the structure of the original and sampled networks, we compute externally studentized residuals, where we include the value of considered property of the original network in computing the mean over different sampling techniques. We compare the number of groups and the parameter $\left<\tau\right>$ for the original networks and their samples (Fig.~\ref{fig:stauo}). The results prove that the original networks contain a significantly larger number of groups with significantly smaller $\left<\tau\right>$ than the sampled networks. Yet, larger parameter $\tau$ and consequently more community-like groups in sampled social networks and less module-like groups in sampled information networks indicate clear changes in the network structure introduced by sampling. We conclude that these changes occur regardless of the network type or the adopted sampling technique.
\begin{figure}[!t]
\centering
\subfigure[Number of groups \#]{\includegraphics[width=0.45\columnwidth]{studentized-nr}\label{subfig:nr}} \quad
\subfigure[Group parameter $\left< \tau \right>$]{\includegraphics[width=0.45\columnwidth]{studentized-tau}\label{subfig:tau}} \\
\caption{Statistical comparison of \subref{subfig:nr} number of groups and \subref{subfig:tau} mean group parameter $\tau$ for the sampled networks obtained with different sampling techniques (see text). We show externally studentized residuals that measure the consistency of each sampling technique with the rest and expose statistically significant inconsistencies between the techniques with two-tailed Student $t$-test at $P\mbox{-value}$ of $0.1$ (shaded regions correspond to $90\%$ confidence intervals). Notice that sampled networks obtained with RLS and FFS reveal less groups (see \subref{subfig:nr}) with significantly smaller parameter $\tau$ (see \subref{subfig:tau}) than other sampling techniques.
}
\label{fig:stau}
\end{figure}
\begin{figure}[!t]
\centering
\subfigure[\textit{Collab} network]{\includegraphics[width=0.45\columnwidth]{hist-collab}\label{subfig:tca}} \quad
\subfigure[\textit{PGP} network]{\includegraphics[width=0.45\columnwidth]{hist-pgp}\label{subfig:tpgp}} \\
\subfigure[\textit{P2P} network]{\includegraphics[width=0.45\columnwidth]{hist-p2p}\label{subfig:tp2p}} \quad
\subfigure[\textit{Citation} network]{\includegraphics[width=0.45\columnwidth]{hist-citation}\label{subfig:tcit}}
\caption{Distributions of group parameter $\tau$ for the original networks and their sampled representatives obtained with selected sampling techniques (see text). Histograms are derived by standard equidistant binning, while the estimates of a beta distribution for the original (solid lines) and sampled networks (dashed lines) are merely a guide for the eye. Notice that sampled networks are characterized by denser groups with notably larger $\tau$ than the original ones. Groups are more community-like in the case of social networks (see \subref{subfig:tca} and \subref{subfig:tpgp}), while less module-like in the case of information networks (see \subref{subfig:tp2p} and~\subref{subfig:tcit}).
}
\label{fig:tau-hists}
\end{figure}
Notice that the largest $\tau$ and thus the strongest characterization by community-like groups is revealed in the sampled networks obtained with both random selection techniques, RND and RLI. In RND nodes with higher degrees are more likely to be selected to the sample by definition, while RLI is biased in a similar way~\cite{ANK11}. Thus, densely connected groups of nodes have a higher chance of being included in the sampled network, while sparse parts of the networks remain unsampled. On the other hand, BFS and EXS sample the broad neighborhood of a randomly selected seed node and thus the sampled network represents a connected component. In the case of BFS, all nodes and links of some particular part of the original network are sampled. The latter is believed to be representative of the entire network~\cite{KMT10}, yet BFS is biased towards sampling nodes with higher degree~\cite{NW01} and overestimates the clustering coefficient, especially in information networks~\cite{LKJ06}. On the other hand, EXS ensures the smallest partition distance among several other sampling techniques, which means that nodes grouped together in communities of sampled network are also in the same community in the original network~\cite{MBT10}. Therefore, the stronger characterization by community-like groups in sampled networks can also be explained by the definition and behavior of the sampling techniques.
\begin{figure}[!t]
\centering
\subfigure[Number of groups \#]{\includegraphics[width=0.45\columnwidth]{studentized-nr-org}\label{subfig:nro}} \quad
\subfigure[Group parameter $\left\langle \tau \right\rangle$]{\includegraphics[width=0.45\columnwidth]{studentized-tau-org}\label{subfig:tauo}} \\
\caption{Statistical comparison of \subref{subfig:nro} number of groups and \subref{subfig:tauo} mean group parameter $\tau$ for the original networks and their sampled representatives obtained with selected sampling techniques (see text). We show externally studentized residuals that measure the consistency of each network with the rest and expose statistically significant inconsistencies between the networks with two-tailed Student $t$-test at $P\mbox{-value}$ of $0.1$ (shaded regions correspond to $90\%$ confidence intervals). Notice that original networks reveal more groups (see \subref{subfig:nro}) with significantly smaller parameter $\tau$ (see \subref{subfig:tauo}) than the sampled networks.
}
\label{fig:stauo}
\end{figure}
\subsection{\label{subsec:large}Group structure of a large network}
Due to the relatively high time complexity of the node group extraction framework, we consider only networks with a few thousand nodes. However, our previous study~\cite{BSB14} proved that the size of the original network does not affect the accuracy of the sampling. Still, for a complete analysis, we also inspect the changes in node group structure introduced by sampling of a large \textit{NotreDame} network with more than a million links. Due to the simplicity and execution time, we present the analysis for two sampling techniques, RND from random selection and BFS from network exploration category. We also limit the number of groups extracted from the networks to $100$ (i.e., we consider top $100$ most significant groups with respect to the objective function).
The \textit{NotreDame} data are collected from the web pages of the University of Notre Dame -- \textit{nd.edu} domain in $1999$. The network contains $325$,$729$ nodes representing individual web pages, while $1$,$497$,$134$ links denote hyperlinks among them.
Table~\ref{tbl:wnd} shows the properties of groups, found in the original and sampled networks. The samples consist of smaller groups, still the mean size $s$ remains larger than the mean size $t$. The majority of groups extracted from the original network are module-like, which reflects in the parameter $\tau$ slightly larger than $0$. On the other hand, the changes introduced by sampling are clear, since the samples contain less modules, which is revealed by a larger parameter $\tau$. These findings are consistent with the results on smaller networks from previous sections. The \textit{NotreDame} as an information network expectedly consists of densely linked groups similar to modules, while the structure of sampled networks exhibits stronger characterization by community-like groups. That is again irrespective of the adopted sampling technique.
\begin{table}[!t]
\scriptsize
\centering
\caption{Groups of nodes extracted from the original \textit{NotreDame} network and its sampled representatives over $100$ realizations of selected sampling techniques (see text). We report the number of groups \#, the mean group size $s$, the mean pattern size $t$, the mean group parameter $\tau$ and standard deviation, the median group parameter denoted $m_{\tau}$ and the distribution over different types of groups. Notice that sampled networks expectedly consist of smaller groups, but with larger $\tau$ than original network (see $\left\langle \tau \right\rangle$ and $m_{\tau}$).}
\label{tbl:wnd}
\begin{tabular}{crrrrrrrr}
\hline\noalign{\smallskip}
\multicolumn{1}{c}{Sampling} & \multicolumn{5}{c}{Group} & \multicolumn{1}{c}{Community} & \multicolumn{1}{c}{Mixture} & \multicolumn{1}{c}{Module}\\
& \multicolumn{1}{c}{\#} & \multicolumn{1}{c}{$\left\langle s \right\rangle$} & \multicolumn{1}{c}{$\left\langle t \right\rangle$} & \multicolumn{1}{c}{$\left\langle \tau \right\rangle$} & \multicolumn{1}{c}{$m_{\tau}$}& \multicolumn{3}{c}{Distribution \%} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
/ & $100$ & $876.8$ & $403.6$ & \multicolumn{1}{l}{$0.030$} & $0.028$ & $0.0 \%$ & $99.0 \%$ & $1.0 \%$ \\
\noalign{\smallskip}\hline\noalign{\smallskip}
RND & $100$ & $302.5$ & $271.7$ & $0.369\pm0.010$ & $0.364$ & $0.0 \%$ & $ 100.0\%$ & $ 0.0\%$ \\
BFS & $100$ & $411.6$ & $251.7$ & $0.135\pm0.030$ & $0.119$ & $0.0 \%$ & $ 99.5\%$ & $ 0.5\%$ \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
\section{\label{sec:conclusion}Conclusion}
In this paper, we study the presence of characteristic groups of nodes like communities and modules in different social and information networks. We observe the groups of the original networks and analyze the changes in the group structure introduced by the network sampling.
The results first reveal noticeable differences in the group structure of original social and information networks. Nodes in social networks form smaller community-like groups, while information networks are better characterized by larger modules. After applying network sampling techniques, sampled networks expectedly contain fewer and smaller groups. However, the sampled networks exhibit stronger characterization by community-like groups than the original networks. We have shown that the changes in the node group structure introduced by sampling occur regardless of the network type and consistently across different sampling techniques. Since networks commonly considered in the literature are inevitably just a sampled representative of its real-world analogue, some results, such as rich community structure found in these networks, may be influenced by or are merely an artifact of sampling.
Our future work will mainly focus on larger real-world networks, including other types of networks like biological and technological. Moreover, we will further analyze the changes in the node group structure introduced by sampling and explore techniques that could overcome observed deficiencies.
\section*{Acknowledgment}
This work has been supported in part by the Slovenian Research Agency \textit{ARRS} within the Research Program No. P2-0359, by the Slovenian Ministry of Education, Science and Sport Grant No. 430-168/2013/91, and by the European Union, European Social Fund.
\bibliographystyle{elsarticle-num}
|
{
"timestamp": "2015-04-14T02:12:20",
"yymm": "1504",
"arxiv_id": "1504.03097",
"language": "en",
"url": "https://arxiv.org/abs/1504.03097"
}
|
\section{Introduction}
\emph{Cluster analysis} is a fundamental task in data analysis that aims to partition a set of objects into maximal subsets (called \emph{clusters}) of similar objects. In \emph{graph clustering}, the objects to be clustered are the vertices of a graph and the edges of a graph describe relations between them. These relations may have interpretations for data analysis. For example, if the graph is the friendship graph of a social network, i.e., the vertices are the users of a social network and the edges correspond to friendship relations, edges may indicate that the users are socially related and/or have similar interests. In a co-author graph, where the vertices are authors and edges describe co-authorships, edges may be interpreted as a sign that the authors work in the same scientific community. A \emph{cluster} is then a maximal subset of vertices that are \emph{well-connected} to each other, where the precise meaning of being well-connected can be defined in various ways.
In many cases, once we know the interpretation of a single edge, there is a natural interpretation of clusters. For example, clusters in a friendship graph correspond to social groups or clusters in a co-author graph correspond to scientific communities. For similar reasons, a vast amount of graph clustering methods are applied to many different kinds of social/information/biological networks to reveal hidden cluster structure, etc. (see, e.g., surveys \cite{For10:community,POM09:communities,Sch07:clustering}).
Many efficient algorithms for finding clusters in a graph have been developed. However, with the increasing focus on the study of very large networks, we have to concentrate on new features of the clustering algorithms. For example, if one tries to find clusters in the World Wide Web or in a big social network, even linear time algorithms might be too slow. This is particularly important if one wants to study the temporal development of the clusters, which require to solve the problem on many instances (each for a different point of time). In such cases, we need \emph{sublinear time algorithms}. We develop such an algorithm in this paper. Our algorithm can be used to test, if a given graph has a cluster structure, i.e., is composed of at most $k$ clusters.
We will develop the algorithm in the framework of \emph{Property Testing} for bounded degree graphs \cite{GR02:testing}. In this framework, an algorithm has oracle access to an undirected graph $G=(V,E)$ with a bound $d$ on the maximum degree, with $d$ typically assumed to be constant. An algorithm is called a \emph{property tester for a given property $\Pi$} (in our case, the property of all graphs that have a cluster structure with at most $k$ clusters), if it accepts with probability at least $\frac23$ every graph that has the property $\Pi$ and rejects with probability at least $\frac23$ every graph that is \emph{$\varepsilon$-far from $\Pi$}. Here the notion of $\varepsilon$-far means that one has to change more than $\varepsilon dn$ edges to obtain a graph of maximum degree $d$ that has property $\Pi$. If $G$ is not $\varepsilon$-far from $\Pi$, then it is called \emph{$\varepsilon$-close}. To give a property tester on a bounded degree graph $G$, we assume that $G$ is given as an oracle, which allows us to perform \textit{neighbor queries} to $G$ such that for any input pair $(v,i)$, the oracle returns the $i$th neighbor of vertex $v$ if $i \le d_G(v)$, and a special symbol if $i > d_G(v)$, where $d_G(v)$ is the degree of $v$. This framework of graph property testing was initiated by Goldreich and Ron \cite{GR02:testing}. In this model, it is known that several properties are testable in constant time, such as hyperfinite properties \cite{NS13:hyperfinite} (see also \cite{CGRSSS14,GR02:testing} and the references therein). We now also know that properties such as bipartiteness \cite{GR98:sublinear} and expansion \cite{CS10:expansion,GR00:expansion,KS11:expansion,NS10:expansion} are testable in time $\widetilde{O}(\sqrt{n})$, with a nearly matching lower bound, and we need to perform at least $\Omega(n)$ queries to test $3$-colorability \cite{GR02:testing}. For more results, see recent surveys \cite{Gol11:testing,Ron10:testing}.
There are several ways to assess the cluster structure of a graph, such as $k$-means, cliques, modularity etc. One typically would want to argue that vertices in the same cluster should be well-connected and vertices from different clusters should be poorly-connected. In this paper, we use the concept of \textit{conductance} to measure the quality of the cluster structure of a graph. Given a graph $G = (V,E)$ with maximum degree bounded by $d$, and a subset $S\subseteq V$, the \emph{conductance of $S$} is defined as $\phi_G(S) := \frac{e(S, V \setminus S)}{d|S|}$, where $e(S, V \setminus S)$ denotes the number of edges coming out of $S$. Note that $\phi_G(V) = 0$. The \emph{conductance of the graph $G$}, denoted as $\phi(G)$, is defined as the minimum conductance value over all possible subsets $S$ of $V$ with $|S| \le |V|/2$. (For convenience, we define $\phi(G)=\frac{1}{d}$ if $G$ is the singleton graph, that is, the graph consisting of a single isolated vertex with no edges.) For any $S\subseteq V$, let $G[S]$ be the induced subgraph of $G$ on the vertex set $S$. Define the \textit{inner conductance} of $S$ to be the conductance of subgraph $G[S]$, namely, $\phi(G[S])$. To avoid confusion, we will also call the conductance $\phi_G(S)$ of $S$ in $G$ the \textit{outer conductance} of $S$.
Kannan et al.\ \cite{KVV04:clustering} introduced conductance as an appropriate measure of the quality of a cluster and this notion has been later used in numerous more applied works (see, e.g., \cite{Sch07:clustering}).
Further intuition has been employed to assert that a set $S$ with small outer conductance has few connections to the outside of $S$, and a graph $G$ with large conductance means that the vertices of $G$ are well-connected with each other. Following this intuition, Oveis Gharan and Trevisan \cite{OT14:expander} and Zhu et al.\ \cite{ZLM13:local} proposed to combine both outer conductance and inner conductance of a set $S$ to measure whether $S$ is a good cluster or not. That is, a set $S$ is considered to be a good cluster if $\phi_G(S)$ is small and $\phi(G[S])$ is large. In \cite{OT14:expander}, a graph $G$ is defined to be clusterable if $G$ can be partitioned into a number of disjoint parts so that each of them is such a good cluster. In this paper, we will use a related definition to characterize graphs with cluster structure.
\subsection{Our results}
\label{subsec:result}
We begin with the formal definition characterizing graphs with a cluster structure and state our main results. The following definition is inspired by the work of Oveis Gharan and Trevisan \cite{OT14:expander}.
\begin{definition}
\label{def:clusterable-1}
For a $d$-degree bounded undirected graph $G = (V,E)$ with $n$ vertices and parameters $k,\phi,\varepsilon$, we define $G$ to be \emph{$(k,\phi)$-clusterable} if there exists a partition of $V$ into $h$ sets $C_1, \dots, C_h$ such that $1 \le h \le k$, and for each $i$, $1 \le i \le h$, $\phi(G[C_i]) \ge \phi$ and $\phi_G(C_i) \le c_{d,k} \varepsilon^4 \phi^2$, where for fixed $d,k$, $c_{d,k}$ is a universal constant. We call each $C_i$ a \emph{$\phi$-cluster} and the corresponding $h$-partition an \emph{$(h,\phi)$-clustering}.
\end{definition}
The above definition formalizes the idea that the existence of an edge is an indicator that two vertices are similar, i.e., two persons are friends or two authors belong to the same scientific community, while the lack of an edge is a (weaker) sign of the opposite statement. Therefore, a cluster should be, intuitively, well-connected in the inside and poorly-connected to the outside. (We remark that the gap between the conductance of $C_i$ and $G[C_i]$ in Definition \ref{def:clusterable-1} is a feature of our approach rather than an inherent property of the problem.)
In this paper, we develop an algorithm that with probability at least $\frac23$, accepts every $(k,\phi)$-clusterable graph and rejects every graph that is $\varepsilon$-far from every $(k,\phi^*)$-clusterable graph, where $\phi^*=O_{d,k}(\frac{\phi^2\varepsilon^4}{\log n})$. (Throughout the paper we use the notation $O_{d,k}()$ to describe a function in the Big-Oh notation assuming that $d$ and $k$ are constant.)
Our main result is that we can distinguish such a clusterable graph from all graphs that are far from being clusterable
in sublinear time.
\begin{theorem}
\label{thm:main}
Let $c'_{d,k}$ be a suitable constant depending on $d$ and $k$. There exists an algorithm that accepts every $(k, \phi)$-clusterable graph of maximum degree at most $d$ with probability at least $\frac23$, and rejects every graph of maximum degree at most $d$ that is $\varepsilon$-far from being $(k, \phi^*)$-clusterable with probability at least $\frac23$, if $\phi^* \le c'_{d,k} \frac{\phi^2 \varepsilon^4}{\log n}$. The running time of the algorithm is $\frac{\sqrt{n}}{\phi^2}(k \log n / \varepsilon)^{O(1)}$.
\end{theorem}
One can question whether the gap between $\phi^*$ and $\phi$ in the form $\phi^* = O_{d,k}(\frac{\phi^2 \varepsilon^4}{\log n})$ or similar is really required. We believe that for an algorithm with a somewhat similar time complexity, both the $\log n$ and the $\varepsilon$ factors in the gap between $\phi$ and $\phi^*$ are necessary. For further discussion about this gap size we refer to Section \ref{subsection:Expansion}.
Note also that in our results we allow for clusterings with at most $k$ clusters (rather than with exactly $k$ clusters). This can be justified by the fact that in the property testing framework, every $(k,\phi)$-clusterable graph with exactly $h \le k$ clusters is $\varepsilon$-close to some $(k,\phi^*)$-clusterable graph with exactly $k$ clusters, for any reasonable choice of parameters (one can simply remove all edges that are incident to $k-h$ vertices).
\subsection{Comparison with testing expansion and discussion of the gap size}
\label{subsection:Expansion}
For $k=1$, our problem is equivalent to that of \emph{testing graph expansion}, the problem which has received significant attention in the past.
Goldreich and Ron \cite{GR00:expansion} were the first to study this problem in details and proved a lower bound $\Omega(\sqrt{n})$ on the number of queries for testing graph expansion in the bounded degree model. This result has been complemented by a proposed algorithm, which Goldreich and Ron conjectured to be a property tester for the second largest eigenvalue (denoted by $\eta_2$) of the normalized adjacency matrix of a regularized version of the graph, in the sense that it accepts every graph with $\eta_2 \le \eta$ and rejects every graph that is $\varepsilon$-far from having $\eta_2 \le \eta^{\Theta(\mu)}$ for any $\mu > 0$. Note that by Cheeger's inequality (cf. Theorem \ref{thm:cheeger}), resolving of this conjecture would imply that the algorithm is also a property tester
that accepts any graph with $\phi(G) \ge \phi$ and rejects every graph that is $\varepsilon$-far from being a $\phi^*$-expander for $\phi^* = O(\mu \phi^2)$, where a graph $G$ is called a $\phi$-expander if $\phi(G)\ge \phi$.
Czumaj and Sohler \cite{CS10:expansion} proved a weaker version of this conjecture by showing that the algorithm from \cite{GR00:expansion} can distinguish in time $\widetilde{O}(\sqrt{n})$ any $\phi$-expander graph from graphs that are $\varepsilon$-far from being a $\phi^*$-expander for $\phi^* = O(\frac{\phi^2}{\log n})$. Kale and Seshadhri \cite{KS11:expansion} and Nachmias and Shapira \cite{NS10:expansion} extended this result and proved that in $\widetilde{O}(n^{0.5+\mu})$ time the algorithm accepts graphs with expansion $\phi$ and reject graphs which are $\varepsilon$-far from having expansion $\phi^*= O(\mu\phi^2)$.
Since the best known methods require a gap between $\phi$ and $\phi^*$ already for the special case $k=1$, it is clear that our work will also need a similar gap. It seems to be tempting to conjecture that --- similarly to the case of testing expansion --- it will suffice to reject (in the soundness) graphs that are $\varepsilon$-far from being $(k,\Theta(\mu\phi^2))$-clusterable for any $\mu>0$, instead of having a $\log n$ factor dependency between $\phi$ and $\phi^*$, as in our result. However, we do not think that this is possible and in the following we briefly sketch the differences from testing expansion and argue why the approach that led to a better gap for testing expansion is likely to fail (of course, this does not rule out other approaches, but this points to substantial obstacles to obtain an improved result).
Let $u,v$ be any two vertices in the graph $G$, which for simplicity is now assumed to be $d$-regular and connected. Let $\lambda_i$ be the $i$-th smallest eigenvalue and $\textbf{v}_i$ be the corresponding eigenvector of the (normalized) Laplacian of $G$. It is known that the lazy random walk on $G$ converges to the uniform distribution on its end-vertex. One can write (cf. Section \ref{sec:proofs} for details) the $l_2^2$-distance between the distribution $\textbf{p}_v^\ell$ and $\textbf{p}_u^\ell$ of the endpoints of the lazy random walks on $G$ of length $\ell$ starting at $v$ and $u$, respectively, as
\begin{displaymath}
\norm{\textbf{p}_v^\ell - \textbf{p}_u^\ell}_2^2
=
\sum_{i=1}^n(\textbf{v}_i(u)-\textbf{v}_i(v))^2(1-\frac{\lambda_i}{2})^{2\ell}
\enspace.
\end{displaymath}
Since a lazy random walk on a regular graph converges to the uniform distribution, we have $\textbf{v}_1(u) = \textbf{v}_1(v) = 1/\sqrt{n}$. Therefore, in the case $k=1$, by the fact that $0 = \lambda_1 \le \lambda_2 \le \dots \le \lambda_n \le 1$, we can upper bound $\norm{\textbf{p}_v^\ell - \textbf{p}_u^\ell}_2^2$ by bounding the second smallest eigenvalue and by making a proper choice of the length of the walk $\ell$.
If we want to extend this approach to $k > 1$, then our definition implies (cf. \cite{LOT12:high}) that in a $(k,\phi)$-clusterable graph there is a significant gap between $\lambda_{h}$ and $\lambda_{h+1}$ for some $h$, $1 \le h \le k$, where $h$ corresponds to the number of clusters in the instance. Now, assume for simplicity that $h=k$. Then we obtain that
\begin{displaymath}
\norm{\textbf{p}_v^\ell - \textbf{p}_u^\ell}_2^2
=
\sum_{i=1}^k(\textbf{v}_i(u)-\textbf{v}_i(v))^2(1-\frac{\lambda_i}{2})^{2\ell}
+ \sum_{i=k+1}^n(\textbf{v}_i(u)-\textbf{v}_i(v))^2(1-\frac{\lambda_i}{2})^{2\ell}
\enspace.
\end{displaymath}
We can upper bound $\sum_{i=k+1}^n(\textbf{v}_i(u) - \textbf{v}_i(v))^2 (1-\frac{\lambda_i}{2})^{2\ell}$ by using the bound for $\lambda_{h+1}$ in a similar way we can bound the entire term by bounding $\lambda_2$ in the case $k=1$. However, the critical part is the first summand. It turns out that there are instances where the average $l_2^2$-distance between $u, v$ from the same cluster is $\Omega(\frac{\phi^*}{d^3n})$ for a certain reasonable choice of $\ell$, such that the random walk mixes well in the cluster while does not escape from some non-expanding set containing the cluster too often
(for more details, see discussions below and Appendix \ref{subsec:evidence-tight}). This seems to rule out an approach similar to \cite{KS11:expansion,NS10:expansion}, as this approach requires a significantly smaller distance between $\textbf{p}_v^\ell$ and~$\textbf{p}_u^\ell$.
\subsection{Our techniques}
We develop the first sublinear algorithm for testing if a graph is $(k,\phi)$-clusterable, significantly extending earlier works on testing the expansion of a graph. Our algorithm draws a random sample set and tests for every pair of sample vertices if the distributions of the endpoints of a random walk starting at the two vertices are close in the $l_2^2$-distance. If this is the case, then it connects the two sample vertices by an edge in a \emph{similarity graph}. At the end, the algorithm accepts the input graph if the similarity graph is a collection of at most $k$ connected components.
Our main new contributions are as follows.
\begin{itemize}
\item Our algorithm is the first property tester that directly makes use of testing pairwise closeness of distributions induced by random walks. Previous related algorithms \cite{CS10:expansion,GR00:expansion,KPS13,KS11:expansion,NS10:expansion} tested if the distribution of the endpoints a random walk starting at a vertex $v$ is close to the uniform distribution and then drew their conclusions about the structure of the graph. In our case, we do not know how the distribution looks like (it will be close to uniform \emph{inside} every cluster, but this is not very helpful since the cluster is unknown to us and the support size of a distribution is hard to estimate \cite{RRSS09:support}) and it may have significant distance from the uniform distribution.
\item It is the first property tester that exploits (in the completeness case) a ``somewhat stable'' behaviour of the random walk distribution at a length where it is significantly different from the stationary distribution, i.e., we pick the length of the random walk in such a way that it is almost stable on its own cluster, and most of the probability mass will stay in some non-expanding set containing the cluster
\end{itemize}
In order to test closeness of distribution, we use a recent tester for closeness of distributions in $l_2$-norm by Chan et al.\ \cite{CDVV14:optimal}, which gives slightly better bounds than the corresponding tester of Batu et al.\ \cite{BFRSW13:testdist}. A combination with a necessary condition on the $l_2$-norm of the distribution of the endpoints of the random walk from the sample vertices leads to improved bounds. It is tempting to think of this problem in the setting of $l_1$-norm since, for example, the distance between a random walk starting from different clusters is typically $\Omega(1)$ in $l_1$-norm. But this is misleading. It is known that no \emph{stable} $l_1$-tester exists, i.e., $l_1$-testers cannot distinguish the case that distributions are close from the case that they are not \cite{VV11:linear} ($l_1$-testers can only distinguish between identical (or almost identical) distributions and distributions that are far away from each other). However, as already explained in the previous section, we cannot hope for distributions to be arbitrarily close even if the random walks start in the same cluster.
To address this difficulty, we will use the fact (noted earlier by Batu et al. \cite{BFRSW13:testdist}) that an $l_1$-tester can be reduced to an $l_2$-tester if the probability of every item is $O(n^{-1})$, which is likely to be the case if the graph is $(k,\phi)$-clusterable.
We note that in the $l_2^2$-distance, a typical distance between the distribution of the endpoints of the random walks starting in two vertices from different clusters can be very small. For example, if we have two disconnected expanders (clusters) on $n/2$ vertices each, then for a sufficiently long random walk the distribution of the endpoints of the walk will be (almost) uniform on the cluster of the starting vertex. Therefore the distance between the distributions of the endpoints of random walks starting in different clusters will be $O(1/n)$. Furthermore, as we have argued above, the distance between the distributions of the endpoints of random walks will not be much smaller in the case that they come from the same cluster. Analyzing these two cases is one of the central technical challenges of our paper.
\subsection{Other related work}
In the context of property testing, Alon et al.\ \cite{ADPR03:clustering} studied the problem of testing if a set of points in $\mathbb{R}^d$ is clusterable (see also \cite{CS05}), but both their problem definition and techniques are quite different from ours. Kale et al.\ \cite{KPS13} gave a sublinear expansion reconstruction algorithm that outputs the neighborhood of any input vertex $v$ in a $\Omega(\frac{\phi^2}{\log n})$-expander $G'$ that is $\frac{\phi \varepsilon}{\log n}$-close to the input graph $G$, which is assumed to be $\varepsilon$-close to a $\phi$-expander. In particular, they designed an algorithm that runs in $\widetilde{O}(\sqrt{n})$-time and distinguishes vertices from a large set that induces an expander from vertices that belong to a bad cut, by using uniform averaging random walks and testing if the distribution of endpoints of the walk is close to uniform distribution (in the $l_1$-norm distance) or not. This work does not (directly) compare distributions of the endpoints of the random walks starting from different vertices, as we do in our paper.
Our work is closely related to works on testing distributions. Batu et al.\ \cite{BFRSW00:distribution,BFRSW13:testdist} were the first to give sublinear time algorithms for testing the closeness of two discrete distributions and since then, a large body of work has been devoted to the problem of estimating the properties of distributions from a small number of samples (see the recent survey \cite{Rub12:distribution} and the reference therein). In particular, Levi et al.\ \cite{LRR13:multidistri} gave an algorithm with complexity $\widetilde{O}(n^{2/3})$ to test whether a set of distributions over a domain of size $n$ can be partitioned into $k$ clusters. Very recently, Chan et al.\ \cite{CDVV14:optimal} gave asymptotically optimal testers for the closeness of two distributions under both $l_1$ and $l_2$ settings.
Besides the related works in the literature of property testing, our work is also closely related to the area of graph partitioning and spectral clustering. Ng et al.\ \cite{NJW01:spectral} and Shi et al.\ \cite{SM00:spectral} used the first few eigenvectors of some matrices to partition a graph (or a set of data) into sparsely connected clusters. Different ways of measuring clustering based on intra-cluster density vs. inter-cluster sparsity and some experimental results were given in \cite{BGW07}. Kannan et al.\ \cite{KVV04:clustering} proposed a bicriteria to measure the quality of a clustering, in which a good clustering is defined to be a partition of vertex sets such that each set in the partition has large inner conductance and few edges lying between different sets. They gave spectra based approximation algorithm for finding such a clustering. Lee et al.\ \cite{LOT12:high} and Louis et al.\ \cite{LRTV12:sparse} recently gave theoretical analysis of some spectral algorithms that use the first $k$ eigenvectors of the normalized Laplacian matrix for finding a $k$-partition of a graph such that each part is of small (outer) conductance, without any restriction on the inner conductance of the cluster. Zhu et al.\ \cite{ZLM13:local,OZ14:clustering} gave personal PageRank based and flow based local algorithms for finding a set of large inner conductance and small outer conductance. Makarychev et al.\ \cite{MMV12:semirand} studied a semidefinite programming based algorithm in the semi-random model to find such a set. Tanaka \cite{Tan13:partition} and Oveis Gharan and Trevisan \cite{OT14:expander} recently studied the existence and construction of a $k$-clustering such that each cluster is of large inner conductance and of small outer conductance, under the assumption that there is some gap between $\rho_G(k)$ and $\rho_G(k+1)$, where $\rho_G(k)$ is the minimum conductance of any $k$ disjoint subsets of the graph (cf. Section \ref{subsec:spectral-clusterable}). Dey et al.\ \cite{DRS14:spectral} considered the performance of a spectral clustering algorithm that applies a greedy algorithm for $k$-centers on some embedding induced by the first $k$ eigenvectors of the graph Laplacian. Peng et al.\ \cite{PSZ14:partitioning} studied the eigenvector structures of the Laplacian of well-clustered graphs (which is very related to our definition of clusterable graphs) and the approximation ratio of $k$-means clustering algorithms on these graphs.
\subsection{Organization of the paper}
In Section \ref{sec:pre}, we give notations and definitions used throughout the paper. In Section \ref{sec:algorithm}, we give a formal description of our tester for clusterable graphs. We then present in Section \ref{sec:analysis} some central properties, which we use for proving our main result --- Theorem \ref{thm:main}. The proofs of these central properties are given in Section \ref{sec:proof-section5}. Section \ref{sec:conclusion} has final conclusions.
Finally, in Appendix we will present some auxiliary tools used in the analysis.
\section{Preliminaries}
\label{sec:pre}
Let $G=(V,E)$ be an undirected and unweighted graph with maximum degree bounded by a constant $d$. Let $n:=|V|$. For a vertex $v\in V$, let $d_G(v)$ be the degree of $v$. We assume that $G$ is represented by its \textit{adjacency list} and that we can access $G$ through an oracle, which allows us to perform the \textit{neighbor query} to $G$. That is, when the oracle is given as input a vertex $v$ and an integer $i$, it outputs the $i$-th neighbor of $v$ if $d_G(v)\ge i$, and a special symbol otherwise (in constant time).
As mentioned in the introduction, we will use Definition \ref{def:clusterable-1} of $(k,\phi)$-clusterable graphs and $\phi$-clusters inspired by \cite{OT14:expander} to characterize the cluster structure of graphs and the clusters therein. Note that a $(1,\phi)$-clusterable graph is an expander graph with conductance $\phi$, which we abbreviate as $\phi$-expander (this should not be confused with $\phi$-cluster).
We are interested in testing if a given graph is $(k,\phi)$-clusterable in sublinear time
in the framework of property testing. Formally speaking, we will study the following problem: given parameters $k,\phi,\varepsilon$, and a $d$-degree bounded graph $G$, we want to test if $G$ is $(k,\phi)$-clusterable or $\varepsilon$-far from being $(k,\phi^*)$-clusterable with as few queries as possible, for $\phi^*$ being as close to $\phi$ as possible. We have the following definition of graphs that are $\varepsilon$-far from clusterable graphs.
\begin{definition}
~\label{def:eps-far}
A graph $G$ (of maximum degree at most $d$) is \emph{$\varepsilon$-far from $(k, \phi)$-clusterable} if we have to add or delete more than $\varepsilon d n$ edges to obtain a $(k, \phi)$-clusterable graph of maximum degree at most $d$.
If $G$ is not $\varepsilon$-far from $(k, \phi)$-clusterable then it is \emph{$\varepsilon$-close} to $(k, \phi)$-clusterable.
\end{definition}
\section{The algorithm}
\label{sec:algorithm}
In this section, we describe our algorithm used in Theorem \ref{thm:main}. We first introduce the following \textit{random walk} on a $d$-bounded degree graph $G$ that will be used in our algorithm. In this walk, if we are currently at vertex $v$, then in the next step, we choose randomly an incident edge $(v,u)$ with probability $\frac{1}{2d}$ and move to $u$. With the remaining probability, which is at least $\frac12$, we stay at $v$. Note that if we let $G_\textrm{reg}$ denote the weighted $d$-regular graph that is obtained from $G$ by adding an appropriate number of half-weighted self-loops, then this random walk is exactly a \textit{lazy random walk} on $G_\textrm{reg}$. We will let $\textbf{p}_v^\ell$ denote the distribution of endpoints of such a random walk of length $\ell$ starting at $v$.
Our testing algorithm is given as follows.
\begin{center}
\begin{tabular}{|p{0.9\textwidth}|}
\hline
\textbf{$k$-Cluster-Test} $(G,s,\ell,\sigma,k)$\\
\hline
\begin{enumerate}
\item Sample a set $S$ of $s$ vertices independently and uniformly at random from~$V$.
\item For any $v\in S$, let $\textbf{p}_v^\ell$ be the distribution of endpoints of random walk of length $\ell$ starting at $v$.
\item\label{alg:l2norm} For any $v\in S$, test if $||\textbf{p}_v^\ell||_2^2>\sigma$; if so, then abort and reject.
\item\label{alg:distribution-closeness} For each pair $u, v \in S$: if $l_2$ distribution tester accepts that $\norm{\textbf{p}_u^\ell-\textbf{p}_v^\ell}_2^2\le \frac{1}{4n}$, then add an edge $(u,v)$ in ``similarity graph'' $H$ on vertex set~$S$.
\item If $H$ is the union of at most $k$ connected components, then accept; otherwise, reject.
\end{enumerate}\\
\hline
\end{tabular}
\end{center}
If the graph is $(k,\phi)$-clusterable then we will show that (for the right choice of parameters) the distributions of the endpoints of random walks will be close if they come from the same cluster.
Furthermore, Step~\ref{alg:l2norm} tests a necessary condition for the efficient $l_2$ distribution tester that will be used in~Step~\ref{alg:distribution-closeness}, i.e., $||\textbf{p}_v^\ell||_2^2$ is small, which is satisfied for almost all vertices in a $(k,\phi)$-clusterable graph. The small $l_2^2$-norm property of distributions can then be exploited in the testing for closeness of distributions in Step \ref{alg:distribution-closeness} to obtain a better running time.
\subsection{Implementation of distribution testing}
\label{section:Distributions}
Our algorithm relies on an efficient tester for the $l_2$-closeness of two distributions $\textbf{p}$ and $\textbf{q}$. The tester used in Step \ref{alg:distribution-closeness} of \textbf{$k$-Cluster-Test} was recently proposed by Chan et al.\ \cite{CDVV14:optimal} and is similar to the $l_2$ distance tester in \cite{BFRSW13:testdist} that uses the statistics of collisions in the sample sets from both distributions $\textbf{p}, \textbf{q}$.
The following is a direct corollary of Theorem 1.2 from \cite{CDVV14:optimal}.
\begin{theorem}
\label{cor:distribution}
Let $c_{\ref{cor:distribution}}$ be some appropriate constant $c_{\ref{cor:distribution}} \ge 1$. Let $\delta, \xi > 0$ and let $\textbf{p},\textbf{q}$ be two distributions over a set of size $n$ with $b \ge \max\{\norm{\textbf{p}}_2^2, \norm{\textbf{q}}_2^2\}$. Let $r \ge c_{\ref{cor:distribution}} \cdot \frac{\sqrt{b}}{\xi} \ln\frac{1}{\delta}$. There exists an algorithm, denoted by \textbf{$l_2$-Distribution-Test}, that takes as input $r$ samples from each distribution $\textbf{p},\textbf{q}$, and accepts the distributions if $\norm{\textbf{p}-\textbf{q}}_2^2\le \xi$, and rejects the distributions if $\norm{\textbf{p}-\textbf{q}}_2^2\ge 4\xi$, with probability at least $1-\delta$. The running time of the tester is linear in its sample size.
\end{theorem}
We also need an efficient algorithm to estimate the $l_2^2$-norm of the probability distribution of the endpoints of a random walk in a graph.
In Step \ref{alg:l2norm} of our algorithm \textbf{$k$-Cluster-Test} we will use \textbf{$l_2^2$-norm tester}, the performance of which is guaranteed in the following lemma (the proof follows almost directly from the proof of Lemma 4.2 in \cite{CS10:expansion} that in turn is built on Lemma 1 in \cite{GR00:expansion}, cf. Appendix \ref{subsec:proof-on-distribution} for details).
\begin{lemma}
\label{lem:collision}
Let $G=(V,E)$ with $|V|=n$. Let $v \in V$, $\sigma>0$ and $r \ge 16\sqrt{n}$. Let $t\ge 1$ and let $\textbf{p}_v^t$ be the probability distribution of the endpoints of a random walk of length $t$ from $v$. There exists an algorithm, denoted by \textbf{$l_2^2$-norm tester}, that takes as input $r$ samples from $\textbf{p}_v^t$ and accepts the distribution if $\norm{\textbf{p}_v^t}_2^2\le \sigma/4$ and rejects the distribution if $\norm{\textbf{p}_v^t}_2^2>\sigma$, with probability at least $1-\frac{16\sqrt{n}}{r}$. The running time of the tester is linear in its sample size.
\end{lemma}
\section{Analysis of $k$-Cluster-Test}
\label{sec:analysis}
We outline the proof of our main theorem, Theorem \ref{thm:main}. Our techniques are based on two intuitions. The first intuition is that if two ``typical'' vertices $u, v$ are \emph{from the same large cluster}, then the distributions of the endpoints of two \emph{sufficiently long} random walks starting at $u, v$, respectively, are close; and if $u, v$ are \emph{separated by a non-expanding cut}, then the distributions of the endpoints of two \emph{not so long} random walks from $u, v$, respectively, are far away from each other. If this intuition holds, then we can reduce our problem to the problem of testing the closeness of two distributions, and then use the returned results to decide whether the distributions induced by the random walks from different sampled vertices can be divided into $k$ groups or not. In particular, if our input graph $G$ is $(k,\phi)$-clusterable, then we can get at most $k$ connected components in our ``similarity graph'' $H$. (Actually, as will be seen from our proof, sampled vertices from the same cluster form a clique in $H$.)
On the other hand, if $G$ is far from being $(k,\phi^*)$-clusterable, then we expect that we can get at least $k+1$ \junk{cliques or non-clique} connected components in $H$ . The latter is based on our second intuition that if $G$ is far from being $(k, \phi^*)$-clusterable, then there are at least $k+1$ (large) well separated sparse cuts. We present several lemmas that formalize these intuitions in Section \ref{subsec:keyproperties} and then give the proof of Theorem \ref{thm:main} in Section \ref{sec:proof-main-theorem}.
\subsection{Key properties}
\label{subsec:keyproperties}
In this section, we state several lemmas describing the properties used in our analysis of \textbf{$k$-Cluster-Test}. The proofs of the results are deferred to Section \ref{sec:proof-section5}.
In the following we will formally state these key properties under the definition of a more general class of clusterable graphs, even though our main focus is on the study of properties of $(k,\phi)$-clusterable graphs. To study detailed properties of $(k,\phi)$-clusterable graphs and their dependencies on all parameters, we will use the following, more general definition of $(k,\phi_{in},\phi_{out})$-clusterable graphs, which follows the framework from \cite{OT14:expander}.
\begin{definition}
\label{def:clusterable-old}
For an undirected graph $G$, and parameters $k,\phi_{in},\phi_{out}$, we define $G$ to be \emph{$(k,\phi_{in},\phi_{out})$-clusterable} if there exists a partition of $V$ into $h$ subsets $C_1, \dots, C_h$ such that $1 \le h \le k$ and for each $i$, $1 \le i \le h$, $\phi(G[C_i])\ge \phi_{in}$, $\phi_G(C_i)\le \phi_{out}$. We call each $C_i$ a \emph{$(\phi_{in},\phi_{out})$-cluster} and the corresponding $h$-partition an \emph{$(h,\phi_{in},\phi_{out})$-clustering}.
\end{definition}
We can define a graph $G$ to be $\varepsilon$-far from $(k,\phi_{in},\phi_{out})$-clusterable similarly to Definition~\ref{def:eps-far}. Note that a $(k,\phi)$-clusterable graph from Definition \ref{def:clusterable-1} is exactly a $(k,\phi, c_{d,k} \varepsilon^4 \phi^2)$-clusterable graph from Definition \ref{def:clusterable-old}.
We first show that if the graph is $(k,\phi_{in},\phi_{out})$-clusterable then for any large cluster $C$ with $\phi(G[C])\ge \phi_{in}$, there exists a large subgraph $\widetilde{C}$ such that the distributions of the endpoints of two random walks of length large enough starting from any two vertices $u,v\in \widetilde{C}$ are close in the $l_2$-norm (that is, the $l_2$ distance between $\textbf{p}_u^\ell$ and $\textbf{p}_v^\ell$ is small). The proof of this result relies on spectral properties of clusterable graphs given in Section \ref{subsec:spectral-clusterable}.
\begin{lemma}
\label{lem:smalll2}
Let $0 < \alpha, \beta < \frac12$. If $G = (V,E)$ is $(k, \phi_{in}, \phi_{out})$-clusterable, and $C \subseteq V$ is any subset such that $|C| \ge \beta n$ and $\phi(G[C]) \ge \phi_{in}$, then there exists $\alpha_{\ref{lem:smalll2}} = \alpha_{\ref{lem:smalll2}}(k, \alpha, \beta,d)$ and a universal constant $c_{\ref{lem:smalll2}}>0$ such that for any $t \ge \frac{c_{\ref{lem:smalll2}}k^4 \log n}{\phi_{in}^2}$, $\phi_{out}\le \alpha_{\ref{lem:smalll2}}\phi_{in}^2$,
there exists a subset $\widetilde{C}\subseteq C$ with $|\widetilde{C}| \ge (1-\alpha)|C|$ such that for any $u, v \in \widetilde{C}$, the following holds:
\begin{displaymath}
\norm{\textbf{p}_u^t-\textbf{p}_v^t}_2^2\le \frac{1}{4n}
\enspace.
\end{displaymath}
\end{lemma}
\junk{\textcolor{red}{(Remove this paragraph?) We remark that in the proof of Theorem~\ref{thm:main}, we will choose appropriate parameters $\alpha,\beta$ such that for any $\phi$-cluster $C$, if we let $\phi_{in}=\phi(G[C])=\phi$ and $\phi_{out}=\phi_G(C)\le c_{d,k}\varepsilon^4\phi^2$, then it always holds that $\phi_{out}\le \alpha_{\ref{lem:smalll2}}\phi_{in}^2$ by our definition of $c_{d,k}$.}}
In order to use an efficient distribution tester (e.g., as the one given in Theorem \ref{cor:distribution}), we need to guarantee that for a large fraction of vertices a sufficiently long random walk starting from a typical vertex will induce a distribution of its endpoints with small $l_2$-norms. We will prove the following lemma using spectral analysis of clusterable graphs.
\begin{lemma}
\label{lem:smallhnorm}
Let $0 < \alpha < 1$. If $G$ is $(k, \phi_{in}, \phi_{out})$-clusterable, then there exists $V' \subseteq V$ with $|V'|\ge (1-\alpha)|V|$ such that for any $u \in V'$ and any $t \ge \frac{c_{\ref{lem:smallhnorm}} k^4 \log n}{\phi_{in}^2}$, for some universal constant $c_{\ref{lem:smallhnorm}}>0$,
the following holds:
\begin{displaymath}
\norm{\textbf{p}_u^t}_2^2\le \frac{2k}{\alpha n}
\enspace.
\end{displaymath}
\end{lemma}
Note that the above lemma does not require any assumption about $\phi_{out}$, and thus applies directly to any $(k,\phi)$-clusterable graphs by substituting $\phi$ for $\phi_{in}$ in the lemma.
For the soundness of our algorithm, we need the following lemma that shows that given two well separated sets $A, B \subseteq V$, for any two ``typical'' vertices $u \in A$, $v \in B$, the $l_2$-norm of the difference between the corresponding distributions of endpoints of random walks of short length starting from $u, v$ will be large. Our proof relies on the fact that any
set $A$ with small outer conductance has a large subset $\widehat{A}$ such that the random walk starting from any vertex in $\widehat{A}$ will stay inside $A$ for a relatively long time
\begin{lemma}
\label{lem:largel2}
Let $\alpha$ and $\psi$ be arbitrary with $0 < \alpha, \psi < 1$. Let $A \subseteq V$ be any subset of $G$ such that $\phi_G(A) \le \psi$. Then for any $t \ge 1$, there exists a subset $\widehat{A} \subseteq A$ with $|\widehat{A}| \ge (1-\alpha)|A|$ such that for any $v \in \widehat{A}$, the probability that the random walk of length $t$ starting from vertex $v$ never leaves $A$ in all $t$ steps is at least $1 - \frac{t \psi}{2 \alpha}$.
Furthermore, for any $t$, $1 \le t \le \frac{\alpha}{2 \psi}$, any two disjoint subsets $A, B \subseteq V$ with $\phi_G(A), \phi_G(B) \le \psi$, and any two vertices $u, v$ such that $u \in \widehat{A}, v \in \widehat{B}$, the following holds:
\begin{displaymath}
\norm{\textbf{p}_u^t - \textbf{p}_v^t}_2^2
\ge
\frac{1}{n}
\enspace.
\end{displaymath}
\end{lemma}
\begin{remark}
We note that the above lower bound
is almost tight up to constants. Consider the graph that is composed of two disconnected parts such that each of them is a $\phi_{in}$-expanders of size $n/2$. Then for any two starting vertices $u,v$ from two different parts, for $t=\Theta(\frac{\log n}{\phi_{in}^2})$, both $\textbf{p}_u^t$ and $\textbf{p}_v^t$ will be very close to the uniform distribution on each cluster, and therefore, the $l_2^2$ distance between these two distributions will be $O(1/n)$.
\end{remark}
For the analysis showing that graphs far from clusterable will be rejected, we will use a property that if a graph $G=(V,E)$ is $\varepsilon$-far from any $(k, \phi_{in}^*,\phi_{out}^*)$-clusterable graph, then its vertex set $V$ can be partitioned into $k+1$ subsets $V_1, \dots, V_{k+1}$, each of linear size and of small outer conductance.
\begin{lemma}
\label{lemma:partition-eps-far-improved}
Let $\alpha_{\ref{lemma:partition-eps-far-improved}} = \alpha_{\ref{lemma:partition-eps-far-improved}}(d,k)$ be a certain constant that depends on $d$ and $k$. If $G = (V,E)$ is $\varepsilon$-far from $(k, \phi^*_{in}, \phi^*_{out})$-clusterable with $\phi^*_{in} \le \alpha_{\ref{lemma:partition-eps-far-improved}} \cdot \varepsilon$, then there exist a partition of $V$ into $k+1$ subsets $V_1, \dots, V_{k+1}$ such that for each $i$, $1 \le i \le k+1$, $|V_i| \ge \frac{1}{1152k} \varepsilon^2 |V|$
and $\phi_G(V_i) \le c_{\ref{lemma:partition-eps-far-improved}} \phi^*_{in} \varepsilon^{-2}$, for some constant $c_{\ref{lemma:partition-eps-far-improved}} = c_{\ref{lemma:partition-eps-far-improved}}(d,k)$ and for any $0 \le \phi^*_{out} \le 1$.
\end{lemma}
\junk{
Let us remark here that in Lemma \ref{lemma:partition-eps-far} we do not have any explicit bound for $\phi^*_{out}$ and any $0 \le \phi^*_{out}\le 1$ will do. This is the case under our assumption that $\phi^*_{in} \le O(\varepsilon)$. Actually, we can show that the above claim does not hold when $\phi^*_{in}$ is some large constant and $\phi^*_{out}$ is close to $0$. For example, let $G$ be a graph consisting of $k$ great expanders with $\phi(G[V_i]) =\Theta(1)$ and these $k$ parts are ``evenly'' connected by $\approx \varepsilon n$ edges, which is $\varepsilon$-far from $(k,\Theta(1),0)$-clusterable, while it is unlikely to partition it into $k+1$ large and sparse parts, as required in our proof for soundness.
}
\subsection{Proof of main result --- Theorem \ref{thm:main}}
\label{sec:proof-main-theorem}
We will use Lemmas \ref{lem:smalll2}--\ref{lemma:partition-eps-far-improved} to prove our main result --- Theorem \ref{thm:main}. In the rest of this section, we prove the completeness, soundness and analyze the running time of the tester \textbf{$k$-Cluster-Test}.
In the algorithm \textbf{$k$-Cluster-Test}, we set $s = \frac{1536 k \ln(8(k+1))}{\varepsilon^2}$, $\ell = \frac{\max\{c_{\ref{lem:smalll2}}, c_{\ref{lem:smallhnorm}}\}\cdot k^4 \log n}{\phi^2}$, $\sigma = \frac{192sk}{n}$.
We set $r = 192 c_{\ref{cor:distribution}} s \sqrt{skn} \ln s = O(\frac{k^2 (\ln k/\varepsilon)^{5/2} \sqrt{n}}{\varepsilon^3}), b = \frac{216 s k}{n}$, $\xi = \frac{1}{4 n}$, $\delta = \frac{1}{12 s^2}$ in Theorem \ref{cor:distribution}, and set $r = 192 c_{\ref{cor:distribution}} s \sqrt{skn} \ln s$ and $\sigma=\frac{192sk}{n}$ in Lemma~\ref{lem:collision}.
We specify now the constant $c_{d,k}$ that we used in the definition of a $\phi$-cluster to be $c_{d,k} = \frac{\alpha_{\ref{lem:smalll2}}(k, \frac{1}{24s}, \frac{1}{24ks}, d)}{\varepsilon^4} =\frac{c}{k^5 d^4 \ln^2 (8(k+1))}$ for a universal constant $c$
\subsubsection{Completeness --- accepting $(k, \phi)$-clusterable graphs}
We begin with showing that the algorithm \textbf{$k$-Cluster-Test} will accept $k$-clusterable graphs.
\begin{lemma}
\label{lemma:completeness}
If the input graph $G$ is $(k, \phi)$-clusterable, then with probability at least $\frac23$, the algorithm \textbf{$k$-Cluster-Test} accepts $G$.
\end{lemma}
\begin{proof}
As indicated in the algorithm, we consider random walks of length $\ell$. We apply Lemmas \ref{lem:smalll2} and \ref{lem:smallhnorm} to the $(k,\phi)$-clusterable graph $G$, and we set $\phi_{in} = \phi$, $\phi_{out} = c_{d,k} \varepsilon^4 \phi^2$, $t = \ell$, $\alpha = \frac{1}{24s}$, and $\beta = \frac{1}{24ks}$ in the lemmas. Note that by our definition of $\phi$-cluster, the outer conductance of the cluster is at most $c_{d,k}\varepsilon^4 \phi^2 \le \alpha_{\ref{lem:smalll2}} \phi^2$, since $c_{d,k} \varepsilon^4 = \alpha_{\ref{lem:smalll2}}(k, \frac{1}{24s}, \frac{1}{24ks}, d)$, which implies that the conditions of Lemma \ref{lem:smalll2} are satisfied for any $\phi$-cluster of size at least $\beta n$ in $G$. Since $\ell = \frac{\max\{c_{\ref{lem:smalll2}}, c_{\ref{lem:smallhnorm}}\} \cdot k^4 \log n}{\phi_{in}^2}$, we know that the chosen parameters meet all the preconditions in these lemmas.
Since $G$ is $(k,\phi)$-clusterable, there exists some $h$, $1 \le h \le k$, and a partition of the vertex set of $G$ into $h$ subsets $C_1, \dots, C_h$, such that for every $i$, $1 \le i \le h$, we have $\phi(G[C_i]) \ge \phi$ and $\phi_G(C_i) \le c_{d,k} \varepsilon^4 \phi^2$. For any vertex $v$, define $C(v)$ to be the unique cluster $C_i$ to which $v$ belongs.
We call a vertex $v$ \textit{good} if the following three conditions are satisfied:
\begin{enumerate}
\item \label{item:norm} $\norm{\textbf{p}_v^\ell}_2^2 \le \frac{48sk}{n}$.
\item \label{item:clustersize} $|C(v)| \ge \frac{1}{24ks}n$.
\item \label{item:expandcore} $v \in \widetilde{C(v)}$, where $\widetilde{C(v)} \subseteq C(v)$ is defined as in Lemma \ref{lem:smalll2} by setting $C = C(v)$.
\end{enumerate}
The success probability of the algorithm depends on the random coins of sampling and random walks. We show that with probability at least $\frac78$ (over random coins of sampling), all vertices in the sample set $S$ are good; and if all these vertices are good, then our tester will accept with probability at least $\frac56$ (over random coins of random walks). Together, this means that with probability at least $\frac78 \cdot \frac56
= \frac{35}{48} \ge \frac23$ the tester will accept. This will conclude the proof of the lemma.
\begin{claim}
\label{claim:completeness-1}
With probability at least $\frac78$, all vertices in the sampled set $S$ are good.
\end{claim}
\begin{proof}
Let $v$ be any vertex that is sampled uniformly at random from $V$. By Lemma \ref{lem:smallhnorm}, the probability that $\norm{\textbf{p}_v^\ell}_2^2 > \frac{48sk}{n}$ is at most $\alpha = \frac{1}{24s}$. Since there are at most $k$ clusters, the probability that $v$ belongs to a cluster of size at most $\frac{1}{24ks}n$ is at most the probability that $v$ is one of at most $k \cdot \frac{n}{24ks}$ vertices in these small clusters, which is $\frac{1}{24s}$. In addition, since $|\widetilde{C(v)}|\ge (1-\alpha)|C(v)|$, the probability that $v\notin \widetilde{C(v)}$ is at most $\alpha = \frac{1}{24s}$. Overall, the probability that $v$ is not good is at most $\frac{1}{24s} + \frac{1}{24s} + \frac{1}{24s} = \frac{1}{8s}$. By the above analysis and the union bound, with probability at least $1 - \frac{1}{8s} \cdot s = \frac78$, all sampled vertices in $S$ are good.
\end{proof}
\begin{claim}
\label{claim:completeness-2}
Conditioned on the event that all the sampled vertices $v\in S$ are good, our tester will accept $G$ with probability at least $\frac56$.
\end{claim}
\begin{proof}
Let $v\in S$. Since $v$ is good, then $\norm{\textbf{p}_v^\ell}_2^2\le\frac{48sk}{n}=\frac{\sigma}{4}$. Now by Lemma \ref{lem:collision}, \textbf{$l_2^2$-norm estimator} will reject $v$ with probability at most $\frac{16\sqrt{n}}{r}\le\frac{1}{12s}$. By the union bound, the probability that we get rejected at step \ref{alg:l2norm} of the algorithm is at most $\frac{1}{12}$.
For any two vertices $u,v$ from $S$, if $u,v$ belong to the same large cluster, then by Conditions \ref{item:clustersize}--\ref{item:expandcore} of good vertices and by Lemma \ref{lem:smalll2}, $\norm{\textbf{p}_u^\ell-\textbf{p}_v^\ell}_2^2\le \frac{1}{4n}$.
Now recall that we have set $b=\frac{216sk}{n}, \xi=\frac{1}{4n}, \delta=\frac{1}{12s^2}$ and $r = 192 c_{\ref{cor:distribution}} s \sqrt{skn} \ln s$ in Theorem \ref{cor:distribution}. Then $b\ge\max\{\norm{\textbf{p}_v^\ell}_2^2,\norm{\textbf{p}_u^\ell}_2^2\}$, $r \ge c_{\ref{cor:distribution}} \cdot \frac{\sqrt{b}}{\xi} \ln\frac{1}{\delta}$, and we can ensure that with probability at least $1-\delta$, any call to \textbf{$l_2$-Distribution-Test} will accept the distributions $\textbf{p}_u^t,\textbf{p}_v^t$ if $u,v$ belong to the same large cluster.
By the union bound, the probability that there exist some call such that the distribution tester does not accept $u,v$ if $u,v$ are from the same cluster is at most $s^2\delta\le \frac{1}{12}$. Therefore, the probability that the algorithm does not reject at step \ref{alg:l2norm} and all the calls to the \textbf{$l_2$-Distribution-Test} return the correct answer is at least $1-\frac{1}{12}-\frac{1}{12}=\frac{5}{6}$.
Now note that if for any $u,v\in S$ such that $u,v$ belong to the same cluster, the distribution tester with input $\textbf{p}_u^\ell,\textbf{p}_v^\ell$ accepts, then there will an edge $(u,v)$ in the ``similarity graph'' $H$.
This further implies that all the vertices in $S$ that are in the same cluster will form a clique.
(But note that two sampled vertices from two different clusters might also be connected in $H$.)
Since there are at most $k$ clusters, we will get at most $k$ connected components in $H$, and thus the tester will accepts $G$.
\end{proof}
We can now apply Claims \ref{claim:completeness-1} and \ref{claim:completeness-2} to conclude the proof of Lemma \ref{lemma:completeness}.
\end{proof}
\subsubsection{Soundness --- rejecting graphs $\varepsilon$-far from $(k, \phi^*)$-clusterable}
We present now a proof of the soundness of our tester.
\begin{lemma}
\label{lemma:soundness}
Let $\gamma = \gamma_{d,k} > 0$ be some constant depending on $d, k$. If the input graph $G = (V,E)$ is $\varepsilon$-far from $(k, \phi^*)$-clusterable with $\phi^*\le \frac{\gamma \varepsilon^2}{s \ell}$, then the algorithm \textbf{$k$-Cluster-Test} rejects $G$ with probability at least $\frac23$.
\end{lemma}
\begin{proof}
We will use $\gamma = \min\{\frac{1}{48 c_{\ref{lemma:partition-eps-far-improved}}}, \alpha_{\ref{lemma:partition-eps-far-improved}}\}$. Let us first observe that our choice of $\gamma$ ensures that Lemma \ref{lemma:partition-eps-far-improved} implies the existence of a partition of $V$ into $k+1$ disjoint sets $V_1, \dots, V_{k+1}$ such that for each $i$, $1 \le i \le k+1$, $|V_i| \ge \kappa_1 \varepsilon^2 |V|$ and $\phi_G(V_i) \le \kappa_2 \phi^* \varepsilon^{-2}$, for appropriate parameters $\kappa_1 = \frac{1}{1152k}$ and $\kappa_2 = c_{\ref{lemma:partition-eps-far-improved}}$.
Let $\alpha = \frac{1}{24s}$ (here $\alpha$ corresponds to the parameter $\alpha$ used in Lemma \ref{lem:largel2}). For every set $V_i$, $1 \le i \le k+1$, let $\widehat{V_i} \subseteq V_i$ be the set of vertices $v \in V_i$ such that the probability that the random walk of length $\ell$ starting at $v$ does not leave $V_i$ is at least $1 - \frac{\kappa_2 \phi^* \ell}{2 \alpha \varepsilon^2}$. We observe that since $\phi_G(V_i) \le \kappa_2 \phi^* \varepsilon^{-2}$, we have $|\widehat{V_i}| \ge (1-\alpha) |V_i|$ by Lemma \ref{lem:largel2}. Hence, our assumption that $|V_i| \ge \kappa_1 \varepsilon^2 |V|$ implies that $|\widehat{V_i}| \ge (1-\alpha) \kappa_1 \varepsilon^2 |V|$.
Let us call the sample set $S$ chosen by the algorithm \textbf{$k$-Cluster-Test} to be \emph{representative} if $\widehat{V_i} \cap S \ne \emptyset$ for every $i$, $1 \le i \le k+1$, and $S\subseteq \bigcup_{i=1}^{k+1} \widehat{V_i}$.
\begin{claim}
\label{claim:soundness-1}
The probability that the sample set $S$ is representative is at least $\frac56$.
\end{claim}
\begin{proof}
For any set $X \subseteq V$, $\Pr[X \cap S = \emptyset] = (1-|X|/|V|)^{s} \le e^{- s |X|/|V|}$. Therefore, since $|\widehat{V_i}| \ge (1-\alpha) \kappa_1 \varepsilon^2 |V|$, the probability that $S$ does not contain any element from $\widehat{V_i}$ is smaller than or equal to $e^{- s \widehat{V_i}/|V|} \le e^{- s (1-\alpha) \kappa_1 \varepsilon^2}$. Hence, the union bound implies that the probability that there exists some $i\le k+1$ such that $S$ does not contain any element from $\widehat{V_i}$ is at most $(k+1) \cdot e^{- s (1-\alpha)\kappa_1 \varepsilon^2}$. In addition, the probability that there exists some vertex in $S$ that belongs to $V\setminus(\bigcup_{i=1}^{k+1} \widehat{V_i})$ is at most $s\cdot\alpha$. Therefore, the probability that $S$ is representative is greater than or equal to $1 - (k+1) \cdot e^{- s (1-\alpha) \kappa_1 \varepsilon^2} - s \alpha$. Since $s = \frac{1536k \ln(8(k+1))}{\varepsilon^2}$ and $\alpha = \frac{1}{24s}$, we have $s (1-\alpha) \kappa_1 \varepsilon^2 \ge \ln (8(k+1))$, and hence we can conclude that this probability is at least $\frac56$.
\end{proof}
\begin{claim}
\label{claim:soundness-2}
If $S$ is representative then the algorithm \textbf{$k$-Cluster-Test} rejects $G$ with probability at least $\frac56$.
\end{claim}
\begin{proof}
Let $S_i:=\widehat{V_i}\cap S$. Since $S$ is representative, then $S = \bigcup_{i=1}^{k+1} S_i$. Recall that the algorithm \textbf{$k$-Cluster-Test} rejects $G$ if one of the following two cases happen:
\begin{itemize}
\item there is a $v \in S$ such that \textbf{$l_2^2$-norm estimator} passes the testing of $\norm{\textbf{p}_v^t}_2^2 > \sigma$.
\item for any $1\le i< j\le k+1$, and any vertex pair $u,v$ such that $u\in S_i$ and $v\in S_j$, $(u,v)$
is not an edge in the ``similarity graph'' (because in that case the resulting graph $H$ could not be a union of at most $k$ connected components).
\end{itemize}
If there exists some $v \in S$ with $\norm{\textbf{p}_v^t}_2^2 > \sigma$, then by Lemma \ref{lem:collision}, \textbf{$l_2^2$-norm tester} with rejects $v$ with probability at least $1-\frac{16\sqrt{n}}{r}>\frac{2}{3}$ and we are done. Therefore, we assume in the following that for every $v\in S$, $\norm{\textbf{p}_v^t}_2^2 < \sigma$. Let us now observe that the probability that the algorithm \textbf{$k$-Cluster-Test} would reject $G$ is lower bounded by the probability that for any $1 \le i < j \le k+1$, and any vertex pair $u,v$ such that $u\in S_i$ and $v\in S_j$, \textbf{$l_2$-Distribution-Test} rejects the distributions $\textbf{p}_{u}^\ell,\textbf{p}_{v}^\ell$.
Our definition of sets $\widehat{V_1}, \widehat{V_2}, \dots, \widehat{V_{k+1}}$ and the assumption on $\phi_{in}^*$ (which implies that $\ell \le \frac{\alpha}{2 \kappa_2 \phi^* \varepsilon^{-2}} \le \frac{\alpha}{2 \max_i\{\phi_G(V_i)\}}$) ensure that for any $1 \le i < j \le k+1$, and any vertex pair $u,v$ such that $u\in S_i$ and $v\in S_j$, we can apply Lemma \ref{lem:largel2} to obtain $\norm{\textbf{p}_{u}^{\ell}-\textbf{p}_{v}^{\ell}}_2^2 \ge \frac{1}{n}$. We know, by Theorem \ref{cor:distribution} and our choice of $b, \xi, \delta$ in that theorem, that for every such pair $v_i$, $v_j$, \textbf{$l_2$-Distribution-Test} will accept the distributions $\textbf{p}_{v_i}^\ell,\textbf{p}_{v_j}^\ell$ with probability at most $\delta$. Therefore, the probability that there exists some vertex pair $u,v$ such that $u\in S_i$, $v\in S_j$, $1\le i<j\le k+1$ and $(u,v)$ is selected as an edge in the ``similarity graph'' (which would mean that \textbf{$l_2$-Distribution-Test} will accept $\textbf{p}_{u}^\ell,\textbf{p}_{v}^\ell$) is at most $s^2 \cdot \delta$. Therefore we can conclude that the algorithm \textbf{$k$-Cluster-Test} rejects $G$ with probability at least $1 - s^2 \cdot \delta \ge \frac56$.
\end{proof}
Now, the proof of Lemma \ref{lemma:soundness} follows directly from Claims \ref{claim:soundness-1} and \ref{claim:soundness-2}.
\end{proof}
We set $c' \frac{\phi^2 \varepsilon^4}{\log n} \le \frac{\gamma \varepsilon^2}{s \ell}$ in Theorem \ref{thm:main}. By our choice of $s$ and $\ell$, we can find a constant $c' = c'_{d,k}$ that depends on $d$ and $k$ satisfying this condition, and we then require that $\phi^* \le c' \frac{\phi^2 \varepsilon^4}{\log n}$.
\subsubsection{Running time}
Now we analyze the running time of the algorithm \textbf{$k$-Cluster-Test}. First note that to sample from distributions $\textbf{p}_v^\ell$ for any $v\in V$, we need to perform $r$ random walks of length $\ell$ from $v$ and the corresponding time is $O(\ell r)$. Note that each invocation of either distribution tester runs in time linearly in the number of samples, that is $r$. Since we sampled $s$ vertices, invoked \textbf{$l_2^2$-norm tester} for each vertex in the sample set $S$, and invoked \textbf{$l_2$-Distribution-Test} for each vertex pair in $S$, we know that the total running time of the algorithm is $O(\ell s r + r s + s^2 \frac{\sqrt{b}}{\xi} \ln\frac{1}{\delta}) = O(\frac{\sqrt{n} k^7 (\ln k)^{7/2} \ln\frac{1}{\varepsilon}\ln n}{\phi_{in}^2 \varepsilon^5})$.
This completes the proof of Theorem \ref{thm:main}, which follows directly from Lemmas \ref{lemma:completeness} and \ref{lemma:soundness}, and our analysis of the running time given above.
\section{Proofs of central properties (Lemmas \ref{lem:smalll2} -- \ref{lemma:partition-eps-far-improved})}
\label{sec:proof-section5}
In the following, we will prove Lemmas
\ref{lem:smalll2} -- \ref{lemma:partition-eps-far-improved}.
Before that, we present two spectral property on the eigenvalues of $(k, \phi_{in}, \phi_{out})$-clusterable graphs, which might be of independent interest.
\subsection{Spectral properties of clusterable graphs}
\label{subsec:spectral-clusterable}
Before we state the spectral properties of clusterable graphs, we first observe that it will be sufficient for us to consider weighted $d$-regular clusterable graphs. This is true since our algorithm actually performs the lazy random walk on the (virtual) weighted $d$-regularized version $G_\textrm{reg}$ of the input $d$-bounded degree graph $G$. In addition, under our definition, for any set $S\subseteq V$, the outer conductance $\phi_G(S)$ and inner conductance $\phi(G[S])$ of $S$ in $G$ are the same as outer conductance $\phi_{G_\textrm{reg}}(S)$ and inner conductance $\phi(G_\textrm{reg}[S])$ of $S$ in $G_\textrm{reg}$, respectively. For this reason, in the rest of this section, we will assume that $G$ is a weighted $d$-regular graph.
The proofs of spectral properties of clusterable graphs rely on a recent high-order Cheeger inequality by Lee et al.\ \cite{LOT12:high}. To state the inequality, we first introduce some notations.
Let $\textbf{A}$ denote the adjacency matrix of $G$. Let ${\bf \mathcal{L}} = \textbf{I} - \frac{1}{d} \textbf{A}$ be the Laplacian matrix of $G$, where $\textbf{I}$ is the identity matrix. Let $\lambda_i$ be the $i$th smallest eigenvalue of the Laplacian matrix ${\bf \mathcal{L}}$ and let $\textbf{v}_i$ denote the corresponding (unit) eigenvector. Note that the probability transition matrix of the lazy random walk on $G$ is $\textbf{W} := \frac{\textbf{I} + \frac{1}{d} \textbf{A}}{2}$, and it is straightforward to see that $\{1-\frac{\lambda_i}{2}\}_{1 \le i \le n}$ is the set of eigenvalues of $\textbf{W}$ with corresponding eigenvectors $\{\textbf{v}_i\}_{1 \le i \le n}$ (cf. Appendix \ref{subsec:spectra} for more details).
For a $d$-regular graph $G$, let $\rho_G(k)$ denote the minimum value of the maximum conductance over any possible $k$ disjoint nonempty subsets. That is,
\begin{displaymath}
\rho_G(k)
:=
\min_{\textrm{disjoint $S_1,\dots, S_k$}}\max_{1\le i\le k}\phi_G(S_i)
\enspace.
\end{displaymath}
Lee et al.\ \cite{LOT12:high} proved the following higher-order Cheeger's inequality.
\begin{theorem}[\cite{LOT12:high}]
\label{thm:highcheeger}
For any weighted $d$-regular graph $G$ and any $k \ge 2$, it holds that
\begin{displaymath}
\lambda_k/2
\le
\rho_G(k)
\le
c_{\ref{thm:highcheeger}} k^2 \sqrt{\lambda_k}
\enspace,
\end{displaymath}
where $c_{\ref{thm:highcheeger}}$ is some universal constant.
\end{theorem}
\begin{remark}
Lee et al.\ actually proved a stronger version of the above theorem that applies to any weighted graph, by using a \emph{volume}-based definition of conductance (see Appendix \ref{subsec:volume-definition}). The weaker version given by Theorem \ref{thm:highcheeger} will be enough for our application.
\end{remark}
Now we are ready to state the spectral properties of clusterable graphs, which are given in the following two lemmas. The first lemma says that in a $k$-clusterable graph there is a large gap between $\lambda_h$ and $\lambda_{h+1}$ for some $h \le k$.
\begin{lemma}
\label{lem:eigenvalue
If $G$ is weighted $d$-regular and $(k, \phi_{in}, \phi_{out})$-clusterable, then there exists $h$, $1 \le h \le k$, such that $\lambda_i\le 2\phi_{out}$ for any $i\le h$, and $\lambda_{i} \ge \frac{\phi_{in}^2}{c_{\ref{thm:highcheeger}}^2h^4}$ for any $i \ge h+1$.
\end{lemma}
\begin{proof
Since $G$ is $(k,\phi_{in},\phi_{out})$-clusterable, then for some $h$, $1 \le h \le k$, there exists a partition of $V$ into $h$ sets $C_1, \dots, C_h$, such that $\phi(G[C_i]) \ge \phi_{in}$ and $\phi_G(C_i) \le \phi_{out}$ for any $i \le h$. From the latter, we obtain that $\rho_G(h) \le \max_i \phi_G(C_i) \le \phi_{out}$ and then by Theorem \ref{thm:highcheeger}, $\lambda_h \le 2 \phi_{out}$, and thus for any $i \le h$, $\lambda_i \le \lambda_h\le 2\phi_{out}$.
Next, let us consider an arbitrary $(h+1)$-partition $P_1, \dots, P_{h+1}$ of $V$. We note that there must be at least one set in the partition, say $P_{i_0}$, such that $|P_{i_0} \cap C_j| \le \frac12 |C_j|$ for every $1\le j\le h$. This is true since otherwise, for every $i$, $1 \le i \le h+1$, each $P_i$ would contain more than half of the vertices of some cluster, say $C_{\pi(i)}$, that is, $|P_i \cap C_{\pi(i)}| > \frac12 |C_{\pi(i)}|$. Then, since there are
$h$ clusters $C_1, \dots, C_h$, by the pigeonhole principle there would have to exist two indices $i$ and $j$, $1 \le i < j \le h+1$, such that $\pi(i) = \pi(j)$. This would mean that each of $P_i$ and $P_j$ contain more than half of the vertices from the same cluster $C_{\pi(i)}$, which is a contradiction since $P_i$ and $P_j$ are disjoint. This proves the existence of the set $P_{i_0}$.
Let $P := P_{i_0}$. For every $1 \le i \le h$, let $B_i := P \cap C_i$. Since each cluster $C_i$ has large inner conductance, namely $\phi(G[C_i]) \ge \phi_{in}$, and since $|B_i| \le \frac12 |C_i|$, we have $e(B_i, C_i \setminus B_i) \ge \phi_{in} d |B_i|$ for every $1 \le i \le h$. Hence, $\phi_G(P) = \frac{e(P, V \setminus P)}{d |P|} \ge \frac{\sum_{i=1}^h e(B_i, C_i \setminus B_i)}{d \sum_{i=1}^h |B_i|} \ge \phi_{in}$, and thus $\rho_G(h+1) \ge \phi_{in}$. Therefore Theorem \ref{thm:highcheeger} gives $\phi_{in} \le \rho_G(h+1) \le c_{\ref{thm:highcheeger}} h^2 \sqrt{\lambda_{h+1}}$, which yields $\lambda_{h+1} \ge \frac{\phi_{in}^2}{c_{\ref{thm:highcheeger}}^2 h^4}$.
\end{proof}
The second lemma states that in a $k$-clusterable graph, for any large cluster $C$, the average value of $(\textbf{v}_i(u)-\textbf{v}_i(v))^2$ over all $|C|^2$ vertex pairs $u,v\in C$ is as small as $\Theta_d(\frac{\phi_{out}}{|C|\phi_{in}^2})$,
for any $i\le h\le k$.
\begin{lemma}
\label{lem:clusterable-eigenvector}
Let $G = (V,E)$ be a weighted $d$-regular graph that is $(k, \phi_{in}, \phi_{out})$-clusterable and let $C \subseteq V$ be any subset with $\phi(G[C])\ge\phi_{in}$. Then there is $h$, $1 \le h \le k$ such that for every $i$, $1 \le i \le h$, the following holds:
\begin{displaymath}
\frac{1}{|C|} \sum_{u, v \in C} (\textbf{v}_i(u) - \textbf{v}_i(v))^2
\le
\frac{8 d^4 \phi_{out}}{\phi_{in}^2}
\enspace.
\end{displaymath}
\end{lemma}
\begin{proof
Since $G$ is $(k, \phi_{in}, \phi_{out})$-clusterable, by Lemma \ref{lem:eigenvalue}, there exists $h$, $1 \le h \le k$, such that $\lambda_{h+1} \ge \frac{\phi_{in}^2}{c_{\ref{thm:highcheeger}}^2 h^4}$ and $\lambda_i \le 2 \phi_{out}$ for any $1 \le i \le h$. Hence, for any $i \le h$, by the variational principle of eigenvalues (see Fact \ref{fact:eigenvalue} in Appendix), we have
\begin{eqnarray}
\label{eqn:lambdaphiout}
\lambda_i
=
\frac{\sum_{(u,v) \in E} (\textbf{v}_i(u) - \textbf{v}_i(v))^2}{d}
\le
2 \phi_{out}
\enspace.
\end{eqnarray}
Let us recall a known result (see, e.g., \cite[(1.5), p.~5]{Chu97:spectral}) that for any weighted graph $H = (V_H, E_H)$,\footnote{We remark that in \cite{Chu97:spectral}, the summation in the denominator is over all unordered pairs of vertices, while in our context, the summation is over all possible $|V_H|^2$ vertex pairs. Therefore, a multiplicative factor $2$ appears in the numerator in equation (\ref{ineq:from-Chung}), compared with the form in \cite[(1.5), p.~5]{Chu97:spectral}.}
\begin{equation}
\label{ineq:from-Chung}
\lambda_2(H)
=
\textrm{vol}_H(V_H) \cdot
\min_{f} \left\{\frac{2\cdot \sum_{(u,v) \in E_H} (f(u) - f(v))^2}
{\sum_{u,v \in V_H} (f(u) - f(v))^2 d_H(u) d_H(v)}
\right\}
\enspace,
\end{equation}
where $\lambda_2(H)$ denotes the second smallest eigenvalue of the normalized Laplacian of $H$,
the volume $\textrm{vol}_H(S)$ of a set $S\subseteq V_H$ is the sum of degrees of vertices in $S$, that is, $\textrm{vol}_H(S) := \sum_{v \in S} d_H (v)$.
Let us consider the induced subgraph $H := G[C]$ on $C$. Let $\phi_H^\textrm{vol}(S) := \frac{e(S,H\setminus S)}{\textrm{vol}_H(S)}$ and $\phi^\textrm{vol}(H) := \min_{S: \textrm{vol}_H(S) \le \textrm{vol}_H(V_H)/2}\frac{e(S,H\setminus S)}{\textrm{vol}_H(S)}$ (cf. Appendix \ref{subsec:volume-definition}). Since $\phi(H) \ge \phi_{in}$, then it is straightforward to see that $\phi^\textrm{vol}(H) \ge \frac{\phi_{in}}{d}$\footnote{This can be verified by considering the set $S$ with $\textrm{vol}_H(S) \le \textrm{vol}_H(V_H)/2$ such that $\phi_H^\textrm{vol}(S) = \phi^\textrm{vol}(H)$: if $|S| \le \frac{|V_H|}{2}$, then $\phi_H^\textrm{vol}(S)\ge \phi_H(S)\ge \phi_{in}$; if $|S| > \frac{|V_H|}{2}$, then $\phi_H^\textrm{vol}(S)\ge \frac{e(S,V_H\setminus S)}{d|S|}\ge \frac{\phi_{in}d |V_H\setminus S|}{d|S|}\ge \frac{\phi_{in}}{d}$, where the penultimate inequality follows from the fact that $\phi_H(V_H\setminus S)=\frac{e(S,V_H\setminus S)}{d|V_H\setminus S|}\ge \phi_{in}$ and the last inequality follows from that $|S|\le \textrm{vol}_H(S)\le \textrm{vol}_H(V_H\setminus S)\le d|V_H\setminus S|$.}. Cheeger's inequality (cf. Theorem \ref{thm:cheeger}) yields $\lambda_2 (H) \ge \frac{\phi_{in}^2}{2d^2}$. Therefore, if we apply this bound to inequality (\ref{ineq:from-Chung}), then,
\begin{displaymath}
\textrm{vol}_H(V_H) \cdot \frac{2\cdot\sum_{(u,v) \in E_H}(\textbf{v}_i(u) - \textbf{v}_i(v))^2}
{\sum_{u,v \in V_H} (\textbf{v}_i(u) - \textbf{v}_i(v))^2 d_H(u) d_H(v)}
\ge
\lambda_2(H)
\ge
\frac{\phi_{in}^2}{2d^2}
\enspace.
\end{displaymath}
Combining this with the fact that $\sum_{(u,v) \in E_H}(\textbf{v}_i(u) - \textbf{v}_i(v))^2 \le \sum_{(u,v) \in E_G} (\textbf{v}_i(u) - \textbf{v}_i(v))^2 \le 2 d \phi_{out}$, where the last inequality follows from inequality (\ref{eqn:lambdaphiout}), we have that
\begin{displaymath}
\sum_{u,v \in V_H} (\textbf{v}_i(u) - \textbf{v}_i(v))^2 d_H(u) d_H(v)
\le
\frac{8 d^3 \textrm{vol}_H(V_H) \phi_{out}}{\phi_{in}^2}
\enspace.
\end{displaymath}
Next, since $\phi(H) \ge \phi_{in} > 0$ implies that $d_H(u) \ge 1$ for any $u \in V_H$, and since the fact that for any $u \in V_H$, $d_H(u) \le d$ yields $\textrm{vol}_H(V_H) \le d|V_H| = d|C|$, using the bound above we obtain:
\begin{displaymath}
\sum_{u,v \in V_H} (\textbf{v}_i(u) - \textbf{v}_i(v))^2
\le
\sum_{u,v \in V_H} (\textbf{v}_i(u) - \textbf{v}_i(v))^2 d_H(u) d_H(v)
\le
\frac{8 d^3 \textrm{vol}_H(V_H) \phi_{out}}{\phi_{in}^2}
\le
\frac{8 d^4 |C| \phi_{out}}{\phi_{in}^2}
\enspace.
\end{displaymath}
The completes the proof of Lemma \ref{lem:clusterable-eigenvector}.
\end{proof}
\begin{remark}
In Lemma \ref{lem:counterexample} we show that Lemma \ref{lem:clusterable-eigenvector} is essentially tight for $k=2$ and constant $\phi_{in}$. We prove that there is a $(2,\phi_{in},\phi_{out})$-clusterable graph $G$ with clusters $C_1,C_2$ such that for at least one cluster, say $C_1$, the average value of $(\textbf{v}_2(u)-\textbf{v}_2(u))^2$ between vertices $u,v$ from $C_1$ is $\Omega(\frac{\phi_{out}}{d^3|C_1|})$.
\end{remark}
\subsection{Proofs of Lemmas \ref{lem:smalll2}, \ref{lem:smallhnorm}, \ref{lem:largel2}}
\label{sec:proofs}
In this section, we prove Lemmas \ref{lem:smalll2} -- \ref{lem:largel2}. For a $d$-bounded degree graph $G$, recall that $\textbf{p}_v^t$ is the probability distribution of the endpoints of the lazy random walk of length $t$ starting from $v$ on $G_\textrm{reg}$. Let $\textbf{W}_\textrm{reg}$ be the probability transition matrix of the lazy random walk on $G_\textrm{reg}$ and let $\textbf{1}_v$ be the characteristic vector on vertex $v$. Then $\textbf{p}_v^t = \textbf{1}_v(\textbf{W}_\textrm{reg})^t$.
In this section, let $\lambda_i^{\textrm{reg}}$ denote the $i$th smallest eigenvalue of the normalized Laplacian matrix of the regularized version $G_\textrm{reg}$ of $G$ and let $\textbf{v}_i^{\textrm{reg}}$ be the corresponding unit eigenvector.
Now we prove Lemma \ref{lem:smalll2}, which shows that the $l_2$-norm of the difference of two random walk distributions $\textbf{p}_v^t - \textbf{p}_u^t$ is small for most pairs $u, v$ from the same cluster for $t$ large enough.
\begin{proof}[Proof of Lemma \ref{lem:smalll2}]
For the $d$-bounded degree graph $G$, we apply Lemma \ref{lem:clusterable-eigenvector} to its weighted $d$-regular version $G_\textrm{reg}$.
For the subset $C$, by defining $\ensuremath{\Delta}_{C,i} := \frac{1}{|C|} \sum_{u \in C} \textbf{v}_i^{\textrm{reg}}(u)$, we obtain the following:
\begin{displaymath}
\sum_{u \in C} (\textbf{v}_i^{\textrm{reg}}(u) - \ensuremath{\Delta}_{C,i})^2
=
\frac{1}{|C|} \sum_{u,v \in C} (\textbf{v}_i^{\textrm{reg}}(u) - \textbf{v}_i^{\textrm{reg}}(v))^2
\le
\frac{4 d^4 \phi_{out}}{\phi_{in}^2}
\enspace,
\end{displaymath}
where we used the elementary identity $\frac{1}{n} \sum_{i<j} (a_i - a_j)^2 = \sum_{i=1}^n (a_i - \frac{\sum_{i=1}^n a_i}{n})^2$ for any $a_1, \dots, a_n$.
Therefore, the average of $(\textbf{v}_i^{\textrm{reg}}(u) - \ensuremath{\Delta}_{C,i})^2$ over all vertices in $C$ is at most $\frac{1}{|C|} \cdot \frac{4 d^4 \phi_{out}}{\phi_{in}^2}
$. This implies that for at least $(1-\alpha) |C|$ vertices $u \in C$, we have $(\textbf{v}_i^{\textrm{reg}}(u) - \ensuremath{\Delta}_{C,i})^2 \le \frac{4 k d^4 \phi_{out}}{\alpha |C| \phi_{in}^2}$ for all $i$, $1 \le i \le h \le k$. Let $\widetilde{C} \subseteq C$ denote the set of vertices with this property.
Consider any two vertices $u, v \in \widetilde{C}$.
We observe that for any $i$, $1 \le i \le h$, we have $(\textbf{v}_i^{\textrm{reg}}(u) - \textbf{v}_i^{\textrm{reg}}(v))^2 \le 2 ((\textbf{v}_i^{\textrm{reg}}(u) - \ensuremath{\Delta}_{C,i})^2 + (\textbf{v}_i^{\textrm{reg}}(v) - \ensuremath{\Delta}_{C,i})^2) \le \frac{16 kd^4 \phi_{out}}{\alpha |C| \phi_{in}^2}$, where the first inequality that $(x-y)^2\le 2((x-z)^2+(z-y)^2)$ follows directly from the Cauchy-Schwarz inequality, and the second inequality follows from the property of vertices in $\widetilde{C}$. Next, by Fact \ref{fact:spectra} we have $\textbf{p}_v^t - \textbf{p}_u^t = \sum_{i=1}^n (\textbf{v}_i^{\textrm{reg}}(v) - \textbf{v}_i^{\textrm{reg}}(u))(1 - \frac{\lambda_i^{\textrm{reg}}}{2})^t \textbf{v}_i^{\textrm{reg}}$, and therefore
\begin{eqnarray*}
\norm{\textbf{p}_v^t - \textbf{p}_u^t}_2^2
& = &
\sum_{i=1}^n(\textbf{v}_i^{\textrm{reg}}(u)-\textbf{v}_i^{\textrm{reg}}(v))^2(1-\frac{\lambda_i^{\textrm{reg}}}{2})^{2t}\\
&=&
\sum_{i=1}^h(\textbf{v}_i^{\textrm{reg}}(u)-\textbf{v}_i^{\textrm{reg}}(v))^2(1-\frac{\lambda_i^{\textrm{reg}}}{2})^{2t}+
\sum_{i=h+1}^n(\textbf{v}_i^{\textrm{reg}}(u)-\textbf{v}_i^{\textrm{reg}}(v))^2(1-\frac{\lambda_i^{\textrm{reg}}}{2})^{2t}\\
&\le&
\sum_{i=1}^h(\textbf{v}_i^{\textrm{reg}}(u)-\textbf{v}_i^{\textrm{reg}}(v))^2 +
(1-\frac{\lambda_{h+1}^{\textrm{reg}}}{2})^{2t}\sum_{i=h+1}^n(2\textbf{v}_i^{\textrm{reg}}(u)^2+2\textbf{v}_i^{\textrm{reg}}(v)^2)\\
& \le &
\frac{16 h kd^4 \phi_{out}}{\alpha |C| \phi_{in}^2} + 4 (1-\frac{\phi_{in}^2}
{2c_{\ref{thm:highcheeger}}^2h^4})^{2t}\\
& \le &
\frac{16 k^2 d^4 \phi_{out}}{\alpha \beta n \phi_{in}^2} +
4 (1 - \frac{\phi_{in}^2}{2 c_{\ref{thm:highcheeger}}^2 k^4})^{2t}
\enspace.
\end{eqnarray*}
In the bound above, in the penultimate inequality we use the fact that
$\sum_{i=h+1}^n \textbf{v}_i^{\textrm{reg}}(u)^2 \le \sum_{i=1}^n \textbf{v}_i^{\textrm{reg}}(u)^2 = 1$ for any $u \in V$ (by Fact \ref{fact:spectra}) and $\lambda_{h+1}^{\textrm{reg}} \ge \frac{\phi_{in}^2}{c_{\ref{thm:highcheeger}}^2 h^4}$ (by Lemma \ref{lem:eigenvalue}), and in the last inequality we use that $|C| \ge \beta n$.
Now by defining $\alpha_{\ref{lem:smalll2}}:=\alpha_{\ref{lem:smalll2}}(\alpha,\beta,d,k) =\frac{\alpha\beta}{128k^2d^4}$, $c_{\ref{lem:smalll2}}:=c_{\ref{thm:highcheeger}}^2$ and letting $t \ge \frac{c_{\ref{lem:smalll2}} k^4 \log n}{\phi_{in}^2}$, we can conclude that $\norm{\textbf{p}_v^t-\textbf{p}_u^t}_2^2 \le \frac{1}{4n}$.
\end{proof}
To prove Lemma \ref{lem:smallhnorm}, we again use the eigen-decomposition of vector $\textbf{p}_u^t$ as given in Fact \ref{fact:spectra} and the fact that all eigenvalues of the normalized Laplacian of $G_\textrm{reg}$ are large except for the first few ones. This allows us to bound the $l_2^2$ norm of $\textbf{p}_u^t$ by its projection on the first few eigenvectors.
\begin{proof}[Proof of Lemma \ref{lem:smallhnorm}]
For any vertex $u \in V$, let $\delta(u) := \sum_{i=1}^k \textbf{v}_i^{\textrm{reg}}(u)^2$. Since each eigenvector $\textbf{v}_i^{\textrm{reg}}$ is of unit length, we have
\begin{eqnarray*}
\sum_{u \in V} \delta(u)
=
\sum_{u \in V} \sum_{i}^k \textbf{v}_i^{\textrm{reg}}(u)^2
=
\sum_{i}^k \sum_{u\in V} \textbf{v}_i^{\textrm{reg}}(u)^2
=
k
\enspace.
\end{eqnarray*}
Therefore, the expected value of $\delta(u)$ is at most $\frac{k}{n}$, and by the Markov's inequality, we know that for any $0 < \alpha < 1$, there exists a subset $V' \subseteq V$ such that $|V'| \ge (1-\alpha)|V|$ and that for any $u \in V'$, $\delta(u) \le \frac{k}{\alpha n}$. In addition, by Fact \ref{fact:spectra}, $\textbf{1}_u = \sum_{i=1}^n \textbf{v}_i^{\textrm{reg}}(u) \textbf{v}_i^{\textrm{reg}}$, and $\textbf{p}_u^t = \sum_{i=1}^n \textbf{v}_i^{\textrm{reg}}(u) (1 - \frac{\lambda_i^{\textrm{reg}}}{2})^t \textbf{v}_i^{\textrm{reg}}$. Therefore,
\begin{eqnarray*}
\norm{\textbf{p}_u^t}_2^2
=
\norm{\sum_{i=1}^n \textbf{v}_i^{\textrm{reg}}(u) (1-\frac{\lambda_i^{\textrm{reg}}}{2})^t \textbf{v}_i^{\textrm{reg}}}_2^2
&=&
\sum_{i=1}^n\textbf{v}_i^{\textrm{reg}}(u)^2(1 -\frac{\lambda_{i}^{\textrm{reg}}}{2})^{2t}
\\
&=&
\sum_{i=1}^k\textbf{v}_i^{\textrm{reg}}(u)^2(1- \frac{\lambda_i^{\textrm{reg}}}{2})^{2t}+ \sum_{i=k+1}^n\textbf{v}_i^{\textrm{reg}}(u)^2(1- \frac{\lambda_i^{\textrm{reg}}}{2})^{2t}
\\
& \le &
\sum_{i=1}^k\textbf{v}_i^{\textrm{reg}}(u)^2+ (1-\frac{\lambda_{k+1}^{\textrm{reg}}}{2})^{2t} \sum_{i=k+1}^n\textbf{v}_i^{\textrm{reg}}(u)^2
\\
& \le &
\delta(u)+(1-\frac{\lambda_{k+1}^{\textrm{reg}}}{2})^{2t}
\\
& \le &
\frac{k}{\alpha n} + (1 - \frac{\phi_{in}^2}{2 c_{\ref{thm:highcheeger}}^2 k^4})^{2t}
\enspace,
\end{eqnarray*}
where in the last inequality, we used the fact that $\lambda_{k+1}^{\textrm{reg}} \ge \frac{\phi_{in}^2}{c_{\ref{thm:highcheeger}}^2 k^4}$ by Lemma \ref{lem:eigenvalue}. In particular, the last bound implies that if $t \ge \frac{c_{\ref{lem:smallhnorm}} k^4 \log n}{\phi_{in}^2}$ for $c_{\ref{lem:smallhnorm}}:=c_{\ref{thm:highcheeger}}^2$, then $\norm{\textbf{p}_u^t}_2^2 \le \frac{2 k}{\alpha n}$
\end{proof}
Now we give the proof of Lemma \ref{lem:largel2}, which shows that the $l_2$-norm of the difference of two random walk distributions $\textbf{p}_v^t - \textbf{p}_u^t$ is small for most pairs $u, v$ from the two different clusters if $t$ is not too large. For any vector $\textbf{p}$ and vertex set $S$, let $\textbf{p}(S):=\sum_{v\in S}\textbf{p}(v)$.
\begin{proof}[Proof of Lemma \ref{lem:largel2}]
For any given subset $A \subseteq V$, vertex $v \in A$, and integer $t$, let $\textrm{rem}(v,t,A)$ be the event that the lazy random walk of length $t$ starting at vertex $v$ never leaves $A$ in all $t$ steps. Let $I_A$ be the diagonal matrix such that $I_A(v,v)=1$ if $v\in A$ and $0$ otherwise. Then the probability that the walk stays entirely in $A$ is $(\textbf{1}_v(\textbf{W}_\textrm{reg} I_A)^t)(A)$, that is, $\Pr[\textrm{rem}(v,t,A)] = (\textbf{1}_v(\textbf{W}_\textrm{reg} I_A)^t)(A)$. We will use the following claim.
\begin{claim}[Proposition 2.5 in \cite{ST13:local}]
\label{proposition-2.5-ST13:local}
For any $t\ge 1$ and any subset $A \subseteq V$ such that $\phi_G(A) \le \psi$, we have $\frac{\textbf{1}_v(\textbf{W}_\textrm{reg} I_A)^t(A)}{|A|} \ge 1 - t \phi_G(A)/2 \ge 1 - t \psi/2$.
\end{claim}
Let $Q_A = \{v : \Pr[\textrm{rem}(v,t,A)] \le 1 - \frac{t \psi}{2 \alpha}\}$. Then,
\begin{displaymath}
1-\frac{\textbf{1}_A}{|A|}(\textbf{W}_\textrm{reg} I_A)^t(A)
=
\sum_{v \in A}\frac{1}{|A|}(1 - \textbf{1}_v(\textbf{W}_\textrm{reg} I_A)^{t}(A))
\ge
\sum_{v \in Q_A}\frac{1}{|A|}(1 - \textbf{1}_v(\textbf{W}_\textrm{reg} I_A)^{t}(A))
\ge
\frac{|Q_A|}{|A|}\frac{t \psi}{2 \alpha}
\enspace.
\end{displaymath}
From Claim \ref{proposition-2.5-ST13:local} and the inequality above, we conclude that $|Q_A| \le \alpha |A|$. Therefore, if we set $\widehat{A} = A\setminus Q_A$, then $|\widehat{A}| \ge (1-\alpha) |A|$, and for any $v \in \widehat{A}$, $\Pr[\textrm{rem}(v,t,A)] \ge 1-\frac{t \psi}{2 \alpha}$. This proves the first part of the lemma.
To prove the second claim, we continue similarly and set $Q_B = \{v : \Pr[\textrm{rem}(v,t,B)] \le 1 - \frac{t \psi}{2 \alpha}\}$ and define $\widehat{B} = B \setminus Q_B$, to obtain that $|\widehat{B}| \ge (1-\alpha) |B|$, and for any $v \in \widehat{B}$, $\Pr[\textrm{rem}(v,t,B)] \ge 1-\frac{t \psi}{2 \alpha}$. Hence, for any $t \ge 1$ and $0 < \alpha < 1$, for any $u \in \widehat{A}$ and $v \in \widehat{B}$
\begin{displaymath}
\textbf{p}_u^t(A) \ge \Pr[\textrm{rem}(u,t,A)] \ge 1-\frac{t\psi}{2\alpha}
\quad \text{ and } \quad
\textbf{p}_v^t(B) \ge \Pr[\textrm{rem}(v,t,B)] \ge 1-\frac{t\psi}{2\alpha}
\enspace.
\end{displaymath}
Since $A$ and $B$ are disjoint, we have $\textbf{p}_v^t(A) \le \textbf{p}_v^t(V \setminus B) = 1 - \textbf{p}_v^t(B) \le \frac{t \psi}{2 \alpha}$. Therefore, for any $t \ge 1$,
\begin{eqnarray*}
\norm{\textbf{p}_u^t-\textbf{p}_v^t}_2
& \ge &
\frac{\norm{\textbf{p}_u^t-\textbf{p}_v^t}_1}{\sqrt{n}}
=
\frac{2 \max_{R \subseteq V} |\textbf{p}_u^t(R)-\textbf{p}_v^t(R)|}{\sqrt{n}}
\ge
\frac{2 (\textbf{p}_u^t(A)-\textbf{p}_v^t(A))}{\sqrt{n}}
\\
& \ge &
\frac{2 (1 - \frac{t \psi}{2 \alpha} -\frac{t \psi}{2 \alpha}) }{\sqrt{n}}
=
\frac{2 (1 - \frac{t \psi}{ \alpha})}{\sqrt{n}}
\enspace.
\end{eqnarray*}
In particular, if $t \le \frac{\alpha}{2\psi}$, then $\norm{\textbf{p}_u^t - \textbf{p}_v^t}_2 \ge \frac{1}{\sqrt{n}}$ and therefore $\norm{\textbf{p}_u^t - \textbf{p}_v^t}^2_2 \ge \frac{1}{n}$.
\end{proof}
\begin{remark}
It would be tempting to use in the above proof a somewhat stronger version of Claim \ref{proposition-2.5-ST13:local} that lower bounds the escaping probability by $\Omega(1)\cdot (1-3\psi/2)^t$ (see, for example, \cite[Proposition 3.1]{OT12:local}). However, in our proof we we require the fraction of vertices in $\widehat{A}$ to be as large as $1 - \alpha$ for any small $\alpha > 0$, which we are not aware if it is true
in the stronger version of Claim~\ref{proposition-2.5-ST13:local}.
\end{remark}
\subsection{Partitioning into large sets with small cuts: Proof of Lemma \ref{lemma:partition-eps-far-improved}} \label{sec:epsfarlemma}
\newcommand{h}%\ensuremath{\mathfrak{r}}}{h
In this section, we assume that $\varepsilon \le \frac12$ and we prove Lemma \ref{lemma:partition-eps-far-improved} that asserts that if a graph is far from $k$-clusterable then its vertex set can be partitioned into $k+1$ sets with low outer conductance.
Let $0 < c_{\exp} \le \frac12$ be a constant such that for $d=3$ and every $n$, there exists
a graph $H$ with $n$ vertices and maximum degree $d=3$ that has $\phi(H) \ge c_{\exp}$. The proof of the next lemma follows the ideas from \cite{CS10:expansion}, but it is adapted to edge expansion and works also for $d=3$ (the analysis in \cite{CS10:expansion} requires $d \ge 4$).
\begin{lemma}
\label{lemma:subset1}
Let $\alpha \le \frac{c_{\exp}}{150 d}$. If for a graph $G=(V,E)$ there is $A\subseteq V$ with $|A| \le \frac19 \varepsilon |V|$ such that $\phi(G[V \setminus A]) \ge c_{\ref{lemma:subset1}} \cdot \alpha$ for some sufficiently large constant $c_{\ref{lemma:subset1}}$, then $G$ is not $\varepsilon$-far from every graph $H$ with $\phi(H) \ge \alpha$.
\end{lemma}
\begin{proof}
Let $c_{\ref{lemma:subset1}}$ be a sufficiently large constant whose value will be determined later. Let $G$ be a graph as in the lemma and let $A \subseteq V$ be an arbitrary set such that $A \subseteq V$ with $|A| \le \frac19 \varepsilon |V|$ and $\phi(G[V \setminus A]) \ge c_{\ref{lemma:subset1}} \cdot \alpha$. We will turn $G$ into a graph $H$ by modifying at most $\varepsilon d n$ edges of $G$ and then prove that $\phi(H) \ge \alpha$. This will conclude the proof.
Our construction removes all edges between vertices in $A$ and adds an expander graph with maximum degree $3$ on $A$ that has a constant fraction of vertices of degree $2$. The degree $2$ vertices are
then connected to vertices $V\setminus A$. In order to not violate the degree bound, we have to remove some edges between vertices in $V\setminus A$, which is done using the following construction.
We will first construct an auxiliary set $S$ of size $\lceil |A|/4 \rceil$. Each element of set $S$ is an edge $\{u,v\}$ for some $u, v \in V \setminus A$ (we allow selfloops).
The set $S$ can be constructed by the following algorithm
\begin{center}
\begin{tabular}{|p{0.5\textwidth}|}
\hline
{\sc ConstructS}($G,A)$\\
\hline\\[-0.35in]
\begin{tabbing}
\hspace{0.5cm}\= $Q_L = \{ u \in V\setminus A: d_G(u) \le d-2 \}$ \\
\> $S' = \{\{v,v\}: v \in Q_L\}$\\
\> $U = (V \setminus A) \setminus Q_L$\\
\> \textbf{while} there is $v \in U$ with at least one neighbor in $U$ \textbf{do}\\
\>\hspace{0.5cm}\= let $u \in U$ be a neighbor of $v$\\
\>\> $S' = S' \cup \{\{u,v\}\}$\\
\>\> $U = U \setminus \{u,v\}$\\
\> \textbf{return} set $S$ defined as an arbitrary subset of $S'$ of size $\lceil |A|/4 \rceil$
\end{tabbing}\\
\hline
\end{tabular}
\end{center}
\junk{
\begin{center}
\begin{tabular}{|p{0.5\textwidth}|}
\hline
{\sc ConstructS}($G,A)$\\
\hline\\[-0.35in]
\begin{tabbing}
\hspace{0.5cm}\= $S = \emptyset$\\
\> $U = V-A$\\
\> {\bf for each } $v \in U$ {\bf do} \\
\> \hspace{0.5cm} \= {\bf} if $d_G(v)\le d-2$ {\bf then }\\
\>\>\hspace{0.5cm}\= $ S = S \cup \{\{v,v\}\}$\\
\>\>\> $U = U - \{v\}$\\
\> {\bf for each } $v \in U$ {\bf do}\\
\>\> {\bf if} $d_G(v) \ge d-1$ {\bf then }\\
\>\>\> {\bf if} $v$ has at least one neighbor in $U$ {\bf then } \\
\>\>\>\hspace{0.5cm}\= Let $u$ be such a neighbor of $v$\\
\>\>\>\> $S=S \cup \{\{u,v\}\}$\\
\>\>\>\> $U= U - \{u,v\}$
\end{tabbing}\\
\hline
\end{tabular}
\end{center}
}
We prove that {\sc ConstructS} ensures that $|S'| \ge \frac{1}{6} |V|$, which implies that the last step of the algorithm can always be executed and we get $|S| = \lceil |A|/4 \rceil$.
\begin{claim}
If algorithm {\sc ConstructS} is invoked with $A$ that satisfies $|A| \le \frac19 \varepsilon |V|$, $0 < \varepsilon \le \frac12$,
then the constructed set $S'$ has size at least $\frac16 |V|$.
\end{claim}
\begin{proof}
We first observe that at the end of the algorithm, each vertex in $U$ has degree at least $d-1$ and all the neighbors of vertices in $U$ belong to $V \setminus U$. This implies that the number of edges connecting $U$ and $V \setminus U$ is on one hand, at least $(d-1)|U|$, and on the other hand, it is at most $d |V \setminus U|$. Therefore, $d |V \setminus U| \ge (d-1)|U|$, and since $d \ge 3$, this yields $|V \setminus U| \ge \frac23 |U|$, and thus $|U| \le \frac35 |V|$.
Now, we observe that $|S'| \ge \frac12 |(V \setminus A) \setminus U|$, and therefore $|S'| \ge \frac12 (|V| - |A| - |U|) \ge \frac12 (|V| - \frac{1}{18} |V| - \frac35|V|) = \frac{31}{180} |V| \ge \frac16 |V|$, for every $A$ that satisfies the prerequisites of the claim.
\end{proof}
We next describe our construction of the graph $H$. If $|A| \ge 10$, then we proceed as follows. We partition $A$ into two sets $A'$ and $A''$, with $|A''| = 2 \cdot \lceil |A|/4 \rceil$. Let $H' =(A', E')$ be a graph with degree at most $3$ and
$\phi(H') \ge c_{\exp}$, whose existence follows from our definition of $c_{\exp}$. Since adding edges (while maintaining the degree bound) does not decrease the conductance and since $|A| \ge 10$, we may assume that $H'$ has at least $|A''|$ edges. Let $H^* = (A,E^*)$ be a graph obtained from $H'$ by taking an arbitrary set of $|A''|$ edges from $E'$ and replacing them by a path of length two, whose intermediate vertex is from $A''$ in such a way that every vertex from $A''$ is used exactly once.
If $1< |A| < 10$ we define $H^*=(A,E^*)$ to be a path and choose $A''$ to be an arbitrary subset of $A$ of size $2 \lceil \frac{|A|}{4} \rceil$. If $|A|=1$ we define $H^* = (A,E^*)$ with $E^* = \emptyset$, and set $A' =\emptyset$ and $A'' = A$.
Now we will modify $G$ by changing at most $\varepsilon d n$ edges to construct graph $H$ such that $\phi(H) \ge \alpha$. We first remove in $G$ all edges incident to $A$ and then all edges that connect the sets $s \in S$ in $G$ (i.e., we remove from $E$ all edges $(u,v)$ with $u, v \in s$). Then we add an arbitrary perfect matching between the vertices in $A''$ and $S$ (if a vertex appears twice in $s \in S$ then it will be matched to two vertices of $A''$; if $|A''|=1$, then the vertex $v$ from $A''$ will be match to both vertices from $s \in S$. If, in this case, $s=(u,u)$ we only add the edge $(u,v)$).
Finally, we add all edges $E^*$ from the graph $H^*$ defined above.
Our construction creates a new graph $H$ from $G$ by making at most $(d+1)|A|$ edge deletions and $3|A|$ edge insertions. Hence, we modified at most $(d+4) |A| \le \varepsilon d|V|$ edges, as required.
Next we prove that $\phi(H) \ge \alpha$. We begin with two auxiliary claims about construction of $H$.
\begin{claim}
\label{claim:edges1}
Let $X \subseteq V$ be an arbitrary set of size at most $\frac12 |V|$. Then the following holds:
\begin{displaymath}
e_H(X, V \setminus X)
\ge
\frac{1}{15} c_{\exp} \cdot \min\{|X \cap A| , |A \setminus X|\}
\enspace.
\end{displaymath}
\end{claim}
\begin{proof}
If $|A| = 1$ the claim trivially holds for every set $X$. Thus, we can assume $|A| \ge 2$. Let $X$ be a subset of $V$ of size at most $\frac 12 |V|$. If $|A| < 10$, we get $e_H(X, V \setminus X) \ge e_H(X \cap A, A \setminus X) \ge \frac{1}{10} \cdot \min\{|X \cap A| , |A \setminus X|\}$, since either the minimum is $0$ or there is at least one edge connecting the two sets. Since $c_{exp}\le \frac12$, this implies the claim.
Now we consider the case $|A| \ge 10$. Consider an arbitrary set $Y \subseteq A$ with $|Y| \le \frac12 |A|$. Let $Y' = Y \cap A'$ and $Y'' = Y \cap A''$. Let us first focus on the construction of graph $H^*$ (which is a subgraph of $H$). Let $Y^* \subseteq Y''$ be the set of vertices from $Y''$ with both of its neighbors (in $H^*$) to be in $Y$ (and hence, in fact, in $Y' \subseteq A'$).
We consider two cases. If $|Y'' \setminus Y^*| \ge \frac12 |Y|$ then since each vertex in $Y'' \setminus Y^*$ is adjacent in $H^*$ to at least one vertex not in $Y$, we obtain $e_{H^*}(Y, A \setminus Y) \ge |Y'' \setminus Y^*| \ge \frac12 |Y|$.
Otherwise we have $|Y'' \setminus Y^*| < \frac12 |Y|$, and thus $|Y'| + |Y^*| > \frac12 |Y|$. Since each vertex in $Y'$ has degree at most $3$ in $H^*$ and each vertex in $Y^*$ is adjacent in $H^*$ to exactly two vertices from $Y'$, we have $|Y^*| \le \frac32 |Y'|$. Hence, if we combine the bounds $|Y'| + |Y^*| > \frac12 |Y|$ and $|Y^*| \le \frac32 |Y'|$, then we obtain $|Y'| > \frac15 |Y|$.
Now we make another case distinction.
If $|Y'| \le \frac{9}{10} |A'|$, then $|A' \setminus Y'| \ge \frac{1}{10} |A'| \ge \frac 19 |Y'|$. Note that in our construction of $H^*$ from $H'$, if an edge $(u,v)$ with $u \in A' \setminus Y'$ and $v\in Y'$ is replaced by a path of length $2$ with intermediate vertex $w\in A''$, then at least one of the edges $(u,w)$ and $(v,w)$ lies between $Y$ and $A \setminus Y$ in $H^*$. Therefore,
\begin{displaymath}
e_{H^*}(Y, A \setminus Y)
\ge
e_{H'}(Y', A \setminus Y')
\ge
3 c_{exp} \min\{|Y'|, |A' \setminus Y'|\}
\ge
\tfrac13 c_{exp} |Y'|
\ge
\tfrac{1}{15} c_{exp} |Y|
\enspace.
\end{displaymath}
Otherwise, $|Y'| \ge \frac{9}{10} |A'|$.
In our construction we replace $2 \cdot \lceil |A|/4 \rceil$ edges of $H'$ by paths of length $2$. Since $|A' \setminus Y'| \le \frac {1}{10} |A'| \le\frac {1}{10} |A| $ and since $H'$ has maximum degree $3$, there are at most $\frac{3}{20} |A|$ edges with both endpoints in $A'\setminus Y'$ that are replaced. Therefore, there are $2 \lceil |A|/4 \rceil - \frac{3}{20} |A| \ge \frac{7}{20} |A|$ edges replaced that in $H'$ are incident to a vertex from $Y'$. Thus, in $H^*$ there are at least $\frac{7}{20} |A|$ edges leaving $Y'$.
Since $|Y'| \ge \frac{9}{10} |A'|$ and
$|A'| \ge \frac 25 |A|$, we have $|Y'| \ge \frac{9}{25} |A|$.
Therefore our assumption that $|Y| \le \frac12 |A|$ yields $|Y \setminus Y'| \le \frac{7}{50} |A|$. This gives us
$e_{H^*}(Y, A \setminus Y) \ge \frac{7}{20} |A| - \frac{7}{50} |A| = \frac{21}{100} |A| \ge \frac{1}{5} |A|\ge \frac{1}{5} |Y| \ge \frac{1}{15} c_{\exp} |Y|$.
Therefore, we get $e_{H^*}(Y, A \setminus Y) \ge \frac{1}{15} c_{\exp} \cdot |Y|$ for the case that $|Y'' \setminus Y^*| < \frac12 |Y|$.
If we combine the bounds for these two cases together, then we obtain that for any $Y \subseteq A$ with $|Y| \le \frac12 |A|$, we have $e_{H^*}(Y, A \setminus Y) \ge \min\{\frac12 |Y|, \frac{1}{15} c_{\exp} |Y|\} = \frac{1}{15} c_{\exp} |Y|$. This further implies that for any $Y \subseteq A$, $e_{H^*}(Y, A \setminus Y) \ge \frac{1}{15} c_{\exp}\min\{|Y|, |A \setminus Y|\}$.
Now we will extend the analysis to the graph $H$. We have $e_H(X, V \setminus X) \ge e_{H^*}(X \cap A, A \setminus X) \ge \frac{1}{15} c_{\exp} \min\{|X \cap A|, |A \setminus X|\}$.
\end{proof}
\begin{claim}
\label{claim:edges2}
Let $X \subseteq V$ be an arbitrary set of size at most $\frac12 |V|$, $A \subseteq V$ with $|A| \le \frac19 \varepsilon |V|$ and $\varepsilon \le \frac12$. Then the following holds
\begin{displaymath}
e_H(X, V \setminus X)
\ge
\tfrac45 \cdot c_{\ref{lemma:subset1}} \cdot d \cdot \alpha \cdot |(V \setminus A) \cap X| -
\min\{|X \cap A|, |A \setminus X|\}
\enspace.
\end{displaymath}
\end{claim}
\begin{proof}
For simplicity of notation, let us define $B = V \setminus A$. Using the assumption $|A| \le \frac19 \varepsilon |V|$ and $\varepsilon \le \frac12$, we obtain $|B| \ge (1 - \frac19 \varepsilon) |V| \ge \frac{17}{18} |V|$. Therefore, since $|B \cap X| \le |X| \le \frac12 |V|$, we obtain $|B \cap X| \le \frac{9}{17} \cdot |B|$, and hence $|B \setminus X| = |B| - |B \cap X| \ge \frac{8}{17} \cdot |B|$, what yields $\min\{|B \cap X|, |B \setminus X|\} \ge \frac89 |B \cap X|$.
Next, by the assumption about set $A$ in Lemma \ref{lemma:subset1}, we know that $\phi(G[B]) \ge c_{\ref{lemma:subset1}} \cdot \alpha$. Therefore, $e_{G[B]}(B \cap X, B \setminus X) \ge c_{\ref{lemma:subset1}} \alpha d \min\{|B \cap X|, |B \setminus X|\} \ge \frac89 c_{\ref{lemma:subset1}} \alpha d |B \cap X|$.
The only edges that are removed from $G[B]$ in order to obtain $H$ are the edges between vertices $u,v$ with $u,v \in s$ for all $s \in S$. Consider such an edge $(u,v)$ with $u, v \in s$, $s \in S$. Since we are analysing the size of the cut between $B \cap X$ and $B \setminus X$, we only consider $u \in B \cap X$ and $v \in B \setminus X$. By our construction of $H$, both $u$ and $v$ are connected in $H$ to vertices in $A$. If $u$ is connected to a vertex in $A \setminus X$ or $v$ to a vertex in $A \cap X$, then we get a new cut edge between $B \cap X$ and $B \setminus X$, and thus this will compensate the removal of edge $(u,v)$ from $G[B]$. Therefore, we decrease the number of edges in the cut between $B \cap X$ and $B \setminus X$ only if $u$ is connected to a vertex in $A \cap X$ and $v$ is connected to a vertex in $A \setminus X$. Each vertex in $A$ is adjacent in $H$ to at most one vertex from outside $A$, and therefore the number of such edges is bounded by $\min\{|X \cap A|, |A \setminus X|\}$.
If we summarize this, we obtain $e_H(X, V \setminus X) \ge e_{G[B]}(B \cap X, B \setminus X) - \min\{|X \cap A|, |A \setminus X|\} \ge \frac89 c_{\ref{lemma:subset1}} \alpha d |B \cap X| - \min\{|X \cap A|, |A \setminus X|\} \ge \frac45 c_{\ref{lemma:subset1}} \alpha d |B \cap X| - \min\{|X \cap A|, |A \setminus X|\}$.
\end{proof}
With Claims \ref{claim:edges1} and \ref{claim:edges2} at hand, we are ready to conclude the proof of Lemma \ref{lemma:subset1}. Take an arbitrary set $X \subseteq V$ of size at most $\frac12 |V|$. We will prove that $e_H(X,V \setminus X) \ge \alpha d |X|$, what would immediately imply that $\phi(H) \ge \alpha$
If $\min\{|X \cap A|, |A \setminus X|\} \ge \frac {15 \cdot d \cdot \alpha}{c_{\exp}} \cdot |X|$, then Claim \ref{claim:edges1} gives that $e_H(X,V \setminus X) \ge \alpha d |X|$. Otherwise, we have $\min\{|X \cap A|, |A \setminus X|\} < \frac{15 \cdot d \cdot \alpha}{c_{\exp}} \cdot |X| \le \frac{1}{10} \cdot |X|$ for our choice of $\alpha$. If the minimum is attained by $|X \cap A|$, then we have $|(V \setminus A) \cap X| \ge \frac{9}{10} \cdot |X|$. Thus Claim \ref{claim:edges2} implies that assuming that $c_{\ref{lemma:subset1}} \ge \frac{30}{c_{\exp}}$, we have $e_H(X,V \setminus X) \ge \frac45 \cdot d \cdot c_{\ref{lemma:subset1}} \cdot \alpha \cdot \frac{9|X|}{10} - \frac{15 \cdot d \cdot \alpha}{c_{\exp}}|X| \ge \alpha d\cdot |X|$.
If the minimum is attained by $|A \setminus X|$ we consider two cases. If $|(V\setminus A) \cap X| \le \frac{1}{16} |A|$ then $|X| \le |A| + |(V\setminus A) \cap X| \le \frac{17}{16} |A|$.
In this case, $|X\cap A| = | A \setminus (A\setminus X)| = |A| - |A \setminus X| \ge |A| - |X|/10 \ge \frac{143}{160} |A| \ge \frac45 |A|$.
Since $|A''| \ge \frac12 |A|$ we obtain that $|X \cap A''| \ge |X\cap A| - |A'| \ge \frac45 |A| - \frac12 |A| \ge \frac{3}{10} |A|$.
By construction of $H$ each vertex in $A''$ is connected to a vertex in $V \setminus A$ and each vertex in $V\setminus A$ is connected to at most $2$ vertices in $A''$. Since
$|(V\setminus A) \cap X| \le \frac{1}{16} |A|$ there are at most $\frac18 |A|$ vertices of $(V \setminus A) \cap X$ connected to vertices from $X\cap A''$. Hence, for our choice of $\alpha$
there are at least $\frac{3}{10} |A| - \frac18 |A| \ge \frac{1}{10} |A| \ge \frac{16}{170} |X| \ge \alpha d |X|$ edges leaving $X$.
If $|(V \setminus A) \cap X| > \frac{1}{16} |A|$, then $|(V \setminus A) \cap X| > \frac{1}{16} |A \cap X|$ and thus $|(V \setminus A) \cap X| > \frac{1}{17} |X|$.
Now if $c_{\ref{lemma:subset1}} \ge \frac{350}{c_{\exp}}$, Claim \ref{claim:edges2} gives that $e_H(X,V \setminus X) \ge \frac45 \cdot d\cdot c_{\ref{lemma:subset1}} \cdot \alpha \cdot \frac{|X|}{17} - \frac{15 \cdot \alpha d}{c_{\exp}}|X| \ge \alpha d |X|$. Therefore, Lemma \ref{lemma:subset1} follows with $c_{\ref{lemma:subset1}} = \frac{350}{c_{\exp}}$.
\end{proof}
Lemma \ref{lemma:subset1} can be applied to construct a large set $A$ with a small cut, as in the following lemma.
\begin{lemma}
\label{lemma:subset}
Let $0 < \alpha \le \frac{c_{\exp}}{150d}$ and $0 < \varepsilon \le \frac12$. If $G = (V,E)$ is $\varepsilon$-far from any graph $H$ with $\phi(H) \ge \alpha$, then there is a subset of vertices
$A \subseteq V$ with $\frac{1}{18} \varepsilon |V| \le |A| \le \frac12 |V|$ such that $\phi_G(A) \le c_{\ref{lemma:subset}} \cdot \alpha$, for some sufficiently large constant $c_{\ref{lemma:subset}}$.
In particular, $e(A, V \setminus A) \le c_{\ref{lemma:subset}} \cdot \alpha \cdot d \cdot |A|$.
\end{lemma}
\begin{proof}
Lemma \ref{lemma:subset1} ensures that if $G$ is $\varepsilon$-far from any graph $H$ with $\phi(H) \ge \alpha$, then for all $A' \subseteq V$ with $|A'| \le \frac19 \varepsilon |V|$ we have $\phi(G[V \setminus A']) < c_{\ref{lemma:subset1}} \cdot \alpha$. In particular, in our case, this will mean that there is a set $B \subseteq V \setminus A'$ with $|B| \le \frac12 |V \setminus A'|$ such that
$e(B, (V \setminus (A' \cup B)) < c_{\ref{lemma:subset1}} \cdot \alpha \cdot d \cdot |B|$.
We will now repeatedly apply Lemma \ref{lemma:subset1} to construct a large set $A$ satisfying the requirements of Lemma \ref{lemma:subset}. Let $A_1 = \emptyset$. We apply Lemma \ref{lemma:subset1} with $A' = A_1$ to obtain a set $A_2$ with $|A_2| \le \frac12 |V \setminus A'|$ and $\phi_{G[V \setminus A']}(A_2) \le c_{\ref{lemma:subset1}} \cdot \alpha$. If $|A_1 \cup A_2| \ge \frac19 \varepsilon |V|$ then we are done. Otherwise, we set $A' = A_1 \cup A_2$ and repeat this process. We continue this process until for the first time, we obtain a set $A_i$ such that $|A_1 \cup \dots \cup A_i| \ge \frac19 \varepsilon |V|$. In that moment, if $|A_i| \ge |A_1 \cup \cdots \cup A_{i-1}|$ then we set $A = A_i$ and otherwise, we put $A = A_1 \cup \dots \cup A_i$.
Our construction ensures that since $|A_1 \cup \dots \cup A_i| \ge \frac19 \varepsilon |V|$, then we have $|A| \ge \frac{1}{18} \varepsilon |V|$.
The upper bound on the size of $A$ follows since $|A_i| \le \frac12 |V|$ and $|A_1 \cup \dots \cup A_{i-1}| < \frac19 \varepsilon |V|$.
Our construction ensures that for every $1 \le j \le i$, $e(A_j, V \setminus (A_1 \cup \dots \cup A_j)) \le c_{\ref{lemma:subset1}} \cdot \alpha \cdot d \cdot |A_j|$. Therefore, since we have $e(A_1 \cup \dots \cup A_j, V \setminus (A_1 \cup \dots \cup A_j)) \le \sum_{s=1}^j e(A_s, V \setminus (A_1 \cup \dots \cup A_s))$, we conclude that $e(A_1 \cup \dots \cup A_j, V \setminus (A_1 \cup \dots \cup A_j)) \le c_{\ref{lemma:subset1}} \cdot \alpha \cdot d \cdot |A_1 \cup \dots \cup A_j|$. Hence, if $A = A_1 \cup \dots \cup A_i$ then we obtain $e(A, V \setminus A) \le c_{\ref{lemma:subset1}} \cdot \alpha \cdot d \cdot |A|$, and if $A = A_i$ then we obtain
\begin{eqnarray*}
e(A, V \setminus A)
& = &
e(A_i, A_1 \cup \dots \cup A_{i-1}) +
e(A_i, V \setminus (A_1 \cup \dots \cup A_i))
\\
& \le &
e(A_1 \cup \dots \cup A_{i-1}, V \setminus (A_1 \cup \dots \cup A_{i-1})) +
e(A_i, V \setminus (A_1 \cup \dots \cup A_i))
\\
& \le &
c_{\ref{lemma:subset1}} \cdot \alpha \cdot d \cdot |A_1 \cup \dots \cup A_{i-1}| +
c_{\ref{lemma:subset1}} \cdot \alpha \cdot d \cdot |A_i|
\\
& \le &
2 c_{\ref{lemma:subset1}} \cdot \alpha \cdot d \cdot |A|
\enspace,
\end{eqnarray*}
where in the last inequality we use the fact that $|A| = |A_i| \ge |A_1 \cup \cdots \cup A_{i-1}|$.
This completes the proof by setting $c_{\ref{lemma:subset}} = 2 c_{\ref{lemma:subset1}}$.
\end{proof}
Let us extend the notion $e(U_1,U_2)$ to multiple sets and for disjoint subsets $V_1, \dots, V_h$, let us define $e(V_1, \dots, V_h) = \sum_{1 \le i < j \le h} e(V_i, V_j)$.
\begin{lemma}
\label{lemma:partition}
Let $G = (V,E)$ be $\varepsilon$-far from $(k,\phi_{in}^*,\phi_{out}^*)$-clusterable and $\phi_{in}^* \le c_{\exp}/d$. If there is a partition of $V$ into $h$ sets $V_1, \dots, V_h$ with $1 \le h \le k$, such that $e(V_1, \dots, V_h) = 0$, then there is an index $i$, $1 \le i \le h$, with $|V_i| \ge \frac{1}{8k} \cdot \varepsilon |V|$ such that $G[V_i]$ is $\frac{\varepsilon}{2}$-far from any $H$ on vertex set $V_i$ with maximum degree $d$ and $\phi(H) \ge \phi_{in}^*$.
\end{lemma}
\begin{proof}
Let us renumber the indices of sets $V_1, \dots, V_{h}%\ensuremath{\mathfrak{r}}}$ such that $|V_i| \ge |V_{i+1}|$ for every $i$, $1 \le i < h}%\ensuremath{\mathfrak{r}}$. A set $V_i$ with more than $\frac{1}{8k} \varepsilon |V|$ vertices is called \emph{large} and otherwise it is called \emph{small}. Let $s$ be the largest index such that $V_s$ is large. (Simple counting arguments implies that we must have $|V_1| \ge \frac{|V|}{h}%\ensuremath{\mathfrak{r}}}$ (for otherwise we would have $|V_i| < \frac{|V|}{h}%\ensuremath{\mathfrak{r}}}$ for every $i$, $1 \le i \le \ell$, and thus $\sum_{i=1}^{h}%\ensuremath{\mathfrak{r}}}|V_i| < |V|$, which is a contradiction to the fact that $V_1, \dots, V_{h}%\ensuremath{\mathfrak{r}}}$ is a partition of $V$), and hence $V_1$ is large and $s$ is well-defined.) Next, let us observe that $\sum_{1 \le i \le h}%\ensuremath{\mathfrak{r}}: V_i \text{ is small}} |V_i| \le \frac{1}{8k} k \varepsilon |V| = \frac18 \varepsilon |V|$. This follows from $h}%\ensuremath{\mathfrak{r}} \le k$ and from the fact that for a small set $V_i$ we have $|V_i| \le \frac{1}{8k} \varepsilon |V|$.
Let us construct from $G$ a new graph $G^*$ of maximum degree at most $d$ as follows. Define $U = \bigcup_{i: V_i \text{ is small}} V_i = \bigcup_{i=s+1}^{h}%\ensuremath{\mathfrak{r}}} V_i$, and remove in $G$ all edges incident to any vertex in $U$. Then build a degree $3$ \ $c_{\exp}$-expander on $U$ and add it to the graph. Note that with respect to $d$, this expander is a $\frac{c_{exp}}{d}$-expander. Call the obtained graph $G^*$.
Observe that $G^*$ has been obtained from $G$ by adding/inserting at most $ d |U| + 3 |U|$ edges, where the first term corresponds to the removal of all edges incident to $U$ and the second term corresponds to building the degree $3$ $c_{\exp}$-expander on $U$.
Now, since $|U| = \sum_{1 \le i \le h}%\ensuremath{\mathfrak{r}}: V_i \text{ is small}} |V_i| \le \frac18 \varepsilon |V|$, as we have shown above, we note that $G^*$ is obtained from $G$ by adding/deleting at most $ 2\cdot \frac{d}{8} \varepsilon |V| \le \frac{d}{2} \varepsilon |V|$ edges.
Hence, since $G^*$ has maximum degree at most $d$, $G^*$ is $\frac12 \varepsilon$-far from $(k, \phi^*_{in}, \phi^*_{out})$-clusterable.
Observe the structure of $G^*$: it consists of a $\frac{c_{\exp}}{d}$-expander on $U$ and $s$ disjoint components (not necessarily connected) on vertex sets $V_i$ with each $V_i$ being a large set and $G^*[V_i] = G[V_i]$; further, $\phi_{G^*}(U) = \phi_{G^*}(V_1) = \dots = \phi_{G^*}(V_s) = 0$.
For every $i$, $1 \le i \le s$, let us define $H_i$ to be the graph on vertex set $V_i$ with maximum degree at most $d$, with $\phi(H_i) \ge \phi^*_{in}$, and that is obtained from $G^*[V_i]$ by the minimum number of addition/deletion of the edges; let $\kappa_i$ be the number of addition/deletion of the edges needed to transform $G^*[V_i]$ into $H_i$.
Let us observe that the graph $H$ on $V$ obtained as the union of $G^*[U]$ and $H_1, \dots, H_s$ is $(k, \phi^*_{in}, \phi^*_{out})$-clusterable. Indeed, since we have $H[U] = G^*[U]$, $H[V_i] = H_i$ for every $i$, $1 \le i \le s$, and $\phi_H(U) = \phi_H(V_1) = \dots = \phi_H(V_s) = 0$, for the partition of $V$ into $U$, $V_1, \dots, V_s$, we obtain that $\phi(H[U]) \ge c_{\exp}/d \ge \phi^*_{in}$ for every $i$, $1 \le i \le s$, and $\phi_H(U) = \phi_H(V_1) = \dots = \phi_H(V_s) = 0 \le \phi^*_{out}$.
We now note that $H$ is obtained from $G^*$ by adding $\sum_{i=1}^s \kappa_i$ edges. Therefore, since $G^*$ is $\frac12 \varepsilon$-far from $(k, \phi^*_{in}, \phi^*_{out})$-clusterable, since $H$ is $(k, \phi^*_{in}, \phi^*_{out})$-clusterable, we must have $\sum_{i=1}^s \kappa_i > \frac12 \varepsilon d |V|$, and thus $\sum_{i=1}^s \kappa_i > \frac12 \varepsilon d \sum_{i=1}^s |V_i|$. Therefore, there must be at least one $j$, $1 \le j \le s$, with $\kappa_j > \frac12 \varepsilon d |V_j|$. In that case, for such a $j$, by the definition of $H_j$, $G^*[V_j] = G[V_j]$ must be $\frac12 \varepsilon$-far from any graph $Q$ on vertex set $V_j$ with $\phi(Q) \ge \phi^*_{in}$ (any such a graph $Q$ must be obtained from $G^*[V_i]$ by at least $\kappa_j > \frac12 \varepsilon d |V_j|$ addition/deletion of the edges), as required.
\end{proof}
We are now ready to prove Lemma \ref{lemma:partition-eps-far-improved}. We will set $\alpha_{\ref{lemma:partition-eps-far-improved}} = \min\{ \frac{c_{exp}}{150d} , \frac{1}{2 k c_{\ref{lemma:subset}}}\}$, and thus we have $\phi_{in}^* \le \frac{\varepsilon}{2 k c_{\ref{lemma:subset}}}$.
Our proof is by induction: we will construct a sequence of partitions $\{V_1\}$, $\{V_1, V_2\}, \dots, \{V_1, \dots, V_{k+1}\}$ of $V$ such that each partition $\{V_1, \dots, V_h\}$ satisfies the following properties:
\begin{enumerate}[(a)]
\item\label{case-a} $|V_i| \ge \frac{\varepsilon^2}{1152k} |V|$ for every $i$, $1 \le i \le h$, and
\item\label{case-b} $e(V_1, \dots, V_h) \le (h-1) \cdot c_{\ref{lemma:subset}} \cdot \phi_{in}^* \cdot d \cdot |V|$.
\end{enumerate}
Our first partition is the trivial partition $\{V\}$, which clearly satisfies our properties. We then apply inductively Lemma \ref{lemma:partition}. Let us consider some partition $\{V_1, \dots, V_h\}$ with $1 \le h \le k$ and assume that this partition satisfies (\ref{case-a}) and (\ref{case-b}). We will show how to refine it to obtain a partition $\{V_1, \dots, V_{h+1}\}$ satisfying properties (\ref{case-a}) and (\ref{case-b}).
Let us first remove from $G$ all edges between pairs of all distinct sets $V_i$ and $V_j$, $1 \le i < j \le h$, to obtain a graph $G'$. Since $\phi_{in}^* \le \frac{\varepsilon}{2 k c_{\ref{lemma:subset}}}$, we have removed $e(V_1, \dots, V_h) \le (h-1) \cdot c_{\ref{lemma:subset}} \cdot \phi_{in}^* \cdot d \cdot |V| \le \frac12 \varepsilon d |V|$ edges from $G$, and therefore $G'$ is $\varepsilon/2$-far from $(k,\phi_{in}^*,\phi_{out}^*)$-clusterable and such that our partition satisfies the prerequisites of Lemma \ref{lemma:partition}.
Then, by Lemma \ref{lemma:partition}, there is a set $V_{i^*}$ with $1 \le i^* \le h$, such that $|V_{i^*}| \ge \frac{1}{8k} \cdot \frac{\varepsilon}{2} \cdot |V|$ and $G'[V_{i^*}] = G[V_{i^*}]$ is $\frac{\varepsilon}{4}$-far from any $H$ on vertex set $V_{i^*}$ with maximum degree $d$ and $\phi(H) \ge \phi_{in}^*$. Next, we apply Lemma \ref{lemma:subset} on $V_{i^*}$ to obtain a set $A \subseteq V_{i^*}$ with
$\frac{\varepsilon/4}{18} \cdot |V_{i^*}| \le |A| \le \frac12 |V_{i^*}|$ such that $e(A, V_{i^*} \setminus A) \le c_{\ref{lemma:subset}} \phi_{in}^* d |V_{i^*}| \le c_{\ref{lemma:subset}} \phi_{in}^* d |V|$. This gives us our new partition $\{V_1, \dots, A, V_{i^*} \setminus A, \dots, V_h\}$.
Using the bound for the size of $V_{i^*}$, we have
$|A| \ge \frac{\varepsilon/4}{18} \cdot |V_{i^*}| \ge \frac{\varepsilon^2}{1152k} \cdot |V|$ and $|V_{i^*} \setminus A| \ge \frac12 |V_{i^*}| \ge \frac{\varepsilon}{32k} \cdot |V|$, and therefore by the induction hypothesis, our new partition satisfies (\ref{case-a}).
In order to prove (\ref{case-b}), we observe the following
\begin{eqnarray*}
e(V_1, \dots, A, V_{i^*} \setminus A, \dots, V_h)
& \le &
e(V_1, \dots, V_{i^*}, \dots, V_h) +
e(A, V_{i^*} \setminus A)
\\
& \le &
(h-1) \cdot c_{\ref{lemma:subset}} \cdot \phi_{in}^* \cdot d \cdot |V| +
c_{\ref{lemma:subset}} \cdot \phi_{in}^* \cdot d \cdot |V|
\\
& = &
h \cdot c_{\ref{lemma:subset}} \cdot \phi_{in}^* \cdot d \cdot |V|
\enspace,
\end{eqnarray*}
where the second inequality follows from our induction hypothesis and the bound above.
In summary, we have proven by induction the existence of a partition of $V$ into $k+1$ sets $V_1, \dots, V_{k+1}$ such that properties (\ref{case-a}) and (\ref{case-b}) are satisfied. Note that since property (\ref{case-b}) implies that for every $i$, $1 \le i \le k+1$, $e(V_i, V \setminus V_i) \le k \cdot c_{\ref{lemma:subset}} \cdot \phi_{in}^* \cdot d \cdot |V|$, we have
\begin{eqnarray*}
\phi_G(V_i)
& = &
\frac{e(V_i, V \setminus V_i)}{d |V_i|}
\, \le \,
\frac{k \cdot c_{\ref{lemma:subset}} \cdot \phi_{in}^* \cdot d \cdot |V|}
{d \cdot |V_i|}
\, \le \,
\frac{k \cdot c_{\ref{lemma:subset}} \cdot \phi_{in}^* \cdot |V|}
{\frac{\varepsilon^2 |V|}{1152k}}
\, = \,
\frac{1152 \cdot k^2 \cdot c_{\ref{lemma:subset}}}{\varepsilon^2}
\cdot \phi_{in}^*
\enspace.
\end{eqnarray*}
Therefore, Lemma \ref{lemma:partition-eps-far-improved} follows by setting
$c_{\ref{lemma:partition-eps-far-improved}} = 1152 \cdot k^2 \cdot c_{\ref{lemma:subset}}$.
\qed
\section{Conclusion}
\label{sec:conclusion}
We presented the first study of testing the clusterability of a graph in the bounded degree model, where we used both the inner conductance and outer conductance of a set to measure the quality of a cluster \cite{OT14:expander}. Our main result is an asymptotically optimal (up to polylogarithmic factors) algorithm with running time $\widetilde{O}(\sqrt{n}\cdot \textrm{poly}({d,k,\varepsilon}))$ to test if a graph is $(k,\phi)$-clusterable or is $\varepsilon$-far from $(k, \phi^*)$-clusterable for $\phi^* = O_{d,k}(\frac{\phi^2 \varepsilon^4}{\log n})$. Our tester uses new ideas of testing pairwise closeness of distributions of random walks starting from a pair of sample vertices and draws from that conclusions on the graph structure. One of the key techniques underlying our analysis is a new application of the recent results on higher order Cheeger inequalities \cite{LOT12:high}.
For further research, one of the major open problem is to narrow the gap between $\phi$ and $\phi^*$, or to prove that the current gap is almost optimal for any tester with similar running time. As we discussed in Section \ref{subsection:Expansion}, fundamentally new ideas are needed here.
It would also be very interesting to gain deeper insights of the structure of graphs that are $\varepsilon$-far from $(k,\phi^*)$-clusterable, that is, to improve Lemma \ref{lemma:partition-eps-far-improved}. More specifically, is it possible to get rid of the dependency of $\varepsilon$ of the upper bounds for inner and/or outer conductance in Lemma \ref{lemma:partition-eps-far-improved}?
\bibliographystyle{alphabetic}
|
{
"timestamp": "2015-04-14T02:17:42",
"yymm": "1504",
"arxiv_id": "1504.03294",
"language": "en",
"url": "https://arxiv.org/abs/1504.03294"
}
|
\section{Introduction}
It is not known if the electroweak model can be non-perturbatively quantized.
This requires the convergence of the unexpanded functional integrals over all classical field configurations for the vacuum expectation values of its field operators.
It is assumed that the integrals have been continued to Euclidean space to make mathematical sense out of them and that ultraviolet and volume cutoffs are in place in their integrands.
Their introduction will be discussed later.
Since the quantization is non-perturbative most of the functional integrals cannot be done explicitly.
Therefore, the criteria for the non-perturbative renormalization of the model are not known \textit{ab initio}.
Immediately one is confronted with an external field problem: do the regulated integrands grow slowly enough with large amplitude field variations for the functional integrals to converge? It is the aim of this paper to examine this minimal requirement for the non-perturbative quantization of the electroweak model.
Presumably the order of doing the functional integrals is irrelevant aside from their technical difficulty.
If so, it is reasonable to begin with what is well-known.
Accordingly, we first integrate out the fermions.
Then the answer to the above question partly depends on knowing the strong field behavior of each of the 6 lepton and 3$\times$6 quark determinants obtained by integrating out the three generations of leptons and quarks, including their three colors.
For example, the electron and its associated neutrino field\footnote{The extension of the model to massive neutrinos and their mixing is not considered here as it will not affect the main results of this paper.} contribute the following factor to the Euclidean functional integral representation of any electroweak process after spontaneous symmetry breaking:
\begin{widetext}
\begin{align}
\begin{split}
\det &\left[ \slashed P + m_e + e\slashed A
+ \frac{g}{2\cos\theta_W}\slashed Z \left( \frac{1-\gamma_5}{2}\right)
- \frac{g\sin^2\theta_W}{\cos\theta_W}\slashed Z
+ \frac{gm_e}{2M_W}H\right]\times\\
&\det \left[ \slashed P - \frac{g}{2\cos\theta_W}\slashed Z \left (
\frac{1-\gamma_5}{2}\right )
-\frac{g^2}{2}\slashed W^+
\left ( \frac{1-\gamma_5}{2}\right ) S_e\slashed W^-
\left ( \frac{1-\gamma_5}{2}\right )\right ].
\end{split}
\tag{1.1}
\label{eq:onepointone}
\end{align}
\end{widetext}
Here $A_\mu$, $Z_\mu$, $W^\pm_\mu$, and $H$ are the Maxwell, neutral and charged vector boson and Higgs fields; $S_e$, the inverse of the operator in brackets in the first determinant, is the electron propagator in the presence of the $A$, $Z$ and $H$ fields; $m_e$ and $M_W$ are the electron and $W$-boson masses; $e$ is the positron electric charge; $\theta_W$ is the Weinberg angle and $g = e/\sin\theta_W$.
The result in (\ref{eq:onepointone}) follows by inspection of the electroweak Lagrangian \cite{1} and an elementary integration over the electroweak action quadratic in the fermion fields \cite{2}.
The twenty-four determinants multiply the Gaussian measures $d\mu(A)\,d\mu(Z)\,d\mu(W)\,d\mu(H)$ as does the remainder of the electroweak action denoted by $\exp\left[-\int \mathrm d^4x\, \mathcal L (A, Z, W^\pm, H)\right]$. Considering the complexity of the Feynman rules in the 't Hooft-Feynman gauge a non-perturbative calculation may simplify in the unitary gauge. The absence of the Goldstone bosons $\chi, \varphi^{\pm}$ in the determinants in (\ref{eq:onepointone}) indicates that this gauge has been selected.
An ultraviolet cutoff has to be introduced into the $A, Z, W$ and $H$ field propagators. As these fields are to be integrated over they are assumed to be tempered distributions. In order to calculate the fermion determinants these fields need to be smoothed following the procedure outlined at the beginning of Sec. VII for QED. The smoothing procedure introduces an ultraviolet cutoff in the associated propagators when calculating the fields' covariances with the above Gaussian gauge-fixed measures as in Eq.(\ref{eq:7.2}). Thus the ultraviolet cutoffs are introduced by functionally integrating the electroweak model.
The fermion determinants contain all fermion loops and hence the anomalies. The process for cancelling them in this paper begins by noting that the determinants, such as those in (\ref{eq:onepointone}), are ill-defined as they stand. Mathematical sense can be made of them by subtracting out all loops whose degree of divergence is 2, 1 and 0. The subtraction process is illustrated by (\ref{eq:F1}) in Appendix F for the case of QED. As a representative example consider the
$\gamma W^+ W^-$ triangle graph containing three fermion propagators. Schematically the electron neutrino determinant in (\ref{eq:onepointone}) is subtracted so that
det$\rightarrow \exp[\Pi(ee\nu_e)+\text{other subtractions}]\times \text{det}_R $, where det$_R$ is a well-defined remainder determinant similar to det$_5$ in (\ref{eq:F1}) and (\ref{eq:F2}); $\Pi(ee\nu_e)$ denotes the first generation lepton triangle graph for $\gamma\rightarrow W^+ W^-$. When the 23 remaining determinants are subtracted the exponentiated subtractions combine to give the following result for the sum of all the graphs contributing to the first generation $\gamma W^+ W^-$ triangle anomaly:
\begin{align}
\begin{split}
\exp \{ \Pi(ee\nu_e) &+ 3[\Pi(ddu) + \Pi(uud)] |V_{ud}|^2 \\
&+ 3[\Pi(ssu) + \Pi(uus)] |V_{us}|^2 \\
&+ 3[\Pi(bbu) + \Pi(uub)] |V_{ub}|^2 \\
&+ \text{other subtractions} \} \times \Pi_{i=1}^{24}\text{det}_{R_i}.
\label{eq:onepointtwo}
\end{split}
\tag{1.2}
\end{align}
Here $u, d, s ,b$ refer to quark flavors and $V_{ij}$ is the CKM quark mixing matrix \cite{1}. The anomaly is removed by subtracting out the zero-mass limit of these graphs which we denote by $\Pi_0$ . Then the anomaly bearing graphs reduce to
\begin{equation}
\exp\{\Pi_0(ee\nu_e)+3[\Pi_0(uud)+\Pi_0(ddu)](|V_{ud}|^2+|V_{us}|^2+|V_{ub}|^2)\}
\tag{1.3}
\label{eq:onepointthree}
\end{equation}
since there is no difference between the free u,d,s and b propagators in the massless limit. Noting that the unitarity of the CKM matrix requires the sum of the matrix elements in (\ref{eq:onepointthree}) to be one, the sum of the color weighted $\gamma$ -vertices in (\ref{eq:onepointthree}) results in the cancellation of the first generation $\gamma W^+ W^-$ triangle anomaly. This procedure can be continued until all of the three and four leg anomalies in the three generations cancel as they are known to do. These determinant regularizations should be done before they are inserted into the functional integrals over the gauge and Higgs fields.
Summarizing, it is necessary to define the fermion determinants by removing their ill-defined loops by making subtractions that are then either renormalized or cancelled among themselves. This happens to lead to anomaly cancellation at the three and four external leg level. Of course it has not been proved that the product of the remainder determinants is free of terms that can block the non-perturbative renormalization of the electroweak model \cite{55}.
It is known that when $\Pi_{i=1}^{24}\text{det}_{R_i}$ is loop-expanded it contains an exponentiated sum of absolutely convergent graphs beginning with the pentagon graph. These can be calculated in a manifestly gauge invariant way and cannot contain anomalies. The fact that the perturbative expansion of $\Pi_{i=1}^{24}\text{det}_{R_i}$ is anomaly-free leaves open the possibility that this determinant product may eventually be shown to be part of a non-perturbative, anomaly-free, gauge preserving regularization of the electroweak model.
Assuming the functional integrals converge the process of renormalization follows next with the introduction of counterterms to remove the regulators. Presumably the result is in terms of the physical parameters $e$, $M_W$, $M_Z$, $M_H$, $m_i$ -the charged fermion masses- and the renormalized quark mixing matrix $V_{ij}$ after continuing from an intermediate renormalization scheme in Euclidean space to on-shell renormalization in Minkowski space.
The observation that $\mathcal L$ is no more than quadratic in $A_\mu$, that $A_\mu$ does not couple directly to $H$, that a considerable amount is known about the QED determinant $\det(\slashed P -e\slashed A + m)$, and that the regularization of the electrodynamic sector is straightforward suggests that the next simplest functional integration should be over the Maxwell field.
Supppose this is decided.
Twenty-one of the twenty-four fermion determinants involve the Maxwell field as it appears in the electron's determinant in (\ref{eq:onepointone}) with different charges. Should their combined large amplitude $A$-field variation increase faster than $\exp\left[ce^2 \int \mathrm d^4 x\, F_{\mu\nu}^2 \right]$, $c>0$
then the integration over the Maxwell field with any Gaussian measure would be divergent, and the non-perturbative quantization of the electroweak model would be doubtful.
The $F_{\mu\nu}$-dependence is expected since the determinants are gauge invariant.
It is assumed that the strong Maxwell field behavior of these determinants can be obtained by decoupling them from the electroweak model by setting $g=0$. Future theorems dealing with the assumed sub-dominant growth of the remainder determinants can and should be produced. Noting this, there remains a product of twenty one
determinants of the form det$(\slashed{P} - q\slashed{A} + m)$ so that we need only calculate one of them. Accordingly, this paper considers the the non-perturbative quantization of the electroweak model's electrodynamic sector. It is found that this can be done only under restrictive conditions. If the sub-dominance of the remainder determinants assumed here is valid then these conditions extend to the complete electroweak model.
\section{Preliminaries}
Confining attention to QED, sense has to be made of the infinite dimensional determinant $\det( \slashed P - e\slashed A + m)$, where $e>0$ from here on.
It is first normalized to one when $e=0$ by dividing it by $\det( \slashed P + m)$ to get $\det(1-eS\slashed A)$, where $S$ is the free electron propagator.
To make this well-defined it has to be regularized and made ultraviolet finite by a second order charge renormalization subtraction.
A representation of the regulated and renormalized determinant, denoted by $\det_\mathrm{ren}$, is given by Schwinger's proper time definition \cite{3}
\begin{widetext}
\begin{align}
\ln\det{}_\mathrm{ren}(1-e_0S\slashed A)=\frac{1}{2}\int_0^\infty
\frac{\mathrm d t}{t}
\left( \mathrm{Tr}\left\{ e^{-P^2 t}-\exp\left[ -\left( D^2 + \frac{e_{\mathrm{o}}}{2}\sigma_{\mu\nu}F_{\mu\nu}\right)t\right]\right\}
+\frac{e^2_{\mathrm{o}}\| F\|^2}{24\pi^2}\right)e^{-tm^2_{\mathrm{o}}},
\tag{2.1}
\label{eq:twopointone}
\end{align}
\end{widetext}
where $D_\mu = P_\mu -e_\mathrm{o}A_\mu$, $\sigma_{\mu\nu}=[\gamma_\mu,\,\gamma_\nu]/2i$, $\gamma_\mu^\dagger=-\gamma_\mu$, $\|F\|^2=\int \mathrm d^4x\, F_{\mu\nu}^2$, and
$e_{\mathrm{o}}$, $m_{\mathrm{o}}$ are the unrenormalized charge and mass. The last term in (\ref{eq:twopointone}) results in a second-order charge renormalization subtraction in the one-particle irreducible photon self-energy $\Pi(k^2)$ at zero momentum transfer as in Eq.(\ref{eq:C7}), Appendix C. Therefore, as long as $A_{\mu}$ remains a classical field $e_{\mathrm{o}}$ and $m_{\mathrm{o}}$ are the physical parameters e and m. Quantizing $A_{\mu}$ by integrating over it will require a further charge renormalization subtraction given by $1/e_{\mathrm{o}}^2=1/ {e^2}+\Pi(0,e_{\mathrm{o}}^2D_{\mathrm{o}})$, where $\Pi(0,e_{\mathrm{o}}^2D_{\mathrm{o}})$ is the 1PI photon self-energy at $k^2=0$ with the one-loop contribution omitted. It is a functional of the exact unrenormalized photon propagator $D_{\mathrm{o}}$ with $\Pi(0,0)=0$; it is made finite by the regularization procedure outlined in Sec. VII. As renormalization will not be considered further the subscript o will be dropped in (\ref{eq:twopointone})
with the understanding that e and m are the {\it{unrenormalized}} charge and mass in what follows.
Having defined $\det_\mathrm{ren}$ the effective measure for the Maxwell field integration is
\begin{align}
\mathrm d\mu(A)=Z^{-1}\mathrm d\mu_0(A)\det{}_\mathrm{ren}(1-eS\slashed A)
\tag{2.2}
\label{eq:twopointtwo}
\end{align}
where the gauge-fixed Gaussian measure for the random potential $A_\mu$ is now denoted by $d\mu_0$.
It has mean zero and covariance
\begin{align}
\int \mathrm d\mu_0\, A_\mu(x)A_\nu(y)=D_{\mu\nu}(x-y),
\tag{2.3}
\label{eq:twopointthree}
\end{align}
where $D_{\mu\nu}$ is the photon propagator in a fixed gauge.
The vacuum-vacuum amplitude $Z$ in (\ref{eq:twopointthree}) is
\begin{align}
Z=\int \mathrm d\mu_0\,\det{}_\mathrm{ren},
\tag{2.4}
\label{eq:twopointfour}
\end{align}
so that $\int \mathrm d\mu(A)=1$.
The measure (\ref{eq:twopointtwo}) appears in the non-perturbative calculation of every physical process in QED such as the Euclidean Green function for $2n$ external fermions and $m$ photons,
\begin{widetext}
\begin{align}
\begin{split}
S_{\mu_1\ldots\mu_m}&(x_1,\ldots,x_n;y_1,\ldots,y_n;z_1,\ldots,z_m)\\
&=Z^{-1}\int \mathrm d\mu_0(A)\, \det{}_\mathrm{ren}(1-eS\slashed A)\det\left[S(x_i,y_j|eA)\right]_{i,j=1}^n\prod_{k=1}^mA_{\mu_k}(z_k),
\end{split}
\tag{2.5}
\label{eq:twopointfive}
\end{align}
\end{widetext}
where $S(x,y|eA)$ is the electron propagator in the external potential $A_\mu$.
Any attempt to calculate the integrals in (\ref{eq:twopointfour}) and (\ref{eq:twopointfive}) will encounter ultraviolet divergences that require regularization.
How this regularization is introduced will be discussed in Sec.
VII.
In addition $Z$ requires a volume cutoff that will be discussed in in Sec.
VII as well.
A volume cutoff enters QED solely by its determinant to render the vacuum energy finite when the determinant is integrated.
Assuming that the functional integrations in (\ref{eq:twopointfour}) and (\ref{eq:twopointfive}) converge, there remains the task of removing the ultraviolet regulator and volume cutoff by some as yet unknown non-perturbative renormalization procedure that preserves the unitarity of $S$-matrix elements.
The difficulty of implementing this procedure cannot be overstated.
Whether the functional integrals in (\ref{eq:twopointfour}) and (\ref{eq:twopointfive}) converge depends on $\det_\mathrm{ren}$'s behavior for large amplitude variations of a measurable set of random fields $F_{\mu\nu}$ on $\mathbb R^4$.
Since $e$ always multiplies $F_{\mu\nu}$ it will be sufficient to consider the strong coupling behavior of $\det_\mathrm{ren}$.
This leads to one of the main results of this paper.
Although (\ref{eq:twopointone}) is compact and intuitive it -- and all other representations -- have so far failed to give any explicit information on the strong coupling behavior of $\det_\mathrm{ren}$ for random fields on $\mathbb R^4$.
To remedy this an exact representation of $\ln\det_\mathrm{ren}$ is derived from (\ref{eq:twopointone}) that facilitates its strong coupling analysis.
Noting that in Euclidean space $F_{\mu\nu}$ may be regarded as a static, four-dimensional magnetic field, the new representation breaks $\ln\det_\mathrm{ren}$ into a sum of three terms that expose its competing magnetic properties, namely,
\begin{align}
\begin{split}
\ln\det{}_\mathrm{ren} &= \mbox{diamagnetism} + \mbox{paramagnetism} \\ &\quad + \mbox{charge renormalization}.
\end{split}
\tag{2.6}
\label{eq:twopointsix}
\end{align}
The advantage of representation (\ref{eq:twopointsix}) of $\det{}_\mathrm{ren}$ is that the strong coupling analysis of its separate terms is far easier than their combined form in (\ref{eq:twopointone}).
The derivation of (\ref{eq:twopointsix}) is given in Sec. III.
Suffice it to say here that the sum of the diamagnetic term (Sec.IV) and charge renormalization term (Sec.VI) contribute to $\det_\mathrm{ren}$'s strong coupling growth while the paramagnetic term (Sec.V) slows it down.
Therefore, the non-perturbative quantization of QED critically depends on the paramagnetic term and the class of background fields on which it depends.
{\it Prima facie} evidence is given that zero mode supporting background fields are necessary for the non-perturbative quantization of QED.
The presence of substantial numbers of zero modes in the lattice functional integration of QED in its chirally broken phase has been noted \cite{4,5}.
Our result and this observation suggest that Maxwellian zero modes will play a key role in deciding whether the electroweak model can be non-perturbatively quantized.
Our conclusions are summarized in Secs.VIC and VIII, and the appendices deal with mathematical details.
\section{Representation of $\boldsymbol{\det_\mathrm{ren}}$}
The objective is to obtain an expression for $\det_\mathrm{ren}$ that manifests the interplay of diamagnetism, paramagnetism and charge renormalization in its strong coupling behavior for random, static, four-dimensional magnetic fields.
Rewrite (\ref{eq:twopointone}) as
\begin{widetext}
\begin{align}
\begin{split}
\ln&\,\det{}_\mathrm{ren}\\
&=\frac{1}{2}\int_0^\infty \frac{\mathrm dt}{t}\,e^{-tm^2}
\left[ 4\mathrm{Tr}\left( e^{-P^2t} - e^{-D^2t}\right)
-\frac{e^2\|F\|^2}{48\pi^2}
+ \mathrm{Tr} \left( e^{-D^2t} -\exp\left[-\left(D^2+ \frac{e}{2}\sigma_{\mu\nu}F_{\mu\nu}\right)t\right] \right)
+ \frac{e^2\| F\| ^2}{16\pi^2}\right],
\end{split}
\tag{3.1}
\label{eq:threepointone}
\end{align}
\end{widetext}
where the trace over spin was made in the first term to give a factor of 4.
Then (\ref{eq:threepointone}) becomes
\begin{widetext}
\begin{align}
\ln\det{}_\mathrm{ren}=2\ln\det{}_\mathrm{SQED}
+ \frac{1}{2}\int_0^\infty \frac{\mathrm dt}{t}\,e^{-tm^2}
\left[ \mathrm{Tr}\left( e^{-D^2t}-\exp\left[-\left(D^2+ \frac{e}{2}\sigma_{\mu\nu}F_{\mu\nu}\right)t\right] \right)
+ \frac{e^2\| F\| ^2}{16\pi^2}\right],
\tag{3.2}
\label{eq:threepointtwo}
\end{align}
\end{widetext}
where $\ln\det_\mathrm{SQED}$ is the proper time definition of the formal scalar QED determinant $\ln\det\left\{\left[( P - e A)^2 + m^2 \right]/(P^2 + m^2 )\right\}$ with on-shell charge renormalization:
\begin{align}
\begin{split}
\ln&\det{}_\mathrm{SQED}\\
&=\int_0^\infty \frac{\mathrm dt}{t}
\left[ \mathrm{Tr}\left( e^{-P^2t} - e^{-D^2t}\right)
-\frac{e^2\|F\|^2}{192\pi^2}
\right]e^{-tm^2},
\end{split}
\tag{3.3}
\label{eq:threepointthree}
\end{align}
Alternatively, $\ln\det_\mathrm{SQED} = -S_\mathrm{SQED}$, where $S_\mathrm{SQED}$ is the one-loop effective action of scalar QED.
Now consider the remaining terms in (\ref{eq:threepointtwo}) and use the operator identity
\begin{align}
\begin{split}
e^{-t\left(D^2+\frac{1}{2}e\sigma F\right)} &- e^{-tD^2}\\
&= -\int_0^t\mathrm d s\,e^{-(t-s)\left(D^2+\frac{1}{2}e\sigma F\right)}\frac{1}{2}e\sigma F e^{-sD^2}.
\end{split}
\tag{3.4}
\label{eq:threepointfour}
\end{align}
A derivation of (\ref{eq:threepointfour}) is given in \cite{6}.
Iterating it twice gives
\begin{widetext}
\begin{align}
\begin{split}
&e^{-t\left(D^2+\frac{1}{2}e\sigma F\right)} - e^{-tD^2}\\
= &-\int_0^t\mathrm d s\,e^{-(t-s)D^2}\frac{1}{2}e\sigma F e^{-sD^2}\\
&+\int_0^t\mathrm d s_1\,\int_0^{t-s_1}\mathrm d s_2\,
e^{-(t-s_1-s_2)D^2}\frac{1}{2}e\sigma F
e^{-s_2D^2}\frac{1}{2}e\sigma F e^{-s_1D^2}\\
&-\int_0^t\mathrm ds_1\,\int_0^{t-s_1}\mathrm ds_2\,\int_0^{t-s_1-s_2}\mathrm ds_3\,
e^{-(t-s_1-s_2-s_3)\left(D^2+\frac{1}{2}e\sigma F\right)}
\frac{1}{2}e\sigma F e^{-s_3D^2}
\frac{1}{2}e\sigma F e^{-s_2D^2}
\frac{1}{2}e\sigma F e^{-s_1D^2}.
\end{split}
\tag{3.5}
\label{eq:threepointfive}
\end{align}
\end{widetext}
Define the determinant $\det_3$ by
\begin{widetext}
\begin{align}
\begin{split}
\ln\,\mbox{$\det_3$}\left(1+\Delta_A^{1/2} \frac{1}{2}e\sigma F\Delta_A^{1/2}\right)
&= \int_0^\infty \frac{\mathrm dt}{t}\,e^{-tm^2}\mathrm{Tr}\biggl(\int_0^t \mathrm d s_1\,\int_0^{t-s_1}\mathrm d s_2\,\int_0^{t-s_1-s_2}\mathrm d s_3\ \\
&\times e^{-(t-s_1-s_2-s_3)\left(D^2+\frac{1}{2}e\sigma F\right)}
\frac{1}{2}e\sigma F e^{-s_3D^2}
\frac{1}{2}e\sigma F e^{-s_2D^2}
\frac{1}{2}e\sigma F e^{-s_1D^2} \biggr),
\end{split}
\tag{3.6}
\label{eq:threepointsix}
\end{align}
\end{widetext}
where $\Delta_A^{1/2} = (D^2 + m^2 )^{-1/2}$.
Before proceeding with the derivation of (\ref{eq:twopointsix}) it is important to explain what the left-hand side of (\ref{eq:threepointsix}) means \cite{7,8,9,10,11}.
Thus $\det_3$ is the regularized determinant defined by
\begin{align}
\mbox{$\det_3$}(1+T)=\det\left[(1+T)\exp\left(-T+\frac{1}{2}T^2\right)\right],
\tag{3.7}
\label{eq:threepointseven}
\end{align}
provided $T\in \mathscr I_3$.
The trace ideal $\mathscr I_p$ ($1\le p< \infty$) is defined as those compact operators $T$ with $\|T\|_p^p = \mathrm{Tr}((T^\dagger T)^{p/2})<\infty$ \cite{8,9,10}.
Because $T$ is compact its eigenvalues are discrete and have finite multiplicity.
Therefore, the left-hand side of (\ref{eq:threepointsix}) requires that the operator $\Delta_A^{1/2}\sigma F\Delta_A^{1/2}\in\mathscr I_3$.
This is shown in Appendix A for $F_{\mu\nu}\in \cap_{p>2}L^p (\mathbb R^4)$ and $m \neq 0$.
Note that this allows zero mode supporting potentials $A_\mu(x)$ with their necessary $1/|x|$ fall off for $|x|\rightarrow\infty$.
The equivalence of the two sides of (\ref{eq:threepointsix}) follows from Theorem 7.2 in \cite{7} where an outline of its proof is given.
Because of the inaccessibility of \cite{7} and the importance of $\det_3$ to this paper a proof is given in Appendix B.
More will be said about $\det_3$ in Sec.
V.
But already we anticipate that its presence in $\det_\mathrm{ren}$ will be a calculational advantage as it deals with a self-adjoint operator acting on countable, square-integrable eigenstates.
Put differently, $\det_3$'s calculation reduces to a manageable quantum mechanical problem on bound state energy levels as discussed in Sec.
VB.
Continuing with the derivation of (\ref{eq:twopointsix}), insert (\ref{eq:threepointfive}) and (\ref{eq:threepointsix}) in (\ref{eq:threepointtwo}) to obtain
\begin{widetext}
\begin{align}
\begin{split}
\ln&\,\mbox{$\det{}_\mathrm{ren}$}=2\ln\det{}_\mathrm{SQED} + \frac{1}{2}\ln\mbox{$\det_3$}\left(1+\Delta_A^{1/2}\frac{1}{2}e\sigma F\Delta_A^{1/2}\right)\\
&+\frac{e^2}{8}\int_0^\infty \frac{\mathrm dt}{t}\,e^{-tm^2}
\left(\frac{1}{4\pi^2}\|F\|^2
-\mathrm{Tr}\int_0^t\mathrm ds_1\,\int_0^{t-s_1}\mathrm ds_2
e^{-(t-s_1-s_2)D^2}\sigma Fe^{-s_2D^2}\sigma Fe^{-s_1D^2}\right).
\end{split}
\tag{3.8}
\label{eq:threepointeight}
\end{align}
\end{widetext}
It is shown in Appendix C that the last term in (\ref{eq:threepointeight}) can be simplified to give the promised three-term representation of $\ln\det_\mathrm{ren}$:
\begin{widetext}
\begin{align}
\begin{split}
\ln\mbox{$\det{}_\mathrm{ren}$}
=2\ln\det{}_\mathrm{SQED} + \frac{1}{2}\ln\mbox{$\det_3$}\left(1+\Delta_A^{1/2}\frac{1}{2}e\sigma F\Delta_A^{1/2}\right)
+e^2\int_0^\infty \mathrm dt\,e^{-tm^2}
\left[\frac{1}{32\pi^2t}\|F\|^2
-\frac{1}{2} \mathrm{Tr}
\left(e^{-tD^2}F_{\mu\nu}\Delta_AF_{\mu\nu}\right)\right],
\end{split}
\tag{3.9}
\label{eq:threepointnine}
\end{align}
\end{widetext}
where $\Delta_A = (D^2 + m^2)^{-1}$.
Equation (\ref{eq:threepointnine}) is equivalent to (\ref{eq:twopointone}), and each term is separately well-defined and gauge invariant.
Their order follows that in (\ref{eq:twopointsix}).
The signs of the first two terms and their connection with diamagnetism and paramagnetism are discussed in the following sections.
The last term is connected with charge renormalization and is manifestly positive due to QED's lack of asymptotic freedom.
\section{Strong coupling behavior of $\boldsymbol{\det_\mathrm{SQED}}$}
Let the amplitude of $F_{\mu\nu}(x)$ be set by the parameter $\mathscr{F}$ which has the dimension of $L^{-2}$.
Then break the integral in (\ref{eq:threepointthree}) into $\int_0^{1/e\mathscr{F}}$ and $\int_{1/e\mathscr{F}}^\infty$ and use Kato's inequality in the form \cite{12,13,14,15}
\begin{align}
\mathrm{Tr}\left(e^{-P^2t}-e^{-(P-eA)^2t}\right)\ge0,
\tag{4.1}
\label{eq:fourpointone}
\end{align}
to obtain
\begin{align}
\begin{split}
&\ln\mbox{$\det_\mathrm{SQED}$}\ge\\
&\int_0^{1/e\mathscr{F}}\frac{\mathrm dt}{t}\,
\left[ \mathrm{Tr}\left(e^{-P^2t}-e^{-(P-eA)^2t}\right) - \frac{e^2\|F\|^2}{192\pi^2}\right]
e^{-tm^2}\\
&\quad- \frac{e^2\|F\|^2}{192\pi^2}\int_{1/e\mathscr F}^\infty \frac{\mathrm dt}{t}\,
e^{-tm^2}.
\end{split}
\tag{4.2}
\label{eq:fourpointtwo}
\end{align}
The inequality in (\ref{eq:fourpointone}) reflects the diamagnetism of charged scalar bosons: on average the energy levels of such bosons increase in a magnetic field.
This explains the first term in (\ref{eq:twopointsix}).
The selection of $e\mathscr F$ as the scaling parameter is discussed below.
The first integral in (\ref{eq:fourpointtwo}) is dominated by its small-$t$ behavior for $e\gg1$.
Accordingly, make the heat kernel expansion
\begin{widetext}
\begin{align}
\begin{split}
\mathrm{Tr}\left(e^{-P^2t}-e^{-(P-eA)^2t}\right)
&=\frac{1}{16\pi^2}\int\mathrm d^4x\,
\biggl[\frac{e^2}{12}F_{\mu\nu}^2
+\frac{te^2}{120}F_{\mu\nu}\nabla^2F_{\mu\nu}\\
&+\frac{t^2e^2}{1680}F_{\mu\nu}\nabla^4F_{\mu\nu}
+\frac{t^2e^4}{1440}\left[(^\star F_{\mu\nu}F_{\mu\nu})^2
-7(F_{\mu\nu}^2)^2\right] \biggr]+ \mathrm O(t^3),
\end{split}
\tag{4.3}
\label{eq:fourpointthree}
\end{align}
\end{widetext}
where $^\star F_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}F_{\alpha\beta}$.
The $\mathrm O(F^2)$ terms follow from the result for $\ln\det_\mathrm{SQED}$ in (C6); the $\mathrm O(F^4)$ term is inferred from Schwinger's constant field result for scalar QED \cite{3}
To the author's knowledge there is no proof that QED heat kernel expansions are asymptotic series in $t$ although this is generally assumed.
Referring to (\ref{eq:fourpointthree}) it is evident that continuing the expansion in powers of $t$ requires that $F_{\mu\nu}$ be infinitely differentiable ($C^\infty$).
So this is a necessary condition.
In Sec.VII we will introduce an ultraviolet regulator by convoluting the potential $A_\mu$ with a function of rapid decrease.
The resulting smoothed potential is $C^\infty$.
Anticipating Sec. VII we will now assume the fields in (\ref{eq:fourpointthree}) are $C^\infty$.
With this understanding the expansion in (\ref{eq:fourpointthree}) will now be assumed to be asymptotic so that the truncation error after $N$ terms is
\begin{align}
\begin{split}
&\mathrm{Tr}\left(e^{-P^2t}-e^{-(P-eA)^2t}\right)\\
&\quad -\sum_{n=0}^Na_n(eF)t^n\underset{t\searrow0}{\sim}a_M(eF)t^M,
\end{split}
\tag{4.4}
\label{eq:fourpointfour}
\end{align}
where $a_M$ is the first nonzero coefficient after $a_N$ \cite{16}.
Note that since $[t] = L^2$, the maximum power of $F_{\mu\nu}$ in $a_M$ is $M + 2$ so that the truncation error in (\ref{eq:fourpointtwo}) never exceeds $\mathrm O(e^2)$.
From (\ref{eq:fourpointthree}), (\ref{eq:fourpointfour}) and the result
\begin{align}
\int_{1/e\mathscr F}^\infty \frac{\mathrm dt}{t}\,
e^{-tm^2}=\ln\left(\frac{e\mathscr F}{m^2}\right)-\gamma+R,
\tag{4.5}
\label{eq:fourpointfive}
\end{align}
where $\gamma = 0.5772\ldots$ is Euler's constant and $0<|R|<m^2/(e\mathscr F)$, obtain from (\ref{eq:fourpointtwo}) for $e\gg1$
\begin{align}
\ln\mbox{$\det_\mathrm{SQED}$}\ge
-\frac{e^2\|F\|^2}{192\pi^2}\ln\left(\frac{e\mathscr F}{m^2}\right)
+\mathrm O(e^2).
\tag{4.6}
\label{eq:fourpointsix}
\end{align}
We chose $e\mathscr F$ as the scaling parameter in (\ref{eq:fourpointtwo}).
Why not $e^\alpha\mathscr F$?
We set $\alpha= 1$ firstly because we remarked in Sec.II that $e$ always multiplies $F_{\mu\nu}$ so that large amplitude variations of $F_{\mu\nu}$ can just as well be studied in the strong coupling limit; setting $\alpha\neq1$ breaks this correspondence.
Secondly, if $\alpha >1$ then the lower bound in (\ref{eq:fourpointsix}) would be more negative, hence not optimal.
If $\alpha <1$ one gets a better bound in (\ref{eq:fourpointsix}) but the truncation error in (\ref{eq:fourpointtwo}) increases faster than $e^2$ for terms of $\mathrm O(F^4)$ and higher order.
So $\alpha=1$ is the unique choice. The scaling parameter is further discussed in Sec.VI A.
The lower bound in (\ref{eq:fourpointsix}) is related to and in argeement with the constant magnetic field growth of scalar QED's effective action \cite{17}
\begin{align}
S_\mathrm{SQED}=-\ln\mbox{$\det_\mathrm{SQED}$}=\frac{B^2V}{96\pi^2}e^2\ln\left(\frac{eB}{m^2}\right)+\mathrm O(e^2).
\tag{4.7}
\label{eq:fourpointseven}
\end{align}
where $V$ is a four-dimensional volume cutoff.
This completes the discussion of the growth of the first term in (\ref{eq:twopointsix}) and (\ref{eq:threepointnine}).
We now turn to the all-important second term.
\section{Strong Coupling Behavior of $\boldsymbol{\det_3}$}
\subsection{Paramagnetic property of $\boldsymbol{\det_3}$}
In Appendix A it is shown that $\Delta_A^{1/2}\sigma F\Delta_A^{1/2}\equiv T$ belongs to the trace ideal $\mathscr I_3$ for $F_{\mu\nu}\in\cap_{p>2} L^p (\mathbb R^4)$ and $m>0$.
This means that $T$ is a compact operator that, in our case, maps $L^2(\mathbb R ^4)$ into itself.
Being compact its eigenvalues, $\{\lambda_n\}_{n=1}^\infty$, are discrete, and each has finite multiplicity.
We order the $\lambda_n$ by $|\lambda_1|\ge|\lambda_2|\ge\ldots>0$.
Because $T\in\mathscr I_3$ the eigenvalues $\lambda_n\rightarrow 0$ and satisfy
\begin{align}
\sum_{n=1}^\infty|\lambda_n|^3<\infty.
\tag{5.1}
\label{eq:5.1}
\end{align}
Finally, $\ln\mbox{$\det_3$} (1+T)$ is gauge invariant (Appendix D) and satisfies by (\ref{eq:threepointseven})
\begin{align}
\begin{split}
&\ln\mbox{$\det_3$}\left(1+\Delta_A^{1/2}\frac{1}{2}e\sigma F\Delta_A^{1/2}\right)\\
&\quad= \ln\det\left[ \left( 1+T \right)\exp\left( -T+\frac{1}{2} T^2\right) \right]\\
&\quad= \mathrm{Tr}\left[ \ln(1+T)-T+\frac{1}{2}T^2 \right]\\
&\quad= \sum_{n=1}^\infty\left[ \ln(1+\lambda_n)-\lambda_n+\frac{1}{2}\lambda_n^2 \right].
\end{split}
\tag{5.2}
\label{eq:5.2}
\end{align}
In Appendix D it is shown that for every eigenstate of $T$ with eigenvalue $\lambda_n$ there is another with eigenvalue $-\lambda_n$.
Therefore, (\ref{eq:5.2}) becomes
\begin{align}
\begin{split}
&\ln\mbox{$\det_3$}\left(1+\Delta_A^{1/2}\frac{1}{2}e\sigma F\Delta_A^{1/2}\right)\\
&\quad=\sum_{n=1}^\infty\left[ \ln(1-\lambda_n^2)+\lambda_n^2 \right],
\end{split}
\tag{5.3}
\label{eq:5.3}
\end{align}
where the sum is over positive eigenvalues. We will see in Sec.VII B that the condition on $F_{\mu\nu}$ can be relaxed somewhat.
Since $\ln\det_3$ is real and finite then $\lambda_n<1$ for all $n$.
Hence,
\begin{align}
\ln\mbox{$\det_3$}\left( 1+\Delta_A^{1/2}\frac{1}{2}e\sigma F\Delta_A^{1/2} \right)\le 0,
\tag{5.4}
\label{eq:5.4}
\end{align}
since $\ln(1-x^2 )+x^2 \le0$ for $0\le x<1$.
This inequality has a physical origin.
Referring to (\ref{eq:threepointfive}) and (\ref{eq:threepointsix}) and simplifying exactly as outlined in Appendix C for the function $\Pi$ we obtain
\begin{align}
\begin{split}
\ln\mbox{$\det_3$}&=\int_{0}^{\infty}\frac{\mathrm dt}{t}e^{-tm^2}\mathrm{Tr}\Biggl[ e^{-tD^2}
-e^{-t\left( D^2+\frac{1}{2}e\sigma F \right)}\\
&+\frac{e^2}{8}te^{-tD^2/2}\sigma F\Delta_A^{1/2}\Delta_A^{1/2}\sigma Fe^{-tD^2/2}\Biggr].
\end{split}
\tag{5.5}
\label{eq:5.5}
\end{align}
That $\ln\det_3 < 0$ is now seen as a consequence of the paramagnetism of a charged spin-1/2 fermion in a static, four-dimensional magnetic field $F_{\mu\nu}$: on average its energy levels are lowered by $F_{\mu\nu}$.
This is made more precise by a version of the Peierls-Bogoliubov inequality derived from Klein's inequality \cite{18,19,20}:
\begin{align}
\mathrm{Tr}\Big(e^{-t(P-eA)^2} - e^{-\left[ \left( P-eA \right)^2+\frac{1}{2}e\sigma F \right]t}\Big)\le 0 .
\tag{5.6}
\label{eq:5.six}
\end{align}
The last term in (\ref{eq:5.5}) has been purposely written in the form $U^\dagger U$ and is therefore positive.
Nevertheless, it is dominated by the paramagnetism of charged fermions through (\ref{eq:5.six}) which drives the integral in (\ref{eq:5.5}) to a negative value.
This explains the second term in (\ref{eq:twopointsix}).
\subsection{Lower bound on $\boldsymbol{\ln\det_3}$ in the absence of zero modes}
The eigenvalues in (\ref{eq:5.3}) are obtained from
\begin{align}
\frac{e}{2}\Delta_A^{1/2}\sigma F\Delta_A^{1/2}\varphi_n=-\lambda_n\varphi_n,
\tag{5.7}
\label{eq:5.seven}
\end{align}
where $\varphi_n\in L^2$.
Let $\Delta_A^{1/2}\varphi_n=\psi_n$ and obtain
\begin{align}
\left[ (P-eA)^2+\frac{e}{2\lambda_n}\sigma F \right]\psi_n=-m^2\psi_n,
\tag{5.8}
\label{eq:5.8}
\end{align}
where $\psi_n\in L^2$ as shown at the end of Appendix A.
Eq. (\ref{eq:5.8}) illustrates the role of the eigenvalues $\{ \lambda_n \}_{n=1}^{\infty}$ as coupling constants whose discrete values result in bound states with energy $-m^2$ for a fixed value of $e$.
Because $\gamma_5$ commutes with $\sigma$, an eigenstate $\psi_n$ of (\ref{eq:5.8}) has definite chirality.
In the representation (D7) $\gamma_5$ is diagonal with elements $\pm\mathbbm 1_2$, and so we need only deal with the two-dimensional chirality eigenstates $\psi_n^\pm$.
We note that each eigenvalue $\lambda_n(e)$ is a bounded function of $e$ as required by $|\lambda_n(e)|<1$ for all finite values of of $e$.
This is illustrated by the constant field case:
\begin{align}
|\lambda_n|=\frac{|eB|}{(2n+1)|eB|+m^2},\quad n=0,1,\ldots
\tag{5.9}
\label{eq:5.nine}
\end{align}
Therefore, the series in (\ref{eq:5.3}) will tend to an $e$-independent limit for $e\gg1$ unless the degeneracy of the eigenvalues increases with $e$.
The special case of a zero mode supporting background potential that allows $|\lambda_n|$ to approach $1$ arbitrarily closely for $e\gg1$ will be considered in the next section.
To bound $\ln\det_3$ for $e\gg1$ we will first estimate the eigenvalue degeneracy for the most symmetric case of an $\mathrm O(2)\times\mathrm O(3)$ background field.
This estimate will place an upper bound on the eigenvalue degeneracy of any random field.
The $\mathrm O(2)\times\mathrm O(3)$ symmetric fields have the standard form \cite{21,22,23}
\begin{align}
A_\mu(x)=M_{\mu\nu}x_\nu a(r),
\tag{5.10}
\label{eq:5.10}
\end{align}
where $M_{\mu\nu}$ is the antiself-dual antisymmetric matrix with nonvanishing elements $M_{12} = M_{30} = 1$ and $r^2 = x_\mu^2$.
Alternatively $M$ may be replaced with the self-dual antisymmetric matrix $N$ with nonvanishing elements $N_{03} = N_{12} = 1$.
Choosing the matrix $M$ the eigenstates of (\ref{eq:5.8}) have the form \cite{23}
\begin{widetext}
\begin{align}
\psi_n=r^{-2j-3/2}
\left( \begin{array}{c}
\mathscr D^j_{M-\frac{1}{2},m}(x)\rho_1(r) \\
\mathscr D^j_{M+\frac{1}{2},m}(x)\rho_2(r) \\
(j+m)^{\frac{1}{2}}r\rho_3(r)\mathscr D^{j-\frac{1}{2}}_{M,m-\frac{1}{2}}(x)
- (j-m+1)^{\frac{1}{2}} (\rho_4(r)/r) \mathscr D^{j+\frac{1}{2}}_{M,m-\frac{1}{2}}(x) \\
(j-m)^{\frac{1}{2}}r\rho_3(r)\mathscr D^{j-\frac{1}{2}}_{M,m+\frac{1}{2}}(x)
+ (j+m+1)^{\frac{1}{2}} (\rho_4(r)/r) \mathscr D^{j+\frac{1}{2}}_{M,m+\frac{1}{2}}(x) \\
\end{array}\right),
\tag{5.11}
\label{eq:5.11}
\end{align}
\end{widetext}
where $\mathscr D^j_{m_1m_2}(x)$ are the four-dimensional rotation matrices \cite{23,24,25} normalized so that
\newpage
\begin{align}
\int \mathrm d\Omega_4\,\mathscr D_{m_1m_2}^{j*}(x)\mathscr D_{m_3m_4}^{j'}(x)=\delta_{jj'}\delta_{m_1m_3}\delta_{m_2m_4}\frac{2\pi^2r^{4j}}{2j+1},
\tag{5.12}
\label{eq:5.12}
\end{align}
and where $2j = 0, 1,\ldots$; $-j \le m_i \le j$.
This paper follows the conventions of \cite{23,24}; closely related ones appear in \cite{25}.
The index $n$ has been omitted from $\rho_i$.
Inserting the two positive chirality components of (\ref{eq:5.11}) into (\ref{eq:5.8}) results in the following equations for $\rho_{1,2}$ \cite{24},
\begin{widetext}
\begin{align}
\left[ -\frac{\mathrm d^2}{\mathrm dr^2} + \frac{(2j+1)^2-\frac{1}{4}}{r^2}
+ (4M\mp2)ea + e^2r^2a^2 \pm \frac{e}{\lambda^+_n}(4a+r\frac{\mathrm da}{\mathrm dr})\right]\rho_{1,2}
=-m^2\rho_{1,2},
\tag{5.13}
\label{eq:5.13}
\end{align}
\end{widetext}
where the upper (lower) sign applies to $\rho_1$ ($\rho_2$), and $\lambda_n^+$ denotes a positive chirality eigenvalue.
Since $(P-eA)^2 + \frac{e}{2}\sigma F \ge 0$ it is the $\lambda_n^+$-dependent terms in (\ref{eq:5.13}) that are responsible for bound states at $-m^2$.
There is a sequence of eigenvalues $1>\lambda_1^+ \ge \lambda_2^+ \ge \ldots>0$ dependent on $e$, $j$, $M$, $m$, and the parameters specifying $A_\mu$ that result in bound state solutions of (\ref{eq:5.13}).
They are independent of the quantum number $m$ in (\ref{eq:5.11}), resulting in a $(2j+1)$-fold degeneracy. Inspection of (\ref{eq:5.13}) indicates that in the positive chirality sector
\begin{align}
\begin{split}
\frac{1}{2}(\sigma F)^+&= \left(4a+r\frac{\mathrm da}{\mathrm dr}\right)\sigma_3\\
&\equiv V(r)\sigma_3 .
\end{split}
\tag{5.14}
\label{eq:5.14}
\end{align}
In general the degeneracy of the level at $-m^2$ has contributions from both $\rho_1$ and $\rho_2$. Consider $\rho_1$. Assume that $a$ and $a'$ are bounded functions of $r$. Inclusion of zero modes requires
$\lim_{r\rightarrow\infty}r^2 a = \nu$, where we may assume $\nu>0$ as discussed in Sec.C below. Then $r^2 V(r)$ is a bounded function of r and
\begin{align}
\inf\left[ r^2V(r) \right]=-K_1>-\infty.
\tag{5.15}
\label{eq:5.15}
\end{align}
The $\lambda_n^+$-independent terms on the left-hand side of (\ref{eq:5.13}) form a positive operator whose controlling parameter is $j$ for fixed $e$.
Thus a bound state at $-m^2$ can exist only if
\begin{align}
\left( 2j+1 \right)^2<\frac{e}{\lambda_n^+}K_1+\frac{1}{4}.
\tag{5.16}
\label{eq:5.16}
\end{align}
This is a necessary condition but obviously not a sufficient one.
The maximum allowed value of $j$ for all finite values of $m^2$ and a fixed value of $M$ is $J_1 < \left( \frac{eK_1}{4\lambda_n^+}+\frac{1}{16} \right)^\frac{1}{2}-\frac{1}{2}$.
Hence, the maximum degeneracy $\mu_{1n}^+$ of eigenvalue $\lambda_n^+$ associated with $\rho_1$ for $\frac{eK_1}{\lambda_n^+} \ge 1$ is
\begin{align}
\mu_{1n}^+=\sum_{j=0,\frac{1}{2},\dots}^{J_1}(2j+1) < 2 \left[ \left(\frac{eK_1}{4 \lambda_n^+}\right) ^{\frac{1}{2}} +1 \right]^2 .
\tag{5.17}
\label{eq:5.17}
\end{align}
For the other positive chirality state $\mathscr D_{M+\frac{1}{2},m}^{j}\rho_2/r^{2j+\frac{3}{2}}$ inspection of (\ref{eq:5.13}) indicates that the bound state at $-m^2$ acquires an additional maximal degeneracy $\mu_{2n}^+$ satisfying the bound in (\ref{eq:5.17}) with $K_1$ replaced with $K_2=\sup(r^2V(r))<\infty$.
It may happen that either $\rho_1$ or $\rho_2 $ has no bound states at $-m^2$.
Is the dependence of $\mu_{1n}^+$, $\mu_{2n}^+$ on $\lambda_n^+$ reasonable?
As $\lambda_n^+\searrow0$ the potential wells in $\pm\frac{e}{\lambda_n^+}V(r)$ deepen, increasing the probability that such wells can support a bound state at $-m^2$.
As the wells deepen the centrifugal barrier term in (\ref{eq:5.13}) can increase, thereby allowing larger values of $j$ and hence higher degeneracy, consistent with our result (\ref{eq:5.17}).
In the negative chirality sector
\begin{align}
\frac{1}{2}(\sigma F)^-=\left( \begin{array}{cc}
-\mathscr D_{00}^1 & \sqrt 2\mathscr D_{01}^{1*} \\
\sqrt 2\mathscr D_{01}^1 & \mathscr D_{00}^1
\end{array}\right)
\frac{1}{r}\frac{\mathrm da}{\mathrm dr},
\tag{5.18}
\label{eq:5.8een}
\end{align}
where $\mathscr D_{00}^1=x_0^2+x_3^2-x_1^2-x_2^2$ and $\mathscr D_{01}^1=-\sqrt 2(x_0+ix_3)(x_2-ix_1)$.
Insertion of (\ref{eq:5.8een}) and the two negative chirality components of (\ref{eq:5.11}) in (\ref{eq:5.8}) results in coupled equations for $\rho_3$ and $\rho_4$:
\begin{widetext}
\begin{align}
\left( -\frac{\mathrm d^2}{\mathrm dr^2} + \frac{4j^2-\frac{1}{4}}{r^2}
+ 4Mea + e^2r^2a^2\right)\rho_3
+ \frac{e}{\lambda_n^-}ra'\left(
\sqrt{1-\frac{M^2}{(j+\frac{1}{2})^2}}\rho_4
+ \frac{M}{j+\frac{1}{2}}\rho_3\right)
= -m^2\rho_3
\tag{5.19}
\label{eq:5.19}\\
\left( -\frac{\mathrm d^2}{\mathrm dr^2} + \frac{4(j+1)^2-\frac{1}{4}}{r^2}
+ 4Mea + e^2r^2a^2\right)\rho_4
+ \frac{e}{\lambda_n^-}ra'\left(
\sqrt{1-\frac{M^2}{(j+\frac{1}{2})^2}}\rho_3
- \frac{M}{j+\frac{1}{2}}\rho_4\right)
= -m^2\rho_4 .
\tag{5.20}
\label{eq:5.20}
\end{align}
\end{widetext}
These equations can be decoupled for large j by a unitary transformation $U$ on $\rho_3$, $\rho_4$. Let $U\rho=\varphi$ with $U_{33}=U_{44}=(\frac{1+M}{(j+\frac{1}{2})})^{\frac{1}{2}}/\sqrt2$
and $U_{34}=-U_{43}=(\frac{1-M}{(j+\frac{1}{2})})^{\frac{1}{2}}/\sqrt2$ so that the coupled terms in (\ref{eq:5.19}),(\ref{eq:5.20})proportional to $e/\lambda_n^-$ are transformed to $(e/\lambda_n^-)ra'\sigma_3\varphi$.
Comparing this with (\ref{eq:5.13}) the same analysis used in the positive chirality case applies here. Thus, following (\ref{eq:5.17}) the maximum degeneracies
$\mu_{3n}^-$, $\mu_{4n}^-$associated with the bound states $\varphi_3$, $\varphi_4$ at $-m^2$ are bounded by $eK/\lambda_n^-$, where $K$ is an $e$-independent constant.
This assumes $e/\lambda_n^->>1$ corresponding to large j.
We emphasize that the estimated maximum degeneracies above are for one level at $-m^2$. They are not an estimate of the number of bound states at energy $\le -m^2$ which is expected to vary as $e^2$ for $F_{\mu\nu}\in L^2$ by theorem 2.15 in \cite{26}.
We now have estimates for the maximum degeneracy of eigenvalues $\lambda_n^{\pm}$ obtained from (\ref{eq:5.8}) for the most symmetric admissible background field given by (\ref{eq:5.10}).
The above results place an upper bound on the eigenvalue degeneracy $\mu_n$ of any admissible random field, namely for $e\gg1$
\begin{align}
\mu_n(e)<\frac{ec}{\lambda_n} ,
\tag{5.21}
\label{eq:5.21}
\end{align}
where $\lambda_n$ is one of the random field's eigenvalues obtained from (\ref{eq:5.8}), and $c$ is $e$-independent.
The $1/\lambda_n$ dependence of its right-hand side is important because it results in the convergent series $\sum_{n>N}^\infty \lambda_n^3$ in (\ref{eq:5.23}) below, whatever the field may be.
Consider the series in (\ref{eq:5.3}) and divide it into $\sum_{n=1}^N+\sum_{n>N}^\infty$, where $\lambda_n^2<\frac{1}{2}$ for $N>n$, $N$ sufficiently large.
Note in this case that
\begin{align}
\frac{1}{2}\le\left|\frac{\ln(1-\lambda_n^2)+\lambda_n^2}{\lambda_n^4}\right|
< 1.
\tag{5.22}
\label{eq:5.22}
\end{align}
Thus for any admissible random field, excluding those that support a zero mode, there follows from (\ref{eq:5.3}), (\ref{eq:5.21}), and (\ref{eq:5.22})
\begin{align}
\begin{split}
&\left| \ln\mbox{$\det_3$}\left( 1+\Delta_A^\frac{1}{2}\frac{1}{2}e\sigma F\Delta_A^\frac{1}{2} \right)\right |\\
&<\sum _{n=1}^N \left|\ln(1-\lambda_n^2)+\lambda_n^2\right| + \sum_{n>N}^{\infty}\lambda_n^4\\
&<\sum_{n=1}^N\left|\ln(1-\lambda_n^2)+ \lambda_n^2\right|+ec \underset{\text{no degeneracy}}{\sum_{n>N}^{\infty}}\lambda_n^3 ,
\end{split}
\tag{5.23}
\label{eq:5.23}
\end{align}
where the third line of (\ref{eq:5.23}) is valid when $e\gg1$.
In the absence of zero modes $\lim_{e\rightarrow\infty} \lambda_1< 1$ unlike the zero mode case discussed in Sec.C below.
By (\ref{eq:5.1}) the infinite series on the right converges.
Moreover, the $e\rightarrow\infty$ limit of this series is finite.
Thus, there is a number $M$ such that for $n>M$, $\lambda_n (e) < C_n (e)/n^{1/3+\epsilon}$, $\epsilon>0$ and $C_n$ is a bounded function of $n$ and $e$ with $\lim_{e\rightarrow\infty} C_n (e)< \infty$.
Otherwise $\lambda_n<1$ for any $n$ cannot be satisfied.
Accordingly, the right-hand series in (\ref{eq:5.23}) is uniformly convergent in $e$ by the Weierstrass M test, allowing its $e\rightarrow\infty$ limit to be taken term-by-term and establishing our claim.
The remaining series, $\sum_{n=1}^N|\ln(1 - \lambda_n^2 ) + \lambda_n^2|$, is obviously bounded by $e$ following (\ref{eq:5.21}), excluding zero modes.
Combining (\ref{eq:5.3}), (\ref{eq:5.21}), (\ref{eq:5.22}) and (\ref{eq:5.23}) gives in the absence of zero modes
\begin{align}
0\ge\lim_{e\rightarrow\infty}\ln\mbox{$\det_3$}
\left( 1+\Delta_A^\frac{1}{2}\frac{1}{2}e\sigma F\Delta_A^\frac{1}{2} \right)/e > -C,
\tag{5.24}
\label{eq:5.24}
\end{align}
where $C>0$ is an $e$-independent constant depending on the specific background field. $C$ must be a linear function of $F_{\mu\nu}$ to preserve the correlation $eF_{\mu\nu}$.
\subsection{Zero modes}
Consideration is now given to potentials supporting $L^2$ zero modes of the Dirac operator $\slashed P - e\slashed A$.
It is these potentials that provide the mechanism governing the stability of QED and its non-perturbative quantization.
The relevance of zero modes to stability arises as follows.
Suppose $A_\mu$ supports a zero mode, $\psi_{\mathrm{zero},n}$, where $n$ denotes the quantum numbers required to specify it.
It is an $L^2 $ solution of
\begin{align}
\left[ \left( P-eA \right)^2 + \frac{e}{2}\sigma F \right]\psi_{\mathrm{zero},n}=0,
\tag{5.25}
\label{eq:5.25}
\end{align}
obtained from (\ref{eq:5.8}) by setting $\lambda_n= 1$, $m=0$.
We continue to assume $\lambda_n> 0$ as discussed in Sec. V A.
Then (\ref{eq:5.25}) requires $ \bra{\mathrm{zero},n}\sigma F\ket{\mathrm{zero},n}< 0$.
Refer to (\ref{eq:5.8}) and replace $\lambda_n$ with a general eigenvalue $\lambda$ and denote the corresponding eigenstate by $\psi_{\lambda,n}$.
Assume $\bra{\lambda,n}\sigma F\ket{\lambda,n}<0$.
Then from (\ref{eq:5.8}) and (\ref{eq:5.25}) there follows
\begin{align}
\frac{e}{2}\left( \frac{1}{\lambda}-1 \right)\bra{\mathrm{zero},n}\sigma F\ket{\lambda,n}=-m^2\braket{\textrm{zero},n}{\lambda,n}.
\tag{5.26}
\label{eq:5.26}
\end{align}
There is no \emph{a priori} reason why the two sides of (\ref{eq:5.26}) should vanish if the quantum numbers of the two states are the same.
Based on our limited knowledge of four-dimensional Abelian zero modes \cite{24} they have a distinctive structure, and so the nonvanishing of $\braket{\textrm{zero},n}{\lambda,n}$ distinguishes the eigenstate $\psi_{\lambda,n}$ --and its eigenvalue $\lambda$-- from other eigenstates obtained from (\ref{eq:5.8}).
Divide (\ref{eq:5.26}) by $e$. For $e\gg1$ conclude that $\lambda$ has the form
\begin{align}
\lambda=1-\delta(e,n,m,L,\dots) ,
\tag{5.27}
\label{eq:5.27}
\end{align}
where $0<\delta<1$ and that for fixed $m$, $\delta\searrow 0$ for $e\rightarrow \infty$.
$L$ is a parameter with the dimension of length introduced by $A_\mu$ that can combine with $m$ to form a dimensionless $\delta$.
This result requires that the states $\psi_{\lambda,n}$ be in one-to-one correspondence with the zero modes $\psi_{\text{zero},n}$.
The eigenvalue $\lambda$ will be discussed for an analytically solvable case in Sec. 5 E.
Insertion of (\ref{eq:5.27}) in (\ref{eq:5.3}) gives
\begin{align}
\begin{split}
&\ln\mbox{$\det_3$}=\sum_n\sigma_n\\
&\times\left[ -\ln\left(\frac{1-\delta}{\delta}\right)+\ln\left[(1-\delta)(2-\delta)\right]+(1-\delta^2)\right]+\dots,
\end{split}
\tag{5.28}
\label{eq:5.28}
\end{align}
where the remainder in (\ref{eq:5.28}) is the contribution from eigenvalues bounded away from $1$ discussed in the previous section; $\sigma_n$ is the degeneracy of state $n$..
The sum in (\ref{eq:5.28}) is over the quantum numbers specifying the zero modes of $A_\mu$.
Write (\ref{eq:5.26}) in the form
\begin{align}
\frac{1-\delta}{\delta}= \frac{e}{2m^2}\left|\frac{\langle\text{zero},n|\sigma F|\lambda,n\rangle}{\langle\text{zero},n|\lambda,n\rangle}\right|,
\tag{5.29}
\label{eq:5.29}
\end{align}
where
\begin{align}
\left|\frac{\langle\text{zero},n|\sigma F|\lambda,n\rangle}{\langle\text{zero},n|\lambda,n\rangle}\right|\le K \mathscr F.
\tag{5.30}
\label{eq:5.30}
\end{align}
Eq. (\ref{eq:5.30}) assumes $F_{\mu\nu}(x)$ is a bounded function in which case $K$ is an $e$-independent constant; $\mathscr F$ is the amplitude of $F_{\mu\nu}$ corresponding to the scaling parameter introduced in Sec.IV. Inserting (\ref{eq:5.29}) in (\ref{eq:5.28}) gives for $ e \rightarrow \infty$
\begin{align}
\begin{split}
&\ln\mbox{$\det_3$}= -\sum_n \sigma_n \\
&\times\left[\ln\left(\frac{e\mathscr F}{m^2}\right)+\ln\left|\frac{\langle\text{zero},n|\sigma F|\lambda,n\rangle/\mathscr F}{\langle\text{zero},n|\lambda,n\rangle}\right|-2\ln 2-1\right]\\
&+ \text{O(e)}.
\end{split}
\tag{5.31}
\label{eq:5.31}
\end{align}
The $O(e)$ term is the contribution from the eigenvalues bounded away from 1 discussed in the previous section.
Since
\begin{align}
\sum_n\sigma_n=\#\mbox{ zero modes supported by } A_\mu,
\tag{5.32}
\label{eq:5.32}
\end{align}
if the number of zero modes increases as $e^2$ or faster then the result (\ref{eq:5.31}) will override the bound in (\ref{eq:5.24}) and possibly drive $\ln\det_\textrm{ren}$ in (\ref{eq:threepointnine}) negative.
Clearly, these considerations are highly relevant to QED's non-perturbative quantization.
\subsection{Counting zero modes}
Following (\ref{eq:5.31}) and (\ref{eq:5.32}) it is of exceptional interest to know the maximum number of zero modes a potential can support.
To begin we focus on the most symmetric admissible potentials (\ref{eq:5.10}).
It is assumed that zero mode potentials within the class (\ref{eq:5.10}) will produce the maximum number due to their high symmetry and hence large number of degenerate states $\psi_{\mathrm{zero},n}$.
As pointed out in the previous section, eigenstates $\psi_{\lambda,n}$ of (\ref{eq:5.8}) with eigenvalue $\lambda$ given by (\ref{eq:5.27}) will be in one-to-one correspondence with the states $\psi_{\mathrm{zero},n}$.
We would then expect that zero mode supporting potentials with lesser symmetry will have their zero mode number bounded by this most symmetric result.
It turns out that this reasoning is not completely correct and that potentials with lesser symmetry can compete with those in (\ref{eq:5.10}).
This is a huge advantage for QED's stability.
We will begin with the potentials (\ref{eq:5.10}) and then explain why this reasoning has to be modified.
The zero modes supported by the potentials in (\ref{eq:5.10}) have been discussed in \cite{24}.
We continue to assume that $a$ and $a'$ are bounded functions of $r$ and in addition $\lim_{r\rightarrow\infty} r^2 a = \nu$, $\nu\neq 0$.
That is, $A_\mu$ must have a $1/r$ falloff.
This ensures that the global chiral anomaly $\mathcal A$ is nonvanishing:
\begin{align}
\mathcal A = -\frac{1}{16\pi^2}\int\mathrm d^4x \,^\star F_{\mu\nu}F_{\mu\nu}=\pm\frac{\nu^2}{2},
\tag{5.33}
\label{eq:5.33}
\end{align}
where $^\star FF = \partial_\alpha (\epsilon_{\alpha\beta\mu\nu} A_\beta F_{\mu\nu} )$.
The $+$($-$) sign in (\ref{eq:5.33}) results in the case of matrix $M$ ($N$) defined under (\ref{eq:5.10}).
Without loss of generality we will assume $\nu> 0$.
The nonvanishing of $\mathcal A$ indicates that $F_{\mu\nu}$ is not square-integrable. We repeat here that it is sufficient to assume $F_{\mu\nu}\in \cap_{p>2}L^p$ to define $\det_3$, and therefore it can accommodate zero modes.
Choosing the matrix $M$ in (\ref{eq:5.10}) it is found that only the positive chirality sector has normalizable zero modes \cite{24}.
This is a particular example of a vanishing theorem: all normalizable zero modes of $\slashed D^2$ have only one chirality.
There is no such general theorem in $\text{QED}_4$, unlike the non-Abelian case \cite{27,28} and $\text{QED}_2$ \cite{29}.
Up to a normalization constant these are \cite{24}
\begin{align}
\psi_\mathrm{zero}(x)=\mathscr D_{-j,m}^j(x)
e^{-e\int_{r_0}^{r}\mathrm dr\, ra(r)}
\left(\begin{array}{c}
0\\
1\\
0\\
0
\end{array}\right)
\tag{5.34}
\label{eq:5.34}.
\end{align}
Here $\exp\left[-e \int_{r_0}^{r}\mathrm dr\,ra(r)\right] = \rho_2$ in (\ref{eq:5.11}) when $M = -j-1/2$ and in (\ref{eq:5.13}) when in addition $m^2 = 0$ and $\lambda_n^+= 1$.
Eq. (\ref{eq:5.34}) and the assumption $a(r) \underset{r\rightarrow\infty}{\sim}{\nu}/r^2$ indicate that $\psi^+$ is square-integrable provided $e\nu > 2j+2$.
Following (\ref{eq:5.32}),
\begin{align}
\mbox{\# zero modes}=\sum_{j=0,\frac{1}{2},\dots}^{j_\mathrm{max}}(2j+1)
=\frac{1}{2}[e\nu]^2-\frac{1}{2}[e\nu],
\tag{5.35}
\label{eq:5.35}
\end{align}
where $[x]$ is the greatest integer less than $x$.
Using (\ref{eq:5.33}) for $e\nu \gg1$,
\begin{align}
\mbox{\# zero modes}&=\frac{1}{2}(e\nu)^2+\mathrm O(e\nu)\\
&=\frac{e^2}{16\pi^2}\left| \int\mathrm d^4x\, ^\star F_{\mu\nu}F_{\mu\nu}\right|
+ \mathrm O(e\nu).
\tag{5.36}
\label{eq:5.36}
\end{align}
If the matrix $M$ is replaced with $N$ in (\ref{eq:5.10}) the zero modes shift to the negative chirality sector.
Therefore, (\ref{eq:5.36}) includes this case.
Given another potential with lesser symmetry than $\mathrm O(2)\times\mathrm O(3)$ and having the same chiral anomaly we tentatively conclude that its zero mode number is bounded by the right-hand side of (\ref{eq:5.36}).
This assumes that all of the potential's zero modes have one chirality only.
More information about the zero mode number of less symmetric potentials can be obtained from the index theorem for non-compact Euclidean space-time \cite{30},
\begin{align}
n_+-n_--\frac{1}{\pi}\sum_{l}\left[ \delta_l^+(0) - \delta_l^-(0) \right]
= - \frac{e^2}{16\pi^2}\int\mathrm d^4x\, ^\star F_{\mu\nu}F_{\mu\nu},
\tag{5.37}
\label{eq:5.37}
\end{align}
where $n_\pm$ is the number of positive/negative chirality $L^2$ zero modes; $\delta^\pm_l(0) \in(0, \pi ]$ are the zero energy scattering phase shifts for the Hamiltonians $H_\pm = \frac{1}{2}( 1 \pm \gamma_5)\slashed D^2$, and $l$ denotes the quantum numbers required to specify the phase shifts.
The sum over phase shifts gives the fractional discrepancy between the index and the chiral anomaly.
Consequently the sum in (\ref{eq:5.37}) grows more slowly than $e^2$ for $e\gg1$.
Based on (\ref{eq:5.37}) if there were a general vanishing theorem for $\text{QED}_4$ then the $\mathrm O(2)\times\mathrm O(3)$ result in (\ref{eq:5.36}) would continue to hold for potentials with lesser symmetry.
This perhaps counterintuitive conclusion that two potentials with the same chiral anomaly--one with maximal symmetry, the other with lesser symmetry-- have the same number of zero modes is related to their common asymptotic behavior.
Without a vanishing theorem (\ref{eq:5.37}) implies that the total number of zero modes may exceed the chiral anomaly.
Summarizing,
\begin{align}
\begin{split}
&\mbox{\# zero modes supported by }A_\mu\\&\quad\ge
\frac{e^2}{16\pi^2}\left| \int\mathrm d^4x\, ^\star F_{\mu\nu}F_{\mu\nu}\right|+\Delta,
\end{split}
\tag{5.38}
\label{eq:5.38}
\end{align}
with the inequality applying in the absence of a vanishing theorem and $\Delta/e^2\rightarrow O$ for $e\rightarrow\infty$.
Insertion of (\ref{eq:5.38}) in (\ref{eq:5.31}) gives with (\ref{eq:5.32})
\begin{align}
\begin{split}
&\ln\mbox{$\det_3$}\\
&\quad\le -\frac{1}{16\pi^2}\left|
\int\mathrm d^4x\,^\star F_{\mu\nu}F_{\mu\nu}\right| e^2\ln \left(\frac{e \mathscr F}{m^2}\right) + R,
\end{split}
\tag{5.39}
\label{eq:5.39}
\end{align}
with $R/(e^2 \ln e)\rightarrow 0$ for $e\rightarrow\infty$, in which case the bound in (\ref{eq:5.24}) is overridden.
As noted in Sec.A the negative sign in (\ref{eq:5.39}) is a consequence of the paramagnetism of a charged spin-$\frac{1}{2}$ fermion in a static, four-dimensional magnetic field.
\subsection{Eigenvalue $\lambda$}
Because of the possible far-reaching implications of (\ref{eq:5.39}) for the non-perturbative quantization of QED and the electroweak model it is important to have an analytic calculation of the eigenvalue $\lambda$ in (\ref{eq:5.27}) for a few special cases to show that the formalism outlined in Secs. C and D can be implemented.
We consider a class of maximally symmetric zero mode supporting potentials (\ref{eq:5.10}) with profile function
\begin{align}
a(r)=\begin{cases}
\frac{C}{R^2}\left( \frac{r}{R} \right)^{\epsilon-2} +
\frac{(2-\epsilon)C-2\nu}{R^3}r + \frac{(\epsilon-3)C + 3\nu}{R^2},&r\le R\\
\frac{\nu}{r^2},&r>R
\end{cases}
\tag{5.40}
\label{eq:5.40}
\end{align}
It is constructed so that $a$ and $a'$, and hence $F_{\mu\nu}$ are continuous at $r = R$.
The parameter $\epsilon\ge 2$ to ensure that $F \in\cap_{p>2} L^p$.
The constant $C$ can be positive or negative, and we continue to assume $\nu> 0$.
As noted in Sec. D the $L^2$ zero modes of (\ref{eq:5.25}) reside in the positive chirality sector with $M = -j-\frac{1}{2}$ for the potentials (\ref{eq:5.10}).
A $L^2$ solution of (\ref{eq:5.8}) originating from the zero mode (\ref{eq:5.34}) is
\begin{align}
\psi_\lambda(x)=\mathscr D_{-jm}^j(x)\frac{f(r)}{r^{2j+\frac{3}{2}}}
\left(\begin{array}{c}
0 \\
1 \\
0 \\
0
\end{array}\right),
\tag{5.41}
\label{eq:5.41}
\end{align}
where $f\equiv\rho_2$ in (\ref{eq:5.13}) now satisfies
\begin{widetext}
\begin{align}
\left[ \frac{\mathrm d^2}{\mathrm dr^2}
+ \frac{\frac{1}{4}-(2j+1)^2}{r^2}
+ 4\left(j+\frac{1}{\lambda}\right)ea
-e^2r^2a^2
+ \frac{e}{\lambda}r\frac{\mathrm da}{\mathrm dr}\right]f
=m^2f,
\tag{5.42}
\label{eq:5.42}
\end{align}
with eigenvalue $\lambda$ given (\ref{eq:5.27}) when $e\gg1$.
For $r>R$ let $f = r^{\frac{1}{2}} g$ so that (\ref{eq:5.42}) becomes
\begin{align}
g''+\frac{1}{r}g'
- \left( m^2 + \frac{(2j+1-e\nu)^2+2\left(1-\frac{1}{\lambda}\right)e\nu}{r^2} \right)g
= 0,
\tag{5.43}
\label{eq:5.43}
\end{align}
\end{widetext}
whose decaying solution is the modified Bessel function $K_\alpha (mr)$ with
\begin{align}
\alpha=\left[ (2j+1-e\nu)^2+2\left( 1 - \frac{1}{\lambda} \right)e\nu \right]^{\frac{1}{2}}.
\tag{5.44}
\label{eq:5.44}
\end{align}
The eigenvalue $\lambda$ is fixed by the boundary condition at $r=R$:
\begin{align}
\frac{Rf'(R)}{f(R)}=\frac{1}{2}+\frac{RK_\alpha'(mR)}{K_\alpha(mR)}.
\tag{5.45}
\label{eq:5.45}
\end{align}
The left-hand side of (\ref{eq:5.45}) is calculated from the solution of (\ref{eq:5.42}) for $0\le r\le R$.
The analysis simplifies by assuming $mR\ll1$.
Let $e\nu = N + \Delta$, $N = 2, 3,\dots$; $0< \Delta<1$, $j = 0,\frac{1}{2},\dots,j_\mathrm{max}$ with $j_\mathrm{max} = (N-2)/2$ since $L^2$ zero modes exist only for $e\nu > 2j + 2$.
It is known that $\det_\mathrm{ren}$ has a branch point in $m$ beginning at $m=0$ \cite{24} which is evident by the presence of $K_\alpha$ in (\ref{eq:5.45}).
This leads to the following small mass expansions for $j = 0,\frac{1}{2},\dots, j_\mathrm{max}-\frac{1}{2}$ and $\alpha_0=e\nu -2j - 1 > 2$,
\begin{align}
f&=Bf_0\left(1+m^2f_2+m^4f_4+\mathrm O\left(m^{2\alpha_0}\mbox{ or }m^6\right)\right),
\tag{5.46}
\label{eq:5.46}\\
\lambda &=1-m^2\delta_2 - m^4\delta_4 + \mathrm O\left(m^{2\alpha_0}\mbox{ or }m^6\right);
\tag{5.47}
\label{eq:5.47}
\end{align}
for $j = j_\mathrm{max}$, $1 < \alpha_0 < 2$,
\begin{align}
f&=Bf_0\left( 1+m^2f_2+m^{2\alpha_0}f_{2\alpha_0}+\mathrm O \left( m^4 \right) \right)
\tag{5.48}
\label{eq:5.48}\\
\lambda&=1-m^2\delta_2 - m^{2\alpha_0}\delta_{2\alpha_0} + \mathrm O\left(m^4\right);
\tag{5.49}
\label{eq:5.49}
\end{align}
where $\alpha_0$ is the $m = 0$ term in the expansion of $\alpha$ in (\ref{eq:5.44}), and $B$ is a normalization constant.
The expansion of $\delta$ in (\ref{eq:5.27}), (\ref{eq:5.47})and(\ref{eq:5.49}) in powers of $m$ must begin at $m^2$ to be consistent with the boundary condition (\ref{eq:5.45}).
For all cases there is a $\mathrm O(m^2)$ term in the expansions of $f$ and $\lambda$.
The case $e\nu = 3, 4,\dots$ is commented on in Appendix E.
Here $f_0$ is the solution of (\ref{eq:5.42}) when $m = 0$, $\lambda= 1$ and $0\le r \le R$,
\begin{align}
f_0=r^{2j+\frac{3}{2}}e^{-e\int_{0}^{r}\mathrm ds\,sa(s)} ,
\tag{5.50}
\label{eq:5.50}
\end{align}
With these expansions the two sides of (\ref{eq:5.45}) can be matched in powers of $m$ to obtain $\lambda$.
The calculation is outlined in Appendix E.
For $mR<<1$, $e\nu > 2j + 2$ and $e\gg1$ the calculation in Appendix E gives, following (E11) and (E12),
\begin{align}
\lambda=1-\frac{2m^2/e}{\|(\sigma F(r_0))^+\|_1}
(1+\mathrm O(1/e))
+ \mathrm O\left(\frac{m^4R^4}{e^2}\right),
\tag{5.51}
\label{eq:5.51}
\end{align}
where $(\sigma F)^+$ is the positive chirality component of $\sigma F$ in (\ref{eq:5.14}) that is responsible for the existence of zero modes, and $r_0$ is the unique root in the interval $0<r <R$ of
\begin{align}
4j+3-2er^2a(r)=0.
\tag{5.52}
\label{eq:5.52}
\end{align}
Here $\|(\sigma F)^+\|_1$ is the spin trace norm of $(\sigma F)^+$ defined for an operator $A$ by $\|A\|_1=\mathrm{Tr}(A^\dagger A)^{1/2}$.
Because $(\sigma F)^+$ obtained from (\ref{eq:5.14}) and (\ref{eq:5.40}) is a smooth function, $\lambda$ is a slowly varying function of $j$ since $\mathrm dr_0 /\mathrm dj = \mathrm O(1/e)$ from (\ref{eq:5.52}).
For this special case we can count zero modes following (\ref{eq:5.35}), (\ref{eq:5.36}) and rewrite (\ref{eq:5.39}) as an equality. To leading order in $m^2/e$, $\delta$ in (\ref{eq:5.27}) can be read off from (\ref{eq:5.51}). This fixes the argument of the logarithm in (\ref{eq:5.28}) precisely:
\begin{align}
\begin{split}
&\ln\mbox{$\det_3$}\\
&=-\sum_{j=0}^{j_{\text{max}}}(2j+1)\left[\ln\left(\frac{e \|(\sigma F(r_0(j)))^+\|_1}{2m^2}\right)+\text{O(1)}\right]+R_1 ,
\end{split}
\tag{5.53}
\label{eq:5.53}
\end{align}
where $j_{\text{max}}=[e\nu]/2-1$ and $\text{lim}_{e \rightarrow \infty}R_1/(e^2\ln e)=0$. The remainder $R_1$ includes contributions to $\mbox{$\det_3$}$ from eigenvalues bounded away from 1 as discussed in Sec.B. Defining an average $F_{\mu\nu}, \mathscr F$, by
\begin{align}
\sum_{j=0}^{j_{\text{max}}}(2j+1) \ln \|(\sigma F(r_0(j)))^+\|_1 \left/ {\sum}_{j=0}^{j_{\text{max}}}(2j+1)\right. \equiv \ln \mathscr F
\tag{5.54}
\label{eq:5.54}
\end{align}
obtain from (\ref{eq:5.35}) and (\ref{eq:5.36}) for $e\gg1$
\begin{align}
\begin{split}
&\ln\mbox{$\det_3$}\\
& =-\frac{e^2}{16\pi^2}\left|\int\mathrm d^4x\,^\star F_{\mu\nu}F_{\mu\nu}\right|
\left[\ln\left(\frac{e \mathscr F}{2 m^2}\right)+\text{O(1)}\right]+R_2,
\end{split}
\tag{5.55}
\label{eq:5.55}
\end{align}
where $R_2$ contains a $\text{O}(\text{e}\nu\ln(e\mathscr F))$ term from the $\text{O}(\text{e}\nu)$ residue in (\ref{eq:5.36}) and satisfies the same limit as $R_1$. The result (\ref{eq:5.55}) overrides the bound (\ref{eq:5.24}).
\def\text{d}t{\text{d}t}
\def\text{d}^4{\text{d}^4}
\def\mathinner{\mathrm{Tr}}{\mathinner{\mathrm{Tr}}}
\section{Charge renormalization term in $\boldsymbol{\ln\det_\mathrm{ren}}$}
\subsection{ Scaling parameter}
Consider the last contribution to $\ln\det_\mathrm{{ren}}$ in \eqref{eq:twopointsix} and \eqref{eq:threepointnine}, here designated as
\begin{align}
\Pi = e^2 \int_0^{\infty} \text{d}t e^{-tm^2} \left[ \frac{\@ifstar{\oldnorm}{\oldnorm*}{F}^2}{32\pi^2 t} - \frac{1}{2} \mathinner{\mathrm{Tr}} \left( e^{-tD^2} F_{\mu \nu} \Delta_A F_{\mu \nu} \right) \right].
\tag{6.1}
\label{eq:6.1}
\end{align}
It is not obvious what to call the right-hand side of \eqref{eq:6.1}, but since $e^2 \@ifstar{\oldnorm}{\oldnorm*}{F}^2 /(32 \pi^2 t)$ is part of the on-shell charge renormalization
subtraction in $\ln\det_\mathrm{{ren}}$ it will be referred to as the charge renormalization term. As in Sec.IV break the integral in \eqref{eq:6.1} into $\int_{0}^{1/e\mathscr{F}}$ and $\int_{1/e\mathscr{F}}^{\infty}$, where $\mathscr{F}$ fixes the scale of the amplitude of $F_{\mu \nu}$.
Then $\Pi = I_1 + I_2 + I_3$, where
\begin{align}
I_1 &= \frac{e^2 \@ifstar{\oldnorm}{\oldnorm*}{F}^2}{32\pi^2} \int_{1/e \mathscr{F}}^{\infty} \frac{\text{d}t}{t} e^{-tm^2},
\tag{6.2}
\label{eq:6.2}\\
\nonumber
I_2 &= \frac{e^2}{32} \int_0^{1/e \mathscr{F}}\text{d}t e^{-tm^2} \\
&\times \left[ \frac{\@ifstar{\oldnorm}{\oldnorm*}{F}^2}{\pi^2 t}-16 \mathinner{\mathrm{Tr}} \left( e^{-tD^2} F_{\mu \nu} \Delta_A F_{\mu \nu} \right)\right],
\tag{6.3}
\label{eq:6.3}\\
I_3 &= -\frac{e^2}{2} \int_{1/e\mathscr{F}}^{\infty} \text{d}t e^{-tm^2} \mathinner{\mathrm{Tr}} \left( e^{-tD^2} F_{\mu \nu} \Delta_A F_{\mu \nu} \right).
\tag{6.4}
\label{eq:6.4}
\end{align}
At this point the choice of scaling parameter in \eqref{eq:6.2}-\eqref{eq:6.4} appears arbitrary. It is not for the following reasons.\\
(a) As remarked in Sec. IV, if the strong coupling behavior of $\det_\mathrm{{ren}}$ is to have anything to do with large amplitude variations of $F_{\mu \nu}$
then $e$ must appear in the combination $e\mathscr F$.\\
(b) The scaling parameter must be universal and not tied to any particular background field. As $m$ is always present in $\det_\mathrm{{ren}}$
it should be considered in the construction of a possible scaling parameter.\\
(c) The scaling parameter should result in the largest possible lower bound on $\Pi$ for $e \mathscr F \gg m^2$.\\
(d) The lower bound should respect what is known about $\ln\det_\mathrm{ren}$'s mass dependence.\\
Based on (a)-(c) and the requirement that the scaling parameter have dimension (length)$^{-2}$ then possible scaling parameters have the
form $(e \mathscr F )^a m^b$, $2a + b = 2$, $a\ne0$. But only $a=1$, $b=0$ are allowed by requirement (d). To see why consider $I_1$ in \eqref{eq:6.2}. Following the result \eqref{eq:fourpointfive} for $e \mathscr F \gg m^2$,
\begin{align}
I_1 = \frac{e^2 \@ifstar{\oldnorm}{\oldnorm*}{F}^2}{32 \pi^2} \ln \left( \frac{e \mathscr F}{m^2} \right) -\gamma + R,
\tag{6.5}
\label{eq:6.5}
\end{align}
where again $\gamma$ is Euler's constant and $0<|R|<m^2 /(e\mathscr F)$. The mass singularity in \eqref{eq:6.5} is induced by the on-shell charge
renormalization of $\ln\det_\mathrm{{ren}}$ in \eqref{eq:twopointone}, the starting point of this analysis. It is shown in Appendix F that for potentials $A_{\mu} \in \underset{r\ge 4- \epsilon}{\cap} L^r ( \mathbb{R}^4)$,
$\epsilon>0$ and arbitrarily small, $\ln\det_\mathrm{{ren}}$ at $m=0$ is finite when it is renormalized off-shell.
Moreover, its $m=0$ limit is continuous. The restriction on $A_{\mu}$ excludes zero modes. Including them would cause lndet$_3$ to
diverge at $m=0$ as found in the results \eqref{eq:5.31} and \eqref{eq:5.39} that are independent of how $\ln\det_\mathrm{{ren}}$ is renormalized.
To define $\det_5$ in \eqref{eq:F1}, and therefore $\det_\mathrm{{ren}}$, it is sufficient to assume $A_{\mu} \in \underset{r>4}{\cap} L^r (\mathbb R ^4)$ \cite{7,31}.
The charge renormalization term $\Pi$ depends only on $D^2$ and is therefore insensitive to zero modes. Without loss of generality we may assume here that $F_{\mu \nu} \in L^2$
and therefore that $A_{\mu} \in L^4$. This follows from the Sobolev inequality for gradients on $\mathbb R^4$ \cite{32}. Hence the restriction on $A_{\mu}$ in the preceding paragraph can be consistently assumed here.
When the first term in \eqref{eq:6.5} is combined with the mass singularity of $\ln\det_{SQED}$ in \eqref{eq:fourpointsix}, multiplied by $2$ as required
by \eqref{eq:threepointnine}, obtain
\begin{align}
\ln\mbox{$\det{}_\mathrm{ren}$} \underset{m \to 0}{\sim} -\frac{e^2 \@ifstar{\oldnorm}{\oldnorm*}{F}^2}{48 \pi^2} \ln m^2 +\text{finite}.
\tag{6.6}
\label{eq:6.6}
\end{align}
The result in Appendix F allows us to state that this is the only divergent mass singularity of $\ln\det_\mathrm{{ren}}$ in the absence of zero
modes. If $\ln\det_\mathrm{{ren}}$ were subtracted off-shell by adding to \eqref{eq:twopointone}
the term
\begin{align}
\frac{e^2 \@ifstar{\oldnorm}{\oldnorm*}{F}^2}{48 \pi^2} \int_0^{\infty} \frac{\text{d}t}{t} \left( e^{-t\mu^2} - e^{-tm^2} \right) = \frac{e^2 \@ifstar{\oldnorm}{\oldnorm*}{F}^2}{48 \pi^2} \ln \left( \frac{m^2}{\mu^2} \right),
\tag{6.7}
\label{eq:6.7}
\end{align}
then $\ln\det_\mathrm{{ren}}$ would be finite at $m=0$. This freedom to renormalize off-shell must be respected by the scaling parameter. Indeed, if
the scaling parameter $(e\mathscr F)^a m^b$, $b\ne0$ were chosen in \eqref{eq:fourpointtwo} and \eqref{eq:6.2}-\eqref{eq:6.4} then \eqref{eq:6.6} would become
\begin{align}
\ln\mbox{$\det{}_\mathrm{ren}$} \underset{m \to 0}{\sim} \left( \frac{b}{96} - \frac{1}{48} \right) \frac{e^2 \@ifstar{\oldnorm}{\oldnorm*} F^2}{\pi^2} \ln m^2 + \text{finite}.
\tag{6.8}
\label{eq:6.8}
\end{align}
This introduces a spurious $be^2 \@ifstar{\oldnorm}{\oldnorm*} F^2 \ln m^2 /96 \pi^2 $ mass singularity into $\ln\det_\mathrm{{ren}}$'s lower bound when it is renormalized off-shell using
\eqref{eq:6.7}. Therefore, the only acceptable scaling parameter for the strong coupling limit of $\Pi$ in \eqref{eq:6.1} and in $\det_\text{SQED}$ in \eqref{eq:fourpointtwo} is $e\mathscr F$. This further
justifies the choice of scaling parameter in Sec.IV.\\
\subsection{ Estimates}
Consider $I_2$ in \eqref{eq:6.3}. The trace in its last term can be put in the form $\mathinner{\mathrm{Tr}}(A^{\dagger} A)$ using the trace's cyclic property. So the last
term is not negative. Write out the trace term in its original form and note that
\begin{widetext}
\begin{align}
&\int_0^{1/e \mathscr F} \text{d}t e^{-tm^2} \int \text{d}^4 x \text{d}^4 y e^{-tD^2}(x,y) F_{\mu \nu}(y) \Delta_A (y,x) F_{\mu \nu}(x) \notag\\
&\le \int_0^{1/e\mathscr F} \text{d}t e^{-tm^2}\int \text{d}^4 x \left|\left(e^{-tD^2} F_{\mu \nu} \Delta_A \right)(x) \right| \left|F_{\mu \nu}(x)\right| \notag \\
&\le \int_0^{1/e\mathscr F} \text{d}t e^{-tm^2} \int \text{d}^4 x \left( e^{-tP^2} \left| F_{\mu \nu} \right| \left| \Delta_A \right| \right)(x) \left|F_{\mu \nu} (x)\right| \notag \\
&\le \int_0^{1/e\mathscr F} \text{d}t e^{-tm^2}\int \text{d}^4 x \left( e^{-tP^2} \left| F_{\mu \nu} \right|\Delta \right) (x) \left| F_{\mu \nu} (x) \right| \notag \\
&= \int_0^{1/e\mathscr F} \text{d}t e^{-tm^2} \int \text{d}^4 x \text{d}^4 y e^{-tP^2}(x,y) \left| F_{\mu \nu}(x) \right| \Delta(y-x) \left| F_{\mu \nu} (x) \right|.
\tag{6.9}
\label{eq:6.9}
\end{align}
\end{widetext}
To obtain these results we used the diamagnetic inequality of Simon \cite{12,33} to go from the second to the third line:
\begin{align}
\left| (e^{-tD^2} f) (x) \right| \le \left( e^{-tP^2} \left|f \right| \right)(x).
\tag{6.10}
\label{eq:6.10}
\end{align}
This holds for all $t>0$ and almost all $x \in \mathbb R^4$ and for potentials that are locally square integrable, as we are assuming. For more recent
comments on \eqref{eq:6.10} see \cite{34}. In addition we used Kato's inequality in the form given by \eqref{eq:A3} to go from the third to the fourth line.
Noting that
\begin{align}
e^{-tP^2} (x,y) = \frac{1}{16 \pi^2 t^2} e^{- \left| x-y \right|^2/4t},
\tag{6.11}
\label{eq:6.11}
\end{align}
insertion of \eqref{eq:6.9} in \eqref{eq:6.3} gives
\begin{widetext}
\begin{align}
I_2 \ge \frac{e^2}{32 \pi^2} \int_0^{1/e\mathscr F} \frac{\text{d}t}{t} e^{-tm^2} \left( \@ifstar{\oldnorm}{\oldnorm*} F^2 -\frac{1}{t} \int \text{d}^4 x \text{d}^4 y \left| F_{\mu \nu}(x) \right| \Delta(x-y) e^{-(x-y)^2/4t} \left| F_{\mu \nu }(y)\right| \right) .
\tag{6.12}
\label{6.12}
\end{align}
\end{widetext}
By Young's inequality in the form \cite{19}
\begin{align}
\left| \int \text{d}^4 x \text{d}^4 y f(x) g(x-y) h(y) \right| \le \@ifstar{\oldnorm}{\oldnorm*} f _p \@ifstar{\oldnorm}{\oldnorm*} g _q \@ifstar{\oldnorm}{\oldnorm*} h _r ,
\tag{6.13}
\label{eq:6.13}
\end{align}
where $1/p + 1/q + 1/r = 2$, $p,q,r\ge1$ and $\@ifstar{\oldnorm}{\oldnorm*} f _p = ( \int \text{d}^4 x \left|f(x)\right|^p )^{1/p}$, etc.,
\begin{align}
I_2 \ge \frac{e^2 \@ifstar{\oldnorm}{\oldnorm*} F ^2}{32 \pi^2} \int_0^{1/e\mathscr F} \frac{\text{d}t}{t} e^{-tm^2}\left( 1 - \frac{1}{t} \int \text{d}^4 x \Delta (x) e^{-x^2/4t}\right).
\tag{6.14}
\label{eq:6.14}
\end{align}
From $\Delta(x) = mK_1 (mx)/(4\pi^2 x)$ and integral 2.16.8.5 of \cite{35} get
\begin{align}
I_2 \ge \frac{e^2 \@ifstar{\oldnorm}{\oldnorm*} F ^2}{32 \pi^2} \int_0^{1/e\mathscr F} \frac{\text{d}t}{t} e^{-tm^2}\left[ 1 - m^2t e^{m^2t} \Gamma (-1, m^2t)\right],
\tag{6.15}
\label{eq:6.15}
\end{align}
where $\Gamma(-1, m^2 t)$ is the incomplete gamma function which we use in the form
\begin{align}
\Gamma(-1, m^2t) = \frac{1}{m^2t}e^{-m^2t}- \int_{m^2t}^{\infty} \frac{\text d z}{z} e^{-z} .
\tag{6.16}
\label{eq:6.16}
\end{align}
Insertion of \eqref{eq:6.16} in \eqref{eq:6.15} and integrating by parts gives for $e\mathscr F \gg m^2$
\begin{align}
I_2 \ge \frac{e^2 \@ifstar{\oldnorm}{\oldnorm*} F ^2}{32 \pi^2} \left[ \frac{m^2}{e\mathscr F} \left( \ln \left( \frac{e \mathscr F}{m^2} \right) -\gamma + R \right) +1 - e^{-m^2/e\mathscr F} \right],
\tag{6.17}
\label{eq:6.17}
\end{align}
with $\gamma$ and $R$ the same as in \eqref{eq:6.5}. Note that the lower bound in \eqref{eq:6.17} is finite at $m=0$ as it should be.
There are no ultraviolet divergences in $I_2$. The small $t$ dependence of the first term in \eqref{eq:6.3} is cancelled by the trace term, as was shown
in the above non-perturbative estimate. So it must be a general property of the trace term that
\begin{align}
16 \mathinner{\mathrm{Tr}} \left( e^{-tD^2} F_{\mu \nu} \Delta_A F_{\mu \nu} \right) \underset{t \to 0}{\sim} \frac{\@ifstar{\oldnorm}{\oldnorm*} F ^2}{\pi^2 t} + \text{less singular in t}.
\tag{6.18}
\label{eq:6.18}
\end{align}
By inspection of \eqref{eq:6.3} we conclude that
\begin{align}
\lim_{e \mathscr F \to \infty} \frac{I_2}{(e \mathscr F)^2 \ln \left( e \mathscr F \right)} = 0.
\tag{6.19}
\label{eq:6.19}
\end{align}
\def\text{d}{\text{d}}
Now consider $I_3$ in \eqref{eq:6.4}. As noted in the case of $I_2$ the trace is positive so that $I_3 \le 0$. Application of the inequality \eqref{eq:6.10} does
not lead to a satisfactory lower bound on $I_3$ . Namely, if it were saturated $I_3$ would cancel the large amplitude growth of $I_1$ in \eqref{eq:6.5},
resulting in a slow $O((e\mathscr F)^2)$ growth of $\Pi$ in \eqref{eq:6.1} and leading to the uninformative bound $\ln\det_\mathrm{{ren}} \ge -e^2 \@ifstar{\oldnorm}{\oldnorm*} F ^2 \ln (e\mathscr F/m^2)/96 \pi^2 + O\left((e\mathscr F)^2\right)$ following \eqref{eq:threepointnine} and \eqref{eq:fourpointsix}.
We are confident that $\ln\det_\mathrm{{ren}}$ grows at least as fast as $ce^2 \@ifstar{\oldnorm}{\oldnorm*} F ^2 \ln(e\mathscr F)$, $c>0$, in the absence of zero mode supporting background fields.
This confidence is based on the result \cite{36} for the growth of $\ln\det_\mathrm{{ren}}$ for random, square-integrable, time-independent, non-zero mode supporting magnetic fields $\mathbf{B} (x)$ on $\mathbb R ^3$ ,
\begin{align}
\lim_{e \to \infty} \frac{\ln\mbox{$\det{}_\mathrm{ren}$}}{e^2 \ln e} = \frac{\@ifstar{\oldnorm}{\oldnorm*}{\mathbf{B}} ^2 T}{24 \pi^2},
\tag{6.20}
\label{eq:6.20}
\end{align}
where $\@ifstar{\oldnorm}{\oldnorm*}{\mathbf B}^2 = \int \text{d}^3 x \mathbf B \cdot \mathbf B(x)$ and $T$ is the size of the time box.
Therefore, our estimate of $I_3$ has to be more detailed than in the case of $I_2$. We claim that $\underset{e\to \infty}{\lim} I_3 /(e^2 \ln e) = 0$ for the class of fields considered here.\\
By summing over a complete set of scattering eigenstates $\ket{E, \alpha}$ of $D^2$, $I_3$ can be represented as
\begin{widetext}
\begin{align}
I_3 &= -\frac{e^2}{2} \sum_{\alpha, \beta} \int_{1/e \mathscr F}^{\infty} \text{d} t e^{-tm^2} \int_{0}^{\infty}\text{d} E e^{-tE} \int_{0}^{\infty} \text{d} E' \frac{\bra{E,\alpha} F_{\mu \nu} \ket{E', \beta} \bra{E', \beta} F_{\mu \nu} \ket{E,\alpha}}{E' + m^2} \notag \\
&= -\frac{e^2}{2} \sum_{\alpha, \beta} \int_0^{\infty} \text{d} E \int_0^{\infty} \text{d} E' e^{-(E+m^2)/e\mathscr F} \frac{\left| \bra{E,\alpha} F_{\mu \nu} \ket{E', \beta} \right|^2}{(E+m^2)(E'+m^2)} ,
\tag{6.21}
\label{eq:6.21}
\end{align}
\end{widetext}
where $\alpha$ and $\beta$ are complete sets of angular momentum-like quantum numbers. Due to the above theorem on the $m=0$ limit of $\ln\det_\mathrm{{ren}}$ $I_3$ is finite at $m=0$. So whether $F_{\mu \nu}$ is long or short-ranged is irrelevant to the
growth of $I_3$ with $e$. Without loss of generality we may confine this discussion to fields with compact support. As $F_{\mu \nu}$ was assumed to be
differentiable in previous sections the compactly supported fields are assumed to rapidly and smoothly tend to zero in a narrow zone
near their boundries. In addition we may assume rotational symmetry. Asymmetric, tangled fields will tend to lower the matrix elements
$\left|\bra{E,\alpha}F_{\mu \nu} \ket{E',\beta} \right|$. We will assume maximally symmetric $O(3)XO(2)$ fields to maximize $\left|I_3 \right|$.
For the potentials \eqref{eq:5.10} the equation for the radial part of the scattering states that satisfy $D^2 \psi_{E,\alpha} = E \psi_{E,\alpha} $ is \cite{24}
\begin{widetext}
\begin{align}
\begin{split}
\left( -\frac{\text{d} ^2}{\text{d} r^2} + \frac{(2j+1)^2-1/4}{r^2} + 4m_1 ea + e^2r^2a^2 \right) \phi_{Ejm_1}(r) = E \phi_{Ejm_1}(r),
\end{split}
\tag{6.22}
\label{eq:6.22}
\end{align}
\end{widetext}
where $\psi_{E,\alpha}(x) = r^{-2j-3/2} \phi_{Ejm_1}(r) \mathscr D^j_{m_1 m_2} (x), r= \left| x\right|$, and the four-dimensional rotation matrices $\mathscr D_{m_1 m_2}^j$ are defined in Sec. V.B.
Let $F_{\mu \nu}$ have range $R$. For $r>R$ the normalized wave function is, on setting the chiral anomaly equal to zero in \cite{24},
\begin{align}
\begin{split}
\phi_{Ejm_1}(r) &= \sqrt{\frac{r}{2}}J_{2j+1}(kr) \cos \delta_{j m_1} (k,e) \\
&- \sqrt{\frac{r}{2}}Y_{2j+1}(kr) \sin \delta_{j m_1}(k,e),
\end{split}
\tag{6.23}
\label{eq:6.23}
\end{align}
where $\delta_{j m_1} (k,e)$ is the scattering phase shift in the indicated channel, $E=k^2$, and $Y_n$ is a Bessel function of the second kind.
We assumed in Sec. V.B that $a$ and $ra'$ are bounded functions of $r$. This will be assumed here. Therefore, any admissible $a$ maintains the
small distance behavior $\phi_{Ejm_1}\sim r^{2j+3/2}$ independent of $e$. What $\phi_{Ejm_1}$ does as $r\nearrow R$ is manifested in the exterior wave function \eqref{eq:6.23} through the
phase shifts. From \eqref{eq:6.22}, although $a$ descends rapidly to zero in a zone near $r=R$, it is evident from the $(era)^2$ term in \eqref{eq:6.22} that as
$e\to \infty$ there develops a high barrier at some point $r<R$ that blocks the entry of the exterior wave function \eqref{eq:6.23}, resulting in approximate
hard sphere scattering. This happens however rapidly $F_{\mu \nu}$ varies for $r<R$. So there is no reason why $F_{\mu \nu} = $ constant for $r<R$ and falling
rapidly to zero just before $r=R$ cannot be taken as representative of the general field case for the strong coupling estimate of $I_3$.
Accepting this, refer to \eqref{eq:5.10} and set $a(r) = \lambda/R^2$ for $0 \le r \le R- \epsilon$ and $a(R ) = 0$. Then $F_{\mu \nu} = 2 \lambda M_{\mu \nu} /R^2$ for $0<r<R-\epsilon$.
The parameter $\lambda$ is related to the scaling parameter $\mathscr F$ by $\mathscr F^2 = F_{\mu \nu}^2 = 16 \lambda^2/R^4$ since $M_{\mu \nu}^2 = 4$. Then
\begin{align}
\begin{split}
&\bra{E j m_1} F_{\mu \nu} \ket{E' j' m_1'} \\
&= \frac{4 \pi^2 \lambda M_{\mu \nu}}{2j+1} \delta_{j j'} \delta_{m_1 m_1'} \int_0^R \text{d} r \phi_{E j m_1} \phi_{E' j m_1},
\end{split}
\tag{6.24}
\label{eq:6.24}
\end{align}
where we have taken the limit $\epsilon = 0$ on the right-hand side of \eqref{eq:6.24}. As shown below it follows from \eqref{eq:6.22} that
\begin{align}
\begin{split}
&\left( \phi_{E'jm_1} \phi'_{Ejm_1} - \phi_{Ejm_1}\phi'_{E'jm_1}\right)(R) \\
&= (E' - E) \int_0^R \text{d} r \phi_{Ejm_1} \phi_{E'jm_1}.
\end{split}
\tag{6.25}
\label{eq:6.25}
\end{align}
Then \eqref{eq:6.24} and \eqref{eq:6.25} combined with \eqref{eq:6.21} give
\begin{widetext}
\begin{align}
\begin{split}
I_3 =& -2\pi^4(e\mathscr F)^2 \int_0^{\infty} \text{d} E \int_0^{\infty} \text{d} E' e^{-(E+m^2)/e\mathscr F} \\
&\times\sum_{j=0,1/2,..}^{\infty} \frac{1}{(2j+1)^2} \sum_{m_1,m_2 = -j}^{j}
\frac{\left[ \left( \phi_{E'jm_1} \phi'_{Ejm_1} - \phi_{Ejm_1} \phi'_{E'jm_1} \right)(R) \right]^2}{(E+m^2)(E'-E)^2(E'+m^2)}.
\end {split}
\tag{6.26}
\label{eq:6.26}
\end{align}
\end{widetext}
To obtain \eqref{eq:6.25} from the assumed behavior of $F_{\mu \nu}$ multiply \eqref{eq:6.22} at energy $E$ by $\phi_{E'jm_1}(r) F_{\mu \nu}(r)$, subtract the result with $E\leftrightarrow E'$
and integrate by parts over the interval $0 \le r \le R$. Since $F_{\mu \nu}(R)=0$ and $\phi_{Ejm_1}(0)=0$ this gives
\begin{widetext}
\begin{align}
\int_{R-\epsilon}^R \text{d}& r \left( \phi_{E'jm_1} \phi'_{Ejm_1} - \phi_{Ejm_1} \phi'_{E'jm_1} \right) \frac{\text{d} F_{\mu \nu}(r)}{\text{d} r} \notag \\
=&\left( E- E' \right) \left[ \frac{2 \lambda M_{\mu \nu}}{R^2} \int_0^{R-\epsilon} \text{d} r \phi_{Ejm_1} \phi_{E'jm_1} + \int_{R-\epsilon}^R \text{d} r \phi_{Ejm_1} \phi_{E'jm_1} F_{\mu \nu}(r) \right].
\tag{6.27}
\label{eq:6.27}
\end{align}
\end{widetext}
Assuming $\epsilon/R \ll 1$ and noting that $\int_{R-\epsilon}^R \text{d} r F'_{\mu \nu}(r) = - F_{\mu \nu}(R-\epsilon) = -\frac{2\lambda M_{\mu \nu}}{R^2}$, \eqref{eq:6.25} follows after letting $\epsilon \to 0$.
The phase shifts required to calculate $I_3$ are obtained as follows.
Set $a = \lambda /R^2$ in \eqref{eq:6.22} and let, omitting subscripts,
\begin{align}
\phi (r) = r^{2j + 3/2} f(r) e^{-\lambda e r^2 /2R^2}.
\tag{6.28}
\label{eq:6.28}
\end{align}
Then
\begin{align}
\begin{split}
&f'' + \left( \frac{4j+3}{r} - \frac{2 \lambda e r}{R^2} \right) f' + \left[ k^2 - \frac{4\lambda e}{R^2}(j+m_1+1) \right] f \\
&= 0.
\end{split}
\tag{6.29}
\label{eq:6.29}
\end{align}
The solution of \eqref{eq:6.29} regular at the origin is the confluent hypergeometric function
\begin{align}
f(r) = M\left( j+m_1+1 - \frac{(kR)^2}{4\lambda e}, 2j+2, \frac{\lambda e r^2}{R^2} \right),
\tag{6.30}
\label{eq:6.30}
\end{align}
following the notation of \cite{37}. Joining \eqref{eq:6.23} with \eqref{eq:6.28} at $r=R$ gives
\begin{align}
\begin{split}
&\tan \delta_{j m_1} (k,\lambda \text{e}) = \\
&\frac{(\gamma -1/2)J_{2j+1}(kR)-kR J'_{2j+1}(kR)}{(\gamma-1/2)Y_{2j+1}(kR)-kR Y'_{2j+1}(kR)},
\end{split}
\tag{6.31}
\label{eq:6.31}
\end{align}
where $\gamma = (r \phi' /\phi)_R$. Eqs. \eqref{eq:6.28}, \eqref{eq:6.30} and Eq.(13.4.8) in \cite{37} for $\text{d} M(a,b,z)/\text{d} z$ give
\begin{align}
\gamma = 2j + \frac{3}{2} - \lambda e + \frac{2\lambda e a}{b} \frac{M(a+1, b+1, \lambda e)}{M(a,b,\lambda e)},
\tag{6.32}
\label{eq:6.32}
\end{align}
where $a = j + m_1 + 1 - (kR)^2 /(4 \lambda e), b = 2j + 2$. There are several cases. For $j < \lambda e \gg 1$, fixed $k$,
\begin{align}
\begin{split}
\gamma = &\lambda e + 2m_1 -\frac{1}{2} - \frac{(kR)^2}{2e\lambda} \\
&+ O\left( \frac{j^2}{\lambda e}, \frac{j(kR)^2}{(\lambda e)^2}, \frac{(kR)^4}{(\lambda e)^3} \right).
\end{split}
\tag{6.33}
\label{eq:6.33}
\end{align}
For $j > \lambda e \gg 1$, fixed k,
\begin{align}
\gamma = 2j + \frac{m_1}{j} \lambda e - \frac{(kR)^2}{4j} + O\left( \frac{\lambda e}{j^2}, \frac{m_1 \lambda e}{j^2}, \frac{(kR)^2}{j^2}\right),
\tag{6.34}
\label{eq:6.34}
\end{align}
and for $kR \to \infty$, fixed $j$, $\lambda e$,
\begin{align}
\gamma = - \left[ kR + O\left( \frac{1}{kR} \right) \right] \frac{J_{2j+2}(kR)}{J_{2j+1}(kR)} + 2j - \lambda e + 3/2.
\tag{6.35}
\label{eq:6.35}
\end{align}
These results are obtained using the asymptotic expansions of $M(a,b,z)$ for large $a, b, z$ in \cite{37,38}. Following \eqref{eq:6.35} the phase shifts vanish
at high energy as $\tan \delta \sim (e \lambda /kR)\cos^2 (kR-\left(j+ \frac{1}{2} \right) \pi - \pi /4)$.
In order to estimate $I_3$ for $e\mathscr{F}\rightarrow\infty$ it is convenient to divide the range of the $kR$, $k'R$ integrations in \eqref{eq:6.26} into $[0,2)$, $[2, 2\sqrt{ e \mathscr F R^2} )$,
$[2\sqrt{ e \mathscr F R^2} , 2(e \mathscr F R^2 )^{1-\epsilon}), [2e \mathscr F R^2 , \infty ]$ and the special case
$kR, k'R = O(e \mathscr F R^2 )^{1-\epsilon}$, where $0< \epsilon \ll 1$. To accommodate the joining conditions \eqref{eq:6.33}-\eqref{eq:6.35} the range of $j$ also has to be partitioned.
It is essential not to interchange the large $e\mathscr F$ limit with the sum over $j$. We find that the dominant contributions to \eqref{eq:6.26} come from
$0\le j\le \sqrt{ e \mathscr F R^2} , 2 \le kR \lesssim O( \sqrt{e\mathscr F R^2} )$ and $2 \le k'R \le \infty$. There are many cases
to consider; we outline here a representative case to indicate how the estimates are done.
Consider the contribution to \eqref{eq:6.26} given by
\begin{widetext}
\begin{align}
I \equiv - 8 \pi^4 (e \mathscr F R^2)^2 \sum_{j=0}^{\sqrt{e\mathscr F R^2}}\sum_{m_1, m_2 = -j}^{j} \frac{1}{(2j+1)^2} &\int_{2j+1}^{2\sqrt{e \mathscr F R^2}} \frac{\text{d} (kR)}{kR} e^{-\frac{k^2}{e\mathscr F}}\int_{2\sqrt{e\mathscr F R^2}}^{2(e \mathscr F R^2)^{1-\epsilon}} \frac{\text{d} (k'R)}{k'R} \notag \\
& \times \frac{\left[ \left( \phi_{E'jm_1} \phi'_{Ejm_1} - \phi_{Ejm_1} \phi'_{E'jm_1} \right)(R) \right]^2}{R^4(k'^2-k^2)^2},
\tag{6.36}
\label{eq:6.36}
\end{align}
\end{widetext}
where we have noted above that we can set $m=0$. For the range of $kR$, $k'R$ and $j$ in \eqref{eq:6.36} joining condition \eqref{eq:6.33} applies. From \eqref{eq:6.23}, \eqref{eq:6.31}
and \eqref{eq:6.33} obtain
\begin{widetext}
\begin{align}
\frac{(\phi_{E'}\phi'_{E} - \phi_E \phi'_{E'})(R)}{k'^2-k^2} \underset{\lambda e \gg 1}{\sim} \frac{R^2}{2\pi^2(\lambda e)^3} \left[ \left(J_{2j+1}(kR) - \frac{kR}{\gamma -1/2} J'_{2j+1}(kR) \right)^2 + \left( Y_{2j+1} (kR) - \frac{kR}{\gamma -1/2} Y'_{2j+1}(kR)\right)^2 \right]^{-1/2} \notag \\
\times \Big[ k \to k' \Big]^{-1/2} \sim \frac{R^2}{2\pi^2 (\lambda e)^3}\left(J_{2j+1}^2(kR) + Y^2_{2j+1}(kR) \right)^{-1/2} \left(J_{2j+1}^2(k'R) + Y_{2j+1}^2(k'R) \right)^{-1/2}.
\tag{6.37}
\label{eq:6.37}
\end{align}
\end{widetext}
Hence,
\begin{widetext}
\begin{align}
I \underset{e\lambda \gg 1}{\sim} - \frac{8192}{(e\mathscr F R^2)^4} \sum_{j=0}^{\sqrt{e\mathscr F R^2}} \int_{2j+1}^{2\sqrt{e\mathscr F R^2}} \frac{\text{d} (kR)}{kR} e^{-k^2/e\mathscr F} \frac{1}{J_{2j+1}^2(kR) + Y_{2j+1}^2(kR)} \notag \\ \times \int_{2\sqrt{e\mathscr F R^2}}^{2(2\mathscr F R^2)^{1-\epsilon}} \frac{\text{d} (k'R)}{k'R} \frac{1}{J_{2j+1}^2(k'R) + Y_{2j+1}^2(k'R)},
\tag{6.38}
\label{eq:6.38}
\end{align}
\end{widetext}
where the sums over $m_1$ and $m_2$ have been taken. To estimate \eqref{eq:6.38} use Watson's inequality (Eq.(1), Sec.13.74 of \cite{39})
\begin{align}
\frac{2}{\pi x} < J_n^2(x) + Y_n^2(x) < \frac{2}{\pi}(x^2-n^2)^{-1/2},
\tag{6.39}
\label{eq:6.39}
\end{align}
for $x \ge n \ge 1/2$. This is used repeatedly in our estimates. An easy calculation gives
\begin{align}
I \underset{e\lambda \gg 1}{=} O\left( -(e \mathscr F R^2)^{-2-\epsilon} \right),
\tag{6.40}
\label{eq:6.40}
\end{align}
with $0< \epsilon \ll 1$. The remaining contributions to $I_3$ give
\begin{align}
I_3 \underset{e\lambda \gg 1}{=} O\left( -(e \mathscr F R^2)^{-2} \right),
\tag{6.41}
\label{eq:6.41}
\end{align}
or smaller as in \eqref{eq:6.40}. The dominant estimate in \eqref{eq:6.41} comes from the intervals $0 \le j \le \sqrt{ e \mathscr F R^2 }, 2j+1 \le kR \le O((\sqrt{ e \mathscr F R^2 })$,
$O(e\mathscr F R^2 ) \le k'R \le \infty$.
We have given reasons above why this calculation of the large amplitude growth of $I_3$ is representative. In view of \eqref{eq:6.41} we are
confident that
\begin{align}
\lim_{e\mathscr F \to \infty} \frac{I_3}{(e\mathscr F)^2 \ln (e\mathscr F)} = 0,
\tag{6.42}
\label{eq:6.42}
\end{align}
for all admissible random fields. Combining \eqref{eq:6.1}, \eqref{eq:6.5}, \eqref{eq:6.19} and \eqref{eq:6.42} we obtain for large amplitude variations of admissible
random fields $F_{\mu \nu}$
\begin{align}
\Pi \underset{e\mathscr F \to \infty}{=} \frac{e^2 \@ifstar{\oldnorm}{\oldnorm*} F ^2}{32 \pi^2} \ln\left( \frac{e \mathscr F}{m^2} \right) + R_1,
\tag{6.43}
\label{eq:6.43}
\end{align}
with $\underset{e\mathscr F \to \infty}{\lim} R_1 /[(e \mathscr F )^2 \ln(e \mathscr F)] = 0$. The term "admissible random field" is discussed in Sec. VII.
\subsection{Summary}
In the absence of zero mode supporting random background fields \eqref{eq:threepointnine}, \eqref{eq:fourpointsix}, \eqref{eq:5.24} and \eqref{eq:6.43} give the final result
\begin{align}
\text{lndet}_{ren} \underset{e \mathscr F \to \infty}{\ge} \frac{1}{48 \pi^2} e^2 \@ifstar{\oldnorm}{\oldnorm*} F ^2 \ln \left( e \mathscr F / m^2 \right) + R_2,
\tag{6.44}
\label{eq:6.44}
\end{align}
with $R_2$'s growth bounded as $R_1$'s above. The lnm$^2$ contribution to \eqref{eq:6.44} is due to on-shell charge renormalization. For off-shell renormalization $m^2$ is
replaced with a subtaction parameter $\mu^2$ as discussed in Sec. A above.
If zero mode supporting background fields are included and all of
the zero modes have the same chirality then by \eqref{eq:threepointnine}, \eqref{eq:fourpointsix}, \eqref{eq:5.39}(an equality in this case) and \eqref{eq:6.43}.
\begin{align}
\begin{split}
\ln\mbox{$\det{}_\mathrm{ren}$} \underset{e\mathscr F \to \infty}{\ge}& - \frac{1}{16 \pi^2} \left| \int \text{d} ^4 x ^* F_{\mu \nu} F_{\mu \nu} \right| e^2 \\
&\times\ln(\frac{e \mathscr F}{m^2}) + \frac{1}{48 \pi^2} e^2 \@ifstar{\oldnorm}{\oldnorm*} F ^2 \ln(\frac{e \mathscr F}{m^2}) + R_3,
\end{split}
\tag{6.45}
\label{eq:6.45}
\end{align}
with $R_3$ bounded as $R_1$ and $R_2$ above. Recall that $ \int \text{d} ^4 x^* F_{\mu \nu} F_{\mu \nu} /16\pi^2$ is the
chiral anomaly.
If the zero modes supported by $A_{\mu}$ have both positive and negative chirality there is no counting theorem and \eqref{eq:6.45} is replaced with, following \eqref{eq:5.31} and \eqref{eq:5.32},
\begin{align}
\begin{split}
\ln\mbox{$\det{}_\mathrm{ren}$} \underset{e\mathscr F \to \infty}{\ge}& - \left( \# \text{zero modes supported by }A_{\mu} \right)\\
&\times \ln(\frac{e \mathscr F}{m^2}) + \frac{1}{48 \pi^2}e^2 \@ifstar{\oldnorm}{\oldnorm*} F ^2 \ln(\frac{e \mathscr F}{m^2}) + R_4.
\end{split}
\tag{6.46}
\label{eq:6.46}
\end{align}
The number of zero modes grows at least as fast as $e^2$ following \eqref{eq:5.37}, provided the chiral anomaly is non-zero.. If they grow as $e^2$ or less then $\underset{e \mathscr F \to \infty}{\lim} R_4 /[(e \mathscr F )^2 \ln(e \mathscr F )] = 0$.
Known 4D Abelian zero modes require $F_{\mu \nu} \not\in L^2$. So the $\@ifstar{\oldnorm}{\oldnorm*} F ^2$ terms in \eqref{eq:6.45} and \eqref{eq:6.46} need a volume cutoff that will be discussed in Sec.VII.
Assuming in this section that $F_{\mu \nu} \in L^2$ served its purpose to obtain the structure of the charge renormalization term's large field amplitude contribution to $\ln\det_\mathrm{{ren}}$.
An assumption underlying \eqref{eq:6.46} is that all admissible 4D Abelian zero mode supporting potentials have a $1/|x|$ falloff as $|x|\rightarrow \infty$. If there were zero mode supporting potentials whose falloff is faster than $1/|x|$ the associated chiral anomaly would vanish since $^* F_{\mu \nu} F_{\mu \nu}=\partial_{\alpha}(\epsilon _{\alpha\beta\mu\nu}\text{A}_{\beta}F_{\mu\nu}) $.
The vanishing of the right-hand side of \eqref{eq:5.37} implies $n_+=n_-$. Without being able to place a lower bound on the number of zero modes \eqref{eq:6.46} loses its predictive power in this case. A 4D Abelian vanishing theorem stating that all normalizable zero modes have either positive or negative chirality, as in QCD$_4$, needs to be either proved or falsified by a counterexample.
Further discussion of \eqref{eq:6.44}-\eqref{eq:6.46} appears at the end of Sec. VII.
\section{Regularization}
In principle det$_{ren}$ can be calculated as an explicit function of $F_{\mu \nu}$ before inserting it into the functional integral \eqref{eq:twopointfive}. The
input potentials must correspond to random potentials supported by d$\mu_0(A)$. It is generally accepted that these belong to $\mathscr S '(\mathbb{R}^4)$, the
space of tempered distributions. This is the first requirement.
Throughout we have assumed smooth potentials, including zero mode supporting potentials $A_{\mu} (x)$ with a $1/|x|$ falloff for $|x| \to \infty$.
In Sec.VA it was assumed that $F_{\mu \nu} \in \underset{r>2}{\cap}L^r(\mathbb R^4)$ which we noted may be
too strong a condition. The $L^p (\mathbb R^4)$ Sobolev inequality $\|\nabla f\|_p \ge K\|f\|_q$,
where $K$ is a constant and $q=4p/(4-p),1<p<4$ \cite{32}, implies $A_{\mu} \in \underset{r>4}{\cap} L^r ( \mathbb R^4 )$
when $A_{\mu}$ is once differentiable and $F_{\mu \nu} \in \overset{<4}{\underset{>2}{\cap}} L^r(\mathbb R^4)$. This condition
on $A_{\mu}$ and the weaker condition on $F_{\mu\nu}$ are sufficient to define det$_5$ in \eqref{eq:F1} to ensure that $\ln\det_\mathrm{{ren}}$ is defined when $m\neq 0$ \cite{7,31}. These assumptions constitute the second requirement.
The final requirement is that an ultraviolet cutoff mechanism be introduced.
These three requirements can be satisfied by calculating $\ln\det_\mathrm{{ren}}$ in terms of the potentials
\begin{align}
A_{\mu}^{\Lambda} (x) = \int \text{d}^4 y f_{\Lambda}(x-y) A_{\mu}(y),
\tag{7.1}
\label{eq:7.1}
\end{align}
where $A_{\mu} \in \mathscr S'(\mathbb R^4)$ and $f_{\Lambda} \in \mathscr S(\mathbb R^4)$, the space of functions of rapid
decrease. Then $A_{\mu}^{\Lambda} \in \text{C}^{\infty}$. Besides smoothing $A_{\mu}$, \eqref{eq:7.1} also introduces a
sequence of ultraviolet cutoffs. Thus, from \eqref{eq:twopointthree} conclude that
\begin{align}
\int \text{d} \mu _0 (A) A_{\mu}^{\Lambda}(x) A_{\nu}^{\Lambda}(y) = D_{\mu \nu}^{\Lambda}(x-y),
\tag{7.2}
\label{eq:7.2}
\end{align}
where the Fourier transform of the regularized free photon propagator in a fixed gauge is $\hat D_{\mu \nu}(k)|\hat f_{\Lambda} (k)|^2$ with $\hat{f}_{\Lambda} \in \text{C}_0^{\infty}$, the space of C$^{\infty}$
functions with compact support. For example, one might choose $\hat f_{\Lambda} = 1$, $k^2\le \Lambda ^2$ and $\hat f_{\Lambda} = 0, k^2 \ge n \Lambda^2 , n>1$.
We note that if $A_{\mu}$ is a zero mode supporting potential then so is $A_{\mu}^{\Lambda}$. Thus, if $A_{\mu}$ has a $1/|x|$ falloff then so does $A_{\mu}^{\Lambda}$ .
This follows since the small-$p$ dependence of their Fourier transforms, and hence their large-$x$ dependence, are the same when $\hat f_{\Lambda}$ is chosen as above;
chirality is preserved. Other mappings with the convolution in \eqref{eq:7.1} can be followed with Young's inequality in the form \eqref{eq:A7} with $s=1$; the above conditions on $A_\mu$ and $F_{\mu\nu}$ are preserved.
Summarizing, we are instructed to replace all potentials and fields in this analysis with the smoothed potentials $A_{\mu}^{\Lambda}$ and fields $F_{\mu \nu}^{\Lambda} = \partial_{\mu} A_{\nu}^{\Lambda} - \partial_{\nu}A_{\mu}^{\Lambda}$, including the general representation \eqref{eq:twopointfive}. This allows the assumed restrictions on $A_{\mu}$ and $F_{\mu\nu}$ leading to \eqref{eq:6.44}-\eqref{eq:6.46} to be transferred to $A_{\mu}^{\Lambda}$ and $F_{\mu\nu}^{\Lambda}$ while keeping the underlying rough potentials $A_{\mu}$ in place.
The measure d$\mu_0(A)$ is not modified.
The substitution of $A_\mu^\Lambda$ for $A_\mu$ does not affect the analysis of Secs.V A-D. In particular, in Sec.V B where use is made of \eqref{eq:5.10}
we have
\begin{align}
\begin{split}
\hat{A}_\mu(k)&=M_{\mu\nu} \int \text{d}^4x \text{e} ^{-ikx} x_\nu a (r)\\
&= i M_{\mu\nu} \partial_\nu \hat{a}(|k|).
\end {split}
\tag{7.3}
\label{eq:7.3}
\end{align}
Then
\begin{align}
\begin{split}
A_\mu^\Lambda(x)&= \int \text{d}^4y f_\Lambda(x-y) A_\mu(y)\\
&= (a_\Lambda(r)+ \text{h}_\Lambda(r)) M_{\mu\nu} x_\nu,
\end {split}
\tag{7.4}
\label{eq:7.4}
\end{align}
where
\begin{align}
a_\Lambda(r)=\int \frac{\text{d}^4k}{(2\pi)^4}\text{e}^{ikx}\hat{a}(|k|)\hat{f}_\Lambda(|k|),
\tag{7.5}
\label{eq:7.5}
\end{align}
\begin{align}
h_\Lambda(r)x_\nu=-i \int \frac{\text{d}^4k}{(2\pi)^4} \text{e}^{ikx}\hat{a}(|k|) \partial_\nu \hat{f}_\Lambda(|k|).
\tag{7.6}
\label{eq:7.6}
\end{align}
If $A_\mu$ supports a zero mode then $a_\Lambda(r)\underset{r \rightarrow \infty}{\sim}\nu/r^2 $ since $\hat{f}_\Lambda(|k|)=1$ for $k^2 \le \Lambda^2$.
Hence, the only result of substituting $A_\mu^\Lambda$ for $A_\mu$ is to replace $a$ with $a_\Lambda + h_\Lambda$.
In Sec.V E the profile function $a(r)$ in \eqref{eq:5.40} has a discontinuous second derivative at r=R.
So $a(r)$ for $r\le R$ would have to be smoothed to accommodate a reqularized potential. This does not in any way modify the conclusion of Sec.V E, namely that the
formalism of Secs.V C and D can be implemented.
In Sec.VI B we can not choose $F_{\mu\nu}^\Lambda \in \text{C}_0^\infty$ as we did for $F_{\mu\nu}$. For suppose $F_{\mu\nu}^\Lambda \in \text{C}_0^\infty$.
Then $\hat{F}_{\mu\nu}^\Lambda(k)$ is an entire analytic function of $k_\mu$ \cite{40}. Therefore, we cannot set $\hat{F}_{\mu\nu}^\Lambda(k)=\hat{f}_\Lambda(|k|) \hat{F}_{\mu\nu}(k)$ since $\hat{f}_\Lambda (|k|)$ is not an entire analytic function of $|k|$. Nevertheless, $F_{\mu\nu}^\Lambda(x) = f_\Lambda \\ ^* F_{\mu\nu}(x)$ is a polynomial bounded C$^\infty$ function by Theorem IX.4 in \cite{40}. We are now free to choose a $F_{\mu\nu} \in \mathscr S'$ to make $F_{\mu\nu}^\Lambda(x)$ fall off arbitrarily rapidly for $|x|>R$. So $F_{\mu\nu}^\Lambda$ can be chosen arbitrarily close to a compactly supported field. This should not change our conclusion \eqref{eq:6.42} about the bound on $I_3$ for $e>>1$.
Finally, a volume cutoff must be introduced in $\text{det}_\text{ren}$ -and only $\text{det}_\text{ren}$- in order to regularize the vacuum-vacuum amplitude Z in \eqref{eq:twopointfour}. As $\text{det}_\text{ren}$ is gauge invariant this can be done by letting $F_{\mu\nu}^\Lambda \rightarrow \text{g}F_{\mu\nu}^\Lambda$ , where g is a space cutoff such as $\text{g} \in \text{C}_0^\infty$ or $\text{g}=\chi _{\Gamma}$, the characteristic function of a bounded region $\Gamma \subset \mathbb R^4$. This way of introducing g preserves the gauge invariance of $\text{det}_\text{ren}$ .
The regularization procedure used here is a generalization of that used in the two-dimensional Yukawa model \cite{41}. The main conclusions in this paper obtained without regulators remain valid. Thus, in \eqref{eq:6.44}-\eqref{eq:6.46} it is only required to replace $F_{\mu\nu}$ with $\text{g}F_{\mu\nu}^\Lambda$, which is a
special case of the general substitution $\text{det}_\text{ren}(F_{\mu\nu})\rightarrow\text{det}_\text{ren}(\text{g}F_{\mu\nu}^\Lambda)$. $\mathscr F$ is the amplitude of $F_{\mu\nu}^{\Lambda}$ whose scale is set by the amplitude of the underlying potential $A_\mu \in \mathscr S'$.
It does not matter when the regulators are introduced as long as they are in place when $\text{det}_\text{ren}$ is inserted into \eqref{eq:twopointfive} .
{\it{Interpretation of \eqref{eq:6.44}-\eqref{eq:6.46}}}: Each term in representation \eqref{eq:threepointnine} for $\det_\mathrm{ren}$ is gauge invariant and ultraviolet finite. Therefore, each term is independent of the others. It is noted in \eqref{eq:6.44}-\eqref{eq:6.46}, with $F_{\mu \nu}$ replaced by $F_{\mu \nu}^{\Lambda}$ before introducing $g$, that $F_{\mu \nu}^{\Lambda}$ must be square integrable. Within the class of potentials with falloff at infinity those that support a zero mode decrease as $1/|x|$ as far as presently known. This is incompatible with $F_{\mu\nu}^{\Lambda} \in \text{L}^2$. The
terms in \eqref{eq:6.44}-\eqref{eq:6.46} depending on $||F^{\Lambda}||^2$ come from the first and third terms of \eqref{eq:threepointnine}. These terms were dealt with in Secs. IV and VI where it was assumed that $F_{\mu\nu}^{\Lambda} \in \underset{r\ge 2}{\cap} L^r$. Zero modes reside solely in the second term of \eqref{eq:threepointnine}. As shown in Sec. V it can be defined for $F_{\mu\nu}^{\Lambda} \in \underset{r>2}{\cap} L^r$. So the two terms in \eqref{eq:6.45} and \eqref{eq:6.46} are separately defined, each subject to its foregoing field restriction.
To regulate Z in \eqref{eq:twopointfour} a volume cutoff is inserted into $\det_\mathrm{ren}$ as described above. When zero mode supporting potentials are introduced into $\det_\mathrm{ren}$ by the Maxwell measure $d{\mu}_0(A)$ the terms depending on $||F^{\Lambda}||^2$ now remain finite. Therefore, the interpretation of \eqref{eq:6.44}-\eqref{eq:6.46} is that they represent the asymptotic {\it{form}} of $\det_\mathrm{ren}$ before volume cutoffs are introduced.
For \eqref{eq:6.44}-\eqref{eq:6.46} to be relevant the unregularized random connections $A_{\mu}$, including their assumed falloff at infinity, should have $\mu_0$ measure one. As far as the author knows all known results for the growth at infinity of a set of random fields with measure one are for a Gaussian process whose covariance corresponds to a massive scalar field (see, for example, \cite{52, 53}). The covariance (\ref{eq:twopointthree}) in a general covariant gauge does not include an infrared cutoff photon mass as none is required. To the author's knowledge, then, the behavior at infinity of a set of random Euclidean QED$_4$ connections with $\mu_0$ measure one is still not settled.
\section{Conclusion}
Representations \eqref{eq:twopointsix} and \eqref{eq:threepointnine} for the Euclidean fermion determinant in QED, $\ln\det_\mathrm{{ren}}$, have been obtained that reflect
its competing magnetic properties of diamagnetism and paramagnetism.
This way of viewing $\ln\det_\mathrm{{ren}}$ arises since in Euclidean space $F_{\mu \nu}(x)$ may be regarded as a static, four-dimensional magnetic field. This
decomposition of $\ln\det_\mathrm{{ren}}$ has the advantage of simplifying its strong coupling, large field amplitude analysis for a class of random
potentials/fields. The analysis is made possible by a number of theorems developed in the 1970s and 80s that are applicable
to field-theoretic operators in the presence of external gauge fields.
The main results are summarized by \eqref{eq:6.44}-\eqref{eq:6.46} and are interpreted at the end of Sec. VII. Result \eqref{eq:6.44} for the fast growth of $\ln\det_\mathrm{{ren}}$ for large
field variations raises doubt on whether it is integrable with any Gaussian measure whose support does not include zero mode supporting potentials. Results \eqref{eq:6.45} and \eqref{eq:6.46} indicate that the growth of $\ln\det_\mathrm{{ren}}$ is slowed down or stopped by including zero mode supporting
potentials in the Gaussian measure d$\mu_0(A)$ introduced in Sec.II. This is \textit{prima face} evidence that zero mode supporting potentials are
necessary for the non-perturbative quantization of QED. See \cite{54} for an earlier discussion of the non-perturbative quantization of QED.
Refer back to one of the electroweak fermion determinants such as the first one in \eqref{eq:onepointone}. Suppose after being properly defined its
large amplitude Maxwell field variation coincides with that of $\ln\det_\mathrm{{ren}}$. Then \eqref{eq:6.45} and \eqref{eq:6.46} provide \textit{prima face} evidence that the
non-perturbative quantization of the electroweak model also requires its Maxwell Gaussian measure to have support from zero mode supporting
potentials. This assumes that the Maxwell field integration follows next after integrating out the fermion degrees of freedom.
Given such Gaussian measures are they such that no measurable subset of potentials results in the fast growing charge renormalization
term in \eqref{eq:6.45} and \eqref{eq:6.46} becoming dominant? This is entering unknown territory that needs to be explored.
If the QED determinant grows faster than a quadratic in the Maxwell field for a measurable set of fields then there may be a connection between this and the photon propagator's Landau pole \cite{5,56}. The precise connection, if any, remains to be worked out.
It might be objected that the non-perturbative quantization of the electroweak model is irrelevant since perturbative expansions appear to be adequate at presently available energies. This opinion neglects the fact that the electroweak model is not asymptotically free. At some point the model's non-perturbative content will be required.
The author wishes to acknowledge helpful correspondence with Erhard Seiler.
|
{
"timestamp": "2015-04-14T02:13:06",
"yymm": "1504",
"arxiv_id": "1504.03117",
"language": "en",
"url": "https://arxiv.org/abs/1504.03117"
}
|
\section{Introduction and statement of results}
We consider the problem of finding radial solutions for the fractional Yamabe problem in $\r^n$, $n\geq 2$, with an isolated singularity at the origin. This means to look for positive, radially symmetric solutions of
\begin{equation}\label{equation0}(-\Delta)^{\gamma}w=c_{n,\gamma}w^{\frac{n+2\gamma}{n-2\gamma}}\text{ in }\r^n \setminus\ \{0\},\end{equation}
where $c_{n,\gamma}$ is any positive constant that, without loss of generality, will be normalized as in Proposition \ref{cte}. Unless we state the contrary, $\gamma\in(0,\tfrac{n}{2})$. In geometric terms, given the Euclidean metric $|dx|^2$ on $\mathbb R^n$, we are looking for a conformal metric \begin{equation}\label{conformal-change}g_w=w^{\frac{4}{n-2\gamma}}|dx|^2,\ w>0,\end{equation} with positive constant fractional curvature $Q^{g_w}_\gamma\equiv c_{n,\gamma}$, that is radially symmetric and has a prescribed singularity at the origin.
Because of the well known extension theorem for the fractional Laplacian \cite{CaffarelliSilvestre,CaseChang,MarChang} we can assert that equation \eqref{equation0} for the case $\gamma\in(0,1)$ is equivalent to the boundary reaction problem
\begin{equation}\label{equation1}\left\{
\begin{split}
-\divergence(y^{a}\nabla W)=0&\text{ in } \r^{n+1}_+,\\
W=w&\text{ on }\r^n\setminus\{0\},\\
-\tilde{d}_{\gamma}\lim_{y\rightarrow 0}y^a\partial_yW=c_{n,\gamma}w^{\frac{n+2\gamma}{n-2\gamma}}&\text{ on }\r^n\setminus\{0\}.
\end{split}\right.
\end{equation}
Here the constant $\tilde d_{\gamma}$ is defined in \eqref{dtg}. We note that it is possible to write $W=K_\gamma *_x w$, where $K_\gamma$ is the Poisson kernel \eqref{Poisson-kernel} for this extension problem.
It is known that $w_1(r)=r^{-\frac{n-2\gamma}{2}}$,
together with $W_1=K_\gamma *_x w_1$, is an explicit solution for \eqref{equation1}; this fact will be proved in Proposition \ref{cte} and as a consequence we will obtain the normalization of the constant $c_{n,\gamma}$. Therefore, $w_1$ is the model solution for isolated singularities, and it corresponds to the cylindrical metric.
In the recent paper \cite{CaffarelliJinSireXiong} Caffarelli, Jin, Sire and Xiong characterize all the nonnegative solutions to \eqref{equation1}.
Indeed, let $W$ be any nonnegative solution of \eqref{equation1} in $\r^{n+1}_+$ and suppose that the origin
is not a removable singularity. Then, writing $r=|x|$ for the radial variable in $\mathbb R^n$, we must have that
$$W(x,t)=W(r,t)\text{ and }\partial_r W(r,t)<0\quad \forall\ 0<r<\infty.$$
In addition, they also provide their asymptotic behavior. More precisely, if $w=W(\cdot,0)$ denotes the trace of $W$, then near the origin one must have that
\begin{equation}\label{asymptotics}
c_1r^{-\tfrac{n-2\gamma}{2}}\leq w(x)\leq c_2r^{-\tfrac{n-2\gamma}{2}},
\end{equation}
where $c_1$, $c_2$ are positive constants.
We remark that if the singularity at the origin is removable, all the entire solutions for \eqref{equation1} have been completely classified by Jin, Li and Xiong \cite{JinLiXiong} and Chen, Li and Ou \cite{ChenLiOu}, for instance. In particular, they must be the standard ``bubbles"
\begin{equation}\label{sphere}
w(x)=c\left(\frac{\lambda}{\lambda^2+|x-x_0|}\right)^{\frac{n-2\gamma}{2}},\quad c,\lambda>0, \ x_0\in\mathbb R^n.
\end{equation}
In this paper we initiate the study of positive radial solutions for \eqref{equation0}. It is clear from the above that we should look for solutions of the form
\begin{equation}\label{wv}
w(r)=r^{-\frac{n-2\gamma}{2}}v(r)\text{ on } \r^n\setminus\{0\},
\end{equation}
for some function $0<c_1\leq v\leq c_2$. In the classical case $\gamma=1$, equation \eqref{equation0} reduces to a standard second order ODE. However, in the fractional case \eqref{equation0} becomes a fractional order ODE, so classical methods cannot be directly applied here.
The objective of this paper is two-fold: first, to use the natural interpretation of problem \eqref{equation0} in conformal geometry in order to obtain information about isolated singularities for the operator $(-\Delta)^\gamma$ from the scattering theory definition. And second, to take a dynamical system approach to explore how much of the standard ODE study can be generalized to the PDE \eqref{equation1}. In the forthcoming paper \cite{paper2}, that is a joint paper together also with M. del Pino and J. Wei, we conclude this study by using a variational method directly to construct solutions to \eqref{equation0} that generalize the well known Delaunay (sometimes called Fowler) solutions for the scalar curvature problem. Both papers complement each other.\\
Before stating our results we need to introduce the geometric setting. On a general Riemannian manifold $(M^n,g)$, the fractional curvature $Q^{g}_\gamma$ is defined from the conformal fractional Laplacian $P^{g}_\gamma$ as $Q^{g}_\gamma=P^{g}_\gamma(1)$, and it is a nonlocal version of the scalar curvature (which corresponds to the local case $\gamma=1$). The conformal fractional Laplacian is constructed from scattering theory on the conformal infinity $M^n$ of a conformally compact Einstein manifold $(X^{n+1},g^+)$ as a generalized Dirichlet-to-Neumann operator for the eigenvalue problem
\begin{equation}\label{scattering-introduction}-\Delta_{g^+}U-s(n-s)U=0\ \text{in}\ X, \quad s=\tfrac{n}{2}+\gamma,\end{equation}
and it is a (non-local) pseudo-differential operator of order $2\gamma$.
This construction is a natural one from the point of view of the AdS/CFT correspondence in Physics (\cite{AdS/CFT,Witten}).
The main property of $P^g_\gamma$ is its conformal invariance; indeed, for a conformal change of metric $g_w=w^{\frac{4}{n-2\gamma}}g$, we have that
\begin{equation}\label{conformal-transformation}
P_\gamma^{g_w} (f)=w^{-\frac{n+2\gamma}{n-2\gamma}} P^{g}_\gamma(fw),\quad \text{for all } f\text{ smooth,}
\end{equation}
which, in particular when $f=1$, reduces to the fractional curvature equation
\begin{equation}\label{Qequation}
P_\gamma^{g}(w)=Q_\gamma^{g_w} w^{\frac{n+2\gamma}{n-2\gamma}}.
\end{equation}
The fractional Yamabe problem for equation \eqref{Qequation} on compact manifolds was considered in \cite{MarQing,MarWang}, while the fractional Nirenberg problem was introduced in \cite{JinLiXiong,JinLiXiongII,FangGonzalez}. In addition, the study of the singular version for the fractional Yamabe problem was initiated in \cite{GonzalezMazzeoSire}, where the authors construct model singular solutions. Here we look at the case of an isolated singularity.
We note that, in the Euclidean case, $P^{|dx|^2}_\gamma$ coincides with the standard fractional Laplacian $(-\Delta)^{\gamma}$, and thus, imposing the constant curvature condition in equation \eqref{Qequation} yields our original problem \eqref{equation0}. Moreover, one can check that the extension problem \eqref{equation1} comes from the scattering problem \eqref{scattering-introduction} when we take $g^+$ to be the hyperbolic metric $g^+=\frac{dy^2+|dx|^2}{y^2}$. See Section \ref{section:preliminaries} for a review of the known results on the fractional conformal Laplacian.\\
We present now the natural coordinates for studying isolated singularities of \eqref{equation0}. Let $M=\mathbb R^n\backslash \{0\}$ and use the Emden-Fowler change of variable $r=e^{-t}$, $t\in\mathbb R$; with some abuse of the notation, we write $v=v(t)$. Then, in radial coordinates, $M$ may be identified with the manifold $\mathbb R\times \mathbb S^{n-1}$, for which the Euclidean metric is written as
\begin{equation}\label{g0:introduction}|dx|^2=dr^2+r^2 g_{\mathbb S^{n-1}}=e^{-2t}[dt^2+g_{\mathbb S^{n-1}}]=:e^{-2t}g_0.\end{equation}
Since the metrics $|dx|^2$ and $g_0$ are conformally related, we prefer to use $g_0$ as a background metric and thus any conformal change \eqref{conformal-change} may be rewritten as
$$g_v=w^{\frac{4}{n-2\gamma}}|dx|^2=v^{\frac{4}{n-2\gamma}}g_0,$$
where we have used relation \eqref{wv}.
Looking at the conformal transformation property for $P^g_\gamma$ given in \eqref{conformal-transformation} and relation \eqref{wv} again, it is clear that
\begin{equation}\label{relation-introduction}
P_\gamma^{g_0}(v)=r^{\frac{n+2\gamma}{2}} P^{|dx|^2}_{\gamma}(r^{-\frac{n-2\gamma}{2}} v)=r^{\frac{n+2\gamma}{2}}
(-\Delta)^{\gamma} w,
\end{equation}
and thus the original problem \eqref{equation0} is equivalent to the following one: fixed $g_0$ as a background metric on $\mathbb R\times \mathbb S^{n-1}$, find a conformal metric $g_v=v^{\frac{4}{n-2\gamma}}g_0$ of positive constant fractional curvature $Q^{g_v}_\gamma$, i.e., find a positive smooth solution $v$ for
\begin{equation}\label{equation2}P_\gamma^{g_0}(v)=c_{n,\gamma} v^{\frac{n+2\gamma}{n-2\gamma}}\quad \text{on}\quad \mathbb R\times \mathbb S^{n-1}.\end{equation}
The point of view of this paper is to consider problem \eqref{equation2} instead of \eqref{equation0}, since it allows for a simpler analysis. In our first theorem we compute the principal symbol of the operator $P_\gamma^{g_0}$ on $\mathbb R\times \mathbb S^{n-1}$ using the spherical harmonic decomposition for $\mathbb S^{n-1}$. With some abuse of notation, let $\mu_k=-k(k+n-2)$, $k=0,1,2,...$ be the eigenvalues of $\Delta_{\s^{n-1}}$, repeated according to multiplicity. Then, any function on $\mathbb R\times \mathbb S^{n-1}$ may be decomposed as $\sum_{k} v_k(t) E_k$, where $\{E_k\}$ is a basis of eigenfunctions. We show that the operator $P_\gamma^{g_0}$ diagonalizes under such eigenspace decomposition, and moreover, it is possible to calculate the Fourier symbol of each projection. Let
\begin{equation}\label{fourier}
\hat{v}(\xi)=\frac{1}{\sqrt{2\pi}}\int_{\r}e^{-i\xi \cdot t} v(t)\,dt
\end{equation}
be our normalization for the one-dimensional Fourier transform.
\begin{theorem}\label{th1}
Fix $\gamma\in (0,\tfrac{n}{2})$ and let $P^k_{\gamma}$ be the projection of the operator $P^{g_0}_\gamma$ over each eigenspace $\langle E_k\rangle$. Then
$$\widehat{P_\gamma^k (v_k)}=\Theta^k_\gamma(\xi) \,\widehat{v_k},$$
and this Fourier symbol is given by
\begin{equation}\label{symbol}
\Theta^k_{\gamma}(\xi)=2^{2\gamma}\frac{\left|\Gamma\left(\tfrac{1}{2}+\tfrac{\gamma}{2}
+\tfrac{\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}{2}+\tfrac{\xi}{2}i\right)\right|^2}
{\left|\Gamma\left(\tfrac{1}{2}-\tfrac{\gamma}{2}+\tfrac{\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}{2}
+\tfrac{\xi}{2}i\right)\right|^2}.
\end{equation}
\end{theorem}
Since we are mainly interested in radial solutions $v=v(t)$, in many computations we will just need to consider the symbol for the first eigenspace $k=0$
(that corresponds to the constant eigenfunctions):
\begin{equation*}\label{symbolk0}
\Theta^0_{\gamma}(\xi)=2^{2\gamma}\frac{\left|\Gamma(\tfrac{n}{4}+\tfrac{\gamma}{2}+\tfrac{\xi}{2}i)\right|^2}
{\left|\Gamma(\tfrac{n}{4}-\tfrac{\gamma}{2}+\tfrac{\xi}{2}i)\right|^2}.
\end{equation*}
Now we look at the question of finding smooth solutions $v=v(t)$, $0<c_1\leq v\leq c_2$, for equation \eqref{equation2}, and we expect to have periodic solutions. The local case $\gamma=1$, presented below, provides some motivation for this statement and as we have mentioned, in the forthcoming paper \cite{paper2} we construct such periodic solutions from the variational point of view. Here we look at the geometrical interpretation of such solutions and provide a dynamical system approach for the problem.
First, note that
\begin{equation}\label{equation3}-\Delta w=c_{n,1}w^{\frac{n+2}{n-2}}, \quad w>0,\end{equation}
is the constant scalar curvature equation for the metric $g_w$. It is well known (\cite{Caffarelli-Gidas-Spruck,KMPS}) that positive solutions of equation \eqref{equation3} in $\mathbb R^n\backslash\{0\}$ must be radially symmetric and, if the singularity at the origin is not removable, then the solution must behave as in \eqref{asymptotics}.
After the change \eqref{wv}, our equation \eqref{equation3} reduces to a standard second order ODE for the function $v=v(t)$:
\begin{equation}\label{ODE1}
-\ddot{v}+\tfrac{(n-2)^2}{4}v=\tfrac{(n-2)^2}{4}v^{\frac{n+2}{n-2}},\quad v>0.
\end{equation}
This equation it is easily integrated and the analysis of its phase portrait gives that all bounded solutions must be periodic (see, for instance, the lecture notes \cite{Schoen:notas}). More precisely, the Hamiltonian
\begin{equation}\label{Hamiltonian1}H_1(v,\dot{v}):= \tfrac{1}{2}\dot v^2+\tfrac{(n-2)^2}{4}\left(\tfrac{(n-2)}{2n} v^{\frac{2n}{n-2}}-\tfrac{1}{2}v^2\right)\end{equation}
is preserved along trajectories. Thus looking at its level sets we conclude that there exists a family of periodic solutions $\{v_L\}$ of periods $L\in(L_1,\infty)$. Here
\begin{equation}\label{L1}
L_1=\tfrac{2\pi}{\sqrt{n-2}}
\end{equation}
is the minimal period and it can be calculated from the linearization at the equilibrium solution $v_1\equiv 1$. These $\{v_L\}$ are known as the Fowler (\cite{Fowler}) or Delaunay solutions for the scalar curvature.
Delaunay solutions are, originally, rotationally symmetric surfaces with constant mean curvature and they have been known for a long time (\cite{Delaunay,Eells}). In addition, let $\Sigma\in\mathbb R^3$ be a noncompact embedded constant mean curvature surfaces with $k$ ends. It is known that any of such ends must be asymptotic to one of the Delaunay surfaces (\cite{Korevaar-Kusner-Solomon,Meeks:CCM}), which is very similar to what happens in the constant scalar curvature setting (see, for instance \cite{KMPS}), where any positive solution of the constant scalar curvature equation \eqref{equation3} must be asymptotic to a precise deformation of one of the $v_L$. Delaunay-type solutions have been also shown to exist for the first eigenvalue in Serrin's problem \cite{Sicbaldi}.
Let us comment here that the constant mean curvature case is very related to the constant $Q_{1/2}$ case. This is because, fixed $(X^{n+1},\bar g)$ a compact manifold with boundary $M^n$ such that there exists a defining function $\rho$ so that $g^+=\bar g/\rho^2$ is asymptotically hyperbolic and has constant scalar curvature $R_{g^+}=-n(n+1)$, then the conformal fractional Laplacian on $M$ with respect to the metric $g=\bar g|_{M}$ is given by $P^g_{1/2}=\frac{\partial}{\partial \nu}+\frac{n-1}{2} \mathcal H_g$, where $\frac{\partial}{\partial \nu}$ is the Neumann derivative for the harmonic extension and $\mathcal H_g$ the mean curvature of the boundary (\cite{Guillarmou-Guillope,MarChang}). Thus $Q_{1/2}^g$ coincides in this particular setting with $\mathcal H_g$ up to a multiplicative constant. However, there is a further restriction since in the present paper we consider only rotationally symmetric metrics on the boundary and thus, not allowing full generality for the original constant mean curvature problem.\\
Going back to \eqref{ODE1}, we would like to understand how much of this picture is preserved in the non-local case, so we look for radial solutions of equation \eqref{equation0}, which becomes a fractional order ODE. On the one hand, we formulate the problem through the extension \eqref{equation1}. This point of view has the advantage that the new equation is local (and degenerate elliptic) but, on the other hand, it is a PDE with a non-linear boundary condition. Note that because we will be using the extension from Theorem \ref{equivaexten} for the calculation of the fractional Laplacian, we need to restrict ourselves to $\gamma\in(0,1)$ at this stage.
The first difficulty we encounter with our approach is how to write the original extension equation \eqref{equation1} in a natural way after the change of variables $r=e^{-t}$.
Looking at the construction of the conformal fractional Laplacian from the scattering equation \eqref{scattering-introduction} on a conformally compact Einstein manifold $(X^{n+1},g^+)$, we need to find a parametrization of hyperbolic space in such a way that its conformal infinity $M^n=\{\rho=0\}$ is precisely $(\mathbb R\times\mathbb S^{n-1}, g_0)$. The precise metric on the extension is $g^+=\bar g/\rho^2$
for
\begin{equation}\label{metric_introduction}
\bar{g}=d\rho^2+\left(1+\tfrac{\rho^2}{4}\right)^2dt^2+\left(1-\tfrac{\rho^2}{4}\right)^2 g_{\s^{n-1}},\end{equation}
where $\rho\in(0,2)$ and $t\in\mathbb R$. The motivation for this change of metric will be made clear in Section \ref{relaclas}. In the language of Physics, $g^+$ is the Riemannian version of AdS space, a model solution of Einstein's equations which is important in the setting of the AdS/CFT correspondence since AdS space-time is the right background when studying thermodynamically stable black holes \cite{Hawking-Page,Witten}.
Rewriting the equations in this new metric, our original equation \eqref{equation1}, written in terms of the change \eqref{wv}, is equivalent to the extension problem
\begin{equation}\label{fracyamextv}\left\{
\begin{split}
-\divergence_{\bar{g}}(\rho^a\nabla_{\bar{g}} V)+ E(\rho)V&=0\ \text{ in } (X^{n+1},\bar g),\\
V&=v \text{ on }\{\rho=0\},\\
-\tilde{d}_{\gamma}\lim_{\rho\rightarrow 0}\rho^{a}\partial_{\rho}V&=c_{n,\gamma}v^{\frac{n+2\gamma}{n-2\gamma}}\text{ on }\{\rho=0\},
\end{split}
\right.\end{equation}
where the expression for the lower order term $E(\rho)$ will be given in \eqref{Erho}. We look for solutions $V$ to \eqref{fracyamextv} that only depend on $\rho$ and $t$, and that are bounded between two positive constants.
We show first that equation \eqref{fracyamextv} exhibits a Hamiltonian quantity that generalizes \eqref{Hamiltonian1}.
\begin{theorem}\label{thctthamiltonian}
Fix $\gamma\in(0,1)$. Let $V$ be a solution of \eqref{fracyamextv} only depending on $t$ and $\rho$. Then the Hamiltonian quantity
\begin{equation}\label{Hamiltonian}
H_\gamma(t):=\frac{1}{2}\int_{0}^{2} \rho^{a}\left\{e_1(\rho)(\partial_{t}V)^2
-e(\rho)(\partial_\rho V)^2
-e_2(\rho)V^2\right\}\,d\rho
+C_{n,\gamma}v^{\frac{2n}{n-2\gamma}},
\end{equation}
is constant with respect to $t$. Here we write
\begin{equation}\label{e}
\begin{split}
&e_1(\rho)=\left( 1+\tfrac{\rho^2}{4}\right)^{-1}\left(1-\tfrac{\rho^2}{4}\right)^{n-1},\\
&e_2(\rho)=
\tfrac{n-1+a}{4}\left(1-\tfrac{\rho^2}{4}\right)^{n-2}\left(n-2+n\tfrac{\rho^2}{4}\right),\\
&e(\rho)=\left( 1+\tfrac{\rho^2}{4}\right)\left(1-\tfrac{\rho^2}{4}\right)^{n-1},
\end{split}\end{equation}
and the constant is given by
\begin{equation}\label{constantC}
C_{n,\gamma}=\frac{n-2\gamma}{2n}\,\frac{c_{n,\gamma}}{\tilde{d}_{\gamma}}.
\end{equation}
\end{theorem}
Hamiltonian quantities for fractional problems have been recently developed in the setting of one-dimensional solutions for fractional semilinear equations $(-\Delta)^{\gamma}w+f(w)=0$. The first reference we find a conserved Hamiltonian quantity for this type of non-local equations is the paper by Cabr{\'e} and Sol{\'a}-Morales \cite{CabreSola-Morales} for the particular case $\gamma=1/2$. The general case $\gamma\in(0,1)$ was carried out by Cabr{\'e} and Sire in \cite{paperxavi}. On the one hand, in these two papers \cite{paperxavi,CabreSola-Morales}, the authors impose a nonlinearity coming from a double-well potential
and look for layers (i.e., solutions that are monotone and have prescribed limits at infinity), and they are able to write a Hamiltonian quantity that is preserved. In addition, if one considers the same problem but on hyperbolic space, one finds that the geometry at infinity plays a role and the analogous Hamiltonian is only monotone (see \cite{GonzalezSaezSire}).
On the other hand, if, instead, one looks for radial solutions for semilinear equations, then
Cabr\'e and Sire in \cite{paperxavi} and Frank, Lenzman and Silvestre in \cite{FrankLenzmanSilvestre} have developed a monotonicity formula for the associated Hamiltonian. In the setting of radial solutions with an isolated singularity for the fractional Yamabe problem, our Theorem \ref{thctthamiltonian} states that, if one uses the metric \eqref{metric_introduction} to rewrite the problem, then the associated Hamiltonian \eqref{Hamiltonian} is constant along trajectories.\\
If one insists on performing an ODE-type study for the PDE problem \eqref{fracyamextv}, a possibility is to plot a phase portrait of the boundary values (at $\rho=0$), while keeping in mind that the equation is defined on the whole extension. From this point of view, one can prove the existence of two critical points: the constant solutions $v_0\equiv 0$ and $v_1\equiv 1$. Moreover, there exists an explicit homoclinic solution $v_\infty$, whose precise expression will be given in \eqref{C}; it corresponds to the $n$-sphere \eqref{sphere}.
The next step is to linearize the equation. As we will observe in Section \ref{section:linear}, the classical Hardy inequality, rewritten in terms of the background metric $g_0$, decides the stability of the explicit solutions $v_0$, $v_1$ and $v_\infty$. Stability issues for semilinear fractional Laplacian equations have received a lot of attention recently. Some references are: \cite{Cabre-Cinti} for the half-Laplacian, \cite{RosOton-Serra:extremal-solution} for extremal solutions with exponential nonlinearity, \cite{Fall:fractional-Hardy-potential} for semilinear equations with Hardy potential. In the particular case of the fractional Lane-Emden equation, stability was considered in \cite{Davila-Dupaigne-Wei,Fazly-Wei:stable-solutions}, for $\gamma\in(0,1)$ and \cite{Fazly-Wey:higher} for $\gamma\in(1,2)$. We believe that our methods, although still at their initial stage, would provide tools for a unified approach for all $\gamma\in\left(0,\frac{n}{2}\right)$.\\
Finally, we consider the linearization of equation \eqref{equation2} around the equilibrium $v_1\equiv 1$:
\begin{equation*}
P_\gamma^{g_0} v=\tfrac{n+2\gamma}{n-2\gamma}v\quad \text{on}\quad \mathbb R\times \mathbb S^{n-1},
\end{equation*}
and look at the projection over each eigenspace $\langle E_k\rangle$, $k=0,1,\ldots$,
\begin{equation}\label{linearization-k}P_\gamma^k v_k=\tfrac{n+2\gamma}{n-2\gamma}v_k.\end{equation} Although we will not provide a complete calculation of the spectrum, we can say the following:
\begin{theorem}\label{theorem:linearization}
For the projection $k=0$, equation \eqref{linearization-k} has periodic solutions $v(t)$ with period $L_\gamma=\frac{2\pi}{\sqrt{\lambda_\gamma}}$, where $\lambda_\gamma$ is the unique positive solution of \eqref{eqlambdak}. In addition,
$$\lim_{\gamma\to 1}L_\gamma =L_1,$$
so we recover the classical case \eqref{L1}.
\end{theorem}
We conjecture that $L_{\gamma}$ is the minimal period of the radial periodic solutions for the nonlinear problem \eqref{equation2}.\\
\begin{remark}
We also give some motivation to show that the projection on the $k$-eigenspace of \eqref{linearization-k} does not have periodic solutions if $k=1,2,\ldots$.
\end{remark}
Theorem \ref{theorem:linearization} gives the existence of periodic radial solutions for the linear problem. In addition, the existence of a conserved Hamiltonian hints that the original non-linear problem has periodic solutions too. Moreover, the period $L_\gamma$ behaves well under the limit $\gamma\to 1$. Using the Hamiltonian and the methods in \cite{paperxavi} one can prove that periodic solutions also behave well under this limit. We leave this proof for the forthcoming paper \cite{paper2}.
The construction of Delaunay solutions allows for many further studies. For instance, as a consequence of their construction one obtains the non-uniqueness of the solutions for the fractional Yamabe problem in the positive curvature case, since it gives different conformal metrics on $\mathbb S^1(L)\times \mathbb S^{n-1}$ that have constant fractional curvature. This is well known in the scalar curvature case (see the lecture notes \cite{Schoen:notas} for an excellent review, or the paper \cite{Schoen:number}). In addition, this gives some motivation to define a total fractional scalar curvature functional, which maximizes the standard fractional Yamabe quotient (\cite{MarQing}) across conformal classes. We hope to return to this problem elsewhere.
From another point of view, Delaunay solutions can be used in gluing problems. Classical references are, for instance, \cite{Mazzeo-Pacard:isolated, Mazzeo-Pollack-Uhlenbeck} for the scalar curvature, and \cite{Mazzeo-Pacard:Delaunay-ends,Mazzeo-Pacard-Pollack} for the construction of constant mean curvature surfaces with Delaunay ends.\\
There is an alternative notion of fractional curvature and fractional perimeter defined from the singular integral definition of the fractional Laplacian which gives a different quantity than our $Q^g_\gamma$. \cite{Caffarelli-Roquejoffre-Savin} introduces the notion of nonlocal mean curvature for the boundary of a set in $\r^n$ (see also the review \cite{Valdinoci:review}), and it has also received a lot of attention recently. Finding Delaunay-type surfaces with constant nonlocal mean curvature has just been accomplished in \cite{Cabre-Fall-Sola-Weth}. For a related nonlocal equation, but different than nonlocal mean curvature, the recent paper \cite{Davila-delPino-Dipierro-Valdinoci} establishes variationally the existence of Delaunay-type hypersurfaces.\\
We finally comment that the negative fractional curvature case has not been explored yet, except for the works \cite{Chen-Veron:1,Chen-Veron:2}. They consider singular solutions for the problem $(-\Delta)^\gamma w+|w|^{p-1}w=0$ in a domain $\Omega$ with zero Dirichlet condition on $\partial\Omega$. This setting is very different from the positive curvature case because the maximum principle is valid here. We also cite the work \cite{Quittner-Reichel}, where they consider singular solutions of $\Delta W=0$ in a domain $\Omega$ with a nonlinear Neumann boundary condition $\partial_\nu W=f(x,W)-W$ on $\partial\Omega$.\\
The paper will be structured as follows: in Section \ref{section:preliminaries} we
will recall some standard background on the fractional Yamabe problem. In particular we present the equivalent formulation as an extension problem coming from scattering theory. In section \ref{relaclas} we will give a geometric interpretation of the problem. Next, in section \ref{simbolo} we will analyze the scattering equation to give a proof for theorem \eqref{th1}. That is, we will compute the Fourier symbol for the conformal fractional Laplacian. In section \ref{section:ODE} we face the problem from an ODE-type point of view. This kind of study over the extension problem \eqref{fracyamextv} gives us two equilibria and the existence of a Hamiltonian quantity conserved along the trajectories. Moreover we will find in Section \ref{section:explicit} an explicit homoclinic solution, which corresponds to the $n-$sphere. Finally, in Section \ref{section:linear} we will perform a linear analysis close to the constant solutions which corresponds to the $n$-cylinder.
\section{Preliminaries}\label{section:preliminaries}
The conformal Laplacian operator for a Riemannian metric $g$ on a $n$-dimensional manifold $M$ is defined as
\begin{equation}\label{Conformallaplacian}
L_g=-\Delta_g+c_{n} R_g,\quad\text{where } c_{n}=\tfrac{(n-2)}{4(n-1)},
\end{equation}
and $R_g$ is the scalar curvature. The conformal Laplacian is a conformally covariant operator, indeed, given $g_w$ and $g$ two conformally related metrics with $g_w=w^{\frac{4}{n-2}}g$, then the operator $L_g$ satisfies
\begin{equation*}
L_{g_w}(f)=w^{-\frac{n+2}{n-2}}L_g(wf),
\end{equation*}
for every $f\in \mathcal C^{\infty}(M)$.
In the case $f=1$ we obtain the classical scalar curvature equation
$$-\Delta_g w+c_nR_g w=c_nR_{g_w}w^{\tfrac{n+2}{n-2}},$$
which, in the flat case, is precisely equation \eqref{equation3}.\\
The fractional Laplacian on $\mathbb R^n$ is defined through Fourier transform as
$$\widehat{(-\Delta)^{\gamma}w}=|\xi|^{2 \gamma}\widehat{w},\quad \forall\gamma\in \r.$$
Note that we use the Fourier transform defined by
\begin{equation*}
\widehat{w}(\xi)=(2\pi)^{-n/2}\int_{\r^n}w(x)e^{-i\xi\cdot x}\,dx.
\end{equation*}
Let
$\gamma\in (0,1)$ and $u\in L^{\infty}\cap \mathcal C^2$ in $\r^n$, the fractional Laplacian in $\r^n$ can also be defined by
\begin{equation*}\label{deffraclapla}
(-\Delta)^{\gamma}w(x)=\kappa_{n,\gamma}\text{P.V. }\int_{\r^n}\frac{w(x+y)-w(x)}{|y|^{(n+2 \gamma)}}\,dy,
\end{equation*}
where $P.V. $ denotes the principal value, and the constant $\kappa_{n,\gamma}$ (see \cite{Landkof}) is given by
$$\kappa_{n,\gamma}=\pi^{-\tfrac{n}{2}}2^{2\gamma}
\tfrac{\Gamma\left(\tfrac{n}{2}+\gamma\right)}{\Gamma(1-\gamma)}\gamma.$$
Caffarelli-Silvestre introduced in \cite{CaffarelliSilvestre} a different way to compute the fractional Laplacian in $\r^n$ for $\gamma\in(0,1)$. Take coordinates $x\in\mathbb R^n$, $y\in\mathbb R_ +$. Let $w$ be any smooth function defined on $\r^n$ and consider the extension $W:\r^n\times \r^+\rightarrow\r$ solution of the following partial differential equation:
\begin{equation}\label{Caffarelli-Silvestre-extension}\left\{ \begin{split}
\divergence(y^a \nabla W)&=0,\ \ x\in\r^n,\ y\in\r,\\
W(x,0)&=w(x), \ \ x\in\r^n,
\end{split}\right.\end{equation}
where $a=1-2\gamma$.
Note that we can write $W=K_\gamma *_x w$, where $K_\gamma$ is the Poisson kernel
\begin{equation}\label{Poisson-kernel}
K_\gamma(x,y)=c\frac{y^{1-a}}{(|x|^2+y^2)^{\frac{n+1-a}{2}}},
\end{equation}
and $c=c(n,\gamma)$ is a multiplicative constant which is chosen so that, for all $y>0,$ $\int K_\gamma(x,y)\, dx=1$.
In addition,
$$(-\Delta)^{\gamma}w=-\tilde{d}_{\gamma}\lim_{y\rightarrow 0^+}y^a\partial_y W,$$
for the constant
\begin{equation}\label{dtg}
\tilde{d}_{\gamma}=-\frac{2^{2\gamma-1}\Gamma(\gamma)}{\gamma\Gamma(-\gamma)}.
\end{equation}
One can generalize this construction to the curved setting. Let $X^{n+1}$ be a smooth manifold of dimension $n+1$ with smooth boundary $\partial X=M^n.$ A defining function in $\overline{X}$ for the boundary $M$ is a function $\rho$ which satisfies:
\begin{equation}\label{propdf
\rho>0 \text{ in } X,\quad
\rho=0 \text{ on } M\quad \text{and} \quad
d\rho\neq 0 \text{ on } M.
\end{equation}
A Riemannian metric $g^+$ on $X$ is conformally compact if $(\overline{X},\bar{g})$ is a compact Riemannian manifold with boundary $M$ for a defining function $\rho$ and $\bar{g}=\rho^2g^+$.
Any conformally compact manifold $(X,g^+)$ carries a well-defined conformal structure $[g]$ on the boundary $M$, where $g$ is the restriction of $\bar{g}|_M$. We call $(M,[g])$ the conformal infinity of the manifold $X$.
We usually write these conformal changes on $M$ as $g_w=w^{\frac{4}{n-2\gamma}}g$ for a positive smooth function $w$.
A conformally compact manifold $(X,g^+)$ is called conformally compact Einstein manifold if, in addition, the metric satisfies the Einstein equation $Ric_{g^+}=-ng^+$, where $Ric$ represents the Ricci tensor.
One knows \cite{Graham} that
given a conformally compact Einstein manifold $(X,g^+)$ and a representative $g$ in $[g]$ on the conformal infinity $M^n$, there is an unique defining function $\rho$ such that one can find coordinates on a tubular neighborhood $M\times(0,\varepsilon)$ in $X$ in which $g^+$ has the normal form
\begin{equation}\label{confcomeinm}
g^+=\rho^{-2}(d\rho^2+g_\rho),
\end{equation}
where $g_\rho$ is a family on $M$ of metrics depending on the defining function and satisfying $g_{\rho}|_M=g$
Let $(X,g^+)$ be a conformally compact Einstein manifold with conformal infinity $(M,[g])$.
It is well known from scattering theory \cite{GrahamZorski,MazzeoMelrose,Guillarmou} that, given $w\in \mathcal C^{\infty}(M)$ and $s\in \c $, if $ s(n-s) $ does not belong to the set of $L^2$-eigenvalues of $-\Delta_{g^+}$ then the eigenvalue problem
\begin{equation}\label{eigp}
-\Delta_{g^+}U-s(n-s)U=0 \text{ in } X,
\end{equation}
has a unique solution of the form
\begin{equation}\label{formau}
U=W \rho^{n-s}+ W_1 \rho^s, \quad W,W_1 \in \mathcal C^{\infty}(\overline{X}),\ \ W|_{\rho=0}=w.
\end{equation}
Taking a representative $g$ of the conformal infinity $(M,[g])$ we can define a family of meromorphic pseudo-differential operators $S(s)$, called scattering operators, as
\begin{equation}\label{scattering}
S(s)w=W_1|_M.
\end{equation}
The case that the order of the operator is an even integer was studied in \cite{GrahamZorski}. More precisely, suppose that $m\in\n$ and $m\leq\frac{n}{2}$ if $n$ is even, and that $(\frac{n}{2})^2-m^2$ is not an $L^2-$eigenvalue of $-\Delta_{g^+}$, then $S(s)$
has a simple pole at $s=\frac{n}{2}+m$. Moreover, if $P^{g}_m$ denotes the conformally invariant GJMS-operator on $M$ constructed in \cite{GrahamJenneMasonSparling} then
$$c_mP^{g}_m=-\residue_{s=\frac{n}{2+m}}S(s),\ \ \ c_m=(-1)^m[2^{2m}m!(m-1)!]^{-1},$$
where $\residue_{s=s_0}S(s)$ denotes the residue at $s_0$ of the meromorphic family of operators $S(s)$.
In particular,
if $m=1$ we have the conformal Laplacian $P_1^{g}=L_g$ from \eqref{Conformallaplacian}, and if $m=2$, the Paneitz operator $$P_2^{g}=(-\Delta_{g})^2+\delta(a_nR_{g}+b_nRic_{g})d+\frac{n-4}{2}Q_{g},$$
where $Q_g$ is the $Q$-curvature and $a_n,\ b_n$ are dimensional constants (\cite{Paneitz}).
It is also possible to define conformally covariant fractional powers of the Laplacian in the
case $\gamma\not\in\n$. For the rest of the paper, we set $\gamma\in\left(0,\frac{n}{2}\right)$ not an integer, $s=\frac{n}{2}+\gamma$. In addition, assume that $s(n-s)$ is not an $L^2$-eigenvalue for $-\Delta_{g^+}$ and that the first eigenvalue $\lambda_1(-\Delta_{g^+})>s(n-s)$. In this setting:
\begin{definition}
We define the conformally covariant fractional powers of the Laplacian as
\begin{equation}\label{pdg}
P^g_{\gamma}[g^+,g]=d_{\gamma}S\left(\tfrac{n}{2}+\gamma\right),\text{ where }\ d_{\gamma}=2^{2\gamma}\frac{\Gamma(\gamma)}{\Gamma(-\gamma)}.
\end{equation}
\end{definition}
As a pseudodifferential operator, its principal symbol coincides with the one of $(-\Delta_{g})^{\gamma}$.
In the rest of the paper, once $g^+$ is fixed, we will use the simplified notation:
$$P_\gamma^{g}:=P_\gamma[g^+,g].$$
These operators satisfy the conformal property
\begin{equation}\label{conformproperty}
P_{\gamma}^{g_{w}}f=w^{-\frac{n+2\gamma}{n-2\gamma}}P_{\gamma}^{g}(wf), \quad \forall f\in \mathcal C^{\infty}(M),
\end{equation}
for a change of metric $$g_{w}:=w^{\frac{4}{n-2\gamma}}g,\ w>0.$$
\begin{definition}
We define the fractional order curvature as:
$$Q_\gamma^{g}:=P_\gamma^{g}(1).$$
\end{definition}
Note that up to multiplicative constant, $Q_1$ is the classical scalar curvature and $Q_2$ is the so called $Q$-curvature.
\begin{remark}
Using the previous definition we can express the conformal property \eqref{conformproperty} as
\begin{equation}\label{confQ}
P_{\gamma}^{g}(w)=w^{\frac{n+2\gamma}{n-2\gamma}}Q_{\gamma}^{g_{w}}.
\end{equation}
\end{remark
Explicit formulas for $P_{\gamma}^{g}$ are not known in general, however, Branson \cite{Brason}, gave an explicit formula for the conformal Laplacian on the standard sphere, i.e,
\begin{equation}\label{P_g^g1C}
P_{\gamma}^{g_{\s^n}}=
\frac{\Gamma(B+\gamma+\frac{1}{2})}{\Gamma(B-\gamma+\frac{1}{2})},
\end{equation}
where $B=\sqrt{-\Delta_{g_{\s^n}}+(\frac{n-1}{2})^2}$.
For example
$$P_{1}^{g_{\s^n}}=-\Delta_{g_{\s^n}}+\tfrac{n(n-2)}{4},\quad P_{1/2}^{g_{\s^n}}=\sqrt{-\Delta_{g_{\s^n}}+\left(\tfrac{n-1}{2}\right)^2}.$$
From \eqref{P_g^g1C} we can compute the fractional curvature on the unit sphere as
\begin{equation}\label{fqsphere}
Q_{\gamma}^{g_{\s^n}}=P^{g_{\s^n}}_{\gamma}(1)=\frac{\Gamma(\frac{n}{2}+\gamma)}{\Gamma(\frac{n}{2}-\gamma)}.
\end{equation}
It is proven in \cite{MarChang} (see also the more recent paper \cite{CaseChang}) that the conformal fractional Laplacian is the Dirichlet-to-Neumann operator for an extension problem that generalizes \eqref{Caffarelli-Silvestre-extension}:
\begin{theorem}\label{equivaexten}
Let $\gamma\in(0,1)$ and $(X,g^+)$ be a conformally compact Einstein manifold with conformal infinity $(M,[g])$. For any defining function $\rho$ of $M$ satisfying \eqref{confcomeinm} in $X$, the scattering problem \eqref{eigp}-\eqref{formau}
is equivalent to
\begin{equation}\label{divE}\left\{
\begin{split}
-\divergence(\rho^a\nabla W)+E(\rho)W&=0\text{ in }(X,\bar{g}),\\
W&=w\text{ on } M,
\end{split}\right.
\end{equation}
where
$$\bar{g}=\rho^2g^+,\ \ W=\rho^{s-n}U,\ \ s=\tfrac{n}{2}+\gamma,\ \ a=1-2\gamma.$$
and the derivatives in \eqref{divE} are taken respect to the metric $\bar{g}$. The lower order term is given by
\begin{equation*}\label{E1}
E(\rho)=-\Delta_{\bar{g}}(\rho^{\frac{a}{2}})\rho^{\frac{a}{2}}
+\left(\gamma^2-\tfrac{1}{4}\right)\rho^{-2+a}+\tfrac{n-1}{4n}R_{\bar{g}}\rho^a,
\end{equation*}
or written back in the metric $g^+$,
\begin{equation}\label{E}
E(\rho)=\rho^{-1-s}(-\Delta_{g^+}-s(n-s))\rho^{n-s}.
\end{equation}
In addition, we have the following formula for the calculation of the conformal fractional Laplacian
\begin{equation*}
P_{\gamma}^{g}w=-\tilde{d}_{\gamma}\lim_{\rho\rightarrow 0}\rho^{a}\partial_{\rho} W,
\end{equation*}
where $\tilde{d}_{\gamma}$ is defined as
\begin{equation*}
\tilde{d}_{\gamma}=-\frac{2^{2\gamma-1}\Gamma(\gamma)}{\gamma\Gamma(-\gamma)}.
\end{equation*}
\end{theorem}
\begin{remark}\label{remark:Euclidean}
If $X$ is the hyperbolic space $\mathbb H^{n+1}$, identified with the upper half space $\r^{n+1}_+$ with the metric $g^+=\frac{dy^2+|dx|^2}{y^2}$, then the conformal infinity is simply $M=\r^n$ with the standard Euclidean metric $|dx|^2$ and therefore, problem \eqref{divE} is precisely the extension problem considered by Caffarelli-Silvestre \eqref{Caffarelli-Silvestre-extension}. As a consequence, the conformal fractional Laplacian reduces to the standard fractional Laplacian without curvature terms, i.e., $P^{|dx|^2}_{\gamma}=(-\Delta)^{\gamma}$.
\end{remark}
Now we are going to choose a suitable defining function $\rho^*$, in order to transform the problem \eqref{confcomeinm} into one of pure divergence form. We follow \cite{MarChang,CaseChang}:
\begin{theorem}\label{E0}
Set $\gamma\in(0,1)$. Let $(X,g^+)$ be a conformally compact Einstein manifold with conformal infinity $(M,[g])$, and such that $\lambda_1(-\Delta_{g^+})>\frac{n^2}{4}-\gamma^2$. Assuming that $\rho$ is a defining function satisfying \eqref{propdf}, there exists another (positive) defining function $\rho^*$ on $X$,
such that for the term $E(\rho)$ defined in \eqref{E} we have $$E(\rho^*)=0.$$
The asymptotic expansion of this new defining function is
\begin{equation*}
\rho^*=\rho\left(1+\frac{2 Q_{\gamma}^{g}}{(n-2\gamma)d_{\gamma}}\rho^{2\gamma}+O(\rho^2)\right).
\end{equation*}
In addition, the metric $g^*=(\rho^*)^2g^+$ satisfies $g^*|_{\rho=0}=g$ and has asymptotic expansion
$$g^*=(d\rho^*)^2[1+O((\rho^*)^{2\gamma})]+g[1+O((\rho^*)^{2\gamma})].$$
The scattering problem \eqref{formau}-\eqref{eigp} is equivalent to the following one.
\begin{equation*}\left\{
\begin{split}
-\divergence((\rho^*)^a\nabla W)&=0\text{ in }(X,g^*),\\
W&=w\text{ on }M,
\end{split}\right.
\end{equation*}
where the derivatives are taken with respect to the metric $g^*$ and $W=(\rho^*)^{s-n}U$.
Moreover
\begin{equation}\label{Pgrhostar}
P_{\gamma}^{g}w=-\tilde{d}_{\gamma}\lim_{\rho^*\rightarrow 0}(\rho^*)^a\partial_{\rho^*}W+wQ_{\gamma}^{g}.
\end{equation}
\end{theorem}
The fractional Yamabe problem is, for $\gamma\in\left(0,\tfrac{n}{2}\right)$, to find a new metric $g_w=w^{\frac{4}{n-2\gamma}}g$ on $M$ conformal to $g$, with constant fractional curvature $Q_{\gamma}^{g_{w}}$.
Using the conformal property \eqref{confQ} the Yamabe problem is equivalent to find $w$ a strictly positive smooth function on $M$ satisfying
\begin{equation}\label{fracyamp}
P_{\gamma}^{g}(w)=cw^{\frac{n+2\gamma}{n-2\gamma}},\
\ w>0.
\end{equation}
In this paper we are interested in the positive curvature case, and the constant $c=c_{n,\gamma}$ will be normalized as in Proposition \ref{cte} below.
Thanks to Theorem \ref{equivaexten}, \eqref{fracyamp} is equivalent to the existence of a strictly positive $\mathcal C^\infty$ solution for extension problem:
\begin{equation}\label{fracyamext}\left\{
\begin{split}
-\divergence(\rho^a\nabla W)+E(\rho)W&=0\text{ in }(X,\bar{g}),\\
W&=w\text{ on }M,\\
-\tilde{d}_{\gamma}\lim_{\rho\rightarrow 0}\rho^ a \partial_{\rho}W &=c_{n,\gamma}w ^{\frac{n+2\gamma}{n-2\gamma}}\text{ on }M.
\end{split}\right.
\end{equation}
Using the special defining function from Theorem \ref{E0}, the fractional Yamabe problem \eqref{fracyamext} may be rewritten as
\begin{equation}\label{fracyamE0}\left\{
\begin{split}
-\divergence((\rho^*)^a\nabla W)&=0\text{ in }(X,g^*),\\
W&=w\text{ on }M,\\
-\tilde{d}_{\gamma}\lim_{\rho^*\rightarrow 0}(\rho^*)^ a \partial_{\rho^*}W +wQ^{g}_{\gamma}&=c_{n,\gamma}w ^{\frac{n+2\gamma}{n-2\gamma}}\text{ on }M.
\end{split}\right.
\end{equation}
Indeed we only need to rewrite the equation for the Yamabe problem \eqref{fracyamp} using the expression of $P_{\gamma}^{g}$ from \eqref{Pgrhostar}. Without danger of confusion, note that in general the solutions $W$ \eqref{fracyamext} and \eqref{fracyamE0} are different, but in the sequel they will be denoted by the same letter.
\begin{proposition}\label{cte}
The fractional curvature of the cylindrical metric $g_{w_1}={w_1}^{\frac{4}{n-2\gamma}}|dx|^2$ for
the conformal change
\begin{equation}\label{w1}w_1(x)=|x|^{-\frac{n-2\gamma}{2}},\end{equation} is the constant
\begin{equation*}\label{cng}
c_{n,\gamma}=2^{2\gamma}\left(\frac{\Gamma(\frac{1}{2}(\frac{n}{2}+\gamma))}
{\Gamma(\frac{1}{2}(\frac{n}{2}-\gamma))}\right)^2>0.\end{equation*}
\end{proposition}
\begin{proof}
The value is calculated using the conformal property \eqref{confQ}, as follows
\begin{equation*}
Q_{\gamma}^{g_{w_1}}={w_1}^{-\frac{n+2\gamma}{n-2\gamma}}P_{\gamma}^{|dx|^2}(w_1)
={w_1}^{-\frac{n+2\gamma}
{n-2\gamma}}(-\Delta)^{\gamma}(w_1)=:c_{n,\gamma}.
\end{equation*}
The last equality follows from the calculation of the fractional Laplacian of a power function; it can be found in \cite{GonzalezMazzeoSire,xaviros}.
\end{proof}
\section{Geometric setting}\label{relaclas}
We give now the natural geometric interpretation of problem \eqref{equation0} and the extension formulation \eqref{equation1}. Thanks to Remark \ref{remark:Euclidean} and Theorem \ref{equivaexten}, the initial extension problem \eqref{equation1} is the scattering equation \eqref{eigp} in hyperbolic space, denoted by $X_1=\h^{n+1}$, with the metric $g^+=\frac{dy^2+|dx|^2}{y^2}$. Our point if view is to use the metric $g_0$ from \eqref{g0:introduction} as the representative of the conformal infinity, thus we need to rewrite the hyperbolic metric in a different normal form
\begin{equation}\label{normal-form}g^+=\frac{d\rho^2+g_\rho}{\rho^2} \quad\text{with}\quad g_\rho|_{\rho=0}=g_0,\end{equation}
for a suitable defining function $\rho$. Let us introduce some notation now. The conformal infinity (with an isolated singularity) is $M_1=\mathbb R^n\backslash \{0\}$,
which in polar coordinates can be represented as $M_1=\r^+\times\s^{n-1}$ and $|dx|^2=dr^2+r^2g_{\s^{n-1}}$,
or using this change of variable
$r=e^{-t}$, the Euclidean metric may be written as
\begin{equation}\label{relmetrclas}
|dx|^2=e^{-2t}[dt^2+g_{\s^{n-1}}]=:e^{-2t}g_0.
\end{equation}
We consider now several models for hyperbolic space, identified with the Riemannian version of AdS space-time. These models are well known in cosmology since they provide the simplest background for the study of thermodynamically stable black holes (see \cite{Witten,Hawking-Page}, for instance, or the survey paper \cite{Chang-Qing-Yang}). Thus we write the hyperbolic metric as
\begin{equation}\label{HYmodel}
g^+=d\sigma^2+\cosh^2 \sigma \,dt^2+\sinh^2\sigma\,g_{\s^{n-1}},
\end{equation}
where $t\in\mathbb R$, $\sigma\in(0,\infty)$ $\theta\in\mathbb S^{n-1}$.
Using the change of variable $R=\sinh \sigma$,
\begin{equation*}\label{metrR}
g^+=\frac{1}{1+R^2}\,dR^2+(1+R^2)\,dt^2+R^2g_{\s^{n-1}}.
\end{equation*}
This metric can be written in the normal form \eqref{normal-form} as
\begin{equation}\label{metrica21}
g^+=\rho^{-2}\left[d\rho^2+\left( 1+\tfrac{\rho^2}{4}\right)^2dt^2+\left( 1-\tfrac{\rho^2}{4}\right)^2 g_{\mathbb S^{n-1}}\right],
\end{equation}
for $\rho\in(0,2)$, $t\in \mathbb R$, $\theta\in\mathbb S^{n-1}$. Here we have used the relations
\begin{equation}\label{rhos}
\rho=2e^{-\sigma}\quad \text{and}\quad 1+R^2=\left( \tfrac{4-\rho^2}{4\rho} \right)^2
\end{equation}
Let $\bar g=\rho^2g^+$ be a compactification of $g^+$. Note that the apparent singularity at $\rho=2$ in the metric \eqref{metrica21} is just a consequence of the polar coordinate parametrization and thus the metric is smooth across this point.
We define now $X_2=(0,2)\times \s^1(L)\times\s^{n-1}$, with coordinates $\rho\in(0,2),\ t\in\s^1(L),\ \theta\in \mathbb S^{n-1}$,
and the same metric given by \eqref{metrica21}.
The conformal infinity $\{\rho=0\}$ is $M_2=\s^1(L)\times \s^{n-1}$, with the metric given by $g_0=dt^2+g_{\s^{n-1}}.$
Note that $(X_1,g^+_{\h^{n+1}})$ is a covering of $(X_2,g^+)$. Indeed, $X_2$ is the quotient $X_2=\h^{n+1}/\z\approx \r^n\times\s^1(L)$ with $\z$ the group generated by the translations, if we make the $t$ variable periodic. As a consequence, also $(M_1,|dx|^2)$ is a covering of $(M_2,g_0)$ after the conformal change \eqref{relmetrclas}.\\
Summarizing, we denote $X=(0,2)\times\mathbb R\times\s^{n-1}$ and $M=\mathbb R\times\s^{n-1}$ and recall that the metric $\bar{g}=\rho^2g^+$ is given by
\begin{equation}\label{gbarfrac}
\bar{g}=d\rho^2+\left(1+\tfrac{\rho^2}{4}\right)^2dt^2+\left(1-\tfrac{\rho^2}{4}\right)^2 g_{\s^{n-1}},\quad\text{and}\quad g_0=\bar{g}|_M=dt^2+g_{\s^{n-1}}.
\end{equation}
Equality \eqref{relmetrclas} shows that the metrics $|dx|^2$ and $g_0$ are conformally related and therefore using \eqref{wv},
we can write any conformal change of metric on $M$ as
\begin{equation}\label{hatEw}
g_v:=w^{\frac{4}{n-2\gamma}}|dx|^2=
v^{\frac{4}{n-2\gamma}}g_0.
\end{equation}
Our aim is to to find radial (in the variable $|x|$), positive solutions for \eqref{equation1} with an isolated singularity at the origin. Using $g_0$ as background metric on $M$, and writing the conformal change of metric in terms of $v$ as \eqref{hatEw},
this is equivalent to look for positive solutions $v=v(t)$ to \eqref{equation2} with $0<c_1\leq v\leq c_2$, and we hope to find those that are periodic in $t$.\\
Finally, we check that the background metric $g_0$ given in \eqref{gbarfrac} has constant
fractional curvature $Q^{g_0}_{\gamma}\equiv c_{n,\gamma}$. This is true because of the definition of $c_{n,\gamma}$ given in Proposition \ref{cte}, and the conformal equivalence given in \eqref{relmetrclas}. Thus, by construction, the trivial change $v_1\equiv 1$ is a solution to \eqref{equation2}.
\section{The conformal fractional Laplacian on $\r\times\s^{n-1}$.}\label{simbolo}
In this section we present the proof of Theorem \ref{th1}, i.e, the calculation of the Fourier symbol for the conformal fractional Laplacian on $\r\times\s^{n-1}$. This computation is based on the analysis of the scattering equation given in \eqref{eigp}-\eqref{formau} for the extension metric \eqref{metrica21}. We recall that the scattering operator is defined as
\begin{equation}\label{scattering2}
P_\gamma^{g}w=S(s)w=W_1|_{\rho=0},
\end{equation}
and $s=\frac{n}{2}+\gamma$. The main step in the proof is to reduce \eqref{eigp} to an ODE that can be explicitly solved. Note that this idea of studying the scattering problem on certain Lorentzian models has been long used in Physics papers, but in general it is very hard to obtain explicit expressions for the solution and the majority of the existing results are numeric (see, for example, \cite{Fisicos}).
For the calculations below it is better to use the hyperbolic metric given in the coordinates \eqref{HYmodel}. Then the conformal infinity corresponds to the value $\{\sigma=+\infty\}$.
The scattering equation \eqref{eigp} can be written in terms of the variables $\sigma\in(0,\infty)$, $t\in\r$ and $\theta\in\s^{n-1}$ as
\begin{equation}\label{eqs}
\partial_{{\sigma}{\sigma}}U+Q(\sigma)\partial_{\sigma}U+
\cosh^{-2}({\sigma})\partial_{tt}U+\sinh^{-2}(\sigma)\Delta_{\s^{n-1}}U+\left(\tfrac{n^2}{4}-\gamma^2\right)U=0,
\end{equation}
where $U=U({\sigma},t,\theta)$, and $$Q(\sigma)=\frac{\partial_{\sigma}(\cosh{\sigma}\sinh^{n-1}{\sigma})}{\cosh{\sigma}\sinh^{n-1}{\sigma}}.$$
With the change of variable \begin{equation}\label{cambioz}
z=\tanh(\sigma),\end{equation}
equation \eqref{eqs} reads:
\begin{equation}\label{eqz}
\begin{split}
(1-z^2)^2\partial_{zz}U+\left(\tfrac{n-1}{z}-z\right)(1-z^2)\partial_z U+(1-z^2)\partial_{tt}U&
\\+\left(\tfrac{1}{z^2}-1\right)\Delta_{\s^{n-1}}U+\left(\tfrac{n^2}{4}-\gamma^2\right)U&=0.
\end{split}
\end{equation}
We compute the projection of equation \eqref{eqz} over each eigenspace of $\Delta_{\s^{n-1}}$. Given $k\in\n$, let $U_k(z,t)$ be the projection of $U$ over the eigenspace $\langle E_k\rangle$ associated to the eigenvalue $\mu_k=-k(k+n-2)$. Each $U_k$ satisfies the following equation:
\begin{equation}\label{equk}
(1-z^2)\partial_{zz}U_k+\left(\tfrac{n-1}{z}-z\right)\partial_z U_k+\partial_{tt}U_k+\mu_k\tfrac{1}{z^2}U_k
+\tfrac{\tfrac{n^2}{4}-\gamma^2}{1-z^2}U_k=0.
\end{equation}
Taking the Fourier transform \eqref{fourier} in the variable $t$ we obtain
\begin{equation}\label{equkfou}
(1-z^2)\partial_{zz}\widehat{U_k}+\left(\tfrac{n-1}{z}-z\right)\partial_z \widehat{U_k}
+\left[\mu_k\tfrac{1}{z^2}+\tfrac{\tfrac{n^2}{4}-\gamma^2}{1-z^2}-\xi^2\right]\widehat{U_k}=0.
\end{equation}
Fixed $k$ and $\xi$,
we know that
\begin{equation}\label{u_k}
\widehat{U_k}=\widehat{w_k}(\xi)\varphi_k^{\xi}(z),
\end{equation}
where $\varphi:=\varphi_k^{\xi}(z)$ is the solution of the following ODE problem:
\begin{equation}\label{problemphi}\left\{
\begin{split}
&(1-z^2)\partial_{zz}\varphi+
\left(\tfrac{n-1}{z}-z\right)\partial_z\varphi+\left(\tfrac{\mu_k}{z^2}+\tfrac{\frac{n^2}{4}-\gamma^2}
{1-z^2}-\xi^2\right)\varphi =0,\\
&\text{has the expansion \eqref{formau} with }w\equiv 1\text{ near the conformal infinity }z=1, \\
&\varphi \text{ is regular at }z=0.
\end{split}\right.
\end{equation}
This ODE has only regular singular points $z$. The first equation in \eqref{problemphi} can be explicitly solved,
\begin{equation}\label{varphi1}
\begin{split}
\varphi(z)=&A(1-z^2)^{\frac{n}{4}-\frac{\gamma}{2}}z^{1-\frac{n}{2}+\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}
\Hyperg(a,b;c;z^2)\\
+&B(1-z^2)^{\frac{n}{4}-\frac{\gamma}{2}}z^{1-\frac{n}{2}-\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}\Hyperg(\tilde{a},\tilde{b};\tilde{c};,z^2),
\end{split}
\end{equation}
for any real constants $A,B$, where \begin{itemize}
\item $a=\tfrac{-\gamma}{2}+\tfrac{1}{2}+\tfrac{\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}{2}+i\tfrac{\xi}{2}$,
\item $b=\tfrac{-\gamma}{2}+\tfrac{1}{2}+\tfrac{\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}{2}-i\tfrac{\xi}{2}$,
\item $c=1+\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}$,
\item $\tilde{a}=\tfrac{-\gamma}{2}+\tfrac{1}{2}-\tfrac{\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}{2}+i\tfrac{\xi}{2}$,
\item $\tilde{b}=\tfrac{-\gamma}{2}+\tfrac{1}{2}-\tfrac{\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}{2}-i\tfrac{\xi}{2}$,
\item $\tilde{c}=1-\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}$,
\end{itemize}
and $\Hyperg$ denotes the standard hypergeometric function introduced in Lemma \ref{propiedadeshiperg}. Note that we can write $\xi$ instead of $|\xi|$ in the arguments of the hypergeometric functions because $a=\bar{b}$, $\tilde{a}=\overline{\tilde{b}}$ and property \eqref{prop5} given in Lemma \ref{propiedadeshiperg}.
The regularity at the origin $z=0$ implies $B=0$ in \eqref{varphi1}.
Moreover, property \eqref{prop4} from Lemma \ref{propiedadeshiperg} makes it possible to rewrite $\varphi$ as
\begin{equation}\label{varphiz}
\begin{split}
\varphi(z)
=A&\left[\alpha(1+z)^{\frac{n}{4}-\frac{\gamma}{2}}(1-z)^{\frac{n}{4}-\frac{\gamma}{2}}z^{1-\frac{n}{2}
+\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}} \right.\\ &
\cdot\Hyperg(a,b;a+b-c+1;1-z^2)\\
&+\beta (1+z)^{\frac{n}{4}+\frac{\gamma}{2}}(1-z)^{\frac{n}{4}+\frac{\gamma}{2}}z^{1-\frac{n}{2}
+\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}\\&
\left.\cdot\Hyperg(c-a,c-b;c-a-b+1;1-z^2)\right],
\end{split}
\end{equation}
where \begin{align}
\label{alpha}&\alpha=\tfrac{\Gamma\left(1+\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}\right)\Gamma(\gamma)}
{\Gamma\left(\tfrac{1}{2}
+\tfrac{\gamma}{2}+\tfrac{\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}{2}-i\tfrac{\xi}{2}\right)\Gamma\left(\tfrac{1}{2}
+\tfrac{\gamma}{2}+\tfrac{\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}{2}+i\tfrac{\xi}{2}\right)},\\
\notag&\beta=\tfrac{\Gamma\left(1+\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}\right)
\Gamma(-\gamma)}{\Gamma\left(\tfrac{-\gamma}{2}+\tfrac{1}{2}+\tfrac{\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}{2}
+i\tfrac{\xi}{2}\right)\Gamma\left(\tfrac{-\gamma}{2}+\tfrac{1}{2}+\tfrac{\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}{2}
-i\tfrac{\xi}{2}\right)}.
\end{align}
The constant coefficient $A$ will be fixed from the second statement in \eqref{problemphi}. From the definition of the scattering operator in \eqref{scattering2}, $\varphi$ must have the asymptotic expansion near $\rho=0$
\begin{equation}\label{expansionu}
\varphi(\rho)=\rho^{n-s}(1+...)+\rho^s(\widehat{S^k}(s)1+...),
\end{equation}
where $S^k(s)$ is the projection of the scattering operator $S(s)$ over the eigenspace $\langle E_k\rangle$.
We now use the changes of variable \eqref{cambioz} and \eqref{rhos}, obtaining
\begin{equation}\label{zrho}
z=\tanh(\sigma)=\frac{4-\rho^2}{4+\rho^2}=1-\frac{1}{2}\rho^2+\cdots .
\end{equation}
Therefore, substituting \eqref{zrho} into \eqref{varphiz} we can express $\varphi$ as a function on $\rho$ as follows
\begin{equation*}
\begin{split}
\varphi(\rho)\sim
A&\left[\alpha \rho^{\frac{n}{2}-\gamma}\Hyperg(a,b;a+b-c+1;\rho^2)\right.\\
&\,+\left.\beta \rho^{\frac{n}{2}+\gamma}\Hyperg(c-a,c-b;c-a-b+1;\rho^2)\right],\quad \text{as }\rho\to 0.
\end{split}
\end{equation*}
Using property \eqref{prop2} from Lemma \ref{propiedadeshiperg} below, we have that near the conformal infinity,
\begin{equation}\label{varphirhocero}
\begin{split}
\varphi(\rho)\simeq
A\left[\alpha\rho^{\frac{n}{2}-\gamma}+\beta\rho^{\frac{n}{2}+\gamma}+\ldots\right] .
\end{split}
\end{equation}
Comparing \eqref{varphirhocero} with the expansion of $\varphi$ given in \eqref{expansionu}, we have \begin{equation}\label{A}
A=\alpha^{-1},
\end{equation}
and
\begin{equation}\label{S(s)}
\widehat{S^k}(s)
\beta \alpha^{-1}.
\end{equation}
Recalling the definition of the conformal fractional Laplacian given in \eqref{pdg}, and taking into account \eqref{u_k}, we can assert that the Fourier symbol $\Theta^k_{\gamma}(\xi)$ for the projection $P_\gamma^k$ of the conformal fractional Laplacian $P_{\gamma}^{g_0}$ satisfies
\begin{equation*}\label{relsymbolS}
\Theta^k_{\gamma}(\xi)=\frac{\Gamma(\gamma)}{\Gamma(-\gamma)}2^{2\gamma}\widehat{S^k}(s).
\end{equation*}
From here we can calculate the value of this symbol and obtain \eqref{symbol}; just take \eqref{S(s)} into account and property \eqref{prop1g} from Lemma \ref{propiedadesgamma}.
This completes the proof of Theorem \ref{th1}.\\
\begin{remark}
When $\gamma=m$, an integer, we recover the principal symbol for the GJMS operators $P^{g_0}_m$. Indeed, from Theorem \ref{th1} we have that for any dimension $n>2m$, the Fourier symbol of $P^{g_0}_m$ is given by
\begin{equation*}
\begin{split}
\Theta^k_m(\xi)&=2^{2m}\frac{|\Gamma(\tfrac{1}{2}+\tfrac{m}{2}+\tfrac{\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}{2}
+\tfrac{\xi}{2}i)|^2}
{|\Gamma(\tfrac{1}{2}-\tfrac{m}{2}+\tfrac{\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}{2}+\tfrac{\xi}{2}i)|^2}\\&=
2^{2m}\prod_{j=1}^m\left(\tfrac{[4(m-j)-m+1+\sqrt{(\tfrac{n}{2}-1)^2+k(k+n-1)}]^2}{4}+\tfrac{\xi^2}{4}\right)\\
&=\Psi(m,n,k,\xi,\xi^2,...,\xi^{2m-1})+\xi^{2m},
\end{split}
\end{equation*}
where we have used the property \eqref{prop2g} of the Gamma function given in Lemma \ref{propiedadesgamma}. Note that $\Psi$ is a polynomial function on $\xi$ of degree less than $2m$.
For instance, for the classical case $m=1$,
$$\Theta^k_1(\xi)=\xi^2+\tfrac{(n-2)^2}{4}-\mu_k,\quad k=0,1,\ldots,$$
so we recover the usual conformal Laplacian $P^{g_0}_1$ given by
\begin{equation*}
P^k_1(v)=-\ddot{v}+\left[\tfrac{(n-2)^2}{4}-\mu_k \right]v, \quad k=0,1,\ldots,\end{equation*}
that is the operator appearing in \eqref{ODE1} when applied to radial functions.
\end{remark}
This proof also allows us to explicitly calculate the special defining function $\rho^*$ from Theorem \ref{E0}:
\begin{corollary}\label{corollary:rho*}
We have
$$(\rho^*)^{n-s}=\alpha^{-1} \left(\tfrac{4\rho}{4+\rho^2}\right)^{\frac{n}{2}-{\gamma}}
\Hyperg\left(\tfrac{n}{4}-\tfrac{\gamma}{2},\tfrac{n}{4}-\tfrac{\gamma}{2};
\tfrac{n}{2},\left(\tfrac{4-\rho^2}{4+\rho^2}\right)^2\right),$$
where $\alpha$ is the constant from \eqref{alpha}.
As a consequence, $\rho^*\in(0,\rho_0^*)$ where we have defined $(\rho_0^*)^{n-s}=\alpha^{-1}$.
\end{corollary}
\begin{proof}
From the proof of Theorem \ref{E0}, which corresponds to Lemma 4.5 in \cite{MarChang}, we know that
$$\rho^*=(\varphi_0^0)^{\frac{1}{n-s}}(z),$$
where $\varphi$ is the solution of \eqref{problemphi}. Thus from formula \eqref{varphi1} for $B=0$ and the relation between $z$ and $\rho$ from \eqref{zrho} we arrive at the desired conclusion. The behavior when $\rho\to 2$ can be calculated directly from \eqref{varphi1} and, as a consequence, $(\rho_0^*)^{n-s}=\varphi(0)=\alpha^{-1}$.
\end{proof}
\begin{lemma}\label{propiedadeshiperg}
\textnormal{ \cite{Abramowitz,SlavyanovWolfganglay}} Let $z\in\c$. The hypergeometric function is defined for $|z| < 1$ by the power series
$$ \Hyperg(a,b;c;z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n} \frac{z^n}{n!}=\frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}\sum_{n=0}^\infty \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} \frac{z^n}{n!}.$$
It is undefined (or infinite) if $c$ equals a non-positive integer.
Some properties are
\begin{enumerate}
\item The hypergeometric function evaluated at $z=0$ satisfies
\begin{equation}\label{prop2}
\Hyperg(a+j,b-j;c;0)=1; \ j=\pm1,\pm2,...
\end{equation}
\item If $|arg(1-z)|<\pi$, then
\begin{equation}\label{prop4}
\begin{split}
\Hyperg&(a,b;c;z)=
\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}
\Hyperg\left(a,b;a+b-c+1;1-z\right)
\\
+&(1-z)^{c-a-b}\frac{\Gamma(c)\Gamma(a+b-c)}
{\Gamma(a)\Gamma(b)}\Hyperg(c-a,c-b;c-a-b+1;1-z).
\end{split} \end{equation}
\item The hypergeometric function is symmetric with respect to first and second arguments, i.e
\begin{equation}\label{prop5}
\Hyperg(a,b;c;z)= \Hyperg(b,a;c;z).
\end{equation}
\end{enumerate}
\end{lemma}
\begin{lemma}\label{propiedadesgamma}
\textnormal{ \cite{Abramowitz,SlavyanovWolfganglay}} Let $z\in\c$. Some properties of the Gamma function $\Gamma(z)$ are
\begin{align}
&\Gamma(\bar{z})=\overline{\Gamma(z)},\label{prop1g}\\
&\Gamma(z+1)=z\Gamma(z), \label{prop2g}\\
&\Gamma(z)\Gamma\left(z+\tfrac{1}{2}\right)=2^{1-2z}\sqrt{\pi}\,\Gamma(2z). \label{prop3g}
\end{align}
Let $\psi(z)$ denote the Digamma function defined by $$\psi(z)=\frac{d\ln\Gamma (z)}{dz}=\frac{\Gamma'(z)}{\Gamma(z)}.$$
This function has the expansion
\begin{equation}\label{propdg}
\psi(z)=\psi(1)+\sum_{m=0}^{\infty}\left(\tfrac{1}{m+1}-\tfrac{1}{m+z}\right).
\end{equation}
Let $B(z_1,z_2)$ denote the Beta function defined by
\begin{equation*}
B(z_1,z_2)=\frac{\Gamma(z_1)\Gamma(z_2)}{\Gamma(z_1+z_2)}.
\end{equation*}
If $z_2$ is a fixed number and $z_1>0$ is big enough, then this function behaves
\begin{equation}\label{propbeta}
B(z_1,z_2)\sim \Gamma(z_2)(z_1)^{-z_2}.
\end{equation}
\end{lemma}
We end this section with a remark on the classical fractional Hardy inequality. On Euclidean space $(\mathbb R^n, |dx|^2)$, it is well known that, for all $w\in\mathcal C_0^\infty(\mathbb R^n)$ and $\gamma\in\left(0,\frac{n}{2}\right)$,
\begin{equation}\label{fractional-Hardy}
\begin{split}c_H\int_{\mathbb R^n}\frac{|w|^2}{|x|^{2\gamma}}\,dx&\leq
\int_{\mathbb R^n}|\xi|^{2\gamma} |\hat w(\xi)|^2\,d\xi
\\&=\int_{\mathbb R^n} |(-\Delta)^{\frac{\gamma}{2}}w|^2\,dx=\int_{\mathbb R^n} w(-\Delta)^\gamma w\,dx.\end{split}\end{equation}
Moreover, the constant $c_H$ is sharp (although it is not achieved) and its value is given by
$$c_H=c_{n,\gamma},$$
which is the constant in Proposition \ref{cte}. This is not a coincidence, since the functions that are used in the proof of the sharpness statement are suitable approximations of \eqref{w1}. This constant was first calculated in \cite{Herbst}, but there have been many references \cite{Yafaev,Beckner:Pitts,Frank-Lieb-Seiringer:Hardy}, for instance.
A natural geometric context for the fractional Hardy inequality is obtained by taking $g_0$ as a background metric, and using the changes \eqref{wv} and \eqref{g0:introduction}. Indeed, using the conformal relation given by expression \eqref{relation-introduction}, we conclude that \eqref{fractional-Hardy} is equivalent to the following:
\begin{equation}\label{Hardy-v}
c_{n,\gamma}\int_{\mathbb R\times \mathbb S^{n-1}} v^2\dvol_{g_0}\leq \int_{\mathbb R\times \mathbb S^{n-1}} v (P_\gamma^{g_0} v)\dvol_{g_0},
\end{equation}
for every $v\in\mathcal C^{\infty}_0(\mathbb R\times \mathbb S^{n-1})$.
\section{ODE-type analysis}\label{section:ODE}
In this section we fix $\gamma\in(0,1)$. As we have explained, the fractional Yamabe problem with an isolated singularity at the origin is equivalent to the extension problem \eqref{equation1}. We look for radial solutions of the form \eqref{wv}. Based on our previous study, it is equivalent to consider solutions $v=v(t)$ of the extension problem \eqref{fracyamextv}, for the metric \eqref{metric_introduction}. In this section we perform an ODE-type analysis for the PDE problem \eqref{fracyamextv}.
Firstly we calculate
\begin{equation}\label{divergence-barg}
\begin{split}
\divergence_{\bar{g}}(\rho^{a}\nabla_{\bar{g}} V)=&\sum_{i,j}\frac{1}{\sqrt{|\bar{g}|}}\partial_{i}(\bar{g}^{ij}\rho^{a}\sqrt{|\bar{g}|}\partial_jV)\\
=& \tfrac{1}{e(\rho)}\partial_{\rho}\left(\rho^{a}e(\rho)\partial_{\rho}V\right)+\tfrac{\rho^{a}}
{(1+\frac{\rho^2}{4})^2}\partial_{tt}V+\tfrac{\rho^{a}}{(1-\frac{\rho^2}{4})^2}\Delta_{\s^{n-1}}V,
\end{split}
\end{equation}
where \begin{equation*}
e(\rho)=\left(1+\tfrac{\rho^2}{4}\right)\left(1-\tfrac{\rho^2}{4}\right)^{n-1}.
\end{equation*}
Using the expression given in \eqref{E},
\begin{equation}\label{Erho}
E(\rho)=\tfrac{n-1+a}{4}\rho^a\frac{n-2+n\frac{\rho^2}{4}}{\left(1+\frac{\rho^2}{4}\right)\left( 1-\frac{\rho^2}{4}\right)}.
\end{equation}
\begin{remark}\label{solorho}
Let $V$ be the (unique) solution of \eqref{fracyamextv}. If $v$ does not depend on the spherical variable $\theta\in \mathbb S^{n-1}$, then $V$ does not either. Analogously, if $v$ is independent on $t$ and $\theta$, then $V$ is just a function of $\rho$. The proof is a straightforward computation using that the variables in \eqref{divergence-barg} are separated.
\end{remark}
As a consequence of the previous remark, it is natural to look for solutions $V$ of \eqref{fracyamextv} that only depend on $\rho$ and $t$, i.e. solutions of
\begin{equation}\label{problemaradial}\left\{
\begin{split}
-\frac{1}{e(\rho)}\partial_{\rho}\left(\rho^{a}(e(\rho)\partial_{\rho}V\right)-
\frac{\rho^{a}}{(1+\frac{\rho^2}{4})^2}\partial_{tt}V+E(\rho)V&=0\ \text{ for } \rho\in(0,2),t\in\r,\\
V&=v\ \text{ on }\{\rho=0\},\\
-\tilde{d}_{\gamma}\lim_{\rho\rightarrow 0}\rho^{a}\partial_{\rho}V&=c_{n,\gamma}v^{\frac{n+2\gamma}{n-2\gamma}}\ \text{ on }\{\rho=0\}.
\end{split}
\right.\end{equation}
Now we take the special defining function $\rho^*$ given in Theorem \ref{E0}, whose explicit expression is given in Corollary \ref{corollary:rho*}. Then we can rewrite the original problem \eqref{fracyamextv} in $g^*$, defined on the extension $X^*=\{\rho\in(0,\rho_0^*),t\in\mathbb R,\theta\in \mathbb S^{n-1}\}$, as
\begin{equation}\label{Yamfracspec}\left\{
\begin{split}
-\divergence_{g^*}((\rho^*)^a\nabla_{g^*}V)&=0\ \text{ in }(X^*,g^*),\\
V&=v\ \text{ on }\{\rho^*=0\},\\
-\tilde{d}_{\gamma}\lim_{\rho^*\to 0}(\rho^*)^a\partial_{\rho^*}V +c_{n,\gamma}v&=c_{n,\gamma}v^{\frac{n+2\gamma}{n-2\gamma}}\ \text{ on }\{\rho^*=0\},
\end{split}\right.
\end{equation}
where $g^*=\frac{(\rho^*)^2}{\rho^2}\bar{g}$, for $\rho^*=\rho^*(\rho)$.
Note that Proposition \ref{cte} calculates the value $Q_\gamma^{g_0}\equiv c_{n,\gamma}$. The advantage of \eqref{Yamfracspec} over the original \eqref{fracyamextv} is that it is a pure divergence elliptic problem and has nicer analytical properties.
Next, if we look for radial solutions (that depend only on $t$ and $\rho^*$), then the extension problem \eqref{Yamfracspec} reduces to:
\begin{equation}\label{eqfracesp}\left\{
\begin{split}
\frac{1}{e^*(\rho)}\partial_{\rho^*}\left((\rho^*)^{a}e^*(\rho)\partial_{\rho^*}V\right)
+\frac{(\rho^*)^{a}}
{\left(1+\tfrac{\rho^2}{4}\right)^{2}}\,\partial_{tt}V&=0 \text{ for }t\in\r,\rho^*\in(0,\rho^*_0),\\
v&=V\text{ on }\{\rho^*=0\},\\
-\tilde{d}_{\gamma}(\rho^*)^a\partial_{\rho^*}V +c_{n,\gamma}v&=c_{n,\gamma}v^{\frac{n+2\gamma}{n-2\gamma}}\text{ on }\{\rho^*=0\},
\end{split}\right.
\end{equation}
where
$$e^*(\rho)=\left(\tfrac{\rho^*}{\rho}\right)^2 e(\rho).$$
Summarizing, we will concentrate in problems \eqref{problemaradial} and \eqref{eqfracesp}. In some sense \eqref{eqfracesp} is closer to the local equation \eqref{ODE1} and shares many of its properties. For instance, it has two critical points: $v_0\equiv0$ and $v_1\equiv1$, since these are the only constant solutions of the boundary condition $v=v^{\frac{n+2\gamma}{n-2\gamma}}$ on $\rho^*=0$. Moreover, by uniqueness of the solution and Remark \ref{solorho}, the only critical points in the extension are simply $V_0\equiv 0$ and $V_1\equiv 1$.
\begin{remark}
The calculation of the critical points $v_0\equiv 0$ and $v_1\equiv 1$ also holds for any $\gamma\in\left(0,\frac{n}{2}\right)$, since the corresponding extension problem shares many similarities with \eqref{eqfracesp} (c.f. \cite{MarChang,CaseChang,Ray}).
\end{remark}
The linearization at $V_1\equiv 1$ will be considered in Section \ref{section:linear}.
\subsection{A conserved Hamiltonian}
Here we give the proof of Theorem \ref{thctthamiltonian}. The idea comes from \cite{paperxavi}, where they consider layer solutions for semilinear equations with fractional Laplacian and a double-well potential. Multiply the first equation in \eqref{problemaradial} by $e(\rho)\partial_t V$,
and integrate with respect to $\rho\in(0,2)$, obtaining
\begin{equation*}
\begin{split}
-\int_{0}^{2} \partial_{\rho}\left(\rho^{a}e(\rho)\partial_{\rho}V\right)\partial_t V \,d\rho
-\int_{0}^{2} \rho^{a}e_1(\rho
\partial_{tt}V\partial_{t}V\,d\rho+
\int_{0}^{2} \rho^a e_2(\rho)V\partial_t V\,d\rho=0,
\end{split}
\end{equation*}
where we have defined $e$, $e_1$, $e_2$ as in \eqref{e}.
We realize that $\partial_{tt}V\partial_{t}V=\frac{1}{2}\partial_t((\partial_tV)^2)$ and
$V\partial_t V=\frac{1}{2}\partial_t(V^2)$, thus integrating by parts in the first term above we get
\begin{equation*}
\begin{split}
&\int_{0}^{2} \rho^{a}e(\rho
\partial_{\rho}V\partial_{t\rho} V
\,d\rho+\left(\rho^a e(\rho)\partial_{\rho}V\partial_tV\right)|_{\rho=0}\\&
-\partial_t\left(\tfrac{1}{2}\int_{0}^{2} \rho^{a}e_1(\rho)(\partial_tV)^2\,d\rho\right)+
\partial_t\left(\tfrac{1}{2}\int_{0}^{2}\rho^a e_2(\rho)V^2\,d\rho\right)=0.
\end{split}
\end{equation*}
Here we have used the regularity of $V$ at $\rho=2$.
Again we note that $\partial_{\rho}V\partial_{t\rho} V=\frac{1}{2}\partial_t((\partial_{\rho}V)^2)$ and using the boundary condition, i.e., the third equation in \eqref{problemaradial},
we have
\begin{equation}\label{ham1}
\begin{split}
\tfrac{1}{2}\partial_t\left(\int_{0}^{2} \rho^{a}e(\rho)(\partial_{\rho}V)^2 \,d\rho\right)&
-\tfrac{1}{2}\partial_t\left(\int_{0}^{2} \rho^{a}e_1(\rho)(\partial_tV)^2\,d\rho\right)\\
&+\tfrac{1}{2}\partial_t\left(\int_{0}^{2} \rho^a e_2(\rho)V^2\,d\rho\right)
\frac{c_{n,\gamma}}{\tilde{d}\gamma}v^{\frac{n+2\gamma}{n-2\gamma}}\partial_tv .
\end{split}
\end{equation}
Define
$$G(v)=C_{n,\gamma}v^{\frac{2n}{n-2\gamma}},$$
where the constant is defined in \eqref{constantC}.
In this way, we have from \eqref{ham1} that
\begin{equation*}
\begin{split}
&\tfrac{1}{2}\partial_t\int_{0}^{2} \left\{ \rho^{a}e(\rho)(\partial_{\rho}V)^2
-\rho^{a}e_1(\rho)(\partial_tV)^2
+ \rho^a e_2(\rho)V^2\right\}\,d\rho-\partial_t(G(v))=0.
\end{split}
\end{equation*}
So we can conclude that the Hamiltonian
\begin{equation*}
-H_\gamma(t):=\tfrac{1}{2}\int_{0}^{2} \rho^{a}\left\{e(\rho)(\partial_{\rho}V)^2
-e_1(\rho)(\partial_tV)^2
+ e_2(\rho)V^2\right\}\,d\rho
-G(v),
\end{equation*}
is constant respect to $t$. This concludes the proof of Theorem \ref{thctthamiltonian}.
\begin{remark}
One can rewrite the Hamiltonian in terms of the defining function $\rho^*$. For this, we may follow similar computations as above but starting with equation \eqref{eqfracesp}. Indeed, let $V$ be a solution of \eqref{Yamfracspec}, then the new Hamiltonian quantity
\begin{equation}\label{Hamiltonian*}
\begin{split}
H_\gamma^*(t):=&\frac{c_{n,\gamma}}{\tilde{d}_{\gamma}}\left(\tfrac{n-2\gamma}{2n}v^{\frac{2n}{n-2\gamma}}
-\tfrac{1}{2}v^2\right)\\
&+\tfrac{1}{2}\int_0^{\rho^*_0}(\rho^*)^{a}
\left\{e_1^*(\rho)(\partial_tV)^2-e^*(\rho)(\partial_{\rho^*}V)^2\right\}\,d\rho^*
\end{split}\end{equation}
is constant respect to $t$. Here
$$e^*(\rho)=\left(\tfrac{\rho^*}{\rho}\right)^2,\quad e_1^*(\rho)=\left(\tfrac{\rho^*}{\rho}\right)^2 e_1(\rho).$$
This quantity $H_\gamma^*$ is the natural generalization of \eqref{Hamiltonian1}.
\end{remark}
Now we observe that in the local case, the Hamiltonian \eqref{Hamiltonian1} is a convex function in the domain we are interested, thus its level sets are well defined closed trajectories around the equilibrium $v_1\equiv 1$. We would like to have the analogous result for the Hamiltonian quantity $H^*_\gamma$ from \eqref{Hamiltonian*}. This is a very interesting open question that we conjecture to be true. In any case, the second variation for $H^*_\gamma$ near this equilibrium is:
\begin{equation*}\begin{split}
\left.\frac{d^2}{d\epsilon^2}\right|_{\epsilon=0}H^*_\gamma(V_1+\epsilon V)&=
\tfrac{c_{n,\gamma}}{\tilde d_\gamma}\tfrac{4\gamma}{n-2\gamma}v^2\\
&+\tfrac{1}{2}\int_0^{\rho_0^*} (\rho^*)^a\tfrac{(\rho^*)^2}{\rho^2} \left\{e_1(\rho)(\partial_t V)^2-e(\rho)(\partial_{\rho^*} V)^2\right\}d\rho^*.
\end{split}\end{equation*}
\section{The homoclinic solution}\label{section:explicit}
For this section we will take $\gamma\in\left(0,\tfrac{n}{2}\right)$, since it does not depend on the extension problem \eqref{equation1}. It is clear that the standard bubble \eqref{sphere} is a solution of equation \eqref{equation0} that has a removable singularity at the origin. Note that, because of our choice of the constant $c_{n,\gamma}$, we need to normalize it by a positive multiplicative constant. We prove here that, on the boundary phase portrait, the equilibrium $v_1\equiv 1$ stays always bounded by this homoclinic solution in a boundary phase portrait. More precisely:
\begin{proposition}
The positive function
\begin{equation}\label{C}
v_\infty(t)=C(\cosh t)^{-\frac{n-2\gamma}{2}},\quad \text{ with}\quad C=
\left(c_{n,\gamma}\frac{\Gamma(\frac{n}{2}-\gamma)}{\Gamma(\frac{n}{2}+\gamma)}\right)^{-\frac{n-2\gamma}
{4\gamma}}>1\equiv v_1,
\end{equation}
is a smooth solution of the fractional Yamabe problem \eqref{fracyamextv}. The value of $c_{n,\gamma}$ is given in Proposition \ref{cte}.
\end{proposition}
\begin{proof}
The canonical metric on the sphere, rescaled by a constant, maybe written as
\begin{equation*}\label{sphmetric}
g_C=C^{\frac{4}{n-2\gamma}}g_{\s^n}=[C(\cosh t)^{-\frac{n-2\gamma}{2}}]^{\frac{4}{n-2\gamma}}g_0.
\end{equation*}
We choose $C$ such that the fractional curvature of the standard sphere is normalized to
\begin{equation}\label{Qimpongo}
Q^{g_C}_{\gamma}\equiv c_{n,\gamma}.
\end{equation}
Now we use the conformal property \eqref{confQ} for the operator $P^{g_{\s^n}}_{\gamma}$:
\begin{equation}\label{PgC}
P_{\gamma}^{g_{\s^n}}(C)=C^{\frac{n+2\gamma}{n-2\gamma}}Q_{\gamma}^{g_C}.
\end{equation}
One checks that the fractional curvature is homogeneous of order $\gamma$ under rescaling of the metric. Indeed, because of \eqref{PgC} and the linearity of the operator $P_{\gamma}$
\begin{equation}\label{QC}
\begin{split}
Q_{\gamma}^{g_C}&=C^{-\frac{n+2\gamma}{n-2\gamma}}P_{\gamma}^{g_{\s^n}}(C)=C^{-\frac{(n+2\gamma)}
{n-2\gamma}+1}
P_{\gamma}^{g_{\s^n}}(1)=C^{-\frac{4\gamma}{n-2\gamma}}Q_{\gamma}^{g_{\s^n}}.\\
\end{split}
\end{equation}
Comparing equalities \eqref{Qimpongo} and \eqref{QC}, together with the value of the curvature on the standard sphere \eqref{fqsphere} we find the precise value of $C$ as claimed in \eqref{C}.
Next, let us check that the value of the constant $C$ is larger than one.
Because of Proposition \ref{cte} we have to test that
$$2^{2\gamma}\left(\frac{\Gamma(\frac{1}{2}(\frac{n}{2}+\gamma))}{\Gamma(\frac{1}{2}(\frac{n}{2}-\gamma))}\right)^2
\frac{\Gamma(\frac{n}{2}-\gamma)}{\Gamma(\frac{n}{2}+\gamma)}<1.$$
Using the property \eqref{prop3g} of the Gamma function, given in Lemma \ref{propiedadesgamma}, we only need to verify that
\begin{equation*}
X(n,\gamma):=\frac{\Gamma(\frac{1}{2}(\frac{n}{2}+\gamma))}{\Gamma(\frac{1}{2}(\frac{n}{2}-\gamma))}\frac{\Gamma(\frac{1}{2}(\frac{n}{2}-\gamma)
+\frac{1}{2})}{\Gamma(\frac{1}{2}(\frac{n}{2}+\gamma)+\frac{1}{2})}<1.
\end{equation*}
Thanks to Lemma \ref{Xcrecimiento} below, it is enough to see that
\begin{equation*}
X(n,0)\leq 1\quad \forall n,
\end{equation*}
which holds trivially.
\end{proof}
\begin{lemma}\label{Xcrecimiento}
The function $X(n,\gamma)$ defined as follows
\begin{equation*}\label{Xng}
X(n,\gamma):=\frac{\Gamma(\frac{1}{2}(\frac{n}{2}+\gamma))}
{\Gamma(\frac{1}{2}(\frac{n}{2}-\gamma))}\frac{\Gamma(\frac{1}{2}(\frac{n}{2}-\gamma)+\frac{1}{2})}
{\Gamma(\frac{1}{2}(\tfrac{n}{2}+\gamma)+\tfrac{1}{2})},
\end{equation*}
is increasing in $n$, and decreasing in $\gamma$.
\end{lemma}
\begin{proof}
If we denote $\psi(z)$ the Digamma function from Lemma \ref{propiedadesgamma}, we can use the expansion \eqref{propdg} to study the growth of the function $X(n,\gamma)$ with respect to $n$ and $\gamma$.
First,
\begin{equation*}
\begin{split}\frac{\partial}{\partial n}(\log X(n,\gamma))&=\tfrac{1}{4}\left(\psi(\tfrac{n}{4}+\tfrac{\gamma}{2})+\psi(\tfrac{n}{4}-\tfrac{\gamma}{2}
+\tfrac{1}{2})-\psi(\tfrac{n}{4}-\tfrac{\gamma}{2})-\psi(\tfrac{n}{4}+\tfrac{\gamma}{2}+\tfrac{1}{2})
\right)
\\&=\tfrac{\gamma}{4}\sum_{m=0}^\infty\tfrac{m+\frac{n}{4}+\frac{1}{4}}{\left[\left(m+\frac{n}{4}\right)^2
-\frac{\gamma^2}{4}\right]\left[\left(m+\frac{n}{4}+\frac{1}{2}\right)^2
-\frac{\gamma^2}{4}\right]}>0.
\end{split}\end{equation*}
and
\begin{equation}\label{formulon}
\begin{split}\frac{\partial}{\partial \gamma}(\log X(n,\gamma))&=\tfrac{1}{2}\left(\psi(\tfrac{n}{4}+\tfrac{\gamma}{2})-\psi(\tfrac{n}{4}-\tfrac{\gamma}{2}
+\tfrac{1}{2})+\psi(\tfrac{n}{4}-\tfrac{\gamma}{2})-\psi(\tfrac{n}{4}+\tfrac{\gamma}{2}+\tfrac{1}{2})
\right)\\
&=-\tfrac{1}{2}\sum_{m=0}^{\infty}\left[\tfrac{\left(m+\tfrac{n}{4}+\tfrac{1}{2}\right)
\left(m+\tfrac{n}{4}\right)+\tfrac{\gamma^2}{4}}{\left((m+\tfrac{n}{4}+\tfrac{1}{2})^2-\tfrac{\gamma^2}
{4}\right)\left((m+\tfrac{n}{4})^2-\tfrac{\gamma^2}{4}\right)}\right]<0.
\end{split}\end{equation}
\end{proof}
\section{Linear analysis}\label{section:linear}
Let us say a few words about stability. Let $v_*$ be a solution of \eqref{equation2}. The corresponding linearized equation is
$$P_\gamma^{g_0} v=c_{n,\gamma}\tfrac{n+2\gamma}{n-2\gamma}\, v_*^{\frac{4\gamma}{n-2\gamma}}v.$$
We say that $v_*$ is a stable solution of \eqref{equation2} if
\begin{equation}\label{stability}
\int_M v (P^{g_0}_\gamma v)\dvol_{g_0}-c_{n,\gamma} \tfrac{n+2\gamma}{n-2\gamma}\int_{M}v_*^{\frac{4\gamma}{n-2\gamma}} v^2\dvol_{g_0}\geq 0,\quad \text{for all}\quad v\in\mathcal C_0^\infty(M).
\end{equation}
We observe here that the equilibrium $v_1\equiv 1$ is not a stable solution for \eqref{equation2} just by comparing the constant appearing in \eqref{stability} and in the Hardy inequality \eqref{Hardy-v}. In addition, one easily checks that the equilibrium solution $v_0\equiv 0$ is stable.
But it is more interesting to look at the explicit solution $v_\infty$ given in \eqref{C}. It follows from the Hardy inequality \eqref{Hardy-v} that this explicit solution is not stable. The kernel of the linearization at $v_\infty$ is calculated in \cite{Davila-delPino-Sire}, where they show that, although non-trivial, is non-degenerate, i.e., is generated by translations and dilations of the standard bubble.\\
Let us look more closely at the spectrum of the operator $P^{g_0}_{\gamma}$.
It is well known that $P^{g_0}_{\gamma}$ is self-adjoint (\cite{GrahamZorski}), and then
we can compute its first eigenvalue through the Rayleigh quotient. Thus we minimize
\begin{equation*}
\inf_{v\in\mathcal C_0^\infty(M)}\frac{\int_MvP_{\gamma}^{g_0}v\dvol_{g_0}}{\int_{M}v^2\dvol_{g_0}},
\end{equation*}
where $M=\r\times\mathbb S^{n-1}$. We can apply Theorem 4.2 and Corollary 4.3 in \cite{MarQing} (or the Hardy inequality \eqref{Hardy-v}) to conclude that $P^{g_0}_{\gamma}$ is positive-definite. Moreover, the first eigenspace is of dimension one.\\
Now we consider the linear analysis around the equilibrium solution $v_1\equiv 1$. In order to motivate our results, let us explain what happens in the local case $\gamma=1$ for the linearization (see \cite{Mazzeo-Pacard:isolated,Mazzeo-Pollack-Uhlenbeck:moduli-spaces,KMPS}). In these papers the authors actually characterize the spectrum for the linearization of the equation
$$P_1^{g_0} v=\tfrac{(n-2)^2}{4} v^{\frac{n+2}{n-2}},$$
given by (after projection over each eigenspace $\langle E_k\rangle$, $k=0,1,\ldots$)
$$-\ddot{v}-[n-2+\mu_k]v=0.$$
Note that this equation has periodic solutions only for $k=0$, of period $L_1=\frac{2\pi}{\sqrt{\lambda^0}}$ for $\lambda^0=n-2$. Thus we recover \eqref{L1}. For the rest of $k=1,\ldots,$ the corresponding $\lambda^k=n-2+\mu_k<0$, so we do not get periodic solutions.\\
The linearization of equation \eqref{equation2} around the equilibrium $v_1\equiv 1$ is given by
\begin{equation}\label{conforlinearizada}
\begin{split}
P^{g_0}_{\gamma}v=c_{n,\gamma}\tfrac{n+2\gamma}{n-2\gamma}v
\end{split}
\end{equation}
Here we will calculate the period of solutions for this linearized problem (for the projection $k=0$), as stated in Theorem \ref{theorem:linearization}, by the method of separation of variables. We also conjecture that there are not periodic solutions for the linearized problem \eqref{linearization-k} for the rest of $k=1,...$, as it happens in the classical clase.
Therefore, we consider the projection of equation \eqref{eqs} over each eigenspace $\langle E_k\rangle$, $k=0,1,\ldots$. Let
$$U_k(z,t)=T(t)Z(z),$$
be a solution of \eqref{equk}. Then
\begin{equation*}
(1-z^2)\frac{Z''(z)}{Z(z)}+\left(\tfrac{n-1}{z}-z\right) \frac{Z'(z)}{Z(z)}+\frac{\frac{n^2}{4}-\gamma^2}{1-z^2}+\frac{\mu_k}{z^2}
=-\frac{T''(t)}{T(t)}=\lambda^k,
\end{equation*}
for a constant $\lambda^k:=\lambda^k(\gamma)\in\mathbb R$. We are only interested in the case $\lambda>0$, which is the one that leads to periodic solutions in the variable $t$. The period would be calculated from $L^k:=L^k(\gamma)=\frac{2\pi}{\sqrt {\lambda^k}}$.
Note that the equation for $Z(z)$ is simply \eqref{equkfou} with $\xi^2$ replaced by $\lambda^k$. From the discussion in Section \ref{simbolo}, in particular \eqref{varphiz}, \eqref{A} and \eqref{S(s)} we have that
\begin{equation*}
\begin{split}
Z(z)=&(1+z)^{\frac{n}{4}-\frac{\gamma}{2}}(1-z)^{\frac{n}{4}-\frac{\gamma}{2}}z^{1-\frac{n}{2}
+\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}} \\ &
\Hyperg(a,b;a+b-c+1;1-z^2)\\
+&\kappa(1+z)^{\frac{n}{4}+\frac{\gamma}{2}}(1-z)^{\frac{n}{4}+\frac{\gamma}{2}}z^{1-\frac{n}{2}
-\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}\\&
\Hyperg(c-a,c-b;c-a-b+1;1-z^2),
\end{split}
\end{equation*}
where \begin{align*}
a&=-\tfrac{\gamma}{2}+\tfrac{1}{2}+\tfrac{\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}{2}
+i\tfrac{\sqrt{\lambda^k}}{2}\\
b&=-\tfrac{\gamma}{2}+\tfrac{1}{2}+\tfrac{\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}{2}
-i\tfrac{\sqrt{\lambda^k}}{2},\\
c&=1+\sqrt{(\tfrac{n}{2}-1)^2-\mu_k},\\ \kappa&=\tfrac{\Gamma(-\gamma)\left|\Gamma\large(\tfrac{1}{2}+\tfrac{\gamma}{2}
+\tfrac{\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}{2}
+i\frac{\sqrt{\lambda^k}}{2}\large)\right|^2}{\Gamma(\gamma)\left|\Gamma\large(\tfrac{1}{2}
-\tfrac{\gamma}{2}
+\tfrac{\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}{2}+i\tfrac{\sqrt{\lambda^k}}{2}\large)\right|^2}.
\end{align*}
We use the change of variable \eqref{zrho} to analyze the asymptotic behavior of $Z$ near the conformal infinity $\rho=0$
\begin{equation*}
Z\sim\rho^{\frac{n}{2}-\gamma}+\kappa\rho^{\frac{n}{2}+\gamma}.
\end{equation*}
From the definition of the scattering operator \eqref{formau}, \eqref{scattering}, and the definition of the conformal fractional Laplacian we have that
\begin{equation*}
P^{k}_{\gamma}v_k=2^{2\gamma}\frac{\left|\Gamma(\tfrac{1}{2}+\tfrac{\gamma}{2}
+\tfrac{\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}{2}+\tfrac{\sqrt{\lambda^k}}{2}i)\right|^2}
{\left|\Gamma(\tfrac{1}{2}-\tfrac{\gamma}{2}
+\tfrac{\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}{2}+\tfrac{\sqrt{\lambda^k}}{2}i)\right|^2}
v.
\end{equation*}
Imposing the boundary condition
\eqref{conforlinearizada} and the value of $c_{n,\gamma}$ given in \eqref{cng}, the unknown $\lambda^k$ must be a solution of
\begin{equation}\label{eqlambdak}
\frac{\left|\Gamma(\tfrac{1}{2}+\tfrac{\gamma}{2}+\tfrac{\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}{2}
+\tfrac{\sqrt{\lambda^k}}{2}i)\right|^2}
{\left|\Gamma(\tfrac{1}{2}-\tfrac{\gamma}{2}+\tfrac{\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}{2}
+\tfrac{\sqrt{\lambda^k}}{2}i)\right|^2}
=\frac{n+2\gamma}{n-2\gamma}\frac{\left|\Gamma\left(\frac{1}{2}\left(\frac{n}{2}
+\gamma\right)\right)\right|^2}
{\left|\Gamma\left(\frac{1}{2}\left(\frac{n}{2}-\gamma\right)\right)\right|^2}.
\end{equation}
Note that for the canonical projection $k=0$, equality \eqref{eqlambdak} simplifies to
\begin{equation}\label{eqlambda2}
\frac{\left|\Gamma(\tfrac{n}{4}+\tfrac{\gamma}{2}+\tfrac{\sqrt{\lambda^0}}{2}i)\right|^2}
{\left|\Gamma(\tfrac{n}{4}-\tfrac{\gamma}{2}+\tfrac{\sqrt{\lambda^0}}{2}i)\right|^2}
=\frac{n+2\gamma}{n-2\gamma}\frac{\left|\Gamma\left(\frac{1}{2}\left(\frac{n}{2}+\gamma\right)\right)\right|^2}
{\left|\Gamma\left(\frac{1}{2}\left(\frac{n}{2}-\gamma\right)\right)\right|^2}.
\end{equation}
This equation \eqref{eqlambda2} lets us recover the value of $\lambda^0$ for the classical case $\gamma=1$. Indeed, using property \eqref{prop2g} we get $\lambda^0=n-2$ and we recover \eqref{L1}, where $L_1:=L^0(1).$
Going back to equation \eqref{eqlambdak} we can assert that the value of $\lambda^k$ can not be zero and it is unique for each $k$. Indeed if $\lambda=0$ we get a contradiction, and if $\lambda>0$ we may proceed as follows. Define
$$F(\beta)=\frac{\frac{|\Gamma(\alpha_k+\beta i)|^2}{|\Gamma(\tilde{\alpha}_k+\beta i)|^2}}{\frac{n+2\gamma}{n-2\gamma}\frac{\left|\Gamma\left(\frac{1}{2}\left(\frac{n}{2}+\gamma\right)\right)
\right|^2}{\left|\Gamma
\left(\frac{1}{2}\left(\frac{n}{2}-\gamma\right)\right)\right|^2}},$$
where
$$\alpha_k=\tfrac{1}{2}+\tfrac{\gamma}{2}+\tfrac{\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}{2},\quad
\tilde{\alpha}_k=\tfrac{1}{2}-\tfrac{\gamma}{2}+\tfrac{\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}{2}
\quad\text{and}\quad \beta=\tfrac{\sqrt{\lambda^k}}{2}.$$
Note that equation \eqref{eqlambdak} is written as $F(\beta)=1$, for some $\beta>0$.
We derive this expression with respect to $\beta$,
\begin{equation*}
(\log F(\beta))'=\frac{2}{\frac{n+2\gamma}{n-2\gamma}\frac{\left|\Gamma\left(\frac{1}{2}\left(\frac{n}{2}
+\gamma\right)\right)
\right|^2}{\left|\Gamma
\left(\frac{1}{2}\left(\frac{n}{2}-\gamma\right)\right)\right|^2}}
\,\Im [\psi(\tilde{\alpha}_k+\beta i)-\psi(\alpha_k+\beta i)].
\end{equation*}
Here $\Im$ represents the imaginary part of a complex number and $\psi(z)$ the Digamma function from Lemma \ref{propiedadesgamma}. We can use the expansion \eqref{propdg} to arrive at
\begin{equation*}
(\log F(\beta))'
=c\sum_{m=0}^{\infty}
\tfrac{\gamma\beta\left(2m+1+\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}\right)}
{\left[\left(m+\tfrac{1}{2}+\tfrac{\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}}{2}\right)^2
-{\beta}^2-\tfrac{\gamma^2}{4}\right]^2
+\left[\left(2m+1+\sqrt{(\tfrac{n}{2}-1)^2-\mu_k}\right)\beta\right]^2},
\end{equation*}
for some positive constant $c$. Therefore $F(\beta)$ is an increasing function of $\beta$.
Next, note that
$$\lim_{\beta\to +\infty} F(\beta)=+\infty,$$
for all $k=0,1,\ldots$. This follows easily writing
$$\frac{n+2\gamma}{n-2\gamma} F(\beta)=\frac{B\left(\alpha_k+\beta i, \frac{n}{4}-\frac{\gamma}{2}\right)}{B\left(\tilde \alpha_k+\beta i, \frac{n}{4}+\frac{\gamma}{2}\right)},$$
and the asymptotic behavior for the Beta function \eqref{propbeta} from Lemma \ref{propiedadesgamma}.
Now we look at the projection $k=0$. One immediately calculates
$$F(0)=\frac{n-2\gamma}{n+2\gamma}<1,$$
so there exists (and it is unique) a solution $\lambda^0=\lambda^0(\gamma)>0$ for the equation $F(\beta)=1$. From the proof one also gets that
$$\lim_{\gamma\to 1} \lambda^0(\gamma)=n-2.$$
This concludes the proof of Theorem \ref{theorem:linearization}.\\
We believe that, as in the classical case $F(\beta)=1$ does not have any positive solution for $k=1,2,\ldots$. This is a well supported conjecture that only depends on making more rigorous some numerical analysis. In order to motivate this conjecture, let us try to show that
$f_k>1$ for $k=1,2,\ldots$, where we have defined
$$F(0)=\frac{(n-2\gamma)|\Gamma(\alpha_k)|^2\left|\Gamma
\left(\frac{1}{2}\left(\frac{n}{2}-\gamma\right)\right)\right|^2}
{(n+2\gamma)|\Gamma(\tilde{\alpha}_k)|^2
\left|\Gamma\left(\frac{1}{2}\left(\frac{n}{2}+\gamma\right)\right)
\right|^2}=:f_k.$$
Using the same ideas as above, one checks that $f_k$ is an increasing function of $k$, and it is enough to show that
$$f_1= \frac{(n-2\gamma)\left|\Gamma\left(\frac{1}{2}+\frac{\gamma}{2}+\frac{n}{4}\right) \Gamma\left(\frac{n}{4}-\frac{\gamma}{2}\right) \right|^2}{(n+2\gamma)\left|\Gamma\left(\frac{1}{2}-\frac{\gamma}{2}+\frac{n}{4}\right) \Gamma\left(\frac{n}{4}+\frac{\gamma}{2}\right) \right|^2}=\frac{n-2\gamma}{n+2\gamma}X(n,\gamma)^{-2}>1,$$
where $X(n,\gamma)$ is defined in Lemma \ref{Xcrecimiento}. We have numerically observed that $f_1=f_1(\gamma)$ is an increasing function in $\gamma$. Since for $\gamma=0$ we already have that $f_1(0)=1$, we would conclude that $f_k> f_1\geq 1$, as desired.
\section*{Acknowledgement}
The authors would like to thank the hospitality of Princeton University, where part of this work was carried out. They also thank Xavier Cabr{\'e}, Marco Fontelos, Robin Graham, Xavier Ros-Oton and Joan Sol{\`a}-Morales for their useful conversations and suggestions.
\bibliographystyle{abbrv}
|
{
"timestamp": "2015-04-15T02:07:35",
"yymm": "1504",
"arxiv_id": "1504.03493",
"language": "en",
"url": "https://arxiv.org/abs/1504.03493"
}
|
\section{Model and method}
In this article we study two species of ultracold fermions with creation operators $\Psi_1^\dagger(\vec r)$ and $\Psi_2^\dagger(\vec r)$ captured each in a perfectly harmonic trap in three dimensions. The system is described by
\begin{alignat}{1}\label{hamiltonian}
H &= H_0 + H^{(11)}_{\text{int}} +H^{(22)}_{\text{int}}+ H^{(12)}_{\text{int}}; \\
H_0 &= \sum_{i=1}^2 \int\!d^3r\; \Psi_i^\dagger(\vec r)
\!\left[-\frac{\hbar^2 \nabla^2}{2m_i} + \frac{m_i\omega_i^2}{2}(\vec r-\vec r_i^0)^2 \right]\!
\Psi_i(\vec r) \nonumber \\
H^{(12)}_{\text{int}} &= \frac{4\pi\hbar^2a}{2m_{\text{red}}}
\int\!d^3r\; \Psi_1^\dagger(\vec r) \Psi_2^\dagger(\vec r) \Psi_2(\vec r) \Psi_1(\vec r) \nonumber
\end{alignat}
Here, a shift of the position of the potential minimum $\vec{r}_i^0(t)$ can be used to excite dipolar oscillations.
In general, the two fermion species may have different masses $m_i$ and feel different trap potentials with respective trap frequencies $\omega_1=\bar \omega+\frac{\delta \omega}{2}$, $\omega_2=\bar \omega-\frac{\delta \omega}{2}$. $H^{(ii)}_{\text{int}}$ describes the intra-species interaction which we do not specify here as it does not influence our results in any qualitative way.
Furthermore, for spinless fermions $H^{(ii)}_{\text{int}}$ can safely be neglected.
As we will show, all relaxation arises from the inter-species interaction which we parametrize by the s-wave scattering length $a$ with $m_{\text{red}}=1/(m_1^{-1}+m_2^{-1})$ being the reduced mass (note that we use a pseudopotential to describe the scattering, see, e.g., Ref.~\cite{bloch_many-body_2008}).
For $\delta \omega=0$ the COM oscillations do not decay (see below).
We are therefore mainly interested in the limit $\delta \omega\ll \bar \omega$, where a slow decay of the oscillations can be expected. Experimentally, this can, for example, be realized by using two isotopes with slightly different mass, $m\pm\frac{\delta m}{2}$, but identical trapping potential.
In this case $\frac{\delta \omega}{\bar \omega}=-\frac{\delta m}{2 m}$.
Alternatively, one can use two hyperfine states of the same atom in combination with a spin-dependent potential \cite{bloch_many-body_2008}.
The latter setup has the advantage that one can directly tune the parameter $ \frac{\delta \omega}{\bar \omega}$.
Our theoretical approach is based on the idea that for $\frac{\delta \omega}{\bar \omega} \ll 1$ the dynamics is governed by an approximate dynamical symmetry which
prohibits a fast relaxation of the COM oscillations. Furthermore, in the limit of vanishing inter-species interactions, $a \to 0$, also the COM motion of each atomic species separately decouples. Our central goal is to derive an effective, hydrodynamic description of the slowly relaxing modes. We will therefore focus on the dynamics in the operator space spanned by the center-of-mass coordinates $\vec R_i$ and the total momentum $\vec P_i$ of each of the two species defined by
\begin{alignat}{1}\label{comcoordinates}
\vec R_i &= \frac{1}{N_i} \int\!d^3r\; \Psi_i^\dagger(\vec r)\,\vec r\,\Psi_i(\vec r); \\
\vec P_i &= \int\!d^3r\; \Psi_i^\dagger(\vec r)\,(-i\hbar\nabla)\,\Psi_i(\vec r)\nonumber
\end{alignat}
where $N_i$ is the number of particles of type $i=1,2$.
For weak excitations of the system, it is sufficient to study linear response within the Kubo formalism.
The main goal is thereby to calculate the matrix of retarded susceptibilities
\begin{eqnarray}\label{def-chi}
\chi_{mn}(\omega)= \frac{i}{\hbar} \int_0^\infty\!dt\; e^{i \omega t}\langle [A_m(t),A_n(0)]\rangle_{\text{eq.}}
\end{eqnarray}
where $\langle\cdot\rangle_{\text{eq.}}$ denotes the expectation value in equilibrium for $\vec r_i^0(t)=0\;\forall t$ and $A_n=(R_1^x,R_2^x,P_1^x,P_2^x)$. As for a spherical potential the $x$, $y$ and $z$ components do not mix within linear response, we can focus on the $x$ coordinate only. $\chi_{mn}$ allows to calculate all experiments where the COM oscillations are excited by a shift $\vec r_i^{0}(t)$ of the potential and where the COM and/or the average momenta of the particles are observed.
To calculate $\chi_{mn}(\omega)$ we use the so-called memory matrix formalism \cite{forster_hydrodynamic_1995,mori_transport_1965,zwanzig_ensemble_1960}.
The memory matrix is a matrix of relaxation rates of slow variables, which we evaluate perturbatively in the strength of the inter-species interaction.
This formalism has the advantages that (i) it is easy to evaluate -- without the need to solve the type of integral equations needed for Boltzmann approaches or when vertex corrections are taken into account within the Kubo formalism, (ii) it nevertheless automatically includes the effect of vertex corrections, which are essential to describe momentum conservation, which is also governing the COM oscillations \cite{goetze}, (iii) it is accurate in cases where there is a separation of time scales and all slow modes are included in the memory matrix, (iv) it can be used to treat complicated situations like the expansion around a fully interacting integrable system \cite{jung_transport_2006,jung07} and has recently been used to calculate transport properties of exotic non-Fermi liquids \cite{hartnoll_transport_2014,lucas_scale-invariant_2014,lucas_memory_2015} (v) in the case considered here, where we effectively expand around the non-interacting limit, it is equivalent to a solution of the Boltzmann equation by projection onto the slow modes \cite{belitz_electronic_1984,chiacchiera_dipole_2010}.
In Ref.~\onlinecite{jung_lower_2007} we have argued that the formalism gives always a lower limit for conductivities. The situation investigated here is, however, more complicated compared to the case
considered in Ref.~\onlinecite{jung_lower_2007} as we are studying here effects at finite frequency in a system which is not translationally invariant. This leads to extra dephasing effects discussed in detail in Appendices \ref{appendix-sigma-singular} and \ref{appendix-dephasing}.
We refer to Appendix \ref{appendix-memorymatrix} for a brief review of the memory-matrix method.
It allows to express the matrix $\chi_{mn}(\omega)$ of retarded susceptibilities (cf.~Eqs.(\ref{relationcchi},\ref{cofomegamemory})),
\begin{equation}\label{chiMemoryMatrix}
\chi(\omega) = \left(1- \omega\left(\omega-\Omega + i\Sigma(\omega)\right)^{-1}\right) C_0
\end{equation}
in terms of an equal-time correlation matrix $C_0$, a constant matrix $\Omega$ and a frequency-dependent matrix-valued complex function $\Sigma(\omega)$. The latter two matrices have a similar role as the self-energy: they describe directly the
shift of frequencies and the damping of oscillations. They have the advantage that they can be evaluated
directly in perturbation theory, without the need to resum an infinite series of diagrams. More precisely,
the latter statement holds in the case when all slow modes have been included in the set of observables $A_n$. We will use $A_n=(R_1^x,R_2^x,P_1^x,P_2^x)$ as the slow modes, which is sufficient to describe the regime where interactions dominate. As we discuss in detail in section \ref{appendix-sigma-singular} of the appendix, in the limit of vanishing interactions an infinite set of further slow modes exists, which have to be included to describe details
of the dephasing of oscillations for very weak interactions (ballistic regime) studied in detail in Appendix \ref{appendix-dephasing} but not captured for the above choice of $A_n$.
In the following our goal will be to calculate for weak interactions the frequencies and decay rates
of the center-of-mass oscillations.
In appendix \ref{appendix-evaluation-sigma}, we evaluate the matrices $\Omega$, $\Sigma(\omega)$, and $C_0$ in local density approximation for weak interactions.
To linear order in $a$, using Eqs.~\eqref{omega0} and \eqref{omega1}, we find $\Omega=\Omega^{(0)}+\Omega^{(1)}$ with
\begin{alignat}{1}\label{omega0Text}
\Omega^{(0)} &= \begin{pmatrix}
0 & 0 & i/M_1 & 0 \\
0 & 0 & 0 & i/M_2 \\
-iM_1 \omega_1^2 & 0 & 0 & 0 \\
0 & -iM_2 \omega_2^2 & 0 & 0
\end{pmatrix}
\end{alignat}
and
\begin{alignat}{1}\label{deltaOmega}
\Omega^{(1)} &= i \gamma M_2 \omega_2 \begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 &0 \\
1 & -1 & 0 & 0 \\
-1 & 1 & 0 & 0
\end{pmatrix}.
\end{alignat}
Here, $M_i=N_i m_i$ is the total mass of the fermions of species $i$ and $\gamma$ has the unit of a rate and is linear in the scattering length $a$ but depends in general on temperature and other parameters (see below).
Physically, Eq.~\eqref{omega0Text} describes independent oscillations of the two species in the absence of interactions.
The eigenfrequencies of $\Omega^{(0)}$ are given by the trap frequencies, $\pm \omega_1$ and $\pm \omega_2$.
Eq.~(\ref{deltaOmega}) describes that each species introduces a Hartree potential for the other species.
As we will discuss below, this contribution will shift the oscillation frequencies as long as the two species do not oscillate in parallel.
We obtain within a local density approximation using Eqs.~\eqref{omega1} and \eqref{r1r2scalarproduct} from the appendix,
\begin{alignat}{1}\label{gammaexact}
\gamma &= \frac{a k_B T \omega_1^2 \omega_2}{3\pi\hbar^4} \frac{m_1^{5/2} m_2^{3/2}}{N_2 m_{\text{red}}}
\int_0^\infty\!dr\, r^4 g_1(r)g_2(r)
\end{alignat}
with
\begin{alignat}{1}\label{gammaexactAux}
g_i(r) &= \text{Li}_{\frac12}\!\left(-e^{(\mu_i - \frac12 m_i\omega_i^2 r^2)/(k_BT)}\right)
\end{alignat}
where $\text{Li}_{\frac12}$ is the polylogarithm of order $\frac12$ and $\mu_i$ is the chemical potential for particles of species $i$ in the limit $a\to0$, see the discussion in appendix \ref{appendix-eval-c0}.
Damping, described by $\Sigma(\omega)$, arises only to second order in the interaction strength.
The total momentum is conserved during scattering processes, $\partial_t {\bf P}_1=-\partial_t {\bf P}_2$, which leads to the simple matrix structure
\begin{alignat}{1}\label{mmatrix}
\Sigma(\omega\to 0) &\approx \Gamma \begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & M_2/M_1 & -1 \\
0 & 0 & -M_2/M_1 & 1
\end{pmatrix}
\end{alignat}
with
\begin{alignat}{1}\label{Gammaexact}
\Gamma= &\frac{\pi\hbar}{M_2 k_B T} \left(\frac{4\pi\hbar^2a}{2m_{\text{red}}}\right)^2
\int\!d^3r
\prod_{\substack{i=1,2;\\\alpha=1,2}} \int\!\frac{d^3k_{i\alpha}}{(2\pi)^3} \times \nonumber\\
&\quad\times
\delta(\Delta\epsilon) \delta^{(3)}(\Delta \vec k)
q_x^2\, f_{11}f_{21}(1-f_{12})(1-f_{22})
\end{alignat}
to second order in the interaction strength using again the local density approximation, see Appendix \ref{appendixSigma}.
Here $f_{i\alpha}$ are Fermi functions evaluated at the energy $\epsilon_{i\alpha}=\hbar^2 k_{i \alpha}^2/(2 m_i)+\frac{1}{2} m_i \omega_i^2 {\bf r}^2$ and $\mathbf q=\mathbf k_{11}-\mathbf k_{12}$ is the change of momentum of the first species, while $\Delta \mathbf k$ and $\Delta \epsilon$ is the change of total momentum and energy, respectively.
As the oscillation frequency is assumed to be much smaller than all Fermi energies, we have used the limit $\omega \to 0$.
Furthermore, we ignore all frequency shifts to order $a^2$ (arising from the Kramers-Kronig partner of $\Gamma$). A more subtle issue is that our approach also neglects the coupling of the COM oscillations to other modes oscillating with frequency $\omega_i$ for $a \to 0$.
This is justified as, in the presence of interactions, these modes decay rapidly, but formally breaks down in the limit of vanishing interactions.
As discussed in more detail in the supplement, this approximation gives rise to small, but nominally divergent extra contribution to $\Sigma(\omega)$, which do, however, not affect our results.
Finally, the equal-time correlation matrix $C_0$ in Eq.~\eqref{chiMemoryMatrix} is evaluated in Appendix \ref{appendix-eval-c0}.
To linear order in $a$, we obtain $C_0=C_0^{(0)}+C_0^{(1)}$ where
\begin{alignat}{1}\label{c0}
C_0^{(0)} &= \begin{pmatrix}
1/(M_1 \omega_1^2) & 0 & 0 & 0 \\
0 & 1/(M_2 \omega_2^2) & 0 & 0 \\
0 & 0 & M_1 & 0 \\
0 & 0 & 0 & M_2
\end{pmatrix}
\end{alignat}
and
\begin{equation}
C_0^{(1)} = \frac{\gamma}{M_1 \omega_1^2 \omega_2} \begin{pmatrix}
\frac{M_2\omega_2^2}{M_1\omega_1^2} & -1 & 0 & 0 \\
-1 & \frac{M_1\omega_1^2}{M_2\omega_2^2} & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix},
\end{equation}
where $\gamma\propto a$ is given in Eqs.~\eqref{gammaexact}--\eqref{gammaexactAux}.
\section{Analytic results\label{sectionAnalyticResults}}
Three different regimes have to be distinguished when discussing how the interactions affect the COM oscillations, depending on which of the three quantities $\delta \omega$, $|\gamma|$ and $\Gamma$ is largest.
First, in the {\em ballistic regime} ($\delta \omega \gg |\gamma|, \Gamma$) interaction effects can approximately be ignored and the oscillations of the two species are almost independent.
Second, in the {\em frictionless drag regime} ($|\gamma| \gg \delta \omega, \Gamma$) one species drags the other by the interaction-induced Hartree potential.
Finally, in the {\em friction dominated drag regime} ($\Gamma \gg \delta \omega, |\gamma|$) the two clouds are coupled by friction and only a hydrodynamic COM oscillation with small effective damping survives.
For a quantitative calculation we have evaluated the integrals in Eqs.~\eqref{gammaexact} and \eqref{Gammaexact} numerically, see section \ref{sectionNumericalResults} and Fig.~\ref{fig_gammaplot}.
In the limit of very low or very high temperature, also an analytic calculation is possible.
For low temperatures, $k_BT\ll\epsilon_{F,1}$ and $N_2 \leq N_1$ one finds
\begin{alignat}{1}
\gamma &\approx \frac{128}{35\pi^2}\, k_{F,1} a\, \bar\omega \approx 0.37 \,k_{F,1} a\, \bar\omega \nonumber \\
\Gamma &\approx \frac{8\pi}{9} \frac{(k_B T)^2}{\hbar\, \epsilon_{F,1}} \,(k_{F,1}a)^2
\label{lowTlimits}
\end{alignat}
where $k_{F,i}$ is the Fermi momentum of species $i$ in the center of the trap with Fermi energy $\epsilon_{F,i}=k_{F,i}^2/(2 m_i)$ determined for $T \to 0$.
The analytic formulas have been computed in the limit $\delta \omega \to 0$ and for $m_1=m_2$.
While the prefactor of $\Gamma$ is valid for arbitrary ratios of $N_2$ and $N_1$ as long as $N_2\leq N_1$, the prefactor for $\gamma$ is only exact for $N_1=N_2$ but increases by less than a factor of 2 when $N_2/N_1$ is reduced, see Fig.~\ref{fig_gammaplot}.
Surprisingly, the estimates given in Eq.~\eqref{lowTlimits} are even valid when the temperature is larger than the Fermi energy of the second species.
If the temperature is larger than both Fermi energies, in contrast, the scattering rate $\Gamma$ drops with $1/T$ while $\gamma$ vanishes with $1/T^{5/2}$,
\begin{alignat}{1}
\gamma &\approx \frac{k_{F,1}a\,\bar\omega}{24\sqrt{2\pi}} \left(\frac{\epsilon_{F,1}}{k_B T}\right)^{5/2} = \frac{k_T a\,\bar\omega}{24\sqrt{2\pi}} \left(\frac{\epsilon_{F,1}}{k_B T}\right)^3 \nonumber \\
\Gamma &\approx \frac{(k_{F,1}a)^2}{9\pi} \frac{\epsilon_{F,1}^2}{\hbar\, k_B T} = \frac{(k_Ta)^2}{9\pi} \frac{\epsilon_{F,1}^3}{\hbar\, (k_B T)^2}
\label{highTlimits}
\end{alignat}
where $k_T=\sqrt{2mk_BT}/\hbar$ is the thermal wave vector.
The prefactors for the high-temperature limit of both $\gamma$ and $\Gamma$ are valid for arbitrary ratios of $N_2$ and $N_1$ as long as $N_2\leq N_1$.
In both regimes, $\Gamma$ can be identified with the single-particle scattering rate of a particle of species $2$ in the center of the trap.
In the high-temperature regime, this can be seen by rewriting $\Gamma\sim \sigma v_{\rm th} n_1$ in terms of the scattering cross section $\sigma \sim a^2$, the typical velocity $ v_{\rm th} \sim \sqrt{k_B T /m}$ and the density of particles of species $1$ in the center, $n_1 \sim N_1/(T/m \omega^2)^{3/2} \sim \epsilon_{F,1}^3 (m/T)^{3/2}$.
\begin{figure}
\includegraphics[width=\columnwidth]{gammaplot.pdf}
\caption{\label{fig_gammaplot}
(Color online) Numerical results for the quantities $\gamma$ (Eqs.~\eqref{gammaexact}--\eqref{gammaexactAux}) and $\Gamma$ (Eq.~\eqref{Gammaexact}) for different ratios of $N_2/N_1$.
Dotted gray lines are analytic predictions for $T\ll\epsilon_{F,1}$ and $T\gg\epsilon_{F,1}$, see Eqs.~\eqref{lowTlimits}--\eqref{highTlimits}.
The analytic formula for $\gamma$ in the limit $T\to0$ given in Eq.~\eqref{lowTlimits} is only exact for $N_1=N_2$ and underestimates the value of $\gamma$ for $N_2/N_1\to0$ by a factor of $64/(35\pi)\approx0.58$.
All curves were calculated with $m_1=m_2$, $\frac{\delta\omega}{\bar\omega}=0.1$, and are independent of the total particle number in the chosen units.
}
\end{figure}
\subsection{Ballistic regime}
For very small interactions the two species oscillate approximately independently of each other.
More precisely, we require that the strength $\gamma$ of the effective interaction potential and the single-particle scattering rate $\Gamma$ are both smaller (in magnitude) than the {\em difference} of oscillation frequencies, $|\gamma| \ll \delta \omega$ and $\Gamma \ll \delta\omega$.
While this regime is usually not realized experimentally at low temperatures (without tuning interactions close to zero), we discuss it here for completeness.
Note that this regime is always reached in the limit of high temperatures as long as $\delta\omega \neq 0$.
The remaining weak interactions lead to a small shift of the respective oscillation frequencies relative to the trap frequencies and to a finite, but long, lifetime of the two oscillatory modes.
The complex eigenfrequencies are given by the eigenvalues of $\Omega-i\Sigma(\omega)$, where the matrix of retarded susceptibilities, Eq.~\eqref{chiMemoryMatrix}, has poles.
We find for the eigenfrequencies in the ballistic regime,
\begin{eqnarray}\label{omegaBall}
\omega_i^{\rm ballistic} &\approx& \omega_i -\frac{M_2\omega_2}{M_i\omega_i}\frac{\gamma}{2} - i\frac{M_2}{M_i}\frac{\Gamma}{2}
\end{eqnarray}
where we evaluated both the frequency shift (real part) and the decay rate (imaginary part) to lowest order in the interaction strength $a$.
Both the frequency shift and the decay rate are much smaller than $\delta\omega$ in the regime where Eq.~\eqref{omegaBall} is valid.
For low temperatures, $T \ll \epsilon_{F,1}$, one can use Eq.~\eqref{lowTlimits} to obtain for the frequency shift of the order of
\begin{equation}\label{omegashiftSmalla}
\Delta \omega_i \sim \frac{N_2}{N_i}\, k_{F,1} a\, \bar\omega \ll \delta \omega
\end{equation}
while the decay rate of the oscillations is essentially given by the single-particle scattering rate,
\begin{eqnarray}
\frac{1}{\tau_{\rm osc,i} }
&\sim& \frac{N_2}{N_i} \frac{(k_B T)^2}{\hbar^2\, \epsilon_{F,1}}\, (k_F a)^2 \ll \delta \omega.
\end{eqnarray}
For high temperatures, $T\gg\epsilon_{F,1}$, the frequency shift drops faster than the decay rate and is therefore difficult to observe.
In appendix~\ref{appendix-sigma-singular} we show that in the ballistic regime the memory matrix formalism does not reproduce a dephasing of oscillations which gives rise to an extra effective decay rate linear in the scattering length
$a$. This failure of the approach can be traced back to the fact that in the limit $a \to 0$ an infinite set of further slow modes exists which we did not include into the set of slow modes $A_n$, see appendix~\ref{appendix-sigma-singular} for details.
\begin{figure}
\includegraphics[width=\columnwidth]{realtimeplots.pdf}
\caption{\label{figRealTime}
Response to a constant displacement $\vec r_1^0(t<0)=r_0\hat{\vec e}_x$ of the trap potential for species $1$ that is switched of suddenly at time $t=0$.
The solid black (dashed blue) line shows the expectation value $\langle R_1^x(t)\rangle$ ($\langle R_2^x(t)\rangle$) of the center position of the first (second) atomic cloud, respectively, see Eq.~\eqref{realTime}.
The calculations were done for $N_1=N_2$, $m_1=m_2$, $\frac{\delta\omega}{\bar\omega}=0.1$ and $\gamma$ and $\Gamma$ as specified for each case.
For high friction $\Gamma$ (last plot), the oscillations of the two species synchronize quickly despite the finite difference $\delta\omega$ of the respective trapping frequencies, and the remaining COM oscillation decays only slowly on the time scale $\Gamma/\delta\omega^2$, see Eq.~\eqref{omegaFdd}.
}
\end{figure}
In a cold-atom experiment, one can directly observe the response of the clouds in real time.
From the theory side, the real-time response can be obtained by Fourier transformation of the susceptibility, Eq.~(\ref{chiMemoryMatrix}).
We consider the following setup: for time $t<0$ a constant force is applied to the first species.
Equivalently, we set in the Hamiltionian (Eq.~\eqref{hamiltonian}), $\vec r^0_2(t)=0$ and $\vec r^0_1(t)=\vec r_0= r_0 \hat{\vec e}_x$ with $r_0>0$ for $t<0$ ($\hat{\vec e}_x$ is the unit vector in $x$ direction).
The force is suddenly switched off, $\vec r^0_i(t)=0$, for $t\ge 0$.
In Fig.~\ref{figRealTime} the expectation value
\begin{eqnarray}
\langle \vec R_i(t)\rangle= M_1 \omega_1^2\vec r_0 \int_{-\infty}^0 \chi_{i1}(t-t') dt' \label{realTime}
\end{eqnarray}
is plotted as a function of time for both species, $i=1,2$.
In Fig.~\ref{figRealTime}A an example from the ballistic regime is shown.
Due to the finite interactions the two modes couple and a beating pattern emerges which is characteristic for the superposition of the two frequencies $\omega_1$ and $\omega_2$.
All oscillations decay on a time scale set by $1/\Gamma$.
\subsection{Frictionless drag regime}
In the ballistic regime, the approximate symmetry which protects COM oscillations is of no relevance. This is different
in cases where interactions are sufficiently strong so that the first species drags the second one either directly
by the Hartree potential (frictionless drag) or by dissipative processes (friction dominated drag).
In these regimes, the eigenmodes are characterized by a long-lived mode of COM oscillations, where both atomic clouds oscillate in parallel, and a mode of relative oscillations, which decays more quickly.
If all particles synchronize their oscillation, then one can expect that the COM mode is approximately described by undamped oscillations of a rigid body of total mass $M_{\rm tot}=M_1+M_2$ oscillating in an effective potential $\frac{1}{2} (M_1 \omega_1^2+M_2 \omega_2^2)r^2$.
The oscillation frequency in this limit is given by
\begin{equation}\label{omegaCom0}
\omega_{\rm COM}^{(0)} = \sqrt{\frac{M_1 \omega_1^2+M_2 \omega_2^2}{M_{\rm tot}}}.
\end{equation}
We first consider the frictionless drag regime, which is reached when the frequency shift described by Eq.~\eqref{omegaBall} becomes larger than the difference $\delta\omega$ of the trapping frequencies, $|\gamma|\gg \delta\omega$, and at the same time interaction effects between the two species are dominated by the effective potential rather than scattering, i.e., $|\gamma| \gg \Gamma$.
For low $T$, this regime is obtained for $\frac{\delta \omega}{\omega} \ll k_F a \ll \frac{\hbar \omega \, \epsilon_{F1}}{(k_B T)^2}$.
In this frictionless drag regime we can use perturbation theory in $\delta \omega$ to calculate the frequency shift and lifetime of the COM oscillations. We obtain
\begin{equation}\label{omegaFld}
\omega_{\rm COM} \approx \omega_{\rm COM}^{(0)} + \frac{2M_1^2M_2}{M_{\rm tot}^3} \left(\frac{\delta\omega^2}{\gamma} - i \frac{\delta\omega^2\Gamma}{\gamma^2}\right).
\end{equation}
As expected, $\omega_{\rm COM} \to \omega_{\rm COM}^{(0)}$ for large $\gamma\sim a$, as the increasing drag effect causes the two atomic clouds to oscillate more and more in parallel despite the small difference $\delta\omega$ of their trapping frequencies.
Defining $\Delta \omega$ by the shift relative to $\omega_{\rm COM}^{(0)}$, we obtain for low $T$
\begin{equation}
\Delta \omega \sim \frac{ N_2}{N_{1}} \frac{1}{ k_F a}\frac{\delta\omega^2}{\omega} \ll \delta \omega
\end{equation}
where we used again Eq.~\eqref{lowTlimits}.
While the frequency shift is proportional to $1/k_F a$, the lifetime turns out to be independent of the interaction strength in this regime,
\begin{equation}
\frac{1}{\tau_{\rm COM}} \sim \frac{ N_2}{N_{1}} \left( \frac{\delta\omega}{\omega}\right)^2 \frac{(k_B T)^2}{\hbar^2\, \epsilon_{F,1}} \ll \Delta \omega \ll \delta \omega
\end{equation}
Note that both $\Delta \omega$ and $\frac{1}{\tau_{\rm COM}} $ are proportional $(\delta \omega)^2$ as
frequency shift and decay only arise from the small contributions violating the symmetry which approximately protecs COM oscillations.
For completeness, we mention that the complex frequency of the mode of relative oscillations \cite{bruun} is given by
\begin{equation}
\omega_{\rm rel} \approx \omega_{\rm COM}^{(0)} - \frac{M_{\rm tot}}{2M_1} (\gamma + i\Gamma). \label{decayRelative}
\end{equation}
This mode is damped by the single-particle relaxation time $\Gamma$ and obtains a large frequency shift of the order of $k_F a \, \omega$ for low $T$. As discussed above, the formula above ignores extra dephasing effects, see Appendix \ref{appendix-dephasing}.
In Fig.~\ref{figRealTime}B the real-time response is shown in the frictionless drag regime using again Eq.~(\ref{realTime}).
Due to the strong repulsive interactions the two clouds repel each other such that $\langle R_1^x(t\leq0)\rangle$ is larger than $r_0$ and $\langle R_2^x(t\leq0)\rangle$ is negative.
After the external force has been switched off at $t=0$, the first cloud moves towards the center, first pushing the second cloud further away.
After some time, the relative motion of the two clouds has decayed, the oscillations lock into each other and only the COM oscillations remain.
The decay of the latter is given by the tiny rate $\sim \Gamma ({\delta \omega}/{\gamma})^2$, see Eq.~(\ref{omegaFld}), due to the approximate symmetry.
\subsection{Friction dominated drag regime}
Experimentally, the most important regime is perhaps the hydrodynamic regime, where friction dominates, $\Gamma \gg |\gamma|, \delta \omega$.
For $k_B T \ll \epsilon_{F1}$, this condition is fulfilled for $k_F a \gg \frac{\hbar \omega \, \epsilon_{F,1}}{(k_B T)^2}$ and $k_F a \gg \sqrt{\frac{\hbar\delta \omega \, \epsilon_{F1}}{(k_B T)^2}}$, which is, e.g., realized with realistic experimental parameters of $k_{F,1}a\approx 0.2$, $N_1 \approx N_2 \approx 10^6$, $\frac{\delta\omega}{\bar\omega} \approx 0.1$, and $k_B T \approx 0.3\epsilon_{F,1}$ (the Fermi energies are given by $\epsilon_{F,i}=\hbar\omega_i(6N_i)^{1/3}$).
Note that this regime is always reached in the thermodynamic limit defined by $N_i \to \infty$, $\omega_i \to 0$ with $\epsilon_{F,i} = const$.
Furthermore, we demand as above that $ k_F a \ll 1$.
The complex eigenfrequency of the COM mode is again obtained from perturbation theory in $\delta \omega$ and has the form
\begin{equation}\label{omegaFdd}
\omega_{\rm COM} \approx \omega_{\rm COM}^{(0)} + \frac{2M_2^3}{M_{\rm tot}^3} \left(\frac{\delta\omega^2\gamma}{\Gamma^2} - i \frac{\delta\omega^2}{\Gamma}\right).
\end{equation}
Similar to the frictionless drag regime, interaction effects are suppressed for large $\Gamma$ as the friction synchronizes the oscillations of the two atomic clouds. For low $T$ we obtain the decay rate
\begin{equation}
\frac{1}{\tau_{\rm COM}} \sim \frac{1}{(k_{F,1} a)^2} \left( \frac{\hbar \delta\omega}{k_B T}\right)^2 \frac{\epsilon_{F,1}}{\hbar} \ll \delta \omega
\end{equation}
The frequency shift is in this regime much smaller than the decay rate,
\begin{equation}
\Delta \omega \ll \frac{1}{\tau_{\rm COM}}
\end{equation}
and therefore difficult to observe.
For low $T$ one obtains $\Delta \omega \sim \frac{\hbar^2 (\delta \omega)^2 \epsilon_{F1}^2 \bar\omega}{(k_B T)^4 (k_{F,1} a)^3}$.
Fig.~\ref{figRealTime}C demonstrates how efficient a large friction is to lock the motion of the two clouds into each other on a time scale set by $1/\Gamma$.
After this microscopic time-scale, only the center of mass oscillations remain, which decay very slowly on the time scale $\Gamma/(\delta \omega)^2$ , see Eq.~(\ref{omegaFdd}).
The motion of the two clouds is locked perfectly into each other.
\section{Numerical results\label{sectionNumericalResults}}
\subsection{Protocols and possible experimental setups}
Depending on the setup of the cold-atom experiment, there exist various possibilities to access the different physical regimes described in section \ref{sectionAnalyticResults}.
First, by changing the cooling protocol, it is possible to access a broad range of temperatures. Second, by using an Feshbach resonance one can tune the scattering length.
Third, if one has an experimental realization where the trapping potential of the two species can be varied independently, one can directly tune $\delta \omega$.
In Fig.~\ref{fig_parameterspace} we show how each of these methods leads to a different trajectory in the parameter space spanned by $\gamma/\delta \omega$ and $\Gamma/\delta \omega$.
To illustrate the various regimes, we will plot in the following sections, Figs. \ref{fig_tscan}, \ref{fig_ascan}, and \ref{fig_dwscan}, the imaginary part of
\begin{eqnarray}\label{chicom}
\chi_{\text{COM}}(\omega) = (1,1,0,0)\,\chi(\omega)\,(1,1,0,0)^T.
\end{eqnarray}
This describes the response of the center of mass to forces acting on both species simultaneously.
Experimentally, the susceptibility as function of frequency can, e.g., be obtained by observing the real-time dynamics followed by a Fourier transformation.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{parameterspace.pdf}
\caption{\label{fig_parameterspace}
(Color online) Depending on which one of the quantities $\delta\omega$, $|\gamma|$, and $\Gamma$ is largest, the system is either in the ballistic regime (B), the frictionless drag regime (FLD) or the friction dominated drag regime (FDD).
The arrows show the trajectories of the system in the parameter space when the temperature, the interaction strength $a$, or the difference $\delta\omega$ of the trapping frequencies are increased.
All trajectories are calculated for $N_1=N_2=10^6$ and $m_1=m_2$.
A) $\frac{\delta\omega}{\bar\omega}=0.1$ ($0.01$), $k_{F,1}a=0.13$ ($0.06$), and $\frac{k_B T}{\epsilon_{F,1}}=0.05\ldots10$ ($0.07\ldots30$) for the solid black (dashed purple) trajectory, respectively.
B) $\frac{\delta\omega}{\bar\omega}=0.1$ ($0.01$), $\frac{k_B T}{\epsilon_{F,1}}=0.2$ ($0.1$), and $k_{F,1}a$ runs from $0$ to $0.025$ ($0.12$) for the solid black (dashed purple) trajectory, respectively.
C) $k_{F,1}a=0.1$ ($0.02$), $\frac{k_B T}{\epsilon_{F,1}}=0.2$ ($0.1$), and $\frac{\delta\omega}{\bar\omega}=0.03\ldots0.4$ ($0.002\ldots0.03$) for the solid black (dashed purple) trajectory, respectively.
}
\end{figure}
\begin{figure}[tbp]
\centering
\includegraphics[width=\columnwidth]{tscan.pdf}
\caption{\label{fig_tscan}
(Color online) Imaginary part of $\chi_{\text{COM}}(\omega)$, as defined in Eq.~\eqref{chicom}, as a function of temperature for attractive (left panels) and repulsive (right) interactions. The upper (lower) panels correspond to the solid (dashed) trajectories in Fig.~\ref{fig_parameterspace}A, respectively.
Dashed vertical lines indicate crossover temperatures where $\Gamma=|\gamma|$ or $\Gamma=\delta\omega$, horizontal dotted lines are the analytical predictions of Eqs.~\eqref{omegaBall}, \eqref{omegaCom0}, and \eqref{omegaFld} using Eqs.~\eqref{lowTlimits}--\eqref{highTlimits}.
For the top panels, the system evolves with increasing temperature from the ballistic (B) to the friction dominated drag (FDD) and back to the ballistic regime, while for the lower panels, the frictionless drag regime (FLD) is reached at low $T$.
Parameters: $N_1=N_2=10^6$, $m_1=m_2$, $\frac{\delta\omega}{\bar\omega}$ and $k_{F,1}a$ as stated above each plot.
}
\end{figure}
\begin{figure}[tbp]
\centering
\includegraphics[width=\columnwidth]{decayrates.pdf}
\caption{\label{fig_decayrates}
(Color online) Resonance frequencies $\omega_{\text{res}}$ and decay rates $1/\tau$ of the two eigenmodes of the system as a function of temperature, calculated from the real and imaginary part of the eigenvalues of $\Omega-i\Sigma$, respectively, see Eq.~\eqref{chiMemoryMatrix}.
In all four panels, the blue graph corresponds to the mode with longer life-time $\tau$.
Dashed vertical lines indicate crossover temperatures between the ballistic (B), the frictionless drag (FLD), and the friction dominated drag (FDD) regime.
The left (right) column corresponds to the top (bottom) panel in the left column of Fig.~\ref{fig_tscan} and to the solid (dashed) trajectory in Fig.~\ref{fig_parameterspace}A, respectively.
Parameters: $N_1=N_2=10^6$, $m_1=m_2$.
}
\end{figure}
\subsection{Increasing the temperature}
While $\gamma$ decreases monotonically as a function of temperature, $\Gamma$ vanishes for both $T\to0$ and $T\to\infty$ and has a maximum at $k_B T \sim \epsilon_{F,1}$, see Fig.~\ref{fig_gammaplot} and Eqs.~\eqref{lowTlimits}--\eqref{highTlimits}.
Therefore, two scenarios are possible when one increases $T$ while keeping all other parameters constant.
If interactions are weak, $k_{F,1}|a| \ll \frac{\delta\omega}{\bar\omega}$ (solid black trajectory in Fig.~\ref{fig_parameterspace}A), then the system is in the ballistic regime for low temperatures, may reach the friction dominated drag regime at intermediate temperatures $k_B T\sim \epsilon_{F,1}$ provided that $k_{F,1} |a|\gg \sqrt{\frac{\hbar\delta\omega}{\epsilon_{F,1}}}$ and returns to the ballistic regime for high temperatures.
If, on the other hand $k_{F,1}|a| \gg \frac{\delta\omega}{\bar\omega}$ (dashed purple trajectory in Fig.~\ref{fig_parameterspace}A), then the system is in the frictionless drag regime at low temperatures.
Increasing the temperature to the order of the Fermi energy will typically drive the system into the friction dominated drag regime unless $k_{F,1}|a| \ll \hbar\bar\omega/\epsilon_{F,1} \sim N_1^{-1/3}$.
At high temperatures, the ballistic regime is always realized.
For a quantitative analysis, we consider two concrete systems corresponding to the two trajectories in Fig.~\ref{fig_parameterspace}A.
In both cases, $N_1=N_2=10^6$ and $m_1=m_2$.
In the first system, $\frac{\delta\omega}{\bar\omega}=0.1$ and $k_{F,1}|a|=0.13$, while in the second case we use $\frac{\delta\omega}{\bar\omega}=0.01$ and $k_{F,1}|a|=0.06$.
We evaluate the integrals in Eqs.~\eqref{gammaexact} and \eqref{Gammaexact} numerically for these two systems.
Due to the spherical symmetry of the dispersion relation and the trapping potentials, the 12-dimensional integral in Eq.~\eqref{Gammaexact} can be reduced to a five-dimensional integral, which we evaluate using a Monte Carlo integration.
The different regimes can clearly be identified in plots of $\text{Im}[\chi_{\text{COM}}(\omega)]$, Eq.~(\ref{chicom}), shown in Fig.~\ref{fig_tscan}, describing excitations of the COM motion. The vertical dashed lines in Fig.~\ref{fig_tscan}
correspond to the crossovers from one regime to the other, see Fig.~\ref{fig_parameterspace}A, while the horizontal dotted lines give the analytical predictions for oscillation frequencies.
The upper two (lower two) plots in Fig.~\ref{fig_tscan} correspond to the solid (dashed) line in Fig.~\ref{fig_parameterspace}A. On the left side, we consider attractive, on the right side repulsive interactions.
The ballistic regime is characterized by the presence of two peaks: the two clouds oscillate independently with different frequencies. In contrast, a single peak located approximately at $\omega_{\rm COM}\approx \bar \omega$
characterizes the two drag regimes where the oscillation of the two clouds synchronizes. A second (much broader) mode describing relative oscillations does not show up
as for $\chi_{\rm COM}(\omega)$ we only consider a situation where both clouds are displaced
in the same direction (see Fig.~\ref{fig_decayrates} for a plot of both resonance frequencies as a function of temperature).
Note that for the chosen paramters, the system is not very deep in the ballistic regime for low $T$. This does not only lead to considerable shifts of the oscillation frequencies (see below) but also affects the weight of the two modes: the mode which is in frequency closer to $\omega_{\rm COM}^{(0)}$ clearly dominates.
In the low-temperature ballistic regime, the interactions increase (decrease) the oscillation frequencies as the curvature of the potential increases (decreases) due to the attractive (repulsive) interaction with the other species, respectively.
Interestingly, the effect is opposite for the drag-dominated regimes, best visible for the low-temperature regime in the lower two panels of Fig.~\ref{fig_tscan}.
This higher-order effect, well described by our analytical formulas (\eqref{omegaFld} and \eqref{omegaFdd}, drawn as dotted lines in Fig.~\ref{fig_tscan}),
arises from level repulsion from the mode of relative oscillations.
Fig.~\ref{fig_decayrates} shows the (real part of the) resonance frequencies and the decay rates $1/\tau$ of both eigenmodes of the system for the two cases corresponding to the left panels in Fig.~\ref{fig_tscan}. The maxima of the decay rate of the long-lived mode (lower curve in the lower panels of Fig.~\ref{fig_decayrates}) trace the crossover from one regime to the next. The minimum in the friction dominated drag regime, where $1/\tau_{\text{COM}}$ is proportional to the inverse of the single-particle scattering rate $\Gamma$, thereby arises from the maximum of $\Gamma$ displayed in Fig.~\ref{fig_gammaplot}.
While, in the ballistic regime, both modes have a long life time, in the drag regimes, only one long-lived mode remains and the decay rate of the mode of relative oscillations shoots up.
\begin{figure}[tbp]
\centering
\includegraphics[width=\columnwidth]{ascan.pdf}
\caption{\label{fig_ascan}
(Color online) $\text{Im}[\chi_{\text{COM}}(\omega)]$, Eq.~\eqref{chicom}, as a function of the scattering length $a$ (corresponding to solid line in Fig.~\ref{fig_parameterspace}B).
Dashed vertical lines indicate the scattering length where $\Gamma=\delta\omega$ and separate the ballistic (B) from the friction dominated drag regime (FDD).
The dotted lines are analytic predictions of the eigenfrequencies based on Eqs.~\eqref{omegaBall} and \eqref{omegaFdd}, where we used the low-temperature limit, Eq.~\eqref{lowTlimits}, for the values of $\gamma$ and $\Gamma$.
Parameters: $N_1=N_2=10^6$, $m_1=m_2$, $k_B T=0.2\epsilon_{F,1}$, $\frac{\delta\omega}{\bar\omega}=0.1$.
}
\end{figure}
\subsection{Increasing the interaction strength}
As $\gamma\propto a$ and $\Gamma\propto a^2$, the system evolves on a parabola in the parameter space of Fig.~\ref{fig_parameterspace}B when the interaction strength is increased.
While for weak interactions, the ballistic regime and for strong interactions the friction dominated drag regime is always realized, the frictionless drag regime is only reached if
$\frac{\delta\omega}{\bar\omega}(k_BT/\epsilon_{F,1})^2 N_1^{1/3} \lesssim 0.03$.
Fig.~\ref{fig_ascan} shows numerical results for $\text{Im}[\chi_{\text{COM}}(\omega)]$, c.f.~Eq.~\eqref{chicom}, for the solid black trajectory from Fig.~\ref{fig_parameterspace}B.
Dotted lines are again analytic results of the eigenfrequencies.
The analytic prediction overestimates the slopes of the eigenfrequencies in the ballistic regime since it was made based on the $T\to0$ limit of $\gamma$ given in Eq.~\eqref{lowTlimits}, while the actual value of $\gamma$ at $k_BT=0.2\epsilon_{F,1}$ is by a factor of $0.62$ smaller.
\begin{figure}[tbp]
\centering
\includegraphics[width=\columnwidth]{dwscan.pdf}
\caption{\label{fig_dwscan}
(Color online) $\text{Im}[\chi_{\text{COM}}(\omega)]$, Eq.~\eqref{chicom}, as a function of the frequency difference $\delta\omega$ in the case $(k_BT/\epsilon_{F})^2 \,k_{F}|a| \,N_1^{1/3}=0.4>0.07$ (solid black trajectory in Fig.~\ref{fig_parameterspace}C).
At the dashed vertical line, $\delta\omega=\Gamma$, separating the friction dominated drag regime (FDD) from the ballistic regime (B).
The dotted lines are analytic predictions of the eigenfrequencies based on Eqs.~\eqref{omegaBall} and \eqref{omegaFdd}, where we used the low-temperature limit, Eq.~\eqref{lowTlimits}, for the values of $\gamma$ and $\Gamma$.
Parameters: $N_1=N_2=10^6$, $m_1=m_2$, $k_F a=\pm0.1$, $k_B T=0.2\epsilon_F$.
Here, $\epsilon_F$ and $k_F$ denote the Fermi energy and wave vector evaluated at $\delta\omega\to0$, respectively.
}
\end{figure}
\subsection{Increasing the frequency difference $\delta\omega$}
Since $\gamma$ and $\Gamma$ depend only weakly on $\delta\omega$ for $\frac{\delta\omega}{\bar\omega}\ll1$, the trajectories for increasing $\delta\omega$ in Fig.~\ref{fig_parameterspace}C are almost straight lines crossing at the origin.
For low temperatures and small $\delta\omega$, the friction dominated (frictionless) drag regime is realized if $(k_BT/\epsilon_{F,1})^2 \,k_{F,1}|a| \,N_1^{1/3}$ is larger (smaller) than $0.07$, respectively.
For large $\delta\omega$ (and weak interactions), the system enters the ballistic regime.
Fig.~\ref{fig_dwscan} shows numerical results for $\text{Im}[\chi_{\text{COM}}(\omega)]$, c.f.~Eq.~\eqref{chicom}, corresponding to the solid black trajectory in Fig.~\ref{fig_parameterspace}C.
Dotted lines are analytic predictions of the eigenfrequencies based on the low-temperature limit, Eq.~\eqref{lowTlimits}.
\section{Conclusions}
The presence of approximate symmetries leads to a slow equilibration of a perturbed system.
We suggest that this physics can be studied with high precision experimentally by investigating
the center of mass oscillations of two species of ultracold atoms with different but similar mass.
Alternatively, one can also investigate, e.g., two spin species with the same mass but slightly different harmonic confinement. The mass difference and/or difference in the strength of the parabolic potential
breaks a dynamical symmetry which otherwise protects the center-of-mass oscillations from decay.
The interactions of the two species synchronizes the motion of the two clouds and thereby leads to a partial restoration of the dynamical symmetry: the interacting liquid can approximately be viewed as
having a single average mass and oscillating in a single average potential. As a consequence, the decay rate of the center-of-mass oscillations is strongly reduced and of the order of $\frac{(\delta \omega)^2}{\Gamma}$, where $\delta \omega$ is the difference of the trapping frequencies and $\Gamma$ the scattering-rate of the two species. Compared to other hydrodynamic modes (which can also have decay rates proprotional to the inverse of $\Gamma$) one obtains an extra reduction by the factor $(\delta \omega/\omega)^2$.
As all other modes have much faster decay rates, the approximate symmetry leads to an almost perfect drag of the two clouds: the center-of-masses for each of the two species follow each other
after a few scattering times.
For future investigations two directions are especially interesting: First, one can study the highly non-linear regime, where, for example, initially one species is separated far from the second one and one can study the evolution of the center-of-mass oscillations after the two clouds have violently crashed into each other in a setup similar to the one studied by the Zwierlein group \cite{sommer_universal_2011}. Second, one can investigate the interplay of superfluidity and the approximate symmetry, which is of direct relevance for the experiments of the Salomon group \cite{ferrierbarbut_mixture_2014}. Here, in the center of the cloud and for small relative velocities, the superfluid components move without friction and only the normal components can scatter from each other.
\begin{acknowledgments}
We acknowledge useful discussions with E. Demler, J. Lux and C. Salomon and financial support from the DFG (SFB TR 12) and from Deutsche Telekom Stiftung.
\end{acknowledgments}
|
{
"timestamp": "2015-04-17T02:10:39",
"yymm": "1504",
"arxiv_id": "1504.03375",
"language": "en",
"url": "https://arxiv.org/abs/1504.03375"
}
|
\section{Introduction}
The positron upon annihilation with its anti-particle, the electron, yields unique information about the electronic structure of bulk materials \cite{review_tdft,Tuomisto} and nanostructures \cite{Eijt}.
The electron-positron density functional theory (DFT) \cite{tdft} is used in order to obtain
precise knowledge of the positron wave function and its overlap with the electron orbitals.
The powerful combination of positron annihilation spectroscopy (PAS) and DFT calculations provides
a highly accurate method for advanced characterization of materials \cite{bba_review}.
Within the DFT framework, the generalized gradient approximation (GGA) method to describe electron-positron correlation effects in solids has shown a systematic improvement over the local density approximation (LDA) for positron affinities and annihilation characteristics \cite{gga1,gga2,campillo,ggaboro}.
Until now, a dimensional analysis has been used to determine the form of the lowest-order gradient correction with a semi-empirical coefficient $\alpha$.
So far, the pragmatic approach has been to fit $\alpha$ to large databases of positron lifetimes.
Recently, both the LDA and the GGA \cite{gga4,gga5} have been improved on the basis of new quantum Monte Carlo (QMC) data for the electron-positron correlation problem in a homogenous electron gas (HEG) \cite{qmc}.
However, one could claim that such good fits may be in some cases accidental \cite{stachowiak}.
Moreover, the present gradient corrections may also lead to some unphysical effects in the electron-positron correlation potential near the nuclei; namely, its too large oscillations due to the shell structure of core electrons.
Therefore, here we propose to improve the GGA by extracting and deducing the $\alpha$ parameter from more fundamental physical principles.
This more reliable derivation of $\alpha$ also reveals a gentle dependence of the local density reducing the gradient correction near the nuclei.
Thus, $\alpha$ becomes a function of the local density as well.
In the case of the positron immersed in an electron
gas the Coulomb attraction produces a cusp in the electron
density at the positron site.
The correlation potential describing the positron perturbation represents the electronic polarization due to the positron screening and can be obtained via the Hellmann-Feynman theorem using coupling-constant integration as follows \cite{hodges73}
\begin{equation}
V_{c}(\bm{r}) = -\int_{0}^{1}\!\text{d}Z \int\!\text{d}^3\!\bm{R}\,\,
\frac{\rho(\bm{R})\,[g(\bm{r},\bm{R},Z){-}1]}{|\bm{r-R}|} ~,
\end{equation}
where $\rho(\bm{R})\,[g(\bm{r},\bm{R},Z){-}1]$ is the screening cloud density around a positive particle with charge $Z$ ($g(\bm{r},\bm{R},Z)$ is the particle-electron pair distribution function).
The effect of the density gradient on the correlation energy can be deduced
from the distortion of the polarization cloud due to this gradient.
For this purpose, one can use the dynamical structure factor
$S(\bm{q},\omega)$ \cite{Lindhard,Abbamonte} of the HEG to show that in the high density limit the lowest order gradient correction is proportional to the parameter
$\epsilon=(|\nabla \ln \rho|/q\TF)^2$ (which depends on the ratio of
the Thomas-Fermi length $\lambda\TF = 1/q\TF$ and the inhomogeneity length $1/|\nabla \ln \rho|$).
This correction is given by the expression
\begin{equation}
\Delta V_{c}(\bm{r})= \beta \frac{\epsilon(\bm{r})}{16} ~,
\label{eqcg}
\end{equation}
where the constant $\beta=0.066725$ Hartree is linked to the coefficient of the term $q^2$ in the density response function wave vector expansion.
The coefficient $\beta$ has been calculated by Ma and Brueckner \cite{MB} and has been used by various authors \cite{LM1,LM2,ePBEXC}.
Eq. (\ref{eqcg}) is in fact similar to that used to compute the correlation energy \cite{ePBEXC} for an electron gas with slowly varying density
\footnote{The difference originates from the extra screening in the presence of two types of carriers which produces a rescaling by a factor of 1/4 in the gradient correction coefficient \cite{MB}.
}.
In order to interpolate to the case of rapid density variations (i.e. large $\epsilon$),
we use the formula
\begin{equation}
V_c = \VLDA \exp(-\alpha\:\!\epsilon/3) ~,
\end{equation}
from Ref. \cite{gga1} [see Eq. (7) there].
This formula is based on the scaling relation for the correlation potential, as derived by Nieminen and Hodges \cite{scalingrel}.
But $\alpha$ is now a function of the local density (and thereby position).
When we identify the first order expansion in $\epsilon$ with the Ma and Brueckner's result shown above, we find that
\begin{equation}
\alpha(\bm{r}) = - \frac{3}{16}\, \frac{\beta}{\VLDA(\bm{r})} ~.
\label{PFalpha}
\end{equation}
The quantity $\alpha$ remains a gentle function of the density in the valence electron region and at low density it becomes very close to $0.05$ \footnote{Since $V_c$ approaches 1/4 Hartree for small densities, $\alpha \approx 3\beta/4 \doteq 0.05$ if Hartree units are used for $\beta$.} -- a value found earlier within the empirical GGA \cite{gga4,gga5}.
Interestingly, $\alpha$ happens also to be of the same order as the fraction $Z_c$ of an electron displaced in electron-electron correlation effects which is typically of the order of 1/20 of the electron charge \cite{coulson,bba89}.
Like the potential $V_c$, the positron annihilation rate depends on electron-positron correlation effects and must be enhanced over the independent particle model.
The electron-positron enhancement theory \cite{Makkonen14}
has some features in common with the interaction between a core hole and the
conduction electrons treated both in X-ray emission \cite{carbotte68} and in resonant
inelastic X-ray scattering \cite{hancock14}.
We can relate the correlation energy to the annihilation rate by using the scaling relation \cite{scalingrel}.
Therefore, one obtains an electron-positron enhancement annihilation factor $\gamma$ given by
\begin{equation}
\gamma-1 = (\gLDA-1)\,\exp(-\alpha\:\!\epsilon) ~,
\end{equation}
The enhancement term $\gamma$ is used to calculate the total positron annihilation rate or the inverse lifetime $1/\tau$, which is expressed through the simple relation \cite{bba_review}
\begin{equation}
\frac{1}{\tau}=
\pi r_0^2 c\, \int \text{d}^3\bm{r}\,\gamma(\bm{r})\, \rho(\bm{r}) \,
|\psi_+(\bm{r})|^2 \,,
\end{equation}
where $r_0$ is the classical radius of the electron, $c$ is the speed of light
and $\psi_+(\bm{r})$ is the ground state positron wave function.
In this work, we have used the same accurate computational method described in Refs.
\cite{gga4, gga5}.
Electronic structure calculations for selected materials were carried out using the self-consistent WIEN2k code \cite{wien2k}, which imposes no shape restrictions for the electron density and the potential, while the positron wave function and energy were obtained using a Schr\"odinger equation solver based on a finite difference method.
The exchange-correlation potential for the electrons contains gradient corrections within the scheme proposed by Perdew, Burke and Ernzerhof \cite{ePBEXC}
except in the case of the 4$d$ and 5$d$ elemental metals since some of their calculated properties (e.g. the lattice constant) become inappropriate when gradient corrections are used \cite{bmj90}.
GGA corrections introduce cusps in the electron potential, negligible in the LDA, which reflect the atomic shell structure \cite{bmj90}.
Numerical parameters of the WIEN2k code as well as of the positron solver were tested and optimized in order to obtain calculated positron lifetimes within a precision of 0.1 ps and positron affinities within 0.01 eV.
Here we consider only systems in which the positron density is approaching zero in the limit of an infinite crystal.
\begin{table}[htb]
\vspace*{-3mm}
\caption{Positron affinities (in eV) calculated according to various approaches:
GC = original gradient correction with the Arponen and Pajanne potential \cite{AP} ($\alpha=0.22$),
DB = Drummond {\em et al.} \cite{qmc},
DG = gradient correction with DB ($\alpha=0.05$),
PF = parameter-free gradient correction with DB (varying $\alpha$).
The last column gives experimental values taken from Refs. \cite{Gidley88,Jibaly91,weiss}.
The exceptions are C and Si (see Ref. \cite{gga4} and references therein) and
MgO (Ref. \cite{nagashima}).
In the case of MgO an upper limit is given (see the text).
}
\begin{ruledtabular}
\begin{tabular}[t]{@{\ }l l r r r r l}
System & Structure & GC & DB & DG & PF & Exp. \\
\hline\\[-2mm]
\multicolumn{7}{c}{Elements} \\
Li & bcc & $-7.31$ & $-7.02$ & $-6.95$ & $-6.96$ \\
C & diamond & $-1.33$ & $-2.40$ & $-1.87$ & $-1.93$ & $-2.0$ \\
Na & bcc & $-7.18$ & $-6.89$ & $-6.80$ & $-6.81$ \\
Al & fcc & $-4.21$ & $-4.04$ & $-4.00$ & $-4.01$ & $-4.1$ \\
Si & diamond & $-6.29$ & $-6.47$ & $-6.33$ & $-6.35$ & $-6.2$ \\
Fe & bcc & $-3.40$ & $-3.76$ & $-3.62$ & $-3.67$ & $-3.3$ \\
Cu & fcc & $-3.76$ & $-4.23$ & $-4.05$ & $-4.11$ & $-4.3$ \\
Nb & bcc & $-3.61$ & $-3.75$ & $-3.65$ & $-3.68$ & $-3.8$ \\
Ce & fcc, $\alpha$-Ce& $-4.11$ & $-4.16$ & $-4.07$ & $-4.09$ \\
Ce & fcc, $\gamma$-Ce& $-5.34$ & $-5.31$ & $-5.23$ & $-5.25$ \\
W & bcc & $-1.72$ & $-1.91$ & $-1.82$ & $-1.85$ & $-1.9$ \\
Pt & fcc & $-3.31$ & $-3.77$ & $-3.61$ & $-3.67$ & $-3.8$ \\
\multicolumn{7}{c}{Compounds} \\
MgO & rock salt & $-5.56$ & $-6.46$ & $-6.17$ & $-6.25$ & $-5.2$ \\
Cu$_2$O & cuprite & $-5.88$ & $-6.42$ & $-6.21$ & $-6.26$ \\
CeO$_2$ & fluorite & $-6.55$ & $-7.40$ & $-7.12$ & $-7.18$ \\
YBa$_2$Cu$_3$O$_6$ & tetragonal & $-6.11$ & $-6.65$ & $-6.42$ & $-6.46$ \\
YBa$_2$Cu$_3$O$_7$ & orthorhombic & $-6.02$ & $-6.78$ & $-6.52$ & $-6.58$ \\
PrBa$_2$Cu$_3$O$_7$ & orthorhombic& $-5.81$ & $-6.57$ & $-6.30$ & $-6.36$ \\
\end{tabular}
\end{ruledtabular}
\label{tabA}
\vspace*{-1mm}
\end{table}
\begin{table}[htb]
\caption{Positron lifetimes (in ps) calculated according to various approaches explained in the caption of Table \ref{tabA}.
The last column gives experimental values discussed in Refs. \cite{gga4,gga5}.
The last experimental values for cuprates are extracted from Refs. \cite{shukla,bba91}.
}
\begin{ruledtabular}
\begin{tabular}[t]{@{\ }l l r r r r l}
System & Structure & GC & DB & DG & PF & Exp. \\
\hline\\[-2mm]
\multicolumn{7}{c}{Elements} \\
Li & bcc & 283.2 & 303.8 & 316.2 & 313.5 & 291 \\
C & diamond & 102.8 & 94.6 & 98.9 & 97.7 & 98+ \\
Na & bcc & 337.7 & 343.0 & 364.4 & 360.5 & 338 \\
Al & fcc & 154.2 & 161.0 & 164.1 & 163.0 & 160+ \\
Si & diamond & 222.7 & 208.1 & 217.3 & 215.9 & 216+ \\
Fe & bcc & 109.6 & 102.1 & 106.5 & 104.7 & 105+\\
Cu & fcc & 120.0 & 107.4 & 113.3 & 110.9 & 110+ \\
Nb & bcc & 123.4 & 120.9 & 124.3 & 123.1 & 120+ \\
Ce & fcc, $\alpha$-Ce& 169.5 & 165.0 & 170.5 & 169.0 & 233 \\
Ce & fcc, $\gamma$-Ce& 196.8 & 194.1 & 200.6 & 198.9 & 235 \\
W & bcc & 102.7 & 100.6 & 103.4 & 102.3 & 105 \\
Pt & fcc & 105.2 & 97.4 & 101.3 & 99.8 & 99+ \\
\multicolumn{7}{c}{Compounds} \\
MgO & rock salt & 146.2 & 119.0 & 128.5 & 125.4 & 130 \\
Cu$_2$O & cuprite & 177.4 & 147.3 & 158.4 & 154.8 & $\sim$174 \\
CeO$_2$ & fluorite & 173.7 & 138.2 & 149.1 & 146.0 & $<$187 \\
YBa$_2$Cu$_3$O$_6$ & tetragonal & 224.5 & 175.4 & 190.8 & 186.5 & $\sim$190 \\
YBa$_2$Cu$_3$O$_7$ & orthorhombic & 179.2 & 142.4 & 154.0 & 150.5 & $\sim$165 \\
PrBa$_2$Cu$_3$O$_7$ & orthorhombic& 180.4 & 143.4 & 155.0 & 151.6 & $\sim$165 \\
\end{tabular}
\end{ruledtabular}
\label{tabL}
\vspace*{-1mm}
\end{table}
DFT provides an excellent description of the Si electronic structure both in the solid and liquid phases \cite{Okada12}.
It is therefore natural to start our tests of the parameter-free GGA positron potential in Si. A meaningful observable to check is the positron affinity $A$ defined as the sum of the electron and positron chemical potentials.
In the case of a semiconductor, the electron chemical potential is taken from the position of the top of the valence band.
Recently, Cassidy {\em et al.} \cite{Cassidy11} have shown that the temperature invariant time of flight (TOF) component for Ps emitted from the surface of $p$-doped Si(100) has a kinetic energy equal to $0.6$ eV.
This TOF feature is explained by a bulk positron picking up a valence band electron just beneath the surface to form Ps with a kinetic energy of $K=\EPs+A=0.6$ eV.
Therefore, the experimental affinity for Si can be deduced to be $A=-6.2$ eV.
When we use the GGA for both the electron and positron potentials, we find a theoretical value $A=-6.35$ eV, which is in excellent agreement with the value measured by Cassidy {\em et al.} while the corresponding LDA value shows a clear tendency to overestimate the magnitude of $A$.
This LDA problem can be traced back to the screening effects.
In the GGA, the value of $A$ agrees with the experiment by reducing the screening charge. Calculated positron affinities within LDA and GGA against the corresponding experimental values for different materials are shown in Table \ref{tabA}.
The trends follow those of Si, nevertheless the experimental values of $A$ are often of earlier date and not always reliable.
The corresponding positron lifetimes are presented in Table \ref{tabL}.
Clearly, the trends of the parameter-free GGA are very similar to the empirical GGA \cite{gga4,gga5}.
In particular, one of the best result is given by Al which was problematic in the original GGA scheme \cite{gga1}.
Positron lifetime measurements in Li and Na were performed before the advent of reliable spectrometers and fitting procedures, as discussed in detail in Ref. \cite{gga4}, and may be affected by significant errors.
However, in the present scheme the positron has a slightly larger overlap with the core electrons as illustrated in Figs. \ref{fig1} and \ref{fig2} for Si and Cu, respectively. Some noticeable jumps of $\epsilon$ shown in Fig.~\ref{fig1} (d)
and Fig.~\ref{fig2} (d) result in unphysically large local changes in the
empirical GGA correlation potential depicted in Fig. \ref{fig1} (c) and Fig. \ref{fig2} (c).
These problems are now cured by the variation of the function $\alpha$
in space illustrated by Fig.~\ref{fig1} (b) and Fig.~\ref{fig2} (b).
Interestingly, $\alpha$ given by Eq. (\ref{PFalpha}) seems to vary almost like the Thomas-Fermi length $\lambda\TF$ and becomes very small close to the nuclei.
Therefore, the cusps in the parameter free GGA correlation potential become more damped because of the reduction of the screening length in the core region.
This effect is further documented by $\exp(-\alpha\epsilon/3)$ factor plots (Fig. \ref{fig1} (d) and Fig. \ref{fig2} (d)) which define the reduction of the correlation potential in the core region.
The $\exp(-\alpha\epsilon/3)$ factor anticorrelates with the $\epsilon$ parameter;
i.e. a large inhomogeneity corresponds to a small exponential factor.
The variation of $\alpha$ in the core region should also improve the description of high-momentum annihilation spectra observed in coincidence Doppler broadening spectroscopy \cite{alatalo2,mijnarends98} and in angular correlation measurements \cite{Laverock}.
The positron annihilation lifetime (PAL) provides a way to detect very small amounts of vacancy-defects in crystalline materials.
Since thermalized positrons are trapped by vacancies before annihilating with electrons, their lifetime increases with respect the bulk values given the low electron density at the vacancy.
For this reason, PAL has been widely used to characterize doped semiconducting samples of silicon and other technological relevant materials \cite{Tuomisto}.
As shown by Table \ref{tabL} the positron bulk lifetime of Si is very well described by the present theory.
Therefore, deviations from the theoretical lifetime indicate the presence of imperfections in the sample.
In a post-silicon electronics era, engineered doping of oxide electronics, which is similar to conventional doping in semiconductor technology, offers much greater functionality including electronic control of redox chemistry with applications to batteries, photovoltaics and catalysis. In particular, a well characterized material is MgO, which is a simple binary oxide with rock-salt structure.
In MgO, a magnetic moment can arise from the unpaired 2p electrons at an oxygen site surrounding a cation vacancy with each nearest neighbor oxygen carrying a magnetic moment \cite{araujo10}.
This magnetic property can be fine tuned to optimize spintronics devices.
Concerning PAL studies, Tanaka {\em et al}. \cite{MgOdoping} have shown that MgO lifetime is significantly affected by Ga doping, which results in the creation of Mg vacancies.
However, when the number of Mg vacancies decreases the lifetime converges to the bulk value 130 ps \cite{MgOPAS}, which is in reasonable agreement with the present theory.
A reliable experimental TOF study of MgO \cite{nagashima} reports a Ps emission peak energy of $2.6$ eV.
Since Ps is already formed in the bulk of MgO, the kinetic energy is given in this case by
$K = \EPs + A - E_B + E_G$, where $E_B$ is the Ps binding energy inside the MgO matrix and $E_G=7.8$ eV is the energy gap of MgO.
Using our calculated affinity, we deduce that $E_B=5.75$ eV, which is consistent with typical values of Ps binding energy in the bulk \cite{kolkata}.
In fact, this value must be smaller than $\EPs$ because of screening effects in the bulk.
Ceria \cite{jarlborg14} is another oxide which has attracted considerable interest because of its applications in solid oxide fuel cells. It can be noted that by removing all the oxygen atoms, one recovers the fcc structure of Ce. Experimentally, positron seems only to detect the $\gamma$ phase of Ce because of its stronger affinity with respect to the $\alpha$ phase. Interestingly, the experimental ceria lifetime 189 ps \cite{ceriaLT1} appears to be much closer to theoretical value of $\gamma$-Ce rather than ceria. A possible reason for this discrepancy is that real samples can always contain patches of $\gamma$-Ce which strongly attract the positron because of their higher positron affinity. In this context, we should keep in mind that oxygen is very mobile in ceria.
\begin{figure}
\begin{center}
\includegraphics[width=8.7cm,clip]{Si1D.eps}
\end{center}
\caption{(Color online) One-dimensional profiles of (a) the electron density (including atomic orbitals), (b) the $\alpha$ parameter (and the Thomas-Fermi length $\lambda\TF$), (c) the positron correlation potential, and (d) the exponential factor $\exp(-\alpha\epsilon/3)$ (and $\epsilon$ parameter) along the [100] direction in Si for LDA (DB), the empirical (DG) and the parameter-free (PF) GGA approaches.
Si atoms are located at 0 and 10.26 au along [100].
$\lambda\TF$ and $\epsilon$ are shown for the purpose of observing correlations with corresponding quantities (the scales of $\lambda\TF$ and $\epsilon$ are different than those for $\alpha$ and $\exp(-\alpha\epsilon/3)$, respectively).}
\label{fig1}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=8.7cm,clip]{Cu1D.eps}
\end{center}
\caption{(Color online)
One dimensional profiles for Cu as explained in the caption of Fig. \ref{fig1}.
Cu atoms are located at 0 and 6.83 au along the [100] direction.
}
\label{fig2}
\end{figure}
As an example of advanced characterization, we now show that positron annihilation spectroscopy can be useful to understand the role of oxygen-related defects in high temperature superconductivity \cite{icpa16}.
In practice, by comparing the experimental lifetimes \cite{shukla} to an accurate theory it is possible to deduce that positrons are trapped at oxygen vacancies in the superconducting compound YBa$_2$Cu$_3$O$_{7{-}\delta}$ while this trapping becomes negligible in the non-superconducting compound where Y has been replaced by Pr.
When positrons become completely delocalized for temperatures higher than 400 K, the lifetime becomes almost identical in the YBa$_2$Cu$_3$O$_7$ and PrBa$_2$Cu$_3$O$_7$ compounds in agreement with our calculations reported in Table \ref{tabL}.
Moreover, the calculated lifetime in the tetragonal YBa$_2$Cu$_3$O$_6$ lattice is 36 ps longer than in the orthorhombic YBa$_2$Cu$_3$O$_7$.
Such difference is consistent with experiments \cite{bba91}.
Curiously, the calculated positron affinity seems to indicate that Ps is emitted with about $0.15$ eV higher kinetic energy from YBa$_2$Cu$_3$O$_6$ and PrBa$_2$Cu$_3$O$_7$ than from YBa$_2$Cu$_3$O$_7$.
Nevertheless, since the present DFT calculations fail in describing the insulating phase of YBa$_2$Cu$_3$O$_6$ and PrBa$_2$Cu$_3$O$_7$ we should take the positron affinity calculated values for these two compounds with caution.
In conclusion,
we have demonstrated that the parameter-free GGA truly provides a simple, yet accurate step beyond LDA.
It is also reassuring that the most reliable electron-positron LDA parametrization (based on the QMC simulations) combined with the parameter free gradient correction gives the best results compared with any of the older LDA potentials.
Further studies combining the present approach with well-converged momentum densities calculations \cite{ernsting14} are needed to check if first principle methods can soon improve the agreement over empirical approaches \cite{Laverock}.
\begin{acknowledgments}
We acknowledge fruitful discussions with A.P. Mills and Y. Nagashima.
B.B. is supported by the U.S. Department of Energy (USDOE) Contract No. DE-FG0207ER46352 and has benefited for computer time from Northeastern University's Advanced Scientific Computation Center (ASCC) and USDOE’s NERSC supercomputing center.
J.K. acknowledges the support by the IT4Innovations Centre of Excellence project (CZ.1.05/1.1.00/02.0070), funded by the European Regional Development Fund and the national budget of the Czech Republic via the Research and Development for Innovations Operational Programme, as well as Czech Ministry of Education, Youth and Sports via the project Large Research, Development and Innovations Infrastructures (LM2011033).
\end{acknowledgments}
|
{
"timestamp": "2015-04-15T02:01:20",
"yymm": "1504",
"arxiv_id": "1504.03359",
"language": "en",
"url": "https://arxiv.org/abs/1504.03359"
}
|
\section{Introduction}
Let $k$ be a number field, $\bar{k}$ a fixed algebraic closure, and $\ell$ a rational prime number. We let $G_k$ denote the absolute Galois group $\Gal(\bar{k}/k)$. We define two extensions of the function field $\bar{k}(t)$: $\tilde{M}$ is the maximal extension of $\bar{k}(t)$ which is unramified outside the places $t = 0, 1, \infty$, and $M$ is the maximal sub-extension of $\tilde{M}/\bar{k}(t)$ which is pro-$\ell$. Then the Galois group
\[ \pi := \Gal(M/\bar{k}(t)) \]
may be identified with the pro-$\ell$ algebraic fundamental group of $\P^1_{01\infty} \times_k \bar{k}$. From the tower of function fields $k(t) \subset \bar{k}(t) \subset M$, we have the induced short exact sequence
\[
\begin{tikzcd}
1 \ar{r} & \pi \ar{r} & \Gal\bigl( M/k(t) \bigr) \ar{r} & G_k \ar{r} & 1,
\end{tikzcd}
\]
where $\Gal(\bar{k}(t)/k(t)) \cong G_k$. For any element $\sigma \in G_k$, we may lift to an element in $\Gal(M/k(t))$, and use this lift to conjugate the normal subgroup $\pi$. We may associate to each $\sigma$ an automorphism of $\pi$; since the lift of $\sigma$ is only well-defined up to an element of $\pi$, the resulting automorphism is only defined up to an inner automorphism of $\pi$. This gives the \emph{outer canonical pro-$\ell$ Galois representation}:
\[ \Phi_\ell \colon G_k \longrightarrow \Out(\pi) := \Aut(\pi)/\Inn(\pi). \]
We let $\yama := \yama(k, \ell)$ denote the field fixed by the kernel of $\Phi_\ell$, and let $\ten := \ten(k, \ell)$ denote the maximal pro-$\ell$ extension of $k(\mathbf{\upmu}_{\ell^\infty})$ which is unramified away from $\ell$.
Let us explain the motivation behind this notation: The kanji character $\yama$ (pronounced \emph{yama}) means ``mountain,'' and the character $\ten$ (pronounced \emph{ten}) means ``heaven.'' It is known by work of Anderson and Ihara \cite{Anderson-Ihara:1988} that $\yama \subseteq \ten$. It is natural to ask in which cases $\yama$ and $\ten$ coincide -- Ihara first posed this question for $k = \mathbb{Q}$ in 1986, which may be loosely phrased as: \emph{When does the mountain ($\yama$) reach the heavens ($\ten$)}?
The recent result of Francis Brown \cite{Brown:2012}, which demonstrates the Deligne-Ihara conjecture for any prime $\ell$, implies that $\yama(\mathbb{Q}, \ell) = \ten(\mathbb{Q}, \ell)$ for any odd regular prime $\ell$. This implication was demonstrated in an earlier work by Sharifi \cite{Sharifi:2002}; the interested reader may also find further details in the article \cite{Ihara:2002}.
Let $A/k$ be an abelian variety of dimension $g > 0$. We say $A/k$ is \emph{heavenly at $\ell$} if $k(A[\ell^\infty]) \subseteq \ten(k,\ell)$. By \cite[\S1, Theorem 1]{Serre-Tate:1968}, if $A/k$ is heavenly at $\ell$, then it necessarily has good reduction away from $\ell$. When a curve $C/k$ has a Jacobian variety $J/k$ which is heavenly at $\ell$, it is sometimes possible to demonstrate $k(J[\ell^\infty]) \subseteq \yama$, via a combination of geometric (\cite{Rasmussen:2004}, \cite{Papanikolas-Rasmussen:2005}) or arithmetic (\cite{Rasmussen-Tamagawa:2008}) arguments. This connection is described in greater detail in \cite{Rasmussen-Tamagawa:2008, Rasmussen-Tamagawa:2012}. However, for fixed $k$, $g$ and $\ell$, there are only finitely many $k$-isomorphism classes of such abelian varieties.
For fixed $k$, $g$, and $\ell$, we let $\mathscr{G}(k,g,\ell)$ denote the set of $k$-isomorphism classes of abelian varieties $A/k$ of dimension $g$ which have good reduction away from $\ell$. This set is finite by the Shafarevich Conjecture (more precisely, by Zarhin's extension of Faltings' proof of the Shafarevich Conjecture to unpolarized abelian varieties). We denote by $\mathscr{A}(k,g,\ell)$ the subset of $\mathscr{G}(k,g,\ell)$ of isomorphism classes represented by abelian varieties which are heavenly at $\ell$. Under assumption of GRH, the authors have shown in \cite{Rasmussen-Tamagawa:2012} that for any choice of $k$ and $g$, the set $\mathscr{A}(k,g,\ell)$ is empty for sufficiently large $\ell$.
Curiously, in the case $k = \mathbb{Q}$, $g = 1$, we have the following observations:
\[ \begin{split}
\mathscr{A}(\mathbb{Q},1,2) & = \mathscr{G}(\mathbb{Q},1,2), \\
\mathscr{A}(\mathbb{Q},1,3) & = \mathscr{G}(\mathbb{Q},1,3), \\
\mathscr{A}(\mathbb{Q},1,5) & = \mathscr{G}(\mathbb{Q},1,5) = \varnothing, \\
\mathscr{A}(\mathbb{Q},1,7) & = \mathscr{G}(\mathbb{Q},1,7).
\end{split} \]
(The coincidence is lost already at $\ell = 11$, where $\mathscr{A}(\mathbb{Q},1,11)$ is a proper and nonempty subset of $\mathscr{G}(\mathbb{Q},1,11)$.) In the present note, we show a similar phenomenon holds in dimension $2$ when $\ell = 2$, at least when we restrict attention to principally polarized abelian varieties. To be more precise, we introduce two additional sets of isomorphism classes, as follows:
\[
\begin{matrix}
\mathscr{G}^\mathrm{pp}(k,g,\ell) & \subseteq & \mathscr{G}(k,g,\ell) \\
\rotatebox{90}{$\subseteq$} & & \rotatebox{90}{$\subseteq$} \\
\mathscr{A}^\mathrm{pp}(k,g,\ell) & \subseteq & \mathscr{A}(k,g,\ell) \\
\end{matrix}
\]
The elements of $\mathscr{G}^\mathrm{pp}(k,g,\ell)$ are classes represented by abelian varieties which are principally polarized (over $k$). The set $\mathscr{A}^\mathrm{pp}(k,g,\ell)$ denotes the subset of classes represented by heavenly principally polarized abelian varieties.
\begin{main-thm}
Let $K_0/\mathbb{Q}$ be an extension unramified away from $\{2, \infty \}$ with $[K_0:\mathbb{Q}] \leq 2$.
Every principally polarized abelian surface $A/K_0$ with good reduction away from $2$ is heavenly at $2$. In other words,
\[ \mathscr{A}^\mathrm{pp}(K_0,2,2) = \mathscr{G}^\mathrm{pp}(K_0,2,2). \]
\end{main-thm}
Explicitly, $K_0$ is one of the fields $\mathbb{Q}(\sqrt{d})$, $d \in \{\pm 1, \pm 2 \}$.
The paper proceeds as follows. In \S2, we break the proof into three cases, determined by the structure of $A/K_0$, an abelian variety representing a class in $\mathscr{G}^\mathrm{pp}(K_0, 2, 2)$. In \S3, we collect some facts about extensions of $\mathbb{Q}$ unramified away from $2$, and use these observations to extend a previous result \cite{Rasmussen:2004}. The three cases of the main theorem are handled in detail in \S4; the question in higher dimensions is briefly explored in \S5.
\section{Outline of Proof}
Let $\mathcal{A}_2$ denote the moduli space of principally polarized abelian surfaces, and $\mathcal{M}_2$ the moduli space of curves of genus $2$. The Deligne-Mumford compactification of $\mathcal{M}_2$ contains an intermediate space, $\mathcal{M}_2^*$, corresponding to the locus of compact type. It is known that the locus of compact type surjects onto $\mathcal{A}_2$. Moreover, the extremal locus, $\mathcal{M}_2^* - \mathcal{M}_2$, corresponds exactly to those points in $\mathcal{A}_2$ represented by products of elliptic curves. Consequently, over an algebraically closed field $k$, an abelian surface $A/k$ must be isomorphic to either the Jacobian of a (smooth) genus $2$ curve, or a product of elliptic curves. When $k$ is not algebraically closed, we have the following theorem of Gonz\'alez-Gu\`ardia-Rotger \cite[Theorem 3.1]{Gonzalez-Guardia-Rotger:2005}:
\begin{theorem}
Let $k$ be a number field, and let $A/k$ be a principally polarized abelian surface. Then as a polarized abelian variety, $A$ is isomorphic over $k$ to one of the following:
\begin{enumerate}[(1)]
\item $J$, the Jacobian variety of $C/k$, a smooth curve of genus $2$,
\item $E_1 \times E_2$, where $E_i/k$ are elliptic curves,
\item $W := \mathrm{Res}_{k'/k} E$, where $W$ is the Weil restriction of the elliptic curve $E/k'$, and $k'/k$ is a quadratic extension.
\end{enumerate}
\end{theorem}
Gonz\'alez, Gu\`ardia and Rotger give the explicit polarization on $J$, $E_1 \times E_2$, or $W$; it is always the `natural choice.' In order to prove the main theorem, we consider each of the cases (1), (2), (3). In each case, we appeal to some previously established results on the nature of Galois extensions of $\mathbb{Q}$ unramified away from $\{2, \infty\}$ of small degree. These are mainly due to Harbater, Jones, and Jones-Roberts.
\section{Extensions unramified away from $2$}
The database of number fields of Jones and Roberts \cite{Jones-Roberts:DB}, together with a result of Jones \cite{Jones:2010}, describe all small degree extensions of $\mathbb{Q}$ which are unramified away from $\{2, \infty \}$:
\begin{theorem}\label{theorem:Jones}
Suppose $K/\mathbb{Q}$ is a finite extension unramified away from $\{2, \infty \}$, and $[K:\mathbb{Q}] < 16$. Let $L/\mathbb{Q}$ denote the Galois closure of $K$, and let $G = \Gal(L/\mathbb{Q})$. Then:
\begin{enumerate}[(a)]
\item $[K:\mathbb{Q}] \in \{1, 2, 4, 8 \}$,
\item if $[K:\mathbb{Q}] = 4$, then $G$ is isomorphic to $V$, $\mathbb{Z}/4\mathbb{Z}$, or $D_4$,
\item if $[K:\mathbb{Q}] = 8$, then $G$ is a $2$-group of order at most $128$,
\item there exist $\sigma, \tau \in G$ such that $G = \langle \sigma, \tau \rangle$ and $\tau^2 = 1$.
\end{enumerate}
\end{theorem}
Jones's result incorporates several previous results by Harbater, Mark\v{s}a\u{\i}tis, Brueggeman, and Lesseni. We remark more concretely on (c). Several octic extensions of $\mathbb{Q}$ unramified away from $\{2, \infty \}$ have Galois closures of degree $64$ \cite{Jones-Roberts:DB}; the bound $2^7 = 128$ follows simply from the observation that $\ord_2 |S_8| = 7$. The following lemma improves (c) slightly.
\begin{lemma}\label{lemma:octic}
Suppose $K/\mathbb{Q}$ is an octic extension unramified away from $\{2, \infty \}$. Let $L$ be the Galois closure of $K$ in $\bar{\mathbb{Q}}$, a fixed algebraic closure of $\mathbb{Q}$. Then $[L:\mathbb{Q}] =2^\nu$, with $\nu \leq 6$.
\end{lemma}
\begin{proof}
Let $G = \Gal(L/\mathbb{Q})$. By Theorem \ref{theorem:Jones} (c), we need only eliminate the possibility that $|G| = 128$, which we now suppose for the sake of contradiction. We may identify $G$ with some subgroup of $S_8$; note that $G$ must be a Sylow-$2$ subgroup of $S_8$. Consider the following subgroup $H \leq S_8$:
\[ H := \bigl\langle (1234), (13), (5678), (57), \tau \bigr\rangle, \]
where $\tau = (15)(26)(37)(48)$. (The idea is to `mix' two distinct copies of $D_4$ with $\tau$ to obtain $H$.) The subgroup $H$ is also a Sylow-$2$ subgroup of $S_8$; hence, $G \cong H$. However, a routine calculation verifies that $H$ cannot be generated by $2$ elements, let alone satisfy the condition (d) of Theorem \ref{theorem:Jones}. Thus, $G$ cannot either, and this gives the desired contradiction.
\end{proof}
Finally, we will need one more result on such extensions, in this case due to Harbater \cite[Theorem 2.25]{Harbater:1994}.
\begin{proposition}\label{prop:harbater}
Suppose $L/\mathbb{Q}$ is a Galois extension which is unramified away from $\{2, \infty \}$. If $[L:\mathbb{Q}] < 272$, then $[L:\mathbb{Q}]$ is a power of $2$.
\end{proposition}
As Harbater observes, this is the best possible bound, as there exists a subfield of $\mathbb{Q}(\mathbf{\upmu}_{64})$ whose Hilbert class field gives a Galois extension $L/\mathbb{Q}$ unramified away from $\{2, \infty \}$ of exact degree $272$. (This field is discussed in more detail below.)
In \cite{Rasmussen:2004}, the first author proved $\mathscr{A}^\mathrm{pp}(\mathbb{Q},1,2) = \mathscr{G}^\mathrm{pp}(\mathbb{Q},1,2)$ through a geometric argument, by demonstrating a certain criterion of Anderson and Ihara holds for a representative of each class in $\mathscr{G}^\mathrm{pp}(\mathbb{Q},1,2)$. Before turning to the main theorem, we give a more direct proof of this result, which also holds for some number fields other than $\mathbb{Q}$. We will also use this result in the next section.
\begin{proposition}\label{prop:ell_curve}
Suppose $K_0/\mathbb{Q}$ is an extension unramified away from $\{2, \infty \}$ and $[K_0:\mathbb{Q}] \leq 4$. Then $\mathscr{A}^\mathrm{pp}(K_0,1,2) = \mathscr{G}^\mathrm{pp}(K_0,1,2)$.
\end{proposition}
\begin{proof}
Suppose $E/K_0$ is an elliptic curve with good reduction away from $2$. Let $L$ denote the Galois closure of $K_0(E[2])/\mathbb{Q}$. Since the tower
\[ \mathbb{Q} \subseteq K_0 \subseteq K_0(E[2]) \subset K_0(E[2^\infty]) \]
is unramified away from $\{2, \infty \}$ and the top extension is pro-$2$, it suffices to show that $[L : K_0]$ is a power of $2$. Let $d = [K_0(E[2]):K_0]$. Since the Galois group of this extension is isomorphic to a subgroup of $GL_2(\mathbb{F}_2)$, $d \in \{1, 2, 3, 6 \}$. If $d = 3$ or $d = 6$, then there exists a field $F$ with $K_0 \subset F \subseteq K_0(E[2])$ and $[F : K_0] = 3$. But now $[F:\mathbb{Q}] \leq 12$, and so by Theorem \ref{theorem:Jones}, $[F:\mathbb{Q}]$ is a power of $2$. Since it is also divisible by $3$, this is a contradiction. Thus, $d \leq 2$, and so $[K_0(E[2]):\mathbb{Q}] \leq 8$. By Theorem \ref{theorem:Jones}, we see that $[L:\mathbb{Q}]$ is a power of $2$. Consequently, $E$ is heavenly at $2$ and the result holds.
\end{proof}
\section{Abelian Surfaces}
We now turn to the proof of the Main Theorem. Before considering the three cases outlined in \S2, we treat specifically the case $K_0 = \mathbb{Q}$, where a stronger result may be obtained, and with less effort. Let $M$ be the maximal extension of $\mathbb{Q}$ which is unramified away from $\{2, \infty \}$, and set $\Delta := \Gal(M/\mathbb{Q}) = \pi_1(\mathbb{Z}[\tfrac{1}{2}])$.
\begin{proposition}
Let $\rho \colon \Delta \to GL_4(\mathbb{F}_2)$ be a Galois representation. Then the image of $\rho$ is a $2$-group.
\end{proposition}
\begin{proof}
The representation $\rho$ induces an action of $\Delta$ on $V := \mathbb{F}_2^{\oplus 4}$, and necessarily factors through $G := \Gal(M_0/\mathbb{Q})$ for some finite Galois extension $M_0/\mathbb{Q}$. Selecting $M_0$ minimal, we have $G \cong \rho(\Delta)$. Further, there is an induced faithful action of $G$ on $V^\circ := V - \{0 \}$. For any $v \in V^\circ$, let $G_v$ denote the stabilizer of $v$, $M_v$ the subfield of $M_0$ fixed by $G_v$, and $L_v$ the Galois closure of $M_v/\mathbb{Q}$. As $M_0/\mathbb{Q}$ is Galois, we have $L_v \subseteq M_0$ for all $v$. For any $v \in V^\circ$, an application of the orbit-stabilizer theorem gives
\[ [M_v : \mathbb{Q}] = [G : G_v] = \#\mathrm{Orb}(v) \leq |V^\circ| < 16. \]
By Theorem \ref{theorem:Jones}, $[M_v : \mathbb{Q}] \mid 8$ and $L_v/\mathbb{Q}$ is a $2$-extension. The faithfulness of the action guarantees $\cap_v G_v = \{1 \}$, or in other words, that $M_0$ is the compositum of the $M_v$. Equivalently, $M_0$ is the compositum of the $L_v$, and so $M_0/\mathbb{Q}$ is a $2$-extension and $\rho(\Delta)$ is a $2$-group.
\end{proof}
\begin{corollary}
$\mathscr{G}(\mathbb{Q},2,2) = \mathscr{A}(\mathbb{Q},2,2)$.
\end{corollary}
\begin{proof}
Suppose $[A] \in \mathscr{G}(\mathbb{Q},2,2)$, and let $\rho$ be the induced Galois representation on the $2$-torsion of $A$. Then in the context of the previous proposition, $M_0 = \mathbb{Q}(A[2])$ is a $2$-extension of $\mathbb{Q}$, and hence $[A] \in \mathscr{A}(\mathbb{Q},2,2)$.
\end{proof}
\subsection{Case 1: Jacobians}
In this section, we prove the following proposition:
\begin{proposition}\label{prop:Jacobian}
Let $K_0/\mathbb{Q}$ be an extension unramified away from $\{2, \infty \}$, with $[K_0:\mathbb{Q}] \leq 2$. Let $C/K_0$ be a smooth projective curve of genus $2$, and let $J$ denote the Jacobian variety of $C$. Suppose $[J] \in \mathscr{G}^\mathrm{pp}(K_0,2,2)$. Then $[J] \in \mathscr{A}^\mathrm{pp}(K_0,2,2)$.
\end{proposition}
Let $C/K_0$ be a smooth projective curve of genus $2$. Then $C$ is hyperelliptic, and admits a degree $2$ morphism $C \to \P := \P^1_{K_0}$, which is unique up to coordinate change of $\P^1_{K_0}$. The ramification locus $S \subset C$ for $C \to \P$ (that is, the support of the sheaf $\Omega_{C/\P}$) has degree $6$. Namely, $\bar{S} := S \times_{K_0} \bar{K}_0$ consists of $6$ points. Further, $\bar{S}$ admits a natural action of $G_{K_0} := \Gal(\bar{K}_0/K_0)$, and the set of $G_{K_0}$-orbits of $\bar{S}$ is identified with $S$.
For a finite set $T$ and an integer $n \geq 0$, let $T_n$ denote the collection of subsets of $T$ of cardinality $n$; that is, $T_n := \{U \subseteq T : |U| = n \}$. The natural map $\Aut(T) \to \Aut(T_n)$ between symmetric groups is injective, if $0 < n < |T|$. With this notation established, it is a standard result that the map
\[ \bar{S}_2 \to J[2]-\{0\}, \qquad \{s,s'\} \mapsto \mathrm{cl}(s-s') \]
is well-defined, $G_{K_0}$-equivariant, and bijective (for further details, see \cite{Cassels-Flynn:1996}).
It follows that the compositum $K_0(S)$ of the residue fields $K_0(P)$ for $P\in S$ coincides with $K_0(J[2])$. Thus, under the assumption that $[J]\in\mathscr{G}^{\mathrm{pp}}(K_0,2,2)$, we have great control over the field $K_0(J[2])$ for certain fields $K_0$, by combining the observations of Jones and Harbater:
\begin{proposition}
Let $K_0/\mathbb{Q}$ be unramified away from $\{2, \infty \}$, with $[K_0:\mathbb{Q}] \leq 2$. Let $C/K_0$ be a smooth projective curve of genus $2$, and let $J$ denote the Jacobian of $C$. Let $K' = K_0(J[2])$, and let $L'$ be the Galois closure of $K'/\mathbb{Q}$. Suppose that $J/K_0$ has good reduction away from $2$. Then $L'/\mathbb{Q}$ is a $2$-extension satisfying $[L':\mathbb{Q}] \leq 256$.
\end{proposition}
\begin{proof}
We work in a fixed algebraic closure of $K_0$. Let $S = \{P_1,\dots,P_t\}$ be the ramification locus of the degree $2$ morphism $C \to \P$, as above. For $K_i := K_0(P_i)$, the extension $K_i/K_0$ is contained in $K_0(J[2])$, hence
necessarily unramified away from $\{2, \infty \}$. Let $d_i := [K_i : K_0]$. A priori, $d_i \leq 6$, and so $[K_i:\mathbb{Q}] \leq 2 d_i \leq 12$. By Theorem \ref{theorem:Jones}, we see $d_i \in \{1, 2, 4 \}$. Possibly after reindexing, we may assume $d_1 \geq \cdots \geq d_t$. The possible values of $\mathbf{d} = (d_1, \dots, d_t)$ are given in Table \ref{table:degrees}.
\begin{table}[t!]
\begin{tabular}{lrc}
\toprule
$t\;\;\;$ & $\mathbf{d} = (d_1, \dots, d_t)$ & Bound for $[\tilde{L}:\mathbb{Q}]$ \\
\midrule
$2$ & $(4, 2)$ & $2^8$ \\
$3$ & $(4, 1, 1)$ & $2^6$ \\
$3$ & $(2, 2, 2)$ & $2^7$ \\
$4$ & $(2, 2, 1, 1)$ & $2^5$ \\
$5$ & $(2, 1, 1, 1, 1)$ & $2^3$ \\
$6$ & $(1, 1, 1, 1, 1, 1)$ & $2^1$ \\
\bottomrule
\end{tabular}
\smallskip
\caption{Bounds in case $[K_0:\mathbb{Q}] = 2$.}\label{table:degrees}
\end{table}
Now, let $L_i$ denote the Galois closure of $K_i/\mathbb{Q}$ for each $i$, and let $\tilde{L}$ denote the compositum of $L_1, L_2, \dots, L_t$. As $K'$ is the compositum of the $K_i$, it follows that $L' = \tilde{L}$. Thus, it suffices to prove $[\tilde{L}:\mathbb{Q}]$ divides $2^8$; by Proposition \ref{prop:harbater}, we need only prove $[\tilde{L}:\mathbb{Q}] \leq 2^8$.
If $K_0 = \mathbb{Q}$, by Theorem \ref{theorem:Jones}, we see $[L_i:\mathbb{Q}]$ divides $8$ if $d_i = 4$; otherwise, $[L_i:\mathbb{Q}] = d_i$. Consequently,
\[ [\tilde{L}:\mathbb{Q}] \leq \prod_{i=1}^t [L_i:\mathbb{Q}] \leq 16, \]
by checking each possible list of degrees $\mathbf{d}$.
Now suppose $[K_0:\mathbb{Q}] = 2$. Then $[K_i:\mathbb{Q}] = 2d_i$, and by Theorem \ref{theorem:Jones} and Lemma \ref{lemma:octic}, we see $[L_i:\mathbb{Q}]$ divides $2$, $8$, or $64$, as $d_i$ is $1$, $2$, or $4$, respectively. We obtain
\[ [\tilde{L}:\mathbb{Q}] = [K_0:\mathbb{Q}][\tilde{L}:K_0] \leq 2 \cdot \prod_{i=1}^t [L_i:K_0]. \]
So, for example when $\mathbf{d} = (4,2)$, or $\mathbf{d} = (2,2,2)$, we have
\[ [\tilde{L}:\mathbb{Q}] \leq 2 \cdot 32 \cdot 4 = 2^8, \qquad [\tilde{L}:\mathbb{Q}] \leq 2 \cdot 4^3 = 2^7, \]
respectively. The remaining cases are handled by the same style of argument (all the bounds are given in Table \ref{table:degrees}).
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:Jacobian}]
Suppose $C/K_0$ is a curve of genus $2$ with Jacobian $J$, and $[J] \in \mathscr{G}^\mathrm{pp}(K_0,2,2)$. Then we have
\[ K_0 \subseteq K_0(J[2]) \subset K_0(J[2^\infty]). \]
The tower is unramified away from $2$; the Galois closure of the lower extension is a $2$-extension by the previous proposition, and the upper extension is pro-$2$ generally. Thus, $[J] \in \mathscr{A}^\mathrm{pp}(K_0,2,2)$.
\end{proof}
\subsection{Case 2: Product of elliptic curves}
The argument in this case is straightforward.
\begin{proposition}
Suppose $K_0/\mathbb{Q}$ is unramified away from $\{2, \infty \}$ and $[K_0:\mathbb{Q}] \leq 4$. Suppose $[A] \in \mathscr{G}^\mathrm{pp}(K_0,2,2)$, and $A \cong_{K_0} E_1 \times E_2$ for some elliptic curves $E_i/K_0$. Then $A$ is heavenly at $2$.
\end{proposition}
\begin{proof}
Necessarily, the curves $E_i/K_0$ have good reduction away from $2$. By Proposition \ref{prop:ell_curve}, both curves are heavenly at $2$. Thus, $K_0(E_i[2]) \subseteq \ten$, and so
\[ K_0(A[2]) = K_0(E_1[2]) \cdot K_0(E_2[2]) \subseteq \ten, \]
also.
\end{proof}
\subsection{Case 3: Restriction of scalars}
The situation in this case is slightly more delicate, because the quadratic extension introduced by the restriction of scalars could possibly ramify at a prime other than $2$. We will make use of the following lemmas, which are essentially exercises in Galois theory:
\begin{lemma}\label{lemma:gal-over-gal}
Let $E/F$ and $L/E$ be Galois extensions (inside a fixed algebraic closure of $F$), of degrees $d$ and $m$, respectively. Let $L^*/F$ denote the Galois closure of $L/F$. Then $[L^*:F] \leq d \cdot m^d$.
\end{lemma}
\begin{proof}
Set $G = \Gal(L^*/F)$, $N = \Gal(L^*/E)$, $H = \Gal(L^*/L)$. Then $H \trianglelefteq N \trianglelefteq G$. Since $[G:N] = d$, we may choose $\sigma_1, \dots, \sigma_d \in G$ representing each coset of $G/N$. Set $H_i = \sigma_i H \sigma_i^{-1}$. Then it is routine to verify that each $H_i \trianglelefteq N$. Let $L_i \subseteq L^*$ be the subfield fixed by $H_i$. These $d$ fields give every conjugate of $L/F$ within $L^*$. Consequently, $L^*$ coincides with the compositum over $E$ of the $L_i$, and each $L_i/E$ is Galois. Since $[E:F] = d$ and $[L_i:E] = m$, we obtain $[L^*:F] \leq d \cdot m^d$.
\end{proof}
We let $\mathfrak{S}_3$ denote the symmetric group on $3$ symbols.
\begin{lemma}\label{lemma:S3}
Let $F_1/F_0$ and $F_2/F_0$ be Galois $\mathfrak{S}_3$-extensions. The Galois group $G$ of the compositum $F_1 F_2/F_0$ is one of $\mathfrak{S}_3 \times \mathfrak{S}_3$, $(\mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}) \rtimes \mathbb{Z}/2\mathbb{Z}$, $\mathfrak{S}_3$. In every case, $G$ has a subgroup of exact index $3$.
\end{lemma}
\begin{proof}
Since $\mathfrak{S}_3$ has no normal subgroups of index $3$, the (necessarily Galois) extension $F_1 \cap F_2/F_0$ has degree $1$, $2$, or $6$. The result follows by considering each possibility in turn.
\end{proof}
\begin{proposition}
Suppose $K_0/\mathbb{Q}$ is unramified away from $\{2, \infty \}$ and $[K_0:\mathbb{Q}] \leq 2$. Suppose $[A] \in \mathscr{G}^\mathrm{pp}(K_0, 2, 2)$. If $A$ is isomorphic over $K_0$ to $\Res_{K'/K_0} E_1$ for some elliptic curve $E_1/K'$, then $A$ is heavenly at $2$.
\end{proposition}
\begin{proof}
Let $E_2/K'$ be the Galois twist of $E_1$ with respect to the unique nontrivial element of $\Gal(K'/K_0)$.
As $A/K_0$ has good reduction away from $2$, the same is true for $E_i/K'$, $i=1,2$. If $K'/K_0$ is unramified away from $\{2, \infty \}$, then $K'/\mathbb{Q}$ is a quartic extension also unramified away from $\{2, \infty \}$, and so by Proposition \ref{prop:ell_curve}, each $E_i$ is heavenly at $2$. Thus, the Galois closure of $K'(E_i[2])/\mathbb{Q}$ is a $2$-extension. Since $K_0(A[2])$ is contained in the compositum of these two extensions, the Galois closure of $K_0(A[2])/\mathbb{Q}$ is also a $2$-extension, and so $[A] \in \mathscr{A}^\mathrm{pp}(K_0,2,2)$.
Now, suppose that some prime $\mathfrak{p}$ of $K'$ ramifies in $K'/K_0$, with $\mathfrak{p} \nmid 2\mathcal{O}_{K'}$. Notice that we have the equality of fields
\[ M := K' \cdot K_0(A[2]) = K'(A[2]) = K'(E_1[2] \cup E_2[2]) = K'(E_1[2]) \cdot K'(E_2[2]). \]
The two extensions $K'(E_i[2])/K'$ must have isomorphic Galois groups. Choose $\Gamma \leq \mathfrak{S}_3$ such that $\Gamma \cong \Gal(K'(E_i[2])/K')$, and let $c = |\Gamma|$. We must have that $[M:K']$ divides $c^2$. Moreover, since $K'/K_0$ and $K_0(A[2])/K_0$ are both Galois extensions, $M$ is in fact Galois over $K_0$.
\[
\begin{tikzcd}[row sep=2ex]
{} & M \arrow[-, swap]{dl}{2} \arrow[-]{dd} \\
K_0(A[2]) & \\
& K' \\
K_0 \arrow[-]{uu} \arrow[-, swap]{ur}{2} &
\end{tikzcd}
\]
Since $M/K_0$ is Galois, we see $[M:K_0(A[2])]$ divides $2 = [K':K_0]$, and $[M:K']$ divides $[K_0(A[2]):K_0]$. But since $K'/K_0$ is ramified at $\mathfrak{p}$ and $K_0(A[2])/K_0$ is not ramified at $\mathfrak{p}$, the extension $M/K_0(A[2])$ must reflect this ramification; it cannot be a trivial extension. Thus, $[M:K_0(A[2])] = 2$ and $\Gal(M/K') \cong \Gal(K_0(A[2])/K_0)$.
We claim $c \neq 6$. For otherwise, Lemma \ref{lemma:S3} guarantees that $\Gal(K_0(A[2])/K_0)$ possesses a subgroup of index $3$. Such a subgroup corresponds to a cubic extension $L/K_0$ with $L \subset K_0(A[2])$. Consequently, $L/\mathbb{Q}$ is a degree $6$ extension, unramified away from $\{2, \infty \}$, which contradicts Theorem \ref{theorem:Jones}.
So $c < 6$, which in fact implies $c \leq 3$ and $[K_0(A[2]):K_0] \leq 9$. Thus, by Lemma \ref{lemma:gal-over-gal}, the Galois closure of $K_0(A[2])/\mathbb{Q}$ has degree at most $2 \cdot 9^2 = 162$. By Proposition \ref{prop:harbater}, the Galois closure must be a $2$-extension, and so $[A] \in \mathscr{A}^\mathrm{pp}(K_0,2,2)$.
\end{proof}
\section{Failure in higher dimensions}
At this point, one might na\"{\i}vely guess that in the special case $k = \mathbb{Q}$, $\ell = 2$, $\mathscr{A}^\mathrm{pp}(\mathbb{Q},g,2) = \mathscr{G}^\mathrm{pp}(\mathbb{Q},g,2)$ for all $g \geq 1$. This is not the case, as we now show via Weil restriction.
First, we briefly review the useful Example 2.24 of \cite{Harbater:1994}. Let $\zeta \in \bar{\mathbb{Q}}$ be a primitive $64$-th root of unity, and set $\eta_0 = \zeta^{16}(\zeta + \zeta^{-1})$ and $F_0 = \mathbb{Q}(\eta_0)$. Let $L_0$ be the Hilbert class field of $F_0$. Harbater demonstrates that $L_0/\mathbb{Q}$ is Galois with Galois group $\mathbb{Z}/17\mathbb{Z} \rtimes (\mathbb{Z}/17\mathbb{Z})^\times$, and $L_0/\mathbb{Q}$ is a (unique) Galois extension of degree $272$ unramified away from $2$.
We now demonstrate a non-heavenly element of $\mathscr{G}^\mathrm{pp}(\mathbb{Q},272,2)$.
\begin{proposition}
The set $\mathscr{A}^\mathrm{pp}(\mathbb{Q},272,2)$ is a proper subset of $\mathscr{G}^\mathrm{pp}(\mathbb{Q},272,2)$. That is, there exists an abelian variety $A/\mathbb{Q}$ of dimension $272$ with good reduction away from $2$ which is not heavenly at $2$.
\end{proposition}
\begin{proof}
Select an elliptic curve $E/\mathbb{Q}$ such that $[E] \in \mathscr{A}(\mathbb{Q},1,2)$ and $\mathbb{Q}(E[2]) = \mathbb{Q}$. There are two such curves, up to $\mathbb{Q}$-isomorphism, given by Cremona's labeling as `32a2' and `64a1':
\[ \mathrm{(32a2)}\quad y^2 = x^3 - x, \qquad \qquad \mathrm{(64a1)} \quad y^2 = x^3 - 4x. \]
Let $L_0$ be as in the previous paragraph, and set
\[ A := \Res_{L_0/\mathbb{Q}} (E \times_\mathbb{Q} L_0). \]
This is an abelian variety defined over $\mathbb{Q}$, and $[A] \in \mathscr{G}^\mathrm{pp}(\mathbb{Q},272,2)$.
(Note that $A$ is principally polarized by \cite[Proposition 2]{Diem-Naumann:2003}.)
When we view $A[2]$ as a $G_\mathbb{Q}$-module, we have
\[ A[2] \cong \Ind_{G_{L_0}}^{G_\mathbb{Q}} E[2]. \]
Let $H = \Gal(L_0/\mathbb{Q})$. Viewed only as an abelian group, we have
\[ \Ind_{G_{L_0}}^{G_\mathbb{Q}} E[2] = \bigoplus_{\sigma \in H} E[2] \cong \bigoplus_{\sigma \in H} \mathbb{F}_2^{\oplus 2}. \]
However, $E[2] \subseteq E(\mathbb{Q})$, and as $G_\mathbb{Q}$-modules, we have $E[2] \cong_{G_\mathbb{Q}} \mathbb{F}_2^{\oplus 2}$, where $G_\mathbb{Q}$ acts on $\mathbb{F}_2^{\oplus 2}$ trivially. Thus the action of $G_\mathbb{Q}$ on $\Ind_{G_{L_0}}^{G_\mathbb{Q}} E[2]$ is given simply by the permutation action on the summands indexed by $\sigma \in H$. From this we conclude
\[ L_0 = L_0 \cdot \mathbb{Q}(E[2]) \subseteq \mathbb{Q}(E[2])\bigl( A[2] \bigr) = \mathbb{Q}(A[2]). \]
Since $[L_0:\mathbb{Q}] = 272$, we have $\mathbb{Q}(A[2^\infty]) \not\subseteq \ten$, and so $[A] \not\in \mathscr{A}^\mathrm{pp}(\mathbb{Q},272,2)$.
\end{proof}
\bibliographystyle{halpha}
|
{
"timestamp": "2015-04-14T02:10:39",
"yymm": "1504",
"arxiv_id": "1504.03047",
"language": "en",
"url": "https://arxiv.org/abs/1504.03047"
}
|
\section{Introduction}
Hot subdwarf stars \citep[see a review by ][]{heb2009} are core helium burning stars with very thin hydrogen envelopes
and belong to the extreme horizontal branch (EHB). The mass of most hot subdwarfs is about
0.5\,M$_\odot$.
The origin of EHB stars, i.e, the
hot, hydrogen-rich (sdB) and helium-rich subdwarf (sdO) stars, is
closely linked to binarity. \citet{men1976} first proposed
that sdB stars are formed in close binary systems and \citet{dor1993} inferred
the presence of an extremely thin hydrogen envelope ($<0.001$\,M$_\odot$). \citet{han2002,han2003}
proposed three formation channels for sdB stars
through binary interaction, i.e., common envelope (CE), Roche lobe overflow (RLOF), and binary
merger. \citet{han2003} predict a binary fraction
of 76 - 89 per cent with orbital periods ranging from 0.5~hr
to 500~d. However, they caution that the observed frequency
could be much lower due to selection effects. The proposed formation
channels also predict single sdB stars that form via the
merger of two helium white dwarfs. Approximately 11 - 26 per cent
of subdwarfs are expected to form via this merger channel \citep{han2003}.
Formation channels of helium-rich (He-sdO) stars are not as well
defined. \citet{jus2011} proposed that these objects may form in a
close double degenerate binary with the massive component accreting from a helium
white dwarf companion and initiating helium-shell burning. A small number of sdO
stars are known to exist as companions to Be stars \citep{gie1998,pet2008,pet2013}.
These sdO stars are formed through close binary interaction where the more massive
primary star begins mass transfer onto its less massive companion during its
shell-hydrogen burning phase. The result of this mass transfer leaves a spun up Be
star with an sdO companion \citep{pol1991}.
Cool companions to hot subdwarf stars can be revealed as infrared
excess in the spectral energy distribution (SED). \citet{the1995} and \citet{ull1998}
detected infrared excess in over
20 per cent of the hot subdwarf stars studied in their sample. \citet{gir2012}
explored photometric surveys that cover a wide wavelength
range, from the Galaxy Evolution Explorer ($GALEX$) ultraviolet
survey through to the infrared, the Two Micron All Sky Survey (2MASS) and the UKIRT Infrared Deep Sky Survey (UKIDSS),
and searched for main-sequence companions to hot subdwarf stars. They found that
the most common companions to hot subdwarfs have a spectral type
between F0 and K0, while M-type companions were found to be
much rarer.
Radial velocity surveys \citep[e.g.,][]{max2001,mor2003,cop2011,gei2011a}
of sdB stars have shown that approximately half of all sdB stars reside in
close binary systems with either a cool main-sequence star or a
white dwarf companion. These surveys target binary systems with periods of a few hours to $\approx 30$ days. \citet{nap2004}
reported a binary fraction of 39 per cent of sdB stars
from the ESO Supernovae type Ia Progenitor surveY (SPY). \citet{cop2011}
estimated a higher binary fraction of 46 - 56
per cent from their survey of sdB stars selected from the Palomar-Green
and Edinburgh-Cape surveys.
A few rare sdB stars are found in close orbit with a massive
white dwarf ($M_{\rm WD} \ga 0.9$\,M$_\odot$), making them Type Ia
supernova progenitors. These systems would first evolve to AM
CVn systems before detonating either as a Type Ia
or the less energetic Type~.Ia (Iax) supernova \citep{bil2007,fin2010,sol2010}. The
first such candidate is KPD 1930+2752 \citep{max2000a,gei2007},
with a second candidate, GALEX~J1411$-$3053 (CD$-$30~11223),
discovered as part of our radial velocity survey of $GALEX$
selected hot subdwarf stars \citep{ven2012}.
Some sdB stars in close binary systems have stellar parameters that fall below
the zero-age horizontal branch and probably did not initiate helium burning. Such
objects have very low masses
($\approx 0.2$\,M$_\odot$) and are the progenitors of
extremely low mass (ELM) white dwarfs, which will in time evolve into AM CVn systems.
If the companion to these low mass stars is a massive enough white dwarf, then
the system may become a Type Ia supernova. The first known low mass sdB star,
HD~188112, was discovered by \citet{heb2003}.
\citet{ahm2004} discovered the first double subdwarf binary, PG~1544$+$488.
This helium rich sdB (He-sdB) binary remains, at the present time, unique.
The mass ratio determined from the velocity semi-amplitude of
the components show that they
have a similar mass which suggests that the system emerged from a CE comprised of two nearly identical red giant cores \citep{sen2014}.
Alternatively, \citet{lan2004} interpreted the peculiar atmospheric composition of He-sdB stars, such as PG~1544$+$488,
with evolutionary models involving a delayed
helium-core flash and convective mixing while descending on the white dwarf cooling track.
Similarly, HE~0301$-$3039 is a close binary consisting of two sdO stars \citep{lis2004,str2007} that
may be the outcome of double-core CE
evolution \citep{jus2011}.
Surveys of hot subdwarfs involving photometric time series
have uncovered several more low
mass sdB stars. Kepler observations revealed that KIC 6614501 is another low mass
sdB plus white dwarf system \citep{sil2012}. Also, \citet{max2014} presented 17
eclipsing systems from Wide Angle Search for Planets (WASP) survey that are
likely to contain a pre-helium white dwarf, similar to the system
1SWASP~J024743.37$-$251549.2 \citep{max2011}. Follow-up spectroscopy for six
of these systems confirmed them to be main-sequence A stars with very low
mass ($\approx 0.2$\,M$_\odot$) pre-He white dwarfs currently experiencing hydrogen-shell burning.
Wider binaries (orbital periods $\sim$ years) containing a sdB star with a cool
main-sequence companion were reported by \citet{bar2012,bar2013a} and \citet{vos2013}.
The predicted period distribution by \citet{han2003} is bimodal with some B to F type companions in
the longer-period range: The relative frequency of
short- to long-period binaries depends on the actual value of the critical
mass ratio for stable mass transfer; this ratio may be set with
a study of potential subdwarf plus A-star binaries.
\citet{che2013} showed that these long period binaries are the result of
stable RLOF.
\citet{ven2011} and \citet{nem2012} presented a new sample of sdB stars selected from the
$GALEX$ all-sky survey and we conducted a radial velocity
survey of a subsample of stars from this selection. The first two systems
(GALEX~J0321$+$4727 and GALEX~J2349$+$3844) discovered as part of this survey
were presented by \citet{kaw2010a}, followed by the aforementioned short-period system
GALEX~J1411$-$3053 \citep{ven2012,gei2013a}. Additional spectroscopic and photometric
observations of the first two systems were presented in \citet{kaw2012a} along with
a progress report on the other systems that were observed as part of this
programme. The photometric observations confirmed the reflection effect in
GALEX~J0321$+$4727 originally reported by \citet{kaw2010a} and based on
Northern Sky Variability Survey (NSVS)
photometry. The observations also showed that both GALEX~J0321$+$4727 and
GALEX~J2349$+$3844 are V2093 Her type pulsating subdwarfs \citep{gre2003}.
In this paper, we present spectroscopic and photometric observations of a sample
of $GALEX$-selected hot subdwarf stars with the aim of determining their binary
properties.
Sections 2.1 and 2.2 present details of
our spectroscopic observations, while Section 2.3 present archival photometric time series.
In Section 3 we
present an analysis of stellar properties (3.1), and of binary properties
supplemented by our analysis of photometric time series (3.2).
Finally, we present a review of the properties of known binaries comprising a hot subdwarf star, including
the properties of the components (Section 4.1), the population kinematics (4.2), and the properties of
some outstanding individual cases (4.3), followed by a summary of the present work (4.4).
\begin{table*}
\centering
\begin{minipage}{\textwidth}
\caption{Target summary. \label{tbl_sum}}
\renewcommand{\footnoterule}{\vspace*{-15pt}}
\renewcommand{\thefootnote}{\alph{footnote}}
\begin{tabular}{llcccl}
\hline
& Other names & $T_{\rm eff}$ & $\log{g}$ & $\log{(\rm He/H)}$ & Notes \footnotemark[1]\footnotetext[1]{RV: confirmed radial velocity variable star; IR: SED of the stars shows significant IR excess.} \\
GALEX~J & & (K) & c.g.s. & & \\
\hline
004759.6$+$033742 & BPS BS 17579-0012, PB 6168 & $38620^{+2250}_{-970}$ & $6.14^{+0.22}_{-0.18}$ & $-2.63^{+0.44}_{-1.17}$ & sdB+F6V; IR; nearby star\\
004729.4$+$095855 & HD~4539, HIP 3701 & $24650^{+590}_{-200}$ & $5.38^{+0.03}_{-0.05}$ & $-2.42^{+0.20}_{-0.07}$ & \\
004917.2$+$205640 & PG 0046+207 & $27520^{+500}_{-450}$ & $5.55^{+0.07}_{-0.06}$ & $-2.48^{+0.16}_{-0.23}$ & \\
005956.7$+$154419 & HIP 4666, PG 0057+155, PHL 932 & $33530^{+190}_{-310}$ & $5.83^{+0.04}_{-0.05}$ & $-1.69^{+0.06}_{-0.04}$ & \\
020656.1$+$143900 & CHSS~3497 & $30310^{+660}_{-80}$ & $5.77^{+0.05}_{-0.06}$ & $-2.61^{+0.15}_{-0.24}$ & \\
023251.9$+$441126 & FBS 0229+439 & $33260^{+420}_{-380}$ & $5.73^{+0.09}_{-0.10}$ & $-1.70^{+0.08}_{-0.12}$ & \\
040105.3$-$322348 & CD-32 1567, EC 03591-3232 & $30490^{+250}_{-220}$ & $5.71^{+0.06}_{-0.04}$ & $-1.92^{+0.06}_{-0.04}$ & \\
050018.9$+$091203 & HS 0457+0907 & $36270^{+490}_{-1130}$ & $5.75^{+0.15}_{-0.13}$ & $-1.46^{+0.14}_{-0.15}$ & \\
050735.7$+$034814 & & $23990^{+630}_{-610}$ & $5.42^{+0.08}_{-0.11}$ & $-3.05^{+0.48}_{-0.78}$ & Ca\,H\&K, RV \\
061325.3$+$342053 & & $34250^{+330}_{-390}$ & $5.75^{+0.10}_{-0.06}$ & $-1.28^{+0.04}_{-0.08}$ & RV \\
065736.7$-$732447 & CPD-73 420 & $29940^{+900}_{-160}$ & $5.45^{+0.07}_{-0.15}$ & $< -3.21$ & nearby star \\
070331.5$+$623626 & FBS 0658+627 & $28750^{+370}_{-340}$ & $5.40^{+0.07}_{-0.04}$ & $-2.76^{+0.22}_{-0.26}$ & \\
071646.9$+$231930 & TYC 1909-865-1 & $11140/9310$ & $4.39/3.67$ & ... & close B+A\,V binary, RV \\
075147.1$+$092526 & & $30620^{+490}_{-460}$ & $5.74^{+0.11}_{-0.12}$ & $-2.49^{+0.27}_{-0.30}$ & nearby star (6~arcsec), RV \\
080510.9$-$105834 & TYC 5417-2552-1 & $22320^{+330}_{-280}$ & $5.68^{+0.03}_{-0.06}$ & $ < -3.44$ & ELM WD progenitor, RV \\
081233.6$+$160123 & & $31580^{+440}_{-490}$ & $5.56^{+0.10}_{-0.13}$ & $ < -2.90$ & RV \\
104148.6$-$073034 & TYC 5492-642-1 & $27440^{+620}_{-450}$ & $5.63^{+0.09}_{-0.06}$ & $-2.44^{+0.16}_{-0.23}$ & \\
111422.0$-$242130 & EC 11119-2405, TYC 6649-111-1 & $23430^{+480}_{-450}$ & $5.29^{+0.08}_{-0.07}$ & $-2.46^{+0.19}_{-0.31}$ & \\
135629.2$-$493403 & CD-48 8608, TYC 8271-627-1 & $33070^{+230}_{-660}$ & $5.74^{+0.07}_{-0.16}$ & $-2.75^{+0.25}_{-0.43}$ & sdB+G8V; IR \\
140747.6$+$310318 & BPS BS 16082-0122 & $24900^{+50}_{-3050}$ & $4.25^{+0.03}_{-0.09}$ & $-1.18^{+0.08}_{-0.09}$ & high-$\varv$ early B \\
141133.3$+$703737 & TYC 4406-666-1 & $21170^{+1500}_{-1110}$& $5.55^{+0.31}_{-0.23}$ & $<-2.36$ & sdB+F; IR; ELM WD progenitor? \\
142126.5$+$712427 & TYC 4406-285-1 & $25620^{+320}_{-220}$ & $5.67\pm0.04$ & $<-3.7$ & \\
142747.2$-$270108 & EC 14248-2647, TYC 6740-942-1 & $31880^{+360}_{-290}$ & $5.70^{+0.05}_{-0.08}$ & $-1.71^{+0.05}_{-0.11}$ & \\
143519.8$+$001352 & TYC 325-452-1, PG 1432+004 & $23090^{+780}_{-250}$ & $5.28^{+0.08}_{-0.08}$ & $-2.39^{+0.18}_{-0.20}$ & \\
163201.4$+$075940 & TYC 960-1373-1, PG 1629+081 & $38110^{+570}_{-680}$ & $5.38^{+0.06}_{-0.09}$ & $-2.71^{+0.27}_{-0.29}$ & nearby star, RV \\
173153.7$+$064706 & & $27780^{+1030}_{-470}$ & $5.35^{+0.18}_{-0.07}$ & $<-2.53$ & RV \\
173651.2$+$280635 & TYC 2084-448-1 & $36160^{+6500}_{-4200}$& $5.24^{+0.84}_{-0.84}$ & $-1.09^{+0.69}_{-1.34}$ & sdB+F7V; IR; variable \\
175340.5$-$500741 & & $32430^{+880}_{-570}$ & $5.95^{+0.18}_{-0.18}$ & $-2.25^{+0.31}_{-1.04}$ & sdB+F7V; IR \\
184559.8$-$413826 & & $35930^{+840}_{-4770}$ & $5.23^{+0.27}_{-0.23}$ & $+2.10^{+1.10}_{-0.38}$ & sdO; He\,{\sc i} spectrum \\
190211.7$-$513005 & CD-51 11879, TYC 8386-1370-1, LSE 263 & $72300^{+5380}_{-3260}$& $5.49^{+0.11}_{-0.11}$ & $+0.02^{+2.10}_{-0.03}$ & sdO; He\,{\sc ii} spectrum \\
190302.4$-$352828 & BPS CS 22936-0293 & $32100^{+1760}_{-1260}$ & $5.26^{+0.31}_{-0.30}$ & $<-1.96$ & RV \\
191109.2$-$140651 & TYC 5720-292-1 & $55970^{+4540}_{-1780}$ & $5.69^{+0.71}_{-0.09}$ & $+0.25^{+0.70}_{-0.60}$ & sdO; He\,{\sc ii} spectrum \\
203850.3$-$265750 & TYC 6916-251-1 & $58450^{+4600}_{-7920}$ & $5.04^{+0.39}_{-0.17}$ & $-1.13^{+0.27}_{-0.29}$ & sdO+G3.5III; IR; variable \\
215340.4$-$700430 & EC 21494-7018, TYC 9327-1311-1 & $23720^{+260}_{-230}$ & $5.65^{+0.03}_{-0.02}$ & $-3.22^{+0.13}_{-1.15}$ & ELM WD progenitor?\\
220551.8$-$314105 & TYC 7489-686-1, BPS CS 30337-0074 & $28650^{+930}_{-80}$ & $5.68^{+0.01}_{-0.03}$ & $-2.09^{+0.12}_{-0.03}$ & reflection, RV \\
225444.1$-$551505 & & $31070^{+150}_{-190}$ & $5.80^{+0.04}_{-0.06}$ & $-2.47^{+0.15}_{-0.13}$ & RV \\
233451.7$+$534701 & TYC4000-216-1 & $35680^{+340}_{-250}$ & $5.91^{+0.07}_{-0.06}$ & $-1.43\pm0.07$ & \\
234421.6$-$342655 & CD-35 15910, HE 2341-3443 & $28390^{+410}_{-120}$ & $5.39^{+0.05}_{-0.03}$ & $-3.07^{+0.21}_{-0.26}$ & \\
\hline
& Other names & $T_{\rm eff}$ & $\log{g}$ & $\log{(\rm He/H)}$ & Notes \\
J & & (K) & c.g.s. & & \\
\hline
123723.5$+$250400 & Feige 66 & $34300^{+160}_{-180}$ & $5.82\pm0.04$ & $-1.51^{+0.05}_{-0.07}$ & \\
160011.8$-$643330 & TYC 9044-1653-1 & $34640^{+590}_{-580}$ & $6.02^{+0.08}_{-0.11}$ & $-0.30^{+0.05}_{-0.04}$ & \\
\hline
\end{tabular}
\end{minipage}
\end{table*}
\section{Sample selection and observations}
Table~\ref{tbl_sum} lists the stars originally included in our radial velocity
survey with notable properties described in Section 3.1. The sample includes 38
spectroscopically confirmed hot subdwarf stars, and two objects that were
respectively identified as an early B star and a A\,V+B\,V binary. The early
B star GALEX~J1407+3103 is notable for its high radial velocity, while the
close A\,V+B\,V pair GALEX~J0716+2319 shows significant radial velocity
variations on a short time scale. All except two objects were randomly
selected from our catalogue of $GALEX$/Guide Star Catalogue ultraviolet-excess objects
\citep{ven2011,nem2012}. Briefly, the source catalogue includes bright objects ($N_{UV}<14$) with an ultraviolet excess
($N_{UV}-V<0.5$). The latter criterion still allows for the selection of hot subdwarf plus F/G dwarf pairs \citep[see ][]{ven2011}.
Two additional stars that were not observed by $GALEX$, including a blue-excess object \citep{jim2011},
are listed at the bottom of Table~\ref{tbl_sum} with J2000 coordinates.
The $GALEX$ name corresponds to the coordinates of the
ultraviolet source detected in the near ultraviolet (NUV) band (Section 2.3); for convenience, the
names are abbreviated to 4 digits right ascension
and declination. The ultraviolet coordinates are
generally close to the Guide Star Catalog (GSC2.3.2) optical coordinates ($<$1~arcsec), but, in a few cases, offsets as large as
4 to 9~arcsec occurred (GALEX~J1421+7124, J1427$-$2701, J1902$-$5130, J2344$-$3426).
Despite the offsets, the ultraviolet and optical sources must be one and the same.
These offsets cannot be attributed to a high proper-motion and are most likely
due to a distorted point spread function (PSF) in bright off-centred sources in the $GALEX$ images (Section 2.3).
Throughout this paper we will refer to the hot subdwarf as the primary and its companion as the secondary.
Table~\ref{tbl_sum} lists some notable particularities
such as the presence of a nearby star, whether unrelated or physically associated to the hot subdwarf, a bright main-sequence companion, or photometric variability due to
reflection on a late-type companion or stellar activity (see Section 3.1).
Most stars display H\,{\sc i}-dominated line spectra, but we also
noted the presence of He-rich subdwarfs characterized by He\,{\sc i} and
He\,{\sc ii}-dominated line spectra. The stellar parameters of a handful of subdwarfs locate them below the
zero-age EHB (ZAEHB) and these objects are likely progenitors of ELM white dwarfs (Sections 3.1 and 3.2).
\subsection{Intermediate to high-dispersion spectroscopy for radial velocity measurements}
Our first extensive set of observations was obtained with the Wide Field Spectrograph
\citep[WiFeS,][]{dop2007} attached to the 2.3~m telescope at the Siding Spring
Observatory (SSO). The observations were conducted on UT 2011 July 14 to 18,
UT 2011 December 2 to 3 and UT 2012 April 27 to 30.
We used the B3000 and R7000 gratings with a slit width
of 1~arcsec that provided spectral ranges
of 3200-5900 \AA\ at a resolution of $R=\lambda/\Delta\lambda=3000$ and 5300-7000 \AA\ at $R=7000$,
respectively. The RT560 dichroic beam splitter separated the incoming light
into its red and blue components. WiFeS is an image-slicing spectrograph with 25 slitlets ($38\times1$ arcsec)
and depending on the seeing, the target can cover a few slitlets. The signal-to-noise ratio (S/N) of
each observation was maximized by extracting the spectrum from the most significant ($\la 6$)
traces. Each trace was wavelength and flux calibrated prior to co-addition.
The spectra were wavelength calibrated using NeAr arc spectra that were
obtained either prior to or following each observation.
Next, our second set of observations was obtained using the Ritchey-Chr\'etien Focus (R.-C.) Spectrograph attached
to the 4~m telescope at Kitt Peak National Observatory (KPNO) on UT 4 - 6
January 2012. We used the KPC24 grating in second order combined with the T2KA
CCD to provide a spectral range of 6030 - 6720 \AA\ and a dispersion
of 0.52 \AA\ pixel$^{-1}$. The slit width was set to 1.5~arcsec which provided a
resolution of $\sim$0.9 \AA\ or $R=7000$. Contamination from third order was removed using
the GG495 filter. The spectra were wavelength calibrated using HeNeAr spectra
which were obtained following each observation.
We obtained a third set of observations using the ESO Faint Object Spectrograph
and Camera (EFOSC2) attached to the 3.6~m New Technology Telescope (NTT) at
La Silla Observatory in September 2012. We used grism number 20 centred on
H$\alpha$ providing a spectral range from 6040 to 7140 \AA\ and a dispersion
of $0.55$ \AA\ pixel$^{-1}$. We set the slit width to 0.7~arcsec resulting
in a $2$ \AA\ resolution or $R=3500$.
Next, we obtained additional spectra
with EFOSC2 on the NTT on UT 31 July and 1 August 2014. We used grism number
19 that provided a spectral range from 4435 to 5120 \AA\ and, after binning $2\times2$, a dispersion of
0.67 \AA\ per binned pixel. The
slit width was set to 1~arcsec resulting in a resolution of $\sim$2 \AA\ or $R\approx2000$.
Additional EFOSC2 spectra of GALEX~J1731+0647 were extracted from the ESO archive
(programme 090.D-0012, PI S. Geier). The data were also obtained with grism 19, but
binned $2\times1$ resulting in a dispersion of 0.34 \AA\ pixel$^{-1}$. The slit width
was set to 1~arcsec resulting in a resolution of $\sim$2 \AA.
All spectra were wavelength calibrated using
HeAr arc spectra which were obtained following each observation.
Also, we obtained a fourth set of spectra using the grating spectrograph attached
to the 1.9~m telescope at the South African Astronomical Observatory (SAAO) on
UT 2014 February 11. We used the 1200 lines/mm grating with a blaze
wavelength of 6800 \AA. This arrangement provided a range of 6023 to 6782 \AA\
with a dispersion of 0.439 \AA\ per pixel. The slit width was set to 1.05~arcsec
resulting in $R=7000$, or a resolution of $\approx$1\AA\ at H$\alpha$.
A CuNe comparison arc was obtained following each target
observation.
\begin{figure*}
\includegraphics[viewport=40 45 530 545, clip,width=0.9\textwidth]{fig1.pdf}
\caption{Line profile analysis of the WHT spectrum of Feige~66 (top) and KPNO spectrum of TYC4000-216-1 (bottom), labelled
with best-fitting parameters.
\label{fig_Feige66}}
\end{figure*}
We assembled a fifth data set with observations of the bright objects
HD~4539 (GALEX~J0047$+$0958), the spectro-photometric standard Feige~66, GALEX~J1421$+$7124, GALEX~J1736$+$2806,
and GALEX~J2334$+$5347 using the 2m telescope at Ond\v{r}ejov
Observatory. The observing configuration and procedure are described in
\citet{kaw2010a}. Briefly, for each star
we obtained a series of spectra centred on H$\alpha$.
We used the 830.77 lines mm$^{-1}$ grating with a SITe $2030\times 800$ CCD,
this resulted in a spectral resolution of $R = 13\,000$. Each target exposure
was immediately followed by a ThAr comparison arc.
Finally, and introducing our sixth and most recent observation programme,
we obtained three high-dispersion echelle spectra
of the short period binary GALEX~J2254$-$5515. From UT 24 November to 4 December 2014 we used the
Fiber-fed Extended Range Optical Spectrograph (FEROS) attached to the
2.2~m telescope at La Silla. The spectra range from $\approx$3600 to $\approx$9200 \AA\ at a
resolution of $R\approx 48,000$.
We supplemented our data sets with archival spectra.
We extracted processed FEROS data from the ESO archive.
The spectra were obtained under the programmes 076.D-0355, 077.D-0515, 078.D-0098 (PI: L. Morales-Rueda)
and 086.D-0714 (PI: S. Geier).
We also extracted spectra from the Isaac Newton Group (ING) Archive. The first set of
data (GALEX~J1632+0759 and GALEX~J1731+0647) was obtained with the Intermediate Dispersion Spectrograph (IDS)
attached to the Isaac Newton Telescope (INT) on UT 17 May 2013 (run numbers
984456 and 984458) and on UT 19 May 2013 (run numbers 984760, 984762 and 984763).
The spectra were obtained with the R1200B grating which
resulted in a useful range of 3900 to 5200 \AA\ and a
dispersion of 0.48 \AA\ and delivering a resolution of 1.5 \AA\ assuming a 3-pixel full-width at half-maximum (FWHM).
The spectra were wavelength calibrated using CuAr and CuNe arcs
and adjacent exposures were co-added to obtain the final radial velocity.
A second set of data (GALEX~J1632+0759) was obtained with the William Herschel Telescope (WHT) and
the Intermediate dispersion Spectrograph and Imaging System (ISIS) on
UT 26 August 2010 (run numbers 1483813 and 1483814). The spectra were obtained
with the R600B and R600R gratings and calibrated with CuAr and CuNe arcs resulting in
useful ranges of 3500-5100 \AA\ and 5500-7030 \AA\ and dispersion of 0.88 \AA\ per binned pixel in the blue ($2\times2$)
and 0.49 \AA\ pixel$^{-1}$ in the red (binned $2\times1$), corresponding to
spectral resolutions of 1.7 \AA\ in the blue and 1.5 \AA\ in the red assuming
a 3-pixel FWHM.
On average, a high signal-to-noise ratio was achieved with EFOSC2 on the NTT ($\overline{\rm S/N}\approx 100$),
the R.-C. Spectrograph on the KPNO 4~m telescope ($\overline{\rm S/N}\approx 60$),
and WiFeS on the SSO 2.3~m telescope ($\overline{\rm S/N}\approx 80$). A lower signal-to-noise ratio
was achieved with the coud\'e spectrograph on the Ond\v{r}ejov 2~m telescope, FEROS on the
MPG~2.2~m telescope (La~Silla), and the grating spectrograph on the SAAO 1.9~m telescope ($\overline{\rm S/N}\approx 30$).
The lower S/N achieved at Ond\v{r}ejov and La~Silla is largely compensated by the higher dispersion resulting
in comparable or superior velocity accuracy (see next Section). More than 70 per cent of our spectra had a
S/N$\gtrsim 40$ and spectra with ill-defined hydrogen or helium lines (S/N$\lesssim 15$) were rejected.
\subsubsection{Tests of the wavelength and velocity scales}
We performed a series of tests of the wavelength scale of relevant spectra using the O\,{\sc i}
sky emission lines and atmospheric molecular absorption bands.
Diffuse O\,{\sc i}$\lambda$6300.304 emission helps set the accuracy of the wavelength
scale, particularly in low- to intermediate-dispersion spectra.
A strong emission line is detected in 93 per cent of all usable EFOSC2 spectra,
95 per cent of all KPNO and SSO spectra, and nearly all SAAO spectra. A short exposure time as well
as the appearance of scattered moonlight usually limit the usefulness of this template.
The O\,{\sc i} velocity averaged $\varv$(O\,{\sc i})$=0.0$\,km\,s$^{-1}$\ at KPNO with a dispersion
$\sigma_\varv$(O\,{\sc i})$=2.1$\,km\,s$^{-1}$, $\varv$(O\,{\sc i})$=1.9$\,km\,s$^{-1}$\ with EFOSC2 and
a dispersion $\sigma_\varv$(O\,{\sc i})$=5.4$\,km\,s$^{-1}$,
$\varv$(O\,{\sc i})$=3.7$\,km\,s$^{-1}$\ at SSO and a dispersion $\sigma_\varv$(O\,{\sc i})$=7.4$\,km\,s$^{-1}$,
and $\varv$(O\,{\sc i})$=4.6$\,km\,s$^{-1}$\ at SAAO and a dispersion $\sigma_\varv$(O\,{\sc i})$=4.4$\,km\,s$^{-1}$.
The emission line appeared blended in most spectra obtained during bright time at SSO.
Based on this analysis the expected accuracy should be of the order of 2-7\,km\,s$^{-1}$.
The accuracy of the wavelength scale using coud\'e or echelle spectrographs is normally
of the order of 1\,km\,s$^{-1}$\ or better.
Systematic velocity shifts are expected following an improper placement of the star on the
slit, particularly if the stellar image is much narrower than the slit width.
Excellent seeing conditions are often encountered at La Silla and KPNO.
We cross-correlated telluric absorption features in the KPNO and EFOSC2 spectra
with a telluric template of identical
spectral resolution. We measured an average velocity of $-0.8$\,km\,s$^{-1}$\ with
a dispersion of 13.4\,km\,s$^{-1}$\ in the EFOSC2 spectra and an average
velocity of 2.4\,km\,s$^{-1}$\ with a dispersion of 10.0\,km\,s$^{-1}$\ in the KPNO
spectra. Velocity deviations of up to 50\,km\,s$^{-1}$\ were found in a few well-exposed EFOSC2 spectra:
We corrected the measured stellar velocities at La Silla using the telluric template velocities.
In summary, after applying telluric corrections, we estimate that errors in stellar velocity
measurements due to various systematic effects are better than $\sim$10\,km\,s$^{-1}$\ provided that
the photospheric lines are well defined. Ultimately, the accuracy of the wavelength is verifiable
using actual stellar data and by plotting the velocity dispersion distribution (Section 3.1.3).
\subsection{Low-dispersion spectroscopy for stellar parameter determinations}
For stars not listed in \citet{nem2012}, we obtained additional low dispersion
spectra with the R.-C. spectrograph attached to the 4m telescope at KPNO on
UT 2013 July 12 (GALEX J1421$+$7124) and 2014 May 24 (GALEX J2334$+$5347).
We used the KPC-10A grating and T2KA CCD with a dispersion of 2.77 \AA\ pixel$^{-1}$
in first order and centred on 5875 \AA. We used the order sorting filter
WG360 and set the slit width at 1.5~arcsec resulting in a spectral resolution of
$\approx 5.5$ \AA. The spectra were wavelength calibrated using the HeNeAr arc.
We extracted a set of spectra of the spectro-photometric standard Feige~66 from the ING archive. These spectra were
obtained with ISIS
attached to the WHT (run numbers 133198, 133200, 133201,
133223, 133226, 133227). The spectra were obtained using the R300B grating in
the blue arm providing a dispersion of 1.54 \AA\ pixel$^{-1}$ and a spectral
range from 3620 to 5190 \AA. The slit width was set to 2.4~arcsec for each
observation which corresponds to a resolution of $\approx 7.5$ \AA. The spectra
were wavelength calibrated using a CuAr arc.
Details of the low-dispersion spectroscopy obtained of other objects in the present sample are given by
\citet{ven2011} and \citet{nem2012}.
\subsection{Photometry and imaging}
We compiled available optical and infrared photometric measurements and combined them with
the $GALEX$ NUV and FUV photometry from the all-sky imaging survey (AIS) to build a SED for each object in the sample.
The ultraviolet data were collected from the site {\tt galex.stsci.edu/GalexView/}. \citet{mor2007} present details of the
instrument calibration. Table~\ref{tbl_phot} in the Appendix lists the $GALEX$ magnitudes, along
with the available $V$ magnitudes as well as 2MASS \citep{skr2006} and Wide-field Infrared Survey Explorer \citep[$WISE$,][]{wri2010}
infrared measurements.
The PSF of $WISE$ images ranges from 6 to 12~arcsec in the 3 to 24$\mu$m wavelength range, while the PSF in 2MASS images is close to
2.5~arcsec. Because of its relatively broad PSF, stars located within its range and identified in higher-resolution imaging
are certainly contaminating the SED in the mid-IR range.
Also, we extracted photometric time series from the SuperWASP \citep[SWASP,][]{pol2006} public archive,
NSVS \citep{woz2004},
All Sky Automated Survey
\citep[ASAS;][]{poj1997} and Catalina surveys \citep{dra2009}.
The Catalina photometry is unfiltered. Bright targets ($<12$~mag) are often saturated, but the photometric measurements are
more precise with faint targets ($>14$~mag) than those obtained in the other three surveys consulted.
The SWASP images are filtered (4000-7000 \AA).
Light curve analysis of SWASP data is valuable because of the large number of measurements obtained for individual
targets. We obtained ASAS times series in the V band and the NSVS images are unfiltered.
The calibrated $GALEX$ magnitudes are obtained from the count rates extracted using elliptical apertures (fuv\_flux\_auto, nuv\_flux\_auto)
fitted to the actual stellar profiles and
converted into the AB system.
The average $GALEX$ PSF is matched approximately by Gaussian functions with
FWHM of 5.3 and 4.2~arcsec in NUV and FUV images, respectively, and a positional accuracy of $\approx0.5$~arcsec.
However, several factors affect the reliability of the $GALEX$ photometric magnitudes.
The $GALEX$ imaging quality varies with the detector position with a strong dependency
on the radial distance from the image centre. We recorded the target distance to the
centre of the field of view (fov\_radius), as well as the actual FWHM values in the FUV and NUV
images (fuv\_fwhm\_world, nuv\_fwhm\_world) for each target. Measurements with a radial distance outside of
$0^\circ.4$ combined with a large PSF ($>0^\circ.01$) or measurements with an exceedingly large PSF ($>0^\circ.04$) are marked in Table~\ref{tbl_phot} as possibly unreliable. Finally, bright objects with unreliable non-linearity corrections outside the range
of validity are marked.
Non-linearity effects dominate the photometric error: A 10 per cent loss is observed at $N_{UV}=13.9$ and $F_{UV}=13.7$
so that most measurements in the present selection are affected. \citet{mor2007} and \cite{cam2014} propose correction
algorithms that are nearly identical. \citet{cam2014} presented a calibration sample sufficiently large
to allow us to evaluate the scatter in the synthetic versus measured magnitude relations. For example, this scatter
is of the order of 0.35 and 0.4 mag at $N_{UV}$ and $F_{UV}=13$, respectively. We adopted $GALEX$ magnitudes
adjusted using the correction algorithm of \citet{cam2014} with errors estimated using the scatter in these corrections for a
given magnitude.
The EFOSC2 acquisition images provide a deep, high-spatial resolution view of the
fields surrounding target stars. These images were obtained with the Loral/Lesser 2048$\times$2048
CCD. With a focal plane scale of 8.6~arcsec~mm$^{-1}$ and a pixel size of 15 $\mu$m,
the sky images are sampled with a pixel size of $0.129\times0.129$ arcsec$^2$, or, after binning $2\times2$,
a pixel size of $0.258\times0.258$ arcsec$^2$. The images allow for the identification of physical companions or
unrelated, nearby stars.
Fig.~\ref{fig_sed1}, Fig.~\ref{fig_sed2} and Fig.~\ref{fig_sed3} in the
Appendix compare all available photometry to synthetic spectra computed using
stellar parameters listed in Table~\ref{tbl_sum}.
\section{Analysis}
We present, in order, the properties of the sample including an overview of
the stellar parameters ($T_{\rm eff}$, $\log{g}$, $\log{\rm He/H}$) and evolutionary
history, the characteristics
of the SEDs and photometric time series, and the radial velocity data set.
We identify new binary candidates and present an analysis of individual binary properties
from the combined data sets.
\subsection{Sample properties}
\begin{figure}
\includegraphics[width=1.0\columnwidth]{fig2.pdf}
\caption{Physical properties, luminosity versus effective temperature, of the sample: sdO stars are shown
with full triangles while sdB stars are shown with open squares (assuming 0.47\,M$_\odot$) or
full squares (assuming 0.234\,M$_\odot$).
The zero-age EHB is labelled ``a'' while the terminal-age EHB
is labelled ``b''. Evolutionary tracks computed by \citet{dor1993} with a helium core mass of
0.469\,M$_\odot$\ and hydrogen envelopes of, left to right, 0.002, 0.004, 0.006, and 0.01\,M$_\odot$\ are shown
with full lines. The cooling track from \citet{dri1998} for progenitors of
ELM white dwarfs of 0.234\,M$_\odot$\ is
shown prior to hydrogen shell flashes with a dashed line. Lines of constant radii at
0.01, 0.1, 1, and 10\,R$_\odot$\ are labelled accordingly.
\label{fig_sample}}
\end{figure}
Table~\ref{tbl_sum} lists the atmospheric parameters obtained from \citet{nem2012}.
The Balmer line analysis for three additional objects (Feige 66, GALEX~J1421$+$7124
and GALEX~J2334$+$5347) is based on the model grids of \citet{ven2011}.
Best fitting parameters ($T_{\rm eff}$, $\log{g}$, $\log{\rm He/H}$) are obtained using $\chi^2$ minimization techniques with the
observed line profiles (He\,{\sc i,ii} and H\,{\sc i}) being simultaneously adjusted to
interpolated spectra from the model grid.
Examples of Balmer and helium line analyses are shown in Fig.~\ref{fig_Feige66}.
Fig.~\ref{fig_sample} shows properties of the sample presently investigated. Using
the effective temperature ($T_{\rm eff}$) and surface gravity ($g$) we
determined the total luminosity (in L$_\odot$) by adopting for most objects a
sample-average mass of 0.47\,M$_\odot$\ \citep{fon2012}.
\begin{displaymath}
L = 4\pi R^2 \sigma T_{\rm eff}^4,
\end{displaymath}
where $\sigma$ is the Stefan-Boltzmann constant and the radius ($R$) is calculated using
\begin{displaymath}
R = \sqrt{\frac{GM}{g}},
\end{displaymath}
where $M$ is the subdwarf mass and $G$ is the gravitational constant.
The sdB stars form a sequence of approximately constant luminosity, $L=$10-30\,L$_\odot$ or $M_V=4.3$ ($\sigma=0.9$) mag, and located between
the ZAEHB and the TAEHB while a few ageing sdB stars and all He-rich sdO stars set out on a higher luminosity
excursion beyond the stable He-burning stage. The objects
lying below the ZAEHB ($L<10$\,L$_\odot$) with a low-temperature and a high-gravity, GALEX~J0805$-$1058 and, tentatively,
J1411+7037 and J2153$-$7004,
were singled-out and were attributed a mass of 0.23\,M$_\odot$\ based on their likely evolutionary status \citep{dri1998}.
Most objects lie to the left of the EHB tracks suggesting that their hydrogen layer is thinner than
0.002\,M$_\odot$, or, possibly, that their surface gravity is overestimated.
To investigate the latter possibility, we compared the results of a model
atmosphere analysis using the hydrogen Stark broadening tables of
\citet{lem1997}, employed in the present work, to those of \citet{tre2009}, which include improved treatment of merging atomic energy levels.
We found that
improvements in Stark broadening theory may account for a shift of
$\Delta\log{g}=+0.08$ dex near 30\,000~K \citep[see, e.g., ][]{ost2014,
tel2014b} in agreement with a shift of $\Delta\log{g}=+0.06$ dex
found at 40\,000~K by \citet{kle2011}.
These systematic shifts are notable, but
still cannot explain the model offsets apparent in Fig.~\ref{fig_sample}.
Metallicity has little effect on temperature and gravity below
$T_{\rm eff}$$=35\,000$~K as demonstrated by \citet{lat2014}.
It is worth noting that, while a sdB mass of 0.47\,M$_\odot$\ may be typical, it
does not necessarily apply to all objects (e.g., ELM progenitors).
On the other hand, \citet{sch2014} pointed out that current evolutionary models fail to
reproduce some observed properties of EHB stars, such as the core mass derived
from asteroseismology, and concluded that evolutionary models must be updated to
match observed seismic and spectroscopic stellar parameters.
\citet{sch2014} found that very high convective overshooting would be needed to reproduce the seismic
core mass but that it
would, quite improbably, double the EHB lifetime.
Therefore, they conclude that the general treatment of convection in evolutionary
models needs updating, and that new opacity tables and diffusion
calculations are required.
\subsubsection{Overview of the SEDs}
\begin{figure}
\includegraphics[width=1.0\columnwidth]{fig3.pdf}
\caption{$NUV-V$ versus $V-J$ colour-colour diagram. Stars with a composite IR
excess are shown with full black circles and stars with an IR excess due to a
nearby star are shown with open circles (see Section 3.1.1),
while all others are shown in grey. Models at 24, 28, 32, 36, and
$40\times10^3$ K ($\log{g}=5.7$, $\log{\rm He/H}=-2.5$) are shown, from bottom to top, with open squares linked
by a full line. The effect of interstellar extinction ($E_{B-V}=0.05$) on the colours is shown with an arrow in the
upper left corner.
\label{fig_nmv_vmj}}
\end{figure}
\begin{figure}
\includegraphics[width=1.0\columnwidth]{fig4.pdf}
\caption{Same as Fig.~\ref{fig_nmv_vmj} but showing $V-J$ versus $J-H$.
\label{fig_vmj_jmh}}
\end{figure}
\begin{figure}
\includegraphics[width=1.0\columnwidth]{fig5.pdf}
\caption{Same as Fig.~\ref{fig_nmv_vmj} but showing $J-H$ versus $H-W1$.
\label{fig_jmh_hmw1}}
\end{figure}
Fig.~\ref{fig_nmv_vmj} shows the $NUV-V$ versus $V-J$ colour diagram
for the sample of hot subdwarfs listed in Table~\ref{tbl_phot}. Fig.~\ref{fig_vmj_jmh}
and Fig.~\ref{fig_jmh_hmw1} present the $V-J$ versus $J-H$ and
$J-H$ versus $H-W1$ diagrams, respectively.
The effect of interstellar extinction is evident in the $NUV-V$ colour of
some objects, but many are also affected by large systematic errors in the
$GALEX$ $NUV$ photometry (non-linearity).
For example, a larger extinction than observed in the interstellar line-of-sight is apparent toward GALEX~J1632+0759.
The International Ultraviolet Explorer ($IUE$) spectra that supplement uncertain $GALEX$ photometric measurements
indicate $E_{B-V}=0.4$, largely in excess of that found in the extinction map of \citet{sch1998}, $E_{B-V}=0.08$.
The effect on ultraviolet colours of an extinction coefficient $E_{B-V}=0.05$ (see Fig.~\ref{fig_nmv_vmj}) is relatively
modest, but coefficients in excess of 0.4 would displace colours from the upper left to the reddened, lower right corner
in the vicinity of composite stars.
Individual SEDs may be contaminated by the presence of a nearby star, either physically associated or unrelated to the hot subdwarf.
Inspection of the P82, P83, P85 and P89 EFOSC2 acquisition images obtained by \citet{ven2011} and \citet{nem2012} revealed the presence of nearby companions ($<3$~arcsec) to
J0047$+$0337, J0657$-$7324, and J1632$+$0759 (see below). Visual inspections of the guiding
images displayed at KPNO did not reveal the presence
of a nearby companion to any other objects. An inspection of photographic plate material ({\tt http://surveys.roe.ac.uk/ssa/index.html}) helps locate other, more distant
objects ($>3$~arcsec) that may contaminate photometric measurements with large PSF (e.g., $WISE$). For example,
GALEX~J0751+0925 is accompanied by a faint ($\Delta m\sim$5 mag), nearby star $\sim$6~arcsec away at
a position angle of 200$^\circ$ (Epoch 1993).
The composite nature of the spectra of GALEX~J0047+0337, J1411+7037, J1736+2806, J1753$-$5007, J2038$-$2657 \citep{nem2012}
are confirmed by their IR photometric colours.
The flux contributions in the V band for these hot subdwarfs are offset by $\sim$0.7 (possibly contaminated), $\sim$0.0 (weak secondary detection in the optical), $\sim$1.2, $\sim$1.3, and $\sim$1.3 mag, respectively,
relative to their observed composite $V$ magnitudes.
Although evident in the IR colours of GALEX~J1356$-$4934 (Fig.~\ref{fig_sed2}), the presence of a companion was not detected by \citet{nem2012}. The flux contribution from the hot subdwarf in the V band is offset by
$\sim$0.4 mag relative to the observed composite $V$ magnitude. We re-examined the
spectra of GALEX~J1356$-$4934 and we found weak signatures of a cool main-sequence companion in the blue spectrum used by \citet{nem2012}, and stronger
spectral lines representative of a cool main-sequence star in the red spectra used in this paper. We present our spectral decomposition of this system
in Section 3.2.2.
The IR colours for GALEX~J0047$+$0337, J0657$-$7324, and J1632$+$0759 are almost certainly contaminated by their nearby, resolved
companions.
\subsubsection{Overview of the photometric time series}
Table~\ref{tbl_bin_phot} summarizes the photometric time series analyses.
We included objects showing significant radial velocity variations (e.g., GALEX~J2205$-$3141),
objects with composite optical spectra (e.g., GALEX~J1736+2806), and, finally,
three objects with previously published analyses from the present survey: GALEX~J0321+4727 and GALEX~J2349+3844 \citep{kaw2010a}, and
GALEX~J1411$-$3053 \citep{ven2012}. Photometric variations observed in the sdB plus white dwarf system GALEX~J1411$-$3053 are an example of ellipsoidal variations in this class of objects.
Fourier transform calculations of available light curves (Section 2.3) uncovered
three objects with significant periodic variations. The light curves were analysed using
fast Fourier transform analysis from \citet{pre1992}. Both GALEX~J1736+2806 and GALEX~J2038$-$2657 are binaries
comprising a hot subdwarf with a more luminous optical companion, while GALEX~J2205$-$3141 is composed of a hot
subdwarf and late-type companion. The photometric variations in the latter are clearly timed with the orbital
period (see Section 3.2) and caused by reflection of the primary on the cool secondary. Variations in GALEX~J1736+2806
and GALEX~J2038$-$2657 may be caused by a spot on the surface of the secondary coupled to the rotation period.
The variable, double-peaked H$\alpha$ line profile of GALEX~J2038$-$2657 also implies the presence of surface inhomogeneities (see Section 3.3.1).
\begin{table*}
\centering
\begin{minipage}{\textwidth}
\caption{Photometric time series. \label{tbl_bin_phot}}
\renewcommand{\footnoterule}{\vspace*{-15pt}}
\renewcommand{\thefootnote}{\alph{footnote}}
\begin{tabular}{lccccccc}
\hline
Name & Survey & HJD range & Number & Period range & Semi-amplitude & Average magnitude & Standard deviation \\
& & (2450000+)& & (d) & (mmag) & (mag) & (mmag) \\
\hline
J0047+0337 & ASAS & 1868-5168 & 378 & $>0.02$ & $4.4\pm7.0$ & 12.336 & 94.5 \\
& NSVS & 1382-1549 & 177 & $>0.01$ &$17.8\pm3.4$ & 12.676 & 32.2 \\
J0321+4727 & NSVS & 1373-1630 & 173 & 0.26586 \footnotemark[1]\footnotetext[1]{Spectroscopic period.} &$61.3\pm3.9$ & 12.034 & 56.9 \\
& SWASP & 3196-4458 & 4575 & 0.26586 \footnotemark[1] &$43.5\pm1.0$ & 11.490 & 49.4 \\
J0401$-$3223 & SWASP & 3964-4485 & 14208 & $>0.01$ &$8.4\pm0.1$ & 11.268 & 12.4 \\
& & & & 1.85735 \footnotemark[2]\footnotetext[2]{Possible spectroscopic period.} & $0.8\pm0.1$ & & \\
J0507+0348 & Catalina & 3643-6592 & 347 & $>0.02$ & $14.4\pm1.8$ & 14.172 & 23.8 \\
& & & & 0.52813 $^a$ & $2.1\pm1.8$ & & \\
J0613+3420 & SWASP & 3232-4573 & 4700 & $>0.03$ &$12.7\pm2.2$ & 13.958 & 106.4 \\
J0751+0925 & ASAS & 2623-5131 & 198 & $>0.1$ &$80.3\pm20.3$ & 14.126 & 205.9 \\
& & & & 0.17832 $^a$ &$43.5\pm20.8$ & & \\
& Catalina & 3466-6368 & 119 & $>0.1$ &$12.0\pm3.2$ & 14.168 & 18.7 \\
& & & & 0.17832 $^a$ &$3.6\pm3.0$ & & \\
J0805$-$1058 & ASAS & 1868-5168 & 570 & $>0.1$ &$19.9\pm3.8$ & 12.270 & 65.6 \\
& & & & 0.17370 $^a$ &$11.7\pm4.8$ & & \\
& NSVS & 1488-1630 & 132 & $>0.01$ &$33.0\pm7.4$ & 12.812 & 59.8 \\
& & & & 0.17370 $^a$ &$10.9\pm7.2$ & & \\
J1356$-$4934 & ASAS & 1900-5088 & 729 & $>0.04$ &$3.3\pm3.9$ & 12.269 & 74.2 \\
J1411$-$3053 & ASAS & 1902-5088 & 1060 & 0.02449 \footnotemark[3]\footnotetext[3]{Ellipsoidal variations at half-spectroscopic period.} &$46.8\pm3.8$ & 12.342 & 88.2 \\
& SWASP & 3860-4614 & 13079 & 0.02449 \footnotemark[3] &$51.2\pm2.0$ & 12.723 & 165.6 \\
J1632+0759 & ASAS & 2175-5106 & 399 & $>0.1$ &$23.9\pm5.1$ & 12.763 & 74.5 \\
& & & & 2.9515 $^a$ &$3.9\pm5.2$ & & \\
& Catalina & 3466-6471 & 338 & $>0.02$ &$49.8\pm8.4$ & 12.587 & 121.6 \\
& & & & 2.9515 $^a$ &$7.5\pm9.5$ & & \\
& NSVS & 1275-1417 & 115 & $>0.01$ &$37.6\pm8.6$ & 13.248 & 69.3 \\
& & & & 2.9515 $^a$ &$17.9\pm8.9$ & & \\
J1731+0647 & ASAS & 2727-5009 & 73 & $>0.1$ &$98.7\pm49.6$ & 13.799 & 299.1 \\
& & & & 1.17334 $^a$ &$44.5\pm51.5$ & & \\
& Catalina & 3466-6457 & 105 & $>0.1$ &$74.9\pm4.7$ & 13.825 & 67.0 \\
& & & & 1.17334 $^a$ &$21.4\pm9.2$ & & \\
J1736+2806 & SWASP & 3128-4325 & 9140 & 1.33320 \footnotemark[4]\footnotetext[4]{Photometric period.} & $10.2\pm0.2$ & 11.639 & 17.0 \\
J1753$-$5007 & ASAS & 1947-5137 & 544 & $>0.1$ &$18.6\pm5.7$ & 12.955 & 94.2 \\
J1903$-$3528 & SWASP & 3860-4551 & 7388 & $>0.01$ &$3.6\pm0.6$ & 13.089 & 35.2 \\
J2038$-$2657 & NSVS & 1348-1483 & 42 & 1.860 $^d$ &$56.4\pm8.4$ & 11.950 & 51.1 \\
& SWASP & 3958-4614 & 8332 & 1.87022 $^d$ &$12.9\pm0.4$ & 11.856 & 28.2 \\
J2205$-$3141 & ASAS & 1873-5166 & 521 & 0.34156 $^d$ &$40.0\pm5.0$ & 12.381 & 81.4 \\
& Catalina & 3598-6217 & 252 & 0.34156 $^d$ &$46.2\pm6.2$ & 12.07 & 64.4 \\
& SWASP & 3862-4614 & 22731 & 0.34156 $^d$ &$26.7\pm1.0$ & 12.409 & 110.1 \\
J2254$-$5515 & ASAS & 1869-5168 & 672 & $>0.02$ &$20.0\pm3.8$ & 12.113 & 70.9 \\
& & & & 1.22702 $^a$ &$4.7\pm3.9$ & & \\
& Catalina & 3580-6076 & 78 & $>0.1$ &$98.4\pm16.8$ & 12.442 & 103.1 \\
& & & & 1.22702 $^a$ &$28.1\pm15.3$ & & \\
J2349+3844 & NSVS & 1321-1579 & 261 & $>0.1$ &$19.9\pm3.9$ & 12.287 & 44.7 \\
& & & & 0.46252 $^a$ &$5.0\pm3.9$ & & \\
& SWASP & 3154-4669 & 12175 & $>0.01$ &$1.7\pm0.2$ & 11.640 & 15.6 \\
& & & & 0.46252 $^a$ &$1.1\pm0.2$ & & \\
\hline
\end{tabular}
\end{minipage}
\end{table*}
An analysis of time series helps constrain the nature of the companion \citep{max2002}. For example, a simple geometric
model suggests that the presence of a late-type companion generally leads to detectable photometric variations phased
on the orbital period. This reflection effect scales as $R_2^2$, where $R_2$ is the secondary radius,
but only as $a^{-1/2}$, where $a$ is the orbital separation \citep{max2002}. Interestingly, an application
of these simple relations confirms the results of detailed light curve modelling and most importantly that,
for a given mass function, the effect of a lower binary inclination, which reduces the light contrast between inferior and
superior conjunctions as well as its intensity through increased binary separation, is compensated by the increased mass and radius of the secondary calculated using the
mass function, hence increasing the fraction of intercepted light. Following \citet{max2002} and adding
a slight modification to account for the effect of inclination on the visibility of the exposed hemisphere at inferior
and superior conjunctions, the amplitude of the variations is given by:
\begin{displaymath}
\delta\, m = 2.5 \log\Big{(} \frac{f^+}{f^-}\Big{)},
\end{displaymath}
where
\begin{displaymath}
f^\pm = 1 + \Big{(}\frac{R_2}{R_1}\Big{)}^2 \Big{(}\frac{R_1}{\sqrt{2}\,a}\Big{)}^{1/2} \frac{1\pm\sin{i}}{2},
\end{displaymath}
where $f^+$ is the relative flux at superior conjunction, $f^-$ is at inferior conjunction,
$i$ is the binary inclination and $R_1$ is the primary radius estimated from the measured surface gravity and
assumed mass (0.23 or 0.47\,M$_\odot$). The mass of the putative late-type secondary was estimated using the binary mass
function, the assumed primary mass and by varying the inclination angle: The radius is then estimated following the mass-radius
relation for late-type stars of \citet{cai1990}.
Applications of this approximate formula for $\delta\,m$ lead to an overestimation of the amplitude of a factor of $\approx 3$ when
applied to the well known case of GALEX~J0321+4727 (Table~\ref{tbl_bin_phot}):
The semi-amplitude of the phased light curve of GALEX~J0321+4727 reaches 44 and 61~mmag in the SWASP and NSVS data sets, respectively,
and reveals the presence of an irradiated late-type companion.
This simple model also shows that the amplitude of the variations is more
or less constant ($\pm30$ per cent) when varying the inclination as shown in the detailed models of \citet{max2002}.
Applying a factor of 0.3 to the amplitude calculated with the simple formula for $\delta\,m$ presented above should allow us to
confirm or rule out the presence of a late-type companion in the new binaries. For example, the companion to
GALEX~J2349+3844 \citep{kaw2010a} is almost certainly a white dwarf: The predicted semi-amplitude of variations due to a late-type companion
is $\approx 70$~mmag, while the observed variations are less than 1 and 5~mmag in the SWASP and NSVS phased light curves, respectively
(Table~\ref{tbl_bin_phot}). Based on this insight, the photometric times series will help constrain in the following Sections the nature of the companion in
the new binaries.
\subsubsection{Overview of the radial velocity data set}
We measured the radial velocities by fitting a Gaussian function to the deep and narrow
H$\alpha$ core for most red spectra, or He{\sc i}$\lambda$ 6678.15 in a few instances described below.
In the blue we used the H$\beta$ core,
He{\sc ii}$\lambda$4685.698, or He{\sc i}$\lambda$4471.48 if necessary (see below). All measured velocities are heliocentric corrected
and tabulated in Table~\ref{tbl_rad_vel} in Appendix~B. For each target Table~\ref{tbl_rad_vel} also includes the number of spectra, the average
velocity ($\bar{\varv}$) and velocity dispersion ($\sigma_\varv$).
\begin{figure}
\includegraphics[width=1.00\columnwidth]{fig6.pdf}
\caption{Number of objects per velocity dispersion bin (width$=5$\,km\,s$^{-1}$). The sample includes data
from Table~\ref{tbl_rad_vel} in Appendix~B and published results from the same survey \citep{kaw2010a,ven2012}.
\label{fig_stat}}
\end{figure}
Fig.~\ref{fig_stat} shows the distribution of the measured velocity dispersion.
The sample includes three objects published earlier \citep{kaw2010a,ven2012} and seven new identifications described in the following Section.
Three additional objects show significant radial velocity variations ($\sigma_\varv>10$\,km\,s$^{-1}$), but with insufficient sampling
to determine the orbital parameters. Adding two likely close binaries identified through photometric variations
(GALEX~J1736+2806 and J2038$-$2657, see Section 3.1.2) but for which
we only dispose of radial velocity measurements of the secondary, we estimate
that 15 out of 41 hot subdwarfs presently investigated are in close binaries, or a 37 per cent yield, lower than previously estimated \citep[e.g., ][]{cop2011}.
Our survey strategy aimed at short-period binaries would be insensitive
to long-period, low-amplitude variation ($<10$\,km\,s$^{-1}$) systems. A detailed comparison with the sample of known hot subdwarf binaries
should allow us to secure a global estimate of binarity in this population (Section 4.1).
\subsection{Individual properties}
Section 3.2.1 describes objects with variable radial velocities
suggesting the presence of a close binary companion:
Fig.~\ref{fig_0507}, Fig.~\ref{fig_0751},
Fig.~\ref{fig_0805}, Fig.~\ref{fig_1632}, Fig.~\ref{fig_1731}, Fig.~\ref{fig_2205},
and Fig.~\ref{fig_2254}
show results of the period analysis ($1/\chi^2$ versus frequency) for this group of objects.
The confidence level is set at 1$\sigma$ (66 per cent) for
a four-parameter ($p=4$) analysis with the $\chi^2$ normalized on the best-fitting solution
and the radial velocity measurements phased on the best-fitting period.
Sections 3.2.2, 3.2.3, and 3.2.4 review the properties of the remaining systems, i.e., those with
composite spectra, unresolved radial velocity variations, or photometric variability, respectively, and the likelihood
that they might belong to the close binary population.
Finally, Section 3.3 presents known facts concerning the remaining objects.
In the following Section, the subscript ``1'' refers to the hot subdwarf and the
subscript ``2'' refers to its companion. Similarly, the suffix ``B'' designates the companion.
The binary parameters are listed in Table~\ref{tbl_bin_param} including the number of spectra per object ($N$)
and the dispersion in velocity residuals ($\sigma_{vr}$).
\begin{figure}
\includegraphics[width=1.00\columnwidth]{fig7.pdf}
\caption{(Bottom) period analysis of radial velocity measurements of GALEX~J0507$+$0348 (full line) and 66 per cent confidence level (dashed line). (Top) radial velocity measurements phased on the orbital period ($0.52813$\,d) and
and best-fitting sine curve (full line) with the dispersion in velocity residuals shown in upper-right.
The initial epoch $T_0$ corresponds to inferior conjunction of the sdB star. Details are presented in Section 3.2.1.
\label{fig_0507}}
\end{figure}
\begin{figure}
\includegraphics[width=1.00\columnwidth]{fig8.pdf}
\caption{Same as Fig.~\ref{fig_0507} but for GALEX~J0751$+$0925.
\label{fig_0751}}
\end{figure}
\begin{figure}
\includegraphics[width=1.00\columnwidth]{fig9.pdf}
\caption{Same as Fig.~\ref{fig_0507} but for GALEX~J0805$-$1058.
\label{fig_0805}}
\end{figure}
\begin{figure}
\includegraphics[width=1.00\columnwidth]{fig10.pdf}
\caption{Same as Fig.~\ref{fig_0507} but for GALEX~J1632$+$0759. The tick mark above the best-fitting period
indicates the results of the period analysis of \citet{bar2014},
$P=2.951\pm0.001$\,d.
\label{fig_1632}}
\end{figure}
\begin{figure}
\includegraphics[width=1.00\columnwidth]{fig11.pdf}
\caption{Same as Fig.~\ref{fig_0507} but for GALEX~J1731$+$0647.
\label{fig_1731}}
\end{figure}
\begin{figure}
\includegraphics[width=1.00\columnwidth]{fig12.pdf}
\caption{Same as Fig.~\ref{fig_0507} but for GALEX~J2205$-$3141. The photometric period is marked
close to the peak frequency of the velocity periodogram.
\label{fig_2205}}
\end{figure}
\begin{figure}
\includegraphics[width=1.00\columnwidth]{fig13.pdf}
\caption{Same as Fig.~\ref{fig_0507} but for GALEX~J2254$-$5515.
\label{fig_2254}}
\end{figure}
\subsubsection{Close binaries}
\begin{table*}
\centering
\begin{minipage}{\textwidth}
\caption{Spectroscopic binary parameters. \label{tbl_bin_param}}
\renewcommand{\footnoterule}{\vspace*{-15pt}}
\renewcommand{\thefootnote}{\alph{footnote}}
\begin{tabular}{lccccccc}
\hline
& J0507$+$0348 & J0751$+$0925 & J0805$-$1058 & J1632$+$0759 & J1731$+$0647 & J2205$-$3141 & J2254$-$5515 \\
Parameter & & & & & & & \\
\hline
$P$ (d) & $0.528127$ & $0.178319$ & $0.173703$ & $2.9515$ \footnotemark[1]\footnotetext[1]{Based in part on the period obtained by \citet{bar2014}: $P=2.951\pm0.001$ d.} & $1.17334$ & $0.341543$ & $1.22702$ \\
$\sigma_P$\,(d) & $0.000013$ & $0.000005$ & $0.000002$ & $0.0006$ & $0.00004$ & $0.000008$ & $0.00005$ \\
$T_0$ (HJD) & $2456315.349$ & $2455972.827$ & $2456299.0335$ & $2456150.701$ & $2456313.119$ & $2456313.3387$& $2456444.616$ \\
$\sigma\,(T_0)$ & $ 0.015$ & $ 0.001$ & $ 0.0026$ & $ 0.016$ & $ 0.004$ & $ 0.0005$& $ 0.001$ \\
K (km\,s$^{-1}$) & $68.2\pm2.5$ & $147.7\pm2.2$ & $29.2\pm1.3$ & $54.9\pm4.6$ & $87.7\pm4.1$ & $47.8\pm2.2$ & $79.7\pm2.6$ \\
$\gamma$ (km\,s$^{-1}$) & $96.2\pm1.8$ & $15.5\pm1.6$ & $58.2\pm0.9$ & $-31.6\pm2.7$ & $-39.1\pm3.0$ & $-19.4\pm1.7$ & $4.2\pm2.0$ \\
$f_{\rm sec}$ (M$_\odot$) & $0.017\pm0.002$ & $0.059\pm0.003$ & $(4.4\pm0.6)\times10^{-4}$ & $0.048\pm0.013$ & $0.080\pm0.012$ & $0.0037\pm0.0005$ & $0.063\pm0.006$ \\
$M_1$\ (M$_\odot$) & (0.47) & (0.47) & (0.234) & (0.47) & (0.47) & (0.47) & (0.47) \\
$M_2$\ (M$_\odot$) & $>0.20$ & $>0.34$ & $>0.03$ & $>0.31$ & $>0.39$ & $>0.11$ & $>0.35$ \\
N & 16 & 19 & 23 & 12 & 16 & 13 & 24 \\
$\sigma_{\varv}$ (km\,s$^{-1}$) & 6.5 & 6.4 & 2.8 & 8.6 & 6.8 & 5.4 & 8.2 \\
Notes & probable WD & probable WD & low mass sdB, & probable & probable & reflection, & probable \\
& secondary & secondary & possible BD & WD secondary & WD secondary & dM secondary & WD secondary \\
& & & secondary & & & & \\
\hline
\end{tabular}
\end{minipage}
\end{table*}
The sdB star GALEX~J0507+0348 is part of a newly identified spectroscopic binary.
The star is close to the ZAEHB and may be a low-mass sdB star.
The H$\alpha$ radial velocity measurements are phased on a period of $\sim$0.528\,d (Fig.~\ref{fig_0507}).
The SED of GALEX~J0507$+$0348 shows an infrared excess but in the $WISE$ W3 band only.
The nearby object ($sep.=17$~arcsec) visible on photographic plates is not likely to affect the $WISE$ measurements.
Also, low-dispersion spectra show weak CaH\&K lines with an equivalent width of
$E.W.$(CaK)$=270$\,m\AA. The calcium doublet could indicate the presence of a late-type companion. However, no radial
velocity measurements were obtained in that spectral region and we could not confirm variations in the
line position. Moreover, we could not confirm the presence of other late-type spectral signatures such as Mg{\sc i} lines,
and our composite spectral analysis \citep{nem2012} rejects the presence of a companion with
a flux contribution above 1 percent in the optical range.
A series of high dispersion spectra
is required to determine whether the CaH\&K lines originate from the companion, the interstellar medium (ISM), or in the circumstellar environment.
The mass function allows us to infer $M_2>0.20$\,M$_\odot$\ assuming $M_1=0.47$\,M$_\odot$, or
$M_2>0.15$\,M$_\odot$\ assuming $M_1=0.30$\,M$_\odot$. A late-type (dM,dK) companion would satisfy these constraints.
However, the Catalina time series folded on the orbital period constrains the photometric variations to a semi-amplitude lower than 2~mmag.
Reflection effect on a late-type star with a mass exceeding 0.2\,M$_\odot$\ would result in variations of a semi-amplitude of $\approx60$~mmag as observed in the
case of GALEX~J0321+4727 \citep{kaw2010a}.
We conclude that the companion is most likely a white dwarf: At a binary inclination $i<30^\circ$ the mass function implies a minimum mass of 0.51\,M$_\odot$\ that would be consistent with a normal white dwarf star.
The sdB star GALEX~J0751$+$0925 is part of a close binary with the largest velocity semi-amplitude
measured in the present sample, $K\sim$148\,km\,s$^{-1}$\ (Fig.~\ref{fig_0751}).
The mass function implies the presence of a companion relatively more massive than
in other similar systems with $M_2>0.34$\,M$_\odot$\ assuming $M_1=0.47$\,M$_\odot$.
The SED of this system also appears to show an infrared excess in the W1, W2, and W3 bands
which may be caused, in part, by a
nearby star only 6~arcsec away (particularly in the W3 band).
The Catalina and ASAS light curves do not show significant variations at the orbital period. Again, a comparison with
the photometrically variable system GALEX~J0321+4727 indicates that the companion is probably
a white dwarf.
The semi-amplitude of the phased light curve of GALEX~J0751$+$0925 is limited to 4~mmag in the Catalina observations although
the minimum secondary mass in this shorter-period system
is larger than in GALEX~J0321+4727, i.e., 0.34 versus 0.13\,M$_\odot$, and the predicted semi-amplitude of variations
due to a late-type companion would be $\approx190$~mmag. Therefore, the absence of a reflection effect in GALEX~J0751$+$0925
rules out the presence of a late-type companion leaving only the possibility of a $>0.34$\,M$_\odot$\ white dwarf companion,
or $>0.50$\,M$_\odot$\ if $i<50^\circ$.
The sdB GALEX~J0805$-$1058 clearly lies below the ZAEHB (Fig.~\ref{fig_sample}); following the evolutionary tracks of
\citet{dri1998} the mass of the subdwarf is estimated to be in the range 0.2-0.3\,M$_\odot$. The low velocity
amplitude (Fig.~\ref{fig_0805}) and small mass function imply
a very low mass for the companion, $M_2>0.03$\,M$_\odot$, assuming $M_1=0.234$\,M$_\odot$.
At inclinations higher than $i = 26^\circ$, the secondary mass remains lower than 0.08\,M$_\odot$\ and the object is substellar.
Assuming a probability distribution for the inclination angle $i$ of the form $P_i\,di=\sin{i}\,di$, inclinations higher than
$26^\circ$ have $P(>26)=$89.9 per cent probability of occurring.
At lower inclinations ($10^\circ<i<26^\circ$, i.e., $P(10-26)=$8.6 per cent), the secondary mass does not exceed 0.3\,M$_\odot$\ and the object would be a low-mass M dwarf.
At very low inclinations ($i<7^\circ$, i.e., $P(<7)=$0.7 per cent) the secondary would be a normal white dwarf star ($M>0.5$\,M$_\odot$).
The SED shows a single, hot subdwarf star, i.e., as far as the $WISE$ W2 band, and the faint ($\Delta m\sim$6 mag), nearby ($sep.\sim$11~arcsec) object visible in photographic plates does not appear to contaminate
the SED. The ASAS and NSVS light curves do not show significant variations when folded on the orbital period,
i.e., $\lesssim12$~mmag, while the predicted semi-amplitude of variations due to a substellar object
\citep[$R_2\approx 0.1\,R_\odot$, see a review by][]{cha2009} would be $\approx20$~mmag.
Variations of the order of 10~mmag have been observed in brown dwarf plus hot subdwarf binaries \citep[see, e.g.,][]{sch2014c}
and such variations would be detectable in GALEX~J0805$-$1058 in quality photometric time series.
We conclude that the apparent lack of variations is a consequence of the
small radius of a substellar secondary.
\citet{bar2012} recorded radial velocity variations in spectra of the sdB star GALEX~J1632+0759. Their data
suggested a period ranging from 2 to 11 days.
Our measurements, based on H$\alpha$ and He\,{\sc i}5875.621 in the red,
and H$\beta$ and He\,{\sc i}4685.698 in the blue, also revealed large velocity variations (Fig.~\ref{fig_1632}).
Recently, \citet{bar2014} obtained new radial velocity
measurements and determined a period of $2.951\pm0.001$\,d.
We restricted our period analysis to frequencies between 0.3 and 0.4\,d$^{-1}$ and recovered
an identical period. The mass function implies a secondary mass $M_2>0.31$\,M$_\odot$, assuming $M_1=0.47$\,M$_\odot$.
The SED shows a large flux excess apparent in the 2MASS and $WISE$ bands as well as heavy extinction in
the ultraviolet range. The measured extinction coefficient ($E_{B-V}=0.4$) largely exceeds the coefficient
inferred from the maps of \citet{sch1998}, $E_{B-V}=0.08$.
The additional extinction probably originates in the immediate, possibly dusty,
circumstellar environment of the system.
An inspection of our acquisition images of GALEX~J1632$+$0759 reveals the presence of
a nearby star; we measured a separation of 2.3~arcsec at a position angle of 225$^\circ$.
The 2MASS and $WISE$ photometric measurements are likely
contaminated by this object. \citet{ost2005} also resolved GALEX~J1632$+$0759
and the nearby star and measured a separation of 2.1~arcsec. In addition to GALEX~J1632$+$0759, \citet{bar2014}
obtained radial velocity measurements of the nearby star which they classified
as a late G dwarf or early K dwarf that is itself in a close binary: The radial velocity varied with a period of $1.42\pm0.01$~d.
\citet{bar2014} also found that
the systems share the same systemic velocity
suggesting that this is a quadruple system. The ASAS, Catalina, and NSVS time series constrain photometric variations to
semi-amplitudes lower than 4, 8, and 18~mmag. The predicted semi-amplitude of photometric variations due to
the presence of a late-type companion would be $\approx60$~mmag. We conclude that the secondary star is most probably
a white dwarf with a mass ranging from 1.3 to 0.5\,M$_\odot$\ assuming a low inclination ($24\lesssim i\lesssim 46^\circ$),
or with a peculiar low mass ($0.3-0.5$\,M$_\odot$)
assuming $i\gtrsim 46^\circ$.
The new binary system GALEX~J1731$+$0647 (Fig.~\ref{fig_1731}) harbours the heaviest binary companion identified
in our sample. The mass function implies a mass $M_2>0.39$\,M$_\odot$, assuming $M_1=0.47$\,M$_\odot$.
The field surrounding this subdwarf is relatively crowded but only two objects are found
between 13 and 15~arcsec away and with photographic magnitude differentials
$\Delta m\sim$3 and 5 mag. These objects would not affect the SED which shows a single hot subdwarf.
The lack of photometric variations, $<45$~mmag in ASAS time series and $<21$~mmag in Catalina time series, compared
to expected variations of $\approx90$~mmag due to a relatively
large M or K dwarf suggests that
the companion is most likely a white dwarf.
We infer a mass between 1.3 and 0.5\,M$_\odot$\ assuming an inclination of $29\lesssim i\lesssim 58^\circ$,
or a peculiar low mass ($0.4-0.5$\,M$_\odot$)
assuming $i\gtrsim 58^\circ$.
\begin{figure*}
\includegraphics[width=1.00\textwidth]{fig14.pdf}
\caption{(Left panels) Fourier transform analysis of the variable stars GALEX~J2205$-$3141 (top), J1736+2806 (middle) and J2038$-$2657 (bottom), and
phased light curves (right panels). The identification of the photometric period with the spectroscopic period clearly indicates that the light curve of GALEX~J2205$-$3141
shows the effect of reflection of the bright primary on the secondary star.
The initial epoch $T_0$ in GALEX~J2205$-$3141 corresponds to the passage of the secondary star at superior conjunction.
Without knowing their orbital periods, we can only state that the cool, secondary stars in both GALEX~J1736+2806 and J2038$-$2657 are variable.
The photometric periods are marked with star symbols: $P=0.341563$~d (J2205$-$3141), 1.333204~d (J1736+2806), and 1.870221~d (J2038$-$2657).
\label{fig_swasp}}
\end{figure*}
GALEX~J2205$-$3141 is a close binary with $P\approx 0.34$\,d (Fig.~\ref{fig_2205}) showing a reflection effect in the SWASP time series
(semi-amplitude $\Delta m \approx 27$~mmag). Similar variations were also observed in the
ASAS and Catalina time series. The mass function is consistent with the presence of a late-type
companion ($M_2>0.11$\,M$_\odot$).
Photometric time series from the Catalina survey, SWASP (1SWASP~J220551.98$-$314103.9), and ASAS show
variability with mutually consistent periods of $0.341559\pm0.000003$, $0.341563\pm0.000002$, and
$0.341561\pm0.000002$\,d, respectively. The photometric periods are somewhat longer than
the spectroscopic orbital period $P=0.341543\pm0.000008$\,d.
The radial velocity measurements are based on the He{\sc i} 4471.48 and 6678.15 lines.
We noted that the Balmer H$\alpha$ and H$\beta$ lines cores are asymmetric and are possibly contaminated by emission from the
companion as noted in the case of AA~Dor \citep{vuc2008}.
Fig.~\ref{fig_swasp} shows the SWASP measurements phased on the photometric period.
We identify the initial epoch with the passage of the secondary star at superior conjunction
corresponding to maximum reflected light.
Although the photometric variations are clearly caused by the reflection of the primary light
on a late-type dwarf companion, the phasing error between photometric and spectroscopic ephemeris is $\Delta \Phi \approx 0.1$.
We attribute this error to a large gap between the epoch of the spectroscopic observations and that of the
photometric observations. The SED shows a mild flux excess in the IR to mid-IR range possibly due to
the late-type companion. A star found 9~arcsec away and 4 mag fainter does not affect the SED.
However, renormalizing on the $J$ band rather than the $V$ band nearly eradicates this excess.
Assuming a possible $K$ band contribution from the companion of 15 to 40 per
cent, we estimated for the M dwarf companion $M_{K,2}\approx7.5$ to 6.5 if
$M_{K,1}\approx 5.5$. Bearing in mind that reprocessing of ultraviolet
radiation from the hot primary into the cool secondary
atmosphere should contribute to this IR excess,
the absolute K magnitude of the secondary star corresponds to a spectral type later than M3-4, or a mass $M_2\lesssim 0.24$-0.4\,M$_\odot$\ which
requires an orbital inclination $i\gtrsim 20$-30$^\circ$.
We find possible evidence of extinction in the ultraviolet range in excess of the extinction expected
from the \cite{sch1998} map, although the $GALEX$ NUV dip may be the result of larger uncertainties than estimated.
This system is the only confirmed binary in our sample comprised of a hot subdwarf
and late-type companion.
The sdB GALEX~J2254$-$5515 shows large radial velocity variations (Fig.~\ref{fig_2254}) although
the Catalina and ASAS time series indicate that the star is not photometrically variable with semi-amplitudes lower
than 28 and 5~mmag, respectively.
The minimum mass of the secondary, $M_2>0.35$\,M$_\odot$\ assuming $M_1=0.47$\,M$_\odot$, combined with the lack of photometric
variability when compared to expected variations of 150~mmag caused by a reflection effect on a putative late-type companion
imply that the
companion is a white dwarf.
We neglected the possible effect of orbital eccentricity in the period analysis.
The orbits of post-common envelope binaries is expected to be circular
due to the synchronization during the post-CE phase.
However, eccentric orbits in close binaries containing a subdwarf were reported by
\citet{ede2005} and \citet{kaw2012a}. In these cases the eccentricity was
small ($e < 0.1$). Larger eccentricities were reported for long
period binaries, such as BD$+20^\circ 3070$, BD$+34^\circ 1543$, Feige~87
\citep{vos2013} and PG~1449$+$653 \citep{bar2013a}.
Eccentric orbits may indicate the presence of a circumbinary disc
\citep{art1991}.
Now, we summarize additional constraints on the properties of spectroscopic composites, other
likely systems showing radial velocity variations, and systems displaying photometric variability.
\subsubsection{Composite spectra}
Using spectral decomposition, \citet{nem2012} classified GALEX~J0047$+$0337 as a
binary consisting of a hot sdB and a main-sequence F star.
The radial velocity measurements obtained for GALEX~J0047$+$0337B imply a constant velocity with standard deviation of
only 6.3\,km\,s$^{-1}$\ and include a single measurement
deviating from the average velocity by more than 10\,km\,s$^{-1}$.
The ASAS and NSVS photometry
do not show evidence of significant variations: The ASAS data constrain potential variations to a
semi-amplitude lower than 4.4~mmag for all periods larger than 0.5 hr.
The EFOSC acquisition images revealed a nearby star approximately
$\sim$3~arcsec away at a position angle of 344$^\circ$. The object is about
1.2~mag fainter in $R$ and the quoted $WISE$ and
2MASS magnitudes include both stars since they would not be resolved in either
surveys. Fortunately, our optical spectra were not contaminated by the nearby star and the composite nature of
the object is not affected.
GALEX~J1411+7037 and J1753$-$5007 are sdB stars with F-type companions.
Their SEDs are consistent with the presence of a luminous companion derived from the
spectral decomposition of \citet{nem2012}. The H$\alpha$ line profile in each star
is dominated by the main-sequence star and no significant radial
velocity variations have been found for these objects.
The ASAS times series of GALEX~J1753$-$5007 constrain photometric variations
to a semi-amplitude lower than 19~mmag.
The SED of GALEX~J1356$-$4934 shows significant
infrared excess. An inspection of the acquisition images did not reveal
a resolvable, nearby companion and
radial velocity measurements show only marginal variability with radial velocity
maxima reaching a span of 20\,km\,s$^{-1}$. First, we performed a SED decomposition to estimate
the spectral type of the companion. We adopted the sdB parameters determined by \citet{nem2012} and
calculated sdB absolute magnitudes of $M_K = 5.47$ and $M_V = 4.46$.
Adopting the apparent visual magnitude $V=12.3$ and 2MASS magnitude $K=11.633$ and using the
main-sequence colour and absolute magnitude relations from \citet{pec2013}, we determined the absolute
visual and infrared magnitudes of the late-type companion, $M_{K,2}=3.57$ and $M_{V,2}=5.36$, and a distance of 444 pc.
Consequently, the companion mass is 0.94\,M$_\odot$\ corresponding to a G8V star.
Next, we
performed a spectral decomposition with XTGRID \citep{nem2012} making use of both the blue
and red spectra of GALEX~J1356$-$4934. The spectral decomposition showed that the companion
contributes 27 per cent of the flux at 7000 \AA. The new parameters of the
sdB star are $T_{\rm eff} = 32370^{+230}_{-660}$ K, $\log{g} = 5.72^{+0.07}_{-0.16}$, $\log{\rm He/H} = -2.75^{+0.25}_{-0.43}$
and do not differ significantly from our earlier measurements. The parameters of the companion are
$T_{\rm eff} = 5470$, $\log{g} = 4.47$, [Fe/H] $=0.003$, also corresponding to a G8 main-sequence star.
These values supersede those of \citet{nem2012} for GALEX~J1356$-$4934. The ASAS time series limits the photometric
variations to a semi-amplitude of 3~mmag.
Optical spectra of subdwarf plus early-type F-stars are dominated in the red by the companion.
Because the mass ratio is $\gtrsim$3, high-dispersion spectroscopy is required to detect the
secondary star motion.
\subsubsection{Radial velocity variable}
Other objects, in addition to the confirmed binaries listed in Table~\ref{tbl_bin_param}, are likely close
systems. The measured radial velocity extrema suggest that these subdwarfs are in close orbit
with a companion, but the small number of spectra did not allow us to perform a period analysis.
We measured velocity extrema $\Delta\varv\approx80$\,km\,s$^{-1}$\ for GALEX~J0613$+$3420,
$\Delta\varv\approx28$\,km\,s$^{-1}$\ for GALEX~J0812$+$1601, and
$\Delta\varv\approx35$\,km\,s$^{-1}$\ for GALEX~J1903$-$3528.
The SWASP time series for GALEX~J0613$+$3420 and GALEX~J1903$-$3528 constrain photometric variations
to maximum semi-amplitudes of 13 and 4~mmag, respectively, which exclude the presence of close late-type companions.
Further investigations are required to clarify their binary status.
The SED of each object does not reveal the presence of a companion, however the SED
of GALEX~J0613$+$3420 shows evidence of a large interstellar extinction \citep{sch1998},
and a possible excess ($E_{B-V}=0.64$) above the interstellar value ($E_{B-V}=0.36$).
\subsubsection{Photometrically variable}
The SWASP light curve of the
sdB plus F7V pair GALEX~J1736+2806 (1SWASP~J173651.18+280634.6) varies with a period $P=1.33$\,d and a semi-amplitude
of 11~mmag (Fig.~\ref{fig_swasp}). No significant variations were observed in a nearby comparison object (1SWASP~J173635.80+280902.2).
The grouping of data points observed in the light curve are also observed in the light curve of the nearby object and, therefore,
it must be an artefact of data sampling.
Using the SED we found that the absolute $V$ magnitude of the companion is $\sim$0.72~mag brighter than the sdB star
consistent with a value of $\sim$0.81~mag obtained by \citet{nem2012}.
The absolute magnitude of a late F7 star, $M_V\sim$4, would imply for the hot subdwarf
$M_V\sim$4.7; The atmospheric parameters of the hot subdwarf are very uncertain \citep{nem2012}
but would be reconciled with the companion spectral type at the lowest acceptable temperature (30,000K at $\log{g}=5.7$).
The photometric variability may be caused by irradiation of the exposed hemisphere of the
F star although we failed to detect radial velocity variations at the same period.
GALEX~J2038$-$2657 is a relatively luminous hot sdO star with a G type companion \citep{nem2012}.
Our spectroscopic observations revealed variability in the H$\alpha$ profile
(Fig.~\ref{fig_2038}) on a time-scale of a day or less. However, cross-correlation
measurements in the spectral series dominated by the G companion show little variations with a dispersion $\sigma_v=7.8$\,km\,s$^{-1}$\
comparable to the expected accuracy of the wavelength scale. The measurements imply
that the velocity semi-amplitude of the G8III star does not exceed $\approx16$\,km\,s$^{-1}$.
Fig.~\ref{fig_2038_sed} shows the SED of
GALEX~J2038$-$2657 where the ultraviolet range is dominated by the hot subdwarf
and the optical range by the red giant.
The SWASP time series (1SWASP~J203850.49$-$265754.2) reveals variations of 12~mmag semi-amplitude over a period of 1.87022~d (Fig.~\ref{fig_swasp}) that are confirmed by similar
variations in a short NSVS time series. Again, no significant variations were observed in a nearby comparison object
(1SWASP~J203851.22$-$265943.1).
These variations are most likely linked to the observed spectroscopic variability,
but they cannot yet be clearly associated to a possible orbital period.
The system shares some properties with the sdB plus K\,III-IV system HD~185510 \citep{jef1992,fek1993} and
the sdO plus K0\,III system FF~Aqr \citep{vac2003}. All three systems have an evolved secondary star, from
sub-giant to giant, and all three are photometrically variable. However, the hot subdwarf in HD~185510 is
possibly the progenitor of a low-mass, helium white dwarf \citep[0.3\,M$_\odot$, ][]{jef1997}. Photometric
variations in HD~185510 and FF~Aqr coincide with the orbital period and are caused by irradiation of the
exposed hemisphere of the secondary stars. Moreover, the orbital periods HD~185510 and FF~Aqr are 20.7 and 9.2~d, respectively,
with orbital separations of $\sim$43 and $\sim$25\,R$_\odot$, respectively, and well outside the radius of a
sub-giant or giant star ($R({\rm K0\,III})\approx 16$\,R$_\odot$).
Without an estimate of the orbital period we can only set limits to the orbital parameters, such as the binary separation. The identification of the
late-type giant secondary is based on spectral decomposition \citep{nem2012}: The absolute $V$ magnitude of the hot sdO star
is only about $\sim$1.0\,mag fainter than its companion. Adopting a G8\,III type from the spectral decomposition shown in Fig.~\ref{fig_2038_sed},
the absolute magnitude of the companion is $M_V$(G8\,III)$=0.9$, implying an absolute magnitude $M_V$(sdO)$=1.9$ for the primary in agreement
with the estimate of \citet{nem2012}, $M_V$(sdO)$=2.0$. The minimum orbital period for a systemic mass of 2-3\,M$_\odot$\ and an orbit outside the
G8\,III radius (15\,R$_\odot$) is $P\gtrsim$3.1\,d.
Adopting a radius of 15\,R$_\odot$ for the G8\,III star, the photometric period of 1.87~d implies a rotation velocity $\varv_{\rm rot}\approx350$\,km\,s$^{-1}$.
The narrowest features in the SSO spectra have a width of $\varv_{\rm rot}\sin{i}=130$\,km\,s$^{-1}$, that would enforce
a low inclination $i\lesssim 22^\circ$. High dispersion spectroscopy is necessary to help determine the orbital parameters and
help clarify the origin of
the photometric variations.
The most likely scenario is that the photometric variations are caused by a surface spot coupled to the rotation of the
star, and that the orbital period probably exceeds several days with a low velocity amplitude ($K\lesssim$20\,km\,s$^{-1}$).
\begin{figure}
\includegraphics[width=1.0\columnwidth]{fig15.pdf}
\caption{Spectra of the photometrically variable sdO+G8III star GALEX~J2038$-$2657 obtained at SSO and La Silla and showing short-term variable H$\alpha$ emission.
\label{fig_2038}}
\end{figure}
\begin{figure}
\includegraphics[width=1.0\columnwidth]{fig16.pdf}
\caption{Spectral energy distribution of GALEX~J2038$-$2657 combining a cool G8III secondary
and a sdO primary star (full line). Individual contributions are shown with grey lines. The effect of interstellar extinction ($E_{B-V}=0.08$) is included.
\label{fig_2038_sed}}
\end{figure}
\subsection{Notes on other objects from this survey}
GALEX~J0047$+$0958 (HD 4539) is a well known hot sdB star \citep[see, e.g., ][]{kil1984}. Spectropolarimetric
measurements hint at the presence of a weak magnetic field
\citep[$\sim 0.5$ kG][]{lan2012}. \citet{sch2007} reported line profile
variations and radial velocity variations of a few km\,s$^{-1}$\ that may be due
to g-mode pulsations. \citet{lyn2012} obtained photometric series and measured
variations with a frequency of $9.285\pm0.003$~d$^{-1}$ and an amplitude of
$0.0023\pm0.0003$ mag. This photometric frequency is consistent with one of the
frequencies ($9.2875\pm0.0003$~d$^{-1}$) determined from low-amplitude radial velocity
variations, and both are possibly associated to stellar pulsation.
The SED of GALEX~J0049+2056 (Fig.~\ref{fig_sed1}) shows an IR excess that could be attributed to a yet unidentified companion
or nearby object, or to a dusty environment.
The sdB GALEX~J0059$+$1544 (PHL~932) is
embedded in an emission nebula. However, \citet{fre2010} have shown that
the association is only coincidental, but that PHL~932 does contribute and ionize a dense
region of the ISM surrounding it.
The SED of this object shows, as in the case of GALEX~J0049+2056, a considerable IR excess (Fig.~\ref{fig_sed1}).
Several radial velocity measurements of PHL~932 were reported in literature.
\citet{arp1967} measured $15\pm20$\,km\,s$^{-1}$\ using two low-dispersion spectra.
\citet{ede2003a} measured $18\pm2$\,km\,s$^{-1}$\ using echelle spectra. These
velocities are in agreement with our measurements ($\bar{\varv} = 16.7$,
$\sigma = 3.1$\,km\,s$^{-1}$) and, therefore, it does not appear that PHL~932
is in a close binary.
\citet{gei2012a} report a rotational velocity of $v_{\rm rot} = 9.0\pm1.3$\,km\,s$^{-1}$.
\citet{lan2012} obtained spectropolarimetric measurements of PHL~932 but did not detect a magnetic field with an
upper limit of $\sim$300 G.
\citet{bro2008} classified GALEX~J0206$+$1438 (CHSS~3497) as a hot subdwarf.
Our radial velocity measurements vary only marginally ($\sigma_\varv < 10$\,km\,s$^{-1}$), and we do not dispose of
sufficient data to determine a period. A radial velocity measurement of
$V_{r} = 7\pm16$\,km\,s$^{-1}$\ was obtained by \citet{bro2008} and is
consistent with our measurements ($\bar{\varv}=13.8,\ \sigma_\varv=7.5$\,km\,s$^{-1}$).
GALEX~J0232$+$4411 (FBS 0229$+$439) was classified as a sdB star in the First
Byurakan Survey of blue stellar objects \citep{mic2008}.
\citet{cop2011} presented a set of radial velocity measurements for GALEX~J0401-3223 which suggest
that the sdB star is in a close binary system, but
were unable to determine the orbital period with limited data.
\citet{cop2011} measured an average velocity and dispersion of $\varv\pm\sigma_\varv=55.2\pm4.4$\,km\,s$^{-1}$, consistent with our own measurements.
We have combined the \citet{cop2011} data with ours and conducted a period search. We found a best period of 1.8574~d, however
two significant aliases at $P = 0.64$ and 0.066~d cannot be ruled out. The velocity semi-amplitude at all three
periods does not exceed 10 km~s$^{-1}$ and excludes a white dwarf or late-type companion. The relatively short period and low velocity amplitude imply a minimum mass in the
substellar range, 0.01-0.04\,M$_\odot$. The SWASP data folded on the best period (1.8574~d) constrain photometric variations to a
semi-amplitude of only 1~mmag,
or 8~mmag when folded on any periods larger than 0.01~d. The expected variations due to a substellar companion
would be as low as 6~mmag at the two longest periods or 20~mmag at the shortest, but are all significantly larger than
the SWASP limit. It is not possible to describe the companion with present data, although a substellar companion
is a distinct possibility.
\citet{ost2010a} obtained series of photometric observations of
GALEX~J0500$+$0912 in order to search for pulsations and concluded that
it is not photometrically variable. Our limited radial velocity data set does not
indicate variability.
An inspection of the acquisition images of GALEX~J0657$-$7324 shows a nearby companion
and, therefore, the 2MASS and $WISE$ colours of the hot subdwarf are certainly contaminated.
\citet{hei1992} reported that GALEX~J0657$-$7324 (HEI 714)
is a visual double star with a separation of 1.9~arcsec, and our own acquisition image locates the companion
1.8~arcsec away at a position angle of 270$^\circ$. Also, our optical spectra do not
appear to be contaminated by this object and do not indicate variability.
GALEX~J1845$-$4138 is a relatively cool He-rich subdwarf displaying a strong He\,{\sc i}
line series and weaker Balmer lines. The velocity measurements based on He\,{\sc i}6678.154
($\varv=-59.7\pm3.3$\,km\,s$^{-1}$) are consistent with the measurements based on H$\alpha$ and
do not suggest any variability.
GALEX~J1902$-$5130 is a helium sdO star. \citet{lan2012} obtained spectropolarimetry
of GALEX~J1902$-$5130 with a measurement that shows that this star does
not have a magnetic field down to a few hundred gauss.
Our radial velocity measurements suggest there may be long period, low-amplitude
variations. The measurements are based on He\,{\sc ii} 6560.088\AA.
The object is very hot and
our spectra display He\,{\sc i} emission.
GALEX~J1911$-$1406 is also a very hot He-rich subdwarf. The velocity measurements
are based on He\,{\sc ii}6560.088\AA.
\citet{gei2012a} report a rotational velocity of $v_{\rm rot} = 8.6\pm1.8$\,km\,s$^{-1}$\ for GALEX~J2153-7003.
\citet{cop2011} obtained several radial velocity measurements of this star
and found that it is not variable. Their average velocity and dispersion, $39.4\pm7.5$\,km\,s$^{-1}$,
are in a close agreement with our own measurements ($43.4\pm4.2$\,km\,s$^{-1}$).
GALEX~J2344-3426 is a well known sdB star. \citet{ost2010a} obtained photometric series
of the star and found it to be non-variable. \citet{gei2012a} measured a rotational
velocity of $v_{\rm rot} = 7.3\pm1.0$\,km\,s$^{-1}$. \citet{mat2012} and \citet{lan2012} obtained
spectropolarimetry of the star and constrained the longitudinal field to 261 G and $246\pm232$ G, respectively.
\section{Discussion}
The new binary identifications are placed into context with a compilation of all known
spectroscopically identified binaries (Appendix C).
Table~\ref{tbl_sdb_bin_all} lists the orbital parameters of these systems. The compilation includes hot subdwarfs
with an unseen companion and spectroscopically identified late- to early-type companions in a range
of periods from 0.05\, to 1363\,d. Table~\ref{tbl_sample_kine} lists the properties of the primary
as well as kinematical properties of the systems.
Throughout this discussion we assume for most subdwarfs a mass of 0.47\,M$_\odot$\ with
a few exceptions such as the ELM progenitors (0.23\,M$_\odot$) and hot sdO companions to early-type stars (1\,M$_\odot$).
Using pulsating properties of sdB stars and binary systems for which the sdB mass
was measured \citet{fon2012} found that the average mass of a sdB star is
0.47\,M$_\odot$\ with a standard deviation of 0.031\,M$_\odot$. \citet{zha2009}
found that most sdB stars have a mass between 0.42 and 0.54\,M$_\odot$\ and
an average mass of about 0.50\,M$_\odot$. \citet{zha2009} used evolutionary
models and the parameters of a sample of 164 sdB stars.
\subsection{Properties of known binaries: Period and mass function}
Fig.~\ref{fig_cumul} (top) shows the cumulative distribution of orbital periods in the
population of binaries with a hot subdwarf primary. The derivative of the function with
respect to the logarithm of the period provides an estimate of the period distribution (Fig.~\ref{fig_cumul}, bottom).
Several peaks stand-out,
particularly at 0.1, 0.5-2.0, 10, and 1000 days. The last two peaks are clearly separated by
a gap within which few binaries are known: A few Be stars with a hot subdwarf companion populate the
gap and the distribution includes spectroscopically (UV) confirmed Be+sdO \citep{pet2008,pet2013}.
We noticed a hint of a hierarchy in the period distribution: The shortest periods coincide with dM
companions emerging from a CE phase (low mass ratio $M_2/M_1 < 1/2$),
white dwarfs ($M_2/M_1 \approx 1$), and,
at high mass ratio ($M_2/M_1>2$), subdwarfs with a subgiant/giant or Be companion, and, finally,
subdwarfs with a FGK companion at the longest periods and emerging from a RLOF.
Following a RLOF, the orbital separation increases the least for more massive companions. It is remarkable
that main-sequence A-type star companions are still missing although they are predicted in population
syntheses \citep{han2003}. Subdwarfs with A-type main sequence companions should be detectable as UV excess
objects or as low-amplitude radial velocity variations similar to Be+subdwarf binaries.
In summary, binaries near the main peak are mostly white dwarfs plus subdwarfs after possibly two episodes of mass-transfer.
Note that the orbital parameters of many systems with large velocity amplitudes remain unresolved \citep[see, e.g., ][]{cop2011,gei2011a} and are not included in this analysis.
\begin{figure}
\includegraphics[width=1.00\columnwidth]{fig17.pdf}
\caption{Cumulative function of period (top), $N_<$ versus $\log{P}$, and its first derivative (bottom).
The derivative was smoothed with a Gaussian function ($FWHM=0.1$ dex), and shown in the
upper right corner.
\label{fig_cumul}}
\end{figure}
\begin{figure}
\includegraphics[width=1.00\columnwidth]{fig18.pdf}
\caption{Measured velocity amplitude versus period with a sub-sample of
eclipsing binaries shown with open circles. Full lines are labelled with
the mass of secondary stars which were computed for $i=90^\circ$.
\label{fig_eclipse}}
\end{figure}
Fig.~\ref{fig_eclipse} shows the sample of known binaries in the velocity amplitude versus
period plane. Most eclipsing systems have secondary masses between 0.08 to 0.15\,M$_\odot$. Systems
with known white dwarf secondaries have secondary masses close to or above 0.60\,M$_\odot$.
\begin{figure}
\includegraphics[width=1.00\columnwidth]{fig19.pdf}
\caption{Same as Fig.~\ref{fig_eclipse} but with
reflection binaries shown with open circles. The velocity scale is corrected for
the effect of an average inclination of $57^\circ$. Secondary masses cluster
between 0.08 and 0.30\,M$_\odot$.
\label{fig_reflect}}
\end{figure}
Fig.~\ref{fig_reflect} also shows the sample of known binaries in the velocity amplitude versus
period plane but with the velocity
scale corrected for an average inclination of $57^\circ$. The correction allows to draw class properties but should not be applied
to individual objects. Secondary masses for systems showing a reflection effect range, with the exception of FF~Aqr and HD~185510, from
0.08 to 0.30\,M$_\odot$. Remarkably, secondary masses for most non-reflecting systems cluster near
0.60\,M$_\odot$\ and the unseen objects are probably white dwarfs. Secondary stars in the long-period range
and with masses in excess of 0.60\,M$_\odot$\ are identifiable as G and K stars.
All eclipsing systems with $K<100$\,km\,s$^{-1}$, i.e., with an estimated $M_2<0.3$\,M$_\odot$, also show a reflection effect
indicative of a late-type secondary, while the remaining systems cluster at a higher secondary mass $M_2\approx0.6$\,M$_\odot$\ and
almost certainly harbour a white dwarf secondary.
Fig.~\ref{fig_mass_sec} shows secondary mass distribution assuming average system inclination of $57^\circ$.
This distribution may be described by a superposition of two power laws: A shallow distribution with $M_2>0.08$\,M$_\odot$, i.e., $\alpha=1.3$ between
0.08 and 0.5\,M$_\odot$\ and $\alpha=2.3$ above 0.5\,M$_\odot$\ following the initial mass function $\xi(m)\propto m^{-\alpha}$ of \citet{kro2001}, and
a steeper distribution ($\alpha\approx 6$) with $M_2>0.48$\,M$_\odot$. The former power law encompasses mostly M-type dwarfs, many
of them showing a reflection effect, while the latter encompasses white dwarfs
in the 0.5-1.0\,M$_\odot$\ mass
range. This simplified white dwarf distribution represents well the peak and high-mass tail of the white dwarf mass distribution \citep[see, e.g., ][]{kep2007}
but excludes possible low-mass white dwarfs ($<0.48$\,M$_\odot$).
On the other hand, the late-type stellar mass distribution follows the initial mass function and the expectation of a randomly drawn set of late-type stars.
The reflection effect is common in short-period binaries ($P<0.5$~d) but is relatively rare at longer periods (see Fig.~\ref{fig_reflect}) due to increased binary separations and weaker
photometric variations: The actual late-type mass distribution appears as a scaled-up version of the secondary mass distribution in the
sub-sample of reflection binaries, but it also includes longer period binaries with an indiscernible reflection effect.
Note that a third narrow peak is possibly present at $\approx 0.3$ M$_\odot$.
Since this peak is mostly made up of companions that do not show the reflection
effect, the origin of this peak maybe be low-mass white dwarfs. It may also be
due to an incorrect mass estimate of the subdwarf, for example an ELM
progenitor is assumed rather than a normal subdwarf.
Most objects (60-70 per cent) are low mass main sequence stars, while 30-40 per cent are white dwarfs.
Our own survey delivered a 37 per cent fraction of hot subdwarfs in close binaries (Section 3.1.3).
Our survey strategy was aimed at and successfully uncovered short-period binaries. Fig.~\ref{fig_reflect}
shows that setting our detection threshold at 10~km\,s$^{-1}$\ would have allowed for the detection of any stellar companion with an
orbital period $\lesssim20$~d, any late-type companion with a mass of 0.3\,M$_\odot$\ and $P\lesssim600$~d, or just about any white dwarf companion.
However, a close examination of data sampling shows that 60 per cent were obtained with a span of 2 days or less
with another 30 per cent with a span of 100-400 days, i.e., during a subsequent observing run or season.
Systems with periods of 10-20 days or longer would have only been partially covered and most likely avoided detection. Note that
the longest period detected in our survey is only 3~d long. Setting our detection threshold at 10~d,
i.e., some 155 objects out of 179 known binaries, the total yield including longer period binaries could be $\approx$15 per cent larger for
a total binary fraction of 43 per cent. The four additional hot subdwarfs with composite spectra (see Table~\ref{tbl_sum}) which are likely to have longer orbital
periods ($\gtrsim 10$~d) are such objects.
\begin{figure}
\includegraphics[width=1.00\columnwidth]{fig20.pdf}
\caption{Mass distribution of all known binaries with a hot subdwarf primary star as a function of the secondary mass, assuming
an average inclination of $57^\circ$ (full histogram). The peak distribution
of low-mass stars is marked ``dM'' and that of white dwarfs, ``WD''. Binaries showing reflection effect in their light curves
are shown with a dashed histogram. The full lines show synthetic distributions smoothed to two-bins width for a combination of late-type stars and white dwarfs (double-peaked full line), and
that excluding white dwarfs (dashed line).
\label{fig_mass_sec}}
\end{figure}
\subsection{Properties of known binaries: Kinematics}
We calculated the Galactic velocity components ($U,V,W$), which are relative to the local standard of rest (LSR),
of all known hot subdwarf binary systems (listed in
Appendix~C) using their positions, systemic ($\gamma$) velocities, proper
motions and apparent magnitudes.
We adopted the right-handed system for the velocity components, where $U$ is positive in the direction of the Galactic
centre, $V$ is positive in the direction of Galactic rotation and $W$ is positive toward the North Galactic Pole.
We assumed that the solar motion relative to the LSR is ($U,V,W$) = (10.1,4.0,6.7) km~s$^{-1}$
as determined by \citet{hog2005}.
The distribution of systemic velocities, i.e., radial velocities, follows $N\approx e^{-\mid\gamma\mid/\sigma}$, where
$\sigma_r=41$\,km\,s$^{-1}$. The $\sigma_r$ value is the one-dimension equivalent of the two-dimension transversal
velocity dispersion $\sigma_T=59$\,km\,s$^{-1}$\ measured by \citet{ven2011}, where $\sigma_T=\sqrt{2}\,\sigma_r$.
In their study of the kinematics of EHB stars,
\citet{alt2004} measured significantly larger radial velocities with a distribution following $\sigma_r=65$\,km\,s$^{-1}$\ compared to our sample, but
they measured a
transversal velocity distribution consistent with the present one.
To calculate the Galactic velocity vectors, we employed the method outlined in
\citet{joh1987} using as an input the radial velocity, proper motion \citep{zac2013} and distance measurements.
We determined the distance toward each
subdwarf using the distance modulus $V-M_V=5\log{d}-5$, where the magnitude $V$ is listed in Table~\ref{tbl_sdb_bin_all}.
We estimated the absolute magnitude $M_V$ from the measured stellar parameters, i.e.,
\begin{displaymath}
M_V=-2.5\log{(4\pi\Omega\,\bar{H}_V)},
\end{displaymath}
where $\Omega = r^2/d^2$, with $d=10$~pc and $r^2 = GM/g$. The Eddington flux is averaged ($\bar{H}_V$) over the Johnson $V$ transmission curve.
We assumed $M=0.23$\,M$_\odot$\ for low-mass objects, 0.47\,M$_\odot$\ for normal objects and 1.0\,M$_\odot$\ for subdwarfs in massive binaries,
and the published surface gravity measurements were usually obtained using a spectroscopic method similar to that described in Section 3.1.
Fig.~\ref{fig_UV} shows the $U$ and $V$ velocity components for all
hot subdwarf binary systems. Table~\ref{tbl_kin} lists the ($U,V,W$) velocity
components and dispersions for the sample of known binaries.
The distribution appears asymmetric with several objects trailing at large negative
Galactic $V$ velocity.
We computed the straight average and dispersion (``All'') but excluding the extreme case of
OGLE~BUL-SC16335. We also fitted the distributions with Gaussian functions excluding outliers, i.e., all bins with less than three members (`$N>3$`). The sample velocity dispersion is
significantly smaller than that calculated by \citet{alt2004} for single hot subdwarf stars
($\sigma_U = 74$\,km\,s$^{-1}$, $\sigma_V = 79$\,km\,s$^{-1}$, $\sigma_W = 64$\,km\,s$^{-1}$).
However, our velocity dispersion is in better agreement with the
dispersion ($\sigma_U = 62\pm8$\,km\,s$^{-1}$, $\sigma_V = 52\pm7$\,km\,s$^{-1}$,
$\sigma_W = 59\pm8$\,km\,s$^{-1}$) calculated by \citet{deb1997} based on
a sample of 41 hot subdwarf stars.
Possible explanations for the inflated Galactic velocities of \citet{alt2004}
are that their sample included yet unidentified binaries or that lower dispersion
spectroscopy resulted in larger measurement errors.
In summary, the hot subdwarf population and the confirmed binaries among them are
kinematically indistinguishable and drawn from the
same, general population of EHB stars.
Also, Table~\ref{tbl_kin} compares results for the hot subdwarf population
with that of field white dwarf stars.
\citet{pau2006} list 361 thin disc members
and 27 thick disc members, while \citet{kaw2012b} list 57 old disc white dwarfs,
which is a mix of old thin-disk and thick-disk populations, and at least
one halo white dwarf.
A comparison of the velocity dispersions shows that the binary population may be an even older population than
field white dwarf stars with some population members belonging to the old disc or even the halo.
Four systems display peculiar kinematics. Three of these, SDSS~J1622$+$4730,
PHL~861 and SDSS~J1505+1108, have Galactic velocities that make them halo
candidates with a few additional objects lagging in the $V$ component making them thick-disk candidates.
The fourth object, OGLE~BUL-SC16335, is in a crowded field and
there is a possibility that the proper motion measurements are incorrect.
\citet{bar2013a} calculated kinematics for 5 long period systems and found
that two of these (PG~1449$+$653 and PG~1701$+$359) have kinematics suggesting
that they belong either to the thick disc or halo. They also report that
there is a high probability that PG~1104$+$243 belongs to the thick disc. Our
calculated Galactic velocities are similar to those of \citet{bar2013a}.
Note that \citet{bar2013a} includes the disc rotation (220\,km\,s$^{-1}$)
in their $V$ velocity.
A comparison of the velocity components of the hot subdwarf binary population
to that of \citet{kaw2012b} shows that the dispersion is larger for all
velocity components than that of the white dwarf population, suggesting that the
subdwarf population appears to be older than the white dwarf population. Finally, a comparison
with the work of \citet{pau2006} shows that the hot subdwarf binary population has
a velocity dispersion between the thin and thick disc dispersions for white
dwarfs.
\subsection{Low mass subdwarfs as progenitors of extremely low-mass white dwarfs}
The first known ELM white dwarf progenitor, HD~188112, was discovered by
\citet{heb2003}. As part of their survey of ELM white dwarfs, \citet{kil2011}
found that SDSS~J1625$+$3632 is similar to HD 188112. Other recently
discovered systems are KIC~6614501 \citep{sil2012} and NGC~6121-V46 \citep{oto2006a}.
Our radial velocity survey adds one more object to the small sample of
ELM white dwarf progenitors: \citet{ven2011} showed that GALEX~J0805$-$1058
has atmospheric properties representative of ELM white dwarf progenitors ($M\lesssim$0.3\,M$_\odot$) and,
therefore, it was selected for radial velocity follow-up measurements. New radial velocity measurements
proved that GALEX~J0805$-$1058 is in a close binary,
and, because it also lies below the ZAEHB (see Section 3.1), we conclude that it is a genuine ELM white dwarf progenitor. It is also the first ELM white dwarf progenitor without
a more massive white dwarf companion, and this is likely to have significant implications for the origin and
evolution of ELM white dwarfs.
Two other objects from our sample lie below the ZAEHB, J1411+7037, which is paired with an F star, and J2153$-$7004, although we did
not detect significant radial velocity variations.
\begin{table}
\centering
\begin{minipage}{\textwidth}
\caption{Kinematical properties of the hot subdwarf binary population. \label{tbl_kin}}
\renewcommand{\footnoterule}{\vspace*{-15pt}}
\renewcommand{\thefootnote}{\alph{footnote}}
\begin{tabular}{lcccccc}
\hline
Vel. \footnotemark[1]\footnotetext[1]{All velocities expressed in km\,s$^{-1}$.} & $N>3$\footnotemark[2]\footnotetext[2]{This work, but excluding outliers} & All \footnotemark[3]\footnotetext[3]{This work} & sdB \footnotemark[4]\footnotetext[4]{\citet{alt2004}} & WD \footnotemark[5]\footnotetext[5]{\citet{kaw2012b}} & WD$_{\rm thin}$ \footnotemark[6]\footnotetext[6]{\citet{pau2006}} & WD$_{\rm thick}$ \footnotemark[6] \\
\hline
$\bar{U}$ & $0\pm5$ & $2$ & $-8$ & $-7.8$ & ... & ... \\
$\sigma_U$ & $52\pm3$ & 62 & $74$ & $42.8$ & $34$ & $79$ \\
$\bar{V}$ & $-30\pm2$ & $-32$ & $-37$ & $-40.1$ & ... & $-52$ \\
$\sigma_V$ & $42\pm2$ & 47 & $79$ & $31.9$ & $24$ & $36$ \\
$\bar{W}$ & $-5\pm3$ & $-6$ & $12$ & $-5.9$ & ... & ... \\
$\sigma_W$ & $34\pm2$ & 41 & $64$ & $27.4$ & $18$ & $46$ \\
\hline
\end{tabular}
\end{minipage}
\end{table}
\begin{figure}
\includegraphics[width=1.00\columnwidth]{fig21.pdf}
\caption{Galactic velocity vectors $U$ and $V$ of all known binaries
containing a hot subdwarf. The individual distributions are shown in the upper panel ($V$)
and right panel ($U$). Details of the measurements are shown in Table~\ref{tbl_kin} and
discussed in Section 4.2.
\label{fig_UV}}
\end{figure}
\subsection{Summary}
We presented an analysis of the orbital properties of seven new systems
comprising a hot subdwarf primary. The secondary in one system is a late-type
star showing a reflection effect (GALEX~J2205$-$3141), while we found evidence
that the secondary star in GALEX~J0805$-$1058 is a very low mass M dwarf or
possibly a substellar object. The mass function of the other objects implies the
likely presence of a white dwarf companion.
The period of photometric variability of two additional systems is, in the case
of GALEX~J1736+2806 probably coincident with the orbital period, and, in the
case of J2038$-$2657, probably coincident with the rotation period of the giant
companion. Our survey results, taking into account the survey strategy, imply
an incidence of binarity of $\sim$43 per cent in the hot subdwarf population.
We have compiled a list of all known hot subdwarfs in binary systems
and performed a binary population analysis. We found that systems showing
the reflection effect have components that are of a lower mass
(0.08 to 0.30\,M$_\odot$) than those that do not show the reflection
($\sim$0.6\,M$_\odot$). It is very likely that the companion to the hot subdwarf
in most of the systems not showing the reflection effect in the short period
binaries ($P \la 1$ d) are white dwarfs.
The inferred secondary mass distribution is a superposition of two approximate
power laws, one low-mass power-law ($\gtrsim$0.1\,M$_\odot$) and composed of low-mass
main-sequence stars, and another, high-mass power-law ($\gtrsim$0.5\,M$_\odot$) and primarily
composed of white dwarfs with a few early-type main-sequence stars.
White dwarfs constitute $\approx$30-40 per cent of all binary companions.
We have calculated the Galactic velocity components for all known hot subdwarfs
in binary system and showed that this population may be older than the field
white dwarf population. In this sample, we found three systems that possibly
belong to the halo.
Future work will involve high-dispersion spectroscopic follow-up of low-velocity amplitude
binary candidates, and of binaries comprising a hot subdwarf and an early-type main-sequence, or giant
companion.
\section*{Acknowledgments}
A.K. and S.V. acknowledge support from the Grant Agency of the Czech Republic
(P209/12/0217 and 13-14581S) and Ministry of Education, Youth and Sports (LG14013).
We wish to thank S. Ehlerov\'a and team members for their assistance with
observations obtained with the MPG~2.2-m telescope at La Silla.
We thank the referee for a thorough report and for stimulating further work on the paper.
This work was also supported by the project RVO:67985815 in the Czech Republic.
This paper uses observations made at the South African Astronomical Observatory.
This paper makes use of data obtained from the Isaac Newton Group Archive
which is maintained as part of the CASU Astronomical Data Centre at the
Institute of Astronomy, Cambridge.
This publication makes use of data products from the Wide-field Infrared Survey Explorer,
which is a joint project of the University of California, Los Angeles, and the Jet
Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration.
This publication makes use of data products from the Two Micron All Sky Survey, which
is a joint project of the University of Massachusetts and the Infrared Processing and
Analysis Center/California Institute of Technology, funded by the National Aeronautics
and Space Administration and the National Science Foundation.
|
{
"timestamp": "2015-04-14T02:16:16",
"yymm": "1504",
"arxiv_id": "1504.03241",
"language": "en",
"url": "https://arxiv.org/abs/1504.03241"
}
|
\section{Introduction}
Very recent observations \cite{Ade:2015lrj, Planck:2015xua, Ade:2015tva, Ade:2014xna} are on the threshold of making an important distinction about the details of the inflationary regime that gave birth to our universe, by the precise measurement of the parameter $n_s-1$, the slight scale dependence of the primordial perturbations, and the constraint on $r$, the ratio of the power of tensor-to-scalar fluctuations. The distinction arises because of the slow-roll relations
\begin{equation}
n_s=-3\frac{V'^2}{V^2}+2\frac{V''}{V}\approx 0.97,\quad r=8\frac{V'^2}{V^2}\lesssim 0.11
\end{equation}
written in term of derivatives of the potential driving inflation, and with observed values given by \cite{Ade:2015lrj, Planck:2015xua, Bennett:2012zja}. From this we see that the threshold value $r<8/3(1-n_s)\approx 0.08$ implies that the inflaton field has a negative mass squared term, and is thus \emph{tachyonic}. It is still too early to definitively tell whether this is the case or not, but experiments planned for the immediate future \cite{Bouchet:2011ck,Fraisse:2011xz,Matsumura:2013aja} will be able to distinguish this.
If the inflaton does indeed turn out to be tachyonic, it is worth asking what further consequences this could have for the dynamics of the early universe. The hallmark feature of tachyonic fields is that long wavelength modes are unstable (over and above the typical instabilities of light fields in an expanding universe). The possibility that these long wavelengths can have significant enough buildup to heavily influence the dynamics of the background mode was explored in \cite{Cormier:1998nt, Cormier:1999ia}, where the parallel with a similar phenomenon in condensed matter, the $\emph{spinodal regime}$, was drawn. This is the tachyonic regime, where long wavelength modes experience an instability that leads to exponential growth. In these and in a subsequent work \cite{Albrecht:2014sea} it was found that backreaction can be significant enough to even drive substantial periods of inflation on steeply tachyonic potentials, where the slow-roll conditions do not hold, such as natural inflation \cite{Freese:1990rb, Adams:1992bn, Freese:2014nla} with subplanckian values of the axion decay constant. Admittedly, the region of initial conditions suitable for spinodal effects seems quite tuned. However, if the consequences can be so drastic in this special case, it is worth considering whether smaller effects may be present in other models, especially given that data may soon definitively show that the inflaton is tachyonic.
To this end, we explore whether a spinodal buildup can affect the dynamics of some of the typical models of inflation, and find several general features. The magnitude of the backreaction effect generically depends on the initial amount of power in the quantum fluctuations. Accounting for these fluctuations in the energy density does not, in general, cause them to exponentially dilute away, as in the standard scenario where their effect is neglected. This is actually a consequence of the slow-roll shape of the inflaton potential, which translates into the effective potential governing the strength of the fluctuations being almost equally as flat. If the inflaton field is tachyonic, exponential buildup of long wavelength modes does occur, but the effects can be seen even in nontachyonic models of inflation if the initial strength is great enough. We observe several effects:
\begin{itemize}
\item There can be stages where the long wavelength fluctuations dominate the contribution to the power spectrum of density perturbations. If this is the case, the amplitude is not set by the overall mass scale of the potential, but instead by the initial conditions for the long wavelength modes. Additionally, this shifts the value of $n_s-1$ by an amount that depends on the potential. Typically, this is a few percent for large field models, while in small field models the tilt is driven to be exactly scale invariant. As we will see, this will be enough to alter the predictions of some models to lie within the preferred contour in the $n_s-r$ plane.
\item Though the value of the tilt is only slightly altered, the prediction for the tensor-to-scalar ratio can drastically change if these effects are at play. This is because the scalar power spectrum can be much larger than the regular contribution, while the tensor power remains unchanged. This allows for inflation to proceed at a much lower energy scale, which decreases the tensor fluctuations.
\item The spinodal regime will be an inevitable attractor for purely tachyonic models, though approach may be very slow. In these models, it is possible to differentiate whether inflation lasted only several e-folds longer than what we observe, or if there were a parametrically longer phase preceding this, by noting whether the spinodal regime has been reached.
\item A final effect is that, if features are present in the potential, the contribution of the fluctuations can serve to wash them out, providing a sort of ``low-pass filter" in which the inflaton is influenced only by the large scale trend of the potential. This can even make slow-roll inflation possible on otherwise completely unsuitable potentials with many false vacua and steep peaks.
\end{itemize}
The spinodal phenomenon is in actuality nonperturbative, and so to make any headway the Hartree approximation \cite{Chang:1975dt} is employed, which is a self-consistent loop resummation scheme. The end result of the Hartree approximation is to replace the original model of inflation with a two-field model, where the second field represents the power in the long wavelength modes. We refer to this second field as the \emph{diakyon}, after the Greek word for fluctuation, and study its influence on the behavior of inflation as a function of its initial value. Even though the background dynamics is effectively two-field inflation, isocurvature perturbations are not generated, because at the basic level the second field decays exactly the same way as the inflaton during (p)reheating, so the dominant effects of this other field is to alter the trajectory of the inflaton, and to set the value of density perturbations. What we find is that in spite of the presence of drastically different regimes, where the diakyon field can provide the dominant contribution to the energy density, the perhaps most salient feature of inflation, the production of a nearly scale invariant spectrum of perturbations, remains robust. We expect the alterations we do observe, a drastic decrease in $r$ and a slight shift in $n_s$, to remain true beyond the Hartree approximation, at the very least at the qualitative level.
Our paper is organized as follows: we devote section \ref{Hart} to the technical framework. We begin by using the path integral formalism to derive the leading quantum corrections to the equations of motion. Afterwards, we discuss the Hartree approximation, how the dynamics is equivalent to replacing long wavelength modes with a second classical field, and show how both can be easily implemented by a simple integral transform of the original potential of the theory. The section ends with a discussion of density perturbations in this framework. In section \ref{Tart} we apply these results to some specific models of inflation. We begin with the simplest model of inflation, $m^2\phi^2$, and show that for certain initial values of the diakyon, the tensor-to-scalar ratio can be made much smaller without altering the other predictions of the model. We go on in section \ref{hillinf} to study the simplest model of a tachyonic field, hilltop inflation, and show that there are several regimes alternative to the standard slow-roll behavior, depending sensitively on the initial value of the diakyon. In section \ref{flatinf} we look at flattened potentials ($\phi^{2/3}$ and, briefly, Starobinsky), and show that the tachyonic nature of the inflaton necessarily causes deviations from standard inflation if the inflaton started from the eternal regime. In section \ref{monoinf} we discuss spinodal effects in monodromy inflation. We find that for small values of the axion decay constant $f$, these serve to taper any oscillations in the spectrum. We also use this to highlight how spinodal effects can allow for slow-roll inflation even when the potential has many pocket false minima, and discuss this general phenomenon. We conclude in section \ref{conclusions}.
\section{Hartree Approximation}\label{Hart}
We devote this section to explaining the Hartree approximation as a tool in field theory, and justifying its use in the case at hand. We provide a simple path integral derivation of the Hartree approximation, which allows us to formally assess the validity of the approximation and, if desired, compute systematic corrections to it. Though undertaking these tasks is beyond the scope of the present paper, we plan to return to these in future publications. Afterwards, we distill the Hartree approximation down to a simple integral transform of the potential, allowing for efficient implementation of spinodal effects.
A useful tool in the study of field theories is the background field method. In this method all fields are divided into background quantities, that may or may not depend on space and time, and quantum fluctuations on top of these. One is then typically interested in two things: the evolution of the background field, and correlations between perturbations at different spacetime points. This method is routinely applied in inflationary cosmology, where an inflaton field acquires a time-dependent expectation value driving the expansion of the universe, and the fluctuations source perturbations in the energy density that eventually collapse to form galaxies. To this end, we assume the dynamics of the early universe is governed by a single field rolling down a potential, and perform the background splitting
\begin{equation}
\phi(t,x)=\bar\phi(t)+\psi(t,x).
\end{equation}
This splitting is made unique by the condition that the tadpole of the perturbation vanish,
\begin{equation}
\langle \psi \rangle=0\Longrightarrow\langle\phi\rangle=\bar\phi.
\end{equation}
To lowest order in $\psi$, this condition implies the classical equations of motion for the background field, and a linear equation determining the fluctuation:
\begin{eqnarray}
\partial_t^2\bar\phi+3H\partial_t\bar\phi+V'(\bar\phi)=0,\nonumber\\
\left(-\Box+V''(\bar\phi)\right)\psi=0.
\label{eom}
\end{eqnarray}
The fluctuations are usually very small, and so then this is a good approximation. During inflation, however, even though the fluctuations are small, once a mode of a given wavelength gets stretched to larger than the horizon size it freezes to a constant value. If this occurs for very long time, a large number of modes will exit the horizon and contribute an effective stochastic offset to the equations of motion for the background, that can be encapsulated in the addition of a noise term \cite{Starobinsky:1986fx,Vilenkin:1983xp}, the backreaction of which was recently considered in \cite{Levasseur:2014ska}. This can lead to drastic inhomogeneities on the largest of scales, and produces divergences if inflation lasts for an exponentially long time (for a review see \cite{Seery:2010kh}), but for the purposes of a single observer measuring local correlators, this only contributes as a shift in the local time coordinate, as set by the value of the background part of the field. In contrast, we are considering the influence of long wavelength modes on the dynamics of these equations that does not manifest itself as a source term.
There is a further reason to expect deviations from the lowest order during inflation if the field is tachyonic. In this case wavelengths larger than the inverse mass scale of the field will grow exponentially, which can compensate or even overcome the exponential dilution, and can cause significant backreaction on the value of the background field. In this case the tadpole condition does not reduce to the classical equations of motion, but instead becomes a system depending on an infinite number of the correlators of the fluctuations. Likewise, the equations determining the values for each of these correlators will depend on a number of the others, as well as the background. There is no hope of solving this infinite set of equations for an infinite amount of variables analytically. Fortunately, there is a well-motivated approximation, devised by Hartree, which allows us to truncate this system to a manageable level, and typically yields results that are quite close to the full calculation (see for instance \cite{Chang:1975dt}). The approximation consists of treating all propagators as if they were moving through an external medium set by the effect of the propagators themselves. In this way, we acknowledge that the fluctuations of the fields can yield a large effect on the quantities we are trying to compute. As shown in Fig \ref{loops}, this method entails replacing all propagators by the sum of insertions of virtual loops, with couplings dictated by the bare Lagrangian.
\begin{figure*}[h]
\centering
\includegraphics[height=3.75cm]{loops.pdf}\
\caption{Graphical depiction of the Hartree resummation. Here the dashed lines correspond to the background field, and the solid lines to the fluctuations. The top row is the tadpole condition, beginning with a line ending on a $\times$ representing the classical equations of motion. In the last row, self-consistency demands that once the propagator is substituted with this series, one must iterate the procedure, replacing every propagator in the series itself with the full series of propagators, and so on. The resulting diagrams lead to this sometimes being called the cactus approximation.}
\label{loops}
\end{figure*}
Once this replacement has been made, self-consistency demands that we do the same procedure for propagators flowing through the loops, and so on and so forth, ad infinitum. It should not be difficult to convince the reader that this quickly becomes impossibly hard to manage. Fortunately, there is a more operationally friendly method of enforcing this approximation, as espoused in \cite{Cormier:1999ia}: one need only replace all higher $n$-point functions with products of propagators, multiplying two or fewer powers of the fields. For example, $\psi^{2n}\rightarrow c_0\langle\psi^2\rangle^n+c_1\langle\psi^2\rangle^{n-1}\psi^2$, where the coefficients are combinatoric factors. Similarly, for an odd number of the fields $\psi^{2n+1}\rightarrow c_2\langle\psi^2\rangle^n\psi$. The coefficients will not be important for our purposes, but can be found in \cite{Cormier:1999ia}. In this way, we have eliminated the need for correlators of much higher powers of the fields, and reduced the system to a set of two equations determining $\bar\phi$ and $\langle\psi^2\rangle$:
\begin{eqnarray}
\partial_t^2\bar\phi+3H\partial_t\bar\phi+\sum_{k=0}^\infty\frac{1}{k!}\bar V^{(2k+1)}\left(\frac{\langle\psi^2\rangle}{2}\right)^k=0,\nonumber\\
\left[\Box+\sum_{k=0}^\infty\frac{1}{k!}\bar V^{(2k+2)}\left(\frac{\langle\psi^2\rangle}{2}\right)^k\right]\psi=0.\label{bench}
\end{eqnarray}
These expressions were derived in \cite{Cormier:1998nt} and their dynamics for natural inflation explored in \cite{Cormier:1999ia}. A few brief comments before we go on to derive this approximation from a path integral point of view: the Hartree approximation is concerned with allowing the form of propagators to be altered to account for the background shift. In principle, higher order correlators will also be altered, for example the value of the quartic coupling may receive corrections by inserting interaction loops through different vertices. This effect is deemed unimportant for the study of two-point functions. Secondly, it can be seen in Fig. \ref{loops} that only contact insertions are included in the sum, in the sense that the (dressed) internal line originates and terminates at the same place on the propagator, with no further interactions on its journey. This ignores substructure wherein a second loop might split off from a first and recombine on an entirely different point. Third, if the Lagrangian has interactions of infinitely high order, such as the ones we will consider throughout this paper, then the Hartree approximation includes arbitrarily high loop order. This makes it a bona fide resummation scheme, and not a usual loop expansion. The path integral derivation should dispel any misgiving of giving precedence to extremely high order contact interactions over relatively low order noncontact interactions by showing how this arises naturally from a single insertion of the interaction Hamiltonian.
\subsection{Path Integral Derivation}
Now we turn to a derivation of the Hartree approximation in terms of the path integral. This will provide the crucial insight we rely on in implementing the Hartree approximation in example potentials. We start by computing the tadpole of the fluctuation in the path integral formalism
\begin{equation}
\langle\psi_x\rangle=\int D\psi e^{iS[\bar\phi+\psi]}\psi_x
\end{equation}
The action can then be expanded in powers of the fluctuation $S[\bar\phi+\psi]=\bar S+\bar S^{(1)}\psi+\bar S^{(2)}\psi^2+S_{\text{int}}$, where, since we take the kinetic terms to be quadratic in fields, $S_{\text{int}}=-\int d^4x\sum_{n=3}^\infty\frac{1}{n!}\bar V^{(n)}\psi^n_x$. Here superscripts represent derivatives with respect to the fields, and barred quantities are evaluated on the background. Then
\begin{equation}
\langle\psi_x\rangle=e^{i\bar S}\int D\psi e^{i\bar S^{(1)}\psi+i S_{\text{int}}}\psi_x e^{i \bar S^{(2)}\psi^2}
\end{equation}
So far this expression is exact: we have just factored out the quadratic part of the action in anticipation of approximating this as a Gaussian integral. Now we expand the first exponential
\begin{eqnarray}
\langle\psi_x\rangle &\approx& ie^{i\bar S}\int D\psi\left(\bar S^{(1)}\psi+S_{\text{int}}\right)\psi_x e^{i\bar S^{(2)}\psi^2} \nonumber\\
&=& ie^{i\bar S}\int D\psi\int d^4y\left(\frac{\delta L}{\delta\phi}\psi_y-\sum_{k=1}^\infty\frac{1}{(2k+1)!}\bar V^{(2k+1)}\psi_y^{2k+1}\right)\psi_x e^{i\bar S^{(2)}\psi^2}
\end{eqnarray}
We have used that the integral of an odd number of $\psi$'s vanishes. If we were working to lowest order we would also neglect the interaction term, and this would be equivalent to the classical equations of motion. Now we see the essence of the Hartree approximation: we insert a single interaction potential in the correlators we wish to compute. Higher order corrections to this will simply be the neglected terms in the expansion of the exponentials. We then treat the integrals over the Gaussian as occurring in the free field vacuum,
\begin{equation}
\int D\psi \psi_x\psi_y e^{i\bar S^{(2)}\psi^2}=\langle\psi_x\psi_y\rangle_0\equiv G_{xy},\quad \langle\psi_x\psi_y^{2k+1}\rangle_0=\frac{(2k+1)!}{2^kk!}G_{xy}G_{yy}^k
\end{equation}
The coincident limit of the Green's function will be divergent as always, so we must regularize to arrive at finite answers. We make use of results \cite{Boyanovsky:1997mq} showing that the renormalization procedure in this setting does indeed result in replacing bare couplings with renormalized values. Our investigation will in any case be of the infrared effects of these fields, and so we will not be concerned with the ill-behaved ultraviolet properties of these quantities. Notice that we have used Wick's theorem to express $n$-point functions in the free vacuum in terms of two-point functions. This is the crucial fact that makes the Hartree approximation equivalent to summing cactus diagrams. Enforcing the tadpole condition finally leads to
\begin{equation}
0= ie^{i\bar S}\int d^4y\left(\frac{\delta L}{\delta\phi}+\sum_{k=1}^\infty\frac{1}{k!}\bar V^{(2k+1)}\left(\frac{G_{yy}}{2}\right)^k\right)G_{xy}
\end{equation}
This implies that the quantity in parentheses vanishes, equivalent to the first equation in (\ref{bench}). So far we have only shown half of the equations in the path integral formulation. Before we discuss the other half we discuss how to replace the coincident limit of the correlators that appear in our expressions with a classical field.
\subsection{The Diakyon Field}\label{diakyonLegit}
From (\ref{bench}) it can be seen that the background dynamics of the inflaton can be interpreted as if the field were evolving in a two-field model in the transformed potential
\begin{equation}
V(\bar\phi+\psi)\rightarrow\sum_{k=0}^\infty\frac{1}{k!}\bar V^{(2k)}\left(\frac{\sigma^2}{2}\right)^k.\label{expansh}
\end{equation}
Here the second field $\sigma(y)=\sqrt{G_{yy}}$ represents the power in the fluctuation field. This second field takes on the role of encapsulating the average amount of power stored in the fluctuations, and has dynamics in its own right. Hereafter, we refer to it as the \emph{diakyon} after the Greek word for fluctuation, $\delta \iota\alpha\kappa\upsilon\mu\alpha\nu\sigma\eta$. Thus, $\sigma$ appears as a second classical field in our equations of motion, rendering our potential effectively dependent on two fields.
We first must establish the legitimacy of grouping long wavelength modes into a classical field such as the diakyon, as a suitable measure of the effect of tachyonic modes during inflation. Such a two-field interpretation was initially proposed in \cite{Cormier:1998nt}, in which the authors separated the contribution of the correlator into two, based on whether the mode $q$ is inside or outside the horizon. Then
\begin{equation}
\sigma = \sqrt{\langle \psi(\vec{x},t)^2\rangle_{q < aH}},\label{root}
\end{equation}
\noindent corresponding to the diakyon, was defined in such a way that it only comprises of modes for which $q < aH$. Hence, calling upon the equation of motion of the diakyon means setting an increasing dynamical cutoff in $q$-space, $q_{max}$, as a function of time, the horizon $(aH)^{-1}$ decreasing during inflation. A numerical analysis involving a fixed $q_{max}$ and mode equations (see for instance \cite{Albrecht:2014sea}), on the other hand, means considering the presence of both subhorizon and superhorizon modes, regardless of their tachyonic characteristic.
In \cite{Albrecht:2014sea}, the choice was made not to call upon a condensation of all the unstable modes in a classical diakyon-like field, in favor of a cutoff in $q$-space. Using mode equations, however, one needs to make choices in order to comply with numerical limitations while still accounting for most of the effect due to the presence of tachyonic modes. One choice corresponds to $q_{max}$. Ideally, one should let $q$ run to infinity. The validity of effective field theory (EFT) allows us to have a finite cutoff in $q$. However, the higher the $q_{max}$ the greater the total number of equations in the system, the latter itself depending also on the step size in $q$ of our numerical analysis. As time goes on, more and more modes will exit the horizon and have some participation in the overall spinodal effect. Hence, most of the impact attributed to the spinodal instabilities happens during early times, as a result of the long wavelength modes crossing the horizon \cite{Cormier:1998nt}. At very early times, however, $aH$ is vanishingly low, which in turn implies that the lowest few $q$-modes are responsible for the major part of the spinodal effect. Therefore, for such times, requiring a high $q_{max}$, while seemingly more precise, may also be unnecessary. Encompassing the lowest $q$-modes into a classical field thus looks efficient to incorporate most of the consequences resulting from tachyonic instabilities.
Our goal in this paper is not to give a very precise quantitative picture but a more qualitative one. After all, we are always limited by the accuracy of numerical computations. In \cite{Boyanovsky:1997xt} a lower bound on the total number (beyond 360 depending on the mass of the field) of e-folds necessary to study spinodal effects after grouping the mean field and quantum fluctuations into one classical field, was given, in the large N approximation. An argument was also made as to why such a classicalization is valid after only a few e-folds. The observation that the long wavelength modes are responsible for most of the spinodal effect \cite{Cormier:1998nt}, implies that grouping the modes which exit the horizon the earliest into one classical mode may be a valid approximation, especially if the total number of e-folds in the model is large. Allowing initial conditions that provide enough inflationary e-folds, and considering times 50 to 60 e-folds before the end of inflation should be sufficient to give a consistent qualitative picture of the effects of spinodal instabilities on inflation.
One should also address the choice of initial conditions. In refs. \cite{Cormier:1998nt} and \cite{Boyanovsky:1997xt} an effective initial condition for the diakyon was given,
\begin{equation}
\sigma_0 = \frac{H}{2\pi}, \label{dstemp}
\end{equation}
\noindent corresponding to a zero-temperature vacuum initial state. A vacuum state is a simple but nonetheless valid choice of initial state. Given the previous discussion, how much the spinodal instabilities manifest themselves depends on the initial value for the diakyon field compared to the inflaton. Thus, the number of initial conditions is the same whether we choose to call upon the mode equations, or a two-field picture. In the former framework, one needs to choose $q_{max}$ and $\phi_0$ while in the latter $\sigma_0$ and $\phi_0$ need to be set. Another parameter that needs to be fixed in both scenarios is the scale of inflation. If the scale of inflation is low, this allows the amplitude of $\langle \psi^2 \rangle$ to be large enough that it has a significant contribution to the overall dynamics of our system, while still maintaining the correct predictions for the other inflationary observables.
Having discussed the regime of validity of our classical two-field framework, we can now proceed to discussing how to transform our potential from a function only involving one field, to one suitable within a two-field construction.
\subsection{The Hartree Transform}
Eq. \eqref{expansh} is a cumbersome formula that involves Taylor expanding the potential of the field, then replacing each factorial coefficient by a different number. This is not only laborsome, but can quickly become impossible, even for some of the relatively simple potentials we consider here. Additionally, the convergence properties are worsened by this substitution; even for the simple example of noninteger monomials this transformed expression fails to converge.
Fortunately, the expression (\ref{expansh}) can be reproduced by a very simple integral transform of the potential, circumventing the need to ever Taylor expand in the first place. We define the $\emph{Hartree Transform}$\footnote{this is actually known as a Weierstrass transform, itself just a convolution against a Gaussian} of a function to be
\begin{equation}
\mathbb{H}[V](\bar\phi,\sigma)\equiv V^\mathbb{H}(\bar\phi,\sigma)=\int_{\mathbb{R}} d\psi\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{\psi^2}{2\sigma^2}}V(\bar\phi+\psi)\label{hart}
\end{equation}
It can be easily seen that the result of the Gaussian integration enforces the correct alteration of the coefficients to reproduce the Taylor-expanded result (\ref{expansh}). This transformation also has a very natural interpretation from the path integral perspective, as it exactly coincides with the approximation we made there to insert a single power of the interaction potential in the free-field vacuum (the quadratic term, though not a part of the interaction potential, automatically assembles with the rest of the terms to yield the unsplit potential). In fact, we have the general formula: $\mathbb{H}[f(\psi)]=\langle f(\psi)\rangle_0$, and can make use of the fact that derivatives with respect to background quantities can be pulled out of the brackets to directly rewrite all results in terms of the Hartree transform. The integration calls for the function to be defined everywhere on the real axis, which may not be part of the dynamically accessible regime, as in fractional powers of fields or logarithms. In order to ensure the reality of the result, we choose the nonunique but physically reasonable prescription that the function vanishes in inaccessible domains.
Let us interpret this expression to get an intuitive understanding of what the Hartree approximation actually does. Here we see that we are smearing over the fluctuations with a Gaussian filter, with a variance that is allowed to depend on time as determined by the new equations of motion. This will impose that the value of the new field will be given exactly by the variance of the fluctuations, and will treat the evolution of the background field as occurring in the ``homogenized" replacement field, the diakyon. Statistically speaking, this will give the same results as if we were to calculate the background and mode functions of the fluctuations, and compute the two-point corrleator for those (to the extent that the approximation holds). Additionally, in the limit that the fluctuations do not have a strong influence on the background mode, the field $\sigma$ approaches zero, and
\begin{equation}
\mathbb{H}[V](\bar\phi,0)=\int_{\mathbb{R}}d\psi\delta(\psi)V(\bar\phi+\psi)=V(\bar\phi).
\end{equation}
The distributional nature of the small $\sigma$ limit will make this formula cumbersome for some purposes, but the expansion (\ref{expansh}) is quite easy to handle. For instance, we see that the mass of the diakyon at 0 is $m^2_\sigma=\bar V_{\phi\phi}$, which encapsulates the fact that the strength of fluctuations will only grow if the inflaton is tachyonic. In this sense, applying the Hartree approximation will automatically tell us if the quantum corrections to the equations of motion are important or not. If we find that the diakyon is driven to $0$ (or even the small value $H/(2\pi)$, as the case would be in de Sitter space) for any initial value on a given potential, we can conclude that the usual lowest order approximation is sufficient, while if we uncover that the diakyon is not pushed to small values we conclude that these effects are important. For the slow-roll potentials we consider, the mass of the inflaton is much smaller than the Hubble rate, which implies that the potential in the fluctuation direction is similarly flat. This means that even for positive mass squared the field is not driven to its standard value $\sigma\sim H/(2\pi)$ very quickly; in fact, for chaotic initial conditions we would expect the two values of the fields to be comparable, meaning that the fluctuations would not necessarily reach the attractor value until near the end of inflation. This is in stark contrast to the scenario in which the backreaction is not taken into account, where any initial power is exponentially diluted away and the field equilibrates to the de Sitter temperature. This translates the question of the importance of quantum corrections to the dynamics to a problem of initial conditions for the perturbations. If we begin inflation on a potential that is otherwise suitable for slow-roll, but the initial value of the diakyon is very large, it will continue to play an important role throughout the evolution. In addition, if we envision that inflation started with a fast-roll phase \cite{Linde:2001ae}, in which the inflaton has a large tachyonic mass, then the diakyon quickly gets driven \emph{away} from its standard value, and so we would expect to find that the slow-roll evolution proceeds with a much stronger amount of fluctuations, regardless of when in the inflationary history the fast-roll event took place.
We end this discussion by noting the important identity
\begin{equation}
\boxed{\sigma\partial_\phi^2V^\mathbb{H}(\phi,\sigma)=\partial_\sigma V^\mathbb{H}(\phi,\sigma)}\label{heat}
\end{equation}
which holds at every point on the potential. This can be verified directly, but can also be arrived at by the observation that if we formally replace $\sigma^2=t$, $\phi=x$ then this is the heat equation, and (\ref{hart}) resembles the heat kernel. This equation will be an important tool for studying the behavior of the system, and we make use of it many times in the remainder of this paper. For example, we can use it to find an expression for the mass of the diakyon that is valid for any value of the fields:
\begin{equation}
\partial_\sigma^2V^\mathbb{H}=\partial_\phi^2V^\mathbb{H}+\sigma^2\partial_\phi^4V^\mathbb{H}.
\end{equation}
This recovers that the masses of the two fields are equal for small $\sigma$, but is an exact expression that informs us that the mass of the fluctuation can either be significantly larger or smaller than the inflaton for large field values.
\subsection{Diakyon Dynamics}
We have shown that the background dynamics can be encapsulated by adding an additional diakyon field representing the strength of the fluctuations and replacing the potential by its Hartree transform. We now show that these replacements perfectly capture the behavior of the fluctuations as well. In addition to the background dynamics, the equations governing the fluctuations can be derived in the path integral formalism. To do this, it is convenient to briefly introduce the two-particle irreducible formalism \cite{Luttinger:1960ua,Baym:1961zz,Cornwall:1974vz}, since this is specially designed to study the behavior of the fluctuation spectrum in the regime where quantum effects can become important. This simply entails adding a bilocal source for the fluctuations in addition to the usual source introduced when we are only interested in one-particle irreducible questions, as demonstrated by
\begin{equation}
Z[j_x,k_{xy}]=\int D\psi e^{iS[\bar\phi+\psi]+i\int dxj_x\psi_x-\frac i2\int dxdyk_{xy}\psi_x\psi_y}.
\end{equation}
The effective action is then defined as the Legendre transform of the logarithm of the previous quantity with respect to both $j_x$ and $k_{xy}$. Enforcing that the derivative of $Z[j_x,k_{xy}]$ with respect to $j_x$ vanishes yields the tadpole condition $\langle\psi\rangle=0$ that we explored above. In addition, we impose the gap equation, namely that the derivative with respect to $k_{xy}$ also vanishes. This will yield a nonlinear equation that dictates the spectrum of fluctuations. In the full theory, this will again involve correlators of arbitrarily many fields, and will be unsolvable. We can employ the same Hartree approximation to truncate the series to a finite subsystem, which yields
\begin{equation}
\Big(\Box_x+ V_{\phi\phi}^{\mathbb{H}}\left(\bar\phi,G_{xx}^{1/2}\right)\Big)G_{xy}=i\delta(x-y).
\end{equation}
If the two-point function is small, this leads to the lowest order result that the two-point correlator is just the Green's function of the second variation of the action. This standard result can be significantly altered in regimes where the two-point function is much larger, as explored in \cite{Riotto:2008mv}, where it was shown that self interactions serve to taper secular growth of correlators of massless fields (also see \cite{Youssef:2013by}, where a refinement on the Hartree approximation was recently employed). Since the Green's function is just the time ordered product of the mode functions, $G_{xy}=\mathcal{T}[\psi_x\psi_y]$, this equation is equivalent to the second equation in (\ref{bench}).
To show that the dynamics of these fluctuations can be replaced with the classical evolution of the diakyon field, we simply use (\ref{heat}) to rewrite (\ref{bench}) as
\begin{equation}
\Box\sigma+\partial_\sigma V^\mathbb{H}(\phi,\sigma)=0.
\end{equation}
With this, we have shown that the entire background dynamics can be reproduced as two classical fields in the altered potential $V^\mathbb{H}(\phi,\sigma)$, and have completely recovered the dynamics laid out in \cite{Cormier:1998nt,Cormier:1999ia}\footnote{There is also the Friedmann equation determining $H$ in terms of the background and fluctuations. This, and any other equation for fields not directly involving $\psi$, are trivially reproduced by the Hartree transform.}. Since we have justified to call upon $\sigma$ to encompass the effect of fluctuations, we will omit the bar in $\bar{\phi}$ and refer to $\phi$ as the inflaton, in the rest of the paper.
\subsection{Density Perturbations}
Though the background behavior is encapsulated as a simple two-field model, the diakyon is in actuality a composite of the inflaton field, and so we would expect fluctuations in the energy density to be different from the usual two-field formula. The expression was derived in \cite{Cormier:1999ia} by calculating the perturbation of the time-time component of the stress-energy tensor \emph{before} resumming the modes into the mean field $\sigma$. This is the most convincing procedure for obtaining the spectrum, but here we content ourselves with reproducing their results by a simpler method, where we employ immediate use of the diakyon approximation. Then
\begin{equation}
P_k=\frac16\frac{\langle \left(V(\phi+\psi)-V(\phi)\right)^2\rangle_0}{\dot H^2}=\left(\frac{V^\mathbb{H}}{(V^\mathbb{H}_\phi)^2+(V^\mathbb{H}_\sigma)^2}\right)^2\Big(\mathbb{H}[V^2]-2V\mathbb{H}[V]+V^2\Big)\label{spectrum}
\end{equation}
This expression is fully general. In the limit of small fluctuations, which is the case for instance if they are set by the typical de Sitter fluctuations, the perturbed potential can be Taylor expanded and the expression becomes
\begin{equation}
P_k\rightarrow\left(\frac{V^\mathbb{H}}{(V^\mathbb{H}_\phi)^2+(V^\mathbb{H}_\sigma)^2}\right)^2\langle V_\phi^2\psi^2\rangle_0.
\end{equation}
Using an identity similar to (\ref{heat}) this can be rewritten as
\begin{equation}
P_k\rightarrow\left(\frac{V^\mathbb{H}}{(V^\mathbb{H}_\phi)^2+(V^\mathbb{H}_\sigma)^2}\right)^2(\sigma^2+\sigma^4\partial_\phi^2)\mathbb{H}[V_\phi^2],
\end{equation}
which, upon taking the limit $\sigma\rightarrow 0$ agrees with the result derived in \cite{Cormier:1999ia}.
The $\sigma\rightarrow 0$ limit in this expression appears somewhat pathological, due to the apparent vanishing of the power spectrum. In this limit the standard scenario is in fact recovered, as the power of the large-wavelength modes can never go below the de Sitter value, meaning that $\sigma\rightarrow H/(2\pi)$, where it tracks the value of the potential. Operationally, we make the replacement $\sigma^2\rightarrow\sigma^2+V/(12\pi^2)$ in all our expressions, as justified by the definition in (\ref{root}). If we treat $\sigma^2/M_p^2$ as a small parameter, all but the leading dependence can be neglected in the expression for the power. In this case the Hartree transform becomes the trivial integration against a delta function, and the expression reduces to
\begin{equation}
P_k\rightarrow \frac{V^2}{V_\phi^2}\left(\sigma^2+\frac{V}{12\pi^2}\right)\label{spectro}
\end{equation}
The interpretation of this limit is clear: the $V^2/V_\phi^2$ prefactor is the usual quantity that makes the expression gauge invariant, and the term in parenthesis is the expected value of the two-point function. If the initial power on large scales is negligible, then this is just given by the de Sitter temperature, whereas if the power on large scales is large it is given by the initial value. What is nontrivial about this expression is the fact that now we are working in a consistent framework that dictates the behavior of large scale power for any given value after evolution. One crucial feature to note now is that in the limit where the diakyon dominates the power, the power spectrum does not depend on the overall scale of the potential. This is in great contrast to the usual case, where the overall scale of the inflaton potential is commonly set to give the observed normalization. Here this condition sets the initial value of the diakyon field. This translates the usual practice of trying to understand the amplitude of perturbations in terms of the microphysics dictating the precise potential to a question of why the diakyon took the value that it did. This also allows a set potential to yield a range of values for the fluctuations, making the observed value a potentially environmental quantity.
To match the value we see, we arrive at
\begin{equation}
\frac{\sigma}{M_p}=\sqrt{\epsilon P_k}\ll 1,
\end{equation}
so that our expansion in small $\sigma/M_p$ is indeed justified. Note, however, that even though the diakyon is small compared to the Planck scale, we are still free to be in the regime where it can be larger than the de Sitter temperature. From these considerations, we expect that the diakyon will have its greatest effect in small field models of inflation, where the inflaton is much smaller than the Planck scale as well.
We end this section by discussing the measurable quantities related to the power spectrum. Still in the limit $\sigma\ll M_p$, we arrive at
\begin{equation}
n_s-1=-\epsilon\left(4+\frac{2}{1+y}\right)+2\eta\frac{1}{1+y}\rightarrow \left\{
\begin{array}{rl}
-6\epsilon+2\eta & y\rightarrow 0\\
-4\epsilon & y\rightarrow \infty
\end{array} \right.
\end{equation}
Where we have introduced $y=12\pi^2\sigma^2/V$ as the dimensionless measure of fluctuations on large scales. In the large diakyon limit, the contribution from $\eta$ completely drops out! This tells that for small field models, where $\epsilon\ll1$ \cite{Lyth:1996im}, the spectrum becomes practically scale invariant in this limit. For large field models, whether the tilt increases or decreases depends on whether $\epsilon$ or $\eta$ is larger. In the eventuality $\epsilon=\eta$, the prediction for the tilt remains constant throughout the transition to the spinodal regime. This is the case for quadratic potentials, and we explicitly verify this in section \ref{m2phi2}.
Similarly, we can derive an expression for the tensor-to-scalar ratio, where we simply use the fact that the value of tensor fluctuations are still set by the (Hartree transformed) potential
\begin{equation}
r=\frac{16\epsilon}{1+y}\rightarrow \left\{
\begin{array}{rl}
16\epsilon & y\rightarrow 0\\
0 & y\rightarrow \infty
\end{array} \right.
\end{equation}
which displays the alluded tendency to vanish for large values of the diakyon. Taken in combination, these two observables indicate that a large value of the diakyon tends to push predictions towards the bottom right corner of the $n_s-r$ plot. Simple application of this formula presumes that the values of $\epsilon$ and $\eta$ remain unchanged no matter the value of the diakyon, in particular that the value of the inflaton field 50 or 60 e-folds before the end of inflation is unaltered. This is certainly true for small enough values of the diakyon, but away from this limit the field trajectories can be solved numerically.
From these expressions we notice a general tendency of spinodal effects to push inflationary predictions down and to the right in the $n_s-r$ plane. This motivates these effects to be considered especially in models that traditionally lie to the left of the Planck contour, in the hopes that corrections might bring them back into agreement with observation. We choose to illustrate this in the simplest scenario of hilltop inflation in section \ref{hillinf}, but some other potentials this could potentially be relevant for include inflection-point inflation, Coleman-Weinberg inflation, double well inflation, and pseudonatural inflation \cite{Martin:2014vha}.
\section{Predictions for Specific Potentials}\label{Tart}
We now apply our findings in the previous section to some specific potentials, to see what effects the backreaction might have. We begin with the simplest model of chaotic inflation, a single field in a quadratic potential, in section \ref{m2phi2}. The Hartree transform of this model consists of two noninteracting fields with equal mass. The only influence of the diakyon field in this setup is through the fluctuations, and can serve to drastically decrease the tensor-to-scalar ratio, while leaving the spectral tilt fixed. We then turn to the simplest tachyonic model, quadratic hilltop inflation, in section \ref{hillinf}. We show in this model that the tilt is driven to be exactly scale invariant and the tensor perturbations unobservable, but that it takes typically $\sim100$ e-folds to reach this regime, providing a window agreeing with the observed values that could possibly encompass cosmic microwave background (CMB) scales. A tachyonic large field model is considered in section \ref{flatinf} where we consider a flattened potential. This model eventually attains a spinodal regime for practically all initial conditions, allowing us to infer the difference between inflation descending from the eternal regime and inflation that lasted only marginally longer than the observed amount. These findings are summarized in Fig.\ref{master}.
\begin{figure*}[h]
\centering
\includegraphics[height=9cm]{nsrEDITFillTo0.png}
\caption{The $n_s-r$ plot for the models we consider in this paper, along with the Planck $1\sigma$ and $2\sigma$ contours. Allowing for strong backreaction can serve to place both $m^2\phi^2$ and hilltop inflation predictions firmly in the preferred region, and kicks the flattened model out. This highlights the generic trend that tensor modes become more negligible the stronger the back reaction is.}
\label{master}
\end{figure*}
As a final application, we study monodromy models, where we see that backreaction effects serve to dampen the oscillations of the potential for small enough values of the axion decay constant $f$. This can even provide a route for the inflaton to escape false minima, allowing for slow-roll even in potentials that would not normally allow it.
Before considering specific models, we first exhibit a useful general result, applicable if the value of the diakyon remains small compared to the Planck scale. We remind the reader that we only consider this regime anyway, since otherwise (\ref{spectrum}) would imply that density perturbations are well above unity. In this regime, the field dynamics, in the slow-roll regime, becomes simply
\begin{equation}
V\frac{d\phi}{dN}=-M_p^2V_\phi,\quad V\frac{d\sigma}{dN}=-M_p^2V_\sigma.
\end{equation}
Here $N$ is the number of e-folds. Several identities can be used to rewrite the diakyon equation in a simpler form. For instance, the heat equation identity (\ref{heat}) can be used to express this as a linear equation in $\sigma$
\begin{equation}
V\frac{d\sigma}{dN}=-M_p^2V_{\phi\phi}\sigma,
\end{equation}
and to lowest order $V$ is the original, $\sigma$ independent potential. Additionally, if the inflaton evolution is monotonic, we can express this as $\sigma(\phi(N))$ to arrive at
\begin{equation}
\frac{d\sigma}{d\phi}=\frac{V_{\phi\phi}}{V_{\phi}}\sigma,\label{lindiak}
\end{equation}
which has the solution
\begin{equation}
\sigma(N)=\sigma_0e^{\int_{\phi_0}^{\phi(N)}d\phi\frac{V_{\phi\phi}}{V_{\phi}}}=\sigma_0e^{\int_{V_{\phi_0}}^{V_{\phi(N)}} d(\log{V_\phi})}=\sigma_0\frac{V_{\phi(N)}}{V_{\phi_0}}.\label{integ}
\end{equation}
This allows us to relate the total amount of buildup of large scale power to properties of the potential evaluated at specific points. It is remarkable that this equation is integrable, and so we reflect on why indeed this is the case. In general, one expects an exponential buildup of modes for tachyonic masses to be given by the integral of (\ref{lindiak}), and modes to subside during periods of positive mass squared. If this were all the information we had about the system, it would not be integrable, as there is no way to relate the amount of time spent with any given value of the mass to the endpoints of the evolution. However, since the dynamics is tied to the evolution of the inflaton, periods of large mass squared are passed through more quickly than periods of small mass squared, and the entire evolution can be encapsulated in the difference between beginning and end points. We note the pathological limits of this expression, where the slope of the potential vanishes: if it vanishes initially, inflation lasts an infinitely long time, and the diakyon attains infinite values. Similarly, if we evolve this to the point where the slope vanishes at the end, the diakyon subsides completely.
\subsection{$m^2\phi^2$}\label{m2phi2}
We begin with a study of the simplest model of inflation, where the potential is just a quadratic function of a single inflaton field to show that, even though the field is not tachyonic, the fluctuations can have a large effect on the background dynamics. We stress that large values of the diakyon are not evolved to dynamically, so in order for effects to be important in this scenario the diakyon must start out very large. Thus, we cannot claim the effects we find to be generic predictions for the model as we can in the following sections, but we include this case anyway because its simplicity enables a clear demonstration of how effects can come into play.
This potential is simple enough that the evolution of the fields can be solved exactly in the slow-roll approximation. The Hartree transform yields a two-field potential
\begin{equation}
V^{\mathbb{H}}_{\text{chaotic}}(\phi,\sigma)=\frac12m^2\left(\phi^2+\sigma^2\right),
\end{equation}
which is simply two noninteracting fields with the same mass. Note that for the background dynamics it is possible to exploit the $O(2)$ symmetry to arrange that only one field direction has an initial value. The perturbation spectrum, however, does not respect this symmetry, and so we prefer to work with the physical fields.
Using the slow-roll equations
\begin{equation}
(\phi^2+\sigma^2)\frac{d\phi}{dN}=-2M_p^2\phi,\quad (\phi^2+\sigma^2)\frac{d\sigma}{dN}=-2M_p^2\sigma,
\end{equation}
we can solve these exactly
\begin{equation}
\phi(N)=\phi_0\sqrt{1-\frac{4M_p^2N}{\phi_0^2+\sigma_0^2}},\quad \sigma(N)=\sigma_0\sqrt{1-\frac{4M_p^2N}{\phi_0^2+\sigma_0^2}}.
\end{equation}
Both fields reach $0$ at precisely the same time, $\sigma$ being just a rescaled copy of $\phi$. If we make the approximation that this is the point at which inflation ends, then we can use equation (\ref{spectrum}) to find
\begin{equation}
N_{\text{CMB}}=\frac{\phi_0^2+\sigma_0^2}{4M_p^2},\quad P_k= N_{\text{CMB}}\frac{\sigma^2}{M_p^2}+\mathcal{O}\left(\frac{\sigma}{M_P}\right)^4 ,\quad n_s-1=-\frac2{N_{\text{CMB}}},\quad r=\frac{4}{3\pi^2}\frac{m^2}{\sigma_0^2}.
\end{equation}
In contrast with the usual predictions, now the amplitude of the power spectrum is set by the value of the diakyon field instead of the mass scale of the potential. If we insisted on the standard value, $\sigma=H/(2\pi)$, the usual results would be recovered, and the value of the mass would be needed to be set to reproduce the observed spectrum. If the diakyon is large, however, normalization of the power spectrum implies
\begin{equation}
\sigma_0\approx\sqrt{P_k/N_{\text{CMB}}}M_p\approx 6\times10^{-6}M_p\ll15M_p\approx\phi_0.
\end{equation}
This indicates that the initial value of the inflaton is much larger than the initial value of the diakyon, but the diakyon still has a much larger value than the usual case for generic $m^2$. Evidently, the tilt is still only sensitive to the total number of e-folds, and so its standard predictions remain unaltered. The only parameter that depends on the mass scale here is the tensor-to-scalar ratio, which now decreases when $m^2$ takes small values. This has the consequence that the predictions of this model can be brought into agreement with the preferred values of these parameters, if the mass scale is sufficiently small. The origin of this result resides in the fact that, if backreaction is large, fluctuations are no longer tied to the expansion rate, but can now be much larger. Since the strength of the scalar fluctuations is dictated by the value of $\sigma$, whereas the tensor fluctuations are still prescribed by the overall energy density, the scale of the potential can be lowered arbitrarily, while still maintaining the observed power spectrum.
\subsection{Hilltop Inflation}\label{hillinf}
While the quartic hilltop model fits with the measured values of inflationary parameters well enough to merit inclusion on the Planck 2015 $n_s-r$ plot \cite{Ade:2015lrj}, the simplest model, quadratic hilltop inflation, predicts too small an $n_s$. As such, it is the ideal case to study. We find already in this simple scenario that there are many different regimes, corresponding to the initial value of the diakyon.
The transformed potential in this case is
\begin{equation}
V_{\text{hilltop}}^\mathbb{H}(\phi,\sigma)=V_0-\frac12m^2(\phi^2+\sigma^2)
\end{equation}
and, similar to the previous case of $m^2\phi^2$, near the top of the hill the two fields behave independently of each other, each experiencing exponential growth:
\begin{equation}
\phi(N)=\phi_0e^{|\eta_0|N},\quad \sigma(N)=\sigma_0e^{|\eta_0|N},
\end{equation}
where $\eta_0=-m^2M_p^2/V_0$ is the slow-roll parameter at the top of the hill. For simplicity, we consider the predictions for the case when inflation ends at the point where the potential becomes negative. In actuality, we expect the form of the potential to be altered well before this, but this simplistic setting will be enough to illustrate the effects of the diakyon field. In this scenario the vacuum energy can be approximated by the constant term in the potential, with corrections becoming important only in the last e-fold or so of the evolution. The value of the inflaton at CMB scales is set to be $\phi_{\text{CMB}}=\sqrt{2/\eta_0}e^{-|\eta_0|N_{\text{CMB}}}$. The initial value of the diakyon must be less than this, or else it would spoil inflation before the required amount of e-folds has elapsed.
The power spectrum is given by
\begin{equation}
P_k=\frac{V_0^2}{m^4}\frac{\sigma^2(\phi^2+\frac34\sigma^2)}{(\phi^2+\sigma^2)^2}.
\end{equation}
The previous requirement that $\sigma<\phi$ doubly suits us, because as it can be seen from above it also corresponds to the regime where $P_k\rightarrow 3/(4\eta_o^2)\gg 1$, certainly of no phenomenological relevance. In the limit $\sigma\ll\phi$ the power spectrum becomes
\begin{equation}
P_k\rightarrow \frac{V_0^2}{m^4}\frac{\sigma^2}{\phi^2}
\end{equation}
where it is understood that for small values of the diakyon this expression should recover the standard result by saturating to the de Sitter temperature. This form displays the scale invariant behavior uncovered in the general case, owing to the fact that both fields have the same exponential dependence on e-fold time (making this regime also of no phenomenological relevance). If, instead, the diakyon starts at values much smaller than the de Sitter temperature, we recover the standard predictions
\begin{equation}
n_s-1=6\eta_0e^{2\eta_0(N_{\text{CMB}}-N)}+2\eta_0,\quad r=-16\eta_0e^{2\eta_0(N_{\text{CMB}}-N)}.
\end{equation}
The approximations we have used break down for $\eta_0\ll N_{\text{CMB}}^{-1}$, but in this case the amount of e-folds is so great that the last 60 will occur at the base of the potential, which is practically linear, and does not really resemble hilltop inflation anyway. The interesting regime is when the diakyon starts smaller than the de Sitter temperature and then, through the course of its exponential growth, transitions to the regime of perfect scale invariance and negligible tensor-to-scalar ratio. This transition actually takes place on the order of 50 to 100 e-folds, as shown in Fig \ref{hillns}, so that this is not a singular event that would require tuning to occur within the observable window. This also ensures that the running of the power spectrum remains unobservably small throughout the evolution, in agreement with current observations.
\begin{figure*}[h]
\centering
\includegraphics[height=6.25cm]{hillns.pdf}\
\caption{Evolution of the tilt with two different values for the slow-roll parameter $\eta_0$, the solid curve corresponding to $\eta_0=-1/25$ and the dashed to $\eta_0=-1/30$. Different initial conditions and overall scale of the potential will shift the transition region to the left or right. These curves illustrate that the transition takes a few dozen e-folds to pass through the observable window, a naive theorist's reconstruction of which, based off \cite{Ade:2015lrj}, is shown as the shaded region.}
\label{hillns}
\end{figure*}
We can comment on other versions of hilltop inflation before we conclude this section. For the case of quartic hilltop inflation, which lies well within the best fit region in the Planck data, the spinodal effects produce
\begin{equation}
V^\mathbb{H}_{\text{ quartic hilltop}}(\phi,\sigma)=V_0-\lambda\left(\phi^4+6\sigma^2\phi^2+3\sigma^4\right).
\end{equation}
So that the diakyon direction is actually much steeper than the inflaton direction. This necessitates the initial values to be even more fine tuned for this to match observations.
\subsection{Flattened Potentials}\label{flatinf}
There are certain contexts where integrating out heavy degrees of freedom serves to flatten the inflaton potential from an initially steeper profile for large field values \cite{Dong:2010in}. This can cause the field to be driven by an effective potential that is a fractional power of the field. A typical example of such a potential that will be testable in the coming years is
\begin{equation}
V_{\text{flat}}(\phi)=\mu^{10/3}\phi^{2/3}.
\end{equation}
The standard predictions for this potential are
\begin{equation}
\phi(N)=\sqrt{\frac43(N_0-N)}M_p,\quad n_s-1=-\frac{4}{3N_{\text{CMB}}},\quad r=\frac{8}{3N_{\text{CMB}}},\label{stand23}
\end{equation}
with $N_0$ being the beginning of inflation and $N_{\text{CMB}}$ the amount of enfolds since CMB scales left the horizon. However, on this potential the field is tachyonic, and so we should expect a region of initial conditions where spinodal effects can become important. To see this we apply the Hartree transform to this potential, yielding a result in terms of confluent hypergeometric functions
\begin{equation}
V^\mathbb{H}_{\text{flat}}(\phi,\sigma)=\mu^{10/3}\left[\frac{\Gamma\left(5/6\right)\sigma^{2/3}}{2^{2/3}\pi^{1/2}}{}_1F_1\left(-\frac{1}{3};\frac{1}{2};-\frac{\phi^2}{2\sigma^2}\right)+\frac{\Gamma\left(4/3\right)\phi}{2^{1/6}\pi^{1/2}\sigma^{1/3}}{}_1F_1\left(\frac{1}{6};\frac{3}{2};-\frac{\phi^2}{2\sigma^2}\right)\right].
\end{equation}
This shows the true utility of (\ref{hart}), allowing for an analytic expression for the full potential. The latter is displayed in Fig.[\ref{two thirds}].
\begin{figure*}[h]
\centering
\includegraphics[height=6.25cm]{twothirds.pdf}\
\caption{The full potential for $\phi$, $\sigma$, overlaid with the $\sigma$-independent original potential. For small initial values of the diakyon the inflaton dynamics exactly corresponds to the usual case, and the diakyon experiences an instability towards larger field values.}
\label{two thirds}
\end{figure*}
The previous expression is not very inviting of an analytical treatment, but it is easy enough to work with numerically, and in asymptotic limits. For small values of the diakyon the original potential is recovered, as must be the case. The dynamics of several choices of initial conditions are shown in Fig.[\ref{paths}].
\begin{figure*}[h]
\centering
\includegraphics[height=5.25cm]{paths.pdf}\
\caption{Inflationary trajectories during the last 60 e-folds for different initial field values. The solid lines correspond to the inflaton, and the dashed lines correspond to the diakyon. For small initial values of the diakyon, as in the green curves (solid top and dashed bottom), it never comes to dominate, and inflation proceeds as usual. The purple (intermediate) and orange (coincident at start) curves exhibit extreme values of the diakyon, where it comes to dominate at late times, leading to a few extra e-folds of inflation. These values lead to a very pathological amount of power, so should be treated as exaggerated scenarios demonstrating the effect present in smaller quantities for more modest values.}
\label{paths}
\end{figure*}
However, for Planckian values of the diakyon, the amplitude of the power spectrum is well over unity, which is nowhere close to the observed value. Therefore, we restrict our analysis to values $\sigma\ll M_p<\phi$, for which case the potential can be approximated as
\begin{equation}
V^\mathbb{H}(\phi,\sigma)\rightarrow\mu^{10/3}\left(\phi^{2/3}-\frac19\frac{\sigma^2}{\phi^{4/3}}\right).
\end{equation}
In the approximations we have made, the inflaton trajectory is barely altered from the standard result (\ref{stand23}), and the diakyon trajectory is
\begin{equation}
\sigma(N)=\frac{\sigma_0}{\left(1-\frac{N}{N_0}\right)^{1/6}}.
\end{equation}
This blows up at the end of inflation, but the approximations we have made break down well before that, as can be seen in Fig. \ref{paths}. In the regime where $\sigma\gg\phi$, we can use the fact that ${}_1F_1(a;b;0)=0$ for all $a$, $b$ to find the limiting form of the potential
\begin{equation}
V^\mathbb{H}_{\text{flat}}(0,\sigma)\rightarrow\frac{\Gamma(5/6)}{2^{2/3}\sqrt{\pi}}\mu^{10/3}\sigma^{2/3},\label{.4}
\end{equation}
which is of the exact same form as the original potential, but with an additional numerical factor $\approx 0.4$. Before this regime is reached, though, the diakyon grows with time, characteristic of tachyonic behavior, and directly opposed to the lowest order approximation, where it tracks the value of the potential which decreases with time. This causes an interesting behavior: there is a discernible difference between inflation on this potential that comes from an eternal regime, and inflation that started only several e-folds outside of the observable window. To highlight the difference between these two scenarios, let us assume that inflation began at the eternal regime $P_k\approx1$ \cite{Linde:1986fd}, with the initial value of the diakyon set by the de Sitter temperature (\ref{dstemp}):
\begin{equation}
\phi_0=\left(\frac49\right)^{3/8}\frac{M_p^{9/4}}{\mu^{5/4}},\quad V_0=\sqrt{\frac23}M_p^{3/2}\mu^{5/2},\quad \sigma_0^2=\frac{1}{12\pi^2}\sqrt{\frac23}\frac{\mu^{5/2}}{M_p^{1/2}}.
\end{equation}
For simplicity we use the approximation that the fields start on their slow-roll attractor behavior, with negligible initial inhomogeneities. The total amount of e-folds from this regime is $N_e=(M_p/\mu)^{5/2}/\sqrt{6}$, much larger than the observed 60 or so e-folds. By the time the fields have evolved down to CMB scales, the value of the diakyon is
\begin{equation}
\sigma_{\text{CMB}}^2=\frac{1}{12\pi^2}\frac{2^{1/3}}{3^{2/3}}\frac{1}{N_{\text{CMB}}^{1/3}}\mu^{5/3}M_p^{1/3}=\frac{1}{6^{1/3}}\frac{1}{N_{\text{CMB}}^{2/3}}\left(\frac{M_p}{\mu}\right)^{5/3}\left(\frac{H}{2\pi}\right)^2\gg \left(\frac{H}{2\pi}\right)^2.
\end{equation}
The diakyon is still subplanckian, so our expansion is valid, but it is much larger than its vacuum value. In this regime the fluctuations behave purely spinodally. Normalization of the power spectrum at $N_{\text{CMB}}=60$ then requires $\mu=1.6\times10^{-5}M_p$, and the predictions for the model become
\begin{equation}
n_s=1-\frac{2}{3N_{\text{CMB}}}= 0.989,\quad r=\frac{7.5\times10^{-7}}{N_{\text{CMB}}}=1.3\times10^{-8},
\end{equation}
which are clearly different from (\ref{stand23}), with half the predicted tilt and practically vanishing tensor-to-scalar ratio. These values of the parameters actually lie substantially outside of the observed values, making this regime in disagreement with experiment.
There is a general trend that is indicative of all tachyonic models with an eternal regime: if inflation proceeds for a sufficiently long time, the diakyon takes over. Contrary to the usual predictions for large field models in this class, we would not expect to see tensor modes, and if application of the usual expression for the tilt were applied, it would lead us to the wrong conclusion about the parameters and shape of the potential. If we were confident that inflation took place on a given potential, however, observations would allow us to infer whether inflation lasted parametrically longer than the observed amount by measuring the amount of deviations from the standard predictions.
We would like to conclude our analysis of flattened potentials with a brief remark on another scenario: the Starobinsky ``$R+R^2$'' model \cite{Starobinsky:1980te}. This analysis is particularly relevant here since it is a small field model, so if spinodal effects are present the spectrum looks scale invariant. If we assume that the diakyon starts at the de Sitter temperature, then straightforward application of (\ref{integ}) yields
\begin{equation}
\sigma_{\text{CMB}}=\left(\frac{N_{\text{tot}}}{N_{\text{CMB}}}\right)^2\left(\frac{H}{2\pi}\right)^2
\end{equation}
Hence, if inflation lasted any longer than the minimum required to solve cosmological problems, the backreaction due to the diakyon would be important and engender a shift in $n_s$ towards exact scale invariance.
\subsection{Monodromy Inflation}\label{monoinf}
As a final application of the Hartree transform we consider axion monodromy inflation \cite{Silverstein:2008sg,McAllister:2008hb}. In the simplest realization of this scenario, inflation is driven by an axion field in which stringy effects break the exact periodicity of the potential. This leads to an approximately linear potential with small modulations,
\begin{equation}
V_{\text{mono}}(\phi)=\mu^3\phi+\Lambda^4\cos\left(\frac{\phi}{f}\right).\label{monopot}
\end{equation}
Inflation is driven by the linear term, yielding standard predictions for inflationary parameters: $n_s-1=-3/(2N_{\text{CMB}})$ and $r = 4/N_{\text{CMB}}$. In addition, the sinusoidal correction leads to characteristic oscillations in the power spectrum that are logarithmic in scale \cite{Flauger:2009ab} and have yet to be found at a significant level \cite{Easther:2013kla}, but can be arbitrarily small depending on the parameter $\Lambda$. One is free to take natural values of the axion decay constant $f\sim \mathcal{O}(\frac{1}{10}-\frac{1}{100})M_p$, in which case the field undergoes many oscillations in a single e-fold, or the cycles could be so large to encompass the entire range of observed CMB scales, possibly providing a running spectrum that explains lack of power both on large scales and in the matter power spectrum \cite{Minor:2014xla}.
Monodromy inflation is a large field model, with
\begin{equation}
\phi(N) = \sqrt{2(N_{\text{CMB}}-N)}M_p.\label{phiofn}
\end{equation}
If spinodal effects are not too large, the normalization of the spectrum enforces $\mu = 6 \times 10^{-4} M_p$. The other parameters can be expressed in a convenient combination $b = \Lambda^4/(\mu^3f)$. If this parameter is too large, $b > 1$, the potential will not be monotonically decreasing, but will instead develop pockets that, if too deep, could potentially cause the inflaton to become trapped in a false minimum instead of undergoing slow roll.
We would like to investigate to what extent spinodal effects might change this story. Taking the Hartree transform of the potential (\ref{monopot}) yields
\begin{equation}
V^\mathbb{H}_{\text{mono}}(\phi,\sigma)=\mu^3\phi+\Lambda^4\cos(\phi/f)e^{-\sigma^2/2f^2}.
\end{equation}
The linear term remains untouched, and the cosine, as an eigenfunction of the Hartree transform, is now multiplied by a gaussian. The potential is plotted in Fig [\ref{mono}].
\begin{figure*}[h]
\centering
\includegraphics[height=6.25cm]{mono.pdf}\
\caption{The two-field potential. For large values of the diakyon the oscillations are washed out, and the inflaton direction is strictly linear.}
\label{mono}
\end{figure*}
We see that allowing for tachyonic modes to build up and backreact on the background evolution can serve to taper the size of the oscillations, depending on the initial conditions taken for the inhomogeneity. An important point is that spinodal effects prevent the usual restriction to an exactly homogeneous slow-roll evolution from being an attractor, so that only for a narrow range of field space $\sigma\lesssim f$ will these oscillations occur at all. For generic values of the diakyon, the potential will look exactly linear, with no additional signals. As we have argued in section \ref{Hart}, if inflation is to have come from an eternal regime, then we would expect a much larger initial value of the diakyon field. We note that spinodal effects can provide a route for slow-roll inflation to occur even for large values of $b$, which would have been traditionally excluded.
The line $\sigma=0$ corresponds to the traditional slow-roll path, which gives the strongest possible oscillation signal. From the form of the potential one can expect that during the periods when the inflaton $\phi$ is near the top of a hill the fluctuations roll towards the plateau region, corresponding to the buildup of tachyonic-wavelength modes. Before this effect can get too strong, however, the inflaton goes through a valley region, at which point the diakyon tends towards decreasing values. If the starting point is well out on the tail of the Gaussian, the diakyon will barely change its value at all, and the evolution of the inflaton will look effectively linear. These qualitative effects can be encapsulated in an analytical approximation below.
To get a stronger understanding of how the diakyon behaves during inflation, we make a few simplifying assumptions to make its time dependence analytically solvable in terms of special functions. We first take the time dependence of the inflaton to be given by its unperturbed slow-roll trajectory (\ref{phiofn}), which is valid to lowest order in slow roll and $b$. Secondly, we take $\sigma$ to be in slow roll as well, which is justified because the slope of the Gaussian is $\frac{\partial V}{\partial\sigma}\lesssim b\frac{\partial V}{\partial\phi}$, and so is comparatively shallow to the inflaton direction unless $b\gg 1$. Third, if the Hubble rate is dominated by the linear term in the potential, then we can write
\begin{equation}
\frac{\mu^3}{M_p}\sqrt{2(N_0-N)}\frac{d\sigma}{dN}=\frac{\Lambda^4}{f^2}\cos\left(\frac{M_p}{f}\sqrt{2(N_0-N)}\right)\sigma e^{-\sigma^2/2f^2},
\end{equation}
which we can integrate to get the following expression for $\sigma(N)$ in terms of the exponential integral $\text{Ei}(x)$:
\begin{equation}
\text{Ei}\left(\frac{\sigma(N)^2}{2f^2}\right)-\text{Ei}\left(\frac{\sigma_0^2}{2f^2}\right)=-2 b\sin\left(\frac{M_p}{f}\sqrt{2(N_0-N)}\right).\label{exactsigma}
\end{equation}
From this expression we can remark that the field oscillates between two extremes corresponding to the maxima and minima of the sine function, so that one full cycle brings the $\sigma$-field back to where it originally started. Moreover, the amplitude of this oscillation cycle does not change, and there is no long-term drift of the central value. This curve, along with extremum values for the field, is plotted in Fig. [\ref{expei}] for example values of the parameters.
\begin{figure*}
\centering
\includegraphics[height=5.25cm]{expei.pdf}\
\caption{Implicit solution to equation (\ref{exactsigma}) for example values $b=0.5$ and $\sigma_0=0.7f$. The diakyon oscillates between the two dashed curves during each cycle. Increasing the initial value of the diakyon field corresponds to shifting the horizontal lines upwards, while increasing $b$ corresponds to increasing the vertical width between them. Both for small and large initial values, there is hardly any change in the field at all.}
\label{expei}
\end{figure*}
Notice that the oscillation can be quite lopsided. If we increase the value of $b$, corresponding to more pronounced hills and valleys, the range of $\sigma$ will be greater. For very small initial values, the maximum value of $\sigma$ is negligible, and spinodal effects can be ignored. This corresponds to placing the trajectory at the very top of the hill, where there is no strong slope to provide a force away. Very large values of the field are plotted on the right plot, for which the field does not move very much at all, regardless of the value of $b$. In this regime asymptotic limits can be made in (\ref{exactsigma}) to find the total range of $\sigma$ during one cycle to be
\begin{equation}
\Delta\sigma\rightarrow b\sigma_0e^{-\sigma^2/2f^2}.
\end{equation}
This corresponds to regions far enough out on the Gaussian tail that the potential looks exactly flat, and so
the diakyon is roughly stationary.
We now discuss the parameter values for which the diakyon can effectively screen the small scale perturbations. If the fluctuations in (\ref{spectro}) are dominated by the potential and not the diakyon, we require $\sigma^2\ll V/M_p^2$, which translates into $\sigma\leq 4.5\times10^{-6}M_p$. In the event that the diakyon dominates the fluctuations, which will occur if the scale $\mu$ is smaller than the standard value, correct normalization enforces that this equality be saturated. In this case the oscillating value of $\sigma$ can reintroduce oscillations in the power spectrum, but these will be large only if $\sigma\approx f$, which is an apparently fine-tuned scenario that we disregard. Screening becomes efficient for $\sigma\gtrsim4f$, which means that the mechanism we have described can only be operational for small values of $f$, say $f<10^{-6}M_p$. This value may seem somewhat low from the perspective of high energy physics models, but corresponds to the upper value of the ``classic window'' for axion dark matter \cite{Hertzberg:2008wr}. Monodromy models with different powers of the inflaton are sometimes considered \cite{McAllister:2014mpa}. Taking into account these models does not substantially raise the value of $f$ needed for the diakyon's screening of small scale perturbations to take place. However, the dynamics may be more interesting, as the mass of the inflation causes the diakyon to increase (or decrease) with the evolution, which could cause the oscillations to appear only at early (or late) times.
\section{Conclusions}\label{conclusions}
We have considered the influence of backreaction from inflationary fluctuations for a few typical models of inflation. We showed that the strength of the influence is largely dependent on the initial conditions of the inflaton, and whether the dynamics are tachyonic enough to drive the diakyon to large field values. If the long wavelength modes are initially populated close to the de Sitter equilibrium values, backreaction is negligible and inflation proceeds as normal, unless a very long time elapses since the beginning of inflation. If the modes are initially populated to a much greater extent, then it can significantly alter the inflationary dynamics and predictions, by influencing the strength of the generated perturbations. Though the details of inflationary dynamics can be substantially modified, the basic predictions of the scenario, mainly a stage of near exponential expansion, generating a nearly scale invariant spectrum of perturbations, remains the case. We note four general trends. The predicted value of the spectral tilt deviates slightly from the original value in large field models, and moves strongly towards scale invariance in small field models. Secondly, since the size of the fluctuations is no longer set by the energy scale of the potential during inflation, inflation can proceed at a much lower scale and still produce the observed size of fluctuations. This causes the tensor-to-scalar ratio to drastically decrease. Thirdly, the eventual tendency towards the tachyonic regime allows us to infer the duration of inflation before observable scales. Lastly, if the potential has any sort of features, the strong backreaction regime serves to smear out these features, providing a smoother potential for the inflaton. This can occur even for features so deep that they would have prevented the inflaton from reaching its true minimum at all. This was studied in the simple case of monodromy inflation, but holds in generic potentials, as displayed in Fig. \ref{pocketses}.
\begin{figure*}[h]
\centering
\includegraphics[height=6.25cm]{pocketses.pdf}\
\caption{Spinodal effects in a generic random potential. For small values of the diakyon the shown potential is completely unsuitable for slow-roll inflation, containing many false minima and steep peaks. Backreactions tame these features by smearing over the original potential with a Gaussian filter. This picks out the large scale trend of the potential, providing a route for slow-roll inflation to proceed.}
\label{pocketses}
\end{figure*}
We remark that this washing out of features is not enough to completely solve the $\eta$ problem, which is the question of how the mass of the inflaton can be lighter than the Hubble scale, even after quantum corrections are taken into account. This stems from the fact that the size of the features that are washed out depends on the value of the diakyon. If we Fourier expand a potential, $V(\phi)=\int dn V_n \cos(n\phi/M_p)$, the Hartree transformed potential multiplies each coefficient by a gaussian $e^{-n^2\sigma^2/M_p^2}$, which effectively cuts off the value of the integral at $n\approx M_p/\sigma$. If all coefficients are of the same order, then this yields $\eta\approx M_p^3/\sigma^3$, but since we require $\sigma\lesssim\sqrt{\epsilon P_k}M_p$ we arrive at $\eta\gtrsim 10^{18}\gg1$. The only way to circumvent this conclusion is if the coefficients of the Fourier series become tiny for suitable values of $n$, but this requires working on a potential where the $\eta$ problem was solved originally. As it stands, this mechanism is only suitable for removing features that are much smaller than the typical field range.
The complicated nature of the strong backreaction regime forced us to employ the Hartree method to encapsulate the qualitative aspects of these effects. We expect it to be a reasonable estimate of the dynamics based on its success in other scenarios. The overall effect of this approximation is to replace the original single field dynamics by another field that governs the strength of the fluctuations. It is important to make an estimate of the next order corrections to this approximation to ensure that the calculations we performed are under full control. The path integral approach to the Hartree approximation is a simple way to do this, and we plan to return to this in future publications. Nevertheless, we expect the qualitative features we have uncovered to remain true in the full theory.
There are a number of other issues worth exploring. We employed the Hartree approximation in the uniform curvature gauge, where all perturbations were in the inflaton field. In general, there is no guarantee that resummation schemes are gauge invariant, and so it would be interesting to extend this framework to gauge invariant perturbation theory. The Hartree approximation is also known to not respect global symmetries \cite{Pilaftsis:2013xna}, and in the simplest settings provides spurious masses to Nambu-Goldstone bosons of broken symmetries. As the inflaton can be thought of the Nambu-Goldstone boson of time translation invariance \cite{Cheung:2007st}, it might be worth considering if a modified version of the Hartree approximation such as used in \cite{Pilaftsis:2013xna} is more appropriate. We have also only considered strong backreaction effects on the scalar two-point correlator. It might be interesting to examine if its effects on the tensor power spectrum can be similar, and what effect it might have on nongaussianities. The latter consideration might even necessitate a generalization of the above scenario to a 3-particle irreducible formalism, similarly causing the strength of nongaussianities to be dependent on initial conditions. We leave these as questions for the future.
\section*{Acknowledgements}
\smallskip
We would like to thank Rich Holman for guidance and comments on the initial version of this draft. Additionally, B.R. would like to thank Andy Albrecht for useful discussions. B.R. was supported in part by DOE grant DE-FG03-91ER40674.
\bibliographystyle{unsrt}
|
{
"timestamp": "2015-04-15T02:00:38",
"yymm": "1504",
"arxiv_id": "1504.03332",
"language": "en",
"url": "https://arxiv.org/abs/1504.03332"
}
|
\section*{Acknowledgments}
This interview took place in January 2013, in Berkeley, California. We
are very much indebted to
Frank Samaniego for his encouragement and assistance, and Bonnie Chung
for technical assistance.
\begin{figure}[t]
\includegraphics{492f10.eps}
\caption{Richard Olshen and John Rice in the Fall of 2013, Berkeley, CA.}\label{fig10}
\end{figure}
\end{document}
|
{
"timestamp": "2015-04-14T02:13:35",
"yymm": "1504",
"arxiv_id": "1504.03131",
"language": "en",
"url": "https://arxiv.org/abs/1504.03131"
}
|
\section{Introduction}
Along with the Galactic center and galaxy clusters, the dwarf spheroidal galaxies (dSph) of the Milky Way have been identified as promising targets for indirect dark matter (DM) searches (see e.g. \citealt{2013PhR...531....1S,2015arXiv150306348C}). Their low astrophysical background, high mass-to-light ratio, and proximity make them compelling targets \citep{1990Natur.346...39L,2004PhRvD..69l3501E}. About twenty-five Galactic dSphs were known as of early 2015, and their observation by $\gamma$-ray telescopes has thus far shown no significant emission, leading to stringent constraints on $\langle\sigma_{\rm ann} v\rangle$, the thermally-averaged DM annihilation cross-section \citep{2010ApJ...720.1174A,2011arXiv1110.6775P,2014PhRvD..90k2012A,2014arXiv1410.2242G,2015arXiv150302641F}.
Recently, imaging data from the Dark Energy Survey has led to the discovery of nine new potential Milky-Way satellites in the Southern sky \citep{2015arXiv150302079K,2015arXiv150302584T}. The nearest object, Reticulum~II (Ret~II, $d \sim 30$ kpc), is particularly intriguing, as evidence of $\gamma$-ray emission has been detected in its direction using the public Fermi-LAT Pass 7 data. \citet{2015arXiv150302320G} determined the probability of background processes producing the observed Ret~II gamma-ray signal to be between $p = 0.01\%$ and $p = 1\%$, depending on the background modelling. An analysis of the new objects published simultaneously by \citet{2015arXiv150302632T}, based on the unreleased Pass 8 data set, reported no significant detection, though the strongest hint was for Ret~II with $p = 6\%$. \citet{2015arXiv150306209H} subsequently performed a similar analysis with public Pass 7 data, finding a $p$ value of $0.16\%$.
In any case, a robust determination of Ret~II's DM content is crucial
in order to constrain particle nature of DM. Reticulum~II was
found to be a DM-dominated dSph galaxy from the independent
chemodynamical analyses of \citet{kinematics},
\citet{2015arXiv150402889S} and \citet{2015arXiv150407916K}. Here,
we reconstruct the DM annihilation and decay emission profiles of
Ret~II from a spherical Jeans analysis applied to stellar kinematic
data obtained with the Michigan/Magellan Fiber System (M2FS)
\citep{kinematics}. We use the optimized Jeans analysis setup from
\citet{2015MNRAS.446.3002B,2015arXiv150402048B}, and compute the
astrophysical $J$- and $D$-factors, for annihilating and decaying DM
respectively, from the reconstructed DM density profiles. We
cross-check our results by varying different ingredients of the
analysis and evaluate the ranking of Ret~II among the most promising dSphs for DM indirect detection.
\section{Astrophysical factors, Jeans analysis and data sets}
\label{sec:jeans}
\subsection{Astrophysical factors}
The differential $\gamma$-ray flux coming from DM annihilation
(resp. decay) in a dSph galaxy is proportional to the so-called 'astrophysical factor' $J$ (resp. $D$) \citep{1998APh.....9..137B},
\begin{equation}
\!J \!=\! \!\int\!\!\!\!\!\int\!\! \rho_{\rm DM}^2 (l,\Omega)
\,dld\Omega \!\!\!\!\quad\left({\rm \!resp.~} D \!=\!\!
\int\!\!\!\!\!\int \!\!\rho_{\rm DM}(l,\Omega)
\,dld\Omega\!\!\right)\!,\!\!
\label{eq:J}
\end{equation}
which corresponds to the integration along the line-of-sight (l.o.s.) of the DM
density squared (resp. DM density) and over the solid angle $\Delta\Omega
= 2\pi\times[1-\cos(\alpha_{\rm int})]$, with $\alpha_{\rm int}$ the
integration angle. This quantity depends on both the extent of the DM halo and the mass density distribution, and is essential for constraining the DM particle properties. All calculations of astrophysical factors are done with the {\tt
CLUMPY} code \citep{2012CoPhC.183..656C}, a new module of which has been
specifically developed to perform the Jeans analysis\footnote{This upgrade
will be publicly available in the new version of the software (Bonnivard et al., in prep.).}.
\subsection{Jeans analysis}
\label{subsec:jeans}
Several approaches have been developed to infer the DM density profile
of dSph galaxies from stellar kinematics (see
e.g. \citealt{2013NewAR..57...52B,2013PhR...531....1S,2013pss5.book.1039W}). Here,
we focus on the spherical Jeans analysis, a widely-used approach
for the determination of astrophysical factors
\citep{2007PhRvD..75h3526S,2010PhRvD..82l3503E,2011MNRAS.418.1526C,2012PhRvD..86b3528C,2015ApJ...801...74G,2015arXiv150402048B}. We
refer the reader to \citet{2015MNRAS.446.3002B} for a thorough
description of the analysis setup we use in this work. Here, we summarize the main ingredients.
Assuming steady-state, spherical symmetry, and negligible rotational support, the second-order Jeans equation, obtained from the collisionless Boltzmann equation, reads \citep{2008gady.book.....B}:
\begin{equation}
\frac{1}{\nu}\frac{d}{dr}(\nu \bar{v_r^2})+2\frac{\beta_{\rm ani}(r)\bar{v_r^2}}{r}=-\frac{GM(r)}{r^2},
\label{eq:jeans}
\end{equation}
with $\nu(r)$ the stellar number density, $\bar{v_r^2}(r)$ the radial velocity dispersion, $\beta_{\rm ani}(r)\equiv 1-\bar{v_{\theta}^2}/\bar{v_r^2}$ the velocity anisotropy, and $M(r)$ the mass\footnote{The mass is dominated by DM, and we neglect the stellar component.} enclosed within radius $r$. After solving Eq.~(\ref{eq:jeans}) and projecting along the l.o.s., the (squared) velocity dispersion at the projected radius $R$ reads
\begin{equation}
\sigma_p^2(R)=\frac{2}{\Sigma(R)}\displaystyle \int_{R}^{\infty}\biggl (1-\beta_{\rm ani}(r)\frac{R^2}{r^2}\biggr )
\frac{\nu(r)\, \bar{v_r^2}(r)\,r}{\sqrt{r^2-R^2}}\mathrm{d}r,
\label{eq:jeansproject}
\end{equation}
with $\Sigma(R)$ the surface brightness profile. We compare the l.o.s velocities of the stars to the projected velocity dispersion $\sigma_p$, computed using parametric forms for the unknown velocity anisotropy $\beta_{\rm ani}(r)$ and DM density profile $\rho_{\rm DM}(r)$. We use the following likelihood function \citep{2007PhRvD..75h3526S}
\begin{equation}
\mathcal{L}\!=\! \!\prod_{i=1}^{\!\!N_{\rm stars}}
\! \!\!\frac{(2\pi)^{-1/2}}{\sqrt{\sigma_p^2(R_i)\!+\!\Delta_{v_{ i}}^{2}}}
\exp\!\biggl [\!-\frac{1}{2}\biggl (\!\frac{(v_{\rm i} \!-\!\bar{v})^{2}}{\sigma_p^2(R_i)\!+\!\Delta_{v_{i}}^{2}\!}
\biggr )\biggr ] \!,
\label{eq:likelihood}
\end{equation}
which assumes a Gaussian distribution of l.o.s. stellar velocities $v_i$, centered on the mean stellar velocity $\bar{v}$, with a dispersion of velocities (at the radius $R_i$) coming from both the intrinsic dispersion $\sigma_p(R_i)$ and the measurement uncertainty $\Delta_{v_{i}}$. Probability density functions (PDFs) of the anisotropy and DM parameters are obtained with a Markov Chain Monte Carlo (MCMC) engine\footnote{We use the {\tt GreAT} toolkit \citep{2011ICRC....6..260P,Putze:2014aba}.}, and are used to compute the median and credible intervals (CIs) of the astrophysical factors for any integration angle.
Following the {\it optimized} Jeans analysis setup proposed in \citet{2015MNRAS.446.3002B}, the DM density is described by an Einasto profile (\citealt{2006AJ....132.2685M}), and the anisotropy and light profiles are given by Baes \& van Hese \citep{2007AA...471..419B} and Zhao-Hernquist \citep{1990ApJ...356..359H,1996MNRAS.278..488Z} parametrisations, respectively. The large freedom allowed by these parametrisations was found to mitigate possible biases of the Jeans analysis \citep{2015MNRAS.446.3002B}. Finally, the extent of the DM halo is computed using the tidal radius estimation as in \citet{2015arXiv150402048B}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\linewidth]{fig1.pdf}
\caption{Projected stellar density profile of Ret~II, derived from the photometric catalog of \citet{2015arXiv150302079K}. Overplotted (red line) is the best-fitting model (we note that the fit is to the unbinned data), which is the sum of contributions from Ret~II itself and a constant background (see Section \ref{subsec:data}). Dotted lines enclose 68\% CIs for the projection of $\nu(r)$.}
\label{fig:light}
\end{center}
\end{figure}
\subsection{Data set}
\label{subsec:data}
\paragraph{Surface brightness data}
We fit the stellar number density profile $\nu(r)$ of Ret~II following the procedure that \citet{2015arXiv150402048B} use for `ultrafaint' dSphs (see their section 3.1). We consider a flexible Zhao-Hernquist model for the 3D profile,
\begin{equation}
\nu^{\rm Zhao}(r)=\frac{\nu_s^{\star}}{(r/r_s^{\star})^\gamma
[1+(r/r_s^{\star})^\alpha]^{(\beta-\gamma)/\alpha}}\;,
\label{eq:nu_zhao}
\end{equation}
where the five parameters are the normalization $\nu_s^{\star}$, the
scale radius $r_s^{\star}$, the inner power law index $\gamma$, the outer index $\beta$, and the transition parameter $\alpha$. Along with an additional free parameter $\Sigma_{\rm bkd}$ that represents a uniform background density, these parameters then specify a model for the projected stellar density:
\begin{equation}
\Sigma_{\rm model}(R)\equiv
2\displaystyle\int_{R}^\infty\frac{\nu(r)r}{\sqrt{r^2-R^2}}dr+\Sigma_{\rm
bkd}.
\label{eq:sbpmodel}
\end{equation}
We fit this model to the photometric catalog generated by \citet{2015arXiv150302079K}, which provides positions, colors, and magnitudes of individual stars detected as point sources. From the raw catalog, we first identify possible members of Ret~II as point sources (selected as sources with Sextractor `spread' parameter $<$ 0.01 in the $g$-band) whose extinction-corrected $g-r$ colors place them within 0.25 dex of the Dartmouth isochrone \citep{2008ApJS..178...89D}, calculated for a stellar population with age 12 Gyr, metallicity $\rm{[Fe/H]}=-2.5$, and distance modulus $m-M=17.4$ \citep{2015arXiv150302079K}. To the unbinned distribution of projected positions for the $N=12470$ RGB candidates identified within $1.5^\circ$ of Ret~II's center, we fit 2D projections of $\nu(r)$ according to the likelihood function:
\begin{equation}
\mathcal{L}_2\propto \displaystyle\prod_{i=1}^{N} \Sigma_{\rm model}(R_i).
\end{equation}
As in \citet{2015arXiv150402048B}, the fit is done with the software package MultiNest \citep{2008MNRAS.384..449F,2009MNRAS.398.1601F,2013arXiv1306.2144F}, and we use the samples from the posterior PDFs to propagate the light profile uncertainty into the Jeans analysis. Figure \ref{fig:light} shows the fit to the projected stellar density profile of Ret~II (dashed red line), with the contributions from Ret~II itself and from the constant background (solid black and blue lines respectively).
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{fig2.pdf}
\includegraphics[width=\columnwidth]{fig3.pdf}
\caption{{\em Top:} velocity dispersion profile of Ret~II and reconstructed median and credible intervals (solid and dashed black lines respectively), as well as best fit$^9$ (long dashed red lines). {\em Bottom:} distribution of membership probabilities as a function of the projected radius $R$ and the departure from the mean velocity ($z$-axis, blue to red color) for the eighteen stars with $P_i \neq 0$. The size of the points is proportional to the velocity uncertainty. See text for discussion.}
\label{fig:pmem_radius}
\end{center}
\end{figure}
\paragraph{Kinematic data}
We use the Ret~II stellar kinematic data set from \citet{kinematics}, obtained with M2FS. It consists of projected positions and l.o.s. velocities for 38 individual stars, as well as an estimation of their membership probability $P_i$. The latter, obtained using an expectation maximization algorithm \citep{2009AJ....137.3109W}, quantifies the probability that a given star belongs to the dSph or to the Milky Way foreground.
The top panel of Figure \ref{fig:pmem_radius} presents the velocity dispersion profile of Ret~II, as well as its reconstruction with the Jeans analysis\footnote{The binned data and associated velocity dispersion reconstruction are only shown for illustration purposes. The final results are obtained with an unbinned analysis.}. The bottom panel of Figure \ref{fig:pmem_radius} shows the distribution of membership probabilities as a function of the projected radius $R$ and the departure from the mean velocity (color-coded), for stars with non-zero $P_i$. As pointed out in \citet{2015arXiv150402048B}, a large fraction of stars with both intermediate $P_i$ ($0.1 < P_i < 0.95$) and large departure from the mean velocity hints at Milky Way foreground contamination, which can affect the $J$- and $D$-factor reconstruction. For Ret~II, only one star shows an intermediate $P_i$ (Ret2-142 in the catalog of \citealt{kinematics}, with $P_i = 0.69$), with a very small departure from the mean velocity. Therefore we do not expect a strong sensitivity to foreground contamination. In this study, and as advocated in \citet{2015arXiv150402048B}, we use the data with $P_i > 0.95$ (sixteen likely members, one less than identified by \citealt{kinematics} after exclusion of Ret2-142) as our fiducial setup.
\section{Results}
\label{sec:J}
\begin{table}
\begin{center}
\caption{Astrophysical factors for Ret~II ($d = 30$ kpc). For five different integration angles, the median $J$ (resp $D$)-factors as well as their 68\% and 95\% CIs are given. Note that possible triaxiality of the dSph galaxies adds a systematic uncertainty of $\pm0.4$ (resp.~$\pm0.3$) \citep{2015MNRAS.446.3002B} and is not included in the quoted intervals.
\label{tab:results}}
\begin{tabular}{ccccc} \hline \hline
$\alpha_{\rm int}$ && $\log_{10}(J(\alpha_{\rm int}))$ && $\log_{10}(D(\alpha_{\rm int}))$ \\[0.1cm]
[deg] &&$[J/$GeV$^2 \,$cm$^{-5}]$\tablenote{1~GeV$^2 \,$cm$^{-5} = 2.25\times 10^{-7} M_{\odot}^{2}\,$kpc$^{-5}$} &&$[D/$GeV$ \,$cm$^{-2}]$\tablenote{1~GeV$ \,$cm$^{-2} = 8.55\times 10^{-15} M_{\odot}\,$kpc$^{-2}$}\\[0.05cm]
\hline
0.01 && $17.1_{-0.5(-0.9)}^{+0.5(+1.1)}$ && $15.7_{-0.3(-0.5)}^{+0.6(+1.0)}$ \\[0.2cm]
0.05 && $18.3_{-0.4(-0.8)}^{+0.5(+1.1)}$ && $17.0_{-0.3(-0.6)}^{+0.5(+1.0)}$ \\[0.2cm]
0.1 && $18.8_{-0.5(-0.8)}^{+0.6(+1.2)}$ && $17.6_{-0.4(-0.6)}^{+0.6(+1.1)}$ \\[0.2cm]
0.5 && $19.6_{-0.7(-1.3)}^{+1.0(+1.7)}$ && $18.8_{-0.7(-1.1)}^{+0.7(+1.2)}$ \\[0.2cm]
1 && $19.8_{-0.9(-1.4)}^{+1.2(+2.0)}$ && $19.3_{-0.9(-1.4)}^{+0.8(+1.4)}$ \\[0.1cm]
\hline
\end{tabular}
\end{center}
\end{table}
Figure \ref{fig:J_D} displays the $J$- (top) and $D$-factors (bottom) of Ret~II, reconstructed from the Jeans/MCMC analysis, as a function of the integration angle $\alpha_{\rm int}$. Solid lines represent the median values, while dashed and dash-dot lines symbolize the 68\% and 95\% CIs respectively. Our data-driven Jeans analysis gives large statistical uncertainties due to the small size of the kinematic sample, comparable to those obtained for other `ultrafaint' dSphs by \citet{2015arXiv150402048B} (see also Figure \ref{fig:J_Comp}). Table \ref{tab:results} summarizes our results for the astrophysical factors of Ret~II.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{fig4.pdf}
\includegraphics[width=\columnwidth]{fig5.pdf}
\caption{Median (solid), 68 \% (dashed), and 95\% (dash-dot) CIs of the $J$- (top) and $D$-factors (bottom) of Ret~II, as a function of integration angle, reconstructed from our Jeans/MCMC analysis.}
\label{fig:J_D}
\end{center}
\end{figure}
We cross-check our findings by varying different ingredients of the
Jeans analysis. The resulting $J$-factors are shown in Figure
\ref{fig:J_Comp}. First, we perform a binned Jeans analysis (see
\citealt{2015arXiv150402048B}) of the kinematic data, and find
compatible results. Second, we calculate the bootstrap mean and
dispersion of the $J$-factor \citep{1982jbor.book.....E}. For this
purpose, we generate 500 bootstrap resamples\footnote{The best-fit
DM profile and anisotropy parameters for each sample are obtained
by maximizing the likelihood of Eq.~(\ref{eq:likelihood}). J-factors were then computed for these best fitting profiles.} by drawing with replacement 16 stars among the 16 of the original sample with $P_i > 0.95$. The results are in excellent agreement with the MCMC analysis. Finally, we use all 38 stars of the sample but weight the log-likelihood function of Eq.~(\ref{eq:likelihood}) by the membership probabilities $P_i$ \citep{2015arXiv150402048B}. As only one star shows an intermediate membership probability $0.1 < P_i < 0.95$, we obtain very similar results. These two tests confirm that the reconstruction of the astrophysical factors of Ret~II is not significantly affected by outliers. This is not always the case, notably for Segue~I (Bonnivard, Maurin \& Walker, in prep.).
We note that \citet{2015arXiv150402889S} independently performed
an analysis of the M2FS Ret~II spectroscopic data and found a slightly smaller $J$-factor.
This can be traced to their choice of priors and light profile (L. Strigari, private communication). A detailed comparison will be presented in Geringer-Sameth at al. (in prep.).
\section{Comparison to other dSphs}
\label{sec:Comp}
The same Jeans analysis has been applied to twenty-one other dSphs in
\citet{2015arXiv150402048B}. In Figure \ref{fig:J_Comp}, we compare
the $J$-factors (for $\alpha_{\rm int} = 0.5^{\circ}$) of Ret~II to
the brightest objects identified in
\citet{2015arXiv150402048B}\footnote{Segue~I may have a highly
uncertain $J$-factor (Bonnivard, Maurin \& Walker, in prep.). We
show it only for illustration purposes.}. Ret~II is comparable to
Wilman~I in terms of its median $J$-factor, but slightly below Coma Berenices
and Ursa Major~II. Its CIs are typical of an `ultrafaint' dSph, and significantly larger than the uncertainties of `classical' dSphs.
Interpreting the possible $\gamma$-ray signal in Ret~II in terms of DM annihilation \citep{2015arXiv150302320G,2015arXiv150306209H}, one would expect similar emissions from the dSphs with comparable $J$-factors, such as UMa~II, Coma, and Wil~I. However, no excess was reported from these latter objects \citep{2014arXiv1410.2242G,2015arXiv150302641F}. This could be explained by the large statistical and systematic\footnote{The latter comes from a possible triaxiality of the dSph (0.4 and 0.3 dex for annihilation and decay respectively, see \citealt{2015MNRAS.446.3002B}), and depends on the l.o.s. orientation with respect to the principle axes of the halo.} uncertainties in the J-factors. Moreover, the Jeans analysis assumes all of these objects to be in dynamical equilibrium, but tidal interactions with the Milky Way could artificially inflate the velocity dispersion and therefore the astrophysical factors. UMa~II, and to a lesser extent Coma, appear to be experiencing tidal disturbance \citep{2007ApJ...670..313S,2007MNRAS.375.1171F,2010AJ....140..138M,2013MNRAS.433.2529S}, while Wil~I may show non-equilibrium kinematics \citep{2011AJ....142..128W}. Caution is therefore always advised when interpreting the astrophysical factors of these objects. The dynamical status of Ret~II is not yet clear. Its flattened morphology may signal ongoing tidal disruption. However, the available kinematic data do not exhibit a significant velocity gradient that might be associated with tidal streaming motions \citep{kinematics}.
\section{Conclusion}
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{fig6.pdf}
\caption{Comparison of the $J$-factors at $\alpha_{\rm int} = 0.5^{\circ}$ obtained for Ret~II (red circle) and for the potentially brightest objects from \citet{2015arXiv150402048B} (blue squares), with the same Jeans/MCMC analysis. Ret~II is comparable to Wil~I in terms of $J$-factors, but slightly below Coma and UMa~II. A 0.4 dex systematic uncertainty was added in quadrature to the 68\% CIs to account for possible triaxiality of the DM halo \citep{2015MNRAS.446.3002B}. Also shown are the $J$-factors obtained for Ret~II by varying different ingredients of the analysis - see Section \ref{sec:J}.}
\label{fig:J_Comp}
\end{center}
\end{figure}
We have applied a spherical Jeans analysis to the newly discovered
dSph Ret~II, using sixteen likely members from the kinematic data set
of \citet{kinematics}. We employed the optimized setup of
\citet{2015MNRAS.446.3002B,2015arXiv150402048B}, which was found to
mitigate several biases of the analysis, and checked that our results
are robust against several of its ingredients. We find that Ret~II
presents one of the largest annihilation $J$-factors among the Milky
Way's dSphs, possibly making it one of the best targets to constrain
DM particle properties. However, it is important to obtain follow-up
photometric and spectroscopic data in order to test the assumptions of
dynamical equilibrium as well as to constrain the fraction of binary
stars in the kinematic sample. Nevertheless, the proximity of Ret II
and its apparently large dark matter content place it among the most
attractive targets for dark matter particle searches.
\acknowledgments This work has been supported by the ``Investissements d'avenir, Labex
ENIGMASS", and by the French ANR, Project DMAstro-LHC,
ANR-12-BS05-0006. MGW is supported by National Science Foundation grants AST-1313045, AST-1412999. SMK is supported by DOE DE-SC0010010, NSF PHYS-1417505, and NASA NNX13AO94G. MM is
supported by NSF grants AST-0808043 and AST-1312997. EWO is supported by NSF grant AST-0807498 and AST-1313006.
|
{
"timestamp": "2015-07-29T02:09:09",
"yymm": "1504",
"arxiv_id": "1504.03309",
"language": "en",
"url": "https://arxiv.org/abs/1504.03309"
}
|
\section{\label{sec:intro}Introduction}
Neutron stars are compact stellar objects with masses between around 1
and 2 solar masses and radii of around 10 to
15~km~\cite{blaschke01:trento,page06:review,becker08:a}. They have
magnetic fields up to around $10^{18}$~G
\cite{thompson95:a,thompson96:a,thompson04:a,mereghetti08:a}.
Standard models for neutron stars traditionally assume that these
objects are perfect spheres whose properties are described, in the
framework of general relativity theory, by the well-known
Tolman-Oppenheimer-Volkoff (TOV) equation~\cite{oppenheimer39:a,
tolman39:a}. The TOV equation is a simple first-order differential
equation which can be solved with little numerical effort (see, for
instance, Refs.\ \cite{glenn,weber}).
The assumption of perfect spherical symmetry may not be correct.
It is known that magnetic fields are present in all neutron stars.
In particular, if the magnetic field is strong (up to around
$10^{18}$~Gauss in the core) such as for magnetars~
\cite{thompson95:a,thompson96:a,thompson04:a,mereghetti08:a}, and/or
the pressure of the matter in the cores of neutron stars is
non-isotropic, as predicted by some models of color superconducting
quark matter~\cite{ferr}, then deformation of neutron stars can
occur \cite{chand,ferrv,goose,katz,hask,payne}. We also mention the
recent work conducted by \cite{proto} which shows that high magnetic
fields in proto-quark stars modify quark star masses. The authors of this
study conclude that using the TOV equation would be insufficient for
numerical calculations of the properties of proto-quark stars.
The main goal of our study is to derive a TOV-like stellar-structure
equation for deformed neutron stars whose mathematical form is similar
to the standard TOV equation for spherical neutron stars. This
equation will enable the user to explore the properties of deformed
neutron stars from an equation that can be solved with rather little
numerical effort, complementing more sophisticated numerical studies
such as the one presented very recently in \cite{mallick14:a}.
In contrast to the TOV stars that are composed of spherically
symmetric mass shells, the stellar models considered in our paper are
made of deformed mass shells which are either of oblate or prolate
shape. Strategically, such a treatment is similar to the formalism
developed by Hartle and Thorne \cite{hartle68:a}, which is based on a
quadrupole approximation of the metric of a rotating compact star. The
oblate and prolate shapes are obtained by parametrizing the polar
$(z)$ direction of the metric in terms of the equatorial $(r)$
direction along with a the deformation parameter $\gamma$, described
as $z={\gamma}r$, where we have assumed the symmetry to be axial
symmetric. This parameter is normalized to $\gamma=1$ for a perfect
sphere. An object that is deformed in the equatorial direction
(oblate spheroid) is obtained for $\gamma <1$, while an object
deformed in the polar direction (prolate spheroid) corresponds to
$\gamma > 1$. Using this parametrization will allow us to keep the
energy momentum tensor in spherical form, while maintaining
deformation structure.
The parametrized metric allows us to derive the stellar structure
equation of deformed neutron stars in analytic form. As already
mentioned above, this equation constitutes a generalization of the
well-known Tolman-Oppenheimer-Volkoff equation
\cite{oppenheimer39:a,tolman39:a}, which describes the properties of
perfect spheres in general relativity theory.
The paper is organized as follows. In Sect.\ \ref{sec:eqs},
we discuss the derivation of the stellar structure equation of
deformed neutron stars (mathematical details are presented
in the Appendix). The nuclear equation of state used to solve
this equation is introduced in Sect.\ \ref{sec:eos}. Our equation
of state is based on a relativistic nuclear lagrangian which
describes confined hadronic matter and a nonlocal
Nambu-Jona-Lasinio lagrangian used to model
quark deconfinement. The results are presented in
Sect.\ \ref{sec:results}. They are of generic nature and do not
depend on the particular choice for the nuclear equation of
state. Conclusions are drawn in Sect.\ \ref{sec:conc}.
\section{\label{sec:eqs}Stellar Structure Equations}
The properties of perfectly spherical stars are determined by the TOV
equation, which is based on the Schwarzschild metric given by
\begin{equation}
\label{eq:schw_metric}
ds^{2} = - {\rm e}^{2\Phi(r)} dt^{2} + {\rm e}^{2\Lambda(r)} dr^{2}
+ r^{2} d\theta^{2} + r^{2} \sin^{2}(\theta) d\phi^{2} \, ,
\end{equation}
where $t$ is the time coordinate, $r, \theta, \phi$ are the spatial
coordinates, and $\Phi(r)$ and $\Lambda(r)$ denote metric functions
which are determined from Einstein's field equation of general
relativity theory. The deformed stellar models studied in this paper
are based on a metric that is similar to the one of
Eq.\ (\ref{eq:schw_metric}). However, instead of spherical mass
shells the deformed stellar models are constructed from mass shells
that are either of prolate or oblate shape. The mathematical form of
the metric of such objects reads ($G=c=1$)
\begin{eqnarray}
\label{eq:gmetric}
ds^{2} = g_{\mu\nu} dx^{\mu}dx^{\nu} = -{\rm e}^{2\Phi(r)} \, dt^{2}
+ \left(1-\frac{2m(r)}{r}\right)^{-\gamma} \, dr^{2}
+r^{2} \, d\theta^{2} + r^{2} \sin^{2}(\theta) \, d\phi^{2} \, ,
\end{eqnarray}
where $\gamma$ denotes a constant that determines the degree of
deformation. To derive the hydrostatic equilibrium equation
associated with Eq.\ (\ref{eq:gmetric}), we start with Einstein's field
equation in the mixed tensor representation
\begin{equation}
\label{eq:ein}
G^\mu{}_\nu {\equiv} R^\mu{}_\nu - \frac{1}{2} \, R \, g^\mu{}_\nu =
- 8 {\pi} T^\mu{}_\nu \, .
\end{equation}
Here $G^\mu{}_\nu$ denotes the Einstein tensor, which is given in
terms of the Ricci tensor $R^\mu{}_\nu$, the Ricci scalar $R$ and the
metric tensor $g^\mu{}_\nu= \delta^\mu{}_\nu$. The energy-momentum
tensor
\begin{equation}
\label{eq:enmom}
T^\mu{}_\nu=(\epsilon + P) \, u^\mu \, u_\nu - g^\mu{}_\nu \, P
\end{equation}
is given in terms of the stellar equation of state (pressure, $P$, as
a function of energy density, $\epsilon$) and the matter's
four-velocity $u^\mu= dx^\mu / d\tau$ and $u_\nu= dx_\nu / d\tau$,
with the proper time $\tau$ given by $d\tau^2 = ds^2$. Using
Eqs.~(\ref{eq:gmetric}) through (\ref{eq:enmom}) along with the
equations provided in the Appendix, one arrives at
the stellar structure equation of a deformed neutron star,
\begin{equation}
\label{eq:gtov}
\frac{dP}{dr}=-\frac{(\epsilon+P)\left[\frac{1}{2}r+4{\pi}r^{3}P
-\frac{1}{2}r \left(1-\frac{2m}{r}\right)^{\gamma}\right]}
{r^{2}\left(1-\frac{2m}{r}\right)^{\gamma}} \, .
\end{equation}
In the limiting case when $\gamma=1$, Eq.\ (\ref{eq:gtov}) becomes the
well-known Tolman-Oppenheimer-Volkoff
equation~\cite{oppenheimer39:a,tolman39:a}
\begin{equation}
\label{eq:ogtov}
\frac{dP}{dr}=-\frac{\left(\epsilon+P\right)
\left(m+4{\pi}Pr^{3}\right)} {r^{2}\left(1 - \frac{2m}{r}\right)}
\, ,
\end{equation}
which describes the structure of perfectly spherically symmetric
objects. The gravitational mass of a deformed neutron star is given by
\begin{equation}
\label{eq:gmass}
\frac{dm}{dr}=4 {\pi} r^2 {\epsilon} {\gamma} \, ,
\end{equation}
so that the total gravitational mass, $M$, of a deformed neutron star
with an equatorial radius $R$ follows as~\cite{herr}
\begin{equation}
\label{eq:ggm}
M=\gamma\, {m(R)} \, .
\end{equation}
In the spherical limit, the total mass is given by $M
\equiv m(R) = 4 \pi \gamma \int_0^R dr r^2 \epsilon$. The stellar
radius $R$ is defined by the condition that pressure at the surface of
a neutron stars vanishes, that is, $P(r=R)=0$.
It is important to investigate the space outside the star as well.
For that we need to examine the ${\rm e}^{2\Phi(r)}$ component of
Eq.~(\ref{eq:gmetric}). Using the equations given in the Appendix,
one finds
\begin{equation}
\label{eq:dphidr}
\frac{d{\Phi}}{dr}=\frac{\left[\frac{1}{2}+4{\pi}r^{2}P -\frac{1}{2}
\left(1-\frac{2m}{r}\right)^{\gamma}\right]}{r\left(1 -
\frac{2m}{r}\right)^{\gamma}} \, .
\end{equation}
One can easily see from Eq.~(\ref{eq:dphidr}) that asymptotically
$d{\Phi}/dr \rightarrow 0$, as required.
Now that we are equipped with the stellar structure equations
(\ref{eq:gtov}) and (\ref{eq:ggm}), which are dependent on the
deformation parameter $\gamma$, we solve them for a given equation of
state. The model chosen here assumes that neutron stars are made of
quark-hybrid matter. It is based on a relativistic nuclear lagrangian
to describe confined hadronic matter and a nonlocal Nambu-Jona-Lasinio
lagrangian to model quark matter~\cite{njl2014,njlcp}. Phase
equilibrium in the quark-hadron mixed phase is governed by the Gibbs
condition. Section~\ref{sec:eos} briefly describes the key features
of this equation of state. We stress, however, that the results
presented in Sect.~\ref{sec:results} are generic and do not depend on
the particular choice for the nuclear equation of state.
\section{\label{sec:eos}Equation of State}
\subsection{\label{subsec:eos1}Hadronic Matter}
At densities higher than that of the inner neutron star crust, and
lower than required for quark deconfinement, we model neutron star
matter composed of baryons
($B=\{n,p,\Lambda,\Sigma,\Xi,\Delta,\Omega\}$) and leptons
($\lambda=\{e^-,\mu^-\}$) using the relativistic mean-field
approximation. The Lagrangian is given by
\cite{glenn,weber,selfinteractions}
\begin{eqnarray}
\label{eq:hlag}
\mathcal{L} &=& \sum\limits_{B}
\bar\psi_B\big[\gamma_{\mu}(i\partial^{\mu}
-g_{\omega}\omega^{\mu}-g_{\rho}\vec{\boldsymbol{\tau}}\cdot
\vec{\boldsymbol{\rho}}^{\mu})
-(m_N-g_{\sigma}\sigma)\big]\psi_B
+\frac{1}{2}(\partial_{\mu}\sigma\partial^{\mu}\sigma-
m^2_{\sigma}\sigma^2)\nonumber\\
&&-\frac{1}{3}b_{\sigma}m_N
(g_{\sigma}\sigma)^3-\frac{1}{4}c_{\sigma}
(g_{\sigma}\sigma)^4-\frac{1}{4}\omega_{\mu\nu}
\omega^{\mu\nu}
+\frac{1}{2}m^2_{\omega}\omega_{\mu}\omega^{\mu}+\frac{1}{2}
m^2_{\rho}\vec{\boldsymbol{\rho}}_{\mu}\cdot
\vec{\boldsymbol{\rho}}^{\mu} \nonumber\\
&&-\frac{1}{4}\vec{\boldsymbol{\rho}}_{\mu\nu}\cdot
\vec{\boldsymbol{\rho}}^{\mu\nu}
+\sum\limits_{\lambda}\bar\psi_{\lambda}(i\gamma_{\mu}
\partial^{\mu}-m_\lambda)\psi_{\lambda} \, .
\end{eqnarray}
The interactions between baryons are described by the exchange of
scalar, vector, and isovector mesons
\begin{table}[tbh]
\begin{center}
\begin{tabular}{lc}
Saturation Properties~~~&~~~ NL3 Parametrization \\
\hline
$\rho_0$ (fm$^{-3}$) & 0.148 \\
$E/N$ (MeV) & $-16.3$ \\
$K$ (MeV) & 272 \\
$m^*/m_N$ & 0.60 \\
$a_{sy}$ (MeV) & 37.4 \\
\hline
\end{tabular}
\caption{Parametrization of hadronic matter, where the saturation
properties are baryonic density $\rho_{0}$, energy per baryon $E/N$,
nuclear incompressibility $K$, effective nucleon mass $m^{*}_{N}$,
and asymmetry energy $a_{\rm sy}$.}
\label{table:nl3}
\end{center}
\end{table}
($\sigma,\omega,\rho$)~\cite{walecka}. In the present work we employ
the NL3 parametrization as given in Table \ref{table:nl3}
\cite{nl3}. For further details, see Refs.\ \cite{glenn,weber} and
references therein.
\subsection{\label{subsec:eos2}Quark Matter}
To determine the equation of state of the deconfined quark phase we
use a nonlocal extension of the three-flavor Nambu-Jona-Lasinio model
(see \cite{njl2014} and references therein). This model hosts numerous
improvements over other models of deconfined quark matter, including
but not limited to the treatment of vector interactions among quarks,
reproduction of confinement for proper parametrization, lack of
ultraviolet divergences with the introduction of the nonlocal form
factor $g(\tilde z)$, and momentum dependent dynamical quark masses.
The Euclidean effective action is given by
\begin{eqnarray}
\label{eq:action}
S_E &=&\int d^4x\, \Big\{\bar\psi(x)\big[-i\gamma_{\mu}
\partial^{\mu}+\hat{m}\big]\psi(x)
-\frac{G_S}{2}\big[j_a^S(x)j_a^S(x)-j_a^P(x)j_a^P(x)\big] \\
&&-\frac{H}{4}T_{abc}\big[j_a^S(x)j_b^S(x)j_c^S(x)-
3j_a^P(x)j_b^P(x)j_c^P(x)\big]
-\frac{G_V}{2}j_{V,f}^{\mu}(x)j_{V,f}^{\mu}(x)\Big\} \, ,
\nonumber
\end{eqnarray}
where $\psi=(uds)^T$, $\hat{m}=\mathrm{diag}(m_u,m_d,m_s)$, and $H$,
$G_S$, and $G_V$ are coupling constants. For convenience we assume
$m_u=m_d=\bar{m}$. The scalar, pseudo-scalar, and vector currents are
respectively
\begin{equation}
j_{a}^{S}(x) = \int d^4z\,\tilde{g}(z)\bar\psi\left(x+\frac{z}{2}\right)
\lambda_a\psi\left(x-\frac{z}{2}\right) \, ,
\end{equation}
\begin{equation}
j_{a}^{P}(x) = \int d^4z\,\tilde{g}(z)\bar\psi
\left(x+\frac{z}{2}\right)
i\gamma_5\lambda_a\psi\left(x-\frac{z}{2}\right) \, ,
\end{equation}
\begin{equation}
j_{V}^{\mu}(x) = \int d^4z\,\tilde{g}(z)\bar\psi
\left(x+\frac{z}{2}\right)
\gamma^{\mu}\lambda_a\psi\left(x-\frac{z}{2}\right) \, .
\end{equation}
Applying standard bosonization to (\ref{eq:action}) we derive the
thermodynamic potential in the mean-field approximation at zero
temperature \cite{njl2014}. We use the same parametrization for
the nonlocal NJL model as given in Ref. \cite{njl2014}. The vector
coupling constant ($G_V$) is given in terms of the scalar coupling
constant ($G_S$) and is chosen to be $G_V = 0.09~G_S$.
\subsection{\label{subsec:eos3}Quark-Hadron Mixed Phase}
Phase equilibrium between the hadronic and quark phases of
neutron star matter is governed by the Gibbs condition,
\begin{equation}
p_H(\mu_n,\mu_e,\{\phi\}) = p_Q(\mu_n,\mu_e,\{\psi\}),
\end{equation}
where $\mu_n$ and $\mu_e$ are the neutron and electron chemical
potentials, and $\{\phi\}$ and $\{\psi\}$ are the field variables and
Fermi momenta associated with solutions of the equations of hadronic
and quark matter, respectively. When this condition is initially met
the first order phase transition from hadronic to quark matter
begins. The relaxed condition of global charge neutrality allows the
hadronic matter to become more isospin symmetric by transferring
negative charge from the hadronic to the quark phase, lowering the
asymmetry energy. This results in a mixed phase with coexisting
regions of positively charged hadronic matter and negatively charged
quark matter \cite{glenn,ng92,ngpr}. The equation of state for this
phase is solved by combining the approaches for hadronic and quark
matter under the Gibbs condition, baryon number conservation, and
global electric charge neutrality.
\section{\label{sec:results}Results}
We first calculate the masses and radii of non-spherical neutron stars
by solving Eqs.~(\ref{eq:gtov}) and (\ref{eq:gmass}) numerically using
the Runge Kutta method. The outcome is shown in
Fig.~\ref{fig:njlmvsr}.
\begin{figure}[ht]
\begin{center}
\epsfig{file=m_vs_er.eps, height=3.0in, angle=0}
\caption{(Color online) Mass-radius relationships of deformed neutron
stars ($\gamma<1$: oblate neutron stars, $\gamma >1$: prolate
neutron stars, $\gamma=1$: spherical neutron stars). The solid dots
on each curve represent the maximum-mass star for each stellar
sequence.}
\label{fig:njlmvsr}
\end{center}
\end{figure}
From the results shown in this figure, we see that the maximum mass of
the spherical ($\gamma=1$) neutron star obtained for the equation of
state of this work is $2.3~{\rm M}_{\odot}$. The equatorial radius of
this star is close to 14 kilometers~\cite{njl2014}. Oblate neutron
stars are obtained for $\gamma$ values less than one, since the polar
coordinate obeys $z=\gamma r$. We find that a decrease of $\gamma$ by
10\% results in a $\sim15\% $ increase in gravitational mass and an
increase in equatorial radius by a few kilometers
(Fig.\ \ref{fig:njlmvsr}). If we continue decreasing $\gamma$, the
mass keeps increasing monotonically, ultimately extending into the
mass region of solar-mass black holes. Prolate neutron stars are
obtained for deformation parameters $\gamma~>1$. In this case, as
shown in Fig.~\ref{fig:njlmvsr}, an increase of $\gamma$ by 10\% leads
to a $\sim 12\%$ decrease in gravitational mass and a decrease in
equatorial radius. If one keeps decreasing $\gamma$ further, the
\begin{figure}
\centering
\includegraphics[width=0.70\textwidth]{NJL-pressdeneqr.eps}
\caption{(Color online) (a) Pressure profiles and (b) energy-density
profiles in equatorial direction for the maximum-mass stars shown
Fig.~\ref{fig:njlmvsr}.}
\label{fig:pdvsr}
\end{figure}
maximum mass of a deformed neutron star drops down toward the $1.5~
{\rm M}_\odot$ region. The situation is graphically illustrated in
Figs.~\ref{fig:obmass} and \ref{fig:promass}, which show the
deformations of the maximum-mass neutron stars of
Fig.~\ref{fig:njlmvsr} for $\gamma$ values ranging from 0.8 to 1.0
\begin{figure}
\bigskip
\centering
\includegraphics[width=0.70\textwidth]{NJL-pressdenpor.eps}
\caption{(Color online) Same as Fig.~\ref{fig:pdvsr} but in polar
direction.}
\label{fig:pdvsz}
\end{figure}
(oblate to spherical deformations) to 1.0 to 1.2 (spherical to prolate
deformations). The pressure and energy-density profiles of the
maximum-mass neutron stars of Fig.~\ref{fig:njlmvsr} are shown in
Figs.~\ref{fig:pdvsr} and \ref{fig:pdvsz}.
\begin{figure*}[htb]
\centering
\begin{subfigure}[t]{0.33\textwidth}
\centering
\includegraphics[width=1.00\textwidth]{gama1-00.eps}
\caption{$\gamma=1.00$; ${\rm M}=2.30~{\rm M}_{\odot}$}
\end{subfigure}%
\begin{subfigure}[t]{0.33\textwidth}
\centering
\includegraphics[width=1.00\textwidth]{gama0-90.eps}
\caption{$\gamma=0.90$; ${\rm M}=2.62~{\rm M}_{\odot}$}
\end{subfigure}
\begin{subfigure}[t]{0.33\textwidth}
\centering
\includegraphics[width=1.00\textwidth]{gama0-80.eps}
\caption{$\gamma=0.80$; ${\rm M}=3.02~{\rm M}_{\odot}$}
\end{subfigure}
\caption{Shapes of the oblate maximum-mass stars shown in Fig.~\ref{fig:njlmvsr}.}
\label{fig:obmass}
\end{figure*}
\begin{figure*}[htb]
\centering
\begin{subfigure}[t]{0.33\textwidth}
\centering
\includegraphics[width=1.00\textwidth]{gama1-00.eps}
\caption{$\gamma=1.00$; ${\rm M}=2.30~{\rm M}_{\odot}$}
\end{subfigure}%
\begin{subfigure}[t]{0.33\textwidth}
\centering
\includegraphics[width=1.00\textwidth]{gama1-10.eps}
\caption{$\gamma=1.10$; ${\rm M}=2.03~{\rm M}_{\odot}$}
\end{subfigure}
\begin{subfigure}[t]{0.33\textwidth}
\centering
\includegraphics[width=1.00\textwidth]{gama1-20.eps}
\caption{$\gamma=1.20$; ${\rm M}=1.81~{\rm M}_{\odot}$}
\end{subfigure}
\caption{Shapes of the prolate maximum-mass stars shown in
Fig.~\ref{fig:njlmvsr}.}
\label{fig:promass}
\end{figure*}
The bottom line of all this is that there may be multiple maximum-mass
neutron stars for one and for the same model for the nuclear equation of
state, depending on the type (oblate or prolate) of stellar deformation,
which in the end is linked to the strengths of the magnetic fields of neutron
stars and/or anisotropic pressure gradients in their cores. Moreover, as
indicated by our calculations, the deformation does not need to be very large to
appreciably change the bulk properties of neutron stars. This finding
may be critical to better understand the ever-widening range of
observed neutron star masses and to discriminate neutron stars from
solar-mass black holes.
Next, we calculate the energy loss of photons emitted from the surface
of a deformed neutron star. We consider a photon created at the
surface of the star (emitter) and leaving its gravitational field
toward a detector located at infinity, where space-time is flat. The
photon's frequency at the emitter, $\nu_E$, is given as the inverse of
the proper time between two wave crests, $d \tau_E$, that is, $\nu_E =
1/ d \tau_E = (- g_{\mu\nu} d x^\mu d x^\nu)_E^{-1/2}$, where $d x^1 =
d x^2 = d x^3=0$ because the emitter stays at a fixed position while
emitting the photon. The same expression written down for the receiver
at infinity reads $\nu_\infty = 1 / d \tau_\infty = (- g_{\mu\nu} d
x^\mu d x^\nu)_\infty^{-1/2}$. The ratio of these two frequencies is
given by
\begin{eqnarray}
\frac{\nu_\infty}{\nu_E} = \frac{[(- g_{00})^{1/2} d x^0]_E}{[(-
g_{00})^{1/2} d x^0]_\infty} \, .
\label{eq:freq.infinity}
\end{eqnarray}
If we assume that the coordinate time $d x^0$ between two wave
crests is the same as at the star's surface and the receiver, which is
the case if the gravitational field is static so that whatever the
world-line of one photon is from the star to the receiver, the next
photon follows a congruent path, merely displaced by $d x^0$ at all
points \cite{glenn,weber}, this ratio simplifies to $\nu_\infty/\nu_E
= [(- g_{00})^{1/2}]_E / [(- g_{00})^{1/2}]_\infty$. Making use of
the definition of the gravitational redshift, $z = (\nu_E/\nu_\infty)
- 1$, we obtain
\begin{equation}
z = \Bigl( 1 - \frac{2\, M}{R} \Bigr)^{-\gamma/2} - 1 \, .
\label{eq:redshift}
\end{equation}
Equation (\ref{eq:redshift}) shows that the gravitational redshift
carries important information about the mass, radius, and the
deformation of a neutron star. The $z$ values at the equators of
several deformed $1.5~ {\rm M}_\odot$ neutron stars are shown in
Fig.~\ref{fig:njlmvsr}.
The eccentricities $e$,
\begin{eqnarray}
e \equiv {\rm sign} (R_{\rm eq} - R_{\rm p})\, \sqrt{ 1 - \left(
\frac{R^{<}}{R^{>}} \right)^2 }~,
\label{eq:eccentricity}
\end{eqnarray}
of the neutron stars shown in Figs.~\ref{fig:obmass} and
\ref{fig:promass} are summarized in Table \ref{table:e}. For
spherical neutron stars
$R^<$ (semi-minor axis) and $R^>$ (semi-major~axis) are
equal, so that $e=0$ for such objects. Neutron stars whose $\gamma$
values differ by $\pm 10$\% from the spherical case have
eccentricities of $e=0.43617$ if the deformation is oblate and
$e=-0.41601$ if the deformation is prolate. Rapid rotation also
deforms neutron stars away from spherical symmetry. For the neutron
stars of this paper, we find eccentricities as low as 0.6 for rotation
at the mass shedding frequency (which sets an absolute limit on rapid
rotation), but not smaller. The metric of a rotating neutron star can
also be used to study the structure of deformed non-rotating neutron
stars. This has been done recently in Ref.\ \cite{mallick14:a}. The
results of this paper cannot be directly compared with our results,
however, because of specific assumptions about the energy-momentum
tensor. We note, however, that the eccentricities of the oblate
neutron stars of our study are compatible with those obtained in
\cite{mallick14:a}, depending on the degree of anisotropy generated by
the magnetic field.
\section{\label{sec:conc}Conclusions}
The goal of this work was to investigate the impact of deformation on
the structure of non-rotating neutron stars in the framework of
general relativity. For this purpose we first derived a stellar
structure equation that describes deformed neutron stars. This
equation constitutes a generalization of the well-known
Tolman-Oppenheimer-Volkoff (TOV) equation, which describes the
structure of non-rotating, perfectly spherically symmetric neutron
stars. The mathematical structure of this generalized TOV equation is
such that the deformation of a neutron star (or any other compact
object, such as a hypothetical quark star) is expressed in terms of a
deformation parameter, $\gamma$. By virtue of this parameter, models
of deformed neutron stars can be built from non-spherical (prolate or
oblate) mass shells rather than spherical mass shells. This leads to a
stellar structure equation for deformed neutron stars which is of the
same simple mathematical structure as the standard TOV equation and
thus can be solved with little numerical effort.
The parametrization introduced in our paper allows one to use a model
for the equation of state in the limiting case of isotropy while
maintaining deformation structure. From the results shown in
Fig.~\ref{fig:njlmvsr}, one sees that modest
\begin{table}[htb]
\begin{center}
\begin{tabular}{cccc}
&$\gamma=0.80$ ~~~~~~~&$\gamma=0.90$~~~~~~~ &$\gamma=1.00$ \\ \cline{2-4}
$\epsilon$ &$0.60000$ &$0.43617$ &$0$ \\ \hline
&$\gamma=1.00$ ~~~~~~~&$\gamma=1.10$~~~~~~~ &$\gamma=1.20$ \\ \cline{2-4}
$\epsilon$ &$0$ &$-0.41601$ &$-0.55222$ \\
\hline
\end{tabular}
\caption{Eccentricities, $\epsilon$, of the oblate and prolate neutron
stars shown in Figs.~\ref{fig:obmass} and \ref{fig:promass},
respectively.} \label{table:e}
\end{center}
\end{table}
deformations can lead to appreciable changes in a neutron star's
gravitational mass and radius. In particular, we find that the mass of
a neutron star increases with increasing oblateness, but decreases
with increasing prolateness. This opens up the possibility that,
depending on the degree of stellar deformation, there may exist multiple
maximum-mass neutron stars for one and the same model for the nuclear
equation of state, which is drastically different for spherically
symmetric neutron stars whose mass-radius relationships are
characterized by one and only one maximum-mass star. This finding may
be critical to properly understand the ever widening range of observed
neutron star masses and to discriminate neutron stars from solar-mass
black holes.
\section*{Acknowledgments}
This work is supported through the National Science Foundation under
grants PHY-1411708 and DUE-1259951. A. Romero is supported by NIH
through the Maximizing Access to Research Careers (MARC), grant number
5T34GM008303-25. Computing resources are provided by the
Computational Science Research Center and the Department of Physics at
San Diego State University.
The authors would like to thank Vivian de la Incera and Efrain Ferrer (UTEP)
for their insightful discussions and initial motivation on this work.
\section{Appendix}{\label{sec:mathapp}}
Below, we outline the derivation of the stellar structure equation of
deformed compact objects. For the metric given in
Eq.~(\ref{eq:gmetric}), the non-vanishing Christoffel symbols are
\begin{eqnarray}
\label{eq:chriss}
\Gamma^{r}_{~tt} ={\beta}~{\rm e}^{2\Phi(r)}\Phi^{\prime}(r) \, , ~~
\Gamma^{t}_{~tr} =\Phi^{\prime}(r) \, ,~~
\Gamma^{r}_{~rr}=\dfrac{{\gamma}\left[-m^{\prime}
(r)r+m(r)\right]}{r[-r+2m(r)]}\, , ~~ \Gamma^{r}_{~\theta\theta} =-{\beta}~r \, , ~~
\nonumber \\
\Gamma^{\theta}_{~r\theta} =\Gamma^{\Phi}_{~r\Phi}=\dfrac{1}{r} \, , ~~
\Gamma^{\Phi}_{~\Phi\Phi} =\cot(\theta) \, , ~~
\Gamma^{r}_{~\Phi\Phi} =-{\beta}~r\sin^{2}(\theta) \, ,
\Gamma^{\theta}_{~\theta\theta} = -\sin(\theta)\cos(\theta) \, , ~~
\end{eqnarray}
where primes denote derivatives with respect to the radial coordinate, $r$, and
\begin{equation}
\beta{\equiv}\left(\frac{r-2m(r)}{r}\right)^{\gamma} \, .
\end{equation}
\noindent
The components of the Ricci tensor $R^{\mu}{}_{\nu}$ for the metric of
Eq.~(\ref{eq:gmetric}) are calculated to be
\begin{eqnarray}
\label{eq:riccitt}
R^{t}{}_{t}&=&
\frac{1}{r(r-2m(r))}\Big[\beta\Phi^{\prime}(r)m^{\prime}(r)\gamma{r} -
\Phi^{\prime}(r)m(r)\gamma -\left(\Phi^{\prime}(r)\right)^{2} r\, m(r)
-\Phi^{\prime\prime}(r) r^{2}
\nonumber\\
&&
+ 2\Phi^{\prime\prime}(r)r \, m(r)
-2\Phi^{\prime}(r)r+4\Phi^{\prime}(r)m(r)\Big] \, ,
\end{eqnarray}
\begin{eqnarray}
\label{eq:riccirr}
R^{r}{}_{r}&=&\frac{1}{r^{2}(r-2m(r))}\Big[-\beta\Phi^{\prime\prime}(r)r^{3}
- 2\Phi^{\prime\prime}(r)r^{2}m(r)
+\left(\Phi^{\prime}(r)\right)^{2}r^{3} -
2\left(\Phi^{\prime}(r)\right)^{2}r^{2}m(r)\nonumber
\\ &&-\gamma\Phi^{\prime}(r)m^{\prime}(r)r^{2}
+\gamma\Phi^{\prime}(r)r\, m(r)-2{\gamma}m^{\prime}(r)r+2{\gamma}m(r)\Big] \, ,
\end{eqnarray}
\begin{eqnarray}
\label{eq:riccitheta}
R^{\theta}{}_{\theta} &=& \frac{1}{r^{2}(r-2m(r))}\Big[-{\beta}r^{2}\Phi^{\prime}(r)
+2\beta\gamma{r}\Phi^{\prime}(r)m(r)
+\beta{\gamma}m^{\prime}(r)r-\beta{\gamma}m(r)\nonumber
\\ &&+\, r-2m(r)-\beta\gamma+2{\beta}m(r)\Big] \, ,
\end{eqnarray}
and
\begin{equation}
\label{eq:ricciphi}
R^{\phi}_{\phi}=R^{\theta}_{\theta} \, .
\end{equation}
The Ricci scalar, $R$, is calculated to be
\begin{eqnarray}
\label{eq:ricciscalar}
R &=&\frac{2}{r^{2}(r-2m(r))} \left[\beta\gamma\Phi^{\prime}(r)m^{\prime}(r)r^{2}
- \beta\gamma\Phi^{\prime}(r)m(r)r
-\beta\left(\Phi^{\prime}(r)\right)^{2}r^{3}\right. \nonumber \\
&&+ 2\beta\left(\Phi^{\prime}(r)\right)^{2}r^{2}m(r)-\beta\Phi^{\prime\prime}(r)r^{3}
+2\beta\Phi^{\prime\prime}(r)r^{2}m(r)-2m(r)
-2\beta\gamma^{2}\Phi^{\prime}(r) \nonumber \\ && +
4\beta\Phi^{\prime}(r)m(r)+2\beta{\gamma}m^{\prime}(r)r
-2\beta{\gamma}m(r) +r -2m(r)-\beta\gamma+2{\beta}m(r)\Big] \, . ~~~~~
\end{eqnarray}
Substituting Eqs.~(\ref{eq:riccitt}) to~(\ref{eq:ricciphi}) along with
Eq.~(\ref{eq:ricciscalar}) into Einstein's field equation (\ref{eq:ein}),
one arrives at the general relativistic stellar structure equation
(\ref{eq:gtov}) of deformed compact objects.
|
{
"timestamp": "2015-04-14T02:09:32",
"yymm": "1504",
"arxiv_id": "1504.03006",
"language": "en",
"url": "https://arxiv.org/abs/1504.03006"
}
|
\section{Introduction}
This is the third paper in a series describing the \^G infrared
search for extraterrestrial civilizations with large energy supplies.
The first two papers \citep{WrightDyson1,WrightDyson2} provide the justification
and framework for the search. Here we give a brief summary of those
works, and the purpose of this paper.
\subsection{Justification}
\citet{hart75} argued that the failure of SETI to date was because
humanity is alone in the Milky Way, based on a comparison of likely
colonization timescales for the Milky Way and its age. Hart's
argument also implies that any galaxy with a spacefaring species will
become thoroughly colonized in a time short compared to the galaxy's
age, suggesting that most galaxies should either contain no spacefaring species or
be filled with them.
\citet{kardashev64} parameterized potential alien civilizations by their energy
supply compared to the starlight available to it, with a Type {\sc{i}}
civilizations (K1 in our notation) commanding its planet's entire
stellar insolation, a Type {\sc{ii}} civilization commanding an entire
star's luminosity (i.e.\ a Dyson sphere, K2), and a Type {\sc{iii}}
civilization (K3) commanding most of the stellar luminosity in a
galaxy. Expressed in these terms, Hart's argument is that the
timescale for the appearance of the first K2 to its growth into a
K3 is very short, implying that we should expect many K3
civilizations in the Universe if spacefaring life is common.
Indeed, the technological sophistication required to construct a Dyson sphere
seems far greater than that required for achieving interstellar
travel: while humanity's solar panels currently fall short of complete
coverage of the Sun by a factor of $\sim 10^{17}$, our deepest space
probes today these fall short of the distance to the nearest star by a
factor of only a few thousand.
If Hart's reasoning is sound, then we should expect that, unless
intelligent, spacefaring life
is unique to Earth {\it in the local universe}, other galaxies should
have galaxy-spanning supercivilizations, and a search for K3's may be
fruitful. If there is a flaw in it, then intelligent, spacefaring
life may be endemic to the Milky Way in the form of many K2's, in
which case a search within the Milky Way would be more likely to
succeed. It is prudent, therefore, to pursue both routes.
\subsection{Prior Searches and the Promise of WISE}
\citet{dyson60} and \citet{slysh85} demonstrated that waste heat would
be an inevitable signature of extraterrestrial civilization, and that
such signatures might be detectable to mid-infrared (MIR) instrumentation
for civilizations with energy supplies comparable to the luminosity of
their host star. The first effective all-sky search sensitive to such
namesake ``Dyson spheres'' was performed by {\it IRAS}, but the
infrared cirrus and the poor angular resolution of {\it IRAS} limited
its sensitivity to only the brightest sources.
\citet{Carrigan09a} and \citet{Carrigan09b} used the {\it IRAS} low
resolution spectrometer (LRS) to determine whether candidate Dyson
spheres' SEDs were consistent with blackbodies with $T$ = 100--600K.
\citeauthor{Carrigan09a} concluded from infrared colors
and low resolution spectra that the best of these of the most Dyson sphere candidates were
typically reddened and dusty objects such as heavily
extinguished stars, protostars, Mira variables, AGB stars, and
planetary nebulae (PNs). Nonetheless, of the 11,000 sources he studied,
\citeauthor{Carrigan09a} identified a few weak Dyson
sphere candidates with spectra consistent with carbon stars. One
candidate, IRAS 20369+5131, showed a nearly featureless blackbody
spectrum with $T$ = 376K, but Carrigan concluded it is likely a
distant red giant with no detectable SiC emission.
\citeauthor{jugaku91} performed a series of follow-up searches of sources with
anomalously red $(K-12\mu)$ colors
\citep{jugaku91,jugaku95,jugaku97,jugaku00}, and found no highly
complete Dyson spheres around any of the 365 solar-type stars within
25 pc studied, or another 180 stars within the same distance
\citep{jugaku04}.
To date, the only search for the
waste heat of a K3 civilizations in the peer-reviewed literature has
been that of \citet{annis99a}, who searched for outliers to the
Tully-Fischer relation to identify K3s intercepting a significant
fraction of their starlight. \citet{carrigan12} also suggested
searching for the morphological signatures of K3s, especially in
elliptical galaxies.
The advent of large solid angle, sensitive
MIR surveys makes a waste-heat based K3 search more feasible
today. The {\it Wide-field Infrared Survey Explorer} \citep[{\it WISE},][]{WISE}
performed an all-sky MIR survey at
3.4, 4.62 12 and 22 $\mu$ (the $W1$,$W2$,$W3$, and $W4$ bands) with
superior angular resolution (by a factor of 5) and sensitivity (by a
factor of 1000) than {\it IRAS}. {\it WISE} is thus the first
sensitive survey for both K2's in the Milky Way and K3's among the
approximately $1\times 10^5$ galaxies it resolved. The {\it Spitzer Space
Telescope} is another powerful tool for waste heat searches, having
superior sensitivity and angular resolution. Its survey of the
Galactic plane will be a powerful tool in the search for K2's in the
Milky Way. Since its large area surveys are generally restricted to
star-forming regions and the Galactic Plane, where sensitivity to K3's
is more limited, we have restricted our efforts in this paper to \mbox {\it WISE}.
\subsection{The AGENT Formalism}
In \citet{WrightDyson2}, we developed the AGENT formalism for quantifying
the expected MIR spectra from galaxies hosting K3s in terms
of the energy supply of an alien civilization, and outlined the
methodology of the {\it Glimpsing Heat from Alien Technology} (\^G) search for such civilizations in the local
Universe using the results of the \mbox {\it WISE}\ All-sky MIR
survey. In particular, we argued in that work that extended sources
have the lowest false positive rate because many of the confounding
sources, primarily dusty and extinguished stars and cosmological
sources, would not be present in that sample.
The AGENT formalism parameterizes the power used by an alien
civilization in terms of starlight absorbed (represented by the parameter
$\alpha$), energy generated by other means ($\epsilon$), thermal waste
heat emitted ($\gamma$), and other energy disposal ($\nu$).
Most relevant to the present paper are the parameters $\gamma$ (waste
heat luminosity, expressed as a
fraction of the starlight available to the civilization) and
$T_{\rm waste}$, the characteristic temperature of the waste heat
(which dictates its infrared colors). For values of $T_{\rm waste}$
in the 100--600 K range, values of $\gamma$ near 1 would
imply that most of the luminosity of a galaxy is in the MIR (in the
form of the waste heat from alien engines), while
values near 0 would imply that the alien waste heat was very small
compared to the output of the stars in the galaxy. For dust-free elliptical
galaxies with little of their luminosity in the MIR, values of $\gamma$ of a few percent
would be detectable as an anomalous MIR excess.
\subsection{Scope and Purpose of this Paper}
As an essential step in our waste heat search, we have produced a
clean catalog of the reddest sources {\em resolved} by \mbox {\it WISE}. The
purpose of our focus, in this work, on resolved sources is twofold:
resolved sources present their own challenges of interpretation and
photometry, necessitating this separate effort; and we wish to first deal with
a relatively small and clean sample of nearby galaxies, consistent
with a search for nearby K3's. \mbox {\it WISE}\ resolves approximately $1\times
10^5$
galaxies (see Section~\ref{number}).
Sources unresolved by \mbox {\it WISE}\ include a wide variety of potential false
positives, such as
false detections, data artifacts, cosmologically redshifted objects,
dusty objects at cosmological distances, dusty stars, and heavily extinguished
stars. The study of these sources requires a different sorts of
analysis from those needed for nearby galaxies, which we will describe in
later papers. By contrast, there are many fewer, and more easily
excluded, sources of false positives among the extended sources in the
all sky catalog.
Our primary objective in this paper is to ``map the landscape'' among
galaxies resolved by \mbox {\it WISE}\ by identifying the nature of the very
reddest of these extended sources in the \mbox {\it WISE}\ All-sky survey, using
several metrics for ``red-ness'', including the AGENT parameter
$\gamma$. A byproduct of this effort is a clean catalog of the
reddest extended sources in \mbox {\it WISE}, which we present here.
Our secondary objective is to identify and explain the most extreme
objects in this catalog, which by their superlative nature are
inherently scientifically interesting,
regardless of the origin of their MIR luminosity. In most
cases, these are well-known objects; in many of the remaining cases
their nature seems clear. A few cases, however, are new to the
scientific literature and their nature is uncertain. These sources of uncertain
nature will be natural candidates for followup, and those that appear
consistent with Kardashev civilizations warrant followup by
communication SETI efforts, in particular.
As a tertiary objective, we place a zeroth-order upper limit on the energy
supplies of nearby K3s by identifying the most MIR-luminous galaxies in
our sample. This upper limit can be pushed down to the
degree that these galaxies' MIR emission can be shown have purely
natural origins. A rigorous upper limit will require a more detailed
analysis of these galaxies' SEDs, and a more precise calculation of
the number of galaxies considered in our sample. We save this
exercise for another paper in this series.
Finally, we also illustrate the waste heat approach by performing a quick check of two classes of anomalous
galaxies to confirm that they are not hosts to MIR-bright K3's.
\subsection{Plan}
In Section~\ref{Sample} we describe how we have analyzed
the \mbox {\it WISE}\ All-sky catalog, and how we performed a series of cuts to
select a sample of only real, red, extended sources (our
``Extended Gold Sample'' of 30,808 sources). Sections~\ref{calibrate} describes how we
calibrated the \mbox {\it WISE}\ photometry, and our estimates of our photometric precision.
Section~\ref{classification} describes our efforts to classify the
sources in the Extended Gold Sample, mostly via SIMBAD object types,
so that we could identify previously unstudied sources and reject
many Galactic sources.
Section~\ref{visualgrading} describes our more detailed efforts to
understand the reddest sources in the Extended Gold Sample using
careful visual inspection and literature searches.
Section~\ref{platinum} describes how we performed a second round of
vetting on the Extended Gold Sample, using our calibrated photometry
from Section~\ref{calibrate}, our classifications from
Section~\ref{classification}, and our visual inspections and literature searches
from Section~\ref{visualgrading} to confidently and carefully identify
the reddest sources and determine the best photometry for
them on a case-by-case basis. The result is the 563 source ``Platinum
Sample,'' which we present in our catalog, and whose fields are described
in Table~\ref{FITS}.
Section~\ref{sec:extreme} describes the reddest objects in the
Platinum Sample, for several definitions of ``red'': all six
combinations of the \mbox {\it WISE}\ bandpasses and the AGENT parameter
$\gamma$. Section~\ref{new} describes five sources that are
effectively new to science, having little or no literature presence
(beyond having been detected with {\it IRAS}).
We present our conclusions in Section~\ref{conclusions}, and in the
appendices we use the \mbox {\it WISE}\ imagery to examine two categories
of anomalous galaxies, \ion{H}{1} dark galaxies and so-called
``passive spirals,'' and show that they do not exhibit sufficient
MIR emission to have their anomalous natures explained by the presence
of a K3.
\section{Sample Selection}
\label{Sample}
The \mbox {\it WISE}\ mission began scientific operations on 2010 January 7 at wavelengths of 3.4, 4.6, 12, and 22 $\mu$m,
hereafter referred to as {\it W1, W2, W3}, and {\it W4}, respectively. The All-sky Data Release was subsequently issued on 2012 March 14 and
reached 5$\sigma$ point source sensitivities in unconfused regions to better than 0.08, 0.11, 1 and 6 mJy,
respectively \citep{WISE}. It should be noted that a more recent data release by the \mbox {\it WISE}\ collaboration, dubbed
ALLWISE\footnote{\url{http://wise2.ipac.caltech.edu/docs/release/allwise/}} was issued on 2013 November 13. The added sensitivity and depth, proper motion measurements and
improved flux variability information in the ALLWISE data products means that they supersede the earlier All-sky Data Release
Catalog and Atlas for {\it{most}} uses. The \mbox {\it WISE}\ team suggests that the All-sky Release Catalog may have better photometric
information for objects brighter than saturation limits in {\it W1}
and {\it W2} ({\it W1}$<$ 8.1 mag and {\it W2}$<$6.7 mag). Given that our project
was well underway before the ALLWISE data release was issued, and the
fact that ALLWISE does not add many ($<1\%$) new, bright and
extended sources in W3, we rely on the All-sky Data Release
measurements for this analysis.
\subsection{12$\mu$m Extended Sample}
\label{rchi2}
The following methodology was used to select the full sample of 12$\mu$m-selected sources with extended photometric profiles from the \mbox {\it WISE}\ All-sky catalog.
The instrumental profile-fit reduced $\chi^2_\nu$
(W3RCHI2)\footnote{We refer to fields/nomenclature in the All-sky catalog in all
caps} was used to distinguish sources with extended profiles
(W3RCHI2 $\ge$ 3) from sources with point source profiles (W3RCHI2 $<$3). Given that the majority of sources in the WISE All-sky catalog are
unresolved in W3, i.e. angular sizes $< 6.5^{\prime\prime}$, we adopt a conservative $\chi^2_\nu$ threshold of 3, which excludes only a small number of marginally
resolved sources while admitting a large but manageable number of
point sources with anomalously poor point-spread function (PSF) fits. We also required that the uncertainty in the
W3 ``standard'' aperture magnitude to be measured (i.e.\ not null). A null result means that the W3 ``standard'' aperture magnitude is a limit, or that no aperture measurement was possible. Finally, we applied a cut in the Galactic latitude ($|b| \ge 10$) to remove contamination from nebular emission in the Galactic Plane:
\begin{displaymath}
{\rm W3RCHI2} \ge 3\ {\rm and}\ |b| \ge 10 \ {\rm and}\
{\rm W3SIGM}\ \mbox{ is not null}
\end{displaymath}
\noindent
These search criteria yield a total of 202,851 sources, which composes our parent sample.
\subsection{Photometry}
The \mbox {\it WISE}\ All-sky database provides photometry measurements using a variety of methods: PSF profile fitting, variable aperture photometry (eight circular apertures), curve of growth (COG) aperture photometry, and
elliptical aperture photometry (for sources matched to the 2MASS
Extended Source Catalog (XSC)).
The profile fitting photometry is referred to as W\#MPRO and is defined as the magnitude measured with profile-fitting photometry. In addition to magnitudes this procedure also derives the
signal-to-noise ratio (SNR) and WRCHI2 (i.e., the goodness of fit to
the PSF model of the source) or the goodness of fit to a PSF model of the source.
The COG or ``standard'' aperture photometry is referred to as W\#MAG
. According to the \mbox {\it WISE}\ explanatory supplement ``This is the curve-of-growth
corrected source brightness measured within an 8.25\ensuremath{^{\prime\prime}} radius
circular aperture centered on the source position. The background sky
reference level is measured in an annular region with inner radius of
50\ensuremath{^{\prime\prime}}\ and outer radius of 70\ensuremath{^{\prime\prime}} '' \footnotemark{\ref{wexp}}.
The \mbox {\it WISE}\ pipeline performed nested circular aperture
photometry. They used eight apertures from 5.5\ensuremath{^{\prime\prime}}\ to
24.75\ensuremath{^{\prime\prime}}\ in W1-W3 and 11\ensuremath{^{\prime\prime}}\ to 49.5'\ensuremath{^{\prime\prime}}\ in
W4\footnote{\url{http://wise2.ipac.caltech.edu/docs/release/allsky/expsup/sec2_2a.html} \label{wexp}}. They provide these in the form of parameters named W\#MAG\_1 for the first aperture of 5.5$''$ in W1 through W3 and 11$''$ in W4, W\#MAG\_2 for the 2nd aperture, W\#MAG\_3 for the 3rd aperture and so on.
The elliptical aperture measurements, referred to as W\#GMAG were based on matched
sources between \mbox {\it WISE}\ and the 2MASS Extended Source Catalog (XSC). The shape of the elliptical aperture was determined by utilizing the shape information of the source provided by the 2MASS XSC. We
discuss the uncertainties in our best photometry from calibrating
aperture magnitudes to careful extended source photometry in Section~\ref{calibrate}.
Since our analysis is primarily interested in identifying galaxy-scale extraterrestrial civilizations K3s, we required that photometric measurements of extended
sources in \mbox {\it WISE}\ be as reliable as possible. The PSF profile
fitting photometry is best used when extracting photometry from
point-like sources in the \mbox {\it WISE}\ survey. For the majority of extended
sources in \mbox {\it WISE}, the COG photometry generally provides more reliable measurements, though the strict 8.25$''$ radius circular aperture fails for large and extended galaxies. Measuring reliable
photometry for extended galaxies has historically proven to be a non-trivial and difficult task, the reason being that galaxies come in all shapes and sizes, and a single ``standard'' aperture fails to encompass
the full range of galaxy sizes and structures.
Recently, \citet{Jarrett12} have attempted to extract reliable galaxy photometry for extended sources in the \mbox {\it WISE}\ catalog. They have developed a complex algorithm in order to construct the
\mbox {\it WISE}\ High Resolution Galaxy Atlas and present their initial results in \citet{Jarrett13}. Even more recently, Tom Jarrett has measured reliable photometry for $\sim 67,000$
extended sources in the \mbox {\it WISE}\ catalog contained within the South Galactic Cap (SGC: $b < -60$). We use this analysis to calibrate the aperture magnitudes as presented by the \mbox {\it WISE}\ All-sky database, as we describe in Section~\ref{calibrate}
\subsection{Quality Flags}
In order to remove spurious sources we use the contamination and confusion flags (CC\_FLAG) given for W3. In Figure~\ref{flow}, we describe
the definitions and provide the number counts of objects having the
various CC\_FLAGs. We retain sources with the following
CC\_FLAG: `0',`d',`h', and `o'. We consider sources with CC\_FLAG `P',`p',`D', `H' and `O' to be contaminated and we removed them from
further analysis. Our decision for choosing these particular flags are motivated on empirical examination of a significant fraction of the flagged
entries (i.e.\ we found that the rejected flags have very good
reliability in flagging false sources, but the ones we retain often
appear for real sources). Using these quality flags reduces the
12$\mu$m sample to 132,651 sources.
\begin{figure*}[htp]
\begin{center}
\begin{tabular}{l}
\includegraphics[width=7in,trim=0 2in 0 1in,clip]{ccflagging}
\end{tabular}
\end{center}
\caption[CAPTION]{\label{flow} Description of cc flags, copied verbatim from the \mbox {\it WISE}\ supplementary catalog.\footnote{Available online at
\url{http://wise2.ipac.caltech.edu/docs/release/allsky/expsup/sec2_2a.html\#cc_flags}}}
\end{figure*}
\subsection{Coordinate and Color Cuts}
In addition to removing the Galactic Plane, we also applied three
additional coordinate cuts to remove the most obvious high density
regions of foreground contamination from objects in star-forming
regions. Region 1 is composed of the Galactic Bulge, (i.e., -12 $<$ $l$ $<$ 10 and 20 $<$ $b$ $<$ 10); the total area covered by this region is
$212.23$\ deg$^2$ and contained 9,323 sources. We removed all sources in Region 1 from our sample. Region 2 comprises sources associated with the Large Magellanic Cloud (LMC); this region
contains a total of 13,104 sources. Region 3 is composed of sources
associated with the Orion Nebula and contains a total of 12,733
sources. Instead of imposing a blanket
coordinate cut for these regions of patchy contamination, we found that we could rather reliably identify
foreground sources by rejecting highly clustered sources and retaining
relatively isolated sources using a surface density algorithm. This
reduces our sensitivity to K3s in regions 2 and 3 without rendering
us completely blind to them. After we removed the sources in these
three regions, the 12$\mu$m sample was reduced to 97,491 sources.
We use two simple color cuts to eliminate the most obvious stellar contaminants (it should be noted that these cuts also eliminate a large number of elliptical galaxies, since these generally tend to be blue in the infrared).
We identify the stellar locus in the lower left (blue) corner of the
diagram (since stellar photospheres have neutral MIR colors) and use the following criteria
\begin{displaymath}
{\rm W2MPRO} - {\rm W3MPRO} < 2\ {\rm and}\ {\rm W3MPRO} - {\rm W4MPRO} \le 1
\end{displaymath}
\noindent
to remove them from further analysis.
We are motivated to use the profile fit photometry because these sources are considered to be stellar-like and thus
should have reliable profile fitting photometry.
\begin{figure}[htp]
\begin{center}
\begin{tabular}{l}
\includegraphics[width=3.5in,trim=0 2.5in 0 0,clip]{W3-W4_vs_W2-W3_CLEAN.jpeg}
\end{tabular}
\end{center}
\caption[CAPTION]{\label{fig:3} W3MPRO - W4MPRO versus W2MPRO - W3MPRO for a sample of $\sim$ 70,000 sources. We use this diagram
to identify the stellar locus and remove these sources from further analysis.}
\end{figure}
We visually examined a representative sample of sources in this region
and concluded that they are indeed stellar-like objects with
anomalously high W3RCHI2 values and since we are primarily concerned
with red extended objects, this cut is compatible with our overall search.
These criteria identified a total of 21,645 stellar-like sources,
leaving 75,846 sources in the 12$\mu$m extended sample for further
inspection. We present a color-color map of these sources in Figure~\ref{ExtendedCC}.
\begin{figure}[htp]
\begin{center}
\begin{tabular}{l}
\includegraphics[width=3.5in,trim=1.5in 2in 0 0,clip]{W1MAG-W2MAG_vs_W2MAG-W3MAG_CLEAN_SAMPLE_CORRECTED.jpeg}
\end{tabular}
\end{center}
\caption[CAPTION]{\label{ExtendedCC} Corrected W1CMAG - W2CMAG versus W2CMAG - W3CMAG
{\it (left)} and W3CMAG - W4CMAG {\it (right)} for the W3 Extended Red
Sample (i.e.\ our sample of All-sky entries that appear to be
extended in W3 and with red MIR colors). High density regions are
represented in a logarithmic
greyscale; the total number of sources in each plot is listed in the
plot's corner.}
\end{figure}
\subsection{Visual Classification}
\label{visual}
We constructed $2^\prime \times 2^\prime$ \mbox {\it WISE}\ color images ((W1+W2)/2 = Blue, W3 = Green, and W4 = red) for the remaining 75,846 sources and have visually classified them
into five primary groups. The five groups are: {\bf{stellar artifacts}}, {\bf{low coverage artifacts}}, {\bf{nebular}}, {\bf{needs closer inspection}}, and
{\bf{high quality}}. Representative examples are presented in
Figure~\ref{artifact_examples}.
\begin{figure*}[htp]
\begin{center}
\begin{tabular}{l}
\includegraphics[width=7in,trim=0 1in 0 0,clip]{artifact_examples}
\end{tabular}
\end{center}
\caption[CAPTION]{\label{artifact_examples} We present examples of the color images for the visually classified sources. The top 3 rows show cases where the source is considered to be the most obvious examples
of astrophysical and instrumental artifacts. The fourth row shows
sources with low coverage W3M $\le 5$ but which could potentially be real
astrophysical sources. The fifth row shows sources with nominal
coverage but lacking 2MASS associations. Some such objects are real
sources (often blends of multiple sources), and we separated these from artifacts by visual inspection.}
\end{figure*}
The {\bf{stellar artifacts}} comprise 11,033 sources which were caused by bright saturated stars (i.e., halos, streaks, and latents). Most of these did not have any CC\_FLAGs
indicating a problem in the All-sky release.
The {\bf{low coverage artifacts}} are sources which are likely an artifact because they were observed
fewer than 6 times by \mbox {\it WISE}\ under nominally good conditions(i.e.\
W3M $<$ 6, where W3M gives the number of individual 8.8s W3 exposures
on which a profile-fit measurement of the source was possible). To put
this number in perspective, the median W3M for all 202,851 sources was
13. There were a total of 14,595 low coverage sources.
We visually examined this sample and recovered 215 sources, which,
though having low coverage, appear to be real astrophysical sources
and are considered for further analysis.
The {\bf{nebular}} sources comprise 12,989 sources and we determined them to be locally bright regions of large, nebular
networks of Galactic dust, and so not discrete objects. These sources required images with a much larger FOV (20$'\times20'$) to be classifiable. Since these are highly extended
in nature we first identified sources within a 10$'$ radius and constructed $20'\times 20'$ color images. The brightest W3 source within a cluster was used as the center of a single color image for the field.
We recovered 192 sources that appeared to be `nebular' in the 2$'\times 2'$ image, but we reclassified as being discrete objects
after inspecting the $20'\times 20'$ images.
There were 4,727 sources laking a 2MASS association the we labeled
{\bf{for closer inspection}}, since they appear to be legitimate sources
with good coverage. The majority of these sources were duplicates of
sources already included in the {\bf{high quality}} sample.
The sources which were unique were rematched to the 2MASS catalog using a two-fold process: The first matching used a 30$''$ radius and recovered
photometry for 509 sources. The second matching used a 60$''$ search radius and recovered photometry for 44 sources.
Figure~\ref{fig:4} illustrates the typical colors and magnitudes of these categories
of sources, and of the W3 Extended Gold Sample (i.e.\ our sample of
{\it real} sources extended in W3 with red MIR colors).
\begin{figure*}[htp]
\begin{center}
\begin{tabular}{rl}
\includegraphics[scale=0.27,trim=3.5in 4in 1in 2in,clip]{MAG_vs_W2MAG-W3MAG_STAR_ARTIFACT.jpeg}
\includegraphics[scale=0.27,trim=3.5in 4in 0 2in,clip]{MAG_vs_W2MAG-W3MAG_LOWEXP.jpeg}\\
\includegraphics[scale=0.27,trim=3.5in 2in 1in 4in,clip]{MAG_vs_W2MAG-W3MAG_NEBULAR.jpeg}
\includegraphics[scale=0.27,trim=3.5in 2in 0 4in,clip]{MAG_vs_W2MAG-W3MAG_W3_CLEAN_SAMPLE.jpeg}
\end{tabular}
\end{center}
\caption[CAPTION]{\label{fig:4}We show W3 magnitude versus W2 - W3 color for classified sources. Top Left: Stellar artifacts. Top Right: Low exposure sources.
Bottom Left: Nebular type sources. Bottom Right: W3 Extended Gold Sample.}
\end{figure*}
We identified 32,502 sources which had 2MASS associations and appear to be real astrophysical objects based on their color images as {\bf{high quality}} sources.
Some extended objects appear in the catalog multiple times. We removed the duplicate entires (those within 60$''$) of the brightest entry in the catalog.
We identified a total of 1,694 duplicated sources which we removed
from the sample.
The number of galaxies we have in our final sample here makes sense. Most of the high latitude extended
sources in \mbox {\it WISE}\ are galaxies, and as we show in Section~\ref{number},
there are approximately 100,000 galaxies that would be extended in the W3
\mbox {\it WISE}\ band if they had significant W3 emission. We have thus
selected approximately the reddest 1/3 of the galaxies on the sky
larger than the angular resolution of the \mbox {\it WISE}\ W3 band.
We have thus constructed a sample of mostly real,
extended, and discrete sources in the \mbox {\it WISE}\ All-sky release which we
refer to as the {\it W3 Extended Gold Sample}.
\label{GoldBuild}
We summarize the vetting procedure that produced the W3 Extended Gold
Sample in Figure~\ref{fig:2} and present the sky distribution for this sample in Figure~\ref{fig:5}
\begin{figure*}[htp]
\begin{center}
\begin{tabular}{l}
\includegraphics[width=7in,trim=0 2in 0 1in,clip]{W3_extended_schematic_final}
\end{tabular}
\end{center}
\caption[CAPTION]{\label{fig:2} Schematic diagram of W3 extended
source analysis. Numbers to the left of Figure ~\ref{fig:2} refer to the section numbers in this manuscript.
Extended sources are identified as those with W3RCHI2 $\ge 3$ and nominal detection are those where W3SIGM is not null.}
\end{figure*}
\begin{figure*}[htp]
\begin{center}
\begin{tabular}{l}
\includegraphics[scale=0.35,trim=1.5in 3in 0 2in,clip]{W3_EXTENDED_GOLD_skymap.jpeg}
\end{tabular}
\end{center}
\caption[CAPTION]{\label{fig:5} We present the sky distribution for the W3 Extended Gold Sample.}
\end{figure*}
\subsection{W3 Extended Gold Sample Photometric Calibration}
\label{calibrate}
The W3 Extended Gold Sample comprises 30,808 apparently extended
sources from the {\bf{high quality}} sample, 597 sources from the
{\bf{for closer inspection}} sample and 192 sources from the
{\bf{nebular}} sample, bringing the final sample to 31,597 sources
(shown in the lower right plot of Figure~\ref{fig:4}). As
discussed in Section~\ref{GoldBuild} , the \mbox {\it WISE}\ COG photometry is generally
reliable for most sources in this sample, but some care must be taken
before using these measurements.
Tom Jarrett has kindly provided us with a preliminary version of his
extended source catalog for the south Galactic Cap ($b<-60$) in
advance of this catalog's publication. To further calibrate the
extended source photometry and determine the degree of systematic
errors in the \mbox {\it WISE}\ pipeline extraction for extended sources, we have cross-matched
1,907 \mbox {\it WISE}\ All-sky sources with sources in Jarrett's preliminary
catalog (210 sources did not match, usually because the source was
only marginally extended in one of the catalogs). We found that for our W3 Extended Gold Sample, the distribution of
differences between the All-sky COG magnitudes and Jarrett's magnitudes
were significantly offset from zero (by 0.4, 0.4, 0.1, and -0.15
magnitudes in W1, W2, W3, and W4) and skewed with long tails past 1
magnitude, with the Jarrett magnitudes typically brighter than the COG
magnitudes.
In Figure~\ref{AllSkyvsTJOriginal} we present the comparison between the COG magnitudes
presented in the All-sky release to those measurements presented in Jarrett's preliminary catalog.
\begin{figure*}[htp]
\begin{center}
\begin{tabular}{l}
\includegraphics[width=8in,trim=1.5in 3in 0 2in,clip]{TJ_vs_WISE_original_magnitudes.jpeg}
\end{tabular}
\end{center}
\caption[CAPTION]{\label{AllSkyvsTJOriginal} \mbox {\it WISE}\ WMAG
(curve-of-growth corrected) All-sky photometry
vs.\ aperture photometry for extended sources in the South Galactic Cap carefully extracted by Tom Jarrett. Numbers in
the plots describe the number, mean ($\mu$), and standard deviation ($\sigma$) of Gaussian
fits to the data. The pipeline photometry has offsets and
color-dependent systematic errors for these extended sources.}
\end{figure*}
We estimated the size of the sources using the difference between the
profile, 4\textsuperscript{th}, and 8\textsuperscript{th}\ aperture magnitudes, W$_\mathrm{profile}$,
W\#\_4, and W\#\_8. For unblended point sources these magnitudes are
typically identical, but for extended sources the larger aperture
magnitudes are more faithful, because they measure flux further up the
curve of growth. We formed ``size'' parameters from the quantities
(W\#\_8-W\#\_4) and (W\#\_8-W$_\mathrm{profile}$), and found that by
using the W\#\_8 magnitudes, with quadratic corrections in the size
parameters and constant offsets, we could reproduce the Jarrett
magnitudes to within 0.06 mag in most cases and colors to within 0.08
mag.
The transformation we used to generate accurate extended source
magnitudes W\# from the \mbox {\it WISE}\ pipeline photometry is:
\begin{eqnarray}
{\rm S1} &=& {\rm W\#}\_8-{\rm W\#}\_4 \\
{\rm S2} &=& {\rm W\#}\_8-W_\mathrm{profile} \\
{\rm W\#} &= & a_0 + a_8 {\rm W\#}\_8 + a_{s1,1} {\rm S1} + a_{S2,1}\\
&&{\rm S2} + a_{S1,2} {\rm S1}^2 + a_{S2,2} {\rm S2}^2
\end{eqnarray}
We performed a singular value decomposition to determine the best
values for the coefficients, and report our coefficients in
Table~\ref{correctioncoefficients}:
\begin{center}
\begin{deluxetable}{rcccccc}
\tablewidth{0pt}
\tablecaption{Correction coefficients to produce corrected magnitudes
for extended sources.}
\tablehead{\colhead{band} & $a_0$ & $a_8$ & $a_{s1,1}$ & $a_{S2,1}$ &
$a_{S1,2}$ & $a_{S2,2}$}
\startdata
W1 &-1.083 & 1.093 & 0.107 & -0.591 & 0.021 &-1.114 \\
W2 &-1.211 & 1.103 & 0.246 & -1.104 & 0.062 &-1.540 \\
W3 & -0.419 & 1.050 & 0.030 &-0.449 & 0.049 &-1.130 \\
W4 & -0.588 & 1.106 & 0.006 &-1.021 &-0.228 & 0.676
\label{correctioncoefficients}
\enddata
\end{deluxetable}
\end{center}
We present a comparison of our corrected photometry to that of the
preliminary Jarrett photometry in Figure~\ref{AllSkyvsTJ} \&
\ref{AllSkyvsTJColor}. Because these distributions are roughly
Gaussian but with extended wings, we report the precision of our
photometry in three ways. In our figures we show the best-fit
Gaussians to each of the residual distributions, and report the width
of these Gaussians as $\sigma$; this represents the typical systematic
error due to limitations of the \mbox {\it WISE}\ pipeline for most of our
extended sources. The long wings of the residual distribution
typically represent blended and highly structured sources for which our
simple calibration scheme failed to match Jarrett's more careful
analysis, and a small number of extreme outliers inflate the standard
deviation of these distributions beyond utility. To quantify the
systematic photometric errors including the bulk of the non-Gaussian
wings in a more robust manner, we calculated a ``robust sigma''
including outlier rejection\footnote{Using the IDL routine {\tt
ROBUST\_SIGMA} written by H. Freudenreich, who cites ``Understanding
Robust and Exploratory Data Analysis'' \citep{Hoaglin}}, and
also the value of the 68\textsuperscript{th}\ percentile absolute deviation from the
median for each magnitude and color combination. We present the
results of all three error estimates in Table~\ref{Errors}.
\begin{figure*}[htp]
\begin{center}
\begin{tabular}{l}
\includegraphics[width=7.5in,trim=1.5in 3in 0 2in,clip]{TJ_vs_WISE_corrected_magnitudes.jpeg}
\end{tabular}
\end{center}
\caption[CAPTION]{\label{AllSkyvsTJ} Results of our calibration of
\mbox {\it WISE}\ photometry to careful extended source photometry of sources
in the south Galactic Cap ($b < -60$) provided by Tom Jarrett.
We present the distribution in differences in our corrected \mbox {\it WISE}\
magnitudes in the four \mbox {\it WISE}\ bands to the Jarrett photometry for common sources. Numbers in
the plots describe the number, mean ($\mu$), and standard deviation ($\sigma$) of Gaussian
fits to the data. These ``corrected'' magnitudes have roughly
Gaussian errors and little or no systematics with color or size for
most sources.}
\end{figure*}
\begin{figure*}[htp]
\begin{center}
\begin{tabular}{l}
\includegraphics[width=8in,trim=1in 1.5in 0 1in,clip]{TJ_vs_WISE_corrected_colors_v2.jpeg}
\end{tabular}
\end{center}
\caption[CAPTION]{\label{AllSkyvsTJColor} Results of our calibration of
\mbox {\it WISE}\ photometry to careful extended source photometry of sources
in the south Galactic Cap ($b < -60$) provided by Tom Jarrett. For
all six color combinations of the four \mbox {\it WISE}\ bands (using the
Jarrett magnitudes),
we present the distribution in differences in our corrected
magnitudes to the Jarrett photometry for common sources. Numbers in
the plots describe the number, mean ($\mu$), and standard deviation ($\sigma$) of Gaussian
fits to the data. These ``corrected'' colors have roughly
Gaussian errors and little or no systematics with magnitude or size for
most sources.}
\end{figure*}
\begin{deluxetable}{cccc}
\tablecaption{Three measures of systematic errors in our extended
source photometry.}
\tablehead{\colhead{Quantity} & \colhead{$\sigma$} & \colhead{Robust
st. dev. (\# rejected)} & \colhead{68\%-ile}}
\tablewidth{0pt}
\startdata
W1 & 38 & 54 (148) & 54 \\
W2 & 54 & 70 (115) & 69 \\
W3 & 39 & 49 (121) & 50 \\
W4 & 51 & 81 (142) & 84 \\
W1-W2 & 30 & 38 (41) & 36 \\
W1-W3 & 42 & 56 (132) & 56 \\
W1-W4 & 73 & 100 (132) & 100 \\
W2-W3 & 50 & 67 (103) & 65 \\
W2-W4 & 79 & 110 (114)& 100 \\
W3-W4 & 74 & 100 (73) & 90
\enddata
\tablecomments{Values in mmag, measured as the width of the
distribution of differences between our calibrated magnitudes and
those of a preliminary version of Jarrett's unpublished extended
source photometry. We present three measures: the width of the
best-fit Gaussian ($\sigma$), the ``robust sigma'' calculated as a
standard deviation with outlier rejection (number of rejected sources
in parentheses, out of 1907 total sources), and the 68\textsuperscript{th} percentile
absolute deviation from the median. For Gaussian distributions these
three quantities should be nearly identical.\label{Errors}}
\end{deluxetable}
While a detailed study of individual targets would benefit from the
more precise photometry of Jarrett's final catalog, this precision is
sufficient for the initial exploration of the \mbox {\it WISE}\ data set and
identification of superlative objects.
\section{Characteristics Of The Extended Gold Sample}
\label{classification}
To help us characterize sources in the Extended Gold Sample, we
employed the source classifications presented in the SIMBAD database,
where available, as a starting point. SIMBAD provides over 130 different types of
astrophysical classifications ranging from the stellar to the
cosmological. We recovered source classifications for $\sim$ 87\% of the
Extended Gold Sample. The median distance between the \mbox {\it WISE}\ and SIMBAD source is
of the order $\sim 0\farcs5$. We find that $\sim$93\% of these sources
have been categorized as being external galaxies vs. Galactic
foreground objects We find that the 7\% that are ``stellar" are
typically not point sources but are resolved in the IR for some reason.
These are encouraging results because they validate our selection and vetting procedures.
In the following sections we discuss
the two most commonly identified types of astrophysical sources ---
those of Galactic origin and those of extragalactic origin, the latter
being of primary interest in this study.
\subsection{Extragalactic Sources}
We have grouped the SIMBAD classifications for extragalactic sources into five broad categories:
\begin{itemize}
\item {\bf normal} Galaxy: Includes the following SIMBAD types, Galaxy (G), Galaxy in Pair (GiP), Galaxy in Group (GiG), Galaxy in Cluster (GiC), Cluster of Galaxies (GlC), Group of Galaxies (GrG), Brightest galaxy in a Cluster (BiC), and Compact Group of Galaxies (CGG)
\item {\bf active} Galaxy: LINERS (LIN), QSO, Radio Galaxy (rG), Active Galactic Nuclei (AGN), Seyfert Galaxies (Sy1 + Sy2), BL Lac - type ob ect (BLL), and Blazar (Bla))
\item {\bf star-forming} Galaxy: Emission Galaxy (EmG), HII Galaxy (H2G), Starburst Galaxy (SBG), and Blue Compact Galaxy (bcG)
\item {\bf interacting} Galaxy: Interacting Galaxy (IG), and Pair of Galaxies (PaG)
\item {\bf low surface brightness} Galaxy: low surface brightness galaxy (LSB)
\end{itemize}
To characterize the galaxy population we construct a simple
color-magnitude diagram (W2-W3 versus W3). Figure~\ref{GoldCMD} present W2-W3
versus W3 for a variety of (mostly extragalactic) sources. The two
most striking features in this figure are the almost complete absence of
galaxies with W3 $> 10.0$, and W2-W3 $\ge 4.5$.
\begin{figure*}[htp]
\begin{center}
\begin{tabular}{l}
\includegraphics[width=7in,trim=0 2in 0 0,clip]{W2-W3_vs_W3_SIMBAD_Galaxies_Corrected.jpeg}
\end{tabular}
\end{center}
\caption[CAPTION]{\label{GoldCMD} Color-magnitude diagrams for a variety
of extragalactic sources in the Extended Gold Sample.}
\end{figure*}
The faint limit is significantly above the \mbox {\it WISE}\ detection limit in W3
of $\sim 11.2$ (for point sources), and so requires explanation. This
limit originates in our requirement that a galaxy be both detected
{\it and extended} in \mbox {\it WISE}\ in the W3 band. We measure extended-ness with W3RCHI2,
and it appears that this parameter rarely takes values above
3 for sources with W3 $\gtrsim$ 9, no matter how extended
the object is. This lower limit is thus an artifact of the \mbox {\it WISE}\ data
pipeline and our Extended Gold Sample selection criteria.
The upper left plot shows where the {\it normal} galaxies and the SIMBAD classification
Part of a Galaxy (PoG) reside in this color-magnitude space. These sources
compose $\sim$ 82\% of the total galaxy population. In the upper right
we show the color-magnitude diagram for star forming and LSB galaxies,
which compose $\sim$ 12\% of the population. The lower left shows the
active and interacting galaxies, composing $\sim$ 6\% of the
population. The lower right plot shows the generic SIMBAD
classifications of (IR) and X-ray (X) sources. These last two are
somewhat ambiguous classifications, and we include them
only for completeness.
These results are quite consistent with our expectations validating
the SIMBAD types. The {\it normal} galaxies appear to occupy the overall
parameter space seen in the other populations suggesting this to be a
mixture of a variety of galaxy types. The {\it star-forming} galaxies
appear to congregate at redder W2-W3 colors (W2-W3 $\ge 3$), as
expected. Interestingly, the LSB galaxies appear to tightly congregate on the fainter
end of the {\it{star-forming}} population. The
{\it{interacting}} galaxies appear to populate the same parameter
space as seen by the {\it{star-forming}} galaxies, which is an
expected result as a majority of {\it{interacting}} galaxies are
experiencing high star formation rates. The {\it active} galaxies populate
similar parameter space as seen in the {\it normal} galaxy population. The
IR sources appear to be a mixture of both extragalactic sources and
Galactic sources, though we do see a tight clustering occupying
similar parameter space as the LSB galaxies.
\subsection{Galactic Sources}
\label{galactic}
Though {\it Galactic sources} (i.e.\ sources within the Milky Way
Galaxy) compose only $\sim$ 7\% of our sample it
is important to understand how they might produce contamination in our
investigation. SIMBAD provides classifications for over 70 different
types of these Galactic sources, which we have reduced to six primary types:
$normal$ stars, Young Stellar Objects (YSO), Variables, Evolved, Evolved +
IR, and stars in clusters. Our grouping methodology is as follows:
\begin{itemize}
\item {\bf normal} Star: Includes the following types, Star (*), Emission-line Star (Em*), Peculiar Star (Pe*), High proper-motion Star (PM*), Star in double system (*i*), Be Star (Be*), and Eclipsing binary (EB*)
\item {\bf YSO}: T Tau-type Star (TT*), Variable Star of FU Ori type (FU*), Herbig-Haro Object (HH), Pre-main sequence Star (pr*), and Young Stellar Object (Y*O)
\item {\bf variable} Star: Variable Star (V*), Variable Star of beta Cep type (bC*), Variable Star of Orion Type (Or*), Semi-regular pulsating Star (sr*), Variable Star of W Vir type (WV*), Long-period variable star (LP*),
CV DQ Her type (intermediate polar) (DQ*), Variable Star of delta Sct type (dS*), Variable Star of alpha2 CVn type (a2*), Cepheid variable Star (Ce*), Variable Star with rapid variations (RI*), Variable Star of RV Tau type (RV*),
Pulsating variable Star (Pu*), Variable Star of Mira Cet type (Mi*), Variable Star of R CrB type (RC*), Eruptive variable Star (Er*), Variable Star of irregular type (Ir*), and Cataclysmic Variable Star (CV*)
\item {\bf evolved} Star: Asymptotic Giant Branch Star (He-burning) (AB*), Red Giant Branch star (RG*), Wolf-Rayet Star (WR*), and S Star (S*)
\item {\bf evolved+IR} Star: Post-AGB Star (proto-PN) (pA*), Star with envelope of OH/IR type (OH*), and Carbon Star (C*)
\item {\bf Star in Cluster}: Star in Cluster (*iC), Cluster of Stars (Cl*), and Star in Nebula (*iN)
\end{itemize}
For completeness we must also consider two other types of Galactic
sources, PN and {\bf Solar System Objects} (SSOs) such as comets, asteroids, and planets. The SSOs are identified by utilizing the \mbox {\it WISE}\ All-sky Known Solar System Object Possible Association List. We also identify two IR bright planets, Neptune and Uranus, which are included in the sample.
To characterize the \mbox {\it WISE}\ high latitude Galactic population we
present Figure~\ref{GoldCMD2} , which shows the color-magnitude
diagrams for the various Galactic classifications. As before we find
that the majority of the Galactic population are relatively rare at W3
$> 10$ and W2-W3 $\ge 4.5$, though we do find sources with very extreme W2-W3 colors, i.e.\ W2-W3 $\ge 6$. We also find, as was expected, that the two types of sources which confuse and contaminate our search the most are the YSO's and planetary nebulae (PNe). While we do find contaminating sources within the Galactic sample, the scarcity of astrophysical sources with W2-W3 $\ge 4.5$ should make our search relatively straightforward for high $\gamma$ K3's.
\begin{figure*}[htp]
\begin{center}
\begin{tabular}{l}
\includegraphics[width=7in,trim=0 2in 0 0,clip]{W2-W3_vs_W3_SIMBAD_Galactic_Corrected.jpeg}
\end{tabular}
\end{center}
\caption[CAPTION]{\label{GoldCMD2} Color-magnitude diagrams for a
variety of galactic sources in the Extended Gold Sample.}
\end{figure*}
\section{Visual Review and Grading of Sources in the W3 Extended Gold Sample}
\label{visualgrading}
A systematic search into the existence and numbers of possible
galaxy-spanning ETIs requires that we carefully inspect the most
interesting subsets of the Extended Gold Sample.
\subsection{AGENT parameterization}
\label{agent}
For every source we apply the methodology described in
\citet{WrightDyson2} to estimate the AGENT parameters $\gamma$ (the
fraction of starlight reradiated as alien waste heat)
and \hbox{\ensuremath{T_{\mbox{\rm \scriptsize waste}}}}\ (the waste heat's characteristic temperature) for each
galaxy, assuming $\nu=0$ (no nonthermal alien power disposal/emission)
and $\alpha=\gamma$ (that is, all of the waste heat originated as
stellar power). We fit all four \mbox {\it WISE}\ bands to the three parameters
$(L/4 \pi d^2)$ (the bolometric flux of the source), $\gamma$, and \hbox{\ensuremath{T_{\mbox{\rm \scriptsize waste}}}}.
For consistency, we modeled every galaxy as having an intrinsic SED
represented by the old elliptical SED model {\it Ell13} of
\citet{Silva98}, of which a fraction $\alpha$ of the starlight is
absorbed and re-emitted as a blackbody at temperature \hbox{\ensuremath{T_{\mbox{\rm \scriptsize waste}}}}. Of
course, many galaxies have a significant blue stellar population and
dust. Because the {\it Ell13} template has the least MIR emission of
all of the \citet{Silva98} galaxy templates, using it will allow us to measure the {\it maximum}
amount of MIR emission from each galaxy that {\it could} be attributed
to ETI's from \mbox {\it WISE}\ broadband photometry alone. Its is thus
consistent with our desire to set upper limits to K3 waste heat emission.
\subsection{Visual inspection}
{\bf Using $\gamma$ as the primary parameter for prioritizing sources we
inspected, reviewed and graded every source in the W3 Extended Gold Sample with $\gamma
\ge 0.25$, ${\sim}4000$ in all.}
To facilitate this visual review, we constructed an interactive GUI
system. This system provided an ``at-a-glance chart'' for each source
(Figure~\ref{fig:finder}) consisting of $2\arcmin \times 2\arcmin$
multiwavelength imaging cutouts (DSS, SDSS, 2MASS, and \mbox {\it WISE}) centered
on the \mbox {\it WISE}\ source position, a variety of hyperlinks to
supplementary imaging data and database queries centered on the
source position (e.g.\ \mbox {\it WISE}\ (L1B + Atlas images, SIMBAD, NED),
and printed information including source coordinates, \mbox {\it WISE}\ catalog colors and
magnitudes, number of
publications listed in SIMBAD, number of times \mbox {\it WISE}\ observed the
source position in each filter (W\#M) and number of times \mbox {\it WISE}\
detected the source in each filter (W\#NM).
\begin{figure*}[htp]
\epsscale{0.9}
\plotone{f9.jpeg}
\caption{
Example of an ``at-a-glance'' chart used for detailed visual review
and classification of sources in the Extended Gold Sample. The red
labels of the images shows the data sources; the scale is the same in
each image and given in the bottom-left-hand image. The magnitudes of
the 2MASS imagery are given in the corresponding images. The
bottom-right-hand image is a color composite from the four \mbox {\it WISE}\ bands
in the bottom row. The SIMBAD name is in the upper right, along with
the file name, J2000 coordinates in sexagesimal and decimal, the
galactic coordinates and ecliptic latitude, the SIMBAD type, and the
number of publications reported by SIMBAD. ``Class'' refers to the
grade of the source (C in this case, reflecting its large number of
associated publications, see Section~\ref{grading}). Our \mbox {\it WISE}\ extended source photometry
appears to the right of the third row, including, for each band, the
RCHI2 parameter (large values indicate the source is well resolved), the signal-to-noise
ratio of the detection, and the magnitude. Also reported are the
various color combinations, the number of visits to the field, and the
number of visits in which the source was detected. Finally, the
contamination and confusion flags (CC{\tt \_}FLAG) are given for the four bands.
This particular source, W122654.61-005239.2,
illustrates what might pass for a high-$\gamma$\ KIII galaxy in
multiwavelength images; it is actually a dusty (and well-studied)
Seyfert 2 galaxy.
}
\label{fig:finder}
\end{figure*}
\subsection{Grading and Literature Search}
\label{grading}
The sample of red, extended objects in \mbox {\it WISE}\ is a combination
of sources, galactic and extragalactic; real and instrumental;
well-studied and new-to-science. We developed a sorting scheme to help
us understand it better and prioritize sources for further
investigation.
We defined a simple alphabetical grading scheme (A-F):
\begin{itemize}
\item A -- Astrophysical red source with no previous publications in SIMBAD, little or no
ancillary survey data. {\em Highest priority candidate for
observational follow-up.}
\item B -- Astrophysical red source with some ancillary data or few
publications, which do {\em not} provide convincing evidence that
its nature is understood. {\em Good candidate for follow-up.}
\item C -- Visual review of source and/or publication list provides
convincing evidence that its nature is understood (although in some cases,
the SIMBAD classification may be incorrect.)
\item D -- Astrophysical source that creates artifacts detected as red
extended sources by \mbox {\it WISE}. Most
frequently associated with a bright star or large region of bright nebulosity.
\item F -- Fake/false source or artifacts, such as a latent image. {\em
Not} a real astrophysical red source. Does not belong in any astronomical catalog.
\end{itemize}
We next provide some details of the decision process leading to the
assignment of the various grades above, working backwards from F to A.
{\em Grade F.}
Persistence artifacts, or ``latents,'' in the W3 band
constitute the most common class of false, extended red source meriting an F grade. Most
latents are culled by the initial visual classification (see Section~\ref{visual}), but some did make it into the Extended Gold Sample. The most
pernicious cases involve W3 latents that happen to fall on the
positions of legitimate W1+W2 sources; these closely resemble our
expectations for a high-$\gamma$ KIII galaxy in the \mbox {\it WISE}\
images. The visual review process readily discards such
sources. They have very low W3NM/W3M, failing to appear in the majority of the
individual \mbox {\it WISE}\ Level 1b frames (planets, comets, and asteroids also
exhibit this behavior, but do so in all bands and appear in the Known
Solar System Object Possible Association List, see Section~\ref{galactic}).
The wider-field \mbox {\it WISE}\ Atlas images typically reveal the bright
star responsible for producing a W3 latent.
{\em Grade D.}
Unlike F-grade sources, sources assigned D grades are real astrophysical
objects, but these are not valid, red extended \mbox {\it WISE}\ sources. There are two
main classes: (1) Saturated stars (W3$\le 3.8$~mag) which would otherwise be point sources, and (2)
bright ``knots'' within larger regions of mid-IR nebulosity (for example, associated with
Galactic foreground emission). Our visual review of the \mbox {\it WISE}\ images
readily identifies such cases.
{\em Grade C.}
For sources that pass the quality control checks associated with F and
D grades above, we next evaluate the citations listed for the
associated SIMBAD source(s).
Sources that receive C grades typically have ${\ga}4$ citations,
which, taken together, convince
us that its astrophysical nature is understood.
If our review of the literature for a given source point toward original object type
listed in SIMBAD, we accept that object type.
In many cases we found either (1) that the preponderance of literature
pointed to a different object type or (2) the nearest SIMBAD source
returned by our automated matching was not the appropriate counterpart
for the \mbox {\it WISE}\ source. If necessary, we manually
overrode the SIMBAD object type and/or matching source for our catalog.
An example of a grade C source is shown in Figure~\ref{fig:finder}.
{\em Grades B and A.}
Our final two grades, B and A, represent real astrophysical extended
red sources with scant (${\la}4$) or zero existing literature citations, respectively;
these sources should be given high priority for further observational
followup to determine their nature.
Grade B sources are cited only in large, survey catalogs containing
minimal interpretation of individual sources. Commonly encountered
catalog papers containing Grade B (and also C) extended red \mbox {\it WISE}\
sources are listed in Table~\ref{tab:refs}.
\begin{deluxetable*}{ll}
\tabletypesize{\footnotesize}
\tablewidth{0 pt}
\tablecaption{ \label{tab:refs}
Frequently Encountered Catalogs Matching Red Extended \mbox {\it WISE}\
Sources}
\tablehead{Reference & Catalog Description}
\startdata
\citet{UGC} & Arcsecond Positions of Uppsala General Catalog (UGC) Galaxies \\
\citet{UZC} & The Updated Zwicky Catalog (UZC) \\
\citet{ESO-Uppsala} & The surface photometry catalogue of the
ESO-Uppsala galaxies \\
\citet{2M++} & The 2M++ galaxy redshift catalogue (69,160 galaxies) \\
\citet{QDOT} & The QDOT all-sky IRAS galaxy redshift survey \\
\citet{FlatGal} & The 2MASS-selected Flat Galaxy Catalog (18,020
disc-like galaxies) \\
\citet{positions} & Positions for 17,124 galaxies including 3301
new companions of UGC galaxies \\
\citet{UZC-SSRS2} & The UZC-SSRS2 Group Catalog (1168 galaxy groups) \\
\citet{PSCz} & The PSCz catalogue (15,411 IRAS galaxies) \\
\citet{KISO} & KISO survey for ultraviolet-excess galaxies.\\
\citet{Imperial} & The Imperial IRAS-FSC Redshift Catalogue (60,303 galaxies) \\
\enddata
\end{deluxetable*}
The most promising candidates for observational follow-up are grade A
sources that are isolated, meaning they are neither part of a cluster of
red objects (for example, a young embedded star cluster or a galaxy
cluster hosting many mergers) nor associated with diffuse mid-IR
nebulosity (a hallmark of embedded star clusters in \mbox{\ion{H}{2}~}~regions or an
indication that they may be associated with a large, extended
galaxy). An example of an isolated ``A'' source and a cluster of red sources is presented
in Figure~\ref{fig:isolated}.
\begin{figure*}[htp]
\centering
\epsscale{0.9}
\plottwo{f10a.jpeg}{f10b.jpeg}
\caption{Left (a): Source W224436.13+372533.7, an example of an isolated red
source given an A grade. Right (b): Source W043329.56+645106.5, an unusual
cluster of sources discovered by our search, also given
an A grade. Top
panels show the color-composite \mbox {\it WISE}\ $2\arcmin \times 2\arcmin$
``postage stamp'' images from our ``at-a-glance'' charts (red = W4, green =
W3, blue = W1+W2), while bottom panels
show the same sources in the wider $30\arcmin \times 30\arcmin$
\mbox {\it WISE}\ atlas images (red = W3, green = W2, blue = W1).
}
\label{fig:isolated}
\end{figure*}
Note that this grading scheme only loosely tracks ``interest'' from a
SETI perspective: a well-studied galaxy might have an anomalously
high MIR luminosity and thus be an outstanding SETI target, while a
new-to-science protostar might be manifestly ordinary and so very low
priority. The ``followup'' of class A sources is thus primarily to
determine their true nature and determine if they warrant further
study from either a natural astrophysics or SETI perspective.
Figure~\ref{Classified} illustrates the colors and magnitudes of the
sources we have graded, and shows where these sources reside in this parameter space.
\begin{figure*}[htp]
\begin{center}
\includegraphics*[width=7in,trim=0 4in 0 5.5in,clip]{W2-W3_vs_W3_Classified_Corrected.jpeg}
\includegraphics*[width=7in,trim=0 4.5in 0 4.5in,clip]{W2-W3_vs_W1-W2_Classified_Corrected.jpeg}
\end{center}
\caption[CAPTION]{\label{Classified} Color-magnitude
diagrams (top) and color-color diagrams (bottom) for W3 Extended
Gold sample sources which
have been carefully vetted and classified as described in
Section~\ref{grading}. Sources with no or little literature presence (grades A and B, respectively) are in the left panels; well-studied sources (grade C) are in the middle; saturated stars and artifacts (grades D and F, respectively) are on the right. Unsurprisingly, grade A and B sources (i.e.\
sources with little presence in the literature) are
typically faint, and grade D and F sources have extreme colors and
magnitudes (indicating that they are artifacts of the instrument or
the analysis pipeline strongly affecting some bands but not others). The density of all W3 Extended Gold Sample sources is
indicated in grayscale in the bottom panels. We use plots such as
this to ensure that we understand the effects of making cuts based
on our grading scheme.}
\end{figure*}
\section{W3 Extended Platinum Sample}
\label{platinum}
\subsection{Construction of the Sample}
The Extended Gold Sample was constructed to be relatively free from
non-extended or non-astrophysical sources, but our curation was
deliberately liberal so that borderline cases could be handled more
carefully on a case-by-case basis. Having constructed the Gold Sample, we
performed this more careful analysis on those objects whose calibrated
photometry indicate that they are especially red.
As discussed in Section~\ref{calibrate}, we have performed magnitude corrections
using the photometry provided
by the \mbox {\it WISE}\ pipeline. For every source in the Extended Gold Sample we
now have photometry using four
different photometric systems (i.e.\ MPRO (profile), MAG
(aperture corrected magnitude), CMAG (Corrected Magnitude, see Section~\ref{calibrate}), and GMAG (elliptical aperture)). While the corrected magnitudes are
reliable for most sources, there are special cases
where a different photometric system may yield more reliable results.
Given this, we have measured $\gamma$, $\hbox{\ensuremath{T_{\mbox{\rm \scriptsize waste}}}}$, and $L/(4\pi d^2)$, (see
Section 4.0, and Paper II)
in each of the four different photometric systems.
To ensure that our choice of photometry does not cause us to miss any
red sources, we selected from the Extended Gold Sample all sources with
$\gamma \ge 0.25$ in any of the photometric systems. This identified a total of 3,145
sources. We used our collected imaging, grading, and literature search
results (Section~\ref{grading}) to carefully and visually vet this sample to identify and remove the
most obvious contaminants, mostly nebulosity and saturated stars that
survived our very liberal curation of the Extended Gold Sample. We identified and removed 366 obvious contaminants, reducing our list to 2779 objects.
We found that in a small percentage of cases, especially with
blended sources, an extended source is broken up into two or more
components in the All-sky catalog. In these cases we choose the entry
coordinates closest to the extended source center and the photometric
system that best captures the true magnitudes of the extended source.
In some cases, the W3RCHI2 value of the catalog entry centered on an
extended source is below 3 because of the details of how the All-sky
catalog breaks up composite sources.
We segregated this sample into four different categories and determined
which photometric system best fits the source in question:
\begin{itemize}
\item Extended Source: Source is truly extended in the \mbox {\it WISE}\ images
\item Point Source Galaxy: Source appears to be a galaxy from
O/IR imaging or other sources, but appears unresolved in the \mbox {\it WISE}\
imagery, despite having a high W3RCHI2 value.
\item Point Source Star: Source appears to be stellar in nature.
\item Junk: Source is a contaminant, should not be in any catalog
of real astrophysical sources.
\end{itemize}
In addition to classifying these sources as described above, we also
determine which photometric system best estimates the true photometry
of the given source. We use the following convention to indicate the
best possible \mbox {\it WISE}\ photometry in the source tables:
\label{photometry}
\begin{itemize}
\item 0: MPRO (Profile fit photometry)
\item 1: MAG (Aperture-corrected magnitude)
\item 2: CMAG (Corrected magnitude)
\item 3: GMAG (2MASS XSC elliptical aperture).
\end{itemize}
Using these conventions we identify 1296(46.7\%) classified as Extended,
974(35\%) classified as unresolved galaxies, 263(9.4\%) classified as stellar, and
246(8.9\%) classified as junk. Of the Extended sources, 563 have
$\gamma > 0.25$ in the photometric system most appropriate for that
source. These compose our W3 Extended Platinum Sample of real,
extended, red sources in \mbox {\it WISE}.
\subsection{Platinum Sample Published Catalog}
We present a catalog of 563 sources deemed to be extended and real in
the \mbox {\it WISE}\ 12$\mu$m filter and identified as the Platinum Sample as a
FITS file associated with this manuscript. This
list of sources have also been selected as those having $\gamma \ge
0.25$ in the preferred photometric system. Table~\ref{FITS} gives the detailed
parameter description for this electronic catalog. For brevity, we
present a minimal number of parameters representing the most important measurements for these
sources, and the \mbox {\it WISE}\ identifier so that users may cross-reference our
catalog to the \mbox {\it WISE}\ All-sky or ALL-WISE catalogs.
Presented in this parameter list are the coordinates, optimal
\mbox {\it WISE}\ photometry and the photometric system used (see Section~\ref{photometry}), the
AGENT parameters, and pertinent information derived from the SIMBAD
database. Tables \ref{reddest}--\ref{extreme} have been derived using this catalog.
\begin{center}
\begin{deluxetable*}{lccl}
\tablewidth{0pt}
\tablecaption{Platinum Sample Published Catalog \label{FITS}}
\tablehead{\colhead{Parameter} & Type & Entry & Description}
\startdata
DESIGNATION & STRING & J190101.24-181215.0 & \mbox {\it WISE}\ Designation \\
NAME & STRING & PNA6651 &
SIMBAD Identifier \\
GRADE & STRING & C & Grade (see Section~\ref{grading}) \\
SIMBAD\_TYPE & STRING & PN & Object Type \\
R.A. & DOUBLE & 285.25520& Right Ascension [degrees]\\
Decl. & DOUBLE & -18.204188& Declination [degrees]\\
W1BEST & FLOAT & 15.4980 & W1 optimal photometry [mag] \\
W2BEST & FLOAT & 13.889&W2 optimal photometry [mag] \\
W3BEST & FLOAT & 9.0770&W3 optimal photometry [mag] \\
W4BEST & FLOAT & 3.0690& W4 optimal photometry [mag]\\
PHOT\_SOURCE & INT & 1& Photometric System, see Section 5.1 \\
TWASTE & FLOAT & 72.923& Waste heat temperature [K]\\
GAMMA & FLOAT & 0.99294& AGENT parameter $\gamma$\\
BOL\_FLUX & FLOAT & 255.83& Bolometric flux [L$_{\odot}$/pc$^2$]
\enddata
\end{deluxetable*}
\end{center}
\section{Extreme Extended Objects in the Platinum Sample}
\label{sec:extreme}
\subsection{Extreme \mbox {\it WISE}\ Colors}
The \mbox {\it WISE}\ filter system allows for the measurement of 6 different
photometric colors, i.e W1-W2, W2-W3, etc. In Table~\ref{reddest}, we present a
list of the 10 reddest (per color) objects contained within the
Extended Platinum Sample. These objects have been ordered by decreasing ``redness'', so
that the 10th object in the list represents the 10th reddest object
within the color explored.
The most extreme MIR objects in the high latitude sky ($|b|
\ge 10$) appear to be dominated by PNs. Other
objects with extreme colors include a pair of comets, three YSOs, a handful of Type {\sc ii} Seyfert galaxies, two
uncataloged sources in the LMC that escaped our mask of that region,
and some objects we discuss in Section~\ref{extremegamma}.
Since this study is primarily interested in finding ETIs of
extra-galactic origin we present Table~\ref{redgal}, which is a list
of the 10 reddest (per color) high latitude extragalactic objects
seen in the Extended Platinum Sample. We find that the colors of
galaxies are in general not as extreme as Galactic sources. Most of the
galaxies on this list are Grade C, meaning that their nature has been
identified in the literature.
Table~\ref{redgal} is dominated by galaxies classified in SIMBAD as
AGN of various stripes, which is unsurprising since those galaxies
are often characterized by their extreme MIR emission. AGN are thus,
as expected, our primary confounders in our waste heat search for K3
civilizations.
\subsection{Extreme $\gamma$}
\label{extremegamma}
In the AGENT formalism (Section~\ref{agent}, \citet{WrightDyson2}) the
parameter $\gamma$ represents the fraction of starlight reemitted in
the MIR, at temperature \hbox{\ensuremath{T_{\mbox{\rm \scriptsize waste}}}}. We have measured maximal values for
this parameter assuming that there is no dust in any of our sources,
and that their underlying stellar population is that of an old
elliptical galaxy (so, virtually dust-free). Since this is the
parameter of interest in searches for alien waste heat, we have sorted
the Extended Platinum Sample by this parameter. We present the top 50
such galaxies in Table~\ref{highgamma}.
The galaxy with the highest measured $\gamma (= 0.85)$ is NGC 4355
(=NGC 4418), which is also has the most extreme colors in four of the six
\mbox {\it WISE}\ color combinations. It is a well-studied Type {\sc ii} Seyfert galaxy in
the Virgo cluster with a (W1-W4) color of 10.19, with the extreme MIR
emission being due to the AGN.
The second source on our list is IRAS
04259-0440, a marginally resolved galaxy with modest presence in the
literature. It has been studied in the context of being a Seyfert
galaxy or LINER \citet{Wu2011}, so we are convinced that the MIR
emission is understood. Nonetheless, given the extreme nature of this galaxy's infrared
emission, this galaxy would appear to warrant more attention than it
has received to date.
The third galaxy in our list is the well-studied Arp 220, the quintessential local starburst
galaxy and a known active galaxy. Fourth and fifth are UGCA 116 and IRAS F20550+1655-SE, both
pairs of interacting galaxies. In all three cases, the extreme MIR colors
are clearly due to star formation triggered by galaxy interactions.
\subsection{Extreme Objects New to Science}
\label{new}
One of the primary objectives of this investigation is to search and
identify the most rare and extreme sources in the high latitude
infrared sky. The majority of the sources in our tables have already been
discovered and discussed in various articles, but there are still a
small number of objects not previously discovered or discussed in the
literature.
In Table~\ref{extreme} we present 3 objects ($\gamma \ge 0.25$)
classified as As, meaning that they are essentially new to science.
In this section we also discuss an extreme and apparently anomalous
object that we gave a B grade, IRAS 15553-1409, and a particularly
interesting lower-$\gamma$ grade A source.
\subsubsection{IRAS 04287+6444: An Unusual Cluster of MIR Sources With
no Optical Counterparts}
Our most unusual objects are associated with IRAS 04287+6444. The
brightest of these sources is slightly blended, which complicated our magnitude
corrections, giving our source an erroneously high $\gamma$ value
before our quality checks corrected the error. This blending also triggered the high
W3RCHI2 value that suggested this was extended source. There appear to be at
least seven very red point sources clustered in this region in all. We
identify and number four fainter sources in Figure~\ref{04287a}
(Bottom right); the other three or more all contribute to the
brighter blend in the NE.
NED reports five entries within 2$^\prime$ of these sources'
positions. The two nearest detections are from
the 2MASS Extended Source Catalog (see below), and the third nearest
entry is the IRAS counterpart to this source. The other two NED
entries are a ROSAT detection \citep[1WGA J0433.4+6451][]{RASS} centered $\sim 33^{\prime\prime}$
away (outside the $20^{\prime\prime}$ astrometric precision of ROSAT), and the 1.4 GHz radio
source NVSS J043322+645120 \citep{NRAOVLA} centered $\sim 0.8^\prime$
away (consistent with being associated with our source).
\citet{IRAS_redshift} included the IRAS center in their optical spectroscopic
survey, and identified it as ``cirrus or dark cloud,'' although they
note that a significant number of such entries may in fact be
galaxies. This non-detection is not surprising given the lack of
optical counterpart to these objects.
\begin{figure*}
\includegraphics[width=6.5in,trim=0 0.75in 0 0,clip]{W0433+6451_paper_figure_a}
\caption{Six views of the extremely red source IRAS 04287+6444 (WISE
J043329.55+645106.5). A red circle indicates the common position of the WISE
emission peak in all six panels. There is no hint of any emission
in the optical (B band, lower left). \mbox {\it WISE}\ reveals a large number of sources
in the region in W1 and W2 (top and bottom middle), and a pair of extremely bright, blended
sources in W4 (upper right). The color composite image (upper left)
shows that there are also four, fainter but also very red objects
to the southwest of the primary pair. We label these four sources
in the W3 image (lower right).
\label{04287a}}
\end{figure*}
We have found a serendipitous archival {\it Spitzer} MIPS image of
these sources, taken because they are within 15$^\prime$ of HD 28495,
a target observed with IRAC as part of the FEPS program \citep{FEPS}.
Figure~\ref{Spitzer} shows how the superior resolution of this 22$\mu$ imagery reveals substructure
to the SE component of the bright blend, and many sources undetected
by \mbox {\it WISE}\ (without color in formation, it is unclear whether these
sources are associated with IRAS 04287+6444).
\begin{figure}
\plotone{fig16_MIPS.jpeg}
\caption{Serendipitous archival MIPS imagery of IRAS 04287+6444. The
bright \mbox {\it WISE}\ source (which shows substantial substructure here) and
the four fainter \mbox {\it WISE}\ sources are all detected in this 22$\mu$ image.\label{Spitzer}}
\end{figure}
Objects \#1 and \#3 are barely detected in the WISE images (not
appearing in the W4 band at all)
and thus provide little clues as to their true nature, but appear to
be fainter versions of the other red objects, with similar
colors. Objects \#2 and \#4 appear
cleanly detected in all four bands.
The angular proximity to HD 28495 is intriguing, but these are likely
unassociated since a common distance at 25 pc \cite{Hipparcos2} would imply
a projected separation of $\sim 2\times 10^4$ AU. Nonetheless, the lack of optical
counterpart complicates efforts to rule out this scenario from proper motion.
One possibility is that this is a previously uncataloged moderate-latitude ($b=11.5^\circ$)
dark cloud, and that these are an embedded cluster of young stellar
objects or protostars. Figure~\ref{04287b} shows the 2MASS imagery
for this region, which has significantly better angular resolution.
This NIR imagery reveals that the brightest source in the \mbox {\it WISE}\
imagery comprises at least three sources, only one of which is evident
in J band. Supporting this interpretation, \citet{YangCO} detected CO (J=1$\rightarrow$0) emission in the
direction of the IRAS source\footnote{The ``Association'' field from
Table 2 of \citet{YangCO} for IRAS 04287+6444 confusingly
reads ``HL Tau''. HL Tau itself appears in the table two
entries prior, where the ``Association'' field reads ``04288+6444,''
apparently a typo for 04287+6444. We presume that \citeauthor{YangCO}
erroneously transposed the ``Association'' values for these entries.
If, instead, it is the target names that are transposed, then the
appropriate LSR radial velocity for IRAS 04287+6444 is 6.91 km/s, the FWHM is
4.3 km/s, and no kinematic distance is available.} with LSR radial velocity -13.27 km/s and
FWHM 3.4 km/s, implying a molecular cloud exists in this direction at
a kinematic distance of 840 pc (and thus a height of $\sim 170$ pc
below the Sun's position in the plane).
\begin{figure*}
\includegraphics[width=7in,trim=0 2in 0 2.5in,clip]{W0433+6451_paper_figure_b}
\caption{Four closer-in, NIR views of the extremely red source IRAS 04287+6444 (WISE
J043329.55+645106.5) from 2MASS. A red circle indicates the common position of the WISE
emission peak in the three single-band panels. Note the change in angular scale with respect
to Figure~\ref{04287a}. The K band image (lower left)
reveals that the bright \mbox {\it WISE}\ source comprises at least three
sources. Of these, only the NE and SW sources are apparent in H
band (upper right), and only the SW shows a J band counterpart
(upper left). The more distant SE object, responsible for making
the \mbox {\it WISE}\ source appear extended, is detected in all three bands. The color composite (lower right) shows that all four
sources have extremely red NIR colors.
\label{04287b}}
\end{figure*}
If the sources are extragalactic, the most natural explanation is
that they are members of a galaxy cluster. \citet{Edelson12}
identified the IRAS source as having a modest chance ($\sim 50\%$) of
being an AGN of some flavor based on the \mbox {\it WISE} , 2MASS, and X-ray
fluxes, however the X-ray detection may be unassociated, and it
appears this probability does not incorporate the fact that the source
has no optical counterpart or that it is not isolated.
The lack of optical counterpart could be due to redshift and internal
extinction. A significant population of high redshift ($z\
> 2$) and more luminous ($L_{IR} > 10^{13}$L$_{\odot}$ ) Dust Obscured
galaxies (called Hot Dust-obscured Galaxies, or ``Hot DOGs") have recently been
identified by the WISE survey, \citep[see][for
a detailed discussion of these types of objects]{HyperLIRGs,
HyperLIRGsubmm,Bridge13,Stern2014}. Indeed, the strong 22 $\mu$m
emission for these objects are reminiscent of Dust Obscured Galaxies (DOGS), either local ($z
\sim 0$) \citep{LocalDOGs} or at high redshift ($z \ge 2$)
\citep{DeyDOGs}.
However, our objects are inconsistent with hot DOGs
since hot DOGs tend to have very little or no emission in the shorter
WISE bands. And such extreme examples of dusty galaxies are not
typically highly clustered as our sources are.
We tentatively favor the interpretation that this is a cluster of
young stellar objects embedded in and heavily extinguished by their
parent molecular cloud. We are intrigued by these objects, and we hope that
spectroscopic observations can and will reveal their true nature in the future.
\subsubsection{WISE J224436.12+372533.6: A new MIR-bright galaxy}
We gave the object WISE J224436.12+372533.6 (shown in
Figure~\ref{W224436}) an A grade because it has
no presence in the astronomical literature beyond having been noted in
the 2MASS Extended Source Catalog. It is MIR bright and red, and
DSS and 2MASS imaging shows what appear to be a galaxy. It is just
barely detected by IRAS (it appears in only two bands in the IRAS
Faint Source catalog \citep{1992iras}), and so may have evaded
prior notice for that reason. It also appears as a 1.4 GHz source in the NRAO VLA radio
survey \citet{NRAOVLA}. This source deserves further study to
understand its superlative nature.
\begin{figure*}
\includegraphics[width=6.5in,trim=0 0.75in 0 0,clip]{W2244+3725_paper_figure.jpeg}
\caption{The previously unstudied but MIR-bright galaxy
WISE J224436.12+372533.6. It was only barely detected by IRAS, but
is easily detected by \mbox {\it WISE}\ as an extremely red MIR source.\label{W224436}
}
\end{figure*}
\subsubsection{IRAS 16329+8252: An MIR-bright galaxy at z=0.04?}
We give this source an A grade because it has virtually no presence in
the literature beyond a single low resolution spectrum by
\citet{Chen11}. If their identification of the emission lines in this
spectrum is correct, then it is a galaxy at $z=0.039$. Its strong MIR
emission suggests large amounts of star formation, possibly triggered
by the disturbance of a nearby neighbor (see Figure~\ref{1627}).
\begin{figure*}
\includegraphics[width=6.5in,trim=0 0.75in 0 0,clip]{W1627+8245_paper_figure.jpeg}
\caption{Six views of the extremely red source IRAS 16329+8252. A red circle indicates the common position of the WISE
emission peak in all six panels. The MIR
morphology (upper left, where red = W4, green = W3, blue = W1+W2) is
consistent with the optical B band (upper middle), although the
shorter \mbox {\it WISE}\ bands (W2, lower right) appear slightly offset to the west, consistent with
the H band imagery (upper right). The W3 and W4 band imagery
appear very slightly extended. This is apparently a very MIR-bright
z=0.039 galaxy, possibly disturbed by a neighbor to the northeast,
or along the line of sight. \label{1627}}
\end{figure*}
\subsubsection{WISE J073504.83-594612.4: a pair of merging galaxies?}
We graded the source WISE J073504.83-594612.4 (Entry 2 in
Table~\ref{extreme}) A because it has virtually no presence in the
literature, appearing only in a handful of photometric catalogs,
including the 2MASS extended source catalog. It has no IRAS
counterpart. It appears to be extragalactic.
The \mbox {\it WISE}\ color image (Figure~\ref{0735}) shows little structure; it is
barely extended in W3 and W4. The H band and W2 imagery, which
presumably trace the stellar populations of the bulges of the
galaxies in this imagery, reveals two sources, one at the position of
the \mbox {\it WISE}\ source and one $\sim 20^{\prime\prime}$ to the north. The DSS B
band image shows significant structure between these positions,
suggesting that this is a disturbed or merging galaxy pair. The
bright W4 emission suggests that this disruption his generating
significant star formation in the southern galaxy.
\begin{figure*}
\includegraphics[width=6.5in,trim=0 0.75in 0 0,clip]{W0735-5946_paper_figure.jpeg}
\caption{Six views of the extremely red source WISE
J073504.83-594612.4. This source has
no known IRAS counterpart. The \mbox {\it WISE}\ color image (upper left, where
red = W4, green = W3, blue = W1+W2) shows little structure; it is
barely extended in W3 and W4 (lower middle and right). The H band image
(upper right) and W2 image (lower left) reveals two sources, one at the position of
the \mbox {\it WISE}\ source and one $\sim 20{^{\prime\prime}}$ to the north, which are
perhaps the bulges of a pair of galaxies. B band shows a possible bridge of
material between the two H band sources, indicating that this may
be a pair of merging galaxies. \label{0735}}
\end{figure*}
\subsubsection{IRAS 15553-1409: a large nebula of dust from a Be shell star?}
\label{sec:15553}
The infrared source IRAS 15553-1409 is the reddest or second reddest
source in four of the six \mbox {\it WISE}\ colors. We give this source a B grade because it has
virtually no presence it the literature, beyond \citet{Carballo92}, who identify
it as a potentially ``evolved Galactic object.'' \mbox {\it WISE}\
imagery reveals it to be a MIR-bright nebula associated with the
classical Be star 48 Librae. Figure~\ref{15553} shows that the MIR
morphology and colors are similar to that
of reflection nebulae, but in this case there is no apparent emission in the
optical.
\begin{figure*}
\includegraphics[width=6.5in,trim=0 0.75in 0 0,clip]{15553.jpeg}
\caption{Six views of the extremely red source IRAS 15553-1409 (WISE
1558-1418). A red circle indicates the common position of the WISE
emission peak in all six panels. The bright point source is the Be star 48 Librae. The MIR
morphology (upper left, where red = W4, green = W3, blue = W1+W2) is
typical of a reflection nebula, but there is no emission obvious in
the optical (B band, upper middle). The NIR (H band, upper right)
and W2 bands (lower left) appear unremarkable. The brightest part
of the cloud is apparent in W3 (lower middle) and large
amount of diffuse emission are obvious around the star in W4 (lower right). \label{15553}}
\end{figure*}
48 Librae is known to be a classical Be shell star
\citep[e.g.][]{48Lib_shell} and a member of the Sco-Cen Assocation.
It has a distance of $\sim 140$ pc \citep{Hipparcos2}, expansion
velocities of 25 km/s \citep{BSCNotes}, and age $< 20$
Myr (Eric Mamajek, private
communication\footnote{\url{https://www.facebook.com/jason.wright.18062/posts/10204670796112674}}
and references therein.) It is sometimes listed in the literature as
a giant star, but this is likely because its rapid rotation gives it an
anomalously low surface gravity, complicating its spectrally derived
luminosity classification.
This source is superlative because of its extreme MIR
colors and extent, although this is a byproduct of its proximity;
similar Be shell stars at greater distances would not be resolved.
\citet{AdamsIR} notes that the infrared excesses of main sequence
stars typically have one of four origins: circumstellar debris disks,
protoplanetary and protostellar disks around very young stars,
``cirrus hot spots'' caused by the illumination of ambient
interstellar dust, and the excretion disks of classical Be stars.
Cirrus hot spots \citep{AdamsIR,vanBuren} can be simple reflection
nebulae \citep[the ``Pleiades phenomenon,''][]{Kalas} or the result of
bow-shocking by the star's wind or radiation as it moves through the
ISM \citep[e.g.][]{vanBuren,Povich08,Everett10}. Inspection of the morphology
of typical reflection nebulae (including that of the Pleiades themselves and
those discovered by \citet{Kalas}) shows that 48 Librae does not
appear to be a typical example. The 48 Librae nebula has no optical
counterpart, suggesting that the 22 micron emission is thermal, and
appears to have symmetries about the position of the star, suggesting
that it has some connection to the star beyond being
illuminated by it (see Figure~\ref{symmetry}).
\begin{figure}
\begin{center}
\includegraphics[width=3in]{BeDust.jpeg}
\caption{A \mbox {\it WISE}\ color composite of the 48 Librae nebula, with
ellipses overlain to illustrate the common symmetries of the arc
structures in the nebula. The ellipses describe concentric rings centered
on the star inclined 19$^\circ$ from edge-on. The physical size of
the nebula is $\sim$ 15,000 AU.\label{symmetry}}
\end{center}
\end{figure}
The most likely explanation of this nebula is that it is therefore a
ring or bow shock. The bow shock and ring interpretations are complicated by the fact
that, despite being a Sco-Cen Association star, 48 Librae does not
appear to be embedded in a region of high density gas and dust, and that
its space motion is not large.
The proper motion of 48 Librae is SW \citep[$(\mu_\alpha, \mu_\delta) = (-12.44,
-16.73)$ mas/yr][]{Hipparcos2}, which is not in the direction of the brightest
sources of emission, but it appears that most of this
apparent motion is actually due to motion of the Sun with respect to
the local standard of rest. Correcting for the solar motion (using radial
velocity $7.5\pm$ 1.8 km/s \citep{48LibRV} and the LSR of \citet{LSR}), the
proper motion is still roughly SW ($(\mu_\alpha, \mu_\delta) = (-2.4,
-0.9)$ mas/yr), corresponding to an LSR space velocity of 4 km/s, with
an error of a few km/s from our LSR correction.
Further complicating the bow shock interpretation is that both sides of the 48 Librae
nebula have similar arcs consistent with being portions of rings
inclined $\sim 20^\circ$ from edge-on, and centered on the star.
Given these difficulties with the typical bow shock and ring models, we suggest that
the nebula have originated with the star itself. Classical and shell
Be stars are not typically considered to be significant sites of dust
formation or sources of MIR excess \citep{Be_review}, although NIR
excesses, presumably due to circumstellar dust, are common, and many unclassified Be
stars are known to have MIR
excesses\citep[e.g.][]{Be_FIR,Be_dust,BE_NIR}. 48 Librae has one of
the larger and stronger disks known among classical Be stars (T.\
Rivinius, private communication).
Since 48 Librae is known to have shells, a time variable disk
\citep{48Librae}, and significant mass loss, it is reasonable that
some of this shell material would condense into dust near the star
before being lost \citep{Be_dust}. The nebula is not too
large for this: the nebula has an angular size of order a few arcminutes, which at the
distance of 48 Librae ($\sim 150$ pc) corresponds to $\sim 15,000$ AU. The expansion
velocity of the shells is 25 km/s, yielding a characteristic timescale
of $\sim 30,000$ yr, significantly shorter than the Be phase of a
star.
So, we may be seeing shocked dust where the shells (really rings) of
excreted material collide, either with the ambient ISM or with
previously ejected material. If so, then we expect the rings we observe
in the nebula to have a rough correspondence to the geometry of the
excretion disk. The rings (which we fit by eye) have an
inclination of 70$^\circ$ (i.e.\ 20$^\circ$ from edge-on), and a
position angle of 72$^\circ$. The actual inclination of the
excretion disk must be $\gtrsim 65^\circ$ (because it shows absorption
lines from the disk, T.\ Rivinius, private communication), and the
actual position angle is known from interferometry to be 50$^\circ
\pm$ 9$^\circ$ \citep{48Librae}, consistent with our rings at 1--2 $\sigma$, within the rough precision
with which we can define them.
Alternatively, we may be seeing excreted dust being illuminated by UV
radiation from 48 Librae itself, although this interpretation is
complicated by to the lack of apparent optical scattered light, and by
the patchiness of the emission.
It is unclear if the nebula contains more dust than could be plausibly
explained by the currently observed mass loss rate, but of course 48 Librae
could have had episodes of higher mass loss rates in the past.
This object may be revealing to us that Be shell stars are, in fact, common sites of
dust generation. This phenomenon warrants further study.
\section{Upper Limits on the Energy Supplies of Type {\sc iii}
Kardashev Civilizations}
\subsection{Limits on energy supplies as a fraction of stellar luminosity}
We can use those sources in our study with the largest amounts of thermal emission
to set an upper limit for waste heat emission among the galaxies we
have surveyed. We have found no sources with $\gamma > 0.85$, and the
50 galaxies we have found with $\gamma > 0.5$ appear to have
natural origins to most of their MIR emission, although we
have not rigorously verified this.
These values for $\gamma$ were calculated under the assumption that our target
galaxies are composed of only two components: an old stellar
population, and alien waste heat originating entirely from
reprocessed starlight. Since real galaxies have other sources of
MIR emission, these numbers are upper limits on the
integrated alien waste heat emitted by these galaxies.
Our assumption that the origin of the alien waste heat is intercepted
starlight (i.e.\ $\gamma = \alpha$, where $\alpha$ is the fraction of
starlight intercepted, see Section~\ref{agent}) means that we have
assumed that the flux in the W1 and W2 bands, which in our model
constrain the total stellar luminosity, does not include the starlight
lost to alien factories. If, instead, we assume that all alien waste
heat is generated by other means and that only a negligible fraction
of starlight is occulted ($\alpha\sim 0$), then we would infer a
higher value for the fraction of starlight emitted as waste heat.
Specifically, our limiting values of $\gamma = 0.85$ and 0.5
correspond to $\gamma_{\alpha=0} = 5.7$ and 1 \citep[see Section
3.2][]{WrightDyson2}.
In other words, we shave shown that there are no galaxies resolved by
\mbox {\it WISE}\ with MIR luminosities consistent with alien energy
supplies in excess of 5.7 times the starlight in their galaxies (i.e.\
we have ruled out $\gamma = \epsilon > 5.7$). If all of 50 galaxies
in our list turn out to have purely natural origins to their emission,
then this upper limit drops to $\gamma = \epsilon < 1$.
If we assume that any large alien energy supply will be based on starlight (that is, $\gamma
\sim \alpha$ and $\epsilon \sim 0$), then our upper limit is much
tighter: no resolved galaxies exist in our search area with more than
85\% of their starlight reprocessed by alien factories, a limit which
will drop to 50\% when our 50 high-$\gamma$ galaxies are more carefully vetted.
Translating these numbers into physical units (erg s$^{-1}$) will
require a more detailed modeling of the stellar and nonstellar
components of the galaxies we have surveyed, a project which is beyond
the scope of this paper. We hope to pursue this in a future paper.
\subsection{Number of galaxies surveyed}
\label{number}
Translating our upper limits into an upper limit on the frequency of
K3's requires knowledge of the number of galaxies we have effectively
surveyed. We cannot use our Gold or Platinum Samples to estimate this
number because they included color cuts to remove stars that also
removed elliptical and other dust-free galaxies. Even if we had
imposed no such cuts, there are many galaxies that would be resolved
in the W3 band if they were MIR bright, but are unresolved
--- or in some cases undetected --- in \mbox {\it WISE}\ because their MIR surface
brightness is below the \mbox {\it WISE}\ detection limits.
To estimate the number of galaxies that {\it would have been included}
in our sample {\it if} they had $\gamma > 0.5$, we can use the number
of sources in the W1 and W2 bands. The W1 band, in particular, has
better angular resolution that W3, and is primarily sensitive to
stellar photospheres, so is in many ways a clean band for estimating
the angular extent of galaxies around which alien factories
might reside.
One concern with using \mbox {\it WISE}\ data for this purpose is that, as we
have seen, the source counts include many non-galaxies (including artifacts), and many point
sources have erroneously large values of RCHI2. To mitigate this, we have used the 2MASS
Extended Source Catalog (XSC), which is relatively clean of point
sources and is composed almost entirely of galaxies.
We first cross-matched \mbox {\it WISE}\ to the 2MASS XSC, and selected only those
sources with $|b| \ge 10$. We then examined the NIR properties of MIR-red galaxies in the 2MASS
XSC by examining the relationship between the WXRCHI2 values for the
W1 and W2 bands and the W3RCHI2 parameter, which we used for the
Platinum sample. We restricted our analysis to 9,589 matched sources
with (W1-W3 $\ge$ 3.8) for the W1RCHI2 analysis, and 14,927
sources with (W2-W3 $\ge$ 3.5) for the W2RCHI2 analysis.
The left hand panels of Figure~\ref{count1}
show the relationship between the W1RCHI2 and W2CHI2 parameters (which
describe the degree to which galaxies are resolved in those bands, see
Section~\ref{rchi2}) and the vs.\ W3RCHI2 parameter we used to define a
source as ``extended'' in our survey.
\begin{figure*}
\includegraphics*[width=6in]{WISE-2MASS_paper_plots_W1RCHI2.jpeg}
\includegraphics*[width=6in]{WISE-2MASS_paper_plots_W2RCHI2.jpeg}
\caption{Correlations among size parameters for red extended sources
in \mbox {\it WISE}\ with 2MASS Extended Source Catalog (XSC) counterparts. The
y-axis in all plots corresponds to the RCHI2 parameter, which
measures the goodness-of-fit of a source to a model for a point
source (large values indicate a poor fit, so an extended source).
The x-axis of the left panels is the RCHI2 parameter in the W3 band,
(we used W3RCHI2$=3$, indicated by the vertical lines, to identify
extended sources in our catalogs). The horizontal lines mark the
approximate RCHI2 value in the W1 and W2 bands for typical red
sources that meet this criterion. The
x-axis of the right panels is the R\_EXT parameter of the XSC, in
units of arcseconds, describing the NIR angular size of the source.
The vertical line shows the value of this parameter
18$^{\prime\prime}$, that best corresponds to the W1 and W2 RCHI2
values typical of barely extended sources in W3. {\it
Top:} correlations between the W1 RCHI2 parameter. {\it Bottom:}
correlations with the W2 RCHI2 parameter. \label{count1}}
\end{figure*}
These panels show that, for the red sources we used to construct this
figure, we can use W1RCHI2 $> 12$, and W2RCHI2 $>3$ as
proxies for our actual criterion W3RCHI2$>3$. Using the full
\mbox {\it WISE}-2MASS XSC cross-matching (that is, not imposing any color cuts), we find
1,589,099 sources common to both catalogs. Of these, 1,463,781 have
$|b| \ge 10$. Of these, 111,617 have W1RCHI2 $\ge 12$ and 104,039 have
W2RCHI2 $\ge 3$. These figures are consistent, suggesting that we
have surveyed $1 \times 10^5$ galaxies. The only previous search for
K3's in the refereed literature, that of \citet{annis99a}, surveyed
163 galaxies.
As a check, we also used the R\_EXT parameter in the XSC, which
corresponds to a measure of the NIR angular size of these galaxies.
The right hand panels in Figure~\ref{count1} show that
R\_EXT corresponds to the angular sizes we
are interested in with good sensitivity (i.e., the test R\_EXT$>18^{\prime\prime}$ has a
low false negative rate), although it is not very specific (i.e.\ it has
roughly a 50\% false positive rate) for these red sources.
A query of the entire XSC with $|b| \ge 10$ and
R\_EXT$\ge 18$ yields 229,813 sources. A random sampling of 100 of
these sources reveals that all are present in the \mbox {\it WISE}\ All-sky
catalog, 45 have W1RCHI2$>12$ and 43 have
W2RCHI2$>3$. These numbers are consistent with the specificity we estimated among the
MIR-red sources in Figure~\ref{count1}. We also tested 100 random
\mbox {\it WISE}\ sources satisfying our extended source criteria in W1 and W2, and
find that 86 and 87 of them, respectively, have
R\_EXT$>18^{\prime\prime}$, also consistent with the sensitivity suggested
in Figure~\ref{count1}.
We conclude that there are $\sim 1\times 10^5$ galaxies with sufficient angular
size that they would have been included in our platinum sample if they
had had significant W3 emission. In our survey of these $\sim 1\times 10^5$ galaxies, we have found
that there are no alien, non-stellar energy supplies in excess of 5.7 times the
stellar luminosity of their host galaxy, and no alien
supercivilizations reprocessing as much as 85\% of their starlight into the
MIR. We have found 50 galaxies consistent with 50\%
reprocessing, all of which are presumably extraordinary, but entirely
natural, star-forming galaxies. Verification of the natural origin of
the MIR flux in these 50 galaxies will thus lower our upper limit to 50\%.
\section{Conclusions}
\label{conclusions}
We have produced a clean catalog of the reddest extended sources in
outside the Zone of Avoidance using in the \mbox {\it WISE}\ All-sky catalog, and
corrected that catalog's photometry of extended sources to be consistent with careful
aperture photometry at the 3--5\% level. We used the point
source goodness-of-fit parameter W\#RCHI2 to identify extended sources,
and various tests (including visual inspection and interrogation of
the Level 1b \mbox {\it WISE}\ data) to clean this sample of instrumental and
data pipeline artifacts and point sources.
We have graded each of our sources in terms of its presence in the
published literature, to determine whether the nature of its MIR
emission is well understood.
Our motivation is to use this catalog to perform the first
extragalactic search for waste heat from galaxy-spanning alien
supercivilizations. To that end, we have used the AGENT formalism of
\citet{WrightDyson2} to interpret the \mbox {\it WISE}\ SEDs of these sources as
ordinary elliptical galaxies with alien waste heat luminosities equal
to a fraction $\gamma$ of the starlight and characteristic temperature
\hbox{\ensuremath{T_{\mbox{\rm \scriptsize waste}}}}. This is an inappropriate model for natural sources,
especially spirals and star-forming galaxies, but it provides a
conservative upper limit on the true $\gamma$ paramater for the
galaxy.
We find that there are no galaxies in our sample of $1\times 10^5$ galaxies with fit values of
$\gamma > 0.85$, meaning that no galaxies resolved by \mbox {\it WISE}\ contain
galaxy-spanning supercivilizations with energy supplies greater than
85\% of the starlight in the galaxy (unless this energy is not
primarily expelled as light in the \mbox {\it WISE}\ bandpasses). We have
further identified all 50 resolved galaxies in our sample with fit
values of $\gamma > 0.5$. More detailed SED modeling of these galaxies, including the
use of other bands, will allow for more stringent upper limits, and we
will perform such modeling in the future.
We also identify 93 sources with $\gamma > 0.25$ but very little study in
the scientific literature. Three of these sources are MIR-bright and red galaxies
that are essentially new to science, having little or no literature
presence beyond bare mentions of a detection by IRAS or other surveys.
Verification that the MIR flux in all of these galaxies is
predominantly from natural sources (e.g., through
SED modeling across many more bands than \mbox {\it WISE}\ offers or
spectroscopy) will push our upper limit on galaxy-spanning alien
energy supplies in our sample of $1\times 10^5$ galaxies down to 50\% of the available starlight. In the meantime,
these are the best candidates in the Local Universe for Type {\sc iii}
Kardashev civilizations. This limit will improve upon the limit of $\alpha < 75\%$ found by
\citet{annis99a} in 57 spiral and 106 elliptical galaxies.
We find that the Be shell star 48 Librae has a large extended MIR
nebula. If the source of this dust is 48 Librae itself, it would suggest,
surprisingly, that dwarf Be shell stars can be sites of significant
dust production.
We have also found a previously unstudied cluster of MIR-bright sources with no
optical counterparts and very red colors. They appear to be Galactic sources
associated with a cloud, and so are likely part of a previously
unstudied star forming region.
In the appendices, we have also illustrated how WISE can be used to rule out broad
classes of K3 civilizations as being responsible for the lack of
emission in so-called \ion{H}{1} dark galaxies and the anomalous
colors and morphologies of ``red'' (or ``passive'') spirals. We find
a sample of five ``red'' spirals with red MIR
and $(NUV-r)$ colors, which are inconsistent with high levels of star
formation but consistent with high levels of alien waste heat.
Significant internal extinction would be a satisfactory natural
explanation for these colors, but until that is ruled out these
galaxies are some of the best candidates for K3's in our search to date.
\acknowledgements
This research is supported entirely by the John Templeton Foundation
through its New Frontiers in Astronomy and Cosmology, administered by
Don York of the University of Chicago. We are grateful for the
opportunity provided by this grant to perform this research.
We thank Jason Young and Sharon X.\ Wang for discussions on the
typical sizes and surface brightnesses of galaxies, especially LSBs.
We thank Caryl Gronwall, and Lea Hagan for their assistance with NED.
We thank Tom Jarrett for sharing a preliminary version of his extended
source catalog. We thank the \mbox {\it WISE}\ team for making the \mbox {\it WISE}\ All-sky survey, and especially Davy Kirkpatrick for useful
discussions and assistance navigating the \mbox {\it WISE}\ data products.
We thank Richard Wade for illuminating the nature of 48 Librae, and
Leisa Townsley, Stan Owocki, Thomas Rivinius, Howard Bond, and Eric Fiegelson for
discussions about the nature of dust around Be stars.
We thank Eric Mamajek for his inordinate efforts in hunting down
photometry and kinematics for our strange no-optical-counterpart
cluster and 48 Librae.
This publication makes use of data products from the Wide-field
Infrared Survey Explorer, which is a joint project of the University
of California, Los Angeles, and the Jet Propulsion
Laboratory (JPL)/California Institute of Technology (Caltech), funded by the National
Aeronautics and Space Administration (NASA). This work is based in part on observations made with the Spitzer
Space Telescope, which is operated by JPL/Caltech under a contract with NASA. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated byJPL/Caltech, under contract with NASA.
The Digitized Sky Surveys were produced at the Space Telescope Science
Institute under U.S. Government grant NAG W-2166. The images of these
surveys are based on photographic data obtained using the Oschin
Schmidt Telescope on Palomar Mountain and the UK Schmidt
Telescope. The plates were processed into the present compressed
digital form with the permission of these institutions. The Second
Palomar Observatory Sky Survey (POSS-II) was made by the California
Institute of Technology with funds from the National Science
Foundation, the National Geographic Society, the Sloan Foundation, the
Samuel Oschin Foundation, and the Eastman Kodak Corporation.
Funding for the Sloan Digital Sky Survey (SDSS) has been provided by
the Alfred P. Sloan Foundation, the Participating Institutions, NASA, the National Science
Foundation, the U.S. Department of Energy, the Japanese
Monbukagakusho, and the Max Planck Society. The SDSS Web site is
\url{http://www.sdss.org/}.
The SDSS is managed by the Astrophysical Research Consortium (ARC) for
the Participating Institutions. The Participating Institutions are The
University of Chicago, Fermilab, the Institute for Advanced Study, the
Japan Participation Group, The Johns Hopkins University, Los Alamos
National Laboratory, the Max-Planck-Institute for Astronomy (MPIA),
the Max-Planck-Institute for Astrophysics (MPA), New Mexico State
University, University of Pittsburgh, Princeton University, the United
States Naval Observatory, and the University of Washington.
Based on observations made with the NASA Galaxy Evolution
Explorer. GALEX is operated for NASA by the California Institute
of Technology under NASA contract NAS5-98034.
The Center for Exoplanets and Habitable Worlds is supported by the Pennsylvania State University, the Eberly College of Science, and the Pennsylvania Space Grant Consortium.
|
{
"timestamp": "2015-04-16T02:10:59",
"yymm": "1504",
"arxiv_id": "1504.03418",
"language": "en",
"url": "https://arxiv.org/abs/1504.03418"
}
|
\section{Introduction}
The LAT instrument onboard the Fermi satellite has detected a high-energy emission at 100 MeV--100 GeV,
extending well after the GBM prompt phase, for dozens of GRB afterglows (first Fermi catalog --
Ackermann et al 2013). The properties of the LAT afterglow emission are: fluence above 100 MeV
$\Phi = 10^{-5\pm 1} \ergcm2$, light-curve peak at 10--20 s after trigger, post-peak flux decay
$\nu F_\nu \propto t^{-1.3\pm 0.3}$ monitored up to 1 ks (sometimes longer), photon spectrum $C_\nu
\propto \nu^{-2.1\pm 0.2}$.
The isotropic energetic output of the brightest LAT afterglows, $E_\gamma = 10^{53 \pm 1}$ erg,
is 10--100 percent of the GRB output at $\mathrel{\rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}} 1$ MeV, the corresponding number of afterglow photons
above $\varepsilon_\gamma \sim 100$ MeV being $N_\gamma = E_\gamma/\varepsilon_\gamma \simeq
10^{57 \pm 1}$. The fraction $f_\pm$ of these photons that form pairs depends strongly on
the Lorentz factor $\Gamma$ of the medium that produced the LAT afterglow emission, because
$\Gamma$ determines the lab-frame collimation of photons and the threshold energy for pair-formation,
and on the source radius $R$, which sets the optical-thickness to photon-photon absorption.
Taking into account that $R \simeq \Gamma^2 ct$, with $t$ the observer time, it follows that
$f_\pm$ has a very strong dependence on $\Gamma$.
From the escape of the higher-energy LAT photons ($\sim 10$ GeV), Abdo et al (2009) have set lower
limits $\Gamma_o \mathrel{\rlap{\raise 0.511ex \hbox{$>$}}{\lower 0.511ex \hbox{$\sim$}}} 200-1000$ for the Lorentz factor of several LAT sources during the prompt emission
phase (burst). Consistent with that, Panaitescu et al (2014) found that, for $\Gamma \mathrel{\rlap{\raise 0.511ex \hbox{$>$}}{\lower 0.511ex \hbox{$\sim$}}} 200$,
photon-photon attenuation does not yield a spectral signature but, for $\Gamma \mathrel{\rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}} 200$,
the attenuation of photons above 1 GeV should be detectable. Additionally, as shown in this article,
for $\Gamma \mathrel{\rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}} 500$, the number of pairs $N_\pm = f_\pm N_\gamma$ is higher than the number
of electrons energized by the forward shock and, for $\Gamma \mathrel{\rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}} 250$, is larger than the number of
ejecta electrons energized by the reverse shock. Therefore, the formation of pairs from LAT photons
is of importance at least for those GRBs/afterglows whose LAT spectrum displays a spectral softening
at GeV energies.
Assuming a single power-law for the LAT spectral component, Panaitescu \& Vestrand (2014) have
calculated {\sl analytically} the emission from the pairs formed only from photons to which the GeV
front is optically-thick, leading to the conclusion that pairs can account for the brightest
optical counterparts (flashes) observed during the prompt phase. A larger number of pairs, but of lower
energy, are formed by the photons for which the LAT emission is optically-thin (to pair-formation).
In this work, we calculate {\sl numerically} the distribution with energy of all pairs formed from
high-energy photons (assuming a broken power-law spectrum), integrate it over the deceleration of
the blast-wave that produces the LAT afterglow, and track numerically the pair radiative cooling,
to obtain accurate light-curves for the optical flashes produced by pairs formed in GRBs and afterglows.
The effects arising from scattering of the high-energy photons on (cold) electrons existing in the
source or on the already-formed pairs are ignored. Such scattering increases the source-frame photon
escape path, which increases the probability that any photon forms a pair and the total number
of pairs formed. As we shall see, for sources that are optically-thin to photon scattering, scattering
on the already-formed pairs occurs with a smaller probability than pair-formation thus, to a good
approximation, the effect of photon scattering on the pair-formation rate can be ignored. The same is true for
scattering on the ejecta electrons, if the Lorentz factor of the GeV source is in the few hundreds.
However, if that Lorentz factor is or exceeds several hundreds, scattering on ejecta electrons should
occur with a higher probability than pair-formation. In this case, the number of pairs formed from
unscattered seed photons, calculated below, underestimates the true number of pairs.
We also ignore the formation of (external) pairs ahead of the afterglow blast-wave, which loads
with leptons the ambient medium and accelerates it (Beloborodov 2002), changing the afterglow dynamics
(Kumar \& Panaitescu 2004).
Consequently, the formalism presented here for the emission from pairs is more pertaining to a GeV
source that is located well behind the forward-shock, so that most pairs form in the shocked fluid
and not ahead of the blast-wave. That condition points to a relativistic reverse-shock as the origin
of the LAT afterglow emission (as could be the case for GRB 130427A - Panaitescu et al 2013).
\section{Pair-formation in a relativistic source}
\subsection{High-energy spectral component}
The number of pairs formed at any observer-frame time $t$ over a dynamical timescale is derived
from the observable 0.1--10 GeV fluence $\Phi$, the spectrum of the high-energy afterglow emission
(with the 0.1-10 GeV spectral slope being the only observational constraint), the source redshift $z$,
and the unknown Lorentz factor $\Gamma$ of the high-energy source. At the redshift of the afterglow source,
the afterglow photon spectrum is assumed to be a broken power-law:
\begin{equation}
\frac{dN_\gamma}{d\eps} = \left. \frac{dN_\gamma}{d\eps}\right|_{\eps_b} \left\{ \begin{array}{ll}
(\eps/\eps_b)^{-\alpha} & \eps < \eps_b \\ (\eps/\eps_b)^{-\beta} & \eps_b < \eps \\
\end{array} \right.
\label{dNg}
\end{equation}
where $\eps_b$ is the spectral-break energy at redshift $z$ (i.e. the peak of the luminosity $\nu L_\nu$
spectrum), $\alpha$ and $\beta$ being the low and high-energy spectral slopes. For synchrotron and
inverse-Compton emissions, $\alpha$ has four possible values: 2/3 for optically-thin synchrotron (sy)
or inverse-Compton (ic) emission from un-cooled electrons, 3/2 for optically-thin sy/ic from cooled
electrons (i.e. with a radiative cooling timescale shorter than the age of the source), -1 for
self-absorbed synchrotron emission (but is unlikely that the source magnetic field is sufficiently
high for self-absorption to be important at MeV), and 0 for the inverse-Compton scattering of
self-absorbed synchrotron emission.
Then, the spectral slope around 2 measured by LAT above 100 MeV indicates that $\veps_b < 100$ MeV
(in the observer frame), $\beta \simeq 2$, and that the low-energy spectrum is not observed,
being dimmer at 10 keV--10 MeV than the GRB spectrum.
The normalization factor of equation (\ref{dNg}) is simply set by the measured fluence $\Phi$
\begin{displaymath}
\Phi (0.1-10 {\rm GeV}) = \frac{(z+1)^3}{4 \pi d_l^2} \int_{0.1GeV}^{10GeV}
d\veps \, \veps \left.\frac{dN_\gamma}{d\eps}\right|_{(z+1)\veps}
\end{displaymath}
\begin{equation}
= \frac{(z+1)^3}{4 \pi d_l^2} f(\beta) \veps_b^2 \left. \frac{dN_\gamma}{d\eps}\right|_{\eps_b}
\label{phi}
\end{equation}
where $d_l \simeq 5.10^{52} (z+1)^2$ cm is the luminosity distance and
\begin{equation}
f(\beta) = \frac{1}{\beta-2} \left[ \left(\frac{\veps_b}{\rm 0.1 GeV} \right)^{\beta-2} -
\left(\frac{\veps_b}{\rm 10 GeV} \right)^{\beta-2} \right]
\end{equation}
\vspace*{2mm}
\subsection{Peak/break energy of the LAT component}
A lower limit on the observer-frame break-energy $\veps_b$ can be set by requiring that the
0.1--10 keV afterglow emission measured during the X-ray light-curve plateau (at 0.3-10 ks) by Swift/XRT,
of about ${\cal F}_{xrt} \simeq 10^{-10}\fluxcgs$ (O'Brien et al 2006), is not dimmer than the
extrapolation ${\cal F}_{lat}$ of the GeV afterglow spectrum, for an afterglow of GeV fluence
$\Phi = 10^{-5} \ergcm2$ at $t=10$ s, decreasing as $\Phi \propto t^{-0.3}$, and with a high-energy
slope $\beta = 1.1$:
\begin{equation}
{\cal F}_{xrt} > {\cal F}_{lat}= 3.10^{-8} \left( \frac{t}{1\, {\rm ks}} \right)^{-1.3}
\left( \frac{\veps_b}{1\, {\rm keV}} \right)^{\alpha - 1.1} \, \fluxcgs
\end{equation}
For $\alpha = 2/3$, the high-energy spectral component does not overshine the X-ray plateau flux if
$\veps_b (1\, ks) \mathrel{\rlap{\raise 0.511ex \hbox{$>$}}{\lower 0.511ex \hbox{$\sim$}}} 50$ keV. For $\alpha = 3/2$, the high-energy spectral component is dimmer
than the X-ray plateau if $\veps_b (1\, ks) \mathrel{\rlap{\raise 0.511ex \hbox{$>$}}{\lower 0.511ex \hbox{$\sim$}}} 10$ MeV, but could be the X-ray plateau if
$\veps_b \simeq 1 (t/1\, ks)^{-1.7 \pm 0.7}$ MeV.
This evolution is consistent with the $t^{-3/2}$ expected for the peak energy of the forward-shock
synchrotron spectrum, but implies that $\veps_b (10 \,s) \mathrel{\rlap{\raise 0.511ex \hbox{$>$}}{\lower 0.511ex \hbox{$\sim$}}} 1$ GeV during the burst, which is
inconsistent with LAT observations, that do not show a high-energy component peaking in the LAT
window.
However, X-ray plateau measurements are often lacking during the early GeV afterglows monitored
by LAT, thus the above low limits on $\veps_b$ cannot be derived for individual afterglows.
The sub-MeV burst light-curve may also set a constraint on the unknown $\veps_b$ in the following way.
For $\veps_b \simeq 10$ MeV and $\alpha=2/3$, the LAT spectral component yields a 100 keV flux
${\cal F}_{lat} = (\Phi/t_{grb}) (0.1\, {\rm MeV}/\veps_b) ^{2-\alpha}= 2.10^{-9} \Phi_{-5}
t_{grb,1}^{-1} \varepsilon_{b,7}^{-4/3} \fluxcgs$ during a $t_{grb} = 10$ s burst
(using the notation $X_n = X(cgs)/10^n$ and measuring photon energies in eV). This emission is
sufficiently below the typical flux of a bright burst, ${\cal F}_{grb} = 10^{-5} \fluxcgs$, that
it does not overshine a fast-decaying $F_{grb} \propto t^{-(2\div4)}$ tail (O'Brien et al 2006).
In contrast, an energy-break $\veps_b$ that falls below 100 keV will produce a burst emission
${\cal F}_{lat} = (\Phi/t_{grb}) (0.1 {\rm MeV}/\veps_b)^{2-\beta}= 10^{-6} \Phi_{-5} t_{grb,1}^{-1}
\fluxcgs$ (independent of $\veps_b$, for a high-energy LAT spectral slope $\beta \simeq 2$) that
rivals that of the prompt emission.
Thus, a bright LAT afterglow following a slowly-fading GRB {\sl may} have a break-energy $\veps_b
< 100$ keV, but one following a burst with a steep decay {\sl must} satisfy $\veps_b \gg 1$ MeV
during the burst tail.
\subsection{Optical thickness to photon-photon absorption}
\begin{figure*}
\centerline{\psfig{figure=tau.eps,height=80mm}}
\figcaption{ Optical-thickness to pair-formation as function of observer-frame
photon energy for a source of high-energy photons with the indicated parameters (fluence in cgs units)
and for three values of the break energy $\veps_b$ .}
\end{figure*}
For an isotropic distribution of photons in the frame of the shocked fluid, which moves at Lorentz factor
$\Gamma$ in the lab-frame (at redshift $z$), the optical thickness to a photon of energy $\eps'_o$ is
\begin{equation}
\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z} (\eps'_o) = \frac{\sigma_e}{4 \pi R^2} \int_0^\pi d\theta' \frac{\sin \theta'}{2}
\int_{\eps'_{th}}^\infty dN_\gamma(\eps') \fgg (\eps'_o\eps',\theta')
\label{taugg}
\end{equation}
where primed quantities are in the shock-frame, $\sigma_e$ is the Thomson cross-section for electron scattering,
\begin{equation}
R (t) \simeq (z+1)^{-1} ct\Gamma^2
\label{R}
\end{equation}
is the source (shock) radius at observer-time $t$ (corresponding to the arrival-time of photons emitted
by the visible edge of the source; equality in equation above holds for an undecelerated source; a factor
4/3 applies to the right-hand side for a blast-wave decelerated by a wind-like medium),
$\theta'$ is the angle of incidence between the test-photon
of energy $\eps'_o$ and a target-photon $\eps'$, $dP/d\theta' = (1/2) \sin \theta'$ is the probability of
two photons interacting at an angle $\theta'$
\begin{equation}
\eps'_{th} (\eps'_o,\theta') = \frac{2 (m_e c^2)^2}{(1-\cos \theta') \eps'_o}
\label{ethr}
\end{equation}
is the threshold-energy for pair-formation, and
\begin{displaymath}
\fgg (\eps'_o\eps',\theta') = \frac{3}{8x^2} \left[ \left( 2 + 2x^{-2} - x^{-4}\right) \ln \left(x + \sqrt{x^2-1}
\right) \right.
\end{displaymath}
\begin{equation}
- \left. \left( 1 + x^{-2} \right) \sqrt{1-x^{-2}}\right] = \frac{\sigma_{\gamma\gamma}}{\sigma_e}
\label{fgg}
\end{equation}
is the cross-section for photon-photon absorption, with
\begin{equation}
x = \sqrt{\frac{1}{2} \frac{\eps'_o \eps'}{m_e^2 c^4} (1-\cos \theta')} =
\sqrt{ \frac{\eps'}{\eps'_{th}(\eps'_o,\theta')} } \geq 1 \;
{\rm for} \; \eps' \geq \eps'_{th}
\label{obs}
\end{equation}
The integral in equation (\ref{taugg}) is calculated numerically; Figure 1 shows $\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z} (\veps)$ for
a photon of observer-frame energy
\begin{equation}
\veps = \frac{\eps}{z+1}=\frac{\Gamma \eps'}{z+1}
\label{zz}
\end{equation}
corresponding to the typical relativistic boost ($\Gamma$) of a photon of shock-frame energy $\eps'$.
To extract the dependence of $\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\veps)$ on the source parameters $\Gamma$, $\Phi$, $z$ and observer
time $t$, an approximation to equation (\ref{taugg}) is needed.
Extending the approximation $\fgg (x) = (3/4) \ln (2x)/x^2$, accurate for $x \gg 1$, to all $x \geq 1$,
allows the second integral in equation (\ref{taugg}) to be calculated analytically, but the resulting
integral over the incidence angle $\theta'$ is not so nice.
The dependence of $\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}$ on source parameters can be obtained by setting $\fgg (x) = const$ and by
approximating
\begin{equation}
\int_{\eps'_{th}}^\infty dN_\gamma(\eps') \simeq \left.\frac{dN_\gamma}{d\eps}\right|_{\eps'_{th}} \eps'_{th}
\end{equation}
which leads to an integral over $\theta'$ that can be calculated easily.
Dropping the integral over $\theta'$ and assuming that most pairs form at threshold are further
simplifications that lead to the correct dependence of $\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}$ on source parameters:
\begin{equation}
\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z} (\eps) \simeq \frac{\sigma_e}{4 \pi R^2} \left.\frac{dN_\gamma}{d\eps}\right|_{\eps_{th}} \eps_{th}(\eps)
\end{equation}
where
\begin{equation}
\eps_{th} (\eps) = 4 \Gamma^2 \frac{(m_e c^2)^2}{\eps}
\end{equation}
is the lab-frame threshold-energy for a lab-frame incidence angle $\theta = \Gamma^{-1} \ll 1$ (the source
motion at $\Gamma$ collimates photons within an angle $\Gamma^{-1}$ around the direction of motion).
Using equations (\ref{dNg}), (\ref{phi}), and (\ref{phi}), one arrives at
\begin{equation}
\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z} (\veps) \propto \frac{\Phi}{t^2} \left\{ \begin{array}{ll}
\left(\frac{\displaystyle z+1}{\displaystyle \Gamma}\right)^{2\beta+2} \veps_b^{\beta-2} \veps^{\beta-1}
& \veps_b < \veps_{th}(\veps) \\
\left(\frac{\displaystyle z+1}{\displaystyle \Gamma}\right)^{2\alpha+2}
\veps_b^{-(2-\alpha)} \veps^{-(1-\alpha)} & \veps_{th}(\veps) < \veps_b \\ \end{array} \right.
\label{tau1}
\end{equation}
having switched to observer-frame photon energies (equation \ref{zz}).
Using equation (\ref{ethr}), the conditions above become
\begin{displaymath}
\left\{ \begin{array}{lll}
\veps_b < \veps_{th}(\veps) & \rightarrow & \veps < \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma} \\
\veps_{th}(\veps) < \veps_b & \rightarrow & \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma} < \veps \\
\end{array} \right.
\end{displaymath}
with $\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma} \equiv [4\Gamma^2 (m_e c^2)^2]/[(z+1)^2 \veps_b]=4.6\,\Z^{-2}\Gamma_{2.3}^2 \veps_{b,6}^{-1}$
GeV, where $\Z \equiv (z+1)/3$.
Equation (\ref{tau1}) shows the obvious fact that $\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}$ is proportional to the photon column density
($N_\gamma/R^2 \propto \Phi/t^2$), and that it has a strong dependence on the source Lorentz factor.
Given that $\alpha \geq 0$ and $\beta \mathrel{\rlap{\raise 0.511ex \hbox{$>$}}{\lower 0.511ex \hbox{$\sim$}}} 2$, equation (\ref{tau1}) also shows that $\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z} (\veps)$ increases
with photon energy for $\veps < \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}$ and decreases with it for $\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma} < \veps$, with the maximal optical
thickness reached at $\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}$.
The coefficients missing in equation (\ref{tau1}) depend on the photon spectrum slopes $\alpha$ and $\beta$.
Figure 1 shows optical thickness for the representative values $\alpha = 2/3$ and $\beta = 2$, obtained
numerically by integrating equation (\ref{taugg}). In the asymptotic power-law regimes, the numerical
approximation is
\begin{equation}
\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z} (\veps) = \left\{ \begin{array}{ll}
0.57 \, \frac{\displaystyle \Z^6 \Phi_{-5}}{\displaystyle \Gamma_{2.3}^6 t_1^2} \,\veps_9 & \veps < \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma} \\
\\ 2.6 \, \frac{\displaystyle \Z^{10/3} \Phi_{-5}}{\displaystyle \Gamma_{2.3}^{10/3} t_1^2 \veps_6^{4/3}}
\,\veps_{11}^{-1/3} & \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma} < \veps \\
\end{array} \right.
\label{tau2}
\end{equation}
The two branches above intersect at
\begin{equation}
\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma} \equiv 10 \, \Z^{-2} \frac{\Gamma_{2.3}^2}{\veps_{b,6}} \; {\rm GeV}
\label{etilde}
\end{equation}
where the optical thickness is maximal:
\begin{equation}
\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}) = 5.7\, \frac{\Z^4 \Phi_{-5}}{\Gamma_{2.3}^4 t_1^2 \veps_{b,6}}
\label{taumax}
\end{equation}
Equation (\ref{tau2}) can now be written as
\begin{equation}
\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z} (\veps) = \tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z} (\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}) \left\{ \begin{array}{ll}
\veps/\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma} & \veps < \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma} \\ (\veps/\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma})^{-1/3} & \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma} < \veps \\
\end{array} \right.
\label{tau3}
\end{equation}
From here, it follows that
\begin{equation}
\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\veps) < 1 \quad {\rm if} \quad \veps_b > \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}_b \equiv
5.7\, \frac{\Z^4 \Phi_{-5}}{\Gamma_{2.3}^4 t_1^2} \, {\rm MeV}
\label{taulow}
\end{equation}
and
\begin{equation}
\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\veps) = \left\{ \begin{array}{ll}
< 1 \;, & \veps < \veps_- \\ > 1 \;, & \veps_- < \veps < \veps_+ \\ < 1 \;, & \veps_+ < \veps \\
\end{array} \right. \quad {\rm if} \quad \veps_b < \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}_b
\label{taucomp}
\end{equation}
where
\begin{equation}
\veps_- \equiv 1.75\, \frac{\Gamma_{2.3}^6 t_1^2}{\Z^6 \Phi_{-5}} \; {\rm GeV} \;,
\veps_+ \equiv 1.82\, \frac{\Z^{10}\Phi_{-5}^3}{\Gamma_{2.3}^{10} t_1^6 \veps_{b,6}^4} \; {\rm TeV}
\label{epm}
\end{equation}
For $\beta = 2$, the optical-thickness to photon-photon absorption is independent of $\veps_b$ for
photons of energy lower than $\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}$ (i.e. photons with threshold energy above $\veps_b$), hence
the lower limit $\veps_-$ above which the photon-front is optically-thick is also independent of
$\veps_b$, as illustrated in Figure 1.
Equation (\ref{taulow}) can be reinterpreted as
\begin{equation}
\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\veps) < 1 \quad {\rm if} \quad \Gamma > \Gamma_\pm \equiv
310\, \frac{\Z \Phi_{-5}^{1/4}}{t_1^{1/2}\veps_{b,6}^{1/4}}
\end{equation}
For $\Gamma < \Gamma_\pm$, the photon-front is optically-thick to photons in the $(\veps_-,\veps_+)$
range, which widens with decreasing $\Gamma$ (see equation \ref{epm}), covering the entire LAT window
if $\veps_- \mathrel{\rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}} 100$ MeV, which is equivalent to $\Gamma \mathrel{\rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}} \Gamma_{lat} \equiv 125\, \Z \Phi_{-5}^{1/6}
t_1^{-1/3}$. For such low Lorentz factors, the LAT emission is heavily absorbed and the afterglow high-energy
emission undetectable. At the other extreme, if $\Gamma \mathrel{\rlap{\raise 0.511ex \hbox{$>$}}{\lower 0.511ex \hbox{$\sim$}}} 2.2\, \Gamma_{lat}$, then $\veps_- \mathrel{\rlap{\raise 0.511ex \hbox{$>$}}{\lower 0.511ex \hbox{$\sim$}}} 10$ GeV
and the LAT emission is weakly absorbed. For $\Gamma_{lat} < \Gamma < 2.2\, \Gamma_{lat}$, the LAT emission
is moderately absorbed, photon-photon absorption rendering a spectrum that curves downward at higher
energies, for a power-law intrinsic spectrum (e.g. figure 2 of Panaitescu et al 2014).
Thus, perfect power-law LAT spectra set only a lower limit on the source Lorentz factor: $\Gamma \mathrel{\rlap{\raise 0.511ex \hbox{$>$}}{\lower 0.511ex \hbox{$\sim$}}} 300\,
\Phi_{-5}^{1/6} t_1^{-1/3}$. Such a weak dependence on the afterglow fluence $\Phi$ and epoch $t$ of
observations suggests that the measurement of curvature in the LAT spectrum would yield a fairly accurate
determination of $\Gamma$. Obviously, the non-detection of the high-energy afterglow emission is not
necessarily proof of high absorption and does not tell us anything about $\Gamma$.
\vspace*{2mm}
\subsection{Total number of pairs}
\label{number}
The total number of pairs is an integral over the photon spectrum of the absorbed fraction
$g[\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\veps)$
\begin{equation}
N_\pm = \int_0^\infty d\veps \frac{dN_\gamma}{d\veps} g[\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\veps)]
\label{N0}
\end{equation}
with the photon spectrum of equation (\ref{dNg}).
To calculate the fraction of absorbed photons corresponding to the optical-thickness to pair-formation
$\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\veps)$ (equation \ref{tau2}), consider a medium of geometrical thickness $\Delta$ and linear
absorption coefficient $\alpha$, and in which the production and absorption of photons is homogeneous
(same at any location).Then, the fraction of photons that are absorbed is
\begin{equation}
g = \int_0^\Delta \frac{dx}{\Delta} (1 - e^{-\alpha (\Delta - x)}) = 1 - \frac{1-e^{-\tau}}{\tau}
\simeq \left\{ \begin{array}{ll} \tau/2 & \tau \ll 1 \\ 1 & \tau \gg 1 \\ \end{array} \right.
\label{g}
\end{equation}
obtained by integrating the photon absorption from the medium inner edge ($x=0$) to its outer boundary
($x=\Delta$), and with $\tau = \alpha \Delta$.
In the case of pair production in a decelerating source, the photons radial distribution is not uniform.
In this case, the fraction of absorbed photons is
\begin{equation}
g = \int_0^1 dy \frac{dn_\gamma}{dy}(1 - e^{-\tau(y \rightarrow 1)}) \;,
\label{gg}
\end{equation}
where $dn_\gamma/dy$ is radial distribution of photons normalized by $\int_0^1 dy (dn_\gamma/dy) =1$ and
\begin{equation}
\tau (y \rightarrow 1) = \int_y^1 \alpha(z) dz = \tau (0 \rightarrow 1) \int_y^1 \frac{dn_\gamma}{dz} dz
\end{equation}
is the absorption optical thickness from coordinate $y = x/\Delta$ to the outer edge at $y=1$,
$\tau (0 \rightarrow 1) \equiv \tau$ being the entire optical thickness of the medium.
Substituting in equation (\ref{gg}), we get
\begin{displaymath}
g = \int_0^1 \frac{dn_\gamma}{dy} dy -
\int_0^1 dy \frac{dn_\gamma}{dy} \exp\left( -\tau \int_y^1 \frac{dn_\gamma}{dz} dz \right)
\end{displaymath}
\begin{equation}
= 1 - \frac{1}{\tau} \int_0^1 dy \frac{d}{dy} \left[
\exp\left( -\tau \int_y^1 \frac{dn_\gamma}{dz} dz \right) \right]
\end{equation}
\begin{displaymath}
= 1 - \frac{1}{\tau} \left[ 1- \exp\left( -\tau \int_0^1 \frac{dn_\gamma}{dy} dy \right) \right]
= 1 - \frac{1-e^{-\tau}}{\tau}
\end{displaymath}
Therefore, as long as the absorption coefficient $\alpha$ is proportional to the density of the
to-be-absorbed photons, the fraction of absorbed photons from a medium depends only on the optical
thickness $\tau = \int \alpha(z) dz$ of that medium and does not "care" about the exact spatial
variation of $\alpha$.
The integral in equation (\ref{N0}) is calculated numerically; for an analytical estimate, we use the
approximation given in equation (\ref{g}).
{\bf Case 1.}
For $\veps_b > \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}_b/2$ (equation \ref{taulow}), the maximal optical thickness (equation \ref{taumax})
satisfies $\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}) < 2$, hence $\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z} (\veps) < 2$ for any photon. With two branches for the photon
spectrum (equation \ref{dNg}) and two for the optical thickness (equation \ref{tau1}), the integral in equation
(\ref{N0}) splits in three integrals
\begin{displaymath}
N_\pm = \int_\veps dN_\gamma \frac{\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\veps)}{2}
\end{displaymath}
\begin{equation}
= \frac{1}{2} \left.\frac{dN_\gamma}{d\veps}\right|_{\veps_b} \tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z} (\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma})
\left[ \int_0^{\veps_b} d\veps \left(\frac{\veps}{\veps_b}\right)^{-\alpha}
\left(\frac{\veps}{\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}}\right)^{\beta-1} + \right.
\label{int1}
\end{equation}
\begin{displaymath}
\left. + \int_{\veps_b}^{\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}} d\veps \left(\frac{\veps}{\veps_b}\right)^{-\beta}
\left(\frac{\veps}{\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}}\right)^{\beta-1}
+ \int_{\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}}^{\infty} d\veps \left(\frac{\veps}{\veps_b}\right)^{-\beta}
\left(\frac{\veps}{\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}}\right)^{\alpha-1} \right]
\end{displaymath}
for the more likely case $\veps_b < \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}$. Using equation (\ref{taumax}), this condition requires
that $\veps_b < 100\, \Z^{-1}\Gamma_{2.3}$ MeV, which is satisfied by LAT spectra and which,
together with the working condition $\veps_b > \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}_b/2$, requires that $\Gamma > \tGamma$ where
\begin{equation}
\tGamma \equiv 100\, \Z \frac{\Phi_{-5}^{1/5}}{t_1^{2/5}}
\label{tGam}
\end{equation}
The scaling of the integrals in equation (\ref{int1}) with $\veps$ is $\veps^{\beta-\alpha}$,
$\ln \veps$, and $\veps^{-(\beta-\alpha)}$, respectively; taking into account that $\beta > \alpha$,
this implies that most pairs are formed from (the second integral, corresponding to) photons with
energy above the spectral break $\veps_b > \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}_b/2 \sim$ few MeV and below the energy for maximal
optical thickness $\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma} \mathrel{\rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}}$ 10 GeV, interacting with photons above threshold energies of about
1 GeV and 1 MeV, respectively.
For $\alpha = 2/3$ and $\beta = 2$, equation (\ref{int1}) yields
\begin{equation}
N_\pm \stackrel{2\veps_b > \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}_b}{=} 10^{55} \left( 0.67 + 0.22 \ln \frac{\Gamma_{2.3}}{\Z\veps_{b,7}}
\right) \Z^8 \frac{\Phi_{-5}^2}{t_1^2 \Gamma_{2.3}^6}
\label{N1}
\end{equation}
Using equation (\ref{R}), the optical-thickness to photon scattering of the pairs is
\begin{equation}
\tau_\pm = \frac{2 \sigma_e N_\pm}{4 \pi R^2} = 0.024 \left( 1 + 0.33 \ln \frac{\Gamma_{2.3}}{\Z\veps_{b,7}}
\right) \Z^{10} \frac{\Phi_{-5}^2}{t_1^4 \Gamma_{2.3}^{10}}
\label{taupairs}
\end{equation}
Ignoring the logarithmic term, this implies that the pairs are optically thin (to photon scattering in the
Thomson regime, because most pairs are cold) for $\Gamma > \Gamma_\tau$ with
\begin{equation}
\Gamma_\tau \equiv 138\, \Z \Phi_{-5}^{1/5} t_1^{-2/5}
\label{Gtau}
\end{equation}
{\bf Case 2.}
For $\veps_b < \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}_b/2$, $\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\veps)$ is relative to 2 as in equation (\ref{taucomp}) but with
$\veps_-$ larger by a factor $2^{1/(\beta-1)}$ than in equation (\ref{epm}) and $\veps_+$ smaller by a
factor $2^{1/(1-\alpha)}$ than in (\ref{epm}). Having two branches for the photon spectrum and three for
the optical thickness, the integral of equation (\ref{N0}) splits in four:
\begin{displaymath}
N_\pm = \int_\veps dN_\gamma \min \left\{\frac{\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\veps)}{2},1\right\}
= \left. \frac{dN_\gamma}{d\veps}\right|_{\veps_b} \times
\end{displaymath}
\begin{displaymath}
\left[ \frac{1}{2}\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z} (\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}) \int_0^{\veps_b} d\veps \left(\frac{\veps}{\veps_b}\right)^{-\alpha}
\left( \frac{\veps}{\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}} \right)^{\beta -1}+ \right.
\end{displaymath}
\begin{equation}
\frac{1}{2} \tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z} (\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}) \int_{\veps_b}^{\veps_-} d\veps \left(\frac{\veps}{\veps_b}\right)^{-\beta}
\left( \frac{\veps}{\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}} \right)^{\beta -1} +
\label{int2}
\end{equation}
\begin{displaymath}
\int_{\veps_-}^{\veps_+} d\veps \left(\frac{\veps}{\veps_b}\right)^{-\beta} +
\left. \frac{1}{2} \tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z} (\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}) \int_{\veps_+}^{\infty} d\veps \left(\frac{\veps}{\veps_b}\right)^{-\beta}
\left( \frac{\veps}{\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}} \right)^{\alpha -1} \right]
\end{displaymath}
for the more likely case $\veps_b < \veps_-$ (for $\Gamma > \tGamma$, this is implied by the working
condition $\veps_b < \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}_b/2$).
The integrals in equation (\ref{int2}) show that most pairs form from (the second integral, corresponding
to) photons above the spectral break $\veps_b < \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}_b/2 \sim 3$ MeV and below $\veps_- \sim 1$ GeV,
for which the photon front is optically thin, interacting with photons above threshold energies $> 1$ GeV
and few MeV, respectively.
For $\alpha = 2/3$ and $\beta =2$, equation (\ref{int2}) leads to
\begin{equation}
N_\pm \stackrel{2\veps_b < \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}_b}{=} 10^{55} \left( 1.1 + 0.11 \ln \frac{\Gamma_{2.3}^6 t_1^2}{\Z^6 \Phi_{5}
\veps_{b,6}} \right) \Z^8 \frac{\Phi_{-5}^2}{t_1^2 \Gamma_{2.3}^6}
\label{N2}
\end{equation}
After calculating the pair optical thickness to photon scattering as done above for $\veps_b > \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}_b/2$,
it can be shown that the minimal Lorentz factor for optical-thinness is close to that in equation (\ref{Gtau}).
Equations (\ref{N1}) and (\ref{N2}) show that the number of pairs formed has a weak dependence on the
(unknown) break $\veps_b$ of the photon spectrum, varies like $\Phi^2$ (as expected for a two-photon
interaction), and has a strong dependence on the source Lorentz factor, resulting in part from the
dependence of the threshold energy for pair formation on $\Gamma$ and in part from the decrease
of the photon density with source radius (which is proportional to $\Gamma^2$).
\vspace*{2mm}
\subsection{Scattering of afterglow photons on internal leptons}
It is worth comparing the number of pairs with that of electrons existing in the two possible
source of GeV afterglow photons, the forward and reverse shocks.
For a Wolf-Rayet GRB progenitor with a mass-loss rate $dM/dt = 10^{-5}\, M_\odot/{\rm yr}$, blowing a wind
of terminal velocity $v_w=10^3 \; {\rm km/s}$, the wind baryon density is
\begin{equation}
n(R) = \frac{dM/dt}{4\pi m_p v_w R^2} = \frac{3.0 \times 10^{35}}{R^2} \; {\rm cm^{-3}}
\label{WR}
\end{equation}
The ratio of the number of formed leptons $N_l = 2 N_\pm$ (eqs \ref{N1} and \ref{N2}) to the electrons
energized by the forward-shock is
\begin{equation}
\frac{N_l}{N_{fs}} = \frac{N_\pm}{\pi n R^3} \simeq 2600 \frac{\Z^9 \Phi_{-5}^2}{t_1^3 \Gamma_{2.3}^8}
\label{ratio}
\end{equation}
taking into account that the wind-like medium is made of elements heavier than hydrogen (with one
electron for two baryons). Thus, for $\Gamma < \Gamma_{fs} \equiv 540\, \Z^{9/8} \Phi_{-5}^{1/4} t_1^{-3/8}$,
the pairs are more numerous than the forward-shock electrons.
If we assume that $i)$ the reverse and forward shock baryons contain about the same (kinetic plus thermal)
energy and $ii)$ the ejecta are normal matter (with one electron per baryon), then the number of ejecta
electrons is at most $\Gamma$ times larger than that of the forward-shock's (as in the case of a
semi-relativistic reverse-shock): $N_{rs} \mathrel{\rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}} \Gamma N_{fs}$. Then, the number of pairs exceeds
that of the ejecta electrons for $\Gamma \mathrel{\rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}} \Gamma_{rs} \equiv 270\, \Z \Phi_{-5}^{2/9} t_1^{-1/3}$.
The above suggest that emission from pairs is of importance for GeV afterglow sources with a Lorentz
factor in the few hundreds, but pairs may radiate at a different energy than the reverse or forward-shock
electrons, where the pairs could dominate the afterglow emission even if they are fewer.
It is also worth investigating if scattering of pair-forming photons on existing (reverse and
forward-shock) electrons or on the already-formed pairs could change significantly the number of pairs
formed from unscattered photons. Most photon scattering occurs on leptons that are cold. That is
certainly the case for the pairs, most of which are born cold (see the distribution of formed
pairs with energy in Figure 2, left panel), and is likely true for the (reverse-shock) ejecta electrons
and the ambient medium electrons (swept-up by the forward-shock) because they should be cooling fast
radiatively if their synchrotron emission were to account for the observed GeV afterglow.
For a scattering optical thickness $\tau_{sc}$, the effective photon-photon attenuation thickness
is $\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z} = \sqrt{\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z} + \tau_{sc})}$, therefore scattering on cold leptons is negligible
when $\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z} (\veps) > \tau_{sc}$. As shown in \S\ref{number}, most pairs form from photons with energy
$\veps < \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}$, for which $\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z} (\veps)$ is that in the first branch of equation (\ref{tau2}).
The optical thickness to Thomson scattering on reverse-shock electrons (which are more numerous than
in the forward-shock) is $\tau_{sc} \mathrel{\rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}} \sigma_e (\Gamma N_{fs})/(4 \pi R^2) \mathrel{\rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}} 5.10^{-3}
\Z/(\Gamma_{2.3} t_1)$. Thus, $\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z} (\veps) > \tau_{sc}$ is satisfied for
$\veps \mathrel{\rlap{\raise 0.511ex \hbox{$>$}}{\lower 0.511ex \hbox{$\sim$}}} \veps_{rs} \equiv 10 (\Gamma_{2.3}/\Z)^5 (t_1/\Phi_{-5})$ MeV.
The optical thickness $\tau_\pm$ to Thomson scattering on (already formed) pairs is that given in equation
(\ref{taupairs}), thus $\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\veps) > \tau_\pm$ is satisfied for $\veps > \veps_{pair}^{(Th)} \equiv 50
(\Z/\Gamma_{2.3})^4 (\Phi_{-5}/t_1^2)$ MeV. Scattering on cold electrons of observer-frame photons with
energy above $\veps_{kn} \equiv \Gamma m_e c^2/(z+1) = 34 \Z^{-1} \Gamma_{2.3}$ MeV occurs in the
Klein-Nishina regime. Given that $\veps_{pair}^{(Th)} \mathrel{\rlap{\raise 0.511ex \hbox{$>$}}{\lower 0.511ex \hbox{$\sim$}}} \veps_{kn}$, it is worth considering scattering
on pairs in the KN regime, when the scattering cross-section $\sigma_{kn} (\veps) \simeq (3/8) \sigma_e
\ln(2x)/x$ with $x = \veps/\veps_{kn}$. In this case, the condition $\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z} (\veps) > \tau_\pm (\veps)$
is satisfied for $\veps > \veps_{pair}^{(kn)} \equiv 25 (\Z/\Gamma_{2.3})^{3/2} (\Phi_{-5}^{1/2}/t_1)$ MeV.
To the above identification of the photon energies for which scattering increases significantly the photon
escape path and attenuation, we add that, according to equations (\ref{int1}) and (\ref{int2}), most pairs
are formed from photons with energy in the range $(\veps_b,\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma})$ (for $\veps_b > \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}_b/2$) or
$(\veps_b,\veps_-)$ (for $\veps_b < \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}_b/2$). This means that scattering is important for photons in
the lower part of those energy intervals and not so important in the upper part.
Equations (\ref{int1}) and (\ref{int2}) also show that each decade of photon energy provides an equal
contribution to the number of pairs. Therefore, scattering has a negligible effect on the number of pairs
if the logarithmic length of the upper interval is larger than that of the lower interval.
Using the expressions for $\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}$, $\veps_-$, $\veps_{rs}$, and $\veps_{pair}^{(kn)}$,
it can be shown that scattering on pairs should not increase much the number of pairs if
$\Gamma > 120$ (for $\veps_b > \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}_b$) and $\Gamma > 140$ (for $\veps_b < \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}_b$), while
scattering on reverse-shock electrons is negligible if $\Gamma < 370$ (for $\veps_b > \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}_b$) and
$\Gamma < 780$ (for $\veps_b < \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}_b$), having left out sub-unity powers of the parameters $\Z,\Phi,t$.
Adding to these that the reverse-shock electrons are more numerous than the pairs if $\Gamma > \Gamma_{rs}
= 270$, it follows that scattering of the pair-forming photons below the LAT range increases the total
number of pairs only through scattering by reverse-shock electrons and only if $\Gamma$ is at least several
hundreds.
\vspace*{2mm}
\subsection{Pair distribution with energy}
The shock-frame energy of a pair $\eps'_p$ depends on the energies of the incident photons, test-photon
$\eps'_o$ and target-photon $\eps'$, the incidence angle $\theta'$, and the center-of-momentum (CoM)
frame angle $\phi''$ at which the electron and positron emerge, measured relative to the direction of
motion of the photons' CoM. In the shocked-fluid frame, the CoM moves at velocity $\vec{\beta'}_{cm} =
(\vec{\eps'}_o + \vec{\eps'})/ (\eps'_o+\eps')$, the corresponding Lorentz factor being
\begin{equation}
\Gamma'_{cm} = \frac{\eps'_o +\eps'}{\sqrt{2 \eps'_o \eps' (1-\cos \theta')}}
\end{equation}
In the CoM frame, the incident photons have the same energy
\begin{equation}
\eps'' = \left[ \frac{1}{2} \eps'_o \eps' (1-\cos \theta') \right]^{1/2} =
\left[ \frac{\eps'}{\eps'_{th}(\eps'_o,\theta')} \right]^{1/2}
\end{equation}
collide head-on, and form an electron and a positron of equal energy $\eps''$, moving in opposite
directions, at angles $\phi''$ and $\pi - \phi''$ relative to $\vec{\beta'}_{cm}$. Then, the shock-frame
electron and positron energies are
\begin{equation}
\eps'_\pm = \Gamma'_{cm} (\eps'' \pm p''c \beta'_{cm} \cos \phi'')
\label{epm1}
\end{equation}
where $p''c = \sqrt{\eps''^2 - m_e^2 c^4}$ is the electron/positron momentum in the CoM frame.
To obtain the distribution of formed leptons with their shock-frame energy, $dN_l/d\eps'_\pm$,
we start from equation (\ref{N0}) in shock-frame photon energy $\eps_o'$
\begin{equation}
\frac{dN_l}{d\eps'_o} = 2 \frac{dN_\gamma}{d\eps'_o} g[\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\eps'_o)]
= \frac{dN_\gamma}{d\eps'_o} \left\{ \begin{array}{ll}
\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\eps'_o) & \tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\eps'_o) \ll 1 \\ 2 & \tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\eps'_o) \gg 1 \\ \end{array} \right.
\label{cici}
\end{equation}
where the factor 2 accounts for two leptons being created from one photon.
The term $\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\eps'_o)$ of the first branch offers a way to calculate $dN_l/d\eps'_\pm$, by expanding
it (as in equation \ref{taugg}), leading to the differential pair-number in the 4-dimensional space
$[\eps'_o,\eps',\theta',\phi'']$
\begin{equation}
\frac{d^4N_l}{d\eps'_o d\Omega' d\eps' d\Omega''} =
\frac{\sigma_e}{4\pi R^2} \frac{dN_\gamma}{d\eps'_o} \frac{dP}{d\Omega'}
\frac{dN_\gamma}{d\eps'} \frac{dP}{d\Omega''} \fgg (\eps'_o\eps',\theta')
\label{num0}
\end{equation}
where $dP/d\Omega'' = \sin \phi''/2$ (because pairs emerge isotropically in the CoM frame), with $\fgg$
the photon-photon absorption cross-section of equation (\ref{fgg}).
For an isotropic distribution of the incident photons ($dP/d\Omega' = \sin \theta'/2$)
\begin{equation}
\frac{d^4N_l}{d\eps'_o d\theta' d\eps' d(\cos \phi'')} =
\frac{\sigma_e}{16\pi R^2} \frac{dN_\gamma}{d\eps'_o} \sin \theta'
\frac{dN_\gamma}{d\eps'} \fgg (\eps'_o\eps',\theta')
\label{num1}
\end{equation}
From equation (\ref{epm1}), $d\eps'_\pm = \pm \Gamma'_{cm}\beta'_{cm}p''c (d\cos \phi'')$, thus
\begin{displaymath}
\frac{d^4N_l}{d\eps'_o d\theta' d\eps' d\eps'_\pm} =
\frac{d^4N_l}{d\eps'_o d\theta' d\eps' d(\cos \phi'')} \left| \frac{d\cos \phi''}{d\eps'_\pm} \right|
\end{displaymath}
\begin{equation}
= \frac{\sigma_e}{16\pi R^2} \frac{dN_\gamma}{d\eps'_o} \sin \theta'
\frac{dN_\gamma}{d\eps'} \frac{\fgg (\eps'_o\eps',\theta')}{\Gamma'_{cm}\beta'_{cm}p''c}
\end{equation}
Then, the distribution of leptons with energy is the integral of the differential pair-number above
over the energies of the incident photons and over the incidence angle
\begin{displaymath}
\frac{dN_l}{d\eps'_\pm} = \frac{\sigma_e}{16\pi R^2}
\int_0^\infty d\eps'_o \frac{dN_\gamma}{d\eps'_o} \int_0^\pi d\theta' \sin \theta'
\end{displaymath}
\begin{equation}
\int_{\eps'_{min}}^{\eps'_{max}} d\eps' \frac{dN_\gamma}{d\eps'}
\frac{\fgg (\eps'_o\eps',\theta')}{\sqrt{(\Gamma_{cm}^{'2} - 1)(\eps''^2 - m_e^2 c^4)}}
\label{num2}
\end{equation}
where the limits $\eps'_{min} (\eps'_\pm)$ and $\eps'_{max} (\eps'_\pm)$ on the integral over the
spectrum of target photons are determined from
\begin{equation}
\Gamma'_{cm} (\eps'' - p''c \beta'_{cm}) \leq \eps'_\pm \leq \Gamma'_{cm} (\eps'' + p''c \beta'_{cm})
\end{equation}
together with $\eps' \geq \eps'_{th} (\eps'_o,\theta')$.
Unfortunately, the term $\Gamma'_{cm}\beta'_{cm}p''$ leads to a fourth degree equation in $\eps'$ that
cannot be solved analytically to obtain the integral limits $\eps'_{min} (\eps'_\pm)$ and $\eps'_{max}
(\eps'_\pm)$.
\begin{figure*}
\centerline{\psfig{figure=pdist.eps,width=180mm}}
\figcaption{ Left panel: distribution with energy (multiplied by $\gamma^\beta$, to show where
the power-law $dN_l/d\gamma \propto \gamma^{-\beta}$ branch occurs) of the pairs formed in a
relativistic source with same parameters as for Fig 1, calculated numerically by integrating equation
(\ref{num1}).
Right panel: comparison between the derivative $d\log (dN_l/d\gamma)/d\log \gamma$ of the
numerical pair distribution and the expected power-law slopes (equations \ref{dN1} and \ref{dN2}).
For $\veps_b > \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}_b$ (red line), the front is optically thin for any photon and the lowest-energy
expected branch is not developed. For $\veps_b \ll \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}_b$, all three branches are seen.}
\end{figure*}
Those limits can be calculated numerically and used to integrate equation (\ref{num2}), with the
following two corrections. First, a multiplicative factor $g[\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\eps'_o)]/ \tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\eps'_o)$
should be applied to the $\eps'_o$ integrand, to account for the correct absorption fraction
$g(\veps'_o)$. That ensures that no more than $dN_\gamma (\eps'_o)$ photons form pairs. Second,
a multiplicative factor $\min \{1, N_\gamma [>\eps'_{th}(\eps'_o)]/[N_\gamma (>\eps'_o)
g(\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\eps'_o))] \}$ ensures that the number of absorbed test photons $N_\gamma (>\eps'_o)
g[\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\eps'_o)]$ does not exceed the number of target photons $N_\gamma [>\eps'_{th}(\eps'_o)]$
above the threshold for pair-formation.
The slopes of the pair distribution with energy $dN_l/d\eps'_\pm$ that results from
integrating equation (\ref{num1}) can be inferred if the crude approximation $\eps'_\pm \simeq \eps'_o/2$
is made. This approximation is suggested by that pairs emerge most likely at an CoM-frame angle
$\phi'' = \pi/2$, hence $\eps'_\pm = \Gamma'_{cm} \eps'' = (\eps'_o + \eps')/2$ (from equation
\ref{epm1}), and by that most pairs are formed from a test-photon of energy $\eps'_o$ larger than
the $\eps'$ of the target-photon. The latter is suggested by that most pairs are formed in interactions
with target-photons close to (but above) the threshold for pair-formation (the integrand in equation
\ref{taugg} shows that $\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z} \propto \eps' (dN_\gamma/d\eps') \fgg (\eps') \propto (\eps')^{1-\alpha}
\ln (2x)/x^2$ with $x^2 \propto \eps'_o \eps'$, thus $\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z} \propto (\eps')^{-\alpha}$ with $\alpha > 0$)
and by that $\eps'_o \gg \eps'_{th}(\eps'_o)$ (optical thickness $\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\veps)$ is maximal at photon
energy $\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma} \gg \veps_b = \veps_{th} (\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma})$ -- equation \ref{taumax}).
The approximation $\eps'_\pm \simeq \eps'_o/2$ implies that $dN_l/d\eps'_\pm \propto dN_\gamma/d\eps'_\pm$.
Then, equation (\ref{cici}) leads to
\begin{equation}
\frac{dN_l}{d\eps'_\pm} \propto \frac{dN_\gamma}{d\eps'_\pm} g[\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\eps'_\pm)] \simeq
\frac{dN_\gamma}{d\eps'_\pm} \min \left\{1,\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\eps'_\pm)\right\}
\end{equation}
with an approximation for the absorption factor $g$ that has the correct dependence on $\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}$.
From here, the slopes of $dN_l/d\eps'_\pm$ can be easily calculated using the photon spectrum
(equation \ref{dNg}) and the optical thickness (equation \ref{tau1}).
For $\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}_b < \veps_b$, we have $\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\veps) < 1$ for any photon, thus
\begin{equation}
\frac{dN_l}{d\gamma} \propto \left\{ \begin{array}{ll}
\gamma^{-1} & \eps'_b < \gamma m_ec^2 < \teps' \\
\gamma^{-(\beta+1-\alpha)} & \teps' < \gamma m_ec^2 \\
\end{array} \right.
\label{dN1}
\end{equation}
with $\teps'$ the shock-frame photon energy for which $\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}$ is maximal (given in equation
\ref{taumax}), $\gamma = \eps'_\pm/(m_ec^2)$ the pair's random Lorentz factor in the shock-frame and
$\eps' = \eps (z+1)/\Gamma$ the shock-frame photon energy corresponding to the observer-frame $\veps$.
A $\gamma m_e c^2 < \eps'_b$ branch does not exist because $\eps'_o < \eps'_b$ photons form pairs in
interaction with photons of energy $\eps' > \eps'_b$ and the corresponding pair energy $\eps'_\pm
\simeq \eps'/2$ is in one of the branches above.
The above distribution was derived for $\eps'_b < \teps'$ (which requires $\Gamma > \tGamma$)
but is also correct for $\eps'_b > \teps'$ (requiring that $\Gamma > \tGamma$) with $\eps'_b$
and $\teps'$ swapped.
For $\veps_b < \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}_b$, we have $\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\veps) > 1$ for $\veps_- < \veps < \veps_+$ and
$\tau_{\gamma\gamma}} \def\fgg{f_{\gamma\gamma}} \def\Z{{\cal Z}(\veps) < 1$ otherwise, thus
\begin{equation}
\frac{dN_l}{d\gamma} \propto \left\{ \begin{array}{ll}
\gamma^{-1} & \eps'_b < \gamma m_ec^2 < \eps'_- \\
\gamma^{-\beta} & \eps'_- < \gamma m_ec^2 < \eps'_+ \\
\gamma^{-(\beta+1-\alpha)} & \eps'_+ < \gamma m_ec^2 \\
\end{array} \right.
\label{dN2}
\end{equation}
for the more likely case $\eps'_b < \eps'_-$, i.e. for $\Gamma > \tGamma$ (equation \ref{tGam}).
If $0.63\, \tGamma < \Gamma < \tGamma$, then $\eps'_- < \eps'_b < \eps'_+$ and the first branch
above is $\gamma^{-\alpha}$ with $\eps'_b$ and $\eps'_-$ swapped. For $\Gamma < 0.63\, \tGamma$,
we have $\eps'_+ < \eps'_b$ and, in addition to the preceding case, the second branch above is
$\gamma^{-1}$, with $\eps'_b$ and $\eps'_+$ swapped.
Equations (\ref{dN1}) and (\ref{dN2}) indicate that the pair distribution with energy has up to
three power-law branches, with four possible values ($-1, -\alpha, -\beta, -\beta -1 +\alpha$)
for the slope of each branch. Figure 2 shows that the pair distribution obtained numerically by
integrating equation (\ref{num1}) displays only the highest-energy branch for $\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}_b < \veps_b$
because the range of energies over which the lowest-energy branch occurs is too narrow, from
$\gamma = 1$ to
\begin{equation}
\gamma (\teps') = \frac{(z+1)\tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}}{2\Gamma m_ec^2} = 15\, \frac{\Gamma_{2.3}}{\Z \veps_{b,7}}
\end{equation}
For $\veps_b \ll \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}_b$, all three branches are found in the numerical pair-distribution.
The lowest-energy branch is short, extending from $\gamma \sim$ few to
\begin{equation}
\gamma (\eps'_-) = \frac{(z+1)\veps_-}{2\Gamma m_ec^2} = 26\, \frac{\Gamma_{2.3}^5 t_1^2}{\Z^5 \Phi_{-5}}.
\end{equation}
while the second branch extends from $\gamma (\eps'_-)$ to
\begin{equation}
\gamma (\eps'_+) = \frac{(z+1)\veps_+}{2\Gamma m_ec^2} = 10^5 \frac{\Z^{11} \Phi_{-5}^3}
{\Gamma_{2.3}^{11} t_1^2 \veps_{b,6}^4}
\end{equation}
The largest photon energy measured by LAT, $\veps \simeq 100$ GeV, corresponds to a pair Lorentz
factor $\gamma \simeq 1500 \Z \veps_{11}/\Gamma_{2.3}$, which is between $\gamma(\eps'_-)$ and
$\gamma(\eps'_+)$ (because $\veps_- < 100\, GeV < \veps_+$). In further calculations, we make the
assumption that the LAT power-law spectrum extends well above 100 GeV, implying that the power-law
distribution of pairs extends to $\gamma > 10^3$, as in Figure 2. A cut-off in the photon spectrum
above 100 GeV does not affect much the total number of pairs formed but reduces the number of higher
energy pairs and, consequently, the optical flux from pairs formed in-between shocks.
\vspace*{2mm}
\subsection{Evolution of pair distribution}
The evolution (with observer time $t$) of the leptons distribution with energy, ${\cal N} (\gamma)
\equiv dN/d\gamma$, is given by
\begin{equation}
\frac{\partial {\cal N}}{\partial t} = {\cal N}_{inj} + \frac{\partial}{\partial \gamma}
\left( {\cal N} \left| \frac{d\gamma}{dt}\right| \right)
\label{kin}
\end{equation}
where
\begin{equation}
{\cal N}_{inj} (\gamma) = \frac{1}{t} \frac{dN_l}{d\gamma}
\end{equation}
is the rate at which leptons are created, calculated numerically from equation (\ref{num2}) that gives
the distribution of leptons formed over a dynamical timescale $t$, and
\begin{displaymath}
\frac{d\gamma}{dt} = \frac{dt'}{dt} \frac{d\gamma}{dt'} = -\frac{2\Gamma}{z+1} \frac{4 \sigma_e}{3m_e c}
(\gamma^2 -1) u'_B (Y+1)
\end{displaymath}
\begin{equation}
= - k_r (\gamma^2 -1) \;,\; k_r \equiv \frac{1}{3 \pi (z+1)} \frac{\sigma_e}{m_e c} \Gamma B^2 (Y+1)
\label{rad}
\end{equation}
is the radiative cooling rate of pairs, $t'$ being the shock-frame time, $u'_B = B^2/8\pi$ the
shock-frame magnetic energy density, and
\begin{equation}
Y = \frac{4}{3} \int_1^\infty \gamma^2 d\tau_{sc} = \frac{\sigma_e}{3 \pi R^2}
\int_1^\infty \gamma^2 {\cal N}(\gamma) d\gamma
\end{equation}
is the Compton parameter (the ratio of the inverse-Compton to synchrotron losses), $\tau_{sc} =
\sigma_e N/(4\pi R^2)$ being the optical thickness to photon scattering by $N$ leptons in a source
of radius $R$.
The lepton distribution ${\cal N}_{inj}$ depends on the properties of the high-energy emission
($\Phi, \veps_b, \alpha, \beta$) and of the photon source ($\Gamma, R$). Figure 3 shows the instantaneous
and integrated injected lepton distributions ${\cal N}_{inj}$ for a source with constant $\Phi, \alpha, \beta$,
a source deceleration corresponding to a blast-wave interacting with a wind-like medium --
$\Gamma = \Gamma_o (t/t_o)^{-1/4}$, and an evolution of the high-energy spectrum break as expected for
the forward-shock emission: $\veps_b = \veps_b(t_o) (t/t_o)^{-3/2}$. Because $\veps_b < \tilde{\varepsilon}} \def\teps{\tilde{\epsilon}} \def\tGamma{\tilde{\Gamma}_b$
(equation \ref{taulow}), the photon front is optically-thick to pair-formation above $\veps_-$
(equation \ref{epm}), thus the pairs have a ${\cal N}_{inj} \propto \gamma^{-\beta}$ distribution
above $\gamma(\eps'_-)$ (equation \ref{dN2}), which increases with time: $\gamma(\eps'_-) \propto
\Gamma^5 t^2 \propto t^{3/4}$.
\begin{figure*}
\centerline{\psfig{figure=Pdist.eps,width=180mm}}
\figcaption{{\sl Solid lines} show the distribution of leptons formed at three epochs (10 s, 100 s, 1 ks -
colored lines) and total distribution at 2 ks (black), integrated over the indicated source evolution
and without radiative cooling ($b=0$). The quasi-instantaneous distributions are normalized to that
integrated up to 2 ks. The source GeV fluence is constant, its deceleration $\Gamma (t)$ and break
energy $\veps_b$ evolutions correspond to a blast-wave interacting with a wind-like external medium.
{\sl Black lines} show time-integrated (up to 2 ks) lepton distribution with synchrotron and inverse-Compton
cooling, for different fractions $b$ of the post-shock energy that is in the magnetic field.
The cooled lepton distributions display a break at the energy for which leptons lose half of their energy
over a dynamical timescale. Above the break, the lepton distribution slope increases by unity (right panel).
The higher the magnetic field parameter $b$, the lower the lepton cooling-break energy. }
\end{figure*}
Figure 3 also shows the lepton distribution resulting from pair-formation with the above properties
and cooled radiatively by a magnetic field
\begin{equation}
\frac{B^2}{8\pi} = 4\, (b n) \, \Gamma^2 m_p c^2
\label{Bn}
\end{equation}
that is a fraction $b$ of the post-shock energy.
The cooled lepton distribution develops a break at an energy $\gamma_c$ that
decreases with time, with the lepton distribution being that injected at $\gamma < \gamma_c$ and
having a slope larger by 1 than that injected for $\gamma > \gamma_c$.
That feature for the radiative cooling of a power-law distribution of particles can be derived from
the kinetic equation for particle cooling (equation \ref{kin}), rewritten as
\begin{equation}
\frac{\partial {\cal N}}{\partial t} = k_i \gamma^{-\beta} + k_r \frac{\partial (\gamma^2 {\cal N})}{\partial \gamma}
\end{equation}
using equation (\ref{rad}). Trying a power-law solution ${\cal N} = a(t) \gamma^{-p}$, leads to
\begin{equation}
k_i \gamma^{-\beta} = \frac{da}{dt} \gamma^{-p} + (p-2) k_r a \gamma^{1-p}
\end{equation}
If the first term on the right-hand side is dominant (i.e. $da/dt \gg k_r a \gamma$ and the
radiative cooling term is negligible), then $p = \beta$ and $da/dt = k_i$, thus $a = \int k_i dt$
and ${\cal N} = (\int k_i dt) \gamma^{-\beta}$, hence the effective distribution is the
integrated injected distribution, at energies $\gamma$ that satisfy $\gamma \ll \gamma_c^{(1)}$, where
\begin{equation}
\gamma_c^{(1)} \equiv \frac{da/dt}{k_r a} = \frac{k_i}{k_r \int k_i dt}
\end{equation}
If the second, radiative cooling term on the rhs is dominant, then $p = \beta + 1$ and $a \simeq k_i/k_r$,
hence ${\cal N} = (k_i/k_r) \gamma^{-(\beta+1)}$. This solution exists for $|da/dt| \gg k_r a \gamma$,
which is equivalent to $\gamma \gg \gamma_c^{(2)}$, where
\begin{equation}
\gamma_c^{(2)} \equiv \frac{|da/dt|}{k_r a} = \frac{1}{k_i} \left| \frac{d}{dt} \frac{k_i}{k_r} \right|
\end{equation}
The above two cooling energies are comparable
\begin{equation}
\frac{\gamma_c^{(1)}}{\gamma_c^{(2)}} = \frac{k_i t}{\int k_i dt} \left/ \frac {d\ln |k_i/k_r|} {d\ln t} \right.
\end{equation}
if $k_i$ does not vary too fast and if $k_i/k_r$ is moderately evolving in time.
Thus, the effective lepton distribution is
\begin{equation}
{\cal N} (\gamma) = \left\{ \begin{array}{ll} (\int k_i dt) \gamma^{-\beta} & \gamma \ll \gamma_c^{(1)} \\
\frac{\displaystyle k_i}{\displaystyle k_r} \gamma^{-(\beta+1)} & \gamma_c^{(2)} \ll \gamma
\end{array}\right.
\label{Np}
\end{equation}
For a constant injection rate $k_i$, the $\gamma_c^{(1)}$ becomes simpler
\begin{equation}
\gamma_c \equiv \frac{1}{k_r t} \propto \frac{1}{\Gamma B^2 (Y+1) t}
\end{equation}
and the effective distribution can be approximated as
\begin{equation}
{\cal N} (\gamma) \simeq {\cal N} (\gamma_c) \left\{ \begin{array}{ll}
\left(\frac{\displaystyle \gamma_c}{\displaystyle \gamma}\right)^\beta & \gamma < \gamma_c \\
\left(\frac{\displaystyle \gamma_c}{\displaystyle \gamma}\right)^{\beta+1} & \gamma_c < \gamma
\end{array}\right. {\cal N} (\gamma_c) \equiv \frac{k_i}{k_r} \gamma_c^{-(\beta+1)}
\end{equation}
We note that the cooled lepton distribution is not calculated from equation (\ref{kin}) because it
is unstable and "suffers" from a Courant-like condition, with the timestep $\delta t$ being upper-limited
by the cooling time $t_c (\gamma) = (k_r \gamma)^{-1}$ of the highest energy leptons $\gamma_{max}$ in the
calculation, which can be $t_c \ll t$ for $bn \gamma_{max} > 10^3 \Gamma_{2.3}^{-3} (Y+1)^{-1}$.
Instead, the lepton distribution is calculated numerically by tracking the flow of particles on a
1-dimensional energy grid, using the cooling law of equation (\ref{rad})
\begin{equation}
\frac{\gamma + 1}{\gamma - 1} = \frac{\gamma_o + 1}{\gamma_o - 1} e^{2 k_r \delta t} \;,\;
\frac{1}{\gamma} \stackrel{\gamma \gg 1}{=} \frac{1}{\gamma_o} + k_r \delta t
\end{equation}
with $\gamma$ the energy of a lepton that had initially an energy $\gamma_o$, after a timestep $\delta t$.
Pair-energy tracking means accounting for that \\
(1) a fraction $\min [1, \delta t/t_c(\gamma)]$ of the $\delta N = {\cal N}(\gamma) \delta \gamma$ leptons
existing in a cell $(\gamma,\gamma + \delta \gamma)$ exit that cell due to their cooling during $\delta t$, \\
(2) a fraction $\max [1, t_c(\gamma)/\delta t]$ of the leptons $\delta N_{inj} = {\cal N}_{inj} (\gamma)
\delta \gamma \delta t$ injected in a cell remain in that cell after cooling for $\delta t$, \\
(3) leptons existing in all cells between energies $\gamma_o (\gamma)$ and $\gamma_o (\gamma +\delta \gamma)$
cool during $\delta t$ to cell $(\gamma,\gamma + \delta \gamma)$, \\
(4) a fraction $[t_c(\gamma_o \rightarrow \gamma) - t_c(\gamma_o \rightarrow \gamma +\delta \gamma)]/\delta t$
of the leptons injected at energy $\gamma_o$ cool to cell $(\gamma,\gamma + \delta \gamma)$ during $\delta t$,
with $t_c(\gamma_o \rightarrow \gamma) \equiv (\gamma^{-1} - \gamma_o^{-1})/k_r$, the cooling time from
energy $\gamma_o$ to $\gamma$. \\
\section{Radiation Emission}
The calculation of the synchrotron self-Compton from pairs three components: synchrotron emission,
synchrotron self-absorption, and inverse-Compton scattering of the self-absorbed synchrotron spectrum.
We consider only the first inverse-Compton scattering, which is appropriate approximation when the Compton
parameter $Y$ is sub-unity and a necessary approximation when leptons radiating at the observing frequency
$\nu$ cool faster than they are created. The former case requires a magnetic field in the shock fluid that
is not much below equipartition ($b \mathrel{\rlap{\raise 0.511ex \hbox{$>$}}{\lower 0.511ex \hbox{$\sim$}}} 10^{-3}$); for the latter case, the argument is that, in an source
that is optically-thin to electron scattering, the lepton distribution can change substantially during the time
it takes a photon to cross the source and be scattered, hence the lepton distribution that produced the
seed photon is not the same as the lepton distribution that upscatters it, an a time-dependent treatment
of upscatterings is needed. For $Y > 1$, ignoring higher-order scatterings leads to an underestimation of
the flux at higher energies (above X-rays) and an overestimation of the synchrotron and first inverse-Compton
flux above the cooling frequency (which would be lower if higher-order scatterings were accounted).
\subsection{Synchrotron Emission}
For a relativistic source moving at Lorentz factor $\Gamma$, the received synchrotron flux from a distribution
of leptons ${\cal N} (\gamma)$ at observer frequency $\nu$ is
\begin{equation}
F_{sy} (\nu) = \frac{z+1}{4 \pi d_l^2} \Gamma \int d\gamma {\cal N} (\gamma) P'_{sy}
\left( \frac{\displaystyle z+1}{\displaystyle \Gamma} \nu, \gamma \right)
\label{Fsy1}
\end{equation}
where the relativistic boost of the comoving-frame emission at frequency $\nu' = (z+1)\nu/\Gamma$ gets
only one power of $\Gamma$ from the contraction of photon arrival time $dt = dt_{lab}/\Gamma^2 = dt'/\Gamma$
relative to the comoving-frame emission time $dt'$ (the boost $\Gamma$ in photon energy is "lost" to that
a comoving energy range $d\nu'$ is stretched into $\Gamma d\nu'$ for the observer, and the angular beaming
boost $\Gamma^2$ is "lost" because, for the observer, that beaming reduces the solid angle of a spherical
source by a factor $\Gamma^2$), and
\begin{equation}
P'_{sy} (\nu', \gamma) = \frac{e^3 B}{m_ec^2} f_{sy} \left( \frac{\nu'}{\nu'_{sy} (\gamma)} \right)
\label{psy}
\end{equation}
is the specific synchrotron power for a lepton, with $f_{sy}$ the "synchrotron function" and
\begin{equation}
\nu'_{sy}(\gamma) = \frac{3}{16} \frac{e}{m_e c} B \gamma^2 = 3.3 \times 10^6 B_o \gamma^2 \; {\rm Hz}
\label{nusy}
\end{equation}
is the synchrotron characteristic frequency at which a lepton of energy $\gamma$ radiates most of its
emission. The synchrotron function is an integral over the modified Bessel function of 5/3 order and has
the following asymptotic behavior
\begin{equation}
f_{sy} (x) \simeq \left\{ \begin{array}{ll} 1.71\, x^{1/3} & x \ll 1 \\
1.25\, x^{1/2} e^{-x} & x \gg 1 \end{array} \right.
\end{equation}
This approximation would be useful if the synchrotron emission at frequency $\nu'$ were produced by
leptons whose characteristic synchrotron frequency $\nu'_{sy}$ is far from $\nu'$, however, the opposite
is true. We approximate the synchrotron function with the asymptotic behaviors given above but with
coefficients such that
$i)$ $f_{sy}$ is continuous at $x=1/2$ (where $x^{1/2} e^{-x}$ has a maximum) and
$ii)$ its integral is equal to that of the exact synchrotron function, $\int f_{sy}(x) dx = (4/3)^3$,
yielding a power-per-lepton $P'(\gamma) = \int P'_{sy}(\nu',\gamma) d\nu' = (4/3) \sigma_e c \gamma^2
(B^2/8\pi)$. The following approximation
\begin{equation}
f_{sy} (x) \simeq \left\{ \begin{array}{ll} 1.50\, x^{1/3} & x < 1/2 \\
2.77\, x^{1/2} e^{-x} & x > 1/2 \end{array} \right.
\label{fsy}
\end{equation}
satisfies the above constraints and has a maximum $f_{sy} (0.5) = 1.2$ (that of the exact synchrotron
function is $f_{sy} (0.3) = 0.92$).
Substituting equations (\ref{psy}) and (\ref{nusy}) in (\ref{Fsy1}), we get
\begin{equation}
F_{sy} (\nu) = \frac{4.3 \times 10^{-56}}{(z+1)^3} B \Gamma \int d\gamma {\cal N}(\gamma)
f_{sy} \left( \frac{\gamma_\nu^2}{\gamma^2} \right) \; {\rm Jy}
\label{Fsy2}
\end{equation}
in cgs units, where
\begin{equation}
\gamma_\nu \equiv 5.5 \times 10^{-4} \left[ \frac{(z+1)\nu}{\Gamma B} \right]^{1/2}
\end{equation}
satisfies $\nu_{sy} (\gamma_\nu) = \nu$. Owing to the exponential cut-off of the synchrotron function
at $x > 1$, only leptons with energy above $\gamma_\nu$ produce the synchrotron emission at frequency $\nu$.
Then, approximating the synchrotron function by only its $x < 1/2$ branch, one obtains
\begin{equation}
F_{sy} (\nu) = \frac{0.92 \times 10^{-57}}{(z+1)^{8/3}} (B^2 \Gamma^2 \nu)^{1/3}
\int_{\sqrt{2}\gamma_\nu}^\infty d\gamma \frac{{\cal N}(\gamma)}{\gamma^{2/3}} \; {\rm Jy}
\end{equation}
For a power-law distribution of particles, ${\cal N}(\gamma) \propto \gamma^{-p}$, this leads to the
well-known spectrum $F_{sy}(\nu) \propto \nu^{-(p-1)/2}$.
\subsection{Synchrotron Self-Absorption}
Starting from equation (6.49) of Rybicki \& Lightman (1949), taking into account that the lepton
column-density is $N/4\pi R^2$, and using the synchrotron emissivity per lepton given in equation (\ref{psy}),
the synchrotron self-absorption optical thickness at observer frequency $\nu = \Gamma \nu'/(z+1)$ is
\begin{equation}
\tau_a (\nu) = - \frac{1}{32 \pi^2 R^2} \left[ \frac{\Gamma}{(z+1) \nu} \right]^2 \frac{e^3 B}{(m_e c)^2}
\end{equation}
\begin{displaymath}
\times \int d\gamma \gamma^2 f_{sy} \left( \frac{\gamma_\nu^2}{\gamma^2} \right)
\frac{\partial}{\partial \gamma} \left( \frac{{\cal N} (\gamma)}{\gamma^2} \right)
\end{displaymath}
Quick progress toward a simpler form can be made if we retain only the $ x < 1/2$ branch of the synchrotron
function (equation \ref{fsy}) and set $f_{sy}(x > 1/2) = 0$ (motivated by the exponential cut-off):
\begin{equation}
\tau_a (\nu) \mathrel{\rlap{\raise 0.511ex \hbox{$>$}}{\lower 0.511ex \hbox{$\sim$}}} - \frac{0.0083}{R^2} \left[ \frac{\Gamma}{(z+1) \nu m_e c} \right]^{5/3} (e^4 B)^{2/3}
\end{equation}
\begin{displaymath}
\times \int_{\sqrt{2}\gamma_\nu}^\infty d\gamma \gamma^{4/3} \frac{\partial}{\partial \gamma}
\left( \frac{{\cal N} (\gamma)}{\gamma^2} \right)
\end{displaymath}
The integral above can be re-written as
\begin{equation}
\tau_a (\nu) \mathrel{\rlap{\raise 0.511ex \hbox{$>$}}{\lower 0.511ex \hbox{$\sim$}}} \frac{4.7}{R^2} \left[ \frac{\Gamma}{(z+1) \nu} \right]^{5/3} B^{2/3}
\label{tauabs}
\end{equation}
\begin{displaymath}
\times \left( \left. \frac{{\cal N}}{\gamma^{2/3}} \right|_{\sqrt{2}\gamma_\nu} +
\frac{4}{3} \int_{\sqrt{2}\gamma_\nu}^\infty d\gamma \frac{{\cal N}}{\gamma^{5/3}} \right)
\end{displaymath}
in cgs units.
For a power-law distribution of leptons with energy, ${\cal N} (\gamma) \propto \gamma^{-p}$, the ratio
of the two terms above is $(3p+2)/4$, where $p = \beta,\beta+1$ (equation \ref{Np} and Figure 3).
For a typical spectrum of the high-energy photons, $\beta \simeq 2$, this ratio is 2 or 11/4,
thus the first term is dominant. Either term is proportional to $\gamma_\nu^{-(p+2/3)}$ and
yields $\tau_a (\nu) \propto \nu^{-(p+4)/2}$.
The emerging synchrotron flux is the intrinsic flux (given in equation \ref{Fsy2}) reduced by the
escaping fraction (equation \ref{g}):
\begin{equation}
F_{sy}^{(obs)} (\nu) = \frac{1-e^{-\tau_a(\nu)}}{\tau_a(\nu)} F_{sy} (\nu)
\label{Fssa}
\end{equation}
Below the self-absorption frequency $\nu_a$, where $\tau_a(\nu_a) = 1$, the received synchrotron flux
satisfies $F_{sy}^{(obs)} (\nu) = F_{sy}(\nu)/\tau_a(\nu)$ which, for a power-law distribution
of particles, is $F_{sy}^{(obs)} (\nu < \nu_a) \propto \nu^{-(p-1)/2}/\nu^{-(p+4)/2} = \nu^{5/2}$,
another well-known result.
\subsection{Inverse-Compton Emission}
For a lepton of energy $\gamma$ scattering a photon of frequency $\nu'_o$ in the Thomson regime,
the inverse-Compton emissivity at photon frequency $\nu'$ is (equation 7.26a in Rybicki \& Lightman 1979)
\begin{equation}
j'_{ic} (\nu') = \frac{3 \sigma_e F'_o}{4 (\gamma \nu'_o)^2}
f_{ic} \left( \frac{\nu'}{4\gamma^2 \nu'_o} \right) \nu'
\end{equation}
where $F'_o$ is the comoving-frame energy flux of $\nu'_o$ photons and
\begin{equation}
f_{ic} (x) = \left\{ \begin{array}{ll} 2x \ln x -2 x^2 + x +1 & (16\gamma^4)^{-1} < x < 1 \\
0 & {\rm otherwise} \\ \end{array} \right.
\end{equation}
By integrating ${\cal N} (\gamma) (\nu'_o) j'_{ic} (\nu')$ over the lepton distribution and over the
incident synchrotron photon spectrum, one obtains the comoving-frame inverse-Compton luminosity,
which is enhanced by a factor $\Gamma$ to yield the lab-frame inverse-Compton luminosity.
Then, the received inverse-Compton emission from pairs at frequency $\nu = \Gamma \nu'/(z+1)$ is
\begin{equation}
F_{ic} (\nu) = \frac{3(z+1)}{8 \pi d_l^2} \sigma_e \Gamma \nu' \int d\gamma {\cal N}(\gamma) \int d\nu'_o
\frac{F'_o (\nu'_o)}{(\gamma \nu'_o)^2} f_{ic} \left( \frac{\nu'}{4\gamma^2 \nu'_o} \right)
\label{mccoy}
\end{equation}
The comoving-frame flux of incident photons $F'_o (\nu'_o)$ in a source of radius $R$ is related to the
received synchrotron flux at photon frequency $\nu_o = \Gamma \nu'_o/(z+1)$ through
\begin{equation}
F_{sy} (\nu_o) = \frac{z+1}{4 \pi d_l^2} \Gamma (4 \pi R^2) F'_o (\nu'_o)
\end{equation}
Substituting in equation (\ref{mccoy}), and changing from comoving-frame to observer-frame photon frequencies,
it follows that the received inverse-Compton flux can be calculated from the received synchrotron flux
(equation \ref{Fsy2}):
\begin{equation}
F_{ic} (\nu) = \frac{3\sigma_e}{16 \pi R^2} \nu \int \frac{d\gamma}{\gamma^2} {\cal N}(\gamma)
\int \frac{d\nu_o}{\nu_o^2} F_{sy}^{(obs)} (\nu_o) f_{ic} \left( \frac{\nu}{4\gamma^2 \nu_o} \right)
\label{Fic1}
\end{equation}
where $F_{sy}^{(obs)}$ is the self-absorbed synchrotron flux (equation \ref{Fssa}), because synchrotron
self-absorption reduces the flux of photons incident on a scattering lepton in the same way that it
affects the received synchrotron flux.
An approximation that changes very little the spectrum of the inverse-Compton pair emission is that
where photons of frequency $\nu$ result from the upscattering by a lepton of energy $\gamma$ of {\sl only}
synchrotron photons of frequency $\nu_o = 3 \nu/(4 \gamma^2)$, which is motivated by that the average
energy of an upscattered photon is $(4/3) \gamma^2 \nu_o$. This is equivalent to approximating the
inverse-Compton function with a $\delta$-function
\begin{equation}
f_{ic} \left( \frac{\nu}{4\gamma^2 \nu_o} \right) = \delta \left( \frac{3\nu}{4\gamma^2 \nu_o} - 1 \right)
\end{equation}
Then, equation (\ref{Fic1}) becomes
\begin{equation}
F_{ic} (\nu) = \frac{\sigma_e}{4\pi R^2} \int d\gamma {\cal N}(\gamma) \left[ F_{sy}^{(obs)}
\left(\frac{3\nu}{4\gamma^2} \right) \right]
\label{Fic2}
\end{equation}
\begin{displaymath}
= \int d\tau_e \left[ F_{sy}^{(obs)} \left( \frac{3\nu}{4\gamma^2} \right) \right]
\end{displaymath}
where $\tau_e = (\sigma_e N_l)/(4\pi R^2)$ is the pairs' optical-thickness to photon scattering.
Equation (\ref{Fic2}) means that the inverse-Compton specific flux is the integral over the lepton
distribution of the scattered synchrotron flux.
\vspace*{6mm}
\section{Emission from pairs formed in the shocked fluid}
\subsection{Approximate dependences}
\begin{figure*}
\centerline{\psfig{figure=lc1.eps,height=8cm,width=180mm}}
\figcaption{ Dependence of pair emission on the fluence of the high-energy LAT afterglow fluence $\Phi$
(left panel) and on the GeV initial Lorentz factor $\Gamma_o$ (right panel), with other parameters
as indicated. Red lines are for synchrotron and blue lines for inverse-Compton. }
\end{figure*}
The most important parameters that determine the pair emission are those that set the number of pairs
-- the blast-wave initial Lorentz factor $\Gamma_o$ and the afterglow high-energy fluence $\Phi$ --
and the magnetic field -- the $nb$ product (equation \ref{Bn}). Less effective, but still relevant,
are three other parameters that determine the number of pairs: the slopes $\alpha$ and $\beta$ of the
high-energy spectrum, and its break energy $\veps_b$.
Equations (\ref{Fsy2}) and (\ref{Fic2}) suggest the following dependences for the synchrotron and
inverse-Compton flux from pairs:
\begin{equation}
F_{sy} (\nu) \propto B\Gamma N_\pm \;,\; F_{ic} (\nu) \propto \frac{N_\pm}{R^2} F_{sy}
\propto \frac{N_\pm^2}{R^2} B \Gamma
\end{equation}
where $R \propto \Gamma^2 t$ (equation \ref{R}), $B \propto \sqrt{nb} \Gamma$ (equation \ref{Bn}),
and $N_\pm \propto \Phi^2/(\Gamma^6 t^2)$ (equations \ref{N1} and \ref{N2}).
Equation (\ref{Fsy2}) actually means that $F_{sy} (\nu) \propto N(>\gamma_\nu)$, hence the use of
$N_\pm$ here is accurate only when the number of pairs above $\gamma_\nu$ is a fixed fraction of the
total number of pairs. That is satisfied only above the pair cooling-energy, where pairs produced
during one cooling timescale reside. With the above substitutions, we find that
\begin{equation}
F_{sy} (\nu) \propto \frac{\Phi^2 (nb)^{1/2}}{\Gamma^4 t^2} \;,\;
F_{ic} (\nu) \propto \frac{\Phi^4 (nb)^{1/2}}{\Gamma^{14} t^6}
\end{equation}
For a wind-like medium, where $n \propto R^{-2}$ and $\Gamma \propto t^{-1/4}$, we arrive at
\begin{equation}
F_{sy} (\nu) \propto \frac{\Phi^2 \sqrt{b}}{\Gamma^6 t^3} \propto \Phi^2 t^{-3/2} \;,\;
F_{ic} (\nu) \propto \frac{\Phi^4 \sqrt{b}}{\Gamma^{16} t^7} \propto \Phi^4 t^{-3}
\label{decay}
\end{equation}
This shows that the inverse-Compton flux has a very strong dependence on the high-energy afterglow fluence
and a super-strong dependence on the Lorentz factor of the GeV afterglow, which suggest that inverse-Compton
emission from pairs could be relevant (i.e could overshine the synchrotron flux) only for the brightest
LAT afterglows ($\Phi \sim 10^{-5} \ergcm2$) and for the slowest GeV sources in which pairs are still
optically-thin ($\Gamma \mathrel{\rlap{\raise 0.511ex \hbox{$>$}}{\lower 0.511ex \hbox{$\sim$}}} 130$ - equation \ref{Gtau}).
\subsection{Light-curves and spectra}
The pair light-curves shown in Figure 4 illustrate the correlation of the pair flux with the observable
LAT fluence $\Phi$ and the unknown source Lorentz factor $\Gamma_o$ at the peak epoch $t_o$ of the LAT
light-curve. Those light-curves were obtained by integrating the synchrotron and inverse-Compton fluxes
given in equations
(\ref{Fsy2}), (\ref{tauabs}), (\ref{Fssa}), and (\ref{Fic1}) (or \ref{Fic2}), over the deceleration of a
blast-wave interacting with a massive-star wind. Although $N_\pm \propto \Phi^2/(\Gamma_o^6 t_o^3)$ is
satisfied by the numerical calculation of the pairs formed, the numerical pair fluxes display a weaker
correlation with $\Phi$ and $\Gamma_o$ (and also with $t_o$) than given in equation (\ref{decay}), which
is due to the use of $N_\pm$ in the derivation of that equation.
Equations (\ref{N1}) and (\ref{N2}) show that the number of pairs is weakly dependent on the
unknown break energy $\veps_b$ of the LAT spectrum (another model {\sl parameter}), with more pairs being
formed for a lower $\veps_b$, because that increases the optical thickness to pair formation (equations
\ref{tau2} and \ref{epm}).
The brightness of the LAT high-energy spectral component at sub-MeV photon energies relative to that
of the burst is the criterion for choosing the two prescriptions given in Figure 5 for the unknown
break-energy: GRBs with a fast-decaying tail require that the $\veps_b$ of a bright LAT component
remains above MeV for the duration of the tail, while bursts with a slowly-decaying tail allow lower
$\veps_b$ (decreasing or not).
As expected, a lower $\veps_b$ yields a brighter pair emission, and a decreasing $\veps_b$ leads to a
slower decay of the pair light-curve. The latter behavior provides a criterion for identifying early
optical afterglows produced by pairs: slowly-dimming pair afterglows (due to a decreasing $\veps_b$)
cannot follow fast-falling bursts (which are incompatible with a decreasing $\veps_b$). However,
fast-falling pair afterglows can follow either type of burst tail (fast or slowly decreasing).
\begin{figure*}
\centerline{\psfig{figure=lcsp.eps,height=8cm,width=180mm}}
\figcaption{ Left panel: optical light-curves from pairs formed in the shocked fluid, for the indicated
parameters, and for the wind-like external medium given in equation (\ref{WR}). Solid lines are
for a constant break-energy $\veps_b$ of the high-energy emission that forms pairs. The synchrotron
flux (red line) exhibits a decay that is slightly steeper than estimated from equation (\ref{decay}),
while the inverse-Compton flux (blue line) has a decay that is significantly slower. Dashed lines
are for a decreasing $\veps_b$, which yields a slower flux decay.
Right panel: synchrotron and inverse-Compton spectra at $t = 100$ s, for a decreasing $\veps_b$.
Dotted lines show spectra without accounting for synchrotron self-absorption (SSA), solid lines are for
absorbed spectra.
Owing to the large magnetic field parameter $b$, the Compton parameter is below unity and the
inverse-Compton emission is dimmer than synchrotron at all frequencies of interest. The synchrotron
cooling break is slightly below the optical. }
\end{figure*}
\subsection{Application to GRB 130427A}
Figure 6 shows a fit to the super-bright optical flash of GRB 130427A (RQD2/Raptor - Vestrand et al 2014)
with the synchrotron emission from internal pairs formed from the high-energy emission monitored by LAT
(Fan et al 2013, Tam et al 2013, Ackermann et al 2014). Observations set the high-energy fluence $\Phi$
and the spectral slope $\beta$ above the unknown break energy $\veps_b$, which is a free model parameter.
The initial Lorentz factor $\Gamma_o$ of the high-energy photons source and the magnetic field parameter
$b$ in the shocked fluid are two other model parameters.
As indicated by equation (\ref{decay}), with the high-energy fluence $\Phi$ set by observations,
the brightness of the optical flash of 130427A constrains the combination $b/\Gamma_o^{12}$.
We find that $\Gamma_o < 300$ is required to match the brightness of the 130427A optical flash because,
for higher Lorentz factors, the number of pairs formed is too small, and the maximal optical flux from
pairs, obtained for a magnetic field that brings the peak of the self-absorbed synchrotron spectrum
in the optical, falls short of the peak brightness of GRB 130427A's optical counterpart.
The decay of the pair synchrotron optical light-curve depends primarily on the LAT light-curve $\Phi(t)$
and on the blast-wave deceleration $\Gamma(t)$. As those quantities are already set, the optical flash decay
($F_o \propto t^{-2}$) constrains the slope of the LAT spectral component below its $\veps_b$ break and
the evolution of $\veps_b$.
For $\alpha=2/3$ (i.e. $\veps_b$ is the peak energy of a synchrotron or inverse-Compton spectrum without cooling),
we find that $\veps_b \propto t^{1/2}$ is required to match the optical light-curve decay at 10--100 s.
This time-dependence is consistent with the behavior of the cooling-break of the synchrotron spectrum
from the forward-shock (and for a wind-like medium), but $\alpha=2/3$ is inconsistent with the expected
value $\alpha = \beta - 1/2$ in that case.
For $\alpha = 3/2$ (when $\veps_b$ would be the injection peak of the sy/ic spectrum with electron cooling)
or for $\alpha = \beta -1/2 = 1.7$ (when $\veps_b$ would be the cooling-break of a sy/ic spectrum),
we find that the optical flash decay requires $\veps_b \propto t^2$, which is consistent with the evolution
of the cooling-break of the inverse-Compton spectrum from the forward-shock (and for a wind-like medium).
Therefore, fits to the decay of the prompt optical emission of GRB 130427A with emission from internal-pairs
sets constraints on the unmeasured peak energy $\veps_b$ of the LAT spectral component that do not elucidate
its shock origin. Furthermore, numerical fits to the multiwavelength emission of this afterglow show comparable
contributions to the LAT emission arising from both synchrotron reverse and forward shocks (Panaitescu et al 2013).
Figure 6 also shows that the X-ray emission from pairs and that LAT component contribution to the X-ray
are below the fluxes measured by Swift, and that the formation of enough pairs to produce a bright optical
flash does not entail a high attenuation of the LAT spectrum above 10 GeV. For the highest Lorentz factor
$\Gamma_o$ that allows a good fit to the optical flash, the intrinsic power-law spectrum above 10 GeV is
attenuated by up to 50 percent at $t=10$ s (when attenuation is maximal), which is not inconsistent with
the detection by LAT of a 70 GeV photon at 18 s. However, Lorentz factors $\Gamma _o \mathrel{\rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}} 200$ are incompatible
with that detection.
Although a good fit with the internal-pair emission for the optical flash of GRB 130427A is obtained,
we do not propose this origin for the optical counterpart of GRB 130427A because modeling of the broadband
(radio, optical, X-ray, and GeV) emission of this afterglow (Panaitescu, Vestrand \& Wozniak 2013),
from 100 s to tens of days, has shown that its wind-like ambient medium must be very tenuous, which leads
to an initial Lorentz factor $\Gamma_o \simeq 750$ that is much higher than allowed by fitting the optical
flash with internal-pairs emission ($\Gamma_o \mathrel{\rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}} 300$).
Synchrotron emission from {\sl external} pairs formed ahead of the blast-wave from {\sl burst MeV}
photons scattered by the ambient medium and, then, accelerated by the forward-shock can also produce
a bright optical flash ($R < 10$) lasting for 100 s, provided that the initial source Lorentz factor
is $\Gamma_o \sim 200$ (figures 4 and 7 of Kumar \& Panaitescu 2004). Vurm, Hascoet \& Beloborodov
(2014) have found that the optical flash of GRB 130427A can be explained with synchrotron emission
from external pairs accelerated by the forward-shock if that shock's Lorentz factor is a low $\Gamma = 200$.
A similar model, but not investigated here, is the emission from the shock-accelerated {\sl external} pairs
formed from {\sl afterglow MeV--TeV} photons ahead of the forward-shock. In one variant of that model --
pair-formation from {\sl unscattered} GeV photons -- the number of pairs is strongly decreasing with the
source Lorentz factor, therefore it requires a low $\Gamma_o \mathrel{\rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}} 300$ to account for the optical flash of
GRB 130427A. In the other variant -- pair-formation from GeV photons {\sl scattered} by the ambient medium
(which decollimates photons sufficiently to lower significantly the pair-formation threshold-energy and
and enriches with pairs the medium ahead of the blast-wave) -- the number of pairs
should be less dependent on $\Gamma_o$. Owing to its similarity to the pair-wind formed from scattered
burst MeV photons, this model may also require a low $\Gamma_o$ to account for the optical flash of GRB 130427A.
If all pair-based models for this flash require low Lorentz factors (for the seed-photon source)
that are incompatible with the afterglow $\Gamma$ inferred from multiwavelength data modeling, the
bright optical flash of GRB 130427A should be attributed to the reverse-shock (M\'esz\'aros \& Rees 1997)
that energizes some
incoming ejecta in an initial injection episode, followed by a quiet period when the ejecta electrons
cool radiatively and yield a fast-decaying flux, followed by a second, longer-lived injection episode,
during which the reverse-shock produces the optical emission measured for the early (up to few ks)
afterglow of GRB 130427A (as proposed by Vestrand et al 2014).
\begin{figure*}
\centerline{\psfig{figure=130427.eps,height=8cm,width=180mm}}
\figcaption{
Left panel: early optical and GeV emission for GRB 130427A and a fit to the optical flash
(up to 100 s) with synchrotron emission from internal pairs formed in the shocked fluid.
LAT fluence (model input) after the $t_o = 10$ s peak is indicated; LAT spectral slope is
$\beta = 2.2$ (Tam et al 2013).
Model parameters are: source Lorentz factor before LAT peak $\Gamma_o = 300$, break-energy of the
LAT spectrum $\veps_b = 500\, (t/t_o)^{1/2}$ keV , magnetic field parameter $b = 10^{-3}$.
The burst ambient medium is that of equation (\ref{WR}),
which sets the source dynamics: $\Gamma (t) = \Gamma_o (t/t_o)^{-1/4}$.
After 100 s, the optical emission is well-fit by the reverse-shock emission (Panaitescu et al 2013).
Mid panel: for the parameters of the optical flash fit, the pair emission is dimmer than the early
X-ray emission of this burst, monitored by Swift's BAT and XRT.
The 10 keV emission from the LAT component with an assumed low-energy slope $\alpha = 2/3$ below
$\veps_b$ is also shown. Part of the burst tail and early X-ray afterglow can be explained with
the forward-shock emission.
Right panel: photon-photon attenuated spectra (dashed lines) at the LAT peak light-curve epoch,
when attenuation is maximal, for the measured LAT spectrum (solid line) and various initial $\Gamma_o$.
For $\Gamma_o = 300$, a moderate absorption occurs above 10 GeV, corresponding to a flux reduction
of at most 50 percent. For smaller $\Gamma_o$, attenuation is stronger.
}
\end{figure*}
\section{Conclusions}
In GRB afterglows, test photons of lab-frame energy above $\sim 10$ MeV form pairs in interactions
with target photons that are above the threshold for pair-formation. The number of pairs depends
moderately on the unknown break-energy $\veps_b$ of the high-energy component (LAT measures only photons
above $\veps_b$), strongly on the afterglow GeV output (which is the observable LAT fluence $\Phi$),
and very strongly on the Lorentz factor $\Gamma$ of the GeV source.
Below the radiative cooling break, the brightness of the synchrotron emission from (internal) pairs, formed
in the shocked fluid (between the reverse and forward shocks), depends on their number (set by one observable
-- $\Phi$ -- and two model parameters -- $\Gamma$ and $\veps_b$) and on the strength of the magnetic field
between shocks (a third model parameter). For an intermediate/low $\Gamma$, pairs produce bright optical
early afterglows even for a magnetic field that is several orders of magnitude below equipartition. In fact,
strong magnetic fields do not warrant a much brighter optical emission because an enhanced radiative
cooling reduces the number of pairs of sufficiently high energy to radiate synchrotron emission in the optical.
The correlation between the number of pairs and the attenuation of the LAT spectrum, induced by the
dependence of these two features on $\Gamma$, provides a way to identify optical counterparts that
originate from internal pairs (formed in the GeV source). For the most relativistic afterglows
($\Gamma \mathrel{\rlap{\raise 0.511ex \hbox{$>$}}{\lower 0.511ex \hbox{$\sim$}}} 500$), the internal-pairs emission should be dim and the LAT spectrum should be an
unattenuated power-law, both because few pairs are formed. $300 \mathrel{\rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}} \Gamma \mathrel{\rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}} 500$ yields a moderately
bright optical flash and no detectable attenuation of the LAT spectrum. For the less relativistic afterglows
($100 \mathrel{\rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}} \Gamma \mathrel{\rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}} 200$), when many pairs are formed, there should be a bright optical emission from
pairs, accompanied by a significant attenuation of the LAT spectrum above 1 GeV.
An additional criterion for identifying optical counterparts from internal-pairs emission arises
from that GRBs with fast-decaying tails require a peak energy $\veps_b \mathrel{\rlap{\raise 0.511ex \hbox{$>$}}{\lower 0.511ex \hbox{$\sim$}}} 10$ MeV of the LAT spectrum,
which yields dimmer and faster-decaying optical emission from pairs. Slowly-decaying GRB tails do not
exclude bright optical flash from pairs, hence there should be some correlation between the speed of the
GRB tail decay and the brightness of the pair optical flash.
|
{
"timestamp": "2015-04-14T02:15:20",
"yymm": "1504",
"arxiv_id": "1504.03205",
"language": "en",
"url": "https://arxiv.org/abs/1504.03205"
}
|
\section{Introduction}
The kinetic Sunyaev-Zeldovich effect \citep[hereafter kSZ,][]{kSZ0,kSZ} describes the Doppler boost experienced by a small fraction of the photon bath of the cosmic microwave background (CMB) radiation when scattering off a cloud of moving electrons.
In the limit of Thomson scattering, where there is no energy exchange between the electrons and the CMB photons, the kSZ effect is equally efficient for all frequencies and gives rise to relative brightness temperature fluctuations in the CMB that are frequency independent. The exact expression for this effect was first written as equation~15 of \citet[][]{kSZveryfirst}, and reads
\begin{equation}
\frac{\delta T}{T_{\rm 0}} (\hat{\mbox{\vec{n}}}) = -\int {\rm d}l\,\sigma_{\rm T} n_{\rm e} \left(\frac{\mbox{\bf {v}}}{c} \cdot \hat{\mbox{\vec{n}}}\right).
\label{eq:kSZ1}
\end{equation}
In this expression, the integral is performed along the line of sight, $n_{\rm e}$ denotes the electron number density, $\sigma_{\rm T}$ is the Thomson cross-section, and $(\mbox{\bf {v}}/c)\cdot \hat{\mbox{\vec{n}}}$ represents the line of sight component of the electron peculiar velocity in units of the speed of light $c$. The equation above shows that the kSZ effect is sensitive to the peculiar {\em momentum} of the free electrons, since it is proportional to both their density and peculiar velocity. A few previous studies \citep{chm05,Bhattacharya08DE,MaZhao13,ks_pksz13,Liangulowhite2014} have proposed that it be used to trace the growth of velocities throughout cosmic history and its connection to dark energy and modified gravity. \citet{Zhang_homog} and \citet{planck2013-XIII} have also used the kSZ effect to test the Copernican Principle and the homogeneity of the Universe. In addition, the impact of the kSZ effect on sub-cluster scales has also been investigated \citep{inogamov_ras, kdolag_ras13}.
With these motivations, there have been previous attempts to detect the kSZ effect in existing CMB data \citep[][]{kashlinsky08,kashlinsky10,lavaux_afshordi12,Handetal2012}. The results of \citet[][]{kashlinsky08,kashlinsky10}, which point to the existence of a bulk flow of large amplitude (800--1000\,km\,s$^{-1}$) extending to scales of at least 800 Mpc, have been disputed by a considerable number of authors \citep[e.g.,][]{keisler09,osborneetal11,modyandamir,planck2013-XIII,snbf2013}. On the other hand, \citet[][]{lavaux_afshordi12} have claimed the detection of the local bulk flow (within $80\,h^{-1}$\,Mpc) by using the kSZ in WMAP data in the direction of nearby galaxies. While these results remain at a low (roughly $1.7\,\sigma$) significance level, they are also in slight tension with the results that we present here. On the other hand, the work of \citet[][]{Handetal2012} constitutes the first clear detection of the kSZ effect (using the ``pairwise'' momentum approach that we describe below), and no other group has confirmed their results to date. In addition, \citet{kSZcso} have provided a first claimed detection of the kSZ effect in a single source. After the first data release of the \textit{Planck}\ mission, some disagreement has been claimed between the cosmological frame set by \textit{Planck}\ and measurements of peculiar velocities as inferred from redshift space distortions \citep[][]{macaulay13}. More recently, \citealt{Mueller2014} forecasted the detectability of the neutrino mass with the kSZ pairwise momentum estimator by using Atacama Cosmology Telescope (ACTPol) and Baryon Oscillation Spectroscopic Survey (BOSS) data.
This work represents the second contribution of the \textit{Planck}\
\footnote{\textit{Planck}\ (\url{http://www.esa.int/Planck}) is a project of the
European Space Agency (ESA) with instruments provided by two scientific
consortia funded by ESA member states and led by Principal Investigators
from France and Italy, telescope reflectors provided through a collaboration
between ESA and a scientific consortium led and funded by Denmark, and
additional contributions from NASA (USA).}
collaboration to the study of the kSZ effect, in which we focus on constraining the ``missing baryons'' with this signal \citep{dedeoetal05,chm08,hoetalkSZ09,chm09,kSZtomography1}. Numerical simulations~\citep{cenostriker2006} have shown that most of the baryons lie outside galactic halos, and in a diffuse phase, with temperature in range of $10^{5}$--$10^{7}$ K. This ``warm-hot'' intergalactic medium is hard to detect in X-ray observations due to the relatively low temperature. Thus, most of the baryons are ``missing" in the sense that they are neither hot enough ($T<10^{8}$~K) to be seen in X-ray observations, nor cold enough ($T>10^{3}$~K) to be made into stars and galaxies. However, as we can see in Eq.~(\ref{eq:kSZ1}), the kSZ signal is proportional to the gas density and peculiar velocity, and thus the gas temperature is irrelevant. Therefore the kSZ effect has been proposed as a ``leptometer," since it is sensitive to the ionized gas in the Universe while most of the baryons remain undetected.
In this work, we measure the kSZ effect through two distinct statistics: the kSZ pairwise momentum, which was used in \citet{Handetal2012} to detect kSZ effect for the first time; and the cross-correlation between the kSZ temperature fluctuations with the reconstructed radial peculiar velocities inferred from a galaxy catalogue. These analyses allow us to probe the amount of gas generating the kSZ signal and provide direct evidence of the elusive missing baryons in the local Universe. In Sect.~\ref{sec:datades} we describe the CMB data and the Central Galaxy Catalogue that will be used in our analysis. In Sect.~\ref{sec:method} we describe two statistical tools we use. Section~\ref{sec:results} presents and explains our results and illustrates the physical meaning of the significance. The conclusions and discussion is presented in the last section. Throughout this work we adopt the cosmological parameters consistent with \citet{planck2014-a15}: $\Omega_{\rm m} = 0.309$; $\Omega_{\Lambda}=0.691$; $n_{\rm s}=0.9608$; $\sigma_{8}=0.809$; and $h=0.68$, where the Hubble constant is $H_0=100\,h$\,km\,s$^{-1}$\,Mpc$^{-1}$.
\section{Data description}
\label{sec:datades}
\subsection{\textit{Planck}\ data}
This work uses \textit{Planck}\ data that are available publicly, both raw frequency maps and the CMB foreground cleaned maps.\footnote{\textit{Planck}'s Legacy Archive: \\\url{http://pla.esac.esa.int/pla/aio/planckProducts.html}} The kSZ effect should give rise to frequency-independent temperature fluctuations and hence constitutes a secondary effect with identical spectral behaviour as for the intrinsic CMB anisotropies. Therefore the kSZ effect should be present in all CMB frequency channels and in all CMB foreground-cleaned maps. However, one must take into account the different effective angular resolutions of each band, particularly when searching for a typically small-scale signal such as the kSZ effect. The raw frequency maps used here are the LFI 70\,GHz map, and the HFI 100, 143, and 217\,GHz maps. These have effective FWHM values of 13.01, 9.88, 7.18, and 4.87\,arcmin for the 70, 100, 143, and 217\,GHz maps, respectively.
The FWHM of the foreground cleaned products is 5\,arcmin for the maps used in this work, namely the {\tt SEVEM}, {\tt SMICA}, {\tt NILC}, and {\tt COMMANDER}\ maps. These maps are the output of four different component-separation algorithms. While the {\tt NILC}\ map is the result of an internal linear combination technique, {\tt SMICA}\ uses a spectral matching approach, {\tt SEVEM}\ a template-fitting method and {\tt COMMANDER}\ a parametric, pixel based Monte Carlo Markov Chain technique to project out foregrounds. We refer to the \textit{Planck}\ component separation papers for details in the production of these maps \citep{planck2014-a11,planck2014-a12}. In passing, we note that the foreground subtraction in those maps is not perfect, and that there exist traces of residuals, particularly on the smallest angular scales, related to radio, dust, and thermal Sunyaev-Zel'dovich [tSZ] emission.
For comparison purposes, we examine 9-year data from the {\it Wilkinson} Microwave Anisotropy Probe (\rm WMAP)\footnote{\rm WMAP: \url{http://map.gsfc.nasa.gov}}. In particular, we downloaded from the LAMBDA site\footnote{LAMBDA: \url{http://lambda.gsfc.nasa.gov}} the \rm WMAP\ satellite 9-year W-band (94\,GHz) map, with an effective FWHM of 12.4\,arcmin.
\subsection{Central Galaxy Catalogue}
\label{sec:cgc}
We define a galaxy sample in an attempt to trace the centres of dark matter halos. Using as a starting point the seventh data release of the Sloan Digital Sky Survey \citep[SDSS/DR7][]{Abazajianetal2009}, the Central Galaxy Catalogue (CGC) is composed of 262\,673 spectroscopic sources brighter than $r=17.7$ (the $r$-band extinction-corrected Petrosian magnitude). These sources were extracted from the SDSS/DR7 New York University Value Added Galaxy Catalogue \citep{Blanton2005} \footnote{NYU-VAGC: \url{ http://sdss.physics.nyu.edu/vagc/ }}, and we have applied the following isolation criterion: no brighter (in $r$-band) galaxies are found within 1.0\,Mpc in the transverse direction and with a redshift difference smaller than 1\,000\,km\,s$^{-1}$. The Sloan photometric sample has been used to remove all possible non-spectroscopic sources that may violate the isolation requirements. By using the ``photometric redshift 2" catalogue \citep[photoz2,][]{Cunhaetal2009}, all spectroscopic sources in our isolated sample with potential photometric companions within 1.0\,Mpc in projected distance, and with more than 10\,\% probability of having a smaller redshift than that of the spectroscopic object, are dropped from the Central Galaxy Catalogue.
Information provided in the NYU-VAGC yields estimates of the stellar mass content in this sample, and this enables us to make a direct comparison to the output of numerical simulations \citep{planck2012-XI}. We refer to this paper for further details on the scaling between total halo mass and stellar mass for this galaxy sample.
We expect most of our CGs to be the central galaxies of their dark matter halos \citep[again see][]{planck2012-XI}, just as bright field galaxies lie at the centres of their satellite systems and cD galaxies lie near the centres of their clusters. They are normally the brightest galaxies in their system. By applying the same isolation criteria to a mock galaxy catalogue based on the Munich semi-analytic galaxy formation model (see Sec.~\ref{sec:sims} for more details), we found that at stellar masses above $10^{11}$\,M$_\odot$, more than 83\,\% of CGs in the mock galaxy catalogue are truly central galaxies. For those CGs that are satellites, we have checked that at $\log( M_\ast/\rm{M}_\odot)>11$ about two-thirds are brighter than the true central galaxies of their halos, while the remainder are fainter, but are considered isolated because they are more than 1\,Mpc (transverse direction) from their central galaxies (60\,\%) or have redshifts differing by more than 1\,000 \,km\,s$^{-1}$ (40\,\%).
A case with stricter isolation criteria has been tested, in which we require no brighter galaxies to be found within 2.0\,Mpc in the transverse direction and with a redshift difference smaller than 2\,000\,km\,s$^{-1}$. Applying these stricter criteria to the same parent SDSS spectroscopic catalogue, we end up with a total of 110\,437 CGs, out of which which 58\,105 galaxies are more massive than $10^{11}$\,M$_\odot$. Thus about 30\,\% of the galaxies with $\log (\rm{M}_\ast/\rm{M}_\odot )>11$ have been eliminated from the sample. This new sample of CGs has a slightly higher fraction of centrals, reaching about 87\,\% at $\log (\rm{M}_\ast/\rm{M}_\odot)>11$. The improvement is small because (as we have checked) most of the satellite galaxies in the 1\,Mpc sample are brighter than the central galaxies of their own halos, and with the stricter criteria they are still included. Considering a balance between the sample size, which directly affects the signal-to-noise ratio in our measurements, and the purity of central galaxies, we choose the 1\,Mpc isolation criteria and the corresponding CGC sample in our analysis from here on. This sample amounts to 262\,673 sources over the DR7 footprint (about 6\,300\,deg$^2$, $f_{\rm sky}=0.15$).
\subsection{Numerical simulations}
\label{sec:sims}
In order to compare measurements derived from observations to theoretical
predictions, we make use of two different numerical simulations
of the large-scale-structure. We first look at the combination
of two hydrodynamical simulations, combining a constrained
realization of the local Universe and a large cosmological
simulation covering 1200\,$h^{-1}$\,Mpc. The simulations were performed
with the {\sc{GADGET-3}} code \citep{2001ApJ...549..681S,2005MNRAS.364.1105S},
which makes use of the entropy-conserving formulation of smoothed-particle hydrodynamics (SPH)
\citep{2002MNRAS.333..649S}.
These simulations include radiative cooling, heating by a uniform
redshift--dependent UV background \citep{1996ApJ...461...20H}, and a treatment of
star formation and feedback processes. The latter is based on a sub-resolution
model for the multiphase structure of the interstellar medium
\citep{2003MNRAS.339..289S} with parameters that have been fixed to obtain
a wind velocity of around $ 350$\,km\,s$^{-1}$. We used the code {\tt SMAC} \citep{dolag2005}
to produce full sky maps from the simulations. For the innermost shell (up to 90\,$h^{-1}$\,Mpc from the observer) we use the simulation of the local Universe, whereas the rest of the shells
are taken from the large, cosmological box. We construct full sky maps
of the thermal and kinetic SZ signals, as well as halo catalogues, by stacking
these consecutive shells through the cosmological boxes taken at the evolution
time corresponding to their distance. In this way, the simulations reach out to
a redshift of 0.22 and contain 13\,058 objects with masses above $10^{14}\,$M$_{\odot}$.
Provided that our CGC lies at a median redshift of 0.12, this combined simulation should be
able to help us interpret the clustering properties of peculiar velocities of highly biased, massive halos, on the largest scales. For more details on the procedure for using the simulations see \citet{kdolag_ras13}. This simulated
catalogue will hereafter be referred to as the {\tt CLUSTER}\ catalogue.
We have also used a mock galaxy catalogue, based on the semi-analytic galaxy formation simulation of \cite{2013MNRAS.428.1351G}, which is implemented on the very large dark matter Millennium simulation \citep{2005Natur.435..629S}. The Millennium simulation follows the evolution of cosmic structure within a box of side 500\,$h^{-1}$\,Mpc (comoving), whose merger trees are complete for subhalos above
a mass resolution of $1.7 \times 10^{10}$\,$h^{-1}$\,M$_\odot$. Galaxies are assigned to dark matter halos following the model recipes described in \cite{2011MNRAS.413..101G}. The rescaling technique of \cite{2010MNRAS.405..143A} has been adopted to convert the Millennium simulation, which is originally based on \rm WMAP-1 cosmology, to the \rm WMAP-7 cosmology. The galaxy formation parameters have been adjusted to fit several statistical observables, such as the luminosity, stellar mass, and correlation functions for galaxies at $z=0$.
We project the simulation box along one axis, and assign every galaxy a redshift based on its line of sight (LOS) distance and peculiar velocity, i.e., parallel to the LOS axis. In this way we can select a sample of galaxies from the simulation using criteria exactly analogous to those used for our CGC based on SDSS. A galaxy is selected if it has no brighter companions within 1\,Mpc in projected distance and 1\ 000\,km\,s$^{-1}$ along the LOS. This sample of simulated galaxies will be referred as the {\tt GALAXY}\ catalogue.
\section{Methodology}
\label{sec:method}
We will search for kSZ signatures in \textit{Planck}\ temperature maps by implementing two different statistics on the CMB data. The first statistic aims to extract the kSZ pairwise momentum by following the approach of \citet{Handetal2012}, which was inspired by \citet{groth89} and \citet{roman98}. The second statistic correlates the kSZ temperature anisotropies estimated from \textit{Planck}\ temperature maps with estimates of the radial peculiar velocities. These velocity estimates are obtained after inverting the continuity equation relating galaxy density with peculiar velocities, as suggested by \citet{dedeoetal05} and \citet{hoetalkSZ09}. The particular inversion methods applied to our data are described in \citet[][hereafter K12]{shuandangulo12}. Throughout this work we use the {\tt HEALPix} software \citep[][]{healpix}\footnote{\url{http://healpix.jpl.nasa.gov}} to deal with the CMB maps.
\subsection{The pairwise kSZ momentum estimator}
\citet{ferreiraetal99} developed an estimator for pairwise momentum with weights depending only on line-of-sight quantities. With this motivation we use a similar weighting scheme on our CMB maps.
The pairwise momentum estimator combines information on the relative spatial distance of pairs of galaxies with their kSZ estimates in a statistic that is sensitive to the gravitational infall of those objects. Specifically we use
\begin{equation}
\hat{p}_{\rm kSZ} (r) = -\frac{\sum_{i<j}(\delta T_{ i} - \delta T_{ j} )\,c_{ i,j}}{\sum_{ i<j} c_{ i,j}^2},
\label{eq:pksz1}
\end{equation}
where the weights $c_{ i,j}$ are given by
\begin{equation}
c_{ i,j} = \hat{\vec{r}}_{ i,j} \cdot \frac{\hat{\vec{r}}_{ i}+\hat{\vec{r}}_{ j}}{2} =
\frac{(r_{ i}-r_{ j})(1+\cos\theta)}{2\sqrt{r_{ i}^2 + r_{ j}^2 - 2r_{ i}r_{ j}\cos\theta}}.
\label{eq:cweight}
\end{equation}
Here ${\vec{r}}_{ i}$ and ${\vec{r}}_{ j}$ are the vectors pointing to the positions of the $i$th and $j$th galaxies on the celestial sphere, $r_{ i}$ and $r_{ j}$ are the comoving distances to those objects, and ${\vec{r}}_{ i,j} = \vec{r}_{ i}-\vec{r}_{ j}$ is the distance vector for this galaxy pair. The {\it hat} symbol ($\hat{\vec{r}}$) denotes a unit vector in the direction of $\vec{r}$, and $\theta$ is the angle separating $\hat{\vec{r}}_{ i}$ and $\hat{\vec{r}}_{ j}$. We also note that $\hat{\vec{r}}_{i,j}$ means $(\vec{r}_i-\vec{r}_j)/|\vec{r}_i-\vec{r}_j|$. The sum is over all galaxy pairs lying a distance $r_{ i,j}$ falling in the distance bin assigned to $r$. The quantity $\delta T_{ i}$ denotes a {\em relative} kSZ temperature estimate at the position of the $i$th galaxy:
\begin{equation}
\delta T_{ i} = T_{\rm AP}(\hat{\mbox{\vec{n}}}_{ i}) - \bar{T}_{\rm AP}(z_i,\sigma_{ z}).
\label{eq:dT}
\end{equation}
In this equation, the symbol $T_{\rm AP}(\hat{\mbox{\vec{n}}}_{ i})$ corresponds to the kSZ amplitude estimate obtained at the angular position of the $i$th galaxy with an aperture photometry (AP) filter, while $\bar{T}_{\rm AP}(z_i,\sigma_{ z})$ denotes the average kSZ estimate obtained from {\em all} galaxies after weighting by a Gaussian of width $\sigma_{ z}$ centred on $z_{ i}$:
\begin{equation}
\bar{T}_{\rm AP}(z_i,\sigma_{ z}) =
\frac{\sum_{ j} {T_{\rm AP}(\hat{\mbox{\vec{n}}}_{ j}) \exp{(-\frac{( z_{ i}-z_{ j} )^2}{2\sigma^2_{ z}} } )}}{\sum_{ j} \exp{(-\frac{( z_{ i}-z_{ j} )^2}{2\sigma^2_{ z}} )} }.
\label{eq:Tav}
\end{equation}
In this AP approach one computes the average temperature within a given angular radius $\theta$, and subtracts from it the average temperature in a surrounding ring of inner and outer radii $[\theta, f\theta]$, with $f>1$. In our case we use $f=\sqrt{2}$ \citep[see, e.g., ][]{planck2013-XIII}. If we knew accurately the spatial gas distribution on these scales there would clearly be room for more optimal approaches, like a matched filter technique \citep{Liangulowhite2014}. But on these relatively small scales the density profile in halos seems to be significantly less regular than the pressure profile \citep[see, e.g.,][]{arnaudetal10},
and this adds to the fact that gas outside halos, not necessarily following any given profile, also contributes to the signal. Furthermore, we would like to conduct an analysis that is as model independent as possible, and thus blind to any specific model for the gas spatial distribution.
The correction by a redshift-averaged quantity introduced in the last equation above was justified by \citet{Handetal2012} through the need to correct for possible redshift evolution of the tSZ signal in the sources. In practice, this correction allows us to measure {\em relative} changes in temperature anisotropies between nearby galaxy pairs after minimizing other noise sources, such as CMB residuals.
\begin{figure}
\centering
\includegraphics[width=9.cm]{./Figures/fig_Tapmono.eps}
\caption[fig:Tapmono]{ Absolute value of $\bar{T}_{\rm AP}(z(r),\sigma_{ z}) $ versus distance to the observer after obtaining AP kSZ temperature estimates on the real positions of CGs (black line) and on a rotated CG configuration (red line). The green line provides the theoretical expectation for the rms of $\bar{T}_{\rm AP}(z(r),\sigma_{ z})$, and the blue histogram displays the radial distribution of the CGs (see text for further details). }
\label{fig:Tapmono}
\end{figure}
We next conduct an analysis into the motivation behind this $z$-dependent monopole correction. In
Fig.~\ref{fig:Tapmono} we display the amplitude of $| \bar{T}_{\rm AP}(z(r),\sigma_{ z})|$ for $\sigma_z=0.01$ versus the comoving distance to the observer $r$. The value of the aperture chosen in this exercise is 8\,arcmin. The black line refers to the real position of the CGs, while the red line refers to a rotated-on-the-sky configuration of the CGs. Since we are plotting absolute values of $\bar{T}_{\rm AP}(z,\sigma_{ z})$, the solid (dashed) parts of these lines refer to positive (negative) values of $\bar{T}_{\rm AP}(z,\sigma_{ z})$. The amplitude of these curves is close to the solid green line, which corresponds to the theoretical prediction for the rms of $\bar{T}_{\rm AP}(z,\sigma_{ z})$ under the assumption that the AP temperature estimates are dominated by CMB residuals. That is, the amplitude of the green curve equals $\sigma [\delta T] / \sqrt{N_{\rm CG}(r)}$, where $\sigma [\delta T] \simeq 20\,\mu$K is the rms of the 8\,arcmin AP kSZ estimates and $N_{\rm CG}(r)$ is the number of CGs effectively falling under the redshift Gaussian window given in Eq.~(\ref{eq:Tav}). While $\sigma [\delta T]\simeq 20\,\mu$K is computed from real data, its amplitude is very close to the theoretical predictions of Fig.~6 in \citet{chm05}, for an angular aperture of 8\,arcmin after considering the CMB exclusively. We can see that the amplitude of $| \bar{T}_{\rm AP}(z,\sigma_{ z})|$ is lowest for those distances where the number density of CGs is highest (as displayed, in arbitrary units, by the blue histogram in Fig.~\ref{fig:Tapmono}).
Therefore Fig.~\ref{fig:Tapmono} shows not only that AP kSZ estimates are dominated by CMB residuals, but also that the difference in $\bar{T}_{\rm AP}(z,\sigma_{ z})$ for CG pair members lying 50--100\,$h^{-1}$\,Mpc away in radial distance typically amounts to few times $0.01\,\mu$K. We shall show below that this amplitude is not completely negligible when compared to the typical kSZ pairwise momentum amplitude between CGs at large distances, and thus subtracting $\bar{T}_{\rm AP}(z,\sigma_{ z})$ becomes necessary. However, this should not be the case when cross-correlating AP kSZ measurements with estimates of radial peculiar velocities (see Sect.~\ref{sec:invertrho} below), since in this case the residual $\bar{T}_{\rm AP}(z,\sigma_{ z})$ will not be correlated with those velocities and should not contribute to this cross-correlation.
Like \citet{Handetal2012}, our choice for $\sigma_z$ is $\sigma_z=0.01$, although results are very similar if we change this by a factor of 2. Adopting higher values introduces larger errors in the peculiar momentum estimates, and smaller values tend to suppress the power on the largest scales; we adopt $\sigma_z=0.01$ as a compromise value. It is worth noting that any effect giving rise to a pair of $\delta T_i, \delta T_j$, whose difference is not correlated to the relative distance of the galaxies $r_i-r_j$, should not contribute to Eq.~(\ref{eq:pksz1}).
\subsection{Recovering peculiar velocities from the CGC galaxy survey}
\label{sec:invertrho}
There is a long history of the use of the density field to generate estimates of peculiar velocities \citep[e.g.,][for some early studies]{dekel93,nusserdavis94,fisherlahavhoffman95,zaroubietal95}.
\citet{dedeoetal05} and \citet{hoetalkSZ09} first suggested inverting the galaxy density field into its peculiar velocity field in the context of kSZ studies. In this section, we use the approach of K12 to obtain estimates of the peculiar velocity field from a matter density tracer. The work of K12 is based upon the analysis of the Millennium simulation, where full access to all dark matter particles was possible. In our case the analysis will obviously be limited to the use of the CGC catalogue, and this unavoidably impacts the performance of the algorithms.
Inspired by K12, in our analysis we conduct three different approaches to obtain the velocity field. The first one is a simple inversion of the linear continuity equation of the density contrast obtained from the galaxies in the CGC: this will be hereafter called the LINEAR approach. It makes use of the continuity equation at linear order,
\begin{equation}
\frac{\partial \delta (\vec{x})}{\partial t} + \nabla\mbox{\bf {v}} (\vec{x} ) = 0,
\label{eq:conteq}
\end{equation}
where $\mbox{\bf {v}} (\vec{x} )$ is the peculiar velocity field and $\delta (\vec{x} )$ is the matter density contrast. In our case, however, we observe the galaxy density contrast $\delta^{\rm g} (\vec{x} )$, and thus the amplitude of the estimated velocity field is modulated by the bias factor $b$ relating $\delta^{\rm g} (\vec{x} )$ and $\delta (\vec{x})$, $\delta^{\rm g} (\vec{x}) = b \delta (\vec{x})$. The bias factor $b$ is assumed to be constant on the scales of interest.
The second approach performs the same inversion of the linear continuity equation, but on a {\em linearized} estimate of the density field. This linearized field is obtained after computing the natural logarithm of unity plus the density contrast of the galaxy number density field and subsequently removing its spatial average \citep[][]{neynricketal09},
\begin{equation}
\delta^{\rm LOG} (\vec{x} ) = \ln{\left(1+ \delta^{\rm g} (\vec{x})\right) } - \langle \ln{\left(1+ \delta^{\rm g}\right) }\rangle_{\rm spatial}.
\label{eq:loglin1}
\end{equation}
In this expression, $\delta^{\rm g}(\vec{x})$ denotes again the density contrast of the galaxy number density at position $\vec{x}$. This approach will be referred to as LOG-LINEAR. Finally, second-order perturbation theory \citep[see, e.g.,][]{bouchetetal95} was applied on this linearized field, yielding a third estimate of the peculiar velocity field (the LOG-2LPT approach). We refer to K12 for details on the implementations of the three approaches. For the sake of simplicity, we discuss results for the LINEAR approach, and leave the corresponding discussion of the two other methods for an appendix. We have also tried using the full SDSS spectroscopic sample (rather than the CGC) on the same volume to recover the peculiar velocity field; the full galaxy sample should be a better proxy for the dark matter density field than the CGC. However, after testing this with an enlarged version of the {\tt GALAXY}\ catalogue (by considering all halos above a mass threshold of $10^{10.8}$\,M$_{\odot}$), we obtain negligible differences with respect to the mock CG. Likewise, we obtain practically the same results when using the real CGC and the full spectroscopic sample. Therefore, for the sake of simplicity, we restrict our analysis to the CGC. All these methods make use of FFTs requiring the use of a 3D spatial grid when computing galaxy number densities. We choose to use a grid of 128$^3$ cells, each cell being 4\,$h^{-1}$\,Mpc on a side. This cell size is well below the scales where the typical velocity correlations are expected (above 40\,$h^{-1}$\,Mpc), and comparable with the positional shifts induced by the redshift space distortions (about 3\,$h^{-1}$\,Mpc for a radial velocity of 300\,km\,s$^{-1}$).
These distortions will be ignored hereafter, since they affect scales much smaller than those of interest in our study (roughly $20\,h^{-1}$\,Mpc and above).
We note that these distortions can be corrected in an iterative fashion (see the pioneering work of \citealt[][]{Yahiletal1991} within the linear approximation; \citealt[][]{Kitauragalleranietal2012} for the lognormal model, or \citealt{Kitauraerdogduetal2012} including non-local tidal fields). Nevertheless, our tests performing such kinds of correction on the mock catalogue yield a very minor improvement on the scales of interest.
\begin{figure}
\centering
\includegraphics[width=10.cm]{./Figures/fig_pecvelrecon.eps}
\caption[fig:velrec]{ Reconstruction of the $x$-component of the peculiar velocity field (arrows) in a narrow slice of the grid containing the CGC over the corresponding density contrast contour 2D plot. }
\label{fig:velrec}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=9.cm]{./Figures/fig_corrcoeff.eps}
\caption[fig:rk]{ Correlation coefficient of the recovered line-of-sight velocities with the actual ones in our {\tt GALAXY}\ mock catalogue. Solid, dotted, and dashed lines refer to the LINEAR, LOG-LINEAR, and LOG-2LPT approaches, respectively. The red lines consider the ideal scenario without any sky mask or selection function, while the black ones are for the same sky mask and selection function present in the real CGC.}
\label{fig:rk}
\end{figure}
After aligning the $X$- and $Z$-axes of the 3D grid with the zero Galactic longitude and zero Galactic co-latitude axes, respectively, we place our grid at a distance vector $\vec{R}_{\rm box}= [-300, -250, 150]$\, $h^{-1}$\,Mpc from the observer. This position vector locates the corner of the 3D grid that constitutes the origin for labelling cells within the box. This choice of $\vec{R}_{\rm box}$ is motivated by a compromise between having as many grid cells in the CGC footprint as possible, and keeping a relatively high galaxy number density. Placing the 3D grid at a larger distance would allow us to have all grid cells inside the CGC footprint, but at the expense of probing distances where the galaxy number density is low (due to the galaxy radial selection function being low as well). Our choice for ${\vec R}_{\rm box}$ results in about 150\,000 CGs being present in the box, and about 82\,\% of the grid cells falling inside the CGC footprint.
The three approaches will provide estimates of the peculiar velocity field in each grid cell, $\mbox{\bf {v}}^{\rm rec}(\vec{x} )$. From these, it is straightforward to compute the radial component as seen by the observer, $v_{\rm los}^{\rm rec} (\vec{x})$, and to assign it to all galaxies falling into that grid cell.
In Fig.~\ref{fig:velrec} arrows show the LINEAR reconstruction of the $x$-component of the peculiar velocity field from the CGC, for a single $z$-slice of data. The coloured contour displays the galaxy density contrast distribution over the same spatial slice.
The methodology outlined in K12 was conducted in the absence of any sky mask or selection function. In our work we address these aspects of the real data by means of a Poissonian data augmentation approach. In a first step, all grid cells falling outside the CGC footprint are populated, via Poissonian realizations, with the average number of galaxies dictated by the CGC radial selection function computed from cells inside the footprint. In this way we fill all holes in the 3D grid. Following exactly the same procedure, we next radially augment the average number of counts in cells in such a way that the radial selection function of the resulting galaxy sample is constant. In this way we avoid radial gradients that could introduce spurious velocities along the line of sight.
In order to test this methodology, we use the {\tt GALAXY}\ mock catalogue, in two different scenarios. The first scenario is an ideal one, with no mask or selection function: we apply the three approaches on a box populated with our mock catalogue, and compare the recovered radial velocities with the real ones provided by the catalogue. When performing this comparison, we evaluate a 3D grid for the original galaxy radial velocity $v_{\rm los}^{\rm orig}$ by assigning to each cell the radial velocity of the galaxy falling nearest to the cell centre, that is, we adopt a {\it nearest particle} (NP) method. For different radial bins of the $\vec{k}$ wavevector, we evaluate the correlation coefficient ($r_k$) computed from the cross-power spectrum between the recovered and the original radial velocity fields and their respective auto-spectra:
\begin{equation}
r_k = \frac{P^{\rm \,orig,\,rec}(k)}{\sqrt{P^{\rm \,orig,\, orig}(k)\, P^{\rm \,rec,\,rec}(k)}}.
\label{eq:rk}
\end{equation}
In this expression, $P^{X,\,Y}(k)$ stands for $\langle v(\vec{k})_{\rm los}^{ X} (v(\vec{k})_{\rm los}^{ Y})^\ast \rangle$, that is, the radially averaged power spectrum of the line of sight velocity modes. The superscripts $\{ X,\,Y\}=\{ {\rm orig, rec}\}$ stand for ``original" and ``recovered" components, respectively, and the asterisk denotes the complex conjugate operation.
In Fig.~\ref{fig:rk} the red lines display the correlation coefficients for the ideal scenario. Solid, dotted, and dashed lines refer to the LINEAR, LOG-LINEAR, and LOG-2LPT approaches, respectively. On large scales (low $k$ modes), the three approaches provide correlation coefficients that are very close to unity, while they seem to lose information on small scales in the same way. We note that a direct comparison to the results of K12 is not possible, since, in our case, we do not use the full dark matter particle catalogue, but only a central halo one.
After the real mask and selection function obtained from the CGC is applied to the {\tt GALAXY}\ mock catalogue, then the recovery of the peculiar velocities from the three adopted approaches worsens considerably. The black lines in Fig.~\ref{fig:rk} display, in general, much lower correlation levels than the red ones. The LINEAR approach seems to be the one that retains most information on the largest scales, and it out-performs the LOG-LINEAR and the LOG-2LPT methods on all scales. We hence expect the LINEAR approach to be more sensitive to the kSZ effect than the other two methods, particularly on the largest scales.
Once the velocity inversion from the CGC density contrast has been performed, we compute the spatial correlation function between the recovered velocities and the kSZ temperature anisotropies,
\begin{equation}
w^{T,v} (r) = \langle \delta T_i v_{\rm los}^{\rm rec} (\vec{x}_j) \rangle_{i,j} (r),
\label{eq:Ctv}
\end{equation}
where $\delta T_i$ is estimated as in Eq.~(\ref{eq:dT}), and the ensemble average is obtained after running through all galaxy pairs $\{i,j\}$ lying a distance $r$ away.
\subsection[]{Template fitting}
When studying the measurements of the kSZ pairwise momentum and kSZ momentum-$v_{\rm los}^{\rm rec}$ correlation, we perform fits to estimates obtained from our numerical simulations. That is, we minimize the quantity
\begin{equation}
\chi^2 = \sum_{ i,j} \bigl(\hat{w}^{ X}(r_{ i}) - A^{ X}\, \tilde{w}^{ X,\,{\rm sim}}(r_{ i})\bigr)\, \tens{C}_{ i j}^{-1} \, \bigl(\hat{w}^{ X}(r_{ j}) - A^{ X}\, \tilde{w}^{ X,\,{\rm sim}}(r_{ j})\bigr),
\label{eq:chis1}
\end{equation}
where the indexes ${i,j}$ run over different radial bins, $\hat{w}^{ X}(r_{ i})$ is the measured quantity in the $i$th radial bin (with ${ X}$ either denoting kSZ pairwise momentum or the kSZ temperature-recovered velocity correlation), and $\tilde{w}^{ X\, {\rm sim}}(r_{ i})$ refers to its counterpart measured in the numerical simulation. The symbol $\tens{C}_{ i j} $ denotes the $i j$ component of the covariance matrix $\tens{C}$ that is computed from the \textit{Planck}\ maps after estimating $\hat{w}^{ X}$ for {\em null} positions where no SZ effect is expected\footnote{These null positions on the \textit{Planck}\ maps correspond to rotated or displaced positions with respect to the original location of the CGs, as will be explained in the next section.}. This minimization procedure provides formal estimates for the amplitude $A^{ X}$ and its associated errors:
\begin{eqnarray}
A^{ X} & = & \frac{ \sum_{ i,j} \hat{w}^{ X}(r_{ i}) \, C_{ i,j}^{-1} \, \tilde{w}^{ X,\,{\rm sim}}(r_{ j}) } {\sum_{ i,j} \tilde{w}^{ X,\, {\rm sim}}(r_{ i}) \, C_{ i,j}^{-1} \, \tilde{w}^{ X,\,{\rm sim}}(r_{ j})};
\label{eq:Aanderror1}
\\
\sigma^2_{A^{ X}} & = & \frac{1}
{\sum_{ i,j} \tilde{w}^{ X,\, {\rm sim}}(r_{ i}) \, C_{ i,j}^{-1} \, \tilde{w}^{ X,\,{\rm sim}}(r_{ j}) }.
\label{eq:Aanderror2}
\end{eqnarray}
Most of the information is located at short and intermediate distances, where the estimated statistic $\hat{w}^{ X}(r)$ differs most from zero, as can be seen in Figs.~\ref{fig:kSZ_cgc} and \ref{fig:clusvsap}. We also test the null hypothesis, that is, we measure the $\chi^2$ statistic (defined in Eq.~\ref{eq:chis1} above) for the particular case of $A^{ X}=0$ and estimate the significance level at which such a value (denoted by $\chi^2_{\rm null}$) is compatible with this null hypothesis. In these cases, we quote the significance as the number of $\sigma$ with which the null hypothesis is ruled out under Gaussian statistics. The array of distance bins on which the covariance matrix in Eq.~(\ref{eq:chis1}) is computed is chosen evenly in the range 0--150\,$h^{-1}$\,Mpc. However, we consider only three separate points centred upon 16, 38, and 81\,$h^{-1}$\,Mpc when computing (conservative) statistical significances. In this way we minimize correlation among radial bins to ensure that the inversion of the covariance matrix is stable. We have checked that, for the adopted set of distance bins, random variations at the level of 10\,\% of the measured $\hat{w}^{ X} (r)$ do not compromise the stability of the recovered significance estimates. In other words, we explicitly check that 10\,\% fluctuations on the measured $\hat{w}^{ X} (r)$ introduces fluctuations at a similar level in the $\chi^2$ estimates (more dramatic changes in the $\chi^2$ estimates would point to singular or quasi-singular inverse covariance matrices).
\section{Results}
\label{sec:results}
\begin{figure*}
\centering
\includegraphics[width=18.cm]{./Figures/kSZpanels_SEVEM.eps}
\caption[fig:kSZ_cgc]{Computation of the kSZ pairwise momentum for the CGC sample. The top row, from left to right, displays the results for different aperture choices on the raw HFI 217\,GHz map, namely 5, 8, 12, and 18\,arcmin. The top row also shows the analysis for the foreground-cleaned {\tt SEVEM}\ map, displayed with blue squares. The fit to the pairwise momentum templates from the {\tt CLUSTER}\ catalogue is also displayed by the solid red line, and the {\tt GALAXY}\ catalogue by the green line. The bottom row presents the results at a fixed aperture of 12\,arcmin for different frequency maps, including \rm WMAP-9 W-band data (red squares). }
\label{fig:kSZ_cgc}
\end{figure*}
\subsection{The kSZ pairwise momentum}
\label{sec:pkSZ}
As mentioned above, the CGC was used previously in \citet{planck2012-XI} to trace the tSZ effect versus stellar mass down to halos of size about twice that of the Milky Way. In this case we use the full CGC to trace the presence of the kSZ signal in \textit{Planck}\ data, since our attempts with the most massive sub-samples of the CGC yield no kSZ signatures.
While the tSZ effect is mostly generated in collapsed structures \citep{Hernandezetal2006a} because it traces gas pressure, the kSZ effect instead is sensitive to {\em all} baryons, regardless of whether they belong to a collapsed gas cloud or not. Thus it is expected that not only will the virialized gas in halos contribute to the kSZ signal, but also the baryons surrounding those halos and moving in the same bulk flow. Rather than using a particular gas density profile, we choose not to make any assumption about the spatial distribution of gas around CGs.
We thus adopt an AP filter of varying apertures around the positions of CGC galaxies. The minimum aperture we consider is close to the resolution of \textit{Planck}\ (5\,arcmin), and we search for signals using increasing apertures of radii 5, 8, 12, and 18\,arcmin. This is equivalent to probing spheres of radius ranging from 0.5 up to 1.8\,$h^{-1}$\,Mpc (in physical units) around the catalogue objects. The result of the kSZ pairwise momentum estimation on the raw 217\,GHz HFI map is displayed in the top row of Fig.~\ref{fig:kSZ_cgc} for the four apertures under consideration. The recovered momenta for the raw frequency maps (displayed by black circles) provide some evidence for kSZ signal for apertures smaller than 12\,arcmin: below a distance of 60\,$h^{-1}$\,Mpc all points are systematically below zero, some beyond the $2\,\sigma$ level. Although the $\chi^2_{\rm null}$ test does not yield significant values (at $0.4\,\sigma, 0.3\,\sigma,0.7\,\sigma$ and $-1.2\,\sigma$ for 5, 8, 12, and 18\,arcmin apertures, respectively), the fits to the {\tt GALAXY}\ peculiar momentum template (displayed by the green solid line in the second-from-the-left panel in the top row) yield S/N $=A^{p_{\rm kSZ}}/\sigma^{p_{\rm kSZ}}=1.7, 1.4, 1.9$ and 0.1 for 5, 8, 12, and 18\,arcmin apertures, respectively. We obtain similar levels for the peculiar momentum template obtained from the {\tt CLUSTER}\ simulation (red line in second-from-the-left panel in top row). When using the {\tt SEVEM}\ clean map, these significances go beyond the $2\,\sigma$ level: S/N $=A^{p_{\rm kSZ}}/\sigma^{p_{\rm kSZ}}=2.5, 1.8, 2.2$ and 0.0 for 5, 8, 12, and 18\,arcmin apertures for the {\tt GALAXY}-derived template, respectively. If we repeat the analysis at 12\,arcmin aperture for the {\tt SMICA}\, {\tt NILC}\ and {\tt COMMANDER}\ maps, we obtain S/N$=A^{p_{\rm kSZ}}/\sigma^{p_{\rm kSZ}}=2.1, 2.2,$ and 2.1, respectively.
\begin{figure}
\centering
\includegraphics[width=9.cm]{./Figures/fig_kDolagkSZ_clus_vs_aperture.eps}
\caption[fig:clusvsap]{ {\it (Filled circles):} Measured kSZ pairwise momentum for the kSZ map derived from the {\tt CLUSTER}\ simulation after considering different radial apertures on a subset of clusters in the mass range (1--2)\,$\times 10^{14}$\, $h^{-1}$\,M$_{\odot}$. {\it (Black triangles):} Pairwise momentum computed from the {\tt GALAXY}\ mock catalogue (filled triangles) after assigning these galaxies a tSZ amplitude following a mass scaling inspired by the tSZ measurements of the CGC given in \citet{planck2012-XI}.}
\label{fig:clusvsap}
\end{figure}
We note at this point that the behaviour of the kSZ evidence for the CGs as displayed in Fig.~\ref{fig:kSZ_cgc} is significantly different to what is found in Fig.~\ref{fig:clusvsap} for the {\tt CLUSTER}\ simulation with halos in the mass range $(1$--$2)\,\times\, 10^{14}\,h^{-1}$\,M$_{\odot}$. In this simulation, most of the kSZ signal is coming from the halos themselves, and thus increasing the aperture to values larger than the virial radius of the clusters results in a dilution of the kSZ pairwise momentum amplitude. For real data, we find that the amplitude of the signal does not show significant changes when increasing the aperture from 5\,arcmin up to 12\,arcmin.
It is worth adding now that the black filled triangles in Fig.~\ref{fig:clusvsap} represent the peculiar momentum measured on the {\tt GALAXY}\ mock catalogue after assigning to these galaxies a tSZ temperature fluctuation following the tSZ versus mass scaling measured in the CGC \citep{planck2012-XI}. This tSZ amplitude assumes an observing frequency of $\nu_{\rm obs} = 100$\,GHz, and hence provides a conservative estimate of the tSZ contamination, which turns out to be at the level of $-3\times 10^{-3}$\,$\mu$K, i.e., about a factor of 30 below the measured amplitude on the real CG sample.
We next investigate the spectral stability of the signal for 12\,arcmin apertures. This choice is motivated by the fact that the LFI 70\,GHz and the HFI 100 and 143\,GHz channels have lower angular resolution, comparable or bigger than 8\,arcmin, and this may compromise the comparison with the 217\,GHz channel. This analysis is shown in the bottom row of Fig.~\ref{fig:kSZ_cgc}. The signal obtained for the raw 217\,GHz channel is found to be remarkably similar to what we obtain at 143, 100\,GHz in HFI: the fits to the kSZ pairwise momentum template from the {\tt GALAXY}\ simulation gives S/N $=A^{\rm p_{kSZ}}/\sigma^{\rm p_{kSZ}}=2.2$ and 2.1, respectively. The LFI 70\,GHz channel, on the other hand, seems to show a much flatter pattern, and this could be due to a larger impact of instrumental noise: the tSZ should also be present at 100 and 143\,GHz, and, as expected, gives rise to negligible changes. The effective full width half maximum (FWHM) for the 70\,GHz channel is close to 13\,arcmin, and this may also contribute to the difference with respect to the other HFI channels.\\
In the bottom row, second panel of Fig.~\ref{fig:kSZ_cgc}, the red squares provide the measurement obtained from the cleaned W-band map of \rm WMAP-9. Surprisingly, the level of anti-correlation for distances below 30\,$h^{-1}$\,Mpc appears higher for \rm WMAP\ data than for the \textit{Planck}\ channels: the first three radial bins lie at the $2.3$--$3.3\,\sigma$ level, and a fit to the {\tt GALAXY}\ template of the pairwise momentum yields S/N $=A^{\rm p_{kSZ}}/\sigma^{\rm p_{kSZ}}=3.3$ as well.
The angular resolution of the W-band in \rm WMAP\ is close to that of the LFI 70\,GHz channel, and the non-relativistic tSZ changes by less than 13\,\% between those channels. If any other frequency-dependent contaminants (which should be absent in the \rm WMAP\ W-band) are responsible for this offset, then they should also introduce more changes at the other frequencies. Thus the reason for the deeper anti-correlation pattern found in the W-band of \rm WMAP-9 remains unclear. Overall, we conclude that the \textit{Planck} results for the pairwise peculiar momentum are compatible with a kSZ signal, at a level ranging between 2 and $2.5\,\sigma$. The output from \rm WMAP-9 W band is also compatible with a kSZ signal, with a statistical significance of $3.3\,\sigma$.
\begin{figure}
\centering
\includegraphics[width=9.cm]{./Figures/fig_dTdv_IV_allvels_rot51_LINEAR.eps}
\caption[fig:dTdv]{Measured cross-correlation function between the kSZ temperature estimates and the recovered radial peculiar velocities, $w^{{\rm T},v_{\rm los}^{\rm rec}} (r)$, according to the LINEAR approach. The recovered velocities are divided with their rms dispersion $\sigma_{\rm v}=310\,$km\,s$^{-1}$, so kSZ temperature estimates are correlated to a quantity of variance unity.
This plot corresponds to an aperture of 8\,arcmin. Filled coloured circles correspond to $w^{{\rm T},v_{\rm los}^{\rm rec}} (r)$ estimates from different CMB maps ({\tt SEVEM}, {\tt SMICA}, {\tt NILC}, {\tt COMMANDER}, and the HFI 217\,GHz map). The dotted lines display the null estimates obtained after computing kSZ temperature estimates for rotated positions on the {\tt SEVEM}\ map, and the thick dashed line displays the average of the dotted lines. Error bars are computed from these null estimates of the correlation function. The solid line provides the best fit of the {\tt SEVEM}\ data to the theoretical prediction for $w^{{\rm T},v_{\rm los}^{\rm rec}} (r)$ obtained from the {\tt GALAXY}\ mock catalogue. These predictions are obtained using only a relatively small number of mock halos, and hence their uncertainty must be considered when comparing to the data.}
\label{fig:dTdv}
\end{figure}
\subsection{Cross-correlation analysis with estimated peculiar velocities}
\label{sec:xcorrvelsTs}
\begin{figure}
\centering
\includegraphics[width=9.cm]{./Figures/fig_dTdv_shuffled.eps}
\caption[fig:shuff]{Same as in Fig.~\ref{fig:dTdv}, but with the null correlation functions (displayed by dotted lines) being estimated after assigning the recovered linear velocity estimate of a given CG to any other CG in the sample (that is, by $v_{\rm los}^{\rm rec}$ ``shuffling"). Only the LINEAR approach is displayed in this plot.
}
\label{fig:shuff}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=9.cm]{./Figures/fig_dTdv_SEVEM_AP_5_8_12_18am_LINEAR_CGCIV_nrot50_smple.eps}
\caption[fig:wvsAPrad]{ Dependence on the aperture radius of the $w^{{\rm T},v_{\rm los}^{\rm rec}} (r)$ correlation function obtained in the LINEAR approach for the {\tt SEVEM}\ map. Error bars are estimated from the rms of the null correlation functions computed from rotated kSZ temperature estimates. }
\label{fig:wvsAPrad}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=9.cm]{./Figures/plot_chisq_vs_AP_III.eps}
\caption[fig:chisqvsap]{ Evaluations of the $\chi^2$ statistic of the $w^{{\rm T},v_{\rm los}^{\rm rec}} (r)$ correlation function (with respect to the null hypothesis) for different angular apertures. Only results for the foreground-cleaned {\tt SEVEM}\ map are shown. There is evidence for kSZ signal in a wide range of apertures above the FWHM, with a local maximum close to 8\,arcmin. This roughly corresponds to a radius of 0.8\,$h^{-1}$\,Mpc from the CG positions at the median redshift of the sample. }
\label{fig:chisqvsap}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=9.cm]{./Figures/fig_dTdv_HFI_renorm5am.eps}
\caption[fig:wTv]{Variation of the recovered $w^{{\rm T},v_{\rm los}^{\rm rec}} (r)$ correlation function for the four lowest frequency raw HFI maps after an effective convolution by a Gaussian beam of FWHM $=5$\,arcmin. Error bars here are computed in the same way as in Figs.~\ref{fig:dTdv} and \ref{fig:wvsAPrad}. }
\label{fig:wTv}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=9.cm]{./Figures/fig_th200hist.eps}
\caption[fig:hist_Theta200]{Histogram of the angle subtended by $R_{200}$ of the CGC members.}
\label{fig:hist_Theta200}
\end{figure}
After reconstructing the CGC density field into a peculiar velocity field in a 3D grid, as explained in Sect.~\ref{sec:invertrho}, we compute the cross-correlation function $w^{{\rm T},v_{\rm los}^{\rm rec}} (r)$ for the three inversion approaches using the \textit{Planck}\ HFI 217\,GHz band, and the {\tt SEVEM}\ , {\tt SMICA}\ , {\tt NILC}\ , and {\tt COMMANDER}\ maps (see Fig.~\ref{fig:dTdv}). Here we are using kSZ temperature estimates obtained with an 8\,arcmin aperture. The three inversion approaches (namely, LINEAR, LOG-LINEAR, and LOG-2LPT) may suffer from different biases that impact the amplitude of the recovered velocities. Thus, in order to avoid issues with the normalization of the recovered velocities and to ease the comparison between different approaches, we normalize the recovered velocities $v_{\rm los}^{\rm rec}$-s by their rms before computing the cross-correlation function. In any case we must keep in mind the different responses of the three approaches on different $k$-scales (see Fig.~\ref{fig:rk}), thus giving rise to different correlation structures. For the sake of clarity, in this section we show results for the LINEAR method only, while in the Appendix we discuss the results for the other two reconstruction methods.
In Fig.~\ref{fig:dTdv}, filled coloured circles display the results obtained after using the $\delta T$ kSZ temperature estimates obtained at the real positions of the CGs on the CMB maps. The null tests were obtained by computing the cross-correlation of the {\em normalized} recovered velocities ($v_{\rm los}^{\rm rec}$) with temperature estimates ($\delta T$) obtained on 50 {\em rotated}\footnote{For each rotation/displacement we use a step of three times the aperture radius adopted.} positions on the CMB maps. In those cases, the $\delta T$s were computed for positions obtained after rotating the real CG angular positions in Galactic longitude. The results from each of these null rotations are displayed by dotted lines in Fig.~\ref{fig:dTdv}, and their average is given by the thick, dashed line (which lies close to zero at all radii).
Both the error bars and the covariance matrix of the correlation function were obtained from these null realizations.
Results at zero-lag rotation (i.e., the real sky) lie far from the distribution of the null rotations. There are several individual distance bins lying more than $3\,\sigma$ (up to $3.8\,\sigma$ on the $16\,h^{-1}$\,Mpc distance bin for the raw 217\,GHz frequency map), and the $\chi^2_{\rm null}$ tests rule out the null hypothesis typically at the level of $2.1$--$2.6\,\sigma$ for the clean maps, and at $3.2\,\sigma$ for the raw 217\,GHz frequency map. Likewise, when fitting the observed correlation function to the $w^{{\rm T},v_{\rm los}^{\rm rec}} (r)$ correlation function obtained from the {\tt GALAXY}\ simulation, we obtain values of S/N $=A^{w^{T,v}}/\sigma^{w^{T,v} }=3.0$--$3.2$ for the clean maps, while for the HFI 217\,GHz raw map we find S/N $=A^{w^{T,v}}/\sigma^{w^{T,v} }=3.8$. We explain this apparent mismatch below.
A clear, large-scale correlation pattern, extending up to about $80\,$ $h^{-1}$\,Mpc, is found in the data.
A complementary systematic test can be conducted by computing the cross-correlation function of the kSZ temperature fluctuations ($\delta T$) with {\em shuffled} estimates of the recovered line-of-sight peculiar velocities, i.e., to each CG we assign a $v_{\rm los}^{\rm rec}$ estimate corresponding to a different, randomly selected CG. The result of performing this test for the LINEAR approach is displayed in Fig.~\ref{fig:shuff}, and shows that the correlation found between the $\delta T$s and the recovered velocities clearly vanishes for all shuffled configurations. By shuffling the recovered velocities we are destroying their coherent, large-scale pattern, which couples with large angle CMB residuals, and generates most of the uncertainty in the measured cross-correlation. This explains the smaller error bars in Fig.~\ref{fig:shuff} when compared to Fig.~\ref{fig:dTdv}.
We further study the dependence of the measured $w^{{\rm T},v_{\rm los}^{\rm rec}} (r)$ correlation function on the aperture radius and show the results in Fig.~\ref{fig:wvsAPrad}. We again restrict ourselves to the LINEAR approach and the {\tt SEVEM}\ foreground-cleaned map, and error bars are computed from the null rotations (rather than from shuffling estimates of $v_{\rm los}^{\rm rec}$). We display results for four apertures ranging from 5 to 18\,arcmin: the lower end of this range is given by the angular resolution of the map, while for the higher end the kSZ signal is found to vanish. We find that, as for the kSZ peculiar momentum, the kSZ amplitude at 8\,arcmin aperture is very close to the one found at 5\,arcmin, and it is still significant at 12\,arcmin aperture.
We then vary the aperture from 5 to 26\,arcmin and calculate the corresponding $\chi^{2}$ values, as shown in Fig.~\ref{fig:chisqvsap}. These results are for the {\tt SEVEM}\ foreground-cleaned map, and provide another view of the angular extent of the signal. For the kSZ peculiar momentum, we have shown in Fig.~\ref{fig:kSZ_cgc} that there is kSZ evidence for apertures as large as 12\,arcmin, in good agreement with what we find now for the $w^{{\rm T},v_{\rm los}^{\rm rec}} (r)$ correlation function. While this statistic seems to have higher significance than the pairwise momentum, for both statistics we find consistently that most of its signal is again coming from gas not locked in the central regions of halos, but in the intergalactic medium surrounding the CG host halos.
We also test the consistency of our results with respect to frequency on raw \textit{Planck}\ raw maps. For that we use the HFI channels ranging from 100\,GHz up to 353\,GHz and now we fix the aperture at 8\,arcmin. Since the 100 and 143\,GHz frequency maps have angular resolution comparable or worse than 8\,arcmin, we choose to deconvolve all HFI maps under consideration by their respective (approximate) Gaussian beams, and convolve them again with a Gaussian beam of FWHM$=5$\,arcmin. While this approach may challenge the noise levels for the HFI maps with coarser beams, we find that this is balanced by the large number of CGs on which we compute the $w^{{\rm T},v_{\rm los}^{\rm rec}} (r)$ correlation function. The results of this analysis are given in Fig.~\ref{fig:wTv}, and show how the $w^{{\rm T},v_{\rm los}^{\rm rec}} (r)$ correlation functions from 100\,GHz up to 217\,GHz agree closely with each other, with no significance dependence on frequency (as is expected for the kSZ effect). However, the result for the 353\,GHz channel is pointing to a significantly higher amplitude of the correlation function, even if the error bars associated with this map are
typically 50--70\,\% larger than for the 100\,GHz map. At the map level, the main difference between the 353\,GHz and lower frequency channels is the significantly larger amount of dust and/or CIB emission in the former map. We have checked that, throughout the rotated configurations, the HFI 353\,GHz map has on average no correlation with the estimated radial velocities of the CGs. Therefore, the excess found in the measured amplitude of the $w^{{\rm T},v_{\rm los}^{\rm rec}} (r)$ correlation function must be due to fortuitous alignment between the estimated radial velocities of the CGs and the dust emission at the position of the CGs. This would also explain the higher amplitude of the $w^{{\rm T},v_{\rm los}^{\rm rec}} (r)$ correlation function found for the raw HFI 217\,GHz map with respect to all other clean maps, as shown in Fig.~\ref{fig:dTdv}: the raw HFI 217\,GHz map should still contain some non-negligible dust contamination when compared to the foreground-cleaned maps {\tt SEVEM}, {\tt SMICA}, {\tt NILC}, and {\tt COMMANDER}.
Another potential issue is related to the fact that, as shown in \citet{planck2012-XI}, about 12\,\% of the CGC members with stellar masses above $10^{10}$\,M$_{\odot}$ are actually not central galaxies, but show some offset with respect to the centres of the host halos. This offset should cause a low bias in the kSZ amplitude estimation. We have simulated the impact of this offset by assuming that 12\,\% of the CGs have uniform probability of lying within a (projected) distance range of 0 to 1\,Mpc from the halo centre. After adopting a NFW profile for the gas distribution, we have found that the LOS-projected kSZ signal should be low biased a 10, 8, and 6\,\% for $\theta_{\rm AP}=5,10$ and 15\,arcmin, respectively. Given the limited precision of our measurements, these biases are relatively small and will be ignored hereafter.
To compare our detection with simulations, the solid line in Fig.~\ref{fig:dTdv} shows the best fit to the prediction inferred from the {\tt GALAXY}\ mock catalogue. This prediction is obtained by applying the LINEAR velocity recovery algorithm on our {\tt GALAXY}\ catalogue after imposing the sky mask and the selection function present in the real CGC. Three pseudo-independent estimates of the correlation function of the recovered LOS velocity $v_{\rm rec}$ and the real LOS velocity $v_{\rm los}$ ($\langle v_{\rm los} v_{\rm rec}\rangle (r) \equiv w^{v_{\rm rec},v}(r)$) are obtained after rotating the 3D grid hosting the mock {\tt GALAXY}\ CGC, so that the side facing the observer is different in each case. The solid line corresponds to the average of these three estimates of $w^{v_{\rm rec},v}(r)$, and the ratio to the observed correlation function can be interpreted as an ``effective" optical depth to Thomson scattering: $w^{{\rm T},v_{\rm los}^{\rm rec}} (r) = -\tau_{\rm
T} \,w^{v_{\rm rec},v}(r)$. We obtain $\tau_{\rm T}=(1.39\pm 0.46)\times 10^{-4}$ (i.e., at the $3\,\sigma$ level) for the {\tt SEVEM}\ map, with very similar values for all other foreground-cleaned maps. We defer the physical interpretation of the kSZ measurements of this paper to an external publication, \citep[]{chm_prl15}.
\section{Discussion and Conclusion}
\label{sec:conclusions}
The roughly $2.2\,\sigma$ detection (varying slightly for the different maps) of the pairwise momentum indicates that the baryonic gas is comoving with the underlying matter flows, even though it may lie outside the virial radius of the halos. The aperture of 8\,arcmin on the CGs (placed at a median redshift of $\bar{z}=0.12$) corresponds to a physical radius of around 1\,Mpc. As we show next, this is considerably higher than the typical virial radius of the CG host halos. Following the same approach as in \citet{planck2012-XI}, we compute the $R_{200}$ radius containing an average matter density equal to 200 times the critical density at the halo's redshift. In Fig.~\ref{fig:hist_Theta200} we display the histogram of the angle subtended by the $R_{200}$ values of the 150\,000 CGs placed in the 3D grid that we use for the velocity recovery. The red vertical solid line indicates the 8\,arcmin aperture, well above the typical angular size of the CGC sources.
The behaviour displayed by the measured kSZ peculiar momentum in the top row panels of Fig.~\ref{fig:kSZ_cgc} differs significantly from the pattern found in Fig.~\ref{fig:clusvsap}. The fact that the measured kSZ pairwise momentum shows a roughly constant amplitude out to an aperture of 12\,arcmin, well above the CG virial size, signals the presence of unbound gas that is contributing to the measurement. The opposite situation is seen in Fig.~\ref{fig:clusvsap}: in this scenario most of the signal comes from the halos rather than from a surrounding gas cloud. This plot displays the kSZ peculiar momentum from a subset of our {\tt CLUSTER}\ catalogue, with sources in the range (1--2)\,$\times 10^{14}$\, $h^{-1}$\,M$_\odot$, after considering different aperture radii. As long as the halo remains unresolved (as is the case for these simulated clusters), then as we increase the aperture size the kSZ signal coming from the halo becomes more diluted and hence the amplitude decreases, contrary to what is found in Fig.~\ref{fig:kSZ_cgc}.
We find a similar situation in the kSZ temperature-peculiar velocity cross-correlation. By cross-correlating the reconstructed peculiar velocity field in a 3D box with the kSZ temperature anisotropies, we find a 3.0\,$\sigma$ detection between the two fields for the {\tt SEVEM}\ map, at an aperture of 8\, arcmin. This again, corresponds to gas clouds with radius roughly $1\,$Mpc, about twice the mean $R_{200}$ radius (we find that $\langle R_{200}\rangle_{\rm CGC}\simeq 0.4\,$Mpc). Since the peculiar velocity is directly related to the underlying matter distribution, our result suggests that gas inside and outside CG host halos are comoving with the matter flows.
One way of quantifying the amplitude of our signal is to ask by what
factor we need to scale the model-based $w^{v_{\rm rec},v}(r)$ in order to match our
measured $w^{{\rm T},v_{\rm los}^{\rm rec}} (r)$. Interpreting this scaling as an ``effective"
optical depth to Thomson scattering we find $\tau_{\rm T}=(1.4\pm 0.5)\times 10^{-4}$,
which is a factor of 3 larger than that expected for the gas in typical CG host halos
alone. This provides another piece of evidence that the kSZ signal found
in \textit{Planck}\ data is generated by gas beyond the virialized regions around the CGs, as
opposed to the tSZ effect, which is mostly generated inside collapsed structures
\citep{Hernandezetal2006a,vanwaerbekeetal14,maetal2014} .
\begin{acknowledgements}
The Planck Collaboration acknowledges the support of: ESA; CNES, and
CNRS/INSU-IN2P3-INP (France); ASI, CNR, and INAF (Italy); NASA and DoE
(USA); STFC and UKSA (UK); CSIC, MINECO, JA and RES (Spain); Tekes, AoF,
and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space
(Denmark); SER/SSO (Switzerland); RCN (Norway); SFI (Ireland);
FCT/MCTES (Portugal); ERC and PRACE (EU). A description of the Planck
Collaboration and a list of its members, indicating which technical
or scientific activities they have been involved in, can be found at
\url{http://www.cosmos.esa.int/web/planck/}. This research was supported by ERC Starting Grant (no.~307209), by the Marie Curie Career Integration Grant CIG 294183 and the by the Spanish Ministerio de Econom\'\i a e Innovaci\'on project AYA2012-30789.
\end{acknowledgements}
\bibliographystyle{aa}
|
{
"timestamp": "2015-11-18T02:00:36",
"yymm": "1504",
"arxiv_id": "1504.03339",
"language": "en",
"url": "https://arxiv.org/abs/1504.03339"
}
|
\section{Introduction}
Failure
detection constitutes a fundamental building block for crafting fault-tolerant distributed systems, and
many researchers have devoted their efforts on this direction during the last decade.
Failure detection protocols are often described by their authors making use of informal
pseudo-codes of their conception. Often these pseudo-codes use syntactical constructs
such as \textsf{repeat periodically}~\cite{ChTo43,Aguilera99,BMS02},
\textsf{at time $t$ send heartbeat}~\cite{Chen02,BMS02},
\textsf{at time $t$ check whether message has arrived}~\cite{Chen02},
or \textsf{upon receive}~\cite{Aguilera99}, together with several
variants (see Table~\ref{t:syntax}).
We observe that such syntactical constructs are not often found in
COTS programming languages such as C or C++, which brings
to the problem of translating the protocol specifications into running software prototypes using
one such standard language.
Furthermore the lack of a formal, well-defined, and standard
form to express failure detection protocols often leads their authors to insufficiently detailed
descriptions. Those informal descriptions in turn complicate the reading process and
exacerbate the work of the implementers, which becomes time-consuming, error-prone
and at times frustrating.
\begin{table*}
{\hbox{\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
Construct &NFD-E~\cite{Chen02} &$\varphi$~\cite{Haya04t}& FD~\cite{BMS02}
&GMFD~\cite{RaTr99} &$\mathcal{D}\in\diamondsuit\mathcal{P}$~\cite{ChTo43}
&$\mathcal{HB}$~\cite{Aguilera99}&$\mathcal{HB}$-pt~\cite{Aguilera99}\\ \hline
Repeat & no & no & yes
& no & yes
& yes & yes \\
periodically& & &
& &
& & \\
Upon $t =$ & yes & no & yes
& yes & no
& no & no \\
current time& & &
& &
& & \\
Upon receive& yes & yes & yes
& yes & yes
& yes & yes \\
message & & &
& &
& & \\
Concurrency & yes & yes & no
& no & yes
& yes & yes \\
management & & &
& &
& & \\ \hline
\end{tabular}}}
\caption{Syntactical constructs used in several failure detector protocols.
$\varphi$ is the accrual failure detector~\cite{Haya04t}.
$\mathcal{D}$ is the eventually perfect failure detector of~\cite{ChTo43}.
$\mathcal{HB}$ is the Heartbeat detector~\cite{Aguilera99}.
$\mathcal{HB}$-pt is the partition-tolerant version of the Heartbeat detector.
By ``Concurrency management'' we mean coroutines, threading or forking.\label{t:syntax}}
\end{table*}
Several researchers and practitioners are currently arguing that failure detection
should be made available as a network service~\cite{Haya04,Vanrenesse98gossipstyle}.
To the best of our knowledge no such service exists to date. Lacking such tool, it is important to devise methods
to express in the application layer of our software even the most complex failure
detection protocols in a simple and natural way.
In the following we introduce one such method---a class of ``time-outs'',
i.e., objects that postpone a certain function call by a
given amount of time. This feature converts time-based events
into non time-based events such as message arrivals and easily expresses
the constructs in Table~\ref{t:syntax} in standard C. In some cases,
our class removes the need of concurrency management requirements
such as coroutines or thread management libraries.
The formal character of our method allows rapid-prototyping of the algorithms
with minimal effort. This is proved through a
Literate Programming~\cite{Knuth92} framework that produces from
a same source file both the description meant for dissemination
and a software skeleton to be compiled in standard C or C++.
The rest of this article is structured as follows:
Section~\ref{s:tool} introduces our tool.
In Sect.~\ref{s:cases} we use it to express three classical
failure detectors.
Section~\ref{s:bb} is a case study describing a software system built with our tool.
Our conclusions are drawn in Sect.~\ref{s:end}.
\section{Time-out Management System}\label{s:tool}
This section briefly describes the architecture
of our time-out management system (TOM).
The TOM class appears to the user as a couple of new types
and a library of functions. Table~\ref{examp} provides an idea of the
client-side protocol of our tool.
\begin{table*}
{\begin{tabular}{ll}
{\bf 1.}& /* declarations */\\
& TOM *tom;\\
& timeout\_{}t t1, t2, t3;\\
& int my\_{}alarm(TOM*), another\_{}alarm(TOM*);\\
{\bf 2.}& /* definitions */\\
& tom $\leftarrow$ tom\_{}init(my\_{}alarm);\\
& tom\_{}declare(\&t1, TOM\_{}CYCLIC, TOM\_{}SET\_{}ENABLE,
TIMEOUT1, SUBID1, DEADLINE1); \\
& tom\_{}declare(\&t2, TOM\_{}NON\_{}CYCLIC, TOM\_{}SET\_{}ENABLE,
TIMEOUT2, SUBID2, DEADLINE2); \\
& tom\_{}declare(\&t3, TOM\_{}CYCLIC, TOM\_{}SET\_{}DISABLE,
TIMEOUT3, SUBID3, DEADLINE3); \\
& tom\_{}set\_{}action(\&t3, another\_{}alarm); \\
{\bf 3.}& /* insertion */\\
& tom\_{}insert(tom, \&t1),
tom\_{}insert(tom, \&t2),
tom\_{}insert(tom, \&t3); \\
{\bf 4.}& /* control */\\
& tom\_{}enable(tom, \&t3); \\
& tom\_{}set\_{}deadline(\&t2, NEW\_{}DEADLINE2); \\
& tom\_{}renew(tom, \&t2); \\
& tom\_{}delete(tom, \&t1); \\
{\bf 5.}& /* deactivation */\\
& tom\_{}close(tom);
\end{tabular}}
\caption{Usage of the TOM class. In {\bf 1.} a time-out
list pointer and three time-out objects are declared, together with two
alarm functions. In {\bf 2.} the time-out list and the time-outs
are initialized, and a new alarm is
associated to time-out {\tt t3}. Insertion is carried out at
point {\bf 3.} At {\bf 4.}
{\tt t3} is enabled and a new deadline
value is specified for {\tt t2}. The latter is renewed
and {\tt t1} is deleted.
The list is finally deactivated in {\bf 5.}\label{examp}}
\end{table*}
To declare a time-out manager,
the user needs to define a pointer to a {\tt TOM} object and then
call function {\tt tom\_{}init}. Argument to this function is
an alarm, i.e., the function to be called when a
time-out expires:
\begin{center}
\begin{tabular}{l}
{\tt int alarm(TOM *);} \ \
{\tt tom = tom\_{}init( alarm );}
\end{tabular}
\end{center}
The first time function {\tt tom\_{}init} is called a custom thread is
spawned. That thread is the actual time-out manager.
Now it is possible to define time-outs.
This is done via
type {\tt timeout\_{}t} and function {\tt tom\_{}declare}; an example follows:
\begin{center}
{\tt timeout\_{}t}~{\tt t}; \ \
{\tt tom\_{}declare(\&t,}{\tt TOM\_{}CYCLIC, TOM\_{}SET\_{}ENABLE,}
{\tt TID, TSUBID, DEADLINE)}.
\end{center}
\noindent
In the above, time-out {\tt t} is declared as:
\begin{itemize}
\item A cyclic time-out (renewed on expiration; as opposed to
{\tt TOM\_{}NON\_{}CYCLIC}, which means ``removed on expiration''),
\item enabled (only enabled time-outs ``fire'', i.e., call their
alarm on expiration; an alarm is disabled with {\tt TOM\_{}SET\_{}DISABLE}),
\item with a deadline of {\tt DEADLINE} local clock ticks before expiration.
\end{itemize}
A time-out {\tt t} is identified as a couple of integers---{\tt TID} and {\tt TSUBID}
in the above example. This is done because
in our experience it is often useful to distinguish instances of \emph{classes\/}
of time-outs. We use then {\tt TID} for the class identifier and {\tt TSUBID}
for the particular instance. A practical example of this is given in Sect.~\ref{s:bb}.
Once defined, a time-out can be submitted to the time-out manager
for insertion in its running list of time-outs---see~\cite{DeFl06} for
further details on this.
From the user point of view, this is managed
by calling function
\begin{center}
{\tt tom\_{}insert( TOM *, timeout\_{}t * )}.
\end{center}
Note that a time-out might be submitted to more than one time-out manager.
After successful insertion
an enabled time-out will trigger the call of the default alarm
function after the specified deadline. If that time-out is declared as
{\tt TOM\_{}CYCLIC} the time-out would then be re-inserted.
Other control functions are available: a time-out can be temporarily suspended
while in the time-out list via function
\begin{center}
{\tt tom\_{}disable( TOM *, timeout\_{}t * )}
\end{center}
\noindent
and (re-)enabled via function
\begin{center}
{\tt tom\_{}enable( TOM *, timeout\_{}t * )}.
\end{center}
Furthermore, the user can specify a new alarm function via
{\tt tom\_{}set\_{}action}) and a
new deadline via {\tt tom\_{}set\_{}deadline}; can
delete a time-out from the list via
\begin{center}
{\tt tom\_{}delete( TOM *, timeout\_{}t * )},
\end{center}
and renew\footnote{Renewing a time-out means removing and re-inserting it.} it via
\begin{center}
{\tt tom\_{}renew( TOM *, timeout\_{}t * )}.
\end{center}
Finally, when the time-out management service is no longer needed, the
user should call function
\begin{center}
{\tt tom\_{}close( TOM * )},
\end{center}
\noindent
which also halts the time-out manager thread should no other client
be still active.
\subsection{System assumptions, building blocks, and algorithms}
This section is to provide the reader with a clear definition of
\begin{itemize}
\item the system assumptions our tool builds upon,
\item the architectural building blocks of our system,
\item the algorithms managing the list of time-outs.
\end{itemize}
\subsubsection{System assumptions}
Our tool is built in C for a generic Unix-like system with threads and
standard inter-process communication facilities. Two implementation exists to date---one
based on Embedded Parix~\cite{Anon96b}, the other using the standard Posix threads library~\cite{Pthreads}.
A fundamental requirement of our model is that processes must have access
to some local physical clock giving them the ability to measure time.
The availability of means to control the priorities of processes is also
an important factor to reducing the chances of late alarm execution.
We also assume that the alarm functions are small grained both in CPU and I/O
usage so as not to interfere ``too much'' with the tasks of the TOM.
Finally, we assume the availability of asynchronous, non-blocking primitives to
send and receive messages.
\subsubsection{Architectural building blocks}
Figure~\ref{buffer} portrays the architecture of our time-outs manager: in
\begin{description}
\item{(1),} the client process sends requests
to the time-out list manager; in
\item{(2),} the time-out list manager
accordingly updates the time-out list with the server-side protocol described
in Sect.~\ref{ss}.
\item{(3)} Each time a time-out reaches its deadline,
a request for execution of the corresponding alarm is sent to a task called alarm scheduler.
\item{(4)} The alarm scheduler allocates an alarm request to the first available process out of those
in a circular list of alarm processes, possibly waiting until one of them becomes available.
\end{description}
Figure~\ref{sequence} shows the sequence diagram corresponding to the initialization of the
system and the management of the first time-out request.
The presence of an alarm scheduler and of the circular list of alarm processes
can have great consequences on performance and on the ability of our
system to fulfil real-time requirements. Such aspects
have been studied in~\cite{DeFl06}. Our system may also operate in a
simpler mode, without the above mentioned two components
and with the time-out list manager taking care of the execution of the alarms.
\subsubsection{Algorithms}\label{ss}
The server-side protocol is run by a component called time-out list
manager (TLM).
The TLM implements a well-known
time-out queuing strategy that is described e.g. in~\cite{Tan96}.
TLM basically checks every {\sf TM\_{}CYCLE} for the occurrence of
one of these two events:
\begin{itemize}
\item A request from a client has arrived. If so, TLM serves that request.
\item One or more time-outs have expired. If so, TLM executes
the corresponding alarms.
\end{itemize}
Each time-out {\sf t} is characterized by its \emph{deadline}
{\sf t.deadline}, a positive integer representing the number of
clock units that must separate the time of insertion or renewal from the
scheduled time of alarm execution. This field can only be set by functions
{\sf tom\_{}declare} and {\sf tom\_{}set\_{}deadline}.
Each time-out {\sf t} holds also a field, {\sf t.running},
initially set to {\sf t.deadline}.
Each time-out list object, say {\sf tom}, hosts a variable representing
the origin of the time axis. This variable, {\sf tom.start\_{}time},
regards in particular the time-out at the top of the time-out list---the idea
is that the top of the list is the only entry whose {\sf running}
field needs to be compared with current time in order to verify
the occurrence of the time-out-expired event.
For the time-outs behind the top one, that field represents relative
values, viz., distances from expiration time of the closest,
preceding time-out.
In other words, the overall time-out list management aims at isolating a
``closest to expiration'' time-out, or head time-out, that is the one
and only time-out to be tracked for expiration, and at keeping
track of a list of ``relative time-outs.''
Let us call {\sf TimeNow} the system function returning the current
value of the clock register. In an ordered, coherent time-out list,
residual time \emph{for the head time-out\/} {\sf t} is given by
\begin{equation}
\hbox{\sf t.running} - (\hbox{\sf TimeNow} - \hbox{\sf tom.start\_{}time}),
\label{eq:1}
\end{equation}
\noindent
that is, residual time minus time already passed by. Let us call
quantity~(\ref{eq:1})
as $r_1$, or head residual.
For time-out $n$, $n>1$, that is for the time-out located $n-1$
entries ``after'' the top block, let us define
\begin{equation}
r_n = r_1 + \sum_{i=2}^n \hbox{\sf t}_i\hbox{\sf .running}
\label{eq:2}
\end{equation}
\noindent
as the $n$-th residual, or residual time for time-out at entry $n$.
If there are $m$ entries in the time-out list, let us define
$r_j=0$ for any $j>m$.
It is now possible to formally define the key operations
on a time-out list: insertion and deletion of an entry.
\paragraph{Insertion}
Three cases are possible, namely insertion on top,
in the middle, and at the end of the list.
\begin{description}
\item{Insertion on top.} In this case we need to insert
a new time-out object, say $t$, such that $t\hbox{\sf .deadline} < r_1$,
or whose deadline is less than the head residual. Let us call $u$
the current top of the list. Then the following operations need to
be carried out:
\[
\left\{
\begin{array}{lll}
t\hbox{\sf .running}&\leftarrow& t\hbox{\sf .deadline} +
\hbox{\sf TimeNow} - \hbox{\sf tom.start\_{}time}
\\
u\hbox{\sf .running}&\leftarrow& r_1 - t\hbox{\sf .deadline}.
\label{ins:2}
\end{array}
\right.
\]
Note that the first operation is needed in order to verify relation
\[
t\hbox{\sf .running} - (\hbox{\sf TimeNow} -
\hbox{\sf tom.start\_{}time}) =
t\hbox{\sf .deadline},
\]
\noindent
while the second operation aims at turning the absolute value
kept in the {\sf running} field of the ``old'' head of the list into
a value relative to the one stored in the corresponding field
of the ``new'' top of the list.
\item{Insertion in the middle.} In this case we need to insert
a time-out $t$ such that
\[ \exists \, j : r_j \leq t\hbox{\sf .deadline} < r_{j+1}. \]
Let us call $u$ time-out $j+1$.
(Note that both $t$ and $u$ exist by hypothesis).
Then the following operations need to be carried out:
\[
\left\{
\begin{array}{ll}
t\hbox{\sf .running}&\leftarrow t\hbox{\sf .deadline} - r_j
\\
u\hbox{\sf .running}&\leftarrow u\hbox{\sf .running} - t\hbox{\sf .running}.
\label{ins:4}
\end{array}
\right.
\]
\begin{obs}
Note how, both in the case of insertion on top and in that of
insertion in the middle of the list, time interval $[0,r_m]$
has not changed its length---only, it has been further subdivided,
and is now to be referred to as $[0,r_{m+1}]$.
\end{obs}
\item{Insertion at the end.} Let us suppose
the time-out list consists of $m>0$ items, and that we need to
insert time-out $t$ such that $t\hbox{\sf .deadline}\ge r_m$.
In this case we simply tail the item and initialize it so that
\[
t\hbox{\sf .running} \leftarrow t\hbox{\sf .deadline} - r_m.
\]
\end{description}
\begin{obs}
Note how insertion at the end of the list is the only way to
prolong the range of action from a certain $[0, r_m]$ to
a larger $[0, r_{m+1}]$.
\end{obs}
\paragraph{Deletion}
The other basic management operation on the time-out list is deletion.
As we had three possible insertions, likewise
we distinguish here deletion from top,
from the middle, and from the end of the list.
\begin{description}
\item{Deletion from top.} If the list is a singleton we are
in a trivial case. Let us suppose there are at least two items in the
list. Let us call $t$ the top of the list and $u$ the next element---the
one that will be promoted to top of the list. From its definition
we know that
\begin{eqnarray}
r_2 & = & u\hbox{\sf .running} + r_1 \nonumber\\
& = & u\hbox{\sf .running} + t\hbox{\sf .running} - (\hbox{\sf TimeNow} - \hbox{\sf tom.start\_{}time}).\label{del:1}
\end{eqnarray}
By (\ref{eq:1}), the bracketed quantity is the elapsed time. Then the amount
of absolute time units that separate current time from the expiration time
is given by $u\hbox{\sf .running} + t\hbox{\sf .running}$.
In order to ``behead'' the list we therefore need to update $t$ as follows:
\[
u\hbox{\sf .running} \leftarrow u\hbox{\sf .running} + t\hbox{\sf .running}.
\]
\item{Deletion from the middle.} Let us say we have two consecutive
time-outs in our list, $t$ followed by $u$, such that $t$ is not the top
of the list. With a reasoning similar to the one just followed we get to
the same conclusion---before physically purging $t$ off the list
we need to perform the following step:
\[
u\hbox{\sf .running} \leftarrow u\hbox{\sf .running} + t\hbox{\sf .running}.
\]
\item{Deletion from the end.} Deletion from the end means deleting
an entry which is not referenced by any further item in the list.
Physical deletion can be performed with no need for updating.
Only, the interval of action is shortened.
\end{description}
\begin{obs}
Variable {\sf tom.start\_{}time} is never set when deleting
from or inserting entries into a time-out list, except when inserting
the first element: in such case, that variable is set to the current value
of {\sf TimeNow}.
\end{obs}
Figure~\ref{tom3} shows the action of the server-side protocol:
In \textbf{1.}, a 330ms time-out
called \textbf{A} is inserted in the list. In \textbf{2.}, after 100ms,
\textbf{A} has been reduced to 230ms and a 400ms time-out, called \textbf{B},
is inserted (its value is 170ms, i.e., 400-230ms). Another 70ms have passed in \textbf{3.}, so
\textbf{A} has been reduced to 160ms. At that point, a 510ms time-out,
\textbf{C} is inserted and goes at the third position. In \textbf{4.}, after 160ms,
time-out \textbf{A} occurs---\textbf{B} becomes then the top of the list;
its decrementation starts. In \textbf{5.} another 20ms have passed and \textbf{B} is
at 150ms---at that point a 230ms time-out, called \textbf{D} is inserted.
Its position is in between \textbf{B} and \textbf{C}, therefore this latter is
adjusted. In \textbf{6.}, after 150ms, \textbf{B} occurs and \textbf{D} goes on top.
\section{Discussion}\label{s:cases}
In this section we show that the syntactical constructs in Table~\ref{t:syntax}
can be expressed in terms of our class of time-outs. We do so by considering three
classical failure detectors and providing their time-out based specifications.
Let us consider the classical formulation of eventually perfect
failure detector $\mathcal{D}$~\cite{ChTo43}.
The main idea of the protocol is to require each task to send a ``heartbeat''
to its fellows and maintain a list of tasks suspected to have failed. A task identifier $q$ enters
the list of task $p$
if no heartbeat is received by $p$ during a certain amount of time, $\hbox{$\Delta_{p}$}(q)$, initially
set to a default value. This value is increased when late heartbeats are received.
The basic structure of $\mathcal{D}$ is that of a
coroutine with three concurrent processes, two of which execute a task
periodically while the third one is triggered by the arrival of a message:
\begin{tabbing}
aa\=aa\=aa\=aa\=aa\=aa\=aa \kill
\> \emph{Every process $p$ executes the following}:\\
\\
\> $\hbox{$\hbox{\it output}_{p}$} \leftarrow 0$\\
\> {\bf for} all $q\in\Pi$\\
\> \> $\hbox{$\Delta_{p}$}(q)\leftarrow$ default time interval\\
\\
\> {\bf cobegin}\\
\> \> || \emph{Task 1:} {\bf repeat periodically}\\
\> \> \> send ``$p$-is-alive'' to all\\
\\
\> \> || \emph{Task 2:} {\bf repeat periodically}\\
\> \> \> {\bf for} all $q\in\Pi$\\
\> \> \> \> {\bf if} $q\not\in\hbox{$\hbox{\it output}_{p}$}$ and $p$ did not receive ``$q$-is-alive'' during\\
\> \> \> \> \> the last $\hbox{$\Delta_{p}$}(q)$ ticks of $p$'s clock {\bf then}\\
\> \> \> \> \> \> $\hbox{$\hbox{\it output}_{p}$} \leftarrow \hbox{$\hbox{\it output}_{p}$} \cup \{q\}$\\
\\
\> \> || \emph{Task 3:} {\bf when} received ``q-is-alive'' for some $q$\\
\> \> \> {\bf if} $q\in\hbox{$\hbox{\it output}_{p}$}$\\
\> \> \> \> $\hbox{$\hbox{\it output}_{p}$} \leftarrow \hbox{$\hbox{\it output}_{p}$} - \{q\}$\\
\> \> \> \> $\hbox{$\Delta_{p}$}(q)\leftarrow \hbox{$\Delta_{p}$}(q) + 1$\\
\> {\bf coend}.
\end{tabbing}
We call the {\bf repeat periodically} in \emph{Task 1} a ``multiplicity 1'' repeat, because
indeed a single action (sending a ``$p$-is-alive'' message) has to be tracked, while
we call ``multiplicity $q$'' repeat the one in \emph{Task 2}, which requires to check $q$
events.
Our reformulation of the above code is as follows:
\begin{tt}
\begin{tabbing}
aa\=aa\=aa\=aa\=aa\=aa \kill
\> \mbox{\emph{Every process $p$ executes the following}:}\\
\\
\> {\bf timeout\_t} \hbox{$t_{\hbox{\small task1}}$}, \hbox{$t_{\hbox{\small task2}}$}[NPROCS];\\
\> {\bf task\_t} $p$, $q$;\\
\> {\bf for} ($q$=0; $q$<NPROCS; $q$++) \{\\
\> \> $\hbox{$\Delta_{p}$}[q]$ = DEFAULT\_TIMEOUT;\\
\> \> $\hbox{$\hbox{\it output}_{p}$}[q]$ = TRUST;\\
\> \}\\
\\
\> /* \emph{``\hbox{$\leadsto$}'' is our symbol for the ``address-of'' operator} */\\
\> tom\_declare(\hbox{$\leadsto$}{}\hbox{$t_{\hbox{\small task1}}$}, TOM\_CYCLIC, TOM\_SET\_ENABLE, $p$, 0, $\hbox{$\Delta_{p}$}[q]$);\\
\> tom\_set\_action(\hbox{$\leadsto$}{}\hbox{$t_{\hbox{\small task1}}$}, action\_Repeat\_Task1);\\
\> tom\_insert(\hbox{$\leadsto$}{}\hbox{$t_{\hbox{\small task1}}$});\\
\\
\> {\bf for} ($q$=0; $q$<NPROCS; $q$++) \{\\
\> \> {\bf if} ($p \neq q$) \{\\
\> \> \> tom\_declare(\hbox{$t_{\hbox{\small task2}}$}$+q$, TOM\_CYCLIC, TOM\_SET\_ENABLE, $q$, 0, $\hbox{$\Delta_{p}$}[q]$);\\
\> \> \> tom\_set\_action(\hbox{$t_{\hbox{\small task2}}$}$+q$, action\_Repeat\_Task2);\\
\> \> \> tom\_insert(\hbox{$\leadsto$}{}\hbox{$t_{\hbox{\small task2}}$});\\
\> \> \}\\
\> \}\\
\\
\> {\bf do} \{ \\
\> \> getMessage(\hbox{$\leadsto$}{}$m$);\\
\> \> {\bf switch} ($m$.\emph{type}) \ \{\\
\> \> \> TASK1;\\
\> \> \> TASK2;\\
\> \> \> TASK3;\\
\> \> \}\\
\> \} {\bf forever};
\end{tabbing}
\end{tt}
where tasks and actions are defined as follows:
\begin{tt}
\begin{tabbing}
aa\=TASK1 $\equiv$ \=aa\=aa\=aa\=aa\=aa \kill
\> TASK1 $\equiv$ \> {\bf case} REPEAT\_TASK1:\\
\> \> \> sendAll(I\_AM\_ALIVE);\\
\> \> {\bf break;}
\\
\> TASK2 $\equiv$ \> {\bf case} REPEAT\_TASK2:\\
\> \> \> $q =$ $m$.\emph{id};\\
\> \> \> {\bf if} ($\hbox{$\hbox{\it output}_{p}$}[q] \equiv$ TRUST)\\
\> \> \> \> $\hbox{$\hbox{\it output}_{p}$}[q]=$ SUSPECT;\\
\> \> {\bf break;}
\\
\> TASK3 $\equiv$ \> {\bf case} I\_AM\_ALIVE:\\
\> \> \> $q =$ $m$.\emph{sender};\\
\> \> \> {\bf if} ($\hbox{$\hbox{\it output}_{p}$}[q] \equiv$ SUSPECT) \ \{\\
\> \> \> \> $\hbox{$\hbox{\it output}_{p}$}[q]=$ TRUST;\\
\> \> \> \> $\hbox{$\Delta_{p}$}(q) = \hbox{$\Delta_{p}$}(q) + 1$;\\
\> \> \> \}\\
\> \> {\bf break;}\\
\\
\> action\_Repeat\_Task1() \{\\
\> \> {\bf message\_t} $m$;\\
\> \> $m$.\emph{type} = REPEAT\_TASK1;\\
\> \> Send($m$, $p$);\\
\> \}\\
\> action\_Repeat\_Task2({\bf timeout\_t} *$t$) \{\\
\> \> {\bf message\_t} $m$;\\
\> \> $m$.\emph{type} = REPEAT\_TASK2;\\
\> \> $m$.\emph{id} = $t$->\emph{id};\\
\> \> Send($m$, $p$);\\
\> \}
\end{tabbing}
\end{tt}
We can draw the following observations:
\begin{itemize}
\item Our syntax is less abstract than the one adopted in the classical
formulation. Indeed we have deliberately chosen a syntax
very similar to that of programming languages such as C or C++.
Behind the lines, we assume also a similar semantics.
\item Our syntax is more strongly typed: we have deliberately chosen
to define (most of) the objects our code deals with.
\item We have systematically avoided set-wise operations such as union,
complement or membership by translating sets into arrays as, e.g., in
$$\hbox{$\hbox{\it output}_{p}$} \leftarrow \hbox{$\hbox{\it output}_{p}$} \cup \{q\},$$
which we changed into
$$\hbox{$\hbox{\it output}_{p}$}[q] = \hbox{\tt PRESENT.}$$
\item We have systematically rewritten the abstract constructs {\tt repeat pe\-ri\-o\-di\-cal\-ly}
as one or more time-outs (depending on their multiplicity). Each of these time-out
has an associated action that sends one message to the client process, $p$. This means that
\begin{enumerate}
\item time-related event ``it's time to send $p$-is-alive to all'' becomes event
``message {\tt REPEAT\_TASK1} has arrived.''
\item time-related events ``it's time to check whether $q$-is-alive has arrived''
becomes event ``message ({\tt REPEAT\_TASK2}, id=$q$) has arrived.''
\end{enumerate}
\item Due to the now homogeneous nature of the possible events (that now are all represented
by message arrivals) a single
process may manage those events through a multiple selection statement (a switch).
In other words, no coroutine is needed anymore.
\end{itemize}
Through the Literate Programming approach and a compliant
tool such as CWEB~\cite{KnLe93,Knuth92}
it is possible to further improve our reformulation. As well known, the CWEB tool allows
a pretty printable \TeX{} documentation
and a C file ready for compilation and testing to be produced from
a single source code.
In our experience this link between
these two contexts can be very beneficial: testing or even simply using the code
provides feedback on the specification of the algorithm, while the improved
specification may reduce the probability of design faults and in general
increase the quality of the code.
Figure~\ref{f:Agui} and Figure~\ref{f:RaTr}
respectively show a reformulation for the $\mathcal{HB}$ failure detector for
partitionable networks~\cite{Aguilera99} and for the group membership failure detector~\cite{RaTr99}
produced with CWEB.
In those reformulations,
symbols such as $\tau$ and ${\mathcal D}_p$ are caught by CWEB and translated into legal C tokens
via its ``@f'' construct~\cite{KnLe93}. Note also that
the expression $m.\hbox{\em path}[q]\leq$\texttt{PRESENT} in Fig.~\ref{f:RaTr} means ``$q$ appears
at most once in \hbox{\emph{path}}''.
A full description of these protocols is out of the scope of this paper---for that we refer the
reader to the above cited articles. The focus here is mainly on the syntactical constructs
used in them and our reformulations, which
include simple translations for the syntactical
constructs in Table~\ref{t:syntax} in terms of our time-out API.
A case worth noting is that of the group membership failure detector: here
the authors mimic the availability of a cyclic time-out service but intrude its management
in their formulation. This management code can be avoided altogether using our approach.
\section{A development experience: the DIR net}\label{s:bb}
What we call ``DIR net''~\cite{DeFl09} is the distributed application at the core of the
software fault tolerance strategy realized through several European projects~\cite{DeFl09,BDDC99b}.
In this section we describe the DIR net and report on how we designed and developed it
by means of the TOM system.
The DIR net is a fault-tolerant network of
failure detectors connected to other peripheral error detectors
(called ``Dtools'' in what follows).
Objective of the DIR net is to ensure consistent fault tolerance
strategies throughout the system and play the role of a backbone
handling information to and from the Dtools~\cite{DeDL00d}.
The DIR net consists of four classes of components. Each processing node
in the system runs an instance of a so-called ``I'm Alive Task'' (IAT) plus
an instance of either a ``DIR Manager'' (\hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}), or a ``DIR Agent'' (\hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal A$}), or
a ``DIR Backup Agent'' (\hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}). A \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal A$}{} gathers all error detection messages
produced by the Dtools on the current processing node and forwards
them to the \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}{} and the \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}'s. A \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{} is a \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal A$}{} which also maintains its messages
into a database located in central memory. It is connected to \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}{} and to the other
\hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{}'s and is eligible for election as a \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}. A \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}{} is a special
case of \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{}. Unique within the system, the \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}{} is the
one component responsible for running error recovery strategies---see~\cite{DeFl09}
for a description of the latter. Let us use DIR-$x$ to address any non-IAT component
(i.e. the \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}{}, or a \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{}, or a \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal A$}.)
An important design goal of the DIR net is that of being tolerant to physical and design faults,
both permanent or intermittent, affecting up to all but one \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}.
This is accomplished
also through a failure detection protocol that we are going to describe in the rest of
this section.
\subsection{The DIR net failure detection protocol}
Our protocol consists of a local part and a distributed part. Each of them is
realized through our TOM class.
\subsubsection{DIR net protocol: local component}
As we already mentioned, each processing node hosts a DIR-$x$ and an IAT. These two
components run a simple algorithm: they share a local Boolean variable, the ``I'm Alive Flag'' (IAF).
The DIR-$x$ has to periodically set the IAF to \texttt{TRUE} while the IAT has to check periodically
that this has indeed occurred and reverts IAF to \texttt{FALSE}. If the IAT finds the IAF set to
\texttt{FALSE} it broadcasts message \hbox{$m_{\hbox{\tiny TEIF}}$}{} (``this entity is faulty'').
The cyclic tasks mentioned above can be easily modeled via two time-outs, \hbox{$t_{\hbox{\tiny IA\_SET}}$}{} and \hbox{$t_{\hbox{\tiny IA\_CLR}}$}, described
in Table~\ref{t:mIAS} and Table~\ref{t:tIAS} (TimeNow being the system function returning the current
value of the clock register.)
\begin{table*}
{\begin{tabular}{|l|l|l|c|}
\hline
\bf Time-out & \bf Caller & \bf Action & \bf Cyclic? \\ \hline
\hbox{$t_{\hbox{\tiny IA\_SET}}$} & DIR-$x$ & On TimeNow + \hbox{$d_{\hbox{\tiny IA\_SET}}$}{} do send $\hbox{$m_{\hbox{\tiny IA\_SET\_ALARM}}$}$ to Caller & Yes\\
\hbox{$t_{\hbox{\tiny IA\_CLR}}$} & IAT & On TimeNow + \hbox{$d_{\hbox{\tiny IA\_CLR}}$}{} do send $\hbox{$m_{\hbox{\tiny IA\_CLR\_ALARM}}$}$ to IAT & Yes\\ \hline
\end{tabular}}{ }
\caption{Description of messages \hbox{$m_{\hbox{\tiny IA\_SET\_ALARM}}$}{} and \hbox{$m_{\hbox{\tiny IA\_CLR\_ALARM}}$}.\label{t:mIAS}}
\end{table*}
\begin{table*}
{\begin{tabular}{|l|l|l|l|}
\hline
\bf Message & \bf Receiver & \bf Explanation & \bf Action \\ \hline
\hbox{$m_{\hbox{\tiny IA\_SET\_ALARM}}$} & DIR-$x$ & Time to set IAF & IAF $\leftarrow$ \texttt{TRUE}\\
\hbox{$m_{\hbox{\tiny IA\_CLR\_ALARM}}$} & IAT $k$ & Time to check IAF & \textbf{if} (IAF $\equiv$ \texttt{FALSE})
SendAll(\hbox{$m_{\hbox{\tiny TEIF}}$}, $k$)\\
& & & \textbf{else} IAF $\leftarrow$ \texttt{FALSE},\\ \hline
\end{tabular}}
\caption{Description of time-outs \hbox{$t_{\hbox{\tiny IA\_SET}}$}{} and \hbox{$t_{\hbox{\tiny IA\_CLR}}$}.\label{t:tIAS}}
\end{table*}
Note that the time-outs' alarm functions do not clear/set the
flag---doing so a hung DIR-$x$ would go undetected. On the contrary, those
functions trigger the transmission of messages that once
received by healthy components trigger the execution of the meant actions.
The following is
a pseudo-code for the IAT{} algorithm:
\begin{tt}
\begin{center}
\begin{tabbing}
aa\=aa\=aa\=aa\=aa\=aa \kill
\> \mbox{\emph{The IAT $k$ executes as follows}:}\\
\\
\> {\bf timeout\_t} \hbox{$t_{\hbox{\tiny IA\_CLR}}$};\\
\> {\bf msg\_t} \emph{activationMessage}, $m$;\\
\\
\> tom\_declare(\hbox{$\leadsto$}{}\hbox{$t_{\hbox{\tiny IA\_CLR}}$}, TOM\_CYCLIC,\\
\> \> \> TOM\_SET\_ENABLE, IAT\_CLEAR\_TIMEOUT, 0, \hbox{$d_{\hbox{\tiny IA\_CLR}}$});\\
\> tom\_set\_action(\hbox{$\leadsto$}{}\hbox{$t_{\hbox{\tiny IA\_CLR}}$}, actionSend\hbox{$m_{\hbox{\tiny IA\_CLR\_ALARM}}$});\\
\> tom\_insert(\hbox{$\leadsto$}{}\hbox{$t_{\hbox{\tiny IA\_CLR}}$});\\
\\
\> Receive(\emph{activationMessage});\\
\\
\> {\bf forever} \{\\
\> \> Receive($m$);\\
\> \> {\bf if} ($m$.\emph{type} $\equiv$ \hbox{$m_{\hbox{\tiny IA\_CLR\_ALARM}}$}) \\
\> \> \> {\bf if} (IAF $\equiv$ \texttt{TRUE}) IAF $\leftarrow$ \texttt{FALSE}; \\
\> \> \> {\bf else} SendAll(\hbox{$m_{\hbox{\tiny TEIF}}$}, $k$); delete\_timeout(\hbox{$\leadsto$}{}\hbox{$t_{\hbox{\tiny IA\_CLR}}$});\\
\> \}\\
\\
\> actionSend\hbox{$m_{\hbox{\tiny IA\_CLR\_ALARM}}$}() \{ Send(\hbox{$m_{\hbox{\tiny IA\_CLR\_ALARM}}$}, IAT $k$); \}
\end{tabbing}
\end{center}
\end{tt}
The time-out formulation of the IAT algorithm is given in next section.
\subsubsection{DIR net protocol: distributed component}\label{s:dirnet.dist}
The resilience of the DIR net to crash faults comes from the \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}{} and
the \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}'s running the following distributed algorithm of failure detection:
\paragraph{Algorithm \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}}\label{dirm}
Let us call {\tt mid} the node hosting the \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}{} and $b$ the number of processing nodes
that host a \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}.
The \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}{} has to send cyclically a \hbox{$m_{\hbox{\tiny MIA}}$}{} (``Manager-Is-Alive'') message
to all the \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}'s each time time-out \hbox{$t_{\hbox{\tiny MIA\_A}}$}{} expires---this
is shown in the right side of Fig.~\ref{f:dadm}. Obviously this is a multiplicity $b$ ``repeat'' construct,
which can be easily managed through a cyclic time-out with an action that signals that a new cycle has begun.
In this case the action is ``send a message of type \hbox{$m_{\hbox{\tiny MIA\_A\_ALARM}}$}{} to the \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}.''
The manager also expects periodically a $(\hbox{$m_{\hbox{\tiny TAIA}}$}, \hbox{\small $i$})$ message (``This-Agent-Is-Alive'')
from each node where a \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{} is expected to be running.
This is easily accomplished through a vector of
$(\hbox{$t_{\hbox{\tiny TAIA\_A}}$}, \hbox{\small $\mathbf i$})$ time-outs.
The left part of Fig.~\ref{f:dadm}
shows this for node $i$. When time-out $(\hbox{$t_{\hbox{\tiny TAIA\_A}}$}, \hbox{\small $p$})$ expires it means that
no $(\hbox{$m_{\hbox{\tiny TAIA}}$}, \hbox{\small $p$})$ message has been received within the current period. In this case
the \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}{} enters what we call a ``suspicion period''. During such period
the manager needs to distinguish the case of a late \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{} from a crashed one.
This is done by inserting a non-cyclic time-out, namely $(\hbox{$t_{\hbox{\tiny TEIF\_A}}$}, \hbox{\small $p$})$.
During the suspicion period only one out of the following three events
may occur:
\begin{enumerate}
\item A late $(\hbox{$m_{\hbox{\tiny TAIA}}$}, \hbox{\small $p$})$ is received.
\item A $(\hbox{$m_{\hbox{\tiny TEIF}}$}, \hbox{\small $p$})$ from IAT{} at node $p$ is received.
\item Nothing comes in and the time-out expires.
\end{enumerate}
In case {1.} we get out of the suspicion period, conclude that \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{} at node $p$
was simply late and go back waiting for the next $(\hbox{$m_{\hbox{\tiny TAIA}}$}, \hbox{\small $p$})$.
It is the responsibility of the user to choose meaningful values for the time-outs'
deadlines. By ``meaningful'' we mean that those values should match the characteristics of the environment
and represent a good trade-off between the following two risks:
\begin{description}
\item[overshooting,] i.e., choosing too large values for the deadlines.
This decreases the probability of false negatives (regarding a slow process as a failed process; this is
known as accuracy in failure detection terminology)
but increases the detection latency;
\item[undershooting,] namely under-dimensioning the deadlines. This may increase considerably
false negatives but reduces the detection latency of failed processes.
\end{description}
Under the hypotheses of properly chosen time-outs' deadlines,
and that of a single, stable environment\footnote{We call an environment ``stable'' when
it does not change drastically its characteristics except under erroneous and exceptional
conditions. Single environments are typical of fixed (non-mobile) applications.},
the occurrences of late $(\hbox{$m_{\hbox{\tiny TAIA}}$}, \hbox{\small $p$})$ messages should be exceptional.
This event would translate in a false deduction uncovered in the next cycle.
Further late messages would postpone a correct assessment, but are considered as an unlikely
situation given the above hypotheses.
An alternative and better approach would be to track the changes in
the environment. For the case at hand this would mean that the time-outs' deadlines
should be adaptively adjusted. This could be possible, e.g., through an approach
such as in~\cite{DB07b}.
If {2.} is the case we assume the remote component has crashed though its node is still
working properly as the IAT{} on that node still gives signs of life. Consequently
we initiate an error recovery step. This includes sending a ``{\tt WAKEUP}''
message to the remote IAT{} so that it spawns another \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{} on that node.
In case {3.} we assume the entire node has crashed and initiate node recovery.
Underlying assumption of our algorithm is that the IAT{} is so simple that
if it fails then we can assume the whole node has failed.
\paragraph{Algorithm \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}}\label{dirb}
This algorithm is also divided into two concurrent tasks. In the first one
\hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{} on node $i$ has to cyclically send
$(\hbox{$m_{\hbox{\tiny TAIA}}$}, \hbox{\small $i$})$ messages to the manager,
either in piggybacking or when time-out
\hbox{$t_{\hbox{\tiny TAIA\_B}}$}{} expires.
This is represented in the right side of Fig.~\ref{dadb}.
The \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{}'s in turn periodically expect a $\hbox{$m_{\hbox{\tiny MIA}}$}$ message from the \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}.
As evident when comparing Fig.~\ref{f:dadm} with Fig.~\ref{dadb},
the \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{} algorithm is very similar to the one of the manager:
also \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{}
enters a suspicion period when its manager does not appear to respond
quickly enough---this period is managed via time-out \hbox{$t_{\hbox{\tiny TEIF\_B}}$},
the same way as in \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}. Also in this case we can get out of this state
in one out of three possible ways: either
\begin{enumerate}
\item a late $(\hbox{$m_{\hbox{\tiny MIA\_B\_ALARM}}$}, \hbox{\small mid})$ is received, or
\item a $(\hbox{$m_{\hbox{\tiny TEIF}}$}, \hbox{\small mid})$ sent by the IAT{} at node mid is received, or
\item nothing comes in and the time-out expires.
\end{enumerate}
In case {1.} we get out of the suspicion period, conclude that the manager
was simply late, go back to normal state and start
waiting for the next $(\hbox{$m_{\hbox{\tiny MIA}}$}, \hbox{\small mid})$ message.
Also in this case, a wrong deduction shall be detected
in next cycles.
If 2. we conclude the manager has crashed though its node is still
working properly, as its IAT{} acted as expected. Consequently
we initiate a manager recovery phase structured similarly to
the \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{} recovery step described in Sect.~\ref{dirm}.
In case 3. we assume the node of the manager has crashed,
elect a new manager among the \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{}'s, and perform
a node recovery phase.
Table~\ref{t:tandm} summarizes the \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}{} and \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{} algorithms.
\begin{table*}
{\vbox{\begin{tabular}{|l|l|l|c|}
\hline
\bf Time-out & \bf Caller & \bf Action & \bf Cyclic? \\ \hline
\hbox{$t_{\hbox{\tiny MIA\_A}}$} & \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$} & Every \hbox{$d_{\hbox{\tiny MIA\_A}}$}{} do send \hbox{$m_{\hbox{\tiny MIA\_A\_ALARM}}$}{} to \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$} & Yes\\
$\hbox{$t_{\hbox{\tiny TAIA\_A}}$}[i]$ & \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$} & Every \hbox{$d_{\hbox{\tiny TAIA\_A}}$}{} do send $(\hbox{$m_{\hbox{\tiny TAIA\_A\_ALARM}}$},i)$ to \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$} & Yes\\
$\hbox{$t_{\hbox{\tiny TEIF\_A}}$}[i]$ & \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$} & On TimeNow + \hbox{$d_{\hbox{\tiny TEIF\_A}}$}{} do send $(\hbox{$m_{\hbox{\tiny TEIF\_A\_ALARM}}$},i)$ to \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$} & No\\
\hbox{$t_{\hbox{\tiny TAIA\_B}}$} & \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{} $j$& Every \hbox{$d_{\hbox{\tiny TAIA\_B}}$}{} do send \hbox{$m_{\hbox{\tiny TAIA\_B\_ALARM}}$}{} to \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{} $j$ & Yes\\
\hbox{$t_{\hbox{\tiny MIA\_B}}$} & \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{} $j$& Every \hbox{$d_{\hbox{\tiny MIA\_B}}$}{} do send \hbox{$m_{\hbox{\tiny MIA\_B\_ALARM}}$}{} to \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{} $j$ & Yes\\
\hbox{$t_{\hbox{\tiny TEIF\_B}}$} & \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{} $j$& On TimeNow + \hbox{$d_{\hbox{\tiny TEIF\_B}}$}{} do send $\hbox{$m_{\hbox{\tiny TEIF\_B\_ALARM}}$}$ to \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{} $j$& No \\ \hline
\end{tabular}\\
\begin{tabular}{|l|l|l|l|}
\hline
\bf Message & \bf Receiver & \bf Explanation & \bf Action \\ \hline
$(\hbox{$m_{\hbox{\tiny TAIA}}$},i)$ & \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$} & \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{} $i$ is OK & (Re-)Insert or renew $\hbox{$t_{\hbox{\tiny TAIA\_A}}$}[i]$ \\
\hbox{$m_{\hbox{\tiny MIA\_A\_ALARM}}$} & \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$} & A new heartbeat is required & Send \hbox{$m_{\hbox{\tiny MIA}}$}{} to all \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}'s \\
\hbox{$m_{\hbox{\tiny TAIA\_A\_ALARM}}$} & \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$} & Possibly \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{} $i$ is not OK & Delete $\hbox{$t_{\hbox{\tiny TAIA\_A}}$}[i]$, insert $\hbox{$t_{\hbox{\tiny TEIF\_A}}$}[i]$\\
$(\hbox{$m_{\hbox{\tiny TEIF}}$},i)$ & \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$} & \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{} $i$ crashed & Declare \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{} $i$ crashed\\
$(\hbox{$m_{\hbox{\tiny TEIF\_A\_ALARM}}$},i)$& \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$} & Node $i$ crashed & Declare node $i$ crashed\\
\hbox{$m_{\hbox{\tiny MIA}}$} & \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{} $j$ & \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}{} is OK & Renew \hbox{$t_{\hbox{\tiny MIA\_B}}$}\\
\hbox{$m_{\hbox{\tiny TAIA\_B\_ALARM}}$} & \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{} $j$ & A new heartbeat is required & Send $(\hbox{$m_{\hbox{\tiny TAIA}}$},j)$ to \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}\\
\hbox{$m_{\hbox{\tiny MIA\_B\_ALARM}}$} & \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{} $j$ & Possibly \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}{} is not OK & Delete \hbox{$t_{\hbox{\tiny MIA\_B}}$}, insert \hbox{$t_{\hbox{\tiny TEIF\_B}}$}\\
\hbox{$m_{\hbox{\tiny TEIF}}$} & \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{} $j$ & \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}{} crashed & Declare \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}{} crashed\\
\hbox{$m_{\hbox{\tiny TEIF\_B\_ALARM}}$} & \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{} $j$ & \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}{}'s node crashed & Declare \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}'s node crashed\\ \hline
\end{tabular}}}
\caption{Time-outs and messages of \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}{} and \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{}.\label{t:tandm}}
\end{table*}
We have developed the DIR net using the Windows TIRAN libraries~\cite{BDDC99b} and the CWEB system
of structured documentation.
\subsection{Special services}
\subsubsection{Configuration}
The management of a large number of time-outs may be an error prone task. To simplify it,
we designed a simple configuration language.
Figure~\ref{f:xxx4} shows an example of configuration script
to specify the structure of the DIR net
(in this case, a four node system with three \hbox{DIR\hskip-1pt-$\hskip-1pt\mathcal B$}{}'s deployed on nodes 1--3 and the \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}{}
on node 0) and of its time-outs.
A translator produces the C header files to properly initialize an instance of the DIR net
(see Fig.~\ref{f:xxx5}).
\subsubsection{Fault injection}
Time-outs may also be used to specify fault injection actions with fixed or pseudo-random deadlines.
In the DIR net this is done as follows. First we define the time-out:
\begin{verbatim}
#ifdef INJECT
tom_declare(&inject, TOM_NON_CYCLIC, TOM_SET_ENABLE,
INJECT_FAULT_TIMEOUT, i, INJECT_FAULT_DEADLINE);
tom_insert(tom, &inject);
#endif
\end{verbatim}
The alarm of this time-out sends the local DIR-$x$ a message of type ``\texttt{IN\-JE\-CT\_FA\-ULT\_TIME\-OUT}''.
Figure~\ref{f:mainloop} shows an excerpt from the actual main loop of the \hbox{DIR\hskip-1pt-$\hskip-2pt\mathcal M$}{} in which
this message is processed.
\subsubsection{Fault tolerance}
A service such as TOM is indeed a single-point-of-failure in that
a failed TOM in the DIR net would result in all components being unable to
perform their failure detection protocols.
Such a case would be indistinguishable from that of a crashed node by the other DIR net components.
As well known from, e.g., \cite{IBR96}, a single design fault in TOM's implementation could bring
the system to a global failure. Nevertheless, the isolation of a \emph{service\/} for time-out management
may pave the way for a cost-effective adoption of multiple-version software fault tolerance techniques~\cite{Lyu98b}
such as the well known recovery block~\cite{RaXu95}, or $N$-version programming~\cite{Avi95}.
Another possibility would be to use the DIR net algorithm to tolerate faults in TOM.
No such technique has been adopted in the current implementation of TOM.
Other factors, such as congestion or malicious attacks might introduce performance
failures that would impact on all modules that depend on TOM to perform their time-based
processing~\cite{DeFl06}.
\section{Conclusions}\label{s:end}
We have introduced a tentative \emph{lingua franca\/} for the expression of failure detection protocols.
TOM has the advantages of being simple, elegant and not ambiguous. Obvious are the
many positive relapses that would come from the adoption of a standard, semi-formal representation with respect to
the current Babel of informal descriptions---easier acquisition of insight, faster verification, and greater ability
to rapid-prototype software systems. The availability of a tool such as TOM is also one of
the requirements of the timed-asynchronous system model~\cite{CrFe99}.
Given the current lack of a network service for failure detection, the availability of standard methods
to express failure detectors in the application layer is an important asset: a tool like the one
described in this paper isolates and crystallizes a part of the complexity required to express failure detection protocols.
This complexity may become transparent of the designer, with tangible savings
in terms of development times and costs, if more efforts will be devoted to
time-outs configuration and automatic adjustments through adaptive
approaches such as the one described in~\cite{DB07b}. Such optimizations will be
the subject of future research.
Future plans also include to port our system to AspectJ~\cite{aspectj} so as to further enhance
programmability and separation of design concerns.
As a final remark we would like to point out how, at the core of our design choices,
is the selection of C and literate programming, which proved to be invaluable tools
to reach our design goals. Nevertheless we must point out how these choices
may turn into intrinsic limitations for the expressiveness of the resulting language.
In particular, they enforce a syntactical and semantic structure, that of the C programming language,
which may be regarded as a limitation by those researchers who are not accustomed to that language.
At the same time we would like to remark also
that those very choices allow us a straightforward translation of our
constructs into a language like Promela~\cite{Hol91}, which resembles very much a C language
augmented with Hoare's CSP~\cite{Hoa78}.
Accordingly, our future work in this framework shall include the adoption of the Promela
extension of Prof. Bo{\v s}na{\v c}ki, which allows the verification of concurrent systems that depend
on timing parameters~\cite{BoDa98}. Interestingly enough, this version of Promela
includes new objects, called discrete time countdown timers, which are basically equivalent to our
non-cyclic time-outs. Our goal is to come up with a tool that generates
from the same literate programming source (1) a pretty printout in \TeX,
(2) C code ready to be compiled and run, and (3) Promela code to verify some properties of the protocol.
\section*{Acknowledgment}
We acknowledge the work by Alessandro Sanchini, who developed
the communication library used by our tool, and the many and valuable comments
of our reviewers.
|
{
"timestamp": "2015-04-15T02:06:25",
"yymm": "1504",
"arxiv_id": "1504.03449",
"language": "en",
"url": "https://arxiv.org/abs/1504.03449"
}
|
\section{Introduction\label{sec:1}}
Orthogonal frequency division multiplexing (OFDM) is adopted by most of the current and future telecommunication standards for high-rate data transmission, particularly in wireless communication systems. Its resilience to multipath channel fading, the spectral efficiency it provides as well as the simplicity of the equalization, has enabled OFDM to remain the most popular modulation scheme. However, OFDM is known to be sensitive to various hardware imperfections, the Dirty-RF effect \cite{Fettweis,Schenk}, originating in the transceiver hardware. \setlength{\parskip}{0pt}
\subsection{Previous Works}
The phase-noise (PHN) phenomenon and its underlying effects on various OFDM systems were studied extensively in \cite{Schenk,Demir, Armada,Petrovic,Ville,Sahai,Pitarokoilis}. The effect of PHN on OFDM systems is classified into two components. The common phase error (CPE), which rotates all the sub-carriers in one OFDM symbol by a common phase distortion, and the inter-carrier interference (ICI), which arises due to the loss of orthogonality between each sub-carrier. Frequency domain approaches to PHN estimation and compensation mainly deal with the CPE and ICI components separately, while time domain approaches attempt to compensate for both jointly. It has been shown that a significant improvement in performance can already be achieved with CPE correction only by treating the ICI term as an additive Gaussian noise \cite{Petrovicc, Sridharan, Nie}. However, CPE correction only is not always sufficient for high rate transmission, therefore ICI compensation is necessary.
Most ICI compensation techniques employ decision-directed feedback (DD-FB) for frequency domain PHN estimation \cite{Petrovic, Bittner,Munier,Corvaja,Yue,Khanzadi}. A vast majority of works assume a known channel frequency response \cite{Petrovic,Bittner,Yue, Khanzadi,Syriala,Tchamov, Linn}. For unknown channel, joint channel and PHN estimation is considered in \cite{Munier} where an ICI reduction scheme over a Rayleigh fading channel is presented in which the PHN process within an OFDM symbol is modeled as a power series. Although the method presented showed a significant bit error rate (BER) improvement, the cost in computation is large. A less complex method is presented in \citep{Corvaja} where the estimate of channel and CPE at pilot subcarriers are interpolated to obtain channel frequency response and CPE followed by a DD-FB loop to estimate ICI components.
A non-iterative compensation scheme in which the CPE between consecutive OFDM symbols are interpolated linearly to estimate the time varying PHN is presented in \cite{Syriala,Tchamov}. In \cite{Chun-Ying}, a method of suppressing the ICI is presented by linearly combining the cyclic-prefix and the corresponding OFDM samples. The linear coefficients are obtained such that the ICI power is minimized. Other methods using the Bayesian framework for joint channel and PHN estimation have been proposed in \cite{Khanzadi2,Septier,Simoens,negusse}. Soft-input maximum a posteriori and extended Kalman smoother are proposed in \citep{Khanzadi2}. In \cite{Septier,Simoens,negusse}, Monte-Carlo methods are employed to approximate the posterior probability distribution of the unknown quantities for PHN tracking and channel estimation. Although they are shown to provide good BER performance, Monte-Carlo methods, such as particle filters, are known for their numerical complexity. The main interest in the paper lies on PHN compensation over unknown channels using pilot sub-carriers. However, the proposed method also uses DD-FB loop for improved performance. For the known channel the proposed method is benchmarked against \cite{Petrovic}. For known and unknown frequency selective Rayleigh fading channels the methods in \cite{Munier,Corvaja} are also used as benchmarks. The problem setting in which the methods presented in \cite{Petrovic}, \cite{Munier,Corvaja} and are employed is identical to the setting the proposed algorithm.
\subsection{Contributions}
In this paper, we employ a simple and novel scheme for PHN cancellation which is shown to have superior performance in terms of BER compared to previously proposed methods. The proposed technique can be applied to any assumed distribution of the PHN process, $\theta(n)$. Additionally, no approximation based on the magnitude of the PHN is taken into account, e.g. $e^{j\theta(n)}\simeq 1 + j\sin(\theta(n))$, as is common in some previous works.
A codebook of $K$ vectors representing a set of trajectories which aim to closely match with the PHN realization is used. An uncountable set of possible PHN realizations is represented by a codebook with $K$ quantized trajectories, which are stored at the receiver.\footnote{The technique is similar to the design of vector quantization codebook where a set of $n$ vectors from some $m$-dimensional space is efficiently represented by a codebook with a smaller set of vectors from $m$-dimensional space \cite{Gersho}.} The complexity of the algorithm is therefore dependent on the number of trajectories in the codebook, $K$. For a moderately small sized codebook, it is shown that the impact of the PHN is significantly reduced (e.g. with $K=27$, there is 60\% reduction in terms of the effective PHN mean square error (MSE), see Section \ref{MSE}).
In this paper, we focus on a solution where the trajectory that minimizes the Euclidean distance between the constellation of the received symbols at the pilot positions and the constellation of the known pilot symbols is chosen. Additionally, a DD-FB technique is employed such that both the estimate of data symbols from the channel decoder and pilot symbols are used to compute the Euclidean distance. The proposed technique can also be used for combined channel estimation and PHN compensation schemes, in which MMSE based channel estimation is employed for each of the vectors in the codebook. The channel estimate corresponding to the trajectory in the codebook which best approximates the PHN realization is chosen. We will show by the simulation that, for AWGN and fading channels with a known frequency response, the proposed scheme outperforms previously proposed PHN compensation techniques. In this scenario, the proposed method does not employ DD-FB while the reference schemes require previously detected symbols for ICI estimation. Simulation results are also shown for unknown channel frequency response, in which the proposed method employs a DD-FB technique for combined channel estimation and PHN compensation. It is shown that the method presented provides improved performance compared to previously proposed methods. Additional results based on a PHN process modelled as Ornstein-Steinbeck is also shown to illustrate the applicability of the proposed method for a PHN model other than Weiner process.
The paper is organized as follows. Sec. \ref{sec:2} introduces the system model and parameters while Sec. \ref{sec:3} presents design of the codebook for Weiner PHN along with some performance analysis. In Sec. \ref{sec:4}, details of implementation of the codebook
for PHN compensation together with channel estimation. Simulation results are shown in Sec. \ref{sec:5} and a computational analysis of the proposed algorithm as well as some of the reference methods is given in Sec. \ref{sec:6}. We end with concluding remarks in \ref{sec:7}.
\section{System Model and Proposed Approach\label{sec:2}}
We consider an OFDM transmission system over a fading time varying channel with $L$ multipath taps employing QAM modulation technique. The OFDM system is assumed to have FFT size of $N$ of which $N_p$ are pilot subcarriers and with a cyclic prefix of length $N_{cp}$ such that $N_{cp} \geq L$. To obtain the $m$-th OFDM symbol of duration $T$, a stream of data bits is divided to $N-N_p$ groups of $M$-bits which are mapped to $2^M$-QAM complex symbols, $S_m(k)$. Assuming perfect frequency and timing synchronization, the baseband representation of the received time domain signal, $y_m(n)$, is given by
\begin{equation}\label{eq1}
y_m(n)=e^{j\theta_m(n)}\bigg(\sum_{\ell=0}^{L-1}h_m(\ell)s_m(n-\ell)\bigg) + w_m(n)
\end{equation}
where $s_m(n)$ is the $N$ point inverse FFT (IFFT) of the complex transmitted symbol, $S_m(k)$, $h_m(i)$ is the fading complex channel pulse response with $L$ propagation paths, $w_m(n)$ is a zero mean complex circular Gaussian channel noise with variance $\sigma^2_w$; and $\theta_m(n)$ is the PHN sample at the receiver at time index $n$ of the $m$-th OFDM symbol. The received signal is sampled at a frequency $f_s = (N+N_{cp})/T$, where $T$ is the OFDM symbol duration.
After removing the cyclic prefix, an FFT operation is applied to the received signal such that the $k$-th subcarrier is given by
\begin{equation}\label{eq2}
\begin{split}
Y_m(k)&=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1}\sum_{\ell=0}^{L-1}h_m(\ell)s_m(n-\ell)e^{j(\theta_m(n)-2\pi k n)}\\
& + \frac{1}{\sqrt{N}}\sum_{n=0}^{N-1}w_m(n)e^{-j2\pi k n}\\
&= S_m(k)H_m(k)\underbrace{A_m(0)}_{\displaystyle \mbox{CPE}} + \underbrace{\sum_{\substack{l=0 \\
l\neq k}}^{N-1} S_m(l)H_m(l)A_m(l-k)}_{\displaystyle \mbox{ICI}} \\ &+ W_m(k)
\end{split}
\end{equation}
where $A_m(k)$ and $W_m(k)$ are respectively the FFT of $e^{j\theta_m(n)}$ and that of the additive complex Gaussian channel noise $w(n)$. $H_m(k)$ is the instantaneous frequency response of the multi-path channel which is assumed to be constant within one OFDM symbol. Figure \ref{fig1_a} shows the block diagram of the system model. Equation \eqref{eq2} can further be written in matrix form as
\begin{figure*}[t]
\centering
\includegraphics[scale=0.63]{figures/figure1}
\caption{Block diagram of the OFDM system.}
\label{fig1_a}
\end{figure*}
\begin{equation}\label{eq3}
\mathbf{Y}_m= \mathbf{A}_m \mathbf{S}^d_m \mathbf{H}_m + \mathbf{W}_m
\end{equation}
where $\mathbf{S}^d_m=\mbox{diag}(S_m(0),..,S_m(N-1))$ is an $N \times N$ matrix representing diagonalization of the symbol vector $\mathbf{S}_m$; and the $N \times N$ matrix $\mathbf{A}_m$ is a unitary circulant matrix containing DFT coefficients of $e^{j\theta_m(n)}$ such that $\mathbf{A}_m(k,l)=A_m(l-k)$. $\mathbf{H}_m$ and $\mathbf{W}_m$ are vectors of the channel frequency response and the complex Gaussian noise respectively. Estimation of the realization of the PHN process, $\mathbf{\theta}_m=[\theta_m(0), \cdots \theta_m(N-1) ]^T$, would enable compensation of CPE as well as ICI.
\subsection{Approximate Model}
We propose a method where the PHN realization at the $m$-th OFDM symbol is approximated by one of a set of $K$ vectors, $\{\mathbf{\phi}^k\}_{k=1}^K$, where $\mathbf{\phi}^k =[\phi_k(0), \phi_k(1), \cdots, \phi_k(N-1)]^T$ plus an initial phase offset, $\psi_m \in [0, 2\pi)$. Associated with each vector, $\mathbf{\phi}_k$, is a corresponding unitary circulant matrix containing DFT coefficients of $\{e^{j\phi_k(l)}\}_{l=0}^{N-1}$ denoted by $\mathbf{\widehat{A}}$. The signal model in \eqref{eq3} is then modified as
\begin{equation}\label{eq4}
\mathbf{\bar{Y}}_m=e^{j\psi_m}\mathbf{\widehat{A}}_{k_m}\mathbf{S}^d_m \mathbf{H}_m + {\mathbf{W}}_m
\end{equation}
which is an approximation of \eqref{eq3}, where the PHN DFT coefficient matrix, $\mathbf{A}_m$, is now replaced by $e^{j\psi_m}\mathbf{\widehat{A}}_{k_m} $, which is a function of an index, $k_m$, that takes on a value between $1$ and $K$ with equal probability. The framework can be extended to unequal probabilities but this is left for future work. The random variable $\psi_m$, which is assumed independent of $k_m$, is uniformly distributed and models the initial phase of the symbol. Both $k_m$ and $\psi_m$ are independent from symbol to symbol. The design of the codebook $\{\phi^k\}_{k=1}^K$ is yet to be discussed in a later section. It should be noted that this approximate model is only used for the purpose of deriving the algorithm. Data from the original model in \eqref{eq3} is used when testing and validating the proposed algorithm by simulation.
\subsection{Cost Function}
Any estimator derived in the absence of PHN can be written as
\begin{equation}\label{eq5}
\widehat{\Omega} = \arg\min_{\Omega}\mathcal{C}(\mathbf{\bar{Y}}_1, \dots, \mathbf{\bar{Y}}_q),
\end{equation}
where $\widehat{\Omega}$ is the estimated parameter, $\mathcal{C}$ is some form of optimization criterion and $q$ is the number of OFDM symbols. It should be noted that $\mathbf{\widehat{A}}_{k_m}$ and its inverse $\mathbf{\widehat{A}}_{k_m}^{-1}$ are unitary circulant matrices. As a result, $\mathbf{\widehat{A}}_{k_m}$ is invertible and $\mathbf{\widehat{A}}_{k_m}^{-1}\mathbf{W}_{m}$ is statistically identical to $\mathbf{W}_{m}$. We may therefore generalize the criterion \eqref{eq5} as
\begin{equation}\label{eq6}
\widehat{\Omega} = \arg\min_{\Omega, k_1,\cdots,k_q,\psi_1,\ldots,\psi_q}\mathcal{C}(e^{-j\psi_1}\mathbf{\widehat{A}}_{k_1}^{-1}\mathbf{\bar{Y}}_1, \dots, e^{-j\psi_q}\mathbf{\widehat{A}}_{k_q}^{-1}\mathbf{\bar{Y}}_{q}).
\end{equation}
for the model \eqref{eq4}. This includes, for instance, maximum-likelihood estimators where now the estimator in \eqref{eq6} is also the maximum likelihood estimator for the model \eqref{eq4}. An example of a use of the proposed new estimator would be in an OFDM system with an equalizer and Viterbi decoder and a pilot subcarrier. In such a system, the optimization function could be the sum path metric. In this case, \eqref{eq6} implies that the entire receiver is run for all possible hypotheses of $k_1 \dots k_q$ (of which there are $K^q$), and selects the one with the lowest path metric (in each symbol optimizing over $\psi$). This implementation is generally not realistic. A practical criterion is hard decoding of the M-QAM symbols of OFDM symbols, where the criterion function separates into a term for each symbol individually. In this case, the criterion function is the Euclidean distance between estimates of the received signal point, $\widehat{S}_m$, at the output of the equalizer and the nearest point in the M-QAM constellation. Here, it is realistic to sequence through a quite large size $K$. Selection of the codebooks is a design problem still to be considered.
For the case where the Wiener PHN model is used, a systematic codebook design is presented in the next subsection. For other PHN models different codebook designs may perform better. However, the design for Wiener PHN may work well for other PHN models as well. This is illustrated in Section \ref{pll} where results based on a PHN modelled as Ornstein-Uhlenbeck process is presented.
\section{Codebook design for Weiner PHN \label{sec:3}}
The model introduced in \cite{Demir} and commonly employed in literature, where the PHN process is described by a Wiener process, is used. Under the Weiner model, the PHN sample at the $m$-th OFDM symbol is given by
\begin{equation}\label{eq7}
\theta_m(n)=\sum_{i=0}^{mN+n}\varepsilon(i) =\psi_m + \sum_{l=0}^{n}\varepsilon(l)
\end{equation}
for $n=0,\ 1, \dots ,N-1$, where $\varepsilon(l)$ is a Gaussian random variable with zero mean and variance $\sigma^2_\varepsilon =2\pi\beta T/N$ (in radians$^2$), and $\beta T$ denotes the rate at which the PHN variance grows in one OFDM symbol.
\begin{figure}
\centering
\psfrag{x}{\scriptsize{Time sample in an OFDM symbol}}
\psfrag{y}{\scriptsize{Phase Error (radians)}}
\psfrag{data1}{$\theta(n)$}
\psfrag{data2}{$\hat{\theta}(n)$}
\includegraphics[scale=0.33]{figures/figure2}
\caption{A single Wiener PHN realization (blue) with $\sigma^2_\epsilon=2\pi(0.01/N)$ where $N=64$ and its approximation according to \eqref{eq8} for $J=8$.}
\label{fig2}
\end{figure}
\begin{figure}
\psfrag{h}{\tiny{$x_1$}}
\psfrag{w}{\tiny{$-x_1$}}
\psfrag{a}{\tiny{$x_2$}}
\psfrag{b}{\tiny{$x_r$}}
\psfrag{t}{\tiny{$-x_{2}$}}
\psfrag{u}{\tiny{$-x_{r}$}}
\psfrag{Q}{\tiny{$R_i$}}
\psfrag{R}{\tiny{$R_1$}}
\psfrag{J}{\tiny{$R_{i-1}$}}
\psfrag{K}{\tiny{$R_{\tiny{i+1}}$}}
\psfrag{D}{\tiny{$R_{\scriptsize{Q}}$}}
\psfrag{c}{$\cdots$}
\psfrag{d}{$\cdots$}
\includegraphics[scale=0.33]{figures/figure3}
\caption{Dividing the Gaussian sample space into $Q$ regions of equal areas, where $Q$ is an odd integer.}
\label{fig3}
\end{figure}
The objective is to construct a codebook containing a finite set of vectors which approximate the trajectories of possible Wiener PHN realizations within one OFDM symbol. The subscript $m$ in $\theta_m(n)$ and $\psi_m(n)$ is dropped for notational simplicity. Let the $N$ samples of the OFDM symbol be divided into $J$ segments with $L=N/J$ samples per segment. Then $\hat{\theta}(n)$ is defined as a process which is constant in each segment. The value of the constants, $\lambda_j$, in each segment are set as the sample average of $\theta(n)$ in the corresponding segment. The PHN process is therefore approximated by $\hat{\theta}(n)$ given by
\begin{equation}\label{eq8}
\begin{split}
\hat{\theta}(n+jL)=\lambda_j&=\frac{1}{L}\sum_{\ell=0}^{L-1}\theta(\ell+jL)\\
&= \frac{1}{L}\sum_{\ell=0}^{L-1} \sum_{i=0}^{jL+\ell}\epsilon(i) + \psi
\end{split}
\end{equation}
for $n=0, \cdots, L-1$ and $j=0, \cdots, J-1$. This is illustrated in Figure \ref{fig2} for $\sigma_\epsilon^2=2\pi(0.01/N)$, $N=64$ and $J=8$, where the samples within each segment are approximated by the sample mean of the elements in the segment. The process, $\hat{\theta}(n)$, is similar to a Weiner process where each step, $X_{j}=\lambda_{j+1}-\lambda_{j}$, occurring every $L$ samples, is given by
\begin{equation}\label{eq9}
\begin{split}
X_{j}&=\frac{1}{L}\sum_{\ell=0}^{L-1}\bigg(\theta((j+1)L+\ell) - \theta(jL+\ell)\bigg)\\
&=\frac{1}{L}\sum_{\ell=0}^{L-1} \sum_{i=\ell+jL+1}^{\ell+(j+1)L}\epsilon(i)
\end{split}
\end{equation}
where the second equality follows from \eqref{eq8}. The increments, $X_j$, are identically distributed Gaussian random variables with zero mean and variance $\sigma_x^2$ given by
\begin{equation}\label{eq17k}
\begin{split}
\sigma_x^2 = E[X^2] &= \frac{1}{L^2}\sum_{\ell=0}^{L-1} \sum_{i=\ell+jL+1}^{\ell+(j+1)L}\sum_{v=0}^{L-1} \sum_{k=v+jL+1}^{v+(j+1)L}E[\epsilon(i)\epsilon(k)]\\
&=\frac{(2L^2+1)}{3L}\sigma_\epsilon^2
\end{split}
\end{equation}
where the auto-correlation $E[\epsilon(\ell)\epsilon(k)] = \sigma_\epsilon^2\delta(k-\ell)$ as well as arithmetic and geometric series are used to obtain the second equality. Note that for $L=1$, that is when the number of segments is equal to the number of samples, $J=N$, then $\sigma_x^2=\sigma_\epsilon^2$. The approximated Weiner PHN process can be re-written in terms of $X_{j}$ as
\begin{equation}\label{eq10}
\hat{\theta}(jL+n)= \lambda_{j-1} + X_{j} +\psi = \sum_{\ell=0}^{j}X_\ell +\psi
\end{equation}
where $\lambda_{-1}=0$ and $X_0=0$ for $j=0$.
Since the Gaussian sampling space is unbounded, the random variable $X$ is instead represented by a set of quantized samples, $\{\hat{x}_i\}_{i=1}^Q$, which represent $Q$ regions, $\{\mathcal{R}_i\}_{i=1}^Q$, in the Gaussian pdf with zero mean and variance $\sigma_x^2$. This implies that the increments at the consecutive segments, $j=1, \cdots, J-1$, are defined by a set $\{\hat{x}_i\}_{i=1}^Q$ which define $Q$ possible trajectories. That is, the second segment, $j=2$, has $Q$ possible increments which set $Q$ possible trajectories. In the next segment, there will be $Q$ possible increments on each trajectory from the previous segment such that there are $Q^{j-1}$ trajectories at the $j$-th segment. Therefore, at the $J$-th segment, there will be $K=Q^{J-1}$ trajectories representing quantized paths of the random walk process each of which are set as a vector entry in the codebook, ${\mathbf{\phi}}_{k=1}^K$. The number of code vectors, $K$, is therefore determined by the predefined set of quantization regions, $Q$, and the number of segments the symbol is divided into, i.e. $J$.
\subsection{Defining Quantization Regions}
The quantization regions can be defined as a set of regions, $\{\mathcal{R}_i\}_{i=1}^Q$, which divides the Gaussian sample space into $Q$ regions. For ease of development, the sample space is divided into a set of regions with equal areas (i.e. equiprobable). The regions are defined as
\begin{equation}\label{eq11}
P\bigg[X\in \mathcal{R}_1\bigg]=\cdots= P\bigg[ X\in \mathcal{R}_Q \bigg]=\frac{1}{Q}
\end{equation}
\begin{figure}
\centering
\psfrag{x}{\scriptsize{Time sample in an OFDM symbol}}
\psfrag{y}{\scriptsize{Phase Error (radians)} }
\psfrag{data1}{{PHN process}}
\psfrag{data2}{\small{Set of trajectories}}
\includegraphics[scale=0.33]{figures/figure4}
\caption{A single Wiener PHN realization (blue) with $\sigma^2_\epsilon=2\pi(0.01/N)$ where $N=64$ and a finite set of $K=2187$ trajectories,* for $J=8$ and $Q=3$, which corresponds to possible trajectories that the PHN realization might take for equiprobable quantization. }
\label{fig4}
\end{figure}
where $P[X\in \mathcal{R}_i]$ implies the probability that the random variable $X$ is in the region $\mathcal{R}_i$. For the case when $Q$ is odd, the sample space is partitioned by $Q+1$ data points denoted by $\{-\infty, x_{-r}, \ldots,x_{-1}, x_{1}, \ldots, x_{r}, \infty \}$, as shown in Figure \ref{fig3}, where $x_{-\ell}=-x_{\ell}$, $\ell=1, \ldots, r$ and $r=(Q-1)/2$. The regions at the tails of the Gaussian pdf are given by $\mathcal{R}_{1}=[-\infty, x_{-r}[$ and $\mathcal{R}_{Q}=[x_{r}, \infty[$, while the regions in the middle are bounded by $[x_{\ell},x_{\ell+1}[$. The bounding points, $x_{\ell}$ are given by
\begin{equation}\label{eq12}
x_\ell = \sqrt{2\sigma_x^2} \mbox{erf}^{-1}\bigg(\frac{2\ell-1}{Q}\bigg), \quad \ell=1, \cdots, \frac{Q-1}{2}
\end{equation}
where $\mbox{erf}^{-1}(\cdot)$ is the inverse error function. On the other hand, when $Q$ is even, the sample space is partitioned by $2r + 1$ data points denoted by $\{-\infty, x_{-r}, \cdots,x_{-1}, x_{0}, x_{1}, \cdots, x_{r}, \infty\}$ where $x_{0}=0$ and $x_{-\ell}=-x_{\ell}$ for $\ell=0, \cdots, r$ where $r=Q/2$. These points are given by
\begin{equation}\label{eq13}
x_\ell = \sqrt{2\sigma_x^2} \mbox{erf}^{-1}\bigg(\frac{2\ell}{Q}\bigg), \quad \ell=0, \cdots, \frac{Q}{2} .
\end{equation}
The plot in Figure \ref{fig4} is shown as an example to illustrate the process for a set of equiprobable quantization regions where $Q=3$ and $J=8$ while Figure \ref{fig4a} shows the trajectory that is closest to the PHN realization. The regions can also be defined as a set which divides the Gaussian sample space uniformly. This implies that the codebook trajectory would have unequal probability. The development of this case is left for future work.
\begin{figure}
\centering
\psfrag{x}{\scriptsize{Time sample in an OFDM symbol}}
\psfrag{y}{\scriptsize{Phase Error (radians)} }
\psfrag{l}{\tiny{PHN process}}
\psfrag{n}{\tiny{Selected trajectory } }
\psfrag{m}{\tiny{Selected trajectory (codebook with uniform quantization) } }
\psfrag{data1}{\tiny{PHN process}}
\psfrag{data2}{\tiny{Selected trajectory}}
\includegraphics[scale=0.33]{figures/figure5}
\caption{The trajectory that best matches the PHN realization.}
\label{fig4a}
\end{figure}
\subsection{Quantization Points}
Given the quantization regions, $\mathcal{R}=\{R_1, \ldots, R_{Q}\}$, the quantization points, $\{\hat{x}_1, \ldots, \hat{x}_Q \}$, which represents each region is defined as the mean point within the region. That is,
\begin{equation}\label{eq14}
\begin{split}
\hat{x}_i =E\bigg[X|X\in R_i\bigg]&=\frac{\int_{R_i}xf_X(x)dx}{\int_{R_i}f_X(x)dx}\\
\end{split}
\end{equation}
for $i=1, \ldots, Q$, where $E[\cdot]$ is the expectation operator and $f(x)$ is the Gaussian distribution function. It can be shown that \eqref{eq14} provides the set which results in a minimum squared error quantization of the random variable $X$ with a pdf $f(x)$, \citep{Gersho}. Moreover, the expected value of quantization error $q_{\tiny{error}}=x-\hat{x}$ is zero, i.e. $E[\hat{x}]=E[x]$. For the Gaussian pdf with zeros mean and variance $\sigma_x^2=(2L^2+1)/3L\sigma_\epsilon^2$, \eqref{eq14} is given by
\begin{equation}\label{eq15}
\hat{x}_i = \frac{-\sigma_x}{0.5\sqrt{2\pi}}\bigg(\frac{\exp(-0.5x_{i+1}^2/\sigma_x^2)-\exp(-0.5x_{i}^2/\sigma_x^2)}{\mbox{erf}(x_{i+1}/\sqrt{2}\sigma_x)-\mbox{erf}(x_{i}/\sqrt{2}\sigma_x)}\bigg).
\end{equation}
It can be seen that the codebook will contain an all zeros entry when $Q$ is odd. This is attractive, since it means no correction will be done if the receiver would have no phase noise and the SNR is high. The plot in Figure \ref{fig4} employs \eqref{eq12} for $Q=3$ to define the quantization regions and \eqref{eq15} to obtain the quantized point for each region.
\begin{figure*}[t]
\centering
\captionsetup{justification=centering,margin=2cm}
\includegraphics[scale=0.63]{figures/figure6}
\caption{Block diagram of the coded OFDM receiver with PHN compensation and channel estimation.}
\label{fig5}
\end{figure*}
\subsection{\mbox{MSE} Analysis \label{MSE}}
{In this section, the $\mbox{MSE}$ for the designed codebook is analysed. The objective is to provide a good indication of the performance of the designed codebook for a given $J$ and $Q$.} The $\mbox{MSE}$ between a random PHN realization and the approximate model introduced is given by
\begin{equation}\label{eq15_a}
\mbox{MSE}=E_{\theta(n)}[ \min_{k,\psi} \sum_n |\theta(n) - \psi- \phi_k(n)|^2 ]
\end{equation}
which is difficult to evaluate analytically. Therefore the MSE is evaluated by simulation. However, an approximate expression for the MSE, derived in Appendix \ref{sec:app}, is given by
\begin{equation}\label{eq17}
\mbox{MSE} = \frac{(N+J)(N-J)}{6J}\sigma_\epsilon^2 + L(J-1)\sigma_q^2
\end{equation}
with $\sigma_q$ given by \eqref{eq17n}.
Table \ref{tab1} shows the $\mbox{MSE}$ values obtained by Monte-Carlo simulations (denoted $\mbox{MSE}_s$) normalized by the $\mbox{MSE}$ for the CPE correction only (i.e. $\mbox{MSE}$ for $J=1$). The number of PHN realizations employed were 5000 for every codebook with various combination of $J$ and $Q$ resulting in $K=Q^{J-1}$. The approximated theoretical $\mbox{MSE}_{\mbox{a}}$ values computed according to \eqref{eq17} are also given. It can be seen that the analytical and simulation results are similar indicating that the reasoning used in the analytical derivation in Appendix \ref{sec:app} is correct. The $\mbox{MSE}$ analysis assumes that the optimum code-book entry is chosen. In order to account also for the effect of noise and decision feedback errors, simulations of the full receiver for various codebook sizes are performed in Sec. \ref{sec:5}. These results confirm that the performance improves with increasing codebook size. The choice of codebook size will ultimately depend on the computational cost and performance requirements for the application at hand. However, it is noted that complexity grows rapidly with $Q$ and $J$, thus moderate numbers such as e.g. $Q=2$, $J=5$ or $Q=3$ and $J=4$ seems to provide a good compromise.
\begin{table*}[t]
\centering
\caption{Approximated $\mbox{MSE}$ for $N=64$ as is given by \eqref{eq17} and simulated $\mbox{MSE}$ denoted by $\mbox{MSE}_s$ for various values of $Q$ and $J$. The values are normalized by the $\mbox{MSE}$ value for CPE correction only given by $\mbox{MSE}=(N-1)(N+1)\sigma_\epsilon^2/6$. \label{tab1}}
\input{tables/table1}
\end{table*}
\section{Implementation on OFDM System \label{sec:4}}
\subsection{Known Channel Response}
The received signal vector after removing cyclic prefix, $\mathbf{y}_m=[y_m(0), \cdots, y_m(N-1)]^T$, is multiplied by each of the $K$ trajectories $e^{-j\mathbf{\phi}_k}$ such that a new set of $K$ de-rotated OFDM signals is obtained at the receiver
\begin{equation}\label{eq18}
\mathbf{\tilde{y}}_m^k=[e^{-j\phi_k(0)}y_m(0), \cdots, e^{-j\phi_k(N-1)}y_m(N-1)]^T.
\end{equation}
An FFT operation on each de-rotated signal vector provides
\begin{equation}\label{eq19}
\begin{split}
\mathbf{\widetilde{Y}}_m^k&=\mathbf{\widehat{A}}_{k}^{-1}\mathbf{Y}_m \\
&=\mathbf{\widehat{A}}_{k}^{-1}(\mathbf{A}_m\mathbf{S}^d_m\mathbf{H}_m + \mathbf{{W}}_m),
\end{split}
\end{equation}
where the $N \times N$ circulant matrix $\mathbf{\widehat{A}}_{k}^{-1}$ is a matrix of DFT coefficients of $e^{-j\phi_k(n)}$. Frequency domain channel equalization is applied on $\mathbf{\tilde{Y}}_m^k$, and the criterion by which the trajectory that best approximates the PHN realization is given by
\begin{equation}\label{eq21}
k^*= \arg\min_{k} \sum_{i=0}^{P-1}\bigg| \widehat{{S}}_m^k(\ell_i) - S_m(\ell_i)\bigg|^2.
\end{equation}
where $\ell_i \in (\ell_0, \dots , \ell_{P-1})$ denotes pilot subcarriers, $\widehat{S}_m^k(\ell_i)$ is an element of the estimated symbol vector for the $k$-th trajectory given by
\begin{equation}\label{eq21a}
\widehat{\mathbf{S}}_m^k=\eta_k^{-1}\mbox{diag}(\mathbf{H}_m)^{-1}\mathbf{\widetilde{Y}}_m^k
\end{equation}
with complex term $\eta_k$ that corrects for the effective CPE which includes the offset $\psi_m$ at each symbol, the DC level of the codebook vector, $\phi^k$, and the accumulated quantization error. It is given by
\begin{equation}\label{eq22}
\eta_k= \frac{\sum_{i=0}^{P-1} S_m^*(\ell_i)\widetilde{Y}_m^k(\ell_i)/H_m^k(\ell_i)}{ \sum_{i=0}^{P-1} |S_m(\ell_i)|^2}.
\end{equation}
In a DD-FB loop in which the output of the channel decoder can be exploited, non-pilot symbols can also be used such that the criterion in \eqref{eq22} takes into account the decoded symbols as well
\begin{equation}\label{eq23}
k^*= \arg\min_{k} \sum_{l=0}^{N-1}\bigg| \widehat{{S}}_m^k(l) - S_m(l)\bigg|^2.
\end{equation}
for $\eta_k$ given by
\begin{equation}\label{eq24}
\eta_k= \frac{\sum_{l=0}^{N-1} S_m^*(l)\widetilde{Y}_m^k(l)/H_m^k(l)}{ \sum_{i=0}^{N-1} |S_m(l)|^2}.
\end{equation}
The symbol sent to the decoder is then given by the index corresponding to $\widehat{\mathbf{S}}_m^{k^*}.$
\subsection{Combined Channel Estimation and ICI Cancellation }
Since the channel is not always known at the receiver, it needs to be estimated.
Assuming the channel remains stationary within the time period of an OFDM symbol, for a codebook of size $K$, there will be $K$ received signal candidates, $\mathbf{\tilde{Y}}^k_m$, which are given by \eqref{eq19} where the channel response, $H_m(k)$, is considered to be unknown. The MMSE estimate of the channel frequency response vector, $\widehat{\mathbf{H}}_m^k$, employing the $k$-th trajectory of the codebook is then given by \cite{Beek}
\begin{equation}\label{eq25}
\widehat{\mathbf{H}}_m^k=E[\mathbf{H}_m(\mathbf{H}_m)^H]\bigg(E[\mathbf{H}_m(\mathbf{H}_m)^H]+\frac{\sigma^2_w}{E_s}\mathbf{I}_{N}\bigg)^{-1}\widehat{\mathbf{H}}^k_{\mbox{\tiny{LS}},m}
\end{equation}
where $E[\mathbf{H}_m (\mathbf{H}_m)^H]$ is the autocorrelation matrix of the channel frequency response for the given statistical model of the channel and $\widehat{\mathbf{H}}_{\mbox{\tiny{LS}},m}^k$ is the least squares estimate of $\mathbf{H}^k_m$ given by
\begin{equation}\label{eq26}
\widehat{\mathbf{H}}^k_{\mbox{\tiny{LS}},m} = \mathbf{S}_m^{-1}\widetilde{\mathbf{Y}}_m^k = \bigg[\frac{\widetilde{Y}^k_m(0)}{S_m(0)}, \cdots, \frac{\widetilde{Y}^k_m(N-1)}{S_m(N-1)}\bigg]^T.
\end{equation}
However, since not all transmitted symbols are known at the receiver, $N_p$ pilot symbols, fitted evenly among the $N$ subcarriers, are employed to obtain a least squares estimate of the channel frequency response at the pilot positions. For the sake of convenience, the LS estimate at pilot positions are denoted by the vector, $\widehat{\mathbf{H}}_{\mbox{\tiny{LS}},m}^{k,p}$. Therefore, the MMSE estimate of the channel frequency response, for $k$-th trajectory in the codebook, is initially obtained based on pilot symbols
\begin{equation}\label{eq27}
\widehat{\mathbf{H}}_m^k=E[\mathbf{H}_m(\mathbf{H}_m^p)^H]\bigg(E[\mathbf{H}_m^{p}(\mathbf{H}_m^{p})^H]+\frac{\sigma^2_w}{E_s}\mathbf{I}_{N_p}\bigg)^{-1}\widehat{\mathbf{H}}_{\mbox{\tiny{LS}},m}^{k,p}
\end{equation}
where $(\cdot)^p$ denotes a vector whose elements are positioned at pilot subcarriers. A set of $K$ channel frequency response estimates, $\{ \widehat{\mathbf{H}}_m^k \}_{k=1}^K$, corresponding to each trajectory in the codebook is available at the receiver. Equation \eqref{eq23} is used to determine the trajectory which closely matches the PHN realization, and the corresponding channel estimate is then used to demodulate and obtain rough decisions on the data symbols. After channel decoding, a decision feedback technique is then employed to obtain \eqref{eq26} based on rough decisions on the symbols containing pilot and decision feedback symbols, $\overline{\mathbf{S}}_m$, which is then used in \eqref{eq25} to compute an MMSE estimate of $\mathbf{H}^k_m$ with improved accuracy.
For deeply faded channels, estimation accuracy during symbol $m$ can be improved by including previously decoded symbols, i.e., $\mathbf{\overline{S}}_{m-1}, \mathbf{\overline{S}}_{m-2}, \cdots, \mathbf{\overline{S}}_{m-D+1}$, in the estimation vector.
Therefore, taking into account $D$ previously decoded OFDM symbols, the MMSE estimator in \eqref{eq27} becomes
\begin{equation}\label{eq28}
\widehat{\mathbf{H}}_m^k=E[\mathbf{H}_m(\mathbf{H})^H]\bigg(E[\mathbf{H}(\mathbf{H})^H]+\frac{\sigma^2_w}{E_s}\mathbf{I}_{DN}\bigg)^{-1}\widehat{\mathbf{H}}_{\mbox{\tiny{LS}}}
\end{equation}
where $\mathbf{H}=[\mathbf{H}_m, \mathbf{H}_{m-1}, \cdots, \mathbf{H}_{m-D+1}]^T$ and $\widehat{\mathbf{H}}_{\mbox{\tiny{LS}}}=[\widehat{\mathbf{H}}_{\mbox{\tiny{LS}},m}^{
k_{m}}, \widehat{\mathbf{H}}_{\mbox{\tiny{LS}},m-1}^{k_{m-1}}, \cdots, \widehat{\mathbf{H}}_{\mbox{\tiny{LS}},m-D+1}^{k_{m-D+1}}]^T$. On the initial run of the $m$-th OFDM symbol, symbols at pilot subcarriers are used in order to compute \eqref{eq27}, after which the criterion in \eqref{eq21} is used to choose the best match index, $k^*$, for the trajectory which closely matches the PHN realization, $\mathbf{\phi}^{k^*}$, as well as the corresponding channel estimate, $\widehat{\mathbf{H}}_m^{k^*}$. Pilot and decision feedback symbols can then be used in the next iteration to obtain a better estimate, $\widehat{\mathbf{H}}_m^k$ using \eqref{eq25} or \eqref{eq28} if using $L$ previous OFDM symbols and \eqref{eq23} to obtain $k^*$. Joint channel equalization and PHN compensation is employed on the received signal $\mathbf{Y}_m$ such that the symbols sent to the decoder are given by
\begin{algorithm}
\centering
\caption{The proposed receiver algorithm with no DD-FB for uncoded system and known channel}\label{alg1}
\begin{algorithmic}[1]
\Procedure{ }{}
\State Received $m$-th OFDM symbol
\For{$k=0:K-1$}
\State $\mathbf{\widetilde{Y}}_m^k=\mathbf{\widehat{A}}_{k}^{-1}\mathbf{Y}_m$
\State Compute $\eta_k$ in \eqref{eq22} using pilot symbols
\State Compute $F(k)=\sum_{i=0}^{P-1}|\widehat{S}_m^k(\ell_i) - S_m(\ell_i)|^2$
\EndFor
\State Compute $k^*= \arg\min_{k} F(k)$ in \eqref{eq21} to obtain $k^*$
\State \textbf{return} $\widehat{\mathbf{S}}_m^{k^*}$
\EndProcedure
\end{algorithmic}
\end{algorithm}
\begin{equation}\label{eq29}
\widehat{\mathbf{S}}_m^{k^*} = \eta_{k^*}^{-1}\mbox{diag}(\widehat{\mathbf{H}}_m^{k^*})^{-1}\widetilde{\mathbf{Y}}_m^{k^*}
\end{equation}
where $k^*$ denotes the index of the most likely trajectory in the codebook obtained from \eqref{eq23}. Figure \ref{fig5} shows the OFDM receiver with the proposed PHN compensation and channel estimation scheme. An outline of the proposed algorithm for a known channel response without DD-FB loop is presented in Algorithm \ref{alg1}. In Algorithm \ref{alg2}, the algorithm for the combined channel frequency response estimation and PHN compensation scheme is presented for a coded frame of $M$ OFDM symbols.
\begin{algorithm}
\centering
\caption{The proposed algorithm for channel estimation and PHN compensation with DD-FB and channel coding}\label{alg2}
\begin{algorithmic}[1]
\Procedure{ }{}
\State Received frame
\Do { \em{Iteration on DD-FB loop} }
\Do { \em{for each OFDM symbol in the frame} }
\For{$k=0:K-1$}
\State $\mathbf{\widetilde{Y}}_m^k=\mathbf{\widehat{A}}_{k}^{-1}\mathbf{Y}_m$
\If{Zeroth iteration}
\State Channel estimation using pilots, \eqref{eq27}
\State Compute $\eta_k$ in \eqref{eq22} using pilot symbols
\State Detected symbol for each $k$ using \eqref{eq29}
\State Compute $F(k)=\sum_{i=0}^{P-1}| \widehat{S}_m^k(\ell_i)-$ \NoNumber{ $S_m(\ell_i)|^2$}
\Else
\State Channel estimation employing \eqref{eq25}
\State Compute $\eta_k$ using \eqref{eq24}
\State Compute $F(k)=\sum_{i=0}^{N-1}|\widehat{S}_m^k(\ell_i) -$ \NoNumber{ $S_m(\ell_i)|^2$}
\EndIf
\EndFor
\State Compute $k^*= \arg\min_{k} F(k)$ in \eqref{eq21} (zeroth iter- \NoNumber{ ation) or \eqref{eq23} to obtain $k^*$}
\State Select $\mathbf{\widehat{S}}_m^{k^*}$
\doWhile{ \textit{End of frame} }
\State Decode frame and deinterleave
\State Interleave, channel coding and modulation frame
\State DD-FB frame $\mathbf{\overline{S}}=[\mathbf{\overline{S}}_{0}, \cdots, \mathbf{\overline{S}}_{m}, \cdots, \mathbf{\overline{S}}_{M-1}]$
\doWhile{ \textit{End of iteration} }
\EndProcedure
\end{algorithmic}
\end{algorithm}
\section{Simulation Results \label{sec:5}}
In this section, simulation results are given to show the BER performance of the proposed technique under Wiener PHN. Two sets of simulation results are presented where in the first set, the performance of the algorithm in Algorithm \ref{alg1} for an uncoded system is presented under AWGN channel. In the second set, performance of the algorithm in Algorithm \ref{alg2} is shown for a Rayleigh fading channel. Simulation results are presented for 16-QAM and 64-QAM modulation schemes with $N=64$, $N_{cp}=16$ and $N_p=8$ pilot subcarriers. The codebook size for the results presented is $K=27$, in which $Q=3$ and $J=4$, which from the analysis in Sec. \ref{MSE} is a good compromise between complexity and performance. Perfect timing and frequency synchronization is assumed at the receiver.
\subsection{Uncoded System, AWGN Channel}
Considering the AWGN channel, the proposed Algorithm \ref{alg1} was compared with the PHN compensation method presented in \cite{Munier}, employing 3 iterations in the DD-FB. In the proposed method no DD-FB iterations are employed. The rate of growth of the PHN variance $\beta T$ is set to $0.01$, and 16-QAM OFDM and 64-QAM OFDM were considered without channel coding. The result is given in Figure \ref{fig6}. For the 64-QAM system (dashed line), the proposed method is $2.5$dB from the ideal PHN free case at a BER of $10^{-2}$, while the method in \cite{Munier} is $9$dB from the ideal case, i.e., an improvement of $6.5$dB. In the 16-QAM system (solid line), the proposed method is $2$dB from the ideal PHN free case at a BER of $10^{-3}$, while the method in \cite{Munier} is $3$dB from the ideal case.
\begin{figure}
\centering
\psfrag{x}{\scriptsize{$E_b/N_0$(dB)} }
\psfrag{y}{\scriptsize{BER}}
\psfrag{data1}{\tiny{\cite{Munier}, 3 DD-FB iterations (64-QAM)} }
\psfrag{data2}{\tiny{Proposed method (64-QAM), no DD-FB}}
\psfrag{data3}{\tiny{AWGN channel no PHN (64-QAM)} }
\psfrag{data4}{\tiny{\cite{Munier}, 3 DD-FB iterations (16-QAM)} }
\psfrag{data5}{\tiny{Proposed method (16-QAM)}, no DD-FB }
\psfrag{data6}{\tiny{AWGN channel no PHN (16-QAM)} }
\includegraphics[scale=0.38]{figures/figure7}
\caption{BER performance of proposed algorithm compared to \cite{Munier} for 16-QAM and 64-QAM uncoded OFDM under AWGN channel and PHN. Corresponding performance plots for an ideal case (i.e. AWGN channel no PHN) is also given. $BT=0.01$. }
\label{fig6}
\end{figure}
\subsection{Coded System with Fading Channel\label{sec:5a}}
A 10 tap Rayleigh fading channel with parameters specified in Table.\ref{tab3} is used. Moreover,
a 1/2 rate convolutional code with a constraint length of 7 is used on the input bit stream, which is then passed on to a bit interleaver over 20 OFDM symbols. A Viterbi decoder is used at the receiver. The PHN variance growth rate, $\beta T$, is set to $0.01$ for 16-QAM OFDM such that the RMS in degrees is $180\sqrt{2\pi\beta T}/\pi=14.4^0$; while for 64-QAM, $\beta T$ is set to $0.005$. Higher order modulation schemes such as 64-QAM are known to be affected particularly worse by PHN. The simulated channel, also employed in \cite{Munier}, is given for a normalized amplitude by \cite{Hoeher}
\begin{table}[t]
\centering
\caption{summary of the Rayleigh channel parameters \label{tab3}}
\input{tables/table2}
\end{table}
\begin{equation}\label{eq30}
H_m(k)=\sum_{i=0}^{L-1}\phi(\tau_i)e^{j(\varphi_i + 2\pi f_i^D m + 2\pi \frac{k\tau_i}{N})}
\end{equation}
where $\alpha_i$, $\varphi_i$, $f_i^D$ and $\tau_i$ are respectively the amplitude, phase, Doppler frequency and time delay of the $i$-th propagation path. The time correlation for the given channel model is defined as
\begin{equation}\label{eq31}
R_t(m-n)=E[H_m(k)H^*_n(k)].
\end{equation}
Assuming uniformly distributed angles of arrival at the mobile station, the correlation follows the well known Jakes model \cite{jakes} such that $R_t(m-n)=J_0(2\pi f_{max}^D(m-n))$, where $J_0(\cdot)$ denotes the zeroth order Bessel function of the first kind, and $f_{max}^D=(vf_c/c)T$ is the maximal Doppler frequency normalized by the OFDM symbol duration, $T$. Assuming an exponentially decaying power delay profile with normalized RMS delay spread $\tau_{\text{rms}}$, \citep{Sandell}, the correlation between different frequency bins is given by
\begin{equation}\label{eq32}
\begin{split}
R_f(k-l)&=E[H_m(k)H^*_m(l)]\\
&=\frac{1-e^{-L(1/\tau_{rms}+j2\pi(k-l))}}{(1-e^{-L/\tau_{rms}})(1+j2\pi(k-l)\tau_{rms})}
\end{split}
\end{equation}
which satisfies $E[|H_m(k)|^2]=1$.
\subsubsection{Channel fully known at the Receiver}
Assuming that the frequency response of the channel is known at the receiver, the performance of the proposed algorithm is presented and compared with previously proposed techniques in \cite{Petrovic} and \cite{Munier}.
In \cite{Petrovic}, a Fourier series representation is used to approximate the PHN realization at the $m$-th OFDM symbol, thereby aiming to suppress all the influence of the PHN. CPE compensation and symbol estimates are initially obtained using pilot subcarriers, after which a DD-FB loop is employed for MMSE estimation of the vector of Fourier coefficients of the PHN realization which is subsequently used for ICI compensation. Similarly, in \cite{Munier}, a two stage cancellation technique is employed in which the CPE is corrected, followed by ICI cancellation using decision feedback symbols. The ICI cancellation technique in \cite{Munier} is derived based on power series expansion of the PHN over an OFDM symbol.
The results in our approach were obtained using Algorithm \ref{alg2}, but where step 8 and 12 are omitted since the channel is already known. The results are shown by the solid curves in Figure \ref{fig7} and Figure \ref{fig8} for 16-QAM and 64-QAM respectively.
First, in Figure \ref{fig7} (solid line), it can be seen that the performance of the proposed algorithm without employing DD-FB is $0.5$dB from the ideal case (no PHN) for a BER of $10^{-4}$. On the other hand, for 64-QAM, Figure \ref{fig8} shows that the performance of the proposed algorithm is $2$dB from the ideal plot for a BER of $10^{-4}$. In both Figures, the proposed method shows performance gain in BER compared to the previously proposed methods in \cite{Petrovic}, which employs a single iteration on the DD-FB, and the method proposed in \cite{Munier}, which is shown for 2 and 3 DD-FB iterations respectively for 16-QAM and 64-QAM. It should be noted that no DD-FB loop is performed in the proposed approach.
\subsubsection{Unknown Channel Response}
For the case where the channel is unknown, joint channel estimation and PHN compensation is employed according to Algorithm \ref{alg2}. Performance of the techniques in \cite{Munier} and \cite{Corvaja} is also presented for comparison.
In \cite{Corvaja}, FFT interpolation is employed using the least squares estimates of the joint channel frequency response and CPE at pilot positions to obtain estimates over all subcarriers. The method also proposes estimating the CPE by averaging the phase displacements on pilot subcarriers which is used to remove its influence from the estimated channel frequency response. Previously decoded symbols are then employed for least squares (LS) estimation of the vector of Fourier coefficients of the PHN realization for ICI compensation using previous channel and symbol estimates. Three sets of results are presented for the method in \cite{Corvaja}. The first two results consider estimation of only one and three Fourier coefficients of the PHN realization closest to the DC for ICI compensation thus requiring less computational complexity. The third result requires inverting a $64\times64$ Toeplitz matrix to estimate the entire vector of Fourier coefficients of the PHN realization for ICI compensation.
The proposed method as well as the method in \cite{Munier} use $D=3$ past OFDM symbols as described in \eqref{eq23} to enhance performance of the channel estimation. Performance plots for the unknown channel case are shown with the dashed curve in Figure \ref{fig7} and Figure \ref{fig8} for 16-QAM and 64-QAM respectively. In both Figure \ref{fig7} and Figure \ref{fig8}, the performance of the proposed technique shows improved performance, in BER, over the previously proposed methods in \cite{Munier}, \cite{Corvaja}.
In Figure \ref{fig7}, the proposed method is $1.5$dB from the ideal case, (no PHN and known channel response) at a BER of $10^{-4}$ employing only 2 iterations on the DD-FB loop. On the other hand, the method in \citep{Munier}, shown for 8 iterations, is $2.5$dB from the ideal case while the method in \citep{Corvaja} is far from the ideal case.
For the 64-QAM system, Figure \ref{fig8} shows that the proposed method provides improved performance compared to \citep{Munier} and the method in \citep{Corvaja}. With 10 iterations, the proposed methods is $3$dB from the ideal case at a BER of $10^{-3}$, while the method in \cite{Munier} is $5$dB from the ideal case for the same number of iterations.
\begin{figure}
\centering
\psfrag{x}{\scriptsize{$E_b/N_0$(dB)} }
\psfrag{y}{\scriptsize{BER}}
\psfrag{data12}{\tiny{Full channel knowledge and no PHN}}
\psfrag{data11}{\tiny{Proposed method, without DD-FB loop (known channel)} }
\psfrag{data10}{\tiny{\cite{Munier}, 2 DD-FB iterations (known channel)} }
\psfrag{data9}{\tiny{ICI-cancellation Method in \cite{Petrovic} (known channel) }}
\psfrag{data8}{\tiny{Proposed method, 2 DD-FB iterations with channel estimation} }
\psfrag{data7}{\tiny{Proposed method without DD-FB with channel estimation} }
\psfrag{data6}{\tiny{\cite{Munier}, 8 DD-FB iterations with channel estimation} }
\psfrag{data5}{\tiny{\cite{Munier}, 2 DD-FB iterations with channel estimation} }
\psfrag{data4}{\tiny{\cite{Munier}, without DD-FB with channel estimation} }
\psfrag{data3}{\tiny{\cite{Corvaja}, full matrix} }
\psfrag{data2}{\tiny{\cite{Corvaja}, partial matrix 3 } }
\psfrag{data1}{\tiny{\cite{Corvaja}, partial matrix 1 } }
\includegraphics[scale=0.37]{figures/figure8}
\caption{BER performance of the proposed algorithm compared to \cite{Munier} and \cite{Corvaja} for 16-QAM OFDM. Performance plot for the ideal case (i.e. perfect channel knowledge and no PHN) is also given. ICI cancellation technique in \cite{Petrovic} is also shown for perfectly known channel. $BT=0.01$}
\label{fig7}
\end{figure}
\begin{figure}
\centering
\psfrag{x}{\scriptsize{$E_b/N_0$(dB)} }
\psfrag{y}{\scriptsize{BER}}
\psfrag{data9}{\tiny{Full channel knowledge and no PHN }}
\psfrag{data8}{\tiny{Proposed method without DD-FB loop (known channel)} }
\psfrag{data7}{\tiny{\cite{Munier} 3 DD-FB iterations (known channel)} }
\psfrag{data6}{\tiny{ICI-cancellation Method in \cite{Petrovic} (known channel) }}
\psfrag{data5}{\tiny{Proposed method, 10 DD-FB iterations with channel estimation} }
\psfrag{data4}{\tiny{Proposed method, 3 DD-FB iterations with channel estimation} }
\psfrag{data3}{\tiny{\cite{Munier}, 10 DD-FB iterations with channel estimation} }
\psfrag{data2}{\tiny{\cite{Munier}, 3 DD-FB iterations with channel estimation} }
\psfrag{data1}{\tiny{\cite{Corvaja}, full matrix } }
\includegraphics[scale=0.38]{figures/figure9}
\caption{BER performance of proposed algorithm compared to \cite{Munier} and \cite{Corvaja} for 64-QAM OFDM. Performance plot for the ideal case (i.e. perfect channel knowledge and no PN) is also given. ICI-cancellation technique in \cite{Petrovic} is also shown for perfectly known channel. $BT=0.005$}
\label{fig8}
\end{figure}
\begin{figure}
\centering
\psfrag{x}{\scriptsize{$E_b/N_0$(dB)} }
\psfrag{y}{\scriptsize{BER}}
\psfrag{data4}{\tiny{$K=27$, $Q=3$, $J=4$}}
\psfrag{data5}{\tiny{$K=625$, $Q=5$, $J=5$}}
\psfrag{data6}{\tiny{$K=2187$, $Q=3$, $J=8$}}
\psfrag{data1}{\tiny{$K=1$, $Q=1$, $J=1$}}
\psfrag{data2}{\tiny{$K=4$, $Q=5$, $J=2$}}
\psfrag{data3}{\tiny{$K=16$, $Q=2$, $J=5$}}
\psfrag{a}{\tiny{with channel estimation}}
\psfrag{b}{\tiny{known channel}}
\psfrag{d}{\tiny{$\beta T=0.01$ (red)}}
\psfrag{c}{\tiny{$\beta T =0.05$ (blue)}}
\includegraphics[scale=0.38]{figures/figure10}
\caption{BER performance of the proposed algorithm on 16-QAM coded OFDM for various number of codebook size.}
\label{fig_aa}
\end{figure}
\subsubsection{Performance with respect to size of $K$}
\indent In Figure \ref{fig_aa}, simulation results are presented showing the BER performance of the proposed algorithm for various codebook size $K$ and for $\beta T=0.01$ and $\beta T=0.05$. The presented result does not employ any DD-FB loop. A 16-QAM OFDM system with the same description as in Sec. \ref{sec:5a} is considered assuming known and unknown Rayleigh fading channel. It can be seen that the BER improves a $K$ grows. However, the relative improvement as $K$ grows beyond $K=27$ is very small. For $\beta T=0.01$, BER improves only slightly at high ${\mbox{SNR}}$ in both cases where the channel is assumed known and where the channel is estimated as $K$ increases from 27. The relative improvement in BER as $K$ increases is more noticeable for $\beta T=0.05$, even though it is still small.
\subsubsection{Sensitivity to input parameters}
\indent The plots in Figure \ref{fig_bb} show the sensitivity of the proposed algorithm and the algorithm in \citep{Munier} to the estimated channel and PHN parameters relative to the true parameters. The rate at which the PHN variance grows is set to $BT=0.03$ and the normalized maximal Doppler frequency is set to $f_{max}^D=0.03$. All other Rayleigh channel parameters are kept the same as in Table \ref{tab3}. Moreover, the size of the codebook was $K=27$ with $Q=3$ and $J=4$ and ${\mbox{SNR}}=20$dB. Both the proposed method and the method in \cite{Munier} use the past two symbols together with the current symbol $D=3$ for channel estimation.
It can be seen that the sensitivity of the proposed algorithm as well as the algorithm in \cite{Munier} to the input values of the normalized maximal Doppler frequency, $\widehat{f}_{max}^D$ is low except for $\widehat{f}_{max}^D=0$. However, for higher value of $D$, the BER performance is expected to be very sensitive to the input value of $\widehat{f}_{max}^D$. However, it can be seen that the performance of both the proposed algorithm and \cite{Munier} is noticeably sensitive to the input values of $\widehat{\beta T}$. On the other hand, the method in \cite{Corvaja} does not depend at all on the channel and PHN statistics and thus the performance remains unaffected by the estimated input parameters.
\begin{figure}
\centering
\psfrag{x}{\scriptsize{$\widehat{f}_{max}^D$} }
\psfrag{y}{\scriptsize{BER}}
\psfrag{data1}{\tiny{$\widehat{\beta T}=0$}}
\psfrag{data2}{\tiny{$\widehat{\beta T}=0.01$}}
\psfrag{data3}{\tiny{$\widehat{\beta T}=0.03$}}
\psfrag{data4}{\tiny{$\widehat{\beta T}=0.05$}}
\psfrag{data5}{\tiny{$\widehat{\beta T}=0.07$}}
\psfrag{a}{\tiny{Proposed method}}
\psfrag{b}{\tiny{Method in \citep{Munier}}}
\psfrag{c}{\tiny{Method in \citep{Corvaja}}}
\psfrag{cc}{\tiny{ $f_{max}^D$}}
\includegraphics[scale=0.38]{figures/figure11}
\caption{BER performance of the proposed algorithm (black line) and the method in \citep{Munier} (red line) as a function of the input doppler frequency $\widehat{f}_{max}^D$ for various input values of $\beta T$ in a 16-QAM OFDM on a Rayleigh channel where the true $f_{max}^D=0.03$, $\beta T=0.03$ and ${\mbox{SNR}}=20$dB.}
\label{fig_bb}
\end{figure}
\subsubsection{PLL Oscillator \label{pll}}
The above results have been provided assuming a free running oscillator in which the PHN is modelled by the Weiner process and based on which the codebook is derived. In order to demonstrate the applicability for a PHN model other than the Weiner process, we employ the proposed method using the Ornstein-Steinbeck process \cite{Gardiner} to model the PHN. The Ornstein-Uhlenbeck process is used to model the PHN at the output of a voltage controlled oscillator (VCO) in a phase-locked loop (PLL) \cite{Petrovic}.
In Figure \ref{fig_smpl}, a PHN realization, $\theta(n)$, for a free running oscillator and a PLL is shown. It can be seen that the PHN from PLL (Ornstein-Uhlenbeck process) is stable around the mean while the PHN from a free running oscillator grows over time.
A detailed analysis of PLL circuit parameters and the associated PHN process generated is given in \citep{Mehrotra}. In Figure \ref{fig_smpl}, the PHN variance of the Weiner process (free running VCO) grows at a rate of $(\beta T)_{vco}=0.05$ while more parameters are required to model the Ornstein-Uhlenbeck process. A Charge pump PLL is considered which is set to have a 3dB bandwidth of $f_{lp}=20$kHz for the low pass filter, $f_{pd}=20$kHz for the phase detector and $f_{pll}=100$kHz for the PLL itself. The PHN variance of the VCO and the reference oscillator grow at a rate of $\beta T_{vco}=0.01$ and $(\beta T)_{ref}=2^{-9}$ respectively. Moreover, the center frequency of the VCO is $f_c=5$GHz while the center frequency of the reference oscillator is $f_{ref}=100$MHz and the sampling frequency is $f_s=25$MHz. A 16-QAM coded OFDM with the same description as in Sec. \ref{sec:5a} is considered assuming unknown Rayleigh fading channel so that channel estimation is employed along PHN compensation.
In Figure \ref{fig_pll}, the performances of the proposed method and the method in \citep{Munier} is compared for various input of the PHN variance of the VCO, which is denoted by $\widehat{\beta T}_{vco}$, for a free running VCO and for PLL at 20dB ${\mbox{SNR}}$. Both methods are derived based on the Weiner PHN assumptions. The proposed method is run in 2 DD-FB loops while the method in \cite{Munier} is run in 5 DD-FB loops. The improved in performance for PLL (dashed line) compared to free running VCO (solid line) can be seen in Figure \ref{fig_pll} for both the proposed as well as the method in \citep{Munier}. It can be seen that the lowest BER for the proposed method in a PLL is achieved around the input $\widehat{\beta T}_{vco}=0.02$ which then starts to increase as $\widehat{\beta T}_{vco}>0.02$. The increasing trend in BER can also be seen in free running VCO when $\widehat{\beta T}_{vco}>0.04$ for the proposed method. The BER performance of the method in \citep{Munier}, on the other hand, seems to be stable around $5\times 10^{-3}$ in a PLL and around $9\times 10^{-3}$ in a free running VCO for an input $\widehat{\beta T}_{vco}$ equal to 0.01 and above. For the range of input $\widehat{\beta T}_{vco}$ values displayed, the proposed method outperforms \cite{Munier} in both cases when a PLL and a free running VCO is used.
\begin{figure}
\centering
\psfrag{x}{\scriptsize{Sample} }
\psfrag{y}{\scriptsize{Phase Error (radians)}}
\psfrag{data1}{\tiny{Free running Oscillator }}
\psfrag{data2}{\tiny{Charge Pump PLL} }
\includegraphics[scale=0.38]{figures/figure12}
\caption{PHN samples, $\theta(n)$, simulated as an out from a free running oscillator (Weiner process) and a charge Pump PLL (Ornstein-Uhlenbeck process).}
\label{fig_smpl}
\end{figure}
\begin{figure}
\centering
\psfrag{x}{\scriptsize{$\widehat{\beta T}_{vco}$} }
\psfrag{y}{\scriptsize{BER}}
\psfrag{data1}{\tiny{Method in \citep{Munier}, 5 DD-FB iteration, Free running VCO,}}
\psfrag{data2}{\tiny{Method in \citep{Munier}, 5 DD-FB iteration, PLL}}
\psfrag{data3}{\tiny{Proposed method, 2 DD-FB iteration, Free running VCO}}
\psfrag{data4}{\tiny{Proposed method, 2 DD-FB iterations, PLL}}
\psfrag{data5}{\tiny{Method in \citep{Corvaja}, Free running VCO}}
\psfrag{data6}{\tiny{Method in \citep{Corvaja}, PLL}}
\includegraphics[scale=0.38]{figures/figure13}
\caption{Comparison of BER performance of the proposed algorithm and the method in \citep{Munier} in a free running VCO and PLL as a function of the input $\widehat{\beta T}_{vco}$ in which the true $(\beta T_{vco})=0.05$. A 16-QAM coded OFDM and a Rayleigh channel where ${\mbox{SNR}}=20$dB used.}
\label{fig_pll}
\end{figure}
\section{Computational analysis \label{sec:6}}
The proposed scheme does incur some computations which are not part of the reference schemes \cite{Munier, Corvaja}. These include the multiplication with the matrix $\mathbf{\widehat{A}}_k^{-1}$ in \eqref{eq19}, and all the other calculations inside the
loop over $k$ in Algorithm \ref{alg2}. It should be noted that the channel estimation in \eqref{eq25}, \eqref{eq27} and \eqref{eq28} only requires the multiplication of the LS channel estimate with a pre-computed matrix. The multiplication with $\mathbf{\widehat{A}}_k^{-1}$ is an operation in the frequency domain. More efficient would be to do the codebook pre-compensation in the time domain and do a separate FFT for each of the $K$ possible received signals. In this case, the number of complex multiplications to obtain $\tilde{\bf Y}^k_m$ for $k=1,...,K$ is on the order of $N(1+\frac{K}{2}\log_2 N)$, rather than $N(KN +\frac{1}{2}\log _2N)$. A summary of the number of complex additions and multiplications required by the most important equations of our algorithm is given in Table \ref{tab3a}.
\begin{table}
\centering
\caption{Summary of total Number of complex operations required per OFDM symbol and DD-FB iteration excluding the Viterbi and a few other computations. \label{tab3a}}
\input{tables/table3}
\end{table}
\begin{table*}
\centering
\caption{Summary of total Number of complex operations required per OFDM symbol excluding the Viterbi and a few other computations. \label{tab5a}}
\input{tables/table4}
\end{table*}
The method in \cite{Munier} involves some computations which are not needed in our algorithm for instance the successive least squares estimation of the coefficients of the power series which approximates the PHN in addition to an $N \times N$ Toeplitz matrix inversion. The overall number of complex additions and multiplications for channel estimation and PHN compensation required for one OFDM symbol by the proposed method as well as the method in \cite{Munier} is given in Table \ref{tab5a}, excluding the Viterbi algorithm and without taking into account the $N \times N$ Toeplitz matrix inversion for \cite{Munier}. The parameter $\Lambda$ in Table \ref{tab5a} refers to the order of the polynomial approximating the ICI coefficients as presented in \cite{Munier}. It can be seen that the complexity of both the proposed algorithm and the method in \cite{Munier} are linearly dependent on $K$ and $\Lambda$ respectively. For $K=27$, $\Lambda=6$
and $N=64$ the proposed algorithm has a computational advantage over \cite{Munier}. This is illustrated in Figure \ref{fig9aa} which compares the measured execution time for various number of DD-FB iterations between the proposed algorithm (Algorithm \ref{alg2}) and the algorithms in \cite{Munier} for 64-QAM OFDM at 24dB SNR. No DD-FB loop or 0 iteration for the method in \cite{Munier} implies channel estimation with CPE compensation which reduces the number of computations significantly. The $N \times N$ Toeplitz matrix inversion in \cite{Munier} has been factored out of the measured execution time (since we did not take into account the Toeplitz structure). Our simulator is a Linux system (Ubuntu 11.04) running on a CPU with Intel-Core i7-2600 and uses the IT++ library in C++ for executing the algorithms. The results in Figure \ref{fig9aa} show that, for the chosen parameter $K=27$, the proposed method provides a significantly better performance-complexity trade-off than the method in \cite{Munier}.
\begin{figure}
\centering
\psfrag{x}{\scriptsize{Averaged Execution Time per Frame (sec)} }
\psfrag{y}{\scriptsize{BER}}
\psfrag{a}{\tiny{$0^*$}}
\psfrag{b}{\tiny{$1^*$}}
\psfrag{c}{\tiny{$2^*$} }
\psfrag{d}{\tiny{$3^*$}}
\psfrag{e}{\tiny{$4^*$}}
\psfrag{f}{\tiny{$5^*$} }
\psfrag{g}{\tiny{$6^*$} }
\psfrag{h}{\tiny{$7^*$} }
\psfrag{i}{\tiny{$8^*$} }
\psfrag{j}{\tiny{$9^*$} }
\psfrag{k}{\tiny{$10^*$} }
\psfrag{data1}{\tiny{Method in \cite{Munier}}}
\psfrag{data2}{\tiny{Proposed method (Algorithm \ref{alg2}))}}
\psfrag{Data 3}{\tiny{No. of iterations}}
\includegraphics[scale=0.38]{figures/figure14}
\caption{BER performance of Algorithm \ref{alg2} and the algorithms in \cite{Munier} on 64-QAM OFDM as a function of averaged execution time per frame (in sec) for $K=27$ and $\Lambda=6$ at $24$dB SNR using various number of iterations (denoted by $i^*$) in the DD-FB loop.}
\label{fig9aa}
\end{figure}
The DD-FB stage in \citep{Petrovic} and \citep{Munier} are not easily parallelized, while the most computationally demanding operations in the proposed algorithm can obviously be split into up to $K$ parallel processors or dedicated hardware. The algorithms in \cite{Petrovic} and \cite{Corvaja} also involve an inverstion of an $N \times N$ (Toeplitz) matrix. However, the method in \cite{Corvaja} also presents a solution in which only the ICI coefficients close to the carrier need to be estimated. This requires solving a less complex linear system, e.g. based on tridiagonal matrix algorithm which only requires $\mathcal{O}(N)$ complex operations. This makes \cite{Corvaja} less computationally complex to implement than the proposed method and \cite{Munier}. In addition [14] has the advantage provided by avoiding the requirement of having to know the channel and PHN statistics. These advantages come at the cost of performance in BER as shown in Figure \ref{fig7} and Figure \ref{fig8}.
\section{Conclusion\label{sec:7}}
Based on a simple codebook table which approximates the phase-noise statistics by a finite number of realizations, a novel PHN compensation approach is introduced. The general idea can be applied to a wide range of PHN scenarios. Herein, we have concentrated on ICI suppression in a convolutional encoded OFDM system with scattered pilots. The complexity of the algorithm is determined by the size of the codebook.
Using a codebook of moderate size, $K=27$, for a 16-QAM uncoded OFDM system in which an AWGN channel and a PHN with $BT=0.01$, which is was considered, the proposed algorithm is 2dB from the ideal PHN free case at a BER of $10^{-3}$. Considering an identical scenario in a 64-QAM uncoded OFDM, the proposed algorithm is 2.5dB from the ideal PHN free case at a BER of $10^{-2}$. By comparison, the method in \citep{Munier} is $9$dB from the ideal case.
For a coded system, the codebook technique is used along with DD-FB loop for combined channel estimation and PHN compensation. For a 16-QAM OFDM system over a Rayleigh fading channel with $BT=0.01$, the proposed method is 1.5dB from the ideal case (PHN free and known channel) at a BER of $10^{-4}$ employing 2 iterations on the DD-FB loop; while the method in \cite{Munier} is $2.5$dB from the ideal case using 8 iterations. For a 64-QAM coded system and PHN with $BT=0.005$, the proposed method provides a $1$dB gain at a BER $10^{-3}$ employing 3 iterations in the DD-FB, compared to the method in \citep{Munier} which uses 10 iterations. The gain obtained by the proposed method is significant compared to the method in \cite{Corvaja}, which uses a pilot interpolation technique for channel estimation. Additional results are also presented for both 16 and 64 QAM coded systems assuming known Rayleigh channel frequency response at the receiver. The proposed method is shown to outperform to the reference schemes without having to employ the DD-FB loop.
The performance of the proposed method with respect to the codebook size is also evaluated showing the relative improvement in BER as $K$ increases. However, for the given example, the relative improvement is very small as $K$ increases above 27. Simulation results are also shown employing PHN from a charge pump PLL, modelled as Ornstein-Uhlenbeck process, is also presented with the proposed method showing improved performance compared to the methods in \citep{Munier} and also demonstrating the applicability of the proposed algorithm for a PHN process other than Weiner process.
Therefore, with a moderately small codebook size and limited number of iterations on DD-FB, an improved performance as well as faster execution time is achieved by the proposed algorithm.
\appendices
\section{Proof of Equation \eqref{eq17} \label{sec:app}}
Let us then first consider the case when the number of quantization regions, $Q$, is infinite. In such a case, each segment of the selected codebook entry will follow the mean of samples of the corresponding segment of the PHN realization. Thus the MSE for infinite $Q$ is given by
\begin{equation}\label{eq16app}
\begin{split}
\mbox{MSE}_{(Q=\infty)}&=\sum_{j=0}^{J-1}\sum_{n=0}^{L-1} E_{\theta(n)}[(\theta(n+jL) - \hat{\theta}(n+jL))^2]\\
& =\sum_{j=0}^{J-1}\sum_{n=0}^{L-1} E[(\theta(n+jL)^2 \\
& -2\theta(n+jL)\hat{\theta}(n+jL) +\hat{\theta}(n+jL)^2]\\
& =\sum_{j=0}^{J-1}\sum_{n=0}^{L-1} E[(\theta(n+jL)^2\\
& - \frac{1}{L}\sum_{j=0}^{J-1}\sum_{n=0}^{L-1}\sum_{l=0}^{L-1}E[\theta(n+jL)\theta(l+jL)]\\
&=\frac{(N-J)(N+J)}{6J}\sigma_\epsilon^2
\end{split}
\end{equation}
in which the last equality follows from the covariance of a random walk process which is $E[\theta(n+\ell)\theta(n)]=n\sigma_\epsilon^2$ for $l\geq0$.
When $Q$ is finite, an additional error is introduced due to the quantization of segment averages. As in the design of the codebook, we assume that $\psi$ is set to the average of the first segment and thus no quantization error occurs in the first segment and the PHN realization crosses the first sample of each segment of the trajectory. Additionally, by assuming that this error in each segment is identically distributed and independent of the difference between the PHN and the codebook trajectory, the MSE due to accumulated quantization error in each segment is $L\sigma_q^2$, where $L=N/J$ and $\sigma_q$ is the standard deviation of quantization error. The MSE due to the quantization error over the entire trajectory is therefore, $\mbox{MSE}_{(Q<\infty)}=(J-1)L\sigma_q^2$, where $(J-1)$ is due to the assumption that the first segment has no quantization error. The approximate MSE of the selected trajectory as an estimator of the PHN realization then becomes
\begin{equation}\label{eq17app}
\mbox{MSE} =\frac{(N+J)(N-J)}{6J}\sigma_\epsilon^2 + L(J-1)\sigma_q^2.
\end{equation}
In order to determine $\sigma_q^2$, we assume that the PHN realization is exactly on one of the codebook entries in the last sample of the previous segment. In this case, the quantization error of the $j$-th segment average is equal to the quantization error of the $j$-th increment, i.e., no influence of previous increments. The variance, $\sigma_q^2$, of the quantization error is then given by
\begin{equation}\label{eq17n}
\begin{split}
\sigma_q^2 &= \sum_{i=1}^Q E[(X-\hat{x})^2|X\in \mathcal{R}_i] \mathcal{P}(X\in \mathcal{R}_i))\\
&= \sum_{i=1}^Q \mathcal{P}(X\in \mathcal{R}_i)(E[X^2|X \in \mathcal{R}_i] -2\hat{x}_iE[X|X \in \mathcal{R}_i] + \hat{x}_i^2)\\
&= \sum_{i=1}^Q \mathcal{P}(X\in \mathcal{R}_i)(E[X^2|X \in \mathcal{R}_i] - \hat{x}_i^2)\\
\end{split}
\end{equation}
where the last equality follows from $E[X|X\in\mathcal{R}_i]=\hat{x}_i$ which is given by \eqref{eq15}. The conditional Expectation $E[X^2|X \in \mathcal{R}_i]$ is given by
\begin{equation}\label{eq14app}
\begin{split}
E\bigg[X^2|X\in R_i\bigg]&=\frac{\int_{R_i}x^2f_X(x)dx}{\mathcal{P}(X\in \mathcal{R}_i)}\\
\end{split}
\end{equation}
where $\mathcal{P}(X\in \mathcal{R}_i)=\int_{R_i}f_X(x)dx$. When $f_X(x)$ a Gaussian function
\begin{equation}\label{eq15aa}
\begin{split}
\int_{R_i}&x^2f_X(x)dx =\sigma_x^2/2\bigg(\mbox{erf}(x_{i+1}/\sqrt{2}\sigma_x)-\mbox{erf}(x_{i}/\sqrt{2}\sigma_x)\bigg)\\
& - \sigma_x\sqrt{2\pi}\bigg(x_{i+1}\exp(-0.5x_{i+1}^2/\sigma_x^2)-x_i\exp(-0.5x_{i}^2/\sigma_x^2)\bigg)
\end{split}
\end{equation}
and
\begin{equation}\label{eq15bb}
\begin{split}
\mathcal{P}(X\in \mathcal{R}_i)&=\int_{R_i}f_X(x)dx\\
&=0.5 \bigg( \mbox{erf}(x_{i+1}/\sqrt{2}\sigma_x)-\mbox{erf}(x_{i}/\sqrt{2}\sigma_x) \bigg).
\end{split}
\end{equation}
Equations \eqref{eq15aa}, \ref{eq15bb} and \eqref{eq15} are then used on \eqref{eq17n} to obtain the variance, $\sigma_q^2$, of the quantization error.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
{\footnotesize
|
{
"timestamp": "2015-04-14T02:16:15",
"yymm": "1504",
"arxiv_id": "1504.03240",
"language": "en",
"url": "https://arxiv.org/abs/1504.03240"
}
|
\section{Introduction}
\label{sec-intro}
\cutsectiondown
Object detection is one of the long-standing and important problems in computer vision.
Motivated by the recent success of deep learning~\cite{lecun1989backpropagation,hinton:science,bengio2007greedy,boureau2008sparse,lee2011unsupervised,bengio:replearn,schmidhuber} on visual object recognition tasks~\cite{alex-imagenet,overfeat,zeiler2014visualizing,vggnet,szegedy2014going}, significant improvements have been made in the object detection problem ~\cite{dnn-det-mask,dnn-scalable-cvpr2014,he2014spatial}.
Most notably, \citet{rcnn-cvpr} proposed the ``regions with convolutional neural network'' (R-CNN) framework for object detection and demonstrated state-of-the-art performance on standard detection benchmarks (e.g., PASCAL VOC~\cite{pascal-voc-2007,pascal-voc-2010}, ILSVRC~\cite{ILSVRCarxiv14}) with a large margin over the previous arts, which are mostly based on deformable part model (DPM)~\cite{dpm-pami}.
There are two major keys to the success of the R-CNN.
First, features matter~\citep{rcnn-cvpr}.
In the R-CNN, the low-level image features (e.g., HOG~\cite{hog}) are replaced with the CNN features, which are arguably more discriminative representations. One drawback of CNN features, however, is that they are expensive to compute.
The R-CNN overcomes this issue by proposing a few hundreds or thousands candidate bounding boxes via the selective search algorithm~\citep{selective-search} to effectively reduce the computational cost required to evaluate the detection scores at all regions of an image.
\commenttext{RUBEN: removed "an" from "caused by an inaccurate localization", "caused by inaccurate localization" is enough}
Despite the success of R-CNN, it has been pointed out through an error analysis~\citep{voc-error} that inaccurate localization causes the most egregious errors in the R-CNN framework~\cite{rcnn-cvpr}.
For example, if there is no bounding box in the close proximity of ground truth among those proposed by selective search, no matter what we have for the features or classifiers, there is no way to detect the correct bounding box of the object.
Indeed, there are many applications that require accurate localization of an object bounding box, such as detecting moving objects (e.g., car, pedestrian, bicycles) for autonomous driving~\cite{KITTI}, detecting objects for robotic grasping or manipulation in robotic surgery or manufacturing~\cite{rss2013deepgrasp}, and many others.
In this work, we address the localization difficulty of the R-CNN detection framework with two ideas.
First, we develop a fine-grained search algorithm to expand an initial set of bounding boxes by proposing new bounding boxes with scores that are likely to be higher than the initial ones.
By doing so, even if the initial region proposals were poor, the algorithm can find a region that is getting closer to the ground truth after a few iterations. We build our algorithm in the Bayesian optimization framework~\cite{mockus1978application,snoek2012practical}, where evaluation of the complex detection function is replaced with queries from a probabilistic distribution of the function values defined with a computationally efficient surrogate model.
Second, we train a CNN classifier with a structured SVM objective that aims at classification and localization simultaneously. \commenttext{RUBEN: this entire sentence does not sound right so I changed it. Here is the original one if you guys want it back:
We define the structured SVM layer of CNN whose objective function is defined with a hinge loss that balances between classification (i.e., determines whether an object exists) and localization (i.e., determines how much it overlaps with the ground truth), and replace the last layer of the CNN, which is usually a softmax or linear SVM layer, with structured SVM layer.
}
We define the structured SVM objective function with a hinge loss that balances between classification (i.e., determines whether an object exists) and localization (i.e., determines how much it overlaps with the ground truth) to be used as the last layer of the CNN.
In experiments, we evaluated our methods on PASCAL VOC 2007 and 2012 detection tasks and compared to other competing methods.
We demonstrated significantly improved performance over the state-of-the-art at different levels of intersection over union (IoU) criteria.
In particular, our proposed method outperforms the previous arts with a large margin at higher IoU criteria (e.g., IoU = $0.7$), which highlights the good localization ability of our method.
\commenttext{Ruben: changed "In sum" to "overall"}
Overall, the contributions of this paper are as follows: 1)~we develop a Bayesian optimization framework that can find more accurate object bounding boxes without significantly increasing the number of bounding box proposals, 2)~we develop a structured SVM framework to train a CNN classifier for accurate localization, 3)~the aforementioned methods are complementary and can be easily adopted to various CNN models, and finally, 4)~we demonstrate significant improvement in detection performance over the R-CNN on both PASCAL VOC 2007 and 2012 benchmarks.
\vspace*{-0.1in}
\vspace*{-0.1in}}\newcommand{\cutsectiondown}{\vspace*{-0.07in}
\section{Related work}
\label{sec-related}
\cutsectiondown
The DPM~\cite{dpm-pami} and its variants~\cite{oc-dpm,seg-dpm} have been the dominating methods for object detection tasks for years.
These methods use image descriptors such as HOG~\cite{hog}, SIFT~\cite{lowe2004distinctive}, and LBP~\cite{lbp} as features and densely sweep through the entire image to find a maximum response region.
With the notable success of CNN on large scale object recognition~\cite{alex-imagenet}, several detection methods based on CNNs have been proposed~\citep{overfeat,sermanet2013pedestrian,dnn-det-mask,dnn-scalable-cvpr2014,rcnn-cvpr}. Following the traditional sliding window method for region proposal, \citet{overfeat} proposed to search exhaustively over an entire image using CNNs, but made it efficient by conducting a convolution on the entire image at once at multiple scales. Apart from the sliding window method, \citet{dnn-det-mask} used CNNs to regress the bounding boxes of objects in the image and used another CNN classifier to verify whether the predicted boxes contain objects.
\citet{rcnn-cvpr} proposed the R-CNN following the ``recognition using regions'' paradigm~\cite{rec-with-region}, which also inspired several previous state-of-the-art methods~\cite{selective-search,regionlets}.
In this framework, a few hundreds or thousands of regions are proposed for an image via the selective search algorithm~\citep{selective-search} and the CNN is finetuned with these region proposals.
Our method is built upon the R-CNN framework using the CNN proposed in~\citep{vggnet}, but with 1) a novel method to propose extra bounding boxes in the case of poor localization, and 2) a classifier with improved localization sensitivity.
The structured SVM objective function in our work is inspired by \citet{struct-localization}, where they trained a kernelized structured SVM on low-level visual features (i.e., HoG~\citep{hog}) to predict the object location.
\citet{struct-cnn} integrated a structured objective with the deep neural network for object detection, but they adopted the branch-and-bound strategy for training as in \citep{struct-localization}.
In our work, we formulate the linear structured objective upon high-level features learned by deep CNN architectures, but our negative mining step is very efficient thanks to the region-based detection framework.
We also present a gradient-based optimization method for training our architecture.
\commenttext{Ruben: Changed "works" for "work"}
There have been several other related work for accurate object localization.
\citet{seg-dpm} incorporated the geometric consistency of bounding boxes with bottom-up segmentation as auxiliary features into the DPM.
\citet{localize-detected} used the structured SVM with color and edge features to refine the bounding box coordinates in DPM framework.
\citet{accurate-cvpr2014} used the height prior of an object.
These auxiliary features to aid object localization can be injected into our framework without modifications.
Localization refinement can be also taken as a CNN regression problem.
\citet{rcnn-cvpr} extracted the middle layer features and linearly regressed the initially proposed regions to better locations.
\citet{overfeat} refined bounding boxes from a grid layout to flexible locations and sizes using the higher layers of the deep CNN architecture.
\citet{dnn-scalable-cvpr2014} jointly conducted classification and regression in a single architecture.
Our method is different in that 1) it uses the information from multiple existing regions instead of a single bounding box for predicting a new candidate region, and 2) it focuses only on maximizing the localization ability of the CNN classifier instead of doing any regression from one bounding box to another.
\begin{figure*}[t]
\centering
\includegraphics[width=1\textwidth]{figures/o6_ver2.pdf}
\vspace*{-0.1in}}\newcommand{\cutcaptiondown}{\vspace*{-0.1in}
\vspace{-0.1in}
\caption{Pipeline of our method.
1) Initial bounding boxes are given by methods such as the selective search~\citep{selective-search} and their detection scores are obtained from the CNN-based classifier trained with structured SVM objective. 2) The box(es) with optimal score(s) in the local regions are found by greedy NMS~\cite{rcnn-cvpr} (shown as ``local optimum'' boxes), and Bayesian optimization takes the neighborhood of each local optimum to propose a new box with a high chance of getting a better
detection score. 3) We evaluate the detection score of the new box, take it as an observation for the next iteration of the Bayesian optimization until convergence. Note that the local optimum and the search region may change in each iteration. 4) All the bounding boxes are fed into the standard post-processing stage (e.g., thresholding and NMS, etc.).}
\label{fig:structured-rcnn}
\cutcaptiondown
\vspace{-0.1in}
\end{figure*}
\vspace*{-0.1in}}\newcommand{\cutsectiondown}{\vspace*{-0.07in}
\section{Fine-grained search for bounding box via Bayesian optimization}
\label{sec-gp}
\cutsectiondown
Let $f(x,y)$ denote a detection score of an image $x$ at the region with the box coordinates $y=(u_{1}, v_{1}, u_{2}, v_{2})\in{\mathcal Y}$.
The object detection problem deals with finding the local maximum of $f(x,y)$ with respect to $y$ of an unseen image $x$.\footnote{When multiple (including zero) objects exist, it involves finding the local maxima that exceed a certain threshold.}
As it requires an evaluation of the score function at many possible regions, it is crucial to have an efficient algorithm to search for the candidate bounding boxes.
A sliding window method has been used as a dominant search algorithm~\cite{hog,dpm-pami}, which exhaustively searches over an entire image with fixed-sized windows at different scales to find a bounding box with a maximum score.
However, evaluating the score function at all regions determined by the sliding window approach is prohibitively expensive when the CNN features are used as the image region descriptor. The problem becomes more severe when flexible aspect ratios are needed for handling object shape variations.
Alternatively, the ``recognition using regions''~\cite{rec-with-region,rcnn-cvpr} method has been proposed, which requires to evaluate significantly fewer number of regions (e.g., few hundreds or thousands) with different scales and aspect ratios, and it can use the state-of-the-art image features with high computational complexity, such as the CNN features~\cite{decaf}.
One potential issue of object detection pipelines based on region proposal is that the correct detection will not happen when there is no region proposed in the proximity of the ground truth bounding box.\footnote{We refer to selective search as a representative method for region proposal.}
To resolve this issue, one can propose more bounding boxes to cover the entire image more densely, but this would significantly increase the computational cost.
In this section, we develop a fine-grained search (FGS) algorithm based on Bayesian optimization that sequentially proposes a new bounding box with a higher \emph{expected} detection score than previously proposed bounding boxes without significantly increasing the number of region proposals.
We first present the general Bayesian optimization framework (Section~\ref{sec-general-bayesian-framework}) and describe the FGS algorithm using Gaussian process as the prior for the score function (Section~\ref{sec-gp-regression}).
We then present the local FGS algorithm that searches over multiple local regions instead of a single global region (Section~\ref{sec-l-FGS}), and discuss the hyperparameter learning of our FGS algorithm (Section~\ref{sec-gp-param}).
\cutsubsectionup
\subsection{General Bayesian optimization framework}
\label{sec-general-bayesian-framework}
\cutsubsectiondown
Let $\{y_{1},\cdots,y_{N}\}$ be the set of solutions (e.g., bounding boxes).
In the Bayesian optimization framework, $f = f(x,y)$ is assumed to be drawn from a probabilistic model: \begin{equation}
p\left(f|\mathcal{D}_{N}\right)\propto p\left(\mathcal{D}_{N}|f\right) p(f),
\end{equation}
where $\mathcal{D}_{N}$ $=$ $\{(y_{j},f_{j})\}_{j=1}^{N}$ and $f_{j}$ $=$ $f(x, y_{j})$.
Here, the goal is to find a new solution $y_{N+1}$ that maximizes the chance of improving the detection score $f_{N+1}$, where the chance is often defined as an acquisition function $a(y_{N+1}|\mathcal{D}_{N})$. Then, the algorithm proceeds by recursively sampling a new solution $y_{N+t}$ from $\mathcal{D}_{N+(t-1)}$, and update the set $\mathcal{D}_{N+t} = \mathcal{D}_{N+(t-1)}\cup\{y_{N+t}\}$ to draw a new sample solution $y_{N+(t+1)}$ with an updated observation.
Bayesian optimization is efficient in terms of the number of function evaluation~\cite{jones2001taxonomy}, and is particularly effective when $f$ is computationally expensive.
When $a(y_{N+1}|\mathcal{D}_{N})$ is much less expensive than $f$ to evaluate, and the computation for $\argmax_{y_{N+1}} a(y_{N+1}|\mathcal{D}_{N})$ requires only a few function evaluations, we can efficiently find a solution that is getting closer to the ground truth.
\commenttext{What do you mean by manageable? Can efficiently be replaced with within few iterations?}
\cutsubsectionup
\subsection{Efficient region proposal via GP regression}
\label{sec-gp-regression}
\cutsubsectiondown
A Gaussian process (GP) defines a prior distribution $p(f)$ over the function $f:\mathcal{Y}\rightarrow\mathbb{R}$. Due to this property, a distribution over $f$ is fully specified by a mean function $m:\mathcal{Y}\rightarrow\mathbb{R}$ and a positive definite covariance kernel $k:\mathcal{Y}\times\mathcal{Y}\rightarrow\mathbb{R}$, i.e., $f$ $\sim$ $\mathcal{GP}(m(\cdot),k(\cdot,\cdot))$.
Specifically, for a finite set $\{y_{j}\}_{j=1}^{N}\subset\mathcal{Y}$, the random vector $[f_{j}]_{1\leq j \leq N}$ follows a multivariate Gaussian distribution $\mathcal{N}\left([m(y_{j})]_{1\leq j \leq N},[k(y_{i},y_{j})]_{1\leq i,j\leq N}\right)$.
A random Gaussian noise with precision $\beta$ is usually added to each $f_j$ independently in practice.
Here, we used the constant mean function $m(y)$ $=$ $m_{0}$ and the squared exponential covariance kernel with automatic relevance determination (SEard) as follows: $k_{\text{SEard}}(y_{i},y_{j};z)=$
\begin{equation}
\eta\exp\left(-\frac{1}{2}\left(\Psi_{z}(y_{i})-\Psi_{z}(y_{j})\right)^{\top}\Lambda\left(\Psi_{z}(y_{i})-\Psi_{z}(y_{j})\right)\right),\nonumber\end{equation}
where $\Lambda$ is a $4\times 4$ diagonal matrix whose diagonal entries are $\lambda_{i}^2, i= 1,\cdots,4$.
These form a $7$-dimensional GP hyperparameter $\theta = (\beta,m_{0},\eta,\lambda_{1}^2,\lambda_{2}^2,\lambda_{3}^2,\lambda_{4}^2)$ to be learned from the training data.
$\Psi_{z}:\mathcal{Y}\rightarrow\mathbb{R}^{4}$ transforms the bounding box coordinates $y$ into a new form:
\begin{equation}
\Psi_{z}(y) = \left[\frac{\bar{u}}{\exp(z)}\; ;\;\frac{\bar{v}}{\exp(z)}\; ;\;\log{w}\; ;\; \log{h} \right],
\end{equation}
where $\bar{u}=\frac{u_{1}+u_{2}}{2}$ and $\bar{v}=\frac{v_{1}+v_{2}}{2}$ denote the center coordinates, $w=u_{2}-u_{1}$ denotes the width, and $h=v_{2}-v_{1}$ denotes the height of a bounding box.
We introduce a latent variable $z$ to make the covariance kernel scale-invariant.\footnote{If the image and the bounding boxes $y_{i}, y_{j}$ are scaled down by a certain factor, we can keep $k_{\text{SEard}}(y_{i},y_{j};z)$ invariant by properly setting $z$.}
We determine $z$ in a data-driven manner by maximizing the marginal likelihood of $\mathcal{D}_{N}$, or
\begin{equation}
\hat{z}=\argmax_{z}p(\{f_j\}_{j=1}^{N}|\{y_j\}_{j=1}^{N};\theta). \label{eq:gp-latent-max}
\end{equation}
The GP regression (GPR) problem tries to find a new argument $y_{N+1}$ given $N$ observations $\mathcal{D}_{N}$ that maximizes the value of acquisition function, which, in our case, is defined with the expected improvement (EI) as: $a_{EI}(y_{N+1}|\mathcal{D}_{N}) =$
\begin{equation}
\int_{\hat{f}_{N}}^{\infty}(f-\hat{f}_{N})\cdot p(f|y_{N+1},\mathcal{D}_{N};\theta)df\label{eq:gp-acq}
\end{equation}
where $\hat{f}_{N} = \max_{1\leq j \leq N} f_{j}$.
The posterior of $f_{N + 1}$ given $(y_{N+1},\mathcal{D}_{N})$ follows Gaussian distribution: $p(f_{N+1}|y_{N+1},\mathcal{D}_{N};\theta)=$
\begin{align}
\mathcal{N}\big(f_{N+1};\mu(y_{N+1}|\mathcal{D}_{N}),\sigma^{2}(y_{N+1}|\mathcal{D}_{N})\big),\label{eq:gp-regression}
\end{align}
with the following mean function and covariance kernels:
\begin{align*}
\mu(y_{N+1}|\mathcal{D}_{N}) & = m_{0}+\mathbf{k}_{N+1}^{\top}\mathbf{K}_{N}^{-1}\left(\big[f_{j} - m_{0}\big]_{1\leq j\leq N}\right), \\
\sigma^{2}(y_{N+1}|\mathcal{D}_{N}) & =k_{N+1}-\mathbf{k}_{N+1}^{\top}\mathbf{K}_{N}^{-1}\mathbf{k}_{N+1}, \\
k_{N+1} & =\beta^{-1}+ k(y_{N+1},y_{N+1}),\\
\mathbf{k}_{N+1} & =\big[k(y_{N+1},y_{j})\big]_{1\leq j\leq N},\\
\mathbf{K}_{N} & =\big[k(y_{i},y_{j})\big]_{1\leq i,j\leq N}+\beta^{-1}\mathbf{I}\;.
\end{align*}
We refer~\cite{GPML} for detailed derivation.
By plugging~\eqref{eq:gp-regression} in~\eqref{eq:gp-acq}, \begin{align}
&a_{EI}(y_{N+1}|\mathcal{D}_{N})=\sigma\left(y_{N+1}|\mathcal{D}_{N}\right)\times\nonumber\\
&\big(\gamma(y_{N+1})F(\gamma(y_{N+1}))+\mathcal{N}(\gamma(y_{N+1});0,1)\big)\label{eq:gp-acq-closed}
\end{align}
where $\gamma(y_{N+1})=\frac{\mu(y_{N+1}|\mathcal{D}_{N})-\hat{f}_{N}}{\sigma\left(y_{N+1}|\mathcal{D}_{N}\right)}$.
$F(\cdot)$ is the cumulative distribution function of standard normal distribution $\mathcal{N}(\cdot)$.
\cutsubsectionup
\subsection{Local fine-grained search}
\label{sec-l-FGS}
\cutsubsectiondown
\begin{algorithm}[t]
\caption{Local fine-grained search (FGS) \label{alg:FGS}}
\begin{algorithmic}[1]
\small {
\Require Image $x$, classifier $f_{\text{CNN}}$, a set of structured labels and classification scores $\mathcal{D}_{N}$ $=$ $\{(y_{j}, f_{j})_{j=1}^{N}\}$, GP hyperparameter $\theta$, maximum number of GP iterations $t_{\text{max}}$, a threshold $f_{\text{prune}}$ to prune out the bounding boxes, different levels of IoU $\rho_{r}, r=1,\ldots,R$ determining the size of local regions.
\Ensure A set of structured labels and classification scores $\mathcal{D}$.
\State $\mathcal{D}\leftarrow\mathcal{D}_{N}$
\For{$t=1,\cdots,t_{\text{max}}$}
\State $\mathcal{D}_{\text{proposal}} = \varnothing$
\State $\mathcal{D}_{\text{prune}} = \{(y,f)\in\mathcal{D}:f>f_{\text{prune}}\}$
\State $\mathcal{D}_{\text{NMS}} = \text{NMS}(\mathcal{D}_{\text{prune}})$
\For{ \textbf{each} $(y_{\text{best}},f_{\text{best}})\in\mathcal{D}_{\text{NMS}}$}
\For{ $r=1,\cdots,R$ }
\State $\mathcal{D}_{\text{local}} = \{(y,f)\in\mathcal{D}:\operatorname{IoU}(y,y_{\text{best}})>\rho_{r}\}$
\State $\hat{z} = {\arg\max}_{z}\, p(\mathcal{D}_{\text{local}};\theta)$ \label{alg-ln:latent}
\hfill (Equation~\eqref{eq:gp-latent-max})
\State $\hat{y} = {\arg\max}_{y}\, a_{EI}(y|\mathcal{D}_{\text{local}};\theta,\hat{z})$ \label{alg-ln:best}
\hfill (Equation~\eqref{eq:gp-acq-closed})
\State $\hat{f} = f_{\text{CNN}}(x,\hat{y})$
\State $\mathcal{D}_{\text{proposal}}\leftarrow\mathcal{D}_{\text{proposal}}\cup\{(\hat{y},\hat{f})\}$
\EndFor
\EndFor
\State $\mathcal{D}\leftarrow\mathcal{D}\cup\mathcal{D}_{\text{proposal}}$
\EndFor
}
\end{algorithmic}
\end{algorithm}
In this section, we extend the GPR-based algorithm for global maximum search to local fine-grained search (FGS).
The local FGS steps are described in Figure~\ref{fig:structured-rcnn}.
We perform the FGS by pruning out easy negatives with low classification scores from the set of regions proposed by the selective search algorithm and sorting out a few bounding boxes with the maximum scores in local regions.
Then, for each local optimum $y_{\text{best}}$ (red boxes in Figure~\ref{fig:structured-rcnn}), we propose a new candidate bounding box (green boxes in Figure~\ref{fig:structured-rcnn}).
Specifically, we initialize a set of local observations $\mathcal{D}_{\text{local}}$ for $y_{\text{best}}$ from the set given by the selective search algorithm, whose localness is measured by an IoU between $y_{\text{best}}$ and region proposals (yellow boxes\footnote{In practice, the local search region associated with $\mathcal{D}_{\text{local}}$ is not a rectangular region around local optimum since we use IoU to determine it.} in Figure~\ref{fig:structured-rcnn}).
$\mathcal{D}_{\text{local}}$ is used to fit a GP model, and the procedure is iterated for each local optimum at different levels of IoU until there is no more acceptable proposal.
We provide a pseudocode of local FGS in Algorithm~\ref{alg:FGS}, where the parameters are set as: $t_{\text{max}}=8$, $(\rho_r)_{r=1}^{R=3}=(0.3,0.5,0.7)$.
In addition to the capability of handling multiple objects in a single image,
better computational efficiency is another factor making local FGS preferable to global search.
As a kernel method, the computational complexity of GPR increases cubically to the number of observations.
By restricting the observation set to the nearby region of a local optimum, the GP fitting and proposal process can be performed efficiently.
In practice, FGS introduces only $<20\%$ computational overhead compared to the original R-CNN.
Please see \suppname, which are also available in our technical report~\citep{fgs-tech}, for more details on its practical efficiency (\suppref{\ref{sup:FGS-efficiency}}).
\cutsubsectionup
\subsection{Learning GP hyperparameter}
\label{sec-gp-param}
\cutsubsectiondown
As we locally perform the FGS, the GP hyperparameter $\theta$ also needs to be trained with observations in the vicinity of ground truth objects.
To this end, for an annotated object in the training set, we form a set of observations with the structured labels and corresponding classification scores of the bounding boxes that are close to the ground truth bounding box. Such an observation set is composed of the bounding boxes (given by selective search and random selection) whose IoU with the ground truth exceed a certain threshold.
Finally, we fit a GP model by maximizing the joint likelihood of such observations:
\begin{equation*}
\hat{\theta} = \argmax_{\theta}\sum_{i\in I_{\text{pos}}}\log p(\{(y,f):y\in\widetilde{\mathcal{Y}}_{i},f=f(x_i,y)\};\theta),
\end{equation*}
where $I_{\text{pos}}$ is the index set for positive training samples (i.e., with ground truth object annotations), and $y_{i}$ is a ground truth annotation of an image $x_i$.\footnote{We assumed one object per image. See Section~\ref{sec-finetuning} for handling multiple objects in training.} We set $\widetilde{\mathcal{Y}}_{i} = \{y=y_{i}\operatorname{or}y\in\mathcal{Y}_{i}\operatorname{or}y\in\bar{\mathcal{Y}}_{i}: \operatorname{IoU}(y,y_{i}) > \rho\}\}$, where $\mathcal{Y}_i$ consists of the bounding boxes given by selective search on $x_i$, $\bar{\mathcal{Y}}_{i}$ is a random subset of $\mathcal{Y}$, and $\rho$ is the overlap threshold.
\commenttext{Generally, we are abusing $\hat{}$ or $\bar{}$ notations.. Such as $\hat{y}=g(x,w)$ (it's used as a max of two different functions in Sec 3.2.1 and Sec 4) $\bar{y}$, $\hat{\mathcal{Y}}$, $\bar{\mathcal{Y}}$... We need to clean up notations..}
The optimal solution $\hat{\theta}$ can be obtained via L-BFGS. Our implementation relies on the GPML toolbox~\cite{GPML}.
\vspace*{-0.1in}}\newcommand{\cutsectiondown}{\vspace*{-0.07in}
\section{Learning R-CNN with structured loss}
\label{sec-structured_loss}
\cutsectiondown
This section describes a training algorithm of R-CNN for object detection using structured loss.
We first revisit the object detection framework with structured output regression introduced by~\citet{struct-localization} in Section~\ref{sec-str_detect}, and extend it to R-CNN pipeline that allows training the network with structured hinge loss in Section~\ref{sec-finetuning}.
\cutsubsectionup
\subsection{Structured output regression for detection}
\label{sec-str_detect}
\cutsubsectiondown
Let $\{x_{1},x_{2},\ldots,x_{M}\}$ be the set of training images and $\{y_{1},y_{2},\ldots,y_{M}\}$ be the set of corresponding structured labels.
The structured label $y\in{\mathcal Y}$ is composed of 5 elements $(l, u_{1}, v_{1}, u_{2}, v_{2})$; when $l=1$, $(u_{1},v_{1})$ and $(u_{2},v_{2})$ denote the top-left and bottom-right coordinates of the object, respectively, and when $l=-1$, it implies that there is no object in $x_i$, and there is no meaning on coordinate elements $(u_{1},v_{1},u_{2},v_{2})$.
Note that the definition of $y_{i}$ is extended from Section~\ref{sec-gp} to indicate the presence of an object $(l)$ as well as its location $(u_{1}, v_{1}, u_{2}, v_{2})$ when exists. When there are multiple objects in an image, we crop an image into multiple positive ($l=1$) images, each of which contains a single object, and a negative image ($l=-1$) that doesn't contain any object.\footnote{We also perform the same procedure for images with a single object during the training.}
Let $\phi(x,y)$ represent the feature extracted from an image $x$ for a label $y$ with $l=1$.
In our case, $\phi(x,y)$ denotes the top-layer representations of the CNN (excluding the classification layer) at location specified by $y$,\footnote{Following~\cite{rcnn-cvpr}, we crop and warp the image patch of $x$ at location given by $y$ to a fixed size (e.g., 224$\times$224) to compute the CNN features.} which are fed into the classification layer. The detection problem is to find a structured label $y\in{\mathcal Y}$ that has the highest score:
\begin{align}
g(x;w) = \argmax_{y\in\mathcal{Y}} f(x,y;w)\label{eq:detector}
\end{align}
where
\begin{align}
f(x,y;w) & = w^{T}\widetilde{\phi}(x,y),\label{eq:score}\\
\widetilde{\phi}(x,y) & = \begin{cases}
\phi(x,y) & ,\;l=+1\;, \\
\mathbf{0} & ,\;l=-1\;.
\end{cases}\label{eq:feature}
\end{align}
Note that \eqref{eq:feature} includes a trick for setting the detection threshold to $0$. The model parameter $w$ is trained to minimize the structured loss $\Delta(\cdot,\cdot)$ between the predicted label $g(x_{i};w)$ and the ground-truth label $y_{i}$:
\begin{equation}
\hat{w} = \argmin_{w} \sum_{i=1}^{M} \Delta(g(x_{i}; w), y_{i}) \label{eq:param_est}
\end{equation}
For the detection problem, the structured loss $\Delta(y, y_i)$ is defined in terms of intersection over union (IoU) of two bounding boxes defined by $y$ and $y_i$ as follows:\begin{equation}
\Delta(y,y_{i})=\begin{cases}
1-\text{IoU}(y,y_{i}) & ,\;\operatorname{if}l=l_{i}=1,\\
0 & ,\;\operatorname{if}l=l_{i}=-1,\\
1 & ,\;\operatorname{if}l\neq l_{i}
\end{cases}\label{eq:strloss}
\end{equation}
where $\text{IoU}(y,y_i) = \frac{\text{Area}(y\cap y_i)}{\text{Area}(y\cup y_i)}$.
In general, the optimization problem~\eqref{eq:param_est} is difficult to solve due to the complicated form of structured loss.
Instead, we formulate the surrogate objective in structured SVM framework~\citep{tsochantaridis2005large} as follows:
\begin{align}
&\min_{w} \;\; \frac{1}{2}\|w\|^{2}+\frac{C}{M}\sum_{i=1}^{M}\xi_{i}\;\;\;\text{, subject to} \label{eq:struct-svm}\\ & \;\; w^{\top}\widetilde{\phi}(x_{i},y_{i})\geq w^{\top}\widetilde{\phi}(x_{i},y)+\Delta(y,y_{i})-\xi_{i},\forall y,\forall i \label{eq:stsvm_const}\\
& \;\; \xi_{i}\geq\;0,\,\forall i
\end{align}
Using~\eqref{eq:feature} and~\eqref{eq:strloss}, the constraint~\eqref{eq:stsvm_const} is written as follows:
\begin{align}
w^{\top}{\phi}(x_{i},y_{i}) \geq &\; w^{\top}{\phi}(x_{i},y)+\Delta^{\text{loc}}(y,y_{i})-\xi_{i},\label{eq:stsvm_const_reform_pos_pos}\\
& \; \forall y\in{\mathcal Y}^{(l=1)}, \forall i \in I_{\text{pos}}\nonumber,\\
w^{\top}{\phi}(x_{i},y_{i}) \geq &\; 1-\xi_{i},\;\forall i \in I_{\text{pos}},\label{eq:stsvm_const_reform_pos_neg}\\
w^{\top}{\phi}(x_{i},y) \leq &\; -1 + \xi_{i}, \;\forall y\in{\mathcal Y}^{(l=1)}, \forall i \in I_{\text{neg}},\label{eq:stsvm_const_reform_neg_pos}
\end{align}
where $\Delta^{\text{loc}}(y,y_{i}) = 1-\text{IoU}(y,y_{i})$, $I_{\text{pos}}$ and $I_{\text{neg}}$ denote the set of indices for positive and negative training examples, respectively, and $\mathcal{Y}^{(l=1)} = \{y\in\mathcal{Y}:l=1\}$.
\cutsubsectionup
\subsection{Gradient-based learning of R-CNN with structured SVM objective}
\label{sec-finetuning}
\cutsubsectiondown
To learn the R-CNN detector with structured loss, we propose to make several modifications to the original structured SVM formulation.
First, we restrict the output space ${\mathcal Y}_{i}\subset{\mathcal Y}$ of $i$th example to regions proposed via selective search.
This results in a change in notation for every ${\mathcal Y}$ in~\eqref{eq:stsvm_const_reform_pos_pos} and~\eqref{eq:stsvm_const_reform_neg_pos} of $i$th example to ${\mathcal Y}_i$.
Second, the constraints (\ref{eq:stsvm_const_reform_pos_pos},~\ref{eq:stsvm_const_reform_pos_neg},~\ref{eq:stsvm_const_reform_neg_pos}) should be transformed into hinge loss to backpropagate the gradient to lower layers of CNN.
Specifically, the objective function~\eqref{eq:struct-svm} is reformulated as follows:
\begin{equation}
\min_{w}\;\; \frac{1}{2}\|w\|^{2}+\frac{1}{M}\big(C_{1}\sum_{i\in I_{\text{pos}}}h_{\text{pos},i} + C_{2}\sum_{i\in I_{\text{neg}}} h_{\text{neg},i}\big)\label{eq:struct-svm-hinge}
\end{equation}
where $h_{\text{pos},i} = h_{\text{pos},i}(w)$, $h_{\text{neg},i} = h_{\text{neg},i}(w)$ are given as:
\begin{align}
&h_{\text{pos},i}(w) = \max\Big\{ 0, 1-w^{\top}\phi(x_{i},y_{i}), \label{eq:hinge-pos}\\
&\quad\max_{y\in\mathcal{Y}_{i}^{(l=1)}}\left(w^{\top}\left(\phi(x_{i},y) - \phi(x_{i},y_{i})\right) + \Delta^{\text{loc}}(y,y_{i})\right)\Big\} \nonumber
\end{align}
\begin{align}
&h_{\text{neg},i}(w) = \max\Big\{ 0, \max_{y\in\mathcal{Y}_{i}^{(l=1)}}(1+w^{\top}\phi(x_{i},y))\Big\} \label{eq:hinge-neg}
\end{align}
Note that we use different $C$ values for positive and negative examples. In experiments, $C_1=2$ and $C_2=1$.
Structured SVM objective may cause a slow convergence in parameter estimation since it utilizes at most one instance $y$ among a large number of instances in the (restricted) output space $\mathcal{Y}_i$, whose size varies from few hundreds to thousands.
To overcome this issue, we alternately perform a gradient-based parameter estimation and hard negative data mining that effectively adapts the number of training examples to be evaluated for updating the parameters (\suppreft{\ref{sup:hard-mining}}).
\commenttext{Ruben: I don't think realize is the correct word here (realize means to become fully aware of (something) as a fact; understand clearly and also to cause (something desired or anticipated) to happen, but the latter is usually used when a dream happens and not for a process like a learning process). I changed it to perform (perform means to carry out, accomplish, or fulfill)}
For model parameter estimation, we use L-BFGS to first learn parameters of the classification layer only.
We found that this already resulted in a good detection performance.
Then, we optionally use stochastic gradient descent to finetune the whole CNN classifiers (\suppreft{\ref{sup:finetuning}}).
\vspace*{-0.1in}}\newcommand{\cutsectiondown}{\vspace*{-0.07in}
\section{Experimental results}
\label{sec-exp}
\cutsectiondown
We applied our proposed methods to standard visual object detection tasks on PASCAL VOC 2007~\cite{pascal-voc-2007} and 2012~\cite{pascal-voc-2012}.
In all experiments, we consider R-CNNs~\citep{rcnn-cvpr} as baseline models.
Following \citep{rcnn-cvpr}, we used the CNN models pretrained on ImageNet database~\citep{imagenet_cvpr09} with $1,000$ object categories~\citep{alex-imagenet,vggnet}, and finetuned the whole network using the target database by replacing the existing softmax classification layer to a new one with a different number of classes (e.g., $20$ classes for VOC 2007 and 2012).
We provide the learning details in \supprefp{\ref{sup:learning-details}}.
Our implementation is based on the Caffe toolbox \citep{caffe}.
Setting the R-CNN as a baseline method, we compared the detection performance of our proposed methods, such as R-CNN with FGS (R-CNN + FGS), R-CNN trained with structured SVM objective (R-CNN + StructObj), and their combination (R-CNN + StructObj + FGS).
Since our goal is to localize the bounding boxes more accurately at the object regions, we also consider the IoU of $0.7$ for an evaluation criterion, which only counts the detection results as correct when the overlap between the predicted bounding box and the ground truth is greater than $70\%$.
This is more challenging than common practices (e.g., IoU $\geq 0.5$), but will be a good indicator for a better localization of an object bounding box if successful.
\begin{figure}[t]
\centering
\vspace{-0.1in}
\includegraphics[width=0.85\columnwidth]{figures/oracle-rp-gp.pdf}
\vspace*{-0.1in}}\newcommand{\cutcaptiondown}{\vspace*{-0.1in}
\protect\caption{\normalsize{mAP at different IoU criteria on PASCAL VOC 2007 test set using an oracle detector. We used different numbers of bounding boxes proposed by selective search (SS)~\cite{selective-search}, Objectness~\cite{objectness}, local random search, and our proposed FTS methods. The average numbers of bounding boxes used for evaluation are specified.}}
\label{fig:oracle-gp}
\cutcaptiondown
\vspace*{-0.15in}
\end{figure}
\begin{table*}[t]
\centering
{\setlength{\extrarowheight}{2pt}\scriptsize
{\begin{tabular}{l <{\hspace{-0.3em}}|>{\hspace{-0.5em}} c <{\hspace{-0.5em}}|>{\hspace{-0.3em}} c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c <{\hspace{-0.3em}}| >{\hspace{-0.3em}}c}
\hline
Model & BBoxReg & aero & bike & bird & boat & bottle & bus & car & cat & chair & cow & table & dog & horse & mbike & person & plant & sheep & sofa & train & tv & mAP\\
\hline
R-CNN (AlexNet) & No & 64.2 & 69.7 & 50.0 & 41.9 & 32.0 & 62.6 & 71.0 & 60.7 & 32.7 & 58.5 & 46.5 & 56.1 & 60.6 & 66.8 & 54.2 & 31.5 & 52.8 & 48.9 & 57.9 & 64.7 & 54.2 \\
R-CNN (VGG) & No & 68.5 & 74.5 & 61.0 & 37.9 & 40.6 & 69.2 & 73.7 & 69.9 & 37.2 & 68.6 & 56.8 & 70.6 & 69.0 & 67.1 & 59.6 & 33.4 & 63.9 & 58.9 & 62.6 & 68.5 & 60.6 \\
+ StructObj & No & 68.7 & 73.5 & 62.6 & 40.6 & 41.5 & 69.6 & 73.5 & 71.1 & 39.9 & 69.6 & 58.1 & 70.0 & 67.5 & 69.8 & 59.8 & 35.9 & 63.6 & 59.0 & 62.6 & 67.7 & 61.2 \\
+ StructObj-FT & No & 69.3 & 75.2 & 62.2 & 39.4 & 42.3 & 70.7 & 74.5 & 74.3 & 40.4 & 71.3 & 59.8 & 72.0 & 69.8 & 69.4 & 60.3 & 35.3 & 64.5 & \bf{62.0} & 63.7 & 69.8 & 62.3 \\
+ FGS & No & 70.6 & 78.4 & 65.7 & 46.2 & 48.8 & \bf{74.6} & 77.0 & 74.3 & 42.7 & 70.8 & 60.9 & 75.1 & 75.8 & 70.7 & 66.3 & 37.1 & 66.3 & 57.6 & 66.6 & 71.0 & 64.8 \\
+ StructObj + FGS & No & \bf{73.4} & \bf{80.9} & 64.5 & \bf{46.7} & 49.1 & 73.9 & 78.2 & 76.8 & 44.8 & \bf{75.3} & \bf{63.0} & 75.3 & 74.2 & 72.7 & \bf{68.5} & 37.0 & 67.5 & 58.1 & \bf{66.9} & 70.5 & 65.9 \\
+ StructObj-FT + FGS & No & 72.5 & 78.8 & \bf{67.0} & 45.2 & \bf{51.0} & 73.8 & \bf{78.7} & \bf{78.3} & \bf{46.7} & 73.8 & 61.5 & \bf{77.1} & \bf{76.4} & \bf{73.9} & 66.5 & \bf{39.2} & \bf{69.7} & 59.4 & 66.8 & \bf{72.9} & \bf{66.5} \\
\hline
R-CNN (AlexNet) & Yes & 68.1 & 72.8 & 56.8 & 43.0 & 36.8 & 66.3 & 74.2 & 67.6 & 34.4 & 63.5 & 54.5 & 61.2 & 69.1 & 68.6 & 58.7 & 33.4 & 62.9 & 51.1 & 62.5 & 64.8 & 58.5 \\
R-CNN (VGG) & Yes & 70.8 & 77.1 & \bf{69.4} & 45.8 & 48.4 & 74.0 & 77.0 & 75.0 & 42.2 & 72.5 & 61.5 & 75.6 & 77.7 & 66.6 & 65.3 & 39.1 & 65.8 & 64.2 & 68.6 & 71.5 & 65.4 \\
+ StructObj & Yes & 73.1 & 77.5 & 69.2 & 47.6 & 47.6 & 74.5 & 78.2 & 75.4 & 44.5 & 76.3 & 64.9 & 76.7 & 76.3 & 69.9 & 68.1 & 39.4 & 67.0 & \bf{65.6} & 68.7 & 70.9 & 66.6 \\
+ StructObj-FT & Yes & 72.6 & 79.4 & \bf{69.4} & 45.2 & 47.8 & 74.4 & 77.8 & 76.5 & 45.4 & 76.3 & 61.4 & \bf{80.2} & 77.1 & 73.8 & 66.8 & 41.1 & 67.8 & 64.7 & 67.9 & 72.3 & 66.9 \\
+ FGS & Yes & \bf{74.2} & 78.9 & 67.8 & \bf{51.6} & 52.3 & 75.7 & 78.7 & 76.6 & 45.4 & 72.4 & 63.1 & 76.6 & \bf{79.3} & 70.7 & 68.0 & 40.3 & 67.8 & 61.8 & \bf{70.2} & 71.6 & 67.2 \\
+ StructObj + FGS & Yes & 74.1 & \bf{83.2} & 67.0 & 50.8 & 51.6 & \bf{76.2} & \bf{81.4} & 77.2 & 48.1 & \bf{78.9} & \bf{65.6} & 77.3 & 78.4 & \bf{75.1} & \bf{70.1} & 41.4 & 69.6 & 60.8 & \bf{70.2} & \bf{73.7} & \bf{68.5} \\
+ StructObj-FT + FGS & Yes & 71.3 & 80.5 & 69.3 & 49.6 & \bf{54.2} & 75.4 & 80.7 & \bf{79.4} & \bf{49.1} & 76.0 & 65.2 & 79.4 & 78.4 & 75.0 & 68.4 & \bf{41.6} & \bf{71.3} & 61.2 & 68.2 & 73.3 & 68.4 \\
\hline
\end{tabular}}}
\vspace*{-0.1in}}\newcommand{\cutcaptiondown}{\vspace*{-0.1in}
\caption{\normalsize{Test set mAP of VOC 2007 with IoU $> 0.5$. The entries with the best APs for each object category are bold-faced.\label{tab-voc2007-iou5}}}
\cutcaptiondown
\end{table*}
\begin{table*}[t]
\centering
{\setlength{\extrarowheight}{2pt}\scriptsize
{\begin{tabular}{l <{\hspace{-0.3em}}|>{\hspace{-0.5em}} c <{\hspace{-0.5em}}|>{\hspace{-0.3em}} c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c >{\hspace{-1.1em}}c <{\hspace{-0.3em}}| >{\hspace{-0.3em}}c}
\hline
Model & BBoxReg & aero & bike & bird & boat & bottle & bus & car & cat & chair & cow & table & dog & horse & mbike & person & plant & sheep & sofa & train & tv & mAP\\
\hline
R-CNN (AlexNet) & No & 32.9 & 40.1 & 19.7 & 18.7 & 11.1 & 39.4 & 40.5 & 26.5 & 14.8 & 29.8 & 24.5 & 26.4 & 23.7 & 31.9 & 18.5 & \bf{13.3} & 27.6 & 25.8 & 26.6 & 39.5 & 26.6 \\
R-CNN (VGG) & No & 40.2 & 43.3 & 23.4 & 14.4 & 13.3 & 48.2 & 44.5 & 36.4 & 17.1 & 34.0 & 27.9 & 36.3 & 26.8 & 28.2 & 21.2 & 10.3 & 33.7 & 36.6 & 31.6 & 48.9 & 30.8 \\
+ StructObj & No & 42.5 & 44.4 & 24.5 & 17.8 & 15.3 & 46.8 & 46.4 & 37.9 & 17.6 & 33.4 & 26.6 & 36.8 & 24.3 & 31.5 & 21.3 & 10.4 & 30.0 & 36.1 & 30.6 & 46.3 & 31.0 \\
+ StructObj-FT & No & 44.1 & 47.1 & 23.4 & 16.6 & 16.4 & 50.1 & 48.7 & 39.7 & 18.4 & 39.4 & 28.6 & 38.6 & 27.5 & 32.4 & 23.6 & 11.1 & 33.1 & \bf{41.0} & 34.3 & 49.6 & 33.2 \\
+ FGS & No & 44.3 & 55.5 & 28.9 & 19.1 & 22.9 & 56.9 & 57.6 & 37.8 & 19.6 & 35.7 & \bf{31.9} & 38.1 & \bf{43.0} & 42.7 & 30.3 & 9.8 & \bf{42.3} & 33.3 & 43.4 & 55.4 & 37.4 \\
+ StructObj + FGS & No & 43.5 & 56.1 & 30.9 & 18.7 & 24.9 & 55.2 & 57.6 & 38.9 & 20.7 & 38.6 & 28.4 & 37.7 & 38.7 & \bf{46.3} & 30.9 & 8.4 & 37.6 & 37.0 & 42.2 & 51.3 & 37.2 \\
+ StructObj-FT + FGS & No & \bf{46.3} & \bf{58.1} & \bf{31.1} & \bf{21.6} & \bf{25.8} & \bf{57.1} & \bf{58.2} & \bf{43.5} & \bf{23.0} & \bf{46.4} & 29.0 & \bf{40.7} & 40.6 & \bf{46.3} & \bf{33.4} & 10.6 & 41.3 & 40.9 & \bf{45.8} & \bf{56.3} & \bf{39.8} \\
\hline
R-CNN (AlexNet) & Yes & 47.6 & 48.7 & 25.3 & \bf{25.0} & 17.3 & 53.4 & 54.6 & 36.8 & 16.7 & 42.3 & 31.6 & 35.8 & 38.0 & 41.8 & 24.5 & 14.3 & 38.8 & 28.9 & 34.0 & 49.0 & 35.2 \\
R-CNN (VGG) & Yes & 45.1 & 48.6 & 26.0 & 18.2 & 21.2 & 57.2 & 52.4 & 37.3 & 20.1 & 33.7 & 31.9 & 38.8 & 39.6 & 36.3 & 26.5 & 9.2 & 37.8 & 33.4 & 39.4 & 50.7 & 35.2 \\
+ StructObj & Yes & 49.4 & 56.5 & \bf{36.5} & 21.3 & 23.3 & 61.0 & 58.1 & 44.3 & 20.8 & 47.4 & 33.3 & 39.8 & 40.7 & 45.9 & 31.0 & \bf{14.7} & 39.6 & 42.9 & 45.7 & 56.9 & 40.5 \\
+ StructObj-FT & Yes & 49.3 & 58.1 & 35.4 & 23.3 & 24.4 & 62.3 & 60.1 & 45.8 & 21.8 & 48.7 & 32.4 & 41.8 & 43.2 & 45.7 & 32.0 & 14.4 & 44.6 & \bf{45.1} & 48.6 & 59.8 & 41.8 \\
+ FGS & Yes & 50.9 & 59.8 & 34.4 & 20.9 & \bf{31.6} & \bf{66.1} & 62.3 & 44.9 & 22.0 & 46.5 & \bf{36.8} & 42.5 & 51.4 & 46.8 & 34.1 & 13.5 & \bf{44.7} & 39.1 & 48.9 & 57.7 & 42.7 \\
+ StructObj + FGS & Yes & \bf{53.6} & 60.7 & 32.1 & 19.9 & 31.3 & 63.2 & 63.2 & 46.4 & 23.6 & \bf{53.0} & 34.9 & 40.4 & \bf{53.6} & \bf{49.9} & 34.6 & 10.2 & 42.2 & 40.1 & 48.3 & 58.3 & 43.0 \\
+ StructObj-FT + FGS & Yes & 47.1 & \bf{61.8} & 35.2 & 18.1 & 29.7 & 66.0 & \bf{64.7} & \bf{48.0} & \bf{25.3} & 50.4 & 34.9 & \bf{43.7} & 50.8 & 49.4 & \bf{36.8} & 13.7 & \bf{44.7} & 43.6 & \bf{49.8} & \bf{60.5} & \bf{43.7} \\
\hline
\end{tabular}}}
\vspace*{-0.1in}}\newcommand{\cutcaptiondown}{\vspace*{-0.1in}
\caption{\normalsize{Test set mAP of VOC 2007 with IoU $> 0.7$. The entries with the best APs for each object category are bold-faced.\label{tab-voc2007-iou7}}}
\cutcaptiondown
\vspace*{-0.05in}
\end{table*}
\begin{table*}[t]
\centering
{\setlength{\extrarowheight}{2pt}\scriptsize
{\begin{tabular}{l <{\hspace{-0.3em}}|>{\hspace{-0.5em}} c <{\hspace{-0.5em}}|>{\hspace{-0.3em}} c >{\hspace{-1em}}c >{\hspace{-1em}}c >{\hspace{-1em}}c >{\hspace{-1em}}c >{\hspace{-1em}}c >{\hspace{-1em}}c >{\hspace{-1em}}c >{\hspace{-1em}}c >{\hspace{-1em}}c >{\hspace{-1em}}c >{\hspace{-1em}}c >{\hspace{-1em}}c >{\hspace{-1em}}c >{\hspace{-1em}}c >{\hspace{-1em}}c >{\hspace{-1em}}c >{\hspace{-1em}}c >{\hspace{-1em}}c >{\hspace{-1em}}c <{\hspace{-0.3em}}| >{\hspace{-0.3em}}c}
\hline
Model & BBoxReg & aero & bike & bird & boat & bottle & bus & car & cat & chair & cow & table & dog & horse & mbike & person & plant & sheep & sofa & train & tv & mAP\\
\hline
R-CNN (AlexNet) & No & 68.1 & 63.8 & 46.1 & 29.4 & 27.9 & 56.6 & 57.0 & 65.9 & 26.5 & 48.7 & 39.5 & 66.2 & 57.3 & 65.4 & 53.2 & 26.2 & 54.5 & 38.1 & 50.6 & 51.6 & 49.6\\
R-CNN (VGGNet) & No & 76.3 & 69.8 & 57.9 & 40.2 & 37.2 & 64.0 & 63.7 & 80.2 & 36.1 & 63.6 & 47.3 & 81.1 & 71.2 & 73.8 & 59.5 & 30.9 & 64.2 & 52.2 & 62.4 & 58.7 & 59.5\\
\hline
R-CNN (AlexNet) & Yes & 71.8 & 65.8 & 52.0 & 34.1 & 32.6 & 59.6 & 60.0 & 69.8 & 27.6 & 52.0 & 41.7 & 69.6 & 61.3 & 68.3 & 57.8 & 29.6 & 57.8 & 40.9 & 59.3 & 54.1 & 53.3 \\
R-CNN (VGGNet) & Yes & 79.2 & 72.3 & 62.9 & 43.7 & 45.1 & 67.7 & 66.7 & 83.0 & 39.3 & 66.2 & 51.7 & 82.2 & 73.2 & 76.5 & 64.2 & 33.7 & 66.7 & 56.1 & 68.3 & 61.0 & 63.0 \\
+ StructObj & Yes & 80.9 & 74.8 & 62.7 & 42.6 & 46.2 & 70.2 & 68.6 & 84.0 & 42.2 & 68.2 & 54.1 & 82.2 & 74.2 & 79.8 & 66.6 & 39.3 & 67.6 & \bf{61.0} & 71.3 & 65.2 & 65.1 \\
+ FGS & Yes & 80.5 & 73.5 & \bf{64.1} & \bf{45.3} & 48.7 & 66.5 & 68.3 & 82.8 & 39.8 & 68.2 & 52.7 & 82.1 & 75.1 & 76.6 & 66.3 & 35.5 & 66.9 & 56.8 & 68.7 & 61.6 & 64.0 \\
+ StructObj + FGS & Yes & \bf{82.9} & \bf{76.1} & \bf{64.1} & 44.6 & \bf{49.4} & \bf{70.3} & \bf{71.2} & \bf{84.6} & \bf{42.7} & 68.6 & \bf{55.8} & \bf{82.7} & \bf{77.1} & \bf{79.9} & \bf{68.7} & \bf{41.4} & \bf{69.0} & 60.0 & \bf{72.0} & \bf{66.2} & \bf{66.4} \\
\hline
NIN~\citep{DBLP:journals/corr/LinCY13} & - & 80.2 & 73.8 & 61.9 & 43.7 & 43.0 & \bf{70.3} & 67.6 & 80.7 & 41.9 & \bf{69.7} & 51.7 & 78.2 & 75.2 & 76.9 & 65.1 & 38.6 & 68.3 & 58.0 & 68.7 & 63.3 & 63.8 \\
\hline
\end{tabular}}}
\vspace*{-0.1in}}\newcommand{\cutcaptiondown}{\vspace*{-0.1in}
\vspace*{0.03in}
\caption{\normalsize{Test set mAP of VOC 2012 with IoU $> 0.5$. The entries with the best APs for each object category are bold-faced.\label{tab-voc2012-iou5}}}
\cutcaptiondown
\vspace*{0.03in}
\end{table*}
\begin{figure*}[t]
\begin{center}
{\renewcommand{\arraystretch}{0.3} \begin{tabular}{c <{\hspace{-0.5em}} >{\hspace{-0.5em}}c c <{\hspace{-0.5em}} >{\hspace{-0.5em}}c c <{\hspace{-0.5em}} >{\hspace{-0.5em}}c c <{\hspace{-0.5em}} >{\hspace{-0.5em}}c c <{\hspace{-0.5em}} >{\hspace{-0.5em}}c}
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/aeroplane/001_009356.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/aeroplane/004_002703.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/bicycle/002_002631.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/bicycle/004_007610.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/bird/002_009696.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/bird/007_005532.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/boat/004_005604.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/boat/016_000295.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/bottle/008_004068.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/bottle/010_007569.pdf} \\
\multicolumn{2}{c}{\footnotesize{aeroplane}} & \multicolumn{2}{c}{\footnotesize{bicycle}} & \multicolumn{2}{c}{\footnotesize{bird}} & \multicolumn{2}{c}{\footnotesize{boat}} & \multicolumn{2}{c}{\footnotesize{bottle}} \\
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/bus/002_008158.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/bus/004_006992.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/car/012_007906.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/car/026_002808.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/cat/003_009310.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/cat/004_009552.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/chair/038_003520.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/chair/093_005412.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/cow/002_008863.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/cow/007_007044.pdf} \\
\multicolumn{2}{c}{\footnotesize{bus}} & \multicolumn{2}{c}{\footnotesize{car}} & \multicolumn{2}{c}{\footnotesize{cat}} & \multicolumn{2}{c}{\footnotesize{chair}} & \multicolumn{2}{c}{\footnotesize{cow}} \\
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/diningtable/002_002674.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/diningtable/005_002809.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/dog/001_002160.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/dog/002_001657.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/horse/003_007587.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/horse/004_007789.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/motorbike/001_008901.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/motorbike/005_004070.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/person/009_007532.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/person/050_000901.pdf} \\
\multicolumn{2}{c}{\footnotesize{diningtable}} & \multicolumn{2}{c}{\footnotesize{dog}} & \multicolumn{2}{c}{\footnotesize{horse}} & \multicolumn{2}{c}{\footnotesize{motorbike}} & \multicolumn{2}{c}{\footnotesize{person}} \\
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/pottedplant/001_002074.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/pottedplant/014_005088.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/sheep/002_003914.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/sheep/003_005578.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/sofa/001_001518.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/sofa/002_001518.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/train/001_000835.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/train/004_008339.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/tvmonitor/005_007203.pdf} &
\includegraphics[width = 0.55in]{figures/voc07-best_vs_base/tvmonitor/008_000571.pdf} \\
\multicolumn{2}{c}{\footnotesize{pottedplant}} & \multicolumn{2}{c}{\footnotesize{sheep}} & \multicolumn{2}{c}{\footnotesize{sofa}} & \multicolumn{2}{c}{\footnotesize{train}} & \multicolumn{2}{c}{\footnotesize{tvmonitor}} \\
\end{tabular}}
\normalsize
\vspace*{-0.1in}}\newcommand{\cutcaptiondown}{\vspace*{-0.1in}
\vspace*{0.03in}
\caption{\normalsize{Detection examples from PASCAL VOC 2007 test set. Two examples from 20 object categories are shown, with the ground truth bounding boxes (green), the boxes obtained by baseline R-CNN (VGGNet) (red), and those obtained by the proposed R-CNN + StructObj + FGS (yellow). The numbers near the bounding boxes denote the IoU with the ground truth.}}
\label{fig-2007examples}
\cutcaptiondown
\vspace*{-0.15in}
\end{center}
\end{figure*}
\cutsubsectionup
\subsection{FGS efficacy test with oracle detector}
\label{exp-oracle}
\cutsubsectiondown
Before reporting the performance of the proposed methods in R-CNN framework, we demonstrate the efficacy of FGS algorithm using an oracle detector.
We design a hypothetical oracle detector whose score function is defined as $f_{\text{ideal}}(x_{i},y)=\operatorname{IoU}(y,y_{i})$, where $y_{i}$ is a ground truth annotation for an image $x_{i}$.
The score function is ideal in the sense that it outputs high scores for bounding boxes with high overlap with the ground truth and vice versa, overall achieving 100\% mAP.
We summarize the results in Figure~\ref{fig:oracle-gp}. We report the performance on the VOC 2007 test set at different levels of IoU criteria ($0.1,\cdots,0.9$) for the baseline selective search (SS; ``fast mode'' in \citep{selective-search}), selective search with objectness~\citep{objectness} (SS + Objectness), selective search with extended super-pixel similarity measurements (SS extended)~\citep{selective-search}, ``quality mode'' of selective search (SS quality)~\cite{selective-search}, local random search,\footnote{Like local FGS, local random search first determine the local search regions by NMS. However, it randomly choose a fixed number of bounding box in those regions rather than sequentially proposing new boxes based on some informed method.} and the proposed FGS method with the baseline selective search.
For low values of IoU ($\leq 0.3$), all methods using the oracle detectors performed almost perfectly due to the ideal score function.
However, we found that the detection performance with different region proposal schemes other than our proposed FGS algorithm start to break down at high values of IoU.
For example, the performance of SS, SS + Objectness, SS extended, and local random search methods, which used around $2,000$ $\sim$ $3,500$ bounding boxes per image in average, significantly dropped at IoU $\geq 0.5$.
SS quality method kept pace with the FGS method until IoU of $0.6$, but again, the performance started to drop at IoU $\geq 0.7$.
On the other hand, the performance of FGS dropped $5\%$ in mAP at IoU of $0.9$ by only introducing approximately $100$ new bounding boxes per image.
Given that SS quality requires $10,000$ region proposals per image, our proposed FGS method is much more computationally efficient ($\sim 80\%$ less bounding boxes) while localizing the bounding boxes much more accurately.
This provides an insight that, if the detector is accurate, our Bayesian optimization framework would limit the number of bounding boxes to a manageable number (e.g., few thousands per image on average) to achieve almost perfect detection results.
We also report similar experimental analysis for the real detector trained with the proposed structured objective in \supprefp{\ref{sup:mAP-diff-pr}}.
\cutsubsectionup
\subsection{PASCAL VOC 2007}
\label{exp-voc2007}
\cutsubsectiondown
In this section, we demonstrate the performance of our proposed methods on PASCAL VOC 2007~\cite{pascal-voc-2007} detection task (comp4), a standard benchmark for object detection problem.
Similarly to the training pipeline of R-CNN~\citep{rcnn-cvpr}, we finetuned the CNN models (with softmax classification layer) pretrained on ImageNet database using images from both train and validation sets of VOC 2007 and further trained the network with linear SVM (baseline) or the proposed structured SVM objective.
We evaluated on the test set using the proposed FGS algorithm.
For post-processing, we performed NMS and bounding box regression~\citep{rcnn-cvpr}.
Figure~\ref{fig-2007examples} shows representative examples of successful detection using our method.
For these cases, our method can localize objects accurately even if the initial bounding box proposals don't have good overlaps with the ground truth.
We show more examples (including the failure cases) in \supprefp{\ref{sup:example-improvement}, \ref{sup:example-fp}, \ref{sup:example-rand}}.
The summary results are in Table~\ref{tab-voc2007-iou5} with IoU criteria of $0.5$ and Table~\ref{tab-voc2007-iou7} with $0.7$.
We report the performance with the AlexNet~\citep{alex-imagenet} and the VGGNet (16 layers)~\citep{vggnet}, a deeper CNN model than AlexNet that showed a significantly better recognition performance and achieved the best performance on object localization task in ILSVRC 2014.\footnote{The 16-layer VGGNet can be downloaded from: \url{https://gist.github.com/ksimonyan/211839e770f7b538e2d8}.}
First of all, we observed the significant performance improvement by simply having a better CNN model.
Building upon the VGGNet, the FGS improved the performance by $4.2\%$ and $1.8\%$ in mAP without and with bounding box regression (Table~\ref{tab-voc2007-iou5}).
It becomes much more significant when we consider IoU criteria of $0.7$ (Table~\ref{tab-voc2007-iou7}), improving upon the baseline model by $6.6\%$ and $7.5\%$ in mAP without and with bounding box regression.
The results demonstrate that our FGS algorithm is effective in accurately localizing the bounding box of an object.
Further improvement has been made by training a classifier with structured SVM objective; we obtained $\mathbf{68.5}\%$ mAP in IoU criteria of $0.5$, which, to our knowledge, is higher than the best published results, and $\mathbf{43.0}\%$ mAP in IoU criteria of $0.7$ with FGS and bounding box regression by training the classification layer only (``StructObj").
By finetuning the whole CNN classifiers (``StructObj-FT''), we observed extra improvement for most cases; for example, we obtained $\mathbf{43.7}\%$ mAP in IOU criteria of $0.7$, which improves by $0.7\%$ in mAP over the method without finetuning.
However, for IoU$>$0.5 criterion, the overall improvement due to finetuning was relatively small, especially when using bounding box regression.
In this case, considering the high computational cost for finetuning, we found that training only the classification layer is practically a sufficient way to learn a good localization-aware classifier.
We provide in-depth analysis of our proposed methods in \suppname{}.
Specifically, we report the precision-recall curves of different combinations of the proposed methods (\suppref{\ref{sup:pr-curves}}), the performance of FGS with different GP iterations (\suppref{\ref{sup:per-iter-FGS}}), the analysis of localization accuracy (\suppref{\ref{sup:loc-distr}}), and more detection examples.
\cutsubsectionup
\subsection{PASCAL VOC 2012}
\label{exp-voc2012}
\cutsubsectiondown
We also evaluate the performance of the proposed methods on PASCAL VOC 2012~\cite{pascal-voc-2012}.
As the data statistics are similar to VOC 2007, we used the same hyperparameters as described in Section~\ref{exp-voc2007} for this experiment.
We report the test set mAP over 20 object categories in Table~\ref{tab-voc2012-iou5}.
Our proposed method shows improvement by $2.1\%$ with R-CNN + StructObj and $1.0\%$ with R-CNN + FGS over baseline R-CNN using VGGNet.
Finally, we obtained $\mathbf{66.4}\%$ mAP by combining the two methods, which significantly improved upon the baseline R-CNN model and the previously published results on the leaderboard.
\vspace*{-0.1in}}\newcommand{\cutsectiondown}{\vspace*{-0.07in}
\section{Conclusion}
\label{sec-conc}
\cutsectiondown
In this work, we proposed two complementary methods to improve the performance of object detection in R-CNN framework with 1) fine-grained search algorithm in a Bayesian optimization framework to refine the region proposals and 2) a CNN classifier trained with structured SVM objective to improve localization.
We demonstrated the state-of-the-art detection performance on PASCAL VOC 2007 and 2012 benchmarks under standard localization requirements.
Our methods showed more significant improvement with higher IoU evaluation criteria (e.g., IoU $=0.7$), and hold promise for mission-critical applications that require highly precise localization, such as autonomous driving, robotic surgery and manipulation.
\vspace*{-0.1in}}\newcommand{\cutsectiondown}{\vspace*{-0.07in}
\section*{Acknowledgments}
\cutsectiondown
This work was supported by Samsung Digital Media and Communication Lab, Google Faculty Research Award, ONR grant N00014-13-1-0762, China Scholarship Council, and Rackham Merit Fellowship. We also acknowledge NVIDIA for the donation of GPUs. Finally, we thank Scott Reed, Brian Wang, Junhyuk Oh, Xinchen Yan, Ye Liu, Wenling Shang, and Roni Mittelman for helpful discussions.
\bibliographystyle{abbrvnat}
\begingroup
{
\small
\setstretch{1}
\setlength{\bibsep}{0pt}
|
{
"timestamp": "2016-01-15T02:04:00",
"yymm": "1504",
"arxiv_id": "1504.03293",
"language": "en",
"url": "https://arxiv.org/abs/1504.03293"
}
|
\section{Introduction}
Several problems in computer vision and image processing, such as object detection/classification, image denoising, inpainting etc., require solving multiple learning tasks at the same time. In such settings a natural question is to ask whether it could be beneficial to solve all the tasks jointly, rather than separately. This idea is at the basis of the field of multi-task learning, where the joint solution of different problems has the potential to exploit tasks relatedness (structure) to improve learning. Indeed, when knowledge about task relatedness is available, it can be profitably incorporated in multi-task learning approaches for example by designing suitable embedding/coding schemes, kernels or regularizers, see \cite{micchelli04,evgeniou05,alvarez12,fergus10,lozano10}.
The more interesting case, when knowledge about the tasks structure is not known a priori, has been the subject
of recent studies. Largely influenced by the success of sparsity based methods, a common approach has been that of considering linear models for each task coupled with suitable parameterization/penalization enforcing task relatedness, for example encouraging the selection of features simultaneously important for all tasks~\cite{argyriou08} or for specific subgroups of related tasks~\cite{jacob08,jayaraman14,zhong12,kang11,hwang11,kumar12}. Other linear methods adopt hierarchical priors or greedy approaches to recover the taxonomy of tasks~\cite{salakhutdinov11,torralba04}. A different line of research has been devoted to the development of non-linear/non-parametric approaches using kernel methods -- either from a Gaussian process~\cite{alvarez12,zhong12} or a regularization perspective~\cite{alvarez12,dinuzzo11}.
This paper follows this last line of research, tackling in particular two issues only partially addressed in previous works. The first is the development of a regularization framework to learn and exploit the tasks structure, which is not only important for prediction, but also for interpretation. Towards this end, we propose and study a family of matrix-valued reproducing kernels, parametrized so to enforce sparse relations among tasks. A novel algorithm dubbed Sparse Kernel MTL is then proposed considering a Tikhonov regularization approach.
The second contribution is to provide a sound computational framework to solve the corresponding minimization problem. While we follow a fairly standard alternating minimization approach, unlike most previous work we can exploit results in convex optimization to prove the convergence of the considered procedure.
The latter has an interesting interpretation where supervised and unsupervised learning steps are alternated: first, given a structure, multiple tasks are learned jointly, then the structure is updated.
We support the proposed method with an experimental analysis both on synthetic and real data, including classification and detection datasets. The obtained results show that Sparse Kernel MTL
can achieve state of the art performances while unveiling the structure describing tasks relatedness.
The paper is organized as follows: in Sec.~\ref{sec:model} we provide some background and notation in order to motivate and introduce the Sparse Kernel MTL model. In Sec.~\ref{sec:optimization} we discuss an alternating minimization algorithm to provably solve the learning problem proposed. Finally, we discuss empirical evaluation in Sec.~\ref{sec:empirical}.
{\bf Notation.} With $S^n_{++} \subset S^n_+ \subset S^n \subset \mathbb{R}^{n \times n}$ we denote respectively the space of positive definite, positive semidefinite (PSD) and symmetric $n \times n$ real-valued matrices. $O^n$ denotes the space of orthonormal $n \times n$ matrices. For any $M\in\mathbb{R}^{n \times m}$, $M^\top$ denotes the transpose of $M$. For any PSD matrix $A\in S_+^n$, $A^\dagger \in S_+^n$ denotes the pseudoinverse of $A$. We denote by $I_n\in S_{++}^n$ the $n \times n$ identity matrix. We use the abbreviation l.s.c. to denote lower semi-continuous functions (i.e. functions with closed sub-level sets)~\cite{boyd04}.
\section{Model}\label{sec:model}
We formulate the problem of solving multiple learning tasks as that of learning a vector-valued function whose output components correspond to individual predictors. We consider the framework originally introduced in~\cite{micchelli04} where the well-known concept of Reproducing Kernel Hilbert Space is extended to spaces of vector-valued functions. In this setting the set of tasks relations has a natural characterization in terms of a positive semidefinite matrix. By imposing a sparse prior on this object we are able to formulate our model, Sparse Kernel MTL, as a kernel learning problem designed to recover the most relevant relations among the tasks.
In the following we review basic definitions and results from the theory of Reproducing Kernel Hilbert Spaces that will allow in Sec.~\ref{sec:sparse_relation_learning} to motivate and introduce our learning framework. In Sec.~\ref{sec:previous_work} we briefly draw connections of our method to previously prosed multi-task learning approaches.
\subsection{Reproducing Kernel Hilbert Spaces for Vector-Valued Functions}\label{sec:RKHSvv}
We consider the problem of learning a function $f: \mathcal{X} \to \mathcal{Y}$ from a set of empirical observations $\{(x_i,y_i)\}_{i=1}^n$ with $x_i \in \mathcal{X}$ and $y_i\in\mathcal{Y} \subseteq \mathbb{R}^T$. This setting includes learning problems such as vector-valued regression ($\mathcal{Y} = \mathbb{R}^T$), multi-label/detection for $T$ tasks ($\mathcal{Y} = \{0,1\}^{T}$) or also $T$-class classification (where we adopt the standard one-vs-all approach mapping the $t$-th class label to the $t$-th element $e_t$ of the canonical basis in $\mathbb{R}^T$). Following the work of Micchelli and Pontil~\cite{micchelli04}, we adopt a Tikhonov regularization approach in the setting of Reproducing Kernel Hilbert Spaces for vector-valued functions (RKHSvv). RKHSvv are the generalization of the well-known RKHS to the vector-valued setting and maintain most of the properties of their scalar counterpart. In particular, similarly to standard RKHS, RKHSvv are uniquely characterized by an operator-valued kernel:
\begin{definition}
Let $\mathcal{X}$ be a set and $(\mathcal{H},\langle\cdot,\cdot\rangle_{\mathcal{H}})$ be a Hilbert space of functions from $\mathcal{X}$ to $\mathbb{R}^T$. A symmetric, positive definite, matrix valued function $\Gamma : \mathcal{X} \times \mathcal{X} \to \mathbb{R}^{T \times T}$ is called a reproducing kernel for $\mathcal{H}$ if for all $x \in \mathcal{X}, c \in \mathbb{R}^T$ and $f \in \mathcal{H}$ we have that $\Gamma(x,\cdot)c \in \mathcal{H}$ and the following reproducing property holds: $\langle f(x), c\rangle_{\mathbb{R}^T} = \langle f,\Gamma(x,\cdot)c\rangle_{\mathcal{H}}.$
\end{definition}
Analogously to the scalar setting, a Representer theorem holds, stating that the solution to the regularized learning problem
\begin{equation}\label{eq:learning_problem}
\underset{f\in\mathcal{H}}{\text{minimize}} \ \ \frac{1}{n} \sum_{i=1}^n V(y_i,f(x_i)) + \lambda \|f\|_\mathcal{H}^2
\end{equation}
is of the form $f(\cdot) = \sum_{i=1}^n \Gamma(\cdot,x_i)c_i$ with $c_i\in\mathbb{R}^T$, $\Gamma$ the matrix-valued kernel associated to the RKHSvv $\mathcal{H}$ and $V:\mathcal{Y} \times \mathbb{R}^T \to \mathbb{R}_+$ a loss function (e.g. least squares, hinge, logistic, etc.) which we assume to be convex. We point out that the setting above can also account for the case where not all task outputs $y_i = (y_{i1},\dots,y_{iT})^\top$ associated to a given input $x_i$ are available in training. Such situation would arise for instance in multi-detection problems in which supervision (e.g. presence/absence of an object class in the image) is provided only for a few tasks at the time.
\subsubsection{Separable Kernels}\label{sec:separable_kernels}
Depending on the choice of operator-valued kernel $\Gamma$, different structures can be enforced among the tasks; this effect can be observed by restricting ourselves to the family of {\it separable} kernels. Separable kernels are matrix-valued functions of the form $\Gamma(x,x') = k(x,x')A$, where $k:\mathcal{X}\times\mathcal{X}\to\mathbb{R}$ is a scalar reproducing kernel and $A \in S_+^T$ a $T \times T$ positive semidefinite (PSD) matrix. Intuitively, the scalar kernel characterizes the individual tasks functions, while the matrix $A$ describes how they are related. Indeed, from the Representer theorem we have that solutions of problem~\eqref{eq:learning_problem} are of the form $f(\cdot) = \sum_{i=1}^n k(\cdot,x_i) A c_i$ with the $t$-th task being $f_t(\cdot) = \sum_{i=1}^n k(\cdot,x_i) \langle A_t, c_t \rangle_{\mathbb{R}^T}$, a scalar function in the RKHS $\mathcal{H}_k$ associated to kernel $k$. As shown in~\cite{evgeniou05}, in this case the squared norm associated to the separable kernel $kA$ in the RKHSvv $\mathcal{H}$, can be written as
\begin{equation}\label{eq:norm}
\|f\|_\mathcal{H}^2 = \sum_{t,s}^T A^{\dagger}_{ts} \langle f_t, f_s \rangle_{\mathcal{H}_k}
\end{equation}
with $A^\dagger_{ts}$ the $(t,s)$-th entry of $A$'s pseudo-inverse.
Eq.~\eqref{eq:norm} shows how $A$ can model the structural relations among tasks by directly coupling predictors: for instance, by setting $A^\dagger= I_T + \gamma (\mathbf{1}\mathbf{1}^\top)/T$, with $\mathbf{1}\in\mathbb{R}^T$ the vector of all $1$s, we have that the parameter $\gamma$ controls the variance $\sum_{t=1}^T \|\bar{f} - f_t\|_{\mathcal{H}_k}^2$ of the tasks with respect to their mean $\bar{f}=\frac{1}{T} \sum_{t=1}^T f_t$. If we have access to some notion of similarity among tasks in the form of a graph with adjacency matrix $W\in S^T$, we can consider the regularizer $\sum_{t,s=1}^T W_{ts} \|f_t - f_s\|_{\mathcal{H}_k}^2 + \gamma \sum_{t}^T \|f_t\|_{\mathcal{H}_k}^2$ which corresponds to setting $A^\dagger = L + \gamma I_T$ with $L$ the graph Laplacian induced by $W$. We refer the reader to~\cite{evgeniou05} for more examples of possible choices for $A$ when the tasks structure is known.
\subsection{Sparse Kernel Multi Task Learning}\label{sec:sparse_relation_learning}
When a-priori knowledge of the problem structure is not available, it is desirable to learn the tasks relations directly from the data. In light of the observations of
Sec.~\ref{sec:separable_kernels}, a viable approach is to parametrize the RKHSvv $\mathcal{H}$ in problem~\eqref{eq:learning_problem} with the
associated separable kernel $kA$ and to optimize jointly with respect to both $f\in \mathcal{H}$ and $A \in S_+^T$. In the following we show how this problem corresponds to that of identifying a set of latent tasks and to combine them in order to form the individual predictors. By enforcing a sparsity prior on the set of such possible combinations, we then propose the Sparse Kernel MTL model, which is designed to recover only the most relevant tasks relations. In Sec.~\ref{sec:previous_work} we discuss, from a modeling perspective, how our framework is related to the current multi-task learning literature.
\subsubsection{Recovering the Most Relevant Relations}\label{sec:model_srl}
From the Representer theorem introduced in Sec.~\ref{sec:RKHSvv} we know that a candidate solution $f: \mathcal{X} \to \mathbb{R}^T$ to problem~\eqref{eq:learning_problem} can be parametrized in terms of the maps $k(\cdot,x_i)$, by a structure matrix $A\in S_{+}^T$ and a set of coefficient vectors $c_1,\dots,c_n \in \mathbb{R}^T$ such that $f(\cdot) = \sum_{i=1}^n k(\cdot,x_i)Ac_i$. If now we consider the $t$-th component of $f$ (i.e. the predictor of the $t$-th task), we have that
\begin{equation}\label{eq:latent}
f_t(\cdot) = \sum_{i=1}^n k(\cdot,x_i) \langle A_t, c_i \rangle_{\mathbb{R}^T} = \sum_{s=1}^T A_{ts} g_s(\cdot)
\end{equation}
where we set $g_s(\cdot) = \sum_{i=1}^n k(\cdot,x_i)c_{is} \in \mathcal{H}_k$ for $s\in\{1,\dots,T\}$ and $c_{is}\in\mathbb{R}$ the $s$-th component of $c_i$. Eq.~\eqref{eq:latent} provides further understanding on how $A$ can enforce/describe the tasks relations: The $g_s$ can be interpreted as elements in a dictionary and each $f_t$ factorizes as their linear combination. Therefore, any two predictors $f_t$ and $f_{t'}$ are implicitly coupled by the subset of common $g_s$.
We consider the setting where the tasks structure is unknown and we aim to recover it from the available data in the form of a structure matrix $A$. Following a denoising/feature selection argument, our approach consists in imposing a sparsity penalty on the set of possible tasks structures, requiring each predictor $f_t$ to be described by a small subset of $g_s$. Indeed, by requiring most of $A$'s entries to be equal to zero, we implicitly enforce the system to recover only the most relevant tasks relations. The benefits of this approach are two-fold: on the one hand it is less sensitive to spurious statistically non-significant tasks-correlations that could for instance arise when few training examples are available. On the other hand it provides us with interpretable tasks structures, which is a problem of interest in its own right and relevant, for example, in cognitive science~\cite{lake10}.
Following the de-facto standard choice of $\ell_1$-norm regularization to impose sparsity in convex settings, the {\it Sparse Kernel MTL} problem can be formulated as
\begin{multline}\label{eq:sparse_relation_learning}
\underset{f\in\mathcal{H}, A\in S_{++}^T}{\text{minimize}} \ \ \frac{1}{n} \sum_{i=1}^n V(y_i,f(x_i)) \ + \\
\lambda (\|f\|_\mathcal{H}^2 + \epsilon \operatorname{tr}(A^{-1}) + \mu \operatorname{tr}(A) + (1-\mu) \ \|A\|_{\ell_1})
\end{multline}
where $\|A\|_{\ell_1}=\sum_{t,s}|A_{ts}|$, $V:\mathcal{Y} \times \mathbb{R}^T \to \mathbb{R}_+$ is a loss function and $\lambda>0$, $\epsilon>0$, and $\mu\in[0,1]$ regularization parameters. Here $\mu \in [0,1]$ regulates the amount of desired entry-wise sparsity of $A$ with respect to the low-rank prior $tr(A)$ (indeed notice that for $\mu = 1$ we recover the low-rank inducing framework of~\cite{argyriou08,zhang10}). This prior was empirically observed (see \cite{argyriou08,zhang10}) to indeed encourage information transfer across tasks; the sparsity term can therefore be interpreted as enforcing such transfer to occur only between tasks that are strongly correlated. Finally the term $\epsilon \operatorname{tr}(A^{-1})$ ensures the existence of a unique solution (making the problem strictly convex), and can be interpreted as a preconditioning of the problem (see Sec.~\ref{sec:unsupervised_step}).
Notice that the term $\|f\|_{\mathcal{H}}^2$ depends on both $f$ and $A$ (see Eq.~\ref{eq:norm}), thus making problem~\eqref{eq:sparse_relation_learning} non-separable in the two variables. However, it can be shown that the objective functional is jointly convex in $f$ and $A$ (we refer the reader to the Appendix for a proof of convexity, which extends results in~\cite{argyriou08} to our setting). This will allow in Sec.~\ref{sec:optimization} to derive an optimization strategy that is guaranteed to converge to a global solution.
\subsubsection{Previous Work on Learning the Relations among Tasks}\label{sec:previous_work}
Several methods designed to recover the tasks relations from the data can be formulated using our notation as joint learning problems in $f$ and $A$. Depending on the expected/desired tasks-structure a set of constraints $\mathcal{A} \subseteq S_{++}^T$ can be imposed on $A$ when solving a joint problem as in~\eqref{eq:sparse_relation_learning}:
\begin{itemize}
\item {\bf Multi-task Relation Learning}~\cite{zhang10}. In~\cite{zhang10}, the relaxation $\mathcal{A} = \{A | \operatorname{tr}(A)\leq1\}$ of the low-rank constraint is imposed, enforcing the tasks $f_t$ to span a low-dimensional subspace in $\mathcal{H}_k$. This method can be shown to be approximately equivalent to~\cite{argyriou08}.
\item {\bf Output Kernel Learning}~\cite{dinuzzo11}. Rather than imposing a hard constraint, the authors penalize the structure matrix $A$ with the squared Frobenius norm $\|A\|_F^2$.
\item {\bf Cluster Multi-task Learning}~\cite{jacob08}. Assuming tasks to be organized into distinct clusters, in~\cite{jacob08} a learning scheme to recover such structure is proposed, which consists of imposing a suitable set of spectral constraints $\mathcal{A}$ on $A$. We refer the reader to the supplementary material for further details.
\item {\bf Learning Graph Relations}~\cite{argyriouboh}. Following the interpretation in~\cite{evgeniou05} reviewed in Sec.~\ref{sec:separable_kernels} of imposing similarity relations among tasks in the form of a graph, in~\cite{argyriouboh} the authors propose a setting where a (relaxed) Graph Laplacian constraint is imposed on $A$.
\end{itemize}
\section{Optimization}\label{sec:optimization}
Due to the clear block variable structure of Eq.~\eqref{eq:sparse_relation_learning} with respect to $f$ and $A$, we propose an alternating minimization approach (see Alg.~\ref{alg:bcd}) to iteratively solve the Sparse Kernel MTL problem by keeping fixed one variable at the time. This choice is motivated by the fact that for a fixed $A$, problem~\eqref{eq:sparse_relation_learning} reduces to the standard multi-task learning problem~\eqref{eq:learning_problem}, for which several well-established optimization strategies have already been considered~\cite{alvarez12,micchelli04,evgeniou05,minh11}. The alternating minimization procedure can be interpreted as iterating between steps of supervised learning (finding the $f$ that best fits the input-output training observations) and unsupervised learning (finding the best $A$ describing the tasks structure, which does not involve the output data).
\subsection{Solving w.r.t. $f$ (Supervised Step)}
Let $A\in S_{++}^T$ be a fixed structure matrix. From the Representer theorem (see Sec.~\ref{sec:RKHSvv}) we know that the solution of problem~\eqref{eq:learning_problem} is of the form $f(\cdot) = \sum_{i=1}^n k(\cdot,x_i) A c_i$ with $c_i \in \mathbb{R}^T$. Depending on the specific loss $V$, different methods can be employed to find such coefficients $c_i$. In particular, for the least-square loss a closed form solution can be derived by taking the coefficient vector $c = (c_1^\top,\dots,c_n^\top)^\top \in \mathbb{R}^{nT}$ to be~\cite{alvarez12}:
\begin{equation}
c = (A \otimes K + \lambda I_{nT})^{-1}y
\end{equation}
where $K \in S_+^n$ is the empirical kernel matrix associated to $k$ the scalar kernel, $y \in \mathbb{R}^{nT}$ is the vector concatenating the training outputs $y_1,\dots,y_n \in \mathbb{R}^T$ and $\otimes$ denotes the Kronecker product. A faster and more compact solution was proposed in~\cite{minh11} by adopting Sylvester's method.
\begin{algorithm}[t]
\caption{\textsc{Alternating Minimization}}
\label{alg:bcd}
\begin{algorithmic}
\State {\bfseries Input:} $K$ empirical kernel matrix, $y$ training outputs, $\delta$ tolerance, $V$ loss, $\lambda,\mu,\epsilon$ hyperparameters, $S$ objective functional of problem~\eqref{eq:sparse_relation_learning}.
\State {\bfseries Initialize:} $f_0=0$, $A_0 = I_T$ and $i=0$
\Repeat
\State $f_{i+1} \gets$ \textsc{SupervisedStep} $(V,K,y,A_{i},\lambda)$
\State $A_{i+1} \gets$ \textsc{SparseKernelMTL}$(K,f_{i+1},\mu,\epsilon)$
\State $i \gets i+1$
\Until{$| S(f_{i+1},A_{i+1})-S(f_{i},A_{i}) | < \delta$}
\end{algorithmic}
\end{algorithm}
\subsection{Solving w.r.t the Tasks Structure (Unsupervised Step)}\label{sec:unsupervised_step}
Let $f$ be known in terms of its coefficents $c_1,\dots,c_n\in\mathbb{R}^T$. Our goal is to find the structure matrix $A\in S_{++}^T$ that minimizes problem~\eqref{eq:sparse_relation_learning}. Notice that each task $f_t$ can be written as $f_t(\cdot) = \sum_{i=1}^n k(\cdot,x_i) \langle A_t,c_i \rangle_{\mathbb{R}^T}= \sum_{i=1}^n k(\cdot,x_i)b_{i,t}$ with $b_{i,t} = \langle A_t,c_i\rangle_{\mathbb{R}^T}$. Therefore, from eq.~\eqref{eq:norm} we have
\begin{equation}\label{eq:eq}
\|f\|_\mathcal{H}^2 = \sum_{t,s}^T A_{ts}^{-1} \langle f_t, f_s \rangle_{\mathcal{H}_k}
= \sum_{t,s}^T \sum_{i,j} A_{ts}^{-1} k(x_i,x_j) b_{it}b_{js}
\end{equation}
where we have used the reproducing property of $\mathcal{H}_k$ for the last equality. Eq.~\eqref{eq:eq} allows to write the norm induced by the separable kernel $kA$ in the more compact matrix notation $\|f\|_\mathcal{H}^2 = \operatorname{tr}(B^\top K B A^{-1})$, where $B\in\mathbb{R}^{n \times T}$ is the matrix with $(i,t)$-th element $B_{it} = b_{it}$.
Under this new notation, problem~\eqref{eq:sparse_relation_learning} with fixed $f$ becomes
\begin{equation}\label{eq:actual_srl}
\underset{A \in S_{++}^T}{\text{min.}} \operatorname{tr}(A^{-1} (B^\top K B+\epsilon I_T) ) + \mu \operatorname{tr}(A) + (1-\mu) \ \|A\|_{\ell_1}
\end{equation}
from which we can clearly see the effect of $\epsilon$ as a preconditioning term for the tasks covariance matrix $B^\top K B$.
By employing recent results from the non-smooth convex optimization literature, in the following we will describe an algorithm to optimize the Sparse Kernel MTL problem.
\subsubsection{Primal-dual Splitting Algorithm}
First order proximal splitting algorithms have been successfully applied to
solve convex composite optimization problems, that can be written as the sum of a smooth
component with nonsmooth ones \cite{BauCom11}.
They proceed by splitting, i.e. by activating each term appearing in the sum individually.
The iteration usually consists of a gradient descent-like step determined by the smooth component,
and various proximal steps induced by the nonsmooth terms \cite{BauCom11}.
In the following we will describe one of such methods, derived in \cite{Bang13,Con13},
to solve the Sparse Kernel MTL problem in eq.~\eqref{eq:actual_srl}.
The proposed method is primal-dual, in the sense that it also provides an additional
dual sequence solving the associated dual optimization problem.
We will rely on the sum structure of the objective function, that can be written as
$G(\cdot) + H_1(\cdot) + H_2(L(\cdot))$, with
$G(A) = \lambda\mu \operatorname{tr}(A)$, $H_1(A) = \lambda(1-\mu)\|A\|_{\ell_1}$ and $H_2(A) =\lambda\epsilon \operatorname{tr}(A^{-1}) + i_{S_{++}^T}(A)$, where $i_{S_{++}^T}$ is the indicator function of a $S_{++}^T$ ($0$ on the set $+\infty$ outside) and enforces the hard constraint $A\in S_{++}^T$.
$L$ is a linear operator defined as $L(A) = M A M$, where we have set $M = (B^\top K B + \epsilon I_T)^{-1/2}$.
We recall here that a square root of a PSD matrix $P \in S_{+}^T$ is a PSD matrix $M\in S_+^T$ such that $P = M M$.
Note that $G$ is smooth with Lipschitz continuous gradient, $L$ is a linear operator and both $H_1$ and
$H_2$ are functions for which the proximal operator can be computed in closed form.
We recall that the proximity operator at a point $y\in\mathbb{R}^m$ of a proper, convex and l.s.c. function $H: \mathbb{R}^m \to \mathbb{R} \cup \{+\infty\}$,
is defined as
\begin{equation}\label{eq:proxdef}
\operatorname{prox}_{H} (y) = \operatornamewithlimits{argmin}_{x \in \mathbb{R}^m} \Big\{H(x) + \frac{1}{2} \| x - y \|^2\Big\}.
\end{equation}
It is well known that for any $\eta>0$, the proximal map of the $\ell_1$ norm $\eta \|\cdot\|_{\ell_1}$ is the so-called {\it soft-thresholding} operator $S_\eta(\cdot)$, which can be computed in closed form. The following result provides an explicit closed-form solution also for the proximal map of $H_2$.
\begin{proposition}\label{prop:prox_h2}
Let $Z\in S^T$ with eigendecomposition $Z = U \Sigma U^\top$ with $U \in O^T$ orthonormal matrix and $\Sigma \in S^T$ diagonal. Then
\begin{equation}\label{eq:prox_h2}
\operatorname{prox}_{H_2}(Z) = \operatornamewithlimits{argmin}_{A \in S_{++}^T} \Big\{\operatorname{tr}(A^{-1}) + \frac{1}{2} \|A-Z\|_F^2\Big\}.
\end{equation}
can be computed in closed form as $\operatorname{prox}_{H_2} (Z) = U \Lambda U^\top$ with $\Lambda\in S_{++}^T$ diagonal matrix with $\Lambda_{tt}$ the only positive root of the polynomial $p(\lambda) = \lambda^3 - \lambda^2 \Sigma_{tt} - 1$ with $\lambda \in \mathbb{R}$.
\end{proposition}
\begin{proof}
Note that $H_2$ is convex and lsc. Therefore the proximity operator is well-defined and the functional
in~\eqref{eq:prox_h2} has a unique minimizer. Its gradient is $-A^{-2} + A - Z$, therefore, the first order condition for a matrix $A$ to be a minimizer is
\begin{equation}\label{eq:first_order_condition}
A^3 - A^2Z - I_T = 0
\end{equation}
We show that it is possible to find $\Lambda \in S_{++}^T$ diagonal such that $A_* = U \Lambda U^\top$ solves eq.~\eqref{eq:first_order_condition}. Indeed, for $A$ with same set of eigenvectors $U$ as $Z$, we have that eq.~\eqref{eq:first_order_condition} becomes $U(\Lambda^3 - \Lambda^2 \Sigma - I_T)U^\top = 0$, which is equivalent to the set of $T$ scalar equations $\lambda^3 - \lambda^2 \Sigma_{tt} - 1 = 0$ for $t\in \{1,\dots,T\}$ and $\lambda\in\mathbb{R}$. Descartes rule of sign~\cite{struik86} assures that for any $\Sigma_{tt} \in \mathbb{R}$ each of these polynomials has exactly one positive root, which can be clearly computed in closed form.
\end{proof}
\begin{algorithm}[t]
\caption{\textsc{Sparse Kernel MTL}}
\label{alg:srl}
\begin{algorithmic}
\State {\bfseries Input:} $K \in S_{+}^n$, $B \in \mathbb{R}^{n \times T}$, $\delta$ tolerance, $0 \leq \mu \leq 1, \epsilon > 0$ hyperparameter.
\State {\bfseries Initialize:} $A_0,D_0 \in S_{++}^T$, $M = (B^\top K B + \epsilon I_T)^{-1/2}$, $\sigma = \| M\|^2$ squared maximum eigenvalue of $M$. $i=0$
\Repeat
\State $A_{i+1} \gets \operatorname{prox}_{\frac{1-\mu}{\sigma} \|\cdot\|_{\ell_1}} (A_i - \frac{1}{\sigma} (\mu I_T + M D_i M) )$
\State $P \gets D_i + \frac{1}{\sigma} M (2A_{i+1} - A_i) M$
\State $D_{i+1} \gets P - \operatorname{prox}_{\sigma H_2} (\sigma P)$
\State $i \gets i+1$
\Until{$ \|A_{i+1} - A_{i} \|_F< \delta$ and $ \|D_{i+1} - D_{i}\|_F < \delta$}
\end{algorithmic}
\end{algorithm}
We have the following result as an immediate consequence.
\begin{theorem}[Convergence of Sparse Kernel MTL, \cite{Bang13,Con13}]\label{thm:bangcon}
Let $k$ be a scalar kernel over a space $\mathcal{X}$, $x_1,\dots,x_n\in\mathcal{X}$ a set of points and $f: \mathcal{X} \to \mathbb{R}^T$ a function characterized by a set of coefficients $b_1,\dots,b_n\in\mathbb{R}^T$ so that $f(\cdot) = \sum_{i=1}^n k(\cdot,x_i) b_i$. Set $K\in S_{+}^n$ to be the empirical kernel matrix associated to $k$ and the points $\{x_i\}_{i=1}^n$ and $B \in \mathbb{R}^{n \times T}$ the matrix whose $i$-th row corresponds to the (transposed) coefficient vector $b_i$.
Then, any sequence of matrices $A_t$ produced by Algorithm~\eqref{alg:srl} converges to a global minimizer of the Sparse Kernel MTL problem~\eqref{eq:sparse_relation_learning} (or, equivalently, to \eqref{eq:actual_srl}) for fixed $f$. Furthermore, the sequence $D_t$ converges to a solution of the dual problem of~\eqref{eq:actual_srl}.
\end{theorem}
\subsection{Convergence of Alternating Minimization}
We additionally exploit the sum structure and the regularity properties of the objective functional
in \eqref{eq:sparse_relation_learning} to prove convergence of the alternating minimization
scheme to a global minimum. We rely on the results in \cite{tseng01}. In particular, the following
result is a direct application of Theorem~4.1 in that paper.
\begin{theorem}\label{thm:tseng} Under the same assumptions as in Theorem~\ref{thm:bangcon}, the
sequence $(f_i,A_i)_{i\in\mathbb{N}}$ generated by Algorithm~\ref{alg:bcd} is a
minimizing sequence for Problem~\ref{eq:sparse_relation_learning} and converges to its unique
solution.
\end{theorem}
\begin{proof}
Let $S$ denote the objective function in \eqref{eq:sparse_relation_learning}.
First note that the level sets of $S$ are compact due to the presence of the term $\epsilon\operatorname{tr}(A^{-1}) + \mu \operatorname{tr}(A)$
and that $S$ is continuous on each level set.
Moreover, since $S$ is regular at each point in the interior of the domain and is convex, \cite[Theorem~4.1(c)]{tseng01}
implies that each cluster point of $(f_{i},A_i)_{i\in\mathbb{N}}$ is the unique minimizer of $S$. Then, the sequence itself is convergent
and is minimizing by continuity.
\end{proof}
\subsubsection{A Note on Computational Complexity \& Times}
Regarding the computational costs/number of iterations required for the convergence of the whole Alg.~\ref{alg:bcd}, up to our knowledge the only results available on rates for Alternating Minimization are in~\cite{beck11}. Unfortunately these results hold only for smooth settings.
Notice however that each iteration of Alg~\ref{alg:srl} is of the order of $O(T^3)$, (the eigendecomposition of A being the most expensive operation) and its convergence rate
is $O(1/k)$ with $k$ equal to the number of iterations.
Hence, Alg.~\ref{alg:srl} is not affected by the number $n$ of training samples. On the contrary, the supervised step in Agl.~\ref{alg:bcd} (e.g. RLS or SVM) typically requires the inversion of the kernel matrix $K$ (or some approximation of its inverse) whose complexity heavily depends on $n$ (order of $O(n^3)$ for inversion). Furthermore, the product $BKB^\top$ costs $O(n^2T)$ which, since $n>>T$, is more expensive than Alg.~\ref{alg:bcd}. Thus, with respect to $n$ SKMTL scales exactly as methods such as [2,7,24].
\section{Empirical Analysis}\label{sec:empirical}
We report the empirical evaluation of SKMTL on artificial and real datasets.
We have conducted experiments on both artificially generated and real dataset to assess the capabilities of the proposed Sparse Kernel MTL method to recover the most relevant relations among tasks and exploit such knowledge to improve the prediction performance.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.8\columnwidth]{sparsity.eps}
\caption{Generalization performance (nMSE and standard deviation) of different multi-task methods with respect to the sparsity of the task structure matrix.}\label{fig:sparsity}
\end{center}
\end{figure}
\subsection{Synthetic Data}
We considered an artificial setting that allows us to control the tasks structure and in particular the actual sparsity of the tasks-relation matrix. We generated synthetic datasets of input-output pairs $(x,y) \in \mathbb{R}^d \times \mathbb{R}^T$ according to linear models of the form $y^\top = x^\top U A + \epsilon$ where $U\in\mathbb{R}^{d \times T}$ is a matrix with orthonormal columns, $A \in S_+^T$ is the task structure matrix and $\epsilon$ is zero-mean Gaussian noise with variance $0.1$. The inputs $x\in\mathbb{R}^d$ were sampled according to a Gaussian distribution with zero mean and identity covariance matrix. We set the input space dimension $d=100$ for our experiments.
In order to quantitatively control the sparsity level of the tasks-relation matrix, we randomly generated $A$ so that the ratio between its support (i.e. the number of non-zero entries) and the total number of entries would vary between $0.1$ ($90\%$ sparsity) and $1$ (no sparsity). A Gaussian noise with zero mean and variance $1/10$ of the mean value of the non-zero entries in $A$ was sampled to corrupt the structure matrix entries (hence, the model $A$ was never ``really'' sparse). This was done to reproduce a more realistic scenario.
We generated multiple models and corresponding datasets for different sparsity ratios and number of tasks $T$ ranging from $5$ to $20$. For each dataset we generated respectively $50$ samples for training and $100$ for test. We performed multi-task regression using the following methods: single task learning (STL) as baseline, Multi-task Relation Learning~\cite{zhang10} (MTRL), Output Kernel Learning~\cite{dinuzzo11} (OKL), our Sparse Kernel MTL (SKMTL) and a fixed task-structure multi-task regression algorithm solving problem~\eqref{eq:learning_problem} using the ground truth (GT) matrix $A$ (after noise corruption) for regularization. We chose least-square loss and performed model selection with five-fold cross validation.
\begin{figure}[t]
\begin{center}
\includegraphics[height=.3\columnwidth]{true_A2}\quad\quad
\includegraphics[height=.3\columnwidth]{learned_A2}
\caption{Structure matrix $A$. True (Left) and recovered by Sparse Kernel MTL (Right). We report the absolute value of the entries of the two matrices. The range of values goes from 0 (Blue) to 1 (Red)}
\label{fig:recovered}
\end{center}
\end{figure}
In Figure~\ref{fig:sparsity} we report the normalized mean squared error (nMSE) of tested method with respect to decreasing sparsity ratios. It can be noticed that knowledge of the true $A$ (GT) is particularly beneficial when the tasks share few relations. This advantage tends to decrease as the tasks structure becomes less sparse. Interestingly, both the MTRL and OKL method do not provide any advantage with respect to the STL baseline since we did not design $A$ to be low-rank (or have a fast eigenvalue decay). On the contrary, the SKMTL method provides a remarkable improvement over the STL baseline.
We point out that the large error bars in the plot are due to the high variability of the nMSE with respect to the different (random) linear models $A$ and number of tasks $T$. The actual improvement of the SKMTL over the other methods is however significant.
The results above suggest that, as desired, our SKMTL method is actually recovering the most relevant relations among tasks. In support of this statement we report in Figure~\ref{fig:recovered} an example of the true (uncorrupted) and recovered structure matrix $A$ in the case of $T=10$ and $50\%$ sparsity. As can be noticed, while the actual values in the entries of the two matrices are not exactly the same, their supports almost coincide, showing that SKMTL was able to recover the correct tasks structure.
\subsection{15-Scenes}
We tested SKMTL in a multi-class classification scenario for visual scene categorization, the $15$-scenes dataset\footnote{http://www-cvr.ai.uiuc.edu/ponce\_grp/data/}. The dataset contains images depicting natural or urban scenes that have been organized in $15$ distinct groups and the goal is to assign each image to the correct scene category. It is natural to expect that categories will share similar visual features. Our aim was to investigate whether these relations would be recovered by the SKMTL method and result beneficial to the actual classification process.
We represented images in the dataset with LLC coding~\cite{wang10}, trained multi-class classifiers on $50$, $100$ and $150$ examples per class and tested them on $500$ samples per class. We repeated these classification experiments $20$ times to account for statistical variability.
In Table~\ref{tab:15scenes} we report the classification accuracy of the multi-class learning methods tested: STL (baseline), Multi-task Feature Learning (MTFL)~\cite{argyriou08}, MTRL, OKL and our SKMTL. For all methods we used a linear kernel and least-squares loss as plug-in classifier. Model selection was performed by five-fold cross-validation.
\begin{table}[t]
\begin{center}
\begin{tabular}{lccc}
& \multicolumn{3}{c}{\bf Accuracy (\%) per} \tstrut \bstrut \\
& \multicolumn{3}{c}{\bf \# tr. samples per class} \tstrut \bstrut \\
& $50$ & $100$ & $150$ \tstrut \bstrut \\
\specialrule{.1em}{.05em}{.0em}
\multirow{2}{*}{\bf STL} & $72.23$ & $76.61$ & $79.23$ \tstrut \bstrut \\
& \cellcolor{gray!35} $\pm 0.04$ & \cellcolor{gray!35} $\pm 0.02$ & \cellcolor{gray!35} $\pm 0.01$ \tstrut \bstrut \\
\multirow{2}{*}{\bf MTFL~\cite{argyriou08}} & $73.23$& $77.24$ & $80.11$ \tstrut \bstrut \\
& \cellcolor{gray!35} $\pm 0.08$& \cellcolor{gray!35} $\pm 0.05$ & \cellcolor{gray!35} $\pm 0.03$ \tstrut \bstrut \\
\multirow{2}{*}{\bf MTRL~\cite{zhang10}} & $73.13$& $77.53$ & $80.21$ \tstrut \bstrut \\
& \cellcolor{gray!35} $\pm 0.08$& \cellcolor{gray!35} $\pm 0.04$ & \cellcolor{gray!35} $\pm 0.05$ \tstrut \bstrut \\
\multirow{2}{*}{\bf OKL~\cite{dinuzzo11}} & $72.25$& $77.06$ & $80.03$ \tstrut \bstrut \\
& \cellcolor{gray!35} $\pm 0.03$& \cellcolor{gray!35} $\pm 0.01$ & \cellcolor{gray!35} $\pm 0.01$ \tstrut \bstrut \\
\multirow{2}{*}{\bf SKMTL} & $\mathbf{73.50}$& $\mathbf{78.23}$ & $\mathbf{81.32}$ \tstrut \bstrut \\
& \cellcolor{gray!35} $\pm 0.11$& \cellcolor{gray!35} $\pm 0.06$ & \cellcolor{gray!35} $\pm 0.08$ \tstrut \bstrut \\
\end{tabular}
\end{center}
\caption{Classification results on the $15$-scene dataset. Four multi-task methods and the single-task baseline are compared.}
\label{tab:15scenes}
\end{table}
As it can be noticed, the SKMTL consistently outperforms all other methods. A possible motivation for this behavior, similarly to the synthetic scenario, is that the algorithm is actually recovering the most relevant relations among tasks and using this information to improve prediction. In support of this interpretation, in Figure~\ref{fig:structure} we report the relations recovered by SKMTL in graph form. An edge between two scene categories $t$ and $s$ was drawn whenever the value of the corresponding entry $A_{ts}$ of the recovered structure matrix was different from zero. Noticeably SKMTL seems to identify a clear group separation between natural and urban scenes. Furthermore, also within these two main clusters, not all tasks are connected: for instance office scenes are not related to scenes depicting the exterior of buildings or mountain scenes are not connected to images featuring mostly flat scenes such as highways or coastal regions.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.8\columnwidth]{structure.pdf}
\caption{Tasks structure graph recovered by the Sparse Kernel MTL (SKMTL) proposed in this work on the $15$-scenes dataset.}\label{fig:structure}
\end{center}
\end{figure}
\subsection{Animals with Attributes}\label{sec:awa}
Animals with Attributes\footnote{http://attributes.kyb.tuebingen.mpg.de/} (AwA) is a dataset designed to benchmark detection algorithms in computer vision. The dataset comprises $50$ different animal classes each annotated with $85$ binary labels denoting the presence/absence of different attributes. These attributes can be of different nature such as color (white, black, etc.), texture (stripes, dots), type of limbs (hands, flippers, etc.), diet and so on. The standard challenge is to perform attribute detection by training the system on a predefined set of $40$ animal classes and testing on the remaining $10$. In the following we will first discuss the performance of multi-task approaches in this setting and then investigate how the benefits of multi-task approaches can sometime be dulled by the so-called ``negative transfer'' and how our Sparse Kernel MTL method seems to be less sensitive to such an issue. For the experiments described in the following we used the DECAF features~\cite{Donahue13} recently made available on the Animals With Attribute website.
\subsubsection{Attribute Detection}
We considered the multi-task problem of attribute detection which consists in $85$ classification (binary) tasks. For each attribute, we randomly sampled $50$, $100$ and $150$ examples for training, $500$ for validation and $500$ for test. Results were averaged over $10$ trials. In Table~\ref{tab:awa_complete} we report the Average Precision (area under the precision/recall curve) of the multi-task classifiers tested. As can be noticed for all multi-task approaches, the effect of sharing information across classifiers seems to have a remarkable impact when few training examples are available (the $50$ or $100$ columns in Table~\ref{tab:awa_complete}). As expected, such benefit decreases as the role of regularization becomes less crucial ($150$).
\begin{table}[t]
\begin{center}
\rowcolors{3}{}{gray!35}
\begin{tabular}{lccc}
& \multicolumn{3}{c}{\bf AUC (\%) per \#tr. samples per class} \tstrut \bstrut \\
& 50 & 100 & 150 \tstrut \bstrut \\
\specialrule{.1em}{.05em}{.0em}
{\bf STL} & $57.26 \pm 1.71$ & $60.73 \pm 1.12$ & $64.37 \pm 1.29$ \tstrut \bstrut \\
{\bf MTFL} & $58.11 \pm 1.23$ & $61.21 \pm 1.14$ & $64.22 \pm 1.10$ \tstrut \bstrut \\
{\bf MTRL} & $58.24 \pm 1.84$ & $61.18 \pm 1.23$ & $64.56 \pm 1.41$ \tstrut \bstrut \\
{\bf OKL} & $\mathbf{58.81 \pm 1.18}$ & $62.07 \pm 1.05$ & $64.26 \pm 1.18$ \tstrut \bstrut \\
{\bf SKMTL} & $58.63 \pm 1.73$ & $\mathbf{63.21 \pm 1.43}$ & $64.51 \pm 1.83$ \tstrut \bstrut \\
\end{tabular}
\end{center}
\caption{Attribute detection results on the Animals with Attributes dataset.}
\label{tab:awa_complete}
\end{table}
\subsubsection{Attribute Prediction - Color Vs Limb Shape}
Multi-task learning approaches ground on the assumption that tasks are strongly related one to the other and that such structure can be exploited to improve overall prediction. When this assumption doesn't hold, or holds only partially (e.g. only {\it some} tasks have common structure), such methods could even result disadvantageous (``negative transfer''~\cite{salakhutdinov11}).
The AwA dataset offers the possibility to observe this effect since attributes are organized into multiple semantic groups~\cite{lampert11,jayaraman14}. We focused on a smaller setting by selecting only two group of tasks, namely {\it color} and {\it limb shape}, and tested the effect of training multi-task methods jointly or independently across such two groups. For all the experiments we randomly sampled for each class $100$ examples for training, $500$ for validation and $500$ for test, averaging the system performance over $10$ trials. Table~\ref{tab:awa_small} reports the average precision separately for the color and limb shape groups.
Interestingly, methods relying on the assumption that all tasks share a common structure, such as MTFL, MTRL or OKL, experience a slight drop in performance when trained on all attribute detection tasks together (right columns) rather than separately (left column). On the contrary, SKMTL remains stable since it correctly separates the two groups.
\begin{table}[t]
\begin{center}
\begin{tabular}{lcccc}
& \multicolumn{4}{c}{\bf Area under PR Curve (\%)} \tstrut \bstrut \\
& \multicolumn{2}{c}{\bf Independent} & \multicolumn{2}{c}{\bf Joint} \tstrut \bstrut \\
& {\bf Color} & {\bf Limb} & {\bf Color} & {\bf Limb} \tstrut \bstrut \\
\specialrule{.1em}{.05em}{.0em}
\multirow{2}{*}{\bf STL} & $74.33$ & $68.13$ & $74.33$ & $68.15$ \tstrut \bstrut \\
& \cellcolor{gray!35} $\pm 0.81 $ & \cellcolor{gray!35} $\pm 0.93 $ & \cellcolor{gray!35} $\pm 0.81 $ & \cellcolor{gray!35} $\pm 0.91 $ \tstrut \bstrut \\
\multirow{2}{*}{\bf MTFL} & $75.21$ & $69.41$ & $74.98$ & $69.71$ \tstrut \bstrut \\
& \cellcolor{gray!35} $\pm 0.73 $ & \cellcolor{gray!35} $\pm 1.01 $ & \cellcolor{gray!35} $\pm 1.18 $ & \cellcolor{gray!35} $\pm 0.81 $ \tstrut \bstrut \\
\multirow{2}{*}{\bf MTRL} & $75.17$ & $69.18$ & $74.92$ & $69.73$ \tstrut \bstrut \\
& \cellcolor{gray!35} $\pm 0.53 $ & \cellcolor{gray!35} $\pm 0.64 $ & \cellcolor{gray!35} $\pm 0.78 $ & \cellcolor{gray!35} $\pm 0.75 $ \tstrut \bstrut \\
\multirow{2}{*}{\bf OKL} & $74.52$ & $68.54$ & $74.31$ & $68.44$ \tstrut \bstrut \\
& \cellcolor{gray!35} $\pm 0.44 $ & \cellcolor{gray!35} $\pm 0.61$ & \cellcolor{gray!35} $\pm 0.54 $ & \cellcolor{gray!35} $\pm 0.22 $ \tstrut \bstrut \\
\multirow{2}{*}{\bf SKMTL} & $75.14$ & $69.21$ & $75.23$ & $69.57$ \tstrut \bstrut \\
& \cellcolor{gray!35} $\pm 0.97 $ & \cellcolor{gray!35} $\pm 0.83 $ & \cellcolor{gray!35} $\pm 0.77 $ & \cellcolor{gray!35} $\pm 0.76 $ \tstrut \bstrut \\
\end{tabular}
\end{center}
\caption{Attribute detection on two subsets of AwA. Comparison between methods trained independently or jointly on the two sets show the effects of negative transfer.}
\label{tab:awa_small}
\end{table}
\section{Conclusions}
We proposed a learning framework designed to solve multiple related tasks while simultaneously recovering their structure. We considered the setting of Reproducing Kernel Hilbert Spaces for vector-valued functions~\cite{micchelli04} and formulated the Sparse Kernel MTL as an output kernel learning problem where both a multi-task predictor and a matrix encoding the tasks relations are inferred from empirical data. We imposed a sparsity penalty on the set of possible relations among tasks in order to recover only those that are more relevant to the learning problem.
Adopting an alternating minimization strategy we were able to devise an optimization algorithm that provably converges to the global solution of the proposed learning problem. Empirical evaluation on both synthetic and real dataset confirmed the validity of the model proposed, which successfully recovered interpretable structures while at the same time outperformed previous methods.
Future research directions will focus mainly on modeling aspects: it will be interesting to investigate the possibility to combine our framework, which identifies sparse relations among the tasks, with recent multi-task linear models that take a different perspective and enforce tasks relations in the form of structured
sparsity penalties on the feature space~\cite{jayaraman14,zhong12}.
\newpage
{\small
\bibliographystyle{ieee}
|
{
"timestamp": "2015-04-14T02:12:37",
"yymm": "1504",
"arxiv_id": "1504.03106",
"language": "en",
"url": "https://arxiv.org/abs/1504.03106"
}
|
\section{Background Research}
\label{section:background}
In this section, we discuss existing relevant work on nonverbal behavior prediction using automatically extracted features.
We particularly focus on the social cues that have been shown to be relevant to job interviews and face-to-face interactions~\cite{Huffcutt-jap01}.
We also discuss previous research on automated conversational systems for job interviews, which is one of the potential applications we envision for the proposed prediction framework.
\subsection{Nonverbal Behavior Recognition}
Nonverbal behaviors are subtle, fleeting, subjective, and sometimes even contradictory.
Even a simple facial expression such as a smile can have different meanings, e.g., delight, rapport, sarcasm, and even frustration~\cite{Hoque-AC12}.
Edward Sapir, in 1927, referred to non-verbal behavior as ``an elaborate and secret code that is written nowhere, known by none, but understood by all''~\cite{sapir1985selected}.
Despite years of research, nonverbal behavior prediction remains a challenging problem.
Gottman et al.~\cite{gottman77,gottman-marriage03} studied verbal and non-verbal interactions between newlywed couples and developed mathematical models to predict marriage stability and chances of divorce.
For example, they found that the greatest predictor of divorce is contempt, which must be avoided for a successful marriage.
Hall et al.~\cite{hall09} studied the non-verbal cues in doctor-patient interaction and showed that doctors who are more sensitive to nonverbal skills received higher ratings of service during patient visits.
Ambady et al.~\cite{ambady93} studied the interactions of teachers with students in a classroom and proposed a framework for predicting teachers' evaluations based on short clips of interactions.
However, these prediction frameworks were based on manually labeled behavioral patterns.
Manually labeling non-verbal behaviors is laborious and time consuming, and is often not scalable to large amounts of data.
To allow for the analysis of larger datasets of social interactions, several automated prediction frameworks have been proposed.
Due to the challenges of collecting and analyzing multimodal data, most of the existing automated prediction frameworks focus on a single behavioral modality, such as prosody~\cite{Soman-ICASSP10,Frick-PsyBultn85,Zechner09}, facial expression~\cite{Sandbach12}, gesture~\cite{Castellano07}, and word usage pattern~\cite{Tausczik-jlsp10}.
Analysis based on a single modality is likely to overlook many critical non-verbal behaviors, and hence there has been a growing interest in analyzing social behaviors in more than a single modality.
Ranganath et al.~\cite{Ranganath-emnlp09,Ranganath-CSL13} studied social interactions in speed-dates using a combination of prosodic and linguistic features.
The analysis is based on the SpeedDate corpus, a spoken corpus of approximately 1000 4-min-speed-dates, where each participant rated his/her date in terms of four different conversational styles (awkwardness, assertiveness, flirtatiousness, and friendliness) on a ten point Likert scale.
Given the speech data, Ranganath et al. proposed a computational framework for predicting these four conversational styles using prosodic and linguistic features only, while ignoring facial expressions.
Stark et al.~\cite{stark14} were able to reliably predict the nature of a telephone conversation (business versus personal, familiar versus unfamiliar) using the lexical and prosodic features extracted from as few as 30 words of speech at the beginning of the conversation.
Kapoor et al.~\cite{Kapoor05} and Pianesi et al.~\cite{Pianesi-ICMI08} proposed systems to recognize different social and personality traits by exploiting only prosody and visual features.
Sanchez et al.~\cite{Sanchez-13} proposed a system for predicting eleven different social moods (e.g., surprise, anger, happiness) from YouTube video monologues, which consist of different social dynamics than face to face interactions.
Perhaps the most relevant, Nguyen et al.~\cite{Nguyen2014} proposed a computational framework to predict the hiring decision using nonverbal behavioral cues extracted from a dataset of 62 interview videos.
Nguyen et al. considered only nonverbal cues, and did not include verbal content in their analysis.
Our work extends the current state-of-the-art and generates new knowledge by incorporating three different modalities (prosody, language, and facial expressions), and fifteen different social traits (e.g., friendliness, excitement, engagement), and quantifies the interplay and relative influences of these different modalities for each of the different social traits.
Furthermore, by analyzing the relative feature weights learned by our regression models, we obtain valuable insights about behaviors that are recommended for success in job interviews (Section~\ref{section:recommendation}).
\subsection{Social Coaching for Job Interviews}
Several automated systems have been proposed for coaching the necessary social skills to succeed in job interviews~\cite{Hoque-ubicomp13,Anderson-ACE13,Baur-HBU13}.
Hoque et al.~\cite{Hoque-ubicomp13} developed MACH (My Automated Conversation coacH), which allows users to improve social skills by interacting with a virtual agent.
The MACH system records videos of the user using a webcam and a microphone, and provides feedback regarding several low level behavioral patterns, e.g., average smile intensity, pause duration, speaking rate, pitch variation, etc.
Anderson et al.~\cite{Anderson-ACE13} proposed an interview coaching system, TARDIS, which presents the training interactions as a scenario-based ``serious game''.
The TARDIS framework incorporates a sub-module named NovA (NonVerbal behavior Analyzer)~\cite{Baur-HBU13} that can recognize several lower level social cues: \emph{hands-to-face}, \emph{looking away}, \emph{postures}, \emph{leaning forward/backward}, \emph{gesticulation}, \emph{voice activity}, \emph{smiles}, and \emph{laughter}.
Using videos that are manually annotated with these ground truth social cues, NovA trains a Bayesian Network that can infer higher-level mental traits (e.g., stressed, focused, engaged, etc.). Automated prediction of higher-level traits remains part of their future work.
Our framework (1) quantifies the relative influences of different low level features on the interview outcome, (2) learns regression models to predict interview ratings and the likelihood of hiring using automatically extracted features, and (3) predicts several other high-level personality traits such as engagement, friendliness, and excitement. One of our objectives is to extend the existing automated conversation systems by providing feedback on the overall interview performance and additional high-level personality traits.
\section{Dataset Description}
\label{section:data}
We used the \emph{MIT Interview Dataset}~\cite{Hoque-ubicomp13}, which consists of 138 audio-visual recordings of mock interviews with internship-seeking students from Massachusetts Institute of Technology (MIT).
The total duration of our interview videos is nearly 10.5 hours (on average, 4.7 minutes per interview, for 138 interview videos).
To our knowledge, this is the largest collection of interview videos conducted by professional counselors under realistic settings.
The following sections provide a brief description of the data collection and ground truth labeling.
\subsection{Data Collection}
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{./setup.eps}
\caption{The experimental setup for collecting audio-visual recordings of the mock interviews. Camera \#1 recorded the video and audio of the interviewee, while Camera \#2 recorded the interviewer.}
\label{fig:interviewSetup}
\end{figure}
\subsubsection{Study Setup}
The mock interviews were conducted in a room equipped with a desk, two chairs, and two wall-mounted cameras, as shown in Figure~\ref{fig:interviewSetup}.
The two cameras with microphones were used to capture the facial expressions and the audio conversations during the interview.
\subsubsection{Participants}
Initially, 90 MIT juniors participated in the mock interviews.
All participants were native English speakers.
The interviews were conducted by two professional MIT career counselors who had over five years of experience. For each participant, two rounds of mock interviews were conducted: before and after interview intervention. For the details of interview intervention, please see~\cite{Hoque-ubicomp13}.
Each individual received \$50 for participating.
Furthermore, as an incentive for the participants, we promised to forward the resume of the top 5\% candidates to several sponsor organizations (Deloitte, IDEO, and Intuit) for consideration for summer internships.
We chose sponsor organizations which are not directly tied to any specific major.
After the data collection, 69 (26 male, 43 female) of the 90 initial participants permitted the use of their video recordings for research purposes.
\subsubsection{Procedure}
During each interview session, the counselor asked interviewees five different questions, which were recommended by the MIT Career Services. These five questions were presented in the following order by the counselors to the participants:
\begin{quote}
\emph{Q1. So please tell me about yourself.}\\
\emph{Q2. Tell me about a time when you demonstrated leadership.}\\
\emph{Q3. Tell me about a time when you were working with a team and faced a challenge. How did you overcome the problem?}\\
\emph{Q4. What is one of your weaknesses and how do you plan to overcome it?}\\
\emph{Q5. Now, why do you think we should hire you?} \\
\end{quote}
\vspace{-8pt}
\begin{comment}
\begin{table}
\small
\renewcommand{\arraystretch}{1.2}
\caption{Five questions asked in the interview} \begin{tabular}{|l|l|}
\hline \hline
\textbf{Number} & \textbf{Question}\\
\hline \hline
1 & Please tell me about yourself. \\
2 & Tell me about a time when you demonstrated\\ & leadership.\\
3 & Tell me about a time when you were working\\ & with a team and faced with a challenge.\\ & How did you overcome the problem?\\
4 & What is one of your weaknesses and how do you plan\\ & to overcome it?\\
5 & Now, why do you think we should\\ & hire you? \\
\hline \hline
\end{tabular}
\label{table:interviewQuestions}
\end{table}
\end{comment}
\begin{table}
\renewcommand{\arraystretch}{1.2}
\footnotesize
\caption{List of questions asked to Mechanical Turk workers. First two questions are related to interviewee performances. Others are on various traits of their behavior}
\begin{tabular}{|| l | l || } \hline \hline
Traits & Description \\ \hline \hline
Overall Rating & The overall performance rating. \\
Recommend Hiring & How likely is he to be hired?\\
Engagement & Did he use engaging tone? \\
Excitement & Did he seem excited?\\
Eye Contact & Did he maintain proper eye contact?\\
Smile & Did he smile appropriately?\\
Friendliness & Did he seem friendly?\\
Speaking Rate & Did he maintain a good speaking rate?\\
No Fillers & Did he use too many filler words?\\
& (1 = too many, 7 = no filler words)\\
Paused & Did he pause appropriately?\\
Authentic & Did he seem authentic?\\
Calm & Did he appear calm?\\
Focused & Did he seem focused?\\
Structured Answers & Were his answers structured?\\
Not Stressed & Was he stressed?\\
& (1 = too stressed, 7 = not stressed)\\
Not Awkward & Did he seem awkward?\\
& (1 = too awkward, 7 = not awkward)\\ \hline \hline
\end{tabular}
\label{table:traits}
\end{table}
\vspace{-5pt}
No job description was given to the interviewees.
The five questions were chosen to assess the interviewee's behavioral and social skills.
The interviewers rated the performances of the interviewees by answering 16 assessment questions on a seven point Likert scale. We list these questions in Table \ref{table:traits}. These questions to the interviewers were selected to evaluate the overall performance and behavioral traits of the interviewees. The first two questions -- ``Overall Rating'' and ``Recommend Hiring'' - represent the overall performance. The remaining questions have been selected to evaluate several high-level behavioral dimensions such as warmth (e.g., ``friendliness", ``smiling"), presence (e.g., ``engagement", ``excitement", ``focused"), competence (e.g. speaking rate), and content (e.g., ``structured").
\subsection{Data Labeling}
The subjective nature of human judgment makes it difficult to collect ground truth for interview ratings.
Due to the nature of the experiment, the counselors interacted with each interviewee twice - before and after intervention, and provided feedback after each session.
The process of feedback and the way the interviewees responded to the feedback may have had an influence on the counselor's ratings. In order to remove the bias introduced by the interaction, we used Amazon Mechanical Turk workers to rate the interview performance.
The Mechanical Turkers used the same questionnaire to assess the ratings as listed in Table~\ref{table:traits}.
Apart from being less affected by bias, the Mechanical Turk workers could pause and replay the video, allowing them to rate more thoroughly.
The Turkers' ratings are more likely to be similar to ``audience'' ratings, as opposed to ``expert ratings".
In order to collect ground truth ratings for interviewee performances, we first selected 10 Turkers out of 25, based on how well they agreed with the career counselors on the five control videos.
Out of these 10 selected Turkers, one did not finish all the rating tasks, leaving us with 9 ratings per video.
We automatically estimated the quality of individual workers using an EM-style optimization algorithm, and estimated a weighted average of their scores as the ground truth ratings, which were used in our prediction framework.
One of our objectives was to model the temporal relationships among the individual interview questions and the overall ratings.
To accomplish this, we performed a second phase of labeling.
We hired a different set of 5 Turkers for rating the performances of an interviewee in each of the five interview question separately.
This was done by splitting each interview video into five different segments, where each segment corresponds to one of the interview questions.
The video segments were shuffled so that each Turker would rate the segments in a random order.
These per-question ratings were used only to analyze the temporal variation in the ratings and measure how the temporal order of the questions correlates with the ratings for entire interview.
\section{Discussion and Conclusion}
\label{section:discussions}
We present an automated prediction framework for quantifying social skills for job interviews.
The proposed model shows encouraging results and predicts human interview ratings with correlation $r > 0.65$ and AUC $\sim 0.80$ (compared to the baseline AUC $= 0.50$).
Several traits such as engagement, excitement, and friendliness were predicted with even higher accuracy ($r \sim 0.75$, AUC $> 0.85$).
One of our immediate next steps will be to integrate the proposed prediction module with existing automated conversational systems such as MACH to allow valuable real-time feedback to the users.
To our knowledge, the interview dataset used in our experiments is the largest collection of job interview videos, collected under reasonably realistic settings.
The interviews are conducted by professional career counselors.
We included the questions that would be relevant in most real-world job interviews.
Despite efforts to record interviews in realistic settings, we do need to acknowledge several caveats and trade-offs.
All the participants in our dataset were MIT undergraduates, all of junior status, which may introduce a selection bias in our data.
In future, we plan to conduct a more comprehensive study over a more general and diverse population group.
We deliberately chose not to specify a job description to encourage larger number of student participants.
At the time of the study, there were nearly 1000 junior students present at MIT, and nearly 30\% were international students.
Out of the remaining 700 native English speaking juniors, we were able to recruit 90, which would have been difficult if we had limited our study to a specific job description.
However, in the absence of a specific job description, the ground truth ratings may not necessarily correspond to actual hiring decisions, and may show a stronger bias towards non-verbal cues, as there is no specific skill requirements.
Furthermore, our mock interviews may lack the stress present in a real job interviews.
Although we promised to forward the resumes of the top 5\% candidates to several sponsor organizations, the incentive was not as strong as an actual job offer.
In the future, we would like to conduct more controlled experiments with a specific job description and with stronger incentives to induce stress and competition.
We aimed to rate each video with multiple independent judges to avoid personal bias.
As a first step, we recruited Turkers as this was scalable, quick, and less expensive.
To ensure reliable ground truth ratings, each video was rated using 9 Mechanical Turk workers, and aggregated using the EM algorithm taking the reliability of each worker into account.
However, Turkers' ratings may not correspond to professional experts.
In future, we plan to collect ratings from a panel of experts, and re-validate the results.
Interestingly, while training regression models using SVR, we obtained better prediction accuracy using the linear kernel, compared to other non-linear kernels (e.g., quadratic, cubic, or Gaussian kernels).
This may indicate that our features do not exhibit complicated non-linear interactions.
However, the features used in the current models were mostly aggregated features, averaged over the entire duration of the video (e.g., average pitch, average smile intensity).
It is plausible that our smile and intonation while uttering a specific word can be a determinant of the final interview decision.
The current aggregated features are incapable of modeling such temporal interactions.
Modeling fine-grained temporal features across multiple modalities is left as our future work.
The outcome of job interviews often depends on a subtle understanding of the interviewee's response.
In our dataset, we noticed interviews in which a momentary mistake (e.g., the use of a swear word) ruined the interview outcome.
Due to the rare occurrences of such events, it is difficult to model these phenomena, and perhaps anomaly detection techniques could be more effective instead.
Extending our prediction framework for quantifying these diverse and complex cues in job interviews can provide valuable insight and understanding regarding job interviews and human behavior in general.
Caveats aside, the results presented in this article show the importance of including multiple modalities while analyzing our social interactions.
The analysis of the feature weights learned by our prediction models provides quantitative insights to the determinants of successful job interviews.
With the knowledge presented in this article, we could train a system to help underprivileged youth receive feedback on job interviews that require a significant amount of social skills.
The framework could also be expanded to help people with social difficulties, train customer service professionals, or even help medical professionals with telemedicine.
\section{Introduction}
Analysis of non-verbal behavior to predict the outcome of a social interaction has been studied for many years in different domains, with predictions ranging from marriage stability based on interactions between newlywed couples~\cite{gottman77,gottman-marriage03}, to patient satisfaction based on doctor-patient interaction~\cite{hall09}, to teacher evaluation by analyzing classroom interactions between a teacher and the students~\cite{ambady93}.
However, many of these prediction frameworks were based on manually labeled behavioral patterns by trained coders, according to carefully designed coding schemes.
Manual labeling of nonverbal behaviors is laborious and time consuming, and therefore often does not scale with large amounts of data.
As a scalable alternative, several automated prediction frameworks have been proposed based on low-level behavioral features, automatically extracted from larger datasets.
Due to the challenges of collecting and analyzing multimodal data, most of these automated methods focused on a single modality of interaction~\cite{Curhan07,Sandbach12,Castellano07,Soman-ICASSP10}.
In this paper, we address the challenge of automated understanding of multimodal human interactions, including facial expression, prosody, and language.
We focus on predicting social interactions in the context of job interviews for college students, which is an exciting and relatively less explored domain.
\begin{figure}
\label{fig:system}
\centering
\includegraphics[width=0.40\textwidth]{./framework-infoFlow.eps}
\caption{Framework of Analysis. Mechanical Turk workers rated interviewee performance by watching videos of job interviews. Various features were extracted from those videos. A framework was built to predict Turker's rating and to gain insight into the characteristics of a good interview.}
\end{figure}
Job interviews are ubiquitous and play inevitable and important roles in our life and career.
Over many years, social psychologists and career coaches have accumulated knowledge and guidelines for success in job interviews~\cite{Huffcutt-jap01,Posthuma-PerPsy02,Macan-HRMR09}.
Studies in social psychology have shown that smiling, using a louder voice, and maintaining eye contact contribute positively to our interpersonal communications~\cite{Huffcutt-jap01,Macan-HRMR09}.
These guidelines are largely based on intuition, experience, and studies involving manual encoding of nonverbal behaviors on a limited amount of data~\cite{Huffcutt-jap01}.
Automated data-driven quantification of both verbal and non-verbal behaviors simultaneously has not been explored in the context of job interviews.
In this paper, we aim to quantify the determinants of a successful job interview using a computational prediction framework based on automatically extracted features, which takes both verbal speech and non-verbal behaviors into account.
Imagine the following scenario in which two students, John and Matt, were individually asked to discuss their leadership skills in a job interview. John responded with the following:
\begin{quote}
``\emph{One semester ago, I was part of a team of ten students [stated in a loud and clear voice].
We worked together to build an autonomous playing robot.
I led the team by showing how to program the robot.
The students did a wonderful job [conveyed excitement with tone]!
In ten weeks, we made the robot play soccer. It was a lot of fun. [concluded with a smile]''. }
\end{quote}
Matt responded with the following:
\begin{quote}
``\emph{Umm ... [paused for 2 seconds] Last semester I led a group in a class project on robot programming. It was a totally crazy experience. The students did almost nothing until the last moment. ... Umm ... Basically, I had to intervene at that point and led them to work hard. Eventually, this project was completed successfully. [looked away from the interviewer]}".
\end{quote}
Who do you think received higher ratings?
Most would agree that the first interviewee, John, provided the more enthusiastic and engaging answer.
We can easily interpret the meaning of our verbal and nonverbal behavior during face-to-face interactions.
However, we often cannot quantify how the combination of these behaviors affects our interpersonal communications.
Previous research~\cite{Kapoor05} shows that the style of speaking, prosody, facial expression, and language reflect valuable information about one's personality and mental states.
Understanding the relative influence of these individual modalities can provide crucial insight regarding job interviews.
In this paper, we attempt to answer the following research questions by analyzing the audio-visual recordings of 138 interview sessions with 69 individuals:
\begin{itemize}
\item Can we automatically quantify verbal and nonverbal behavior, and assess their role in the overall rating of job interviews?
\item Can we build a computational framework that can automatically predict the overall rating of a job interview given the audio-visual recordings?
\item Can we infer the relative importance of language, facial expressions, and prosody (intonation)?
\item Can we make automated recommendations on improving social traits such as excitement, friendliness, and engagement in the context of a job interview?
\end{itemize}
To answer these research questions, we designed and implemented an automated prediction framework for quantifying the ratings of job interviews, given the audio-visual recordings.
The proposed prediction framework (Figure~\ref{fig:system}) automatically extracts a diverse set of multimodal features (lexical, facial, and prosodic), and quantifies the overall interview performance, the likelihood of getting hired, and 14 other social traits relevant to the job interview process.
Our system is capable of predicting the overall rating of a job interview with a correlation coefficient $r > 0.65$ and AUC = 0.81 (baseline 0.50) on average.
We can also predict different social traits such as engagement, excitement, and friendliness with even higher accuracy ($r \geq 0.75$, AUC $> 0.85$).
Furthermore, we investigate the relative weights of the individual verbal and non-verbal features learned by our regression models, and quantify their relative importance in the context of job interviews.
Our prediction model can be integrated with the existing automated interview coaching systems, such as MACH~\cite{Hoque-ubicomp13}, to provide more intelligent and quantitative feedback.
The interview questions asked in our training dataset are chosen to be independent of any job specifications or skill requirements.
Therefore, the ratings predicted by our model are based on social and behavioral skills only, and they may differ from a hiring manager's opinion, given a specific job.
Parts of the research included in this article have been presented in~\cite{Naim-FG15}.
In this article, we present an improved system by including additional facial features and provide more comprehensive results and analysis.
The remaining structure of the article follows. In Section~\ref{section:background}, we discuss the background research on automated quantification of multimodal nonverbal behaviors. Section~\ref{section:data} describes the interview dataset and the data annotation process via Mechanical Turk.
A detailed discussion of the proposed computational framework, feature extraction, and automated prediction is presented in Section~\ref{section:methods}.
We present our detailed results in Section~\ref{section:results}. Finally, we conclude with our findings and discuss our future work in Section~\ref{section:discussions}.
\section{Prediction Framework}
\label{section:methods}
For the prediction framework, we automatically extracted various features from the videos of the interviews. Then we trained two regression algorithms - SVM and LASSO. The objective of this training is twofold: first, to predict the Turker's ratings on the overall performance and each behavioral trait, and second, to quantify and gain meaningful insights on the relative importance of each modality and the interplay among them.
\subsection{Feature Extraction}
We collected three types of features for each interview video: (1) prosodic features, (2) lexical features, and (3) facial features. We selected these features to reflect the behaviors that have been shown to be relevant in job interviews (e.g., smile, intonation, language content, etc.)~\cite{Huffcutt-jap01}, and also based on the past literature on automated social behavior recognition~\cite{Sanchez-13,Ranganath-emnlp09,Ranganath-CSL13,Zechner09}.
For extracting reliable lexical features, we chose not to use automated speech recognition. Instead, we transcribed the videos by hiring Amazon Mechanical Turk workers, who were specifically instructed to include filler and disfluency words (e.g., ``uh'', ``umm'', ``like'') in the transcriptions.
Our lexical features were extracted from these transcripts.
We also collected a wide range of prosodic and facial features.
\subsubsection{Prosodic Features}
\begin{table}
\footnotesize
\caption{List of prosodic features and their brief descriptions}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{ || l | l ||} \hline \hline
Prosodic Feature & Description \\
\hline \hline
Energy & Mean spectral energy.\\
F0 MEAN & Mean F0 frequency. \\
F0 MIN & Minimum F0 frequency. \\
F0 MAX & Maximum F0 frequency.\\
F0 Range & Difference between F0 MAX and F0 MIN.\\
F0 SD & Standard deviation of F0.\\
Intensity MEAN & Mean vocal intensity. \\
Intensity MIN & Minimum vocal intensity . \\
Intensity MAX & Maximum vocal intensity . \\
Intensity Range & Difference between max and \\
& min intensity. \\
Intensity SD & Standard deviation.\\
F1, F2, F3 MEAN & Mean frequencies of the first 3\\ & formants: F1, F2, and F3.\\
F1, F2, F3 SD & Standard deviation of F1, F2, F3. \\
F1, F2, F3 BW & Average bandwidth of F1, F2, F3. \\
F2/F1 MEAN & Mean ratio of F2 and F1.\\
F3/F1 MEAN & Mean ratio of F3 and F1.\\
F2/F1 SD & Standard deviation of F2/F1.\\
F3/F1 SD & Standard deviation of F3/F1.\\
Jitter & Irregularities in F0 frequency. \\
Shimmer & Irregularities in intensity. \\
Duration & Total interview duration. \\
\% Unvoiced & Percentage of unvoiced region. \\
\% Breaks & Average percentage of breaks. \\
maxDurPause & Duration of the longest pause. \\
avgDurPause & Average pause duration.\\
\hline
\end{tabular}
\label{table:prosodic_features}
\end{table}
Prosody reflects our speaking style, particularly the rhythm and the intonation of speech.
Prosodic features have been shown to be effective for social intent modeling~\cite{Soman-ICASSP10,Frick-PsyBultn85,Zechner09}.
To distinguish between the speech of the interviewer and the interviewee, we manually annotated the beginning and end of each of the interviewee's answers.
We extracted and analyzed prosodic features of the interviewee's speech.
Each prosodic feature is first collected over an interval corresponding to a single answer by the interviewee, and then averaged over all her/his five answers. We used the open-source speech analysis tool PRAAT~\cite{Boersma-Praat09} for prosody analysis.
The important prosodic features include pitch information, vocal intensities, characteristics of the first three formants, and spectral energy, which have been reported to reflect our social traits~\cite{Frick-PsyBultn85}.
To reflect the vocal pitch, we extracted the mean and standard deviation of fundamental frequency F0 (F0 MEAN and F0 SD), the minimum and maximum values (F0 MIN, F0 MAX), and the total range (F0 MAX - F0 MIN).
We extracted similar features for voice intensity and the first 3 formants.
Additionally, we collected several other prosodic features such as pause duration, percentage of unvoiced frames, jitter (irregularities in pitch), shimmer (irregularities in vocal intensity), percentage of breaks in speech, etc.
Table~\ref{table:prosodic_features} shows the complete list of prosodic features.
\subsubsection{Lexical features}
Lexical features can provide valuable information regarding the interview content and the interviewee's personality.
One of the most commonly used lexical features is the unigram counts for each individual word. However, treating unigram counts as features often results in sparse high-dimensional feature vectors, and suffers from the ``curse of dimensionality'' problem, especially for a limited sized corpus.
\begin{table}
\footnotesize
\caption{LIWC Lexical features used in our system.}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{|| l || l ||}
\hline \hline
LIWC Category & Examples \\
\hline \hline
I & \emph{I, I'm, I've, I'll, I'd,} etc. \\
We & \emph{we, we'll, we're, us, our,} etc. \\
They & \emph{they, they're, they'll, them,} etc. \\
Non-fluencies &words introducing non-fluency in \\
& speech, e.g., \emph{uh, umm, well}. \\
PosEmotion & words expressing positive emotions, \\
& e.g., \emph{hope, improve, kind, love}. \\
NegEmotion & words expressing negative emotions, \\
& e.g., \emph{bad, fool, hate, lose}. \\
Anxiety & \emph{nervous, obsessed, panic, shy,} etc. \\
Anger & \emph{agitate, bother, confront, disgust,} etc.\\
Sadness & \emph{fail, grief, hurt, inferior,} etc.\\
Cognitive & \emph{cause, know, learn, make, notice,} etc. \\
Inhibition & \emph{refrain, prohibit, prevent, stop,} etc.\\
Perceptual & \emph{observe, experience, view, watch,} etc. \\
Relativity & \emph{first, huge, new,} etc. \\
Work & \emph{project, study, thesis, university,} etc.\\
Swear & Informal and swear words.\\
Articles & \emph{a, an, the,} etc.\\
Verbs & common English verbs.\\
Adverbs & common English adverbs.\\
Prepositions & common prepositions. \\
Conjunctions & common conjunctions. \\
Negations & \emph{no, never, none, cannot, don't,} etc.\\
Quantifiers & \emph{all, best, bunch, few, ton, unique,} etc.\\
Numbers & words related to number, e.g., \\
& \emph{first, second, hundred,} etc.\\
\hline
\end{tabular}
\label{table:liwc}
\end{table}
We address this challenge with two techniques. First, instead of using raw unigram counts, we employed counts of various psycholinguistic word categories defined by the tool ``Linguistic Inquiry Word Count'' (LIWC)~\cite{Pennebaker-liwc01}.
The LIWC categories include words describing negative emotions (sad, angry, etc.), positive emotions (happy, kind, etc.), different function word categories (articles, quantifiers, pronouns, etc.), and various content categories (e.g., anxiety, insight).
We selected 23 such LIWC word categories, which is significantly smaller than the number
of individual words.
The LIWC categories correlate with various psychological traits, and often provide indications about our personality and social skills~\cite{Tausczik-jlsp10}.
Many of these categories are intuitively related to interview performance. Table~\ref{table:liwc} shows the complete list of the LIWC features used in our experiments.
Although the hand coded LIWC lexicon has proven to be useful for modeling many different social behaviors, the lexicon is predefined and may not cover many important aspects of job interviews.
To address this challenge, we aimed to automatically learn a lexicon from the interview dataset.
We apply the Latent Dirichlet Allocation (LDA)~\cite{Blei-jmlr03} method to automatically learn common topics from our interview dataset.
We set the number of topics to 20.
For each interview, we estimate the relative weights of these learned topics, and use these weights as lexical features.
Similar ideas have been exploited by Ranganath et al.~\cite{Ranganath-emnlp09,Ranganath-CSL13} for modeling social traits in speed dating dataset, but they used deep auto-encoders~\cite{Hinton-AAAS06} instead of LDA.
Finally, we collected additional lexical features that correlate to job interview ratings.
These are features related to our linguistic and speaking skills. Table~\ref{table:wpsec} contains the full list.
Similar speaking rate and fluency features were exploited by Zechner et al.~\cite{Zechner09} in the context of automated scoring of non-native speech in TOEFL practice tests.
\begin{table}
\caption{Additional features related to speaking rate and fluency.}
\renewcommand{\arraystretch}{1.2}
\centering
\begin{tabular} { || l || l ||} \hline \hline
feature Name & Description \\ \hline \hline
wpsec & Words per second. \\
upsec & Unique words per second.\\
fpsec & Filler words per second.\\
wc & Total number of words. \\
uc & Total number of unique words. \\ \hline
\end{tabular}
\label{table:wpsec}
\end{table}
\subsubsection{Facial features}
We extracted facial features for the interviewees from each frame in the video.
First, faces were detected using the Shore~\cite{Froba-FG04} framework.
We trained a classifier to distinguish between neutral and smiling faces. The classifier is trained using the AdaBoost algorithm.
The classifier output is normalized in the range [0,100], where 0 represents no smile, and 100 represents full smile.
Finally, we averaged the smile intensities from individual frames, and used this as a feature in our model.
We also extracted head gestures such as nods and shakes as explained in~\cite{Hoque-ubicomp13}.
In addition to the smile intensity and head gestures (nod and shake), we also extracted a number of other facial features using a Constrained Local Model (CLM)~\cite{saragih2009face} based face tracker\footnote{https://github.com/kylemcdonald/FaceTracker}, as illustrated in Fig~\ref{fig:face_annot}.
The face tracker detects 66 interest points on a face image.
It works by fitting the following parametric shape model~\cite{saragih2009face}\cite{Saragih2011}:
\begin{equation}
\mathbf{x}_i = s\mathbf{R}(\mathbf{\bar{x}}_i + \mathbf{\Psi_i}\mathbf{q})+\mathbf{t},
\end{equation}
where $\mathbf{x}_i$ is the coordinate of $i^{\text{th}}$ interest point and $\mathbf{\bar{x}}_i$ denotes its mean location pre-trained from a large collection of hand-labeled training images.
$\mathbf{\Psi_i}$ denotes the bases of local variations for the $i^{\text{th}}$ interest point.
Each element of the vector $\mathbf{q}$ represents a coefficient corresponding to a basis of local variation.
The parameters $s, \mathbf{R},$ and $t$ corresponds to the global transformations associated with scaling, rotation, and translation respectively.
The face tracker adjusts the model parameters $p=\{s,\mathbf{R},\mathbf{q},\mathbf{t}\}$ so that each of the mean interest points ($\mathbf{\bar{x}}_i$) fits best to its corresponding point ($\mathbf{x}_i$) on the test face.
While extracting features from these tracked interest points, we want to disregard the global transformations (translation, rotation, and scaling), and consider only the local transformations, which provide useful information regarding our facial expressions.
After the face tracker converges to an optimal estimate of the parameters, we recalculate each of the interest points $\mathbf{x}_i$ by applying the local transformations only, while disregarding the global transformations ($s, \mathbf{R},$ and $t$).
Mathematically, we calculate the following shape model from the optimal parameters obtained from the face tracker:
\begin{equation}
\mathbf{\hat{x}}_i = (\mathbf{\bar{x}}_i + \mathbf{\Psi_i}\mathbf{q})
\end{equation}
Once we find $\mathbf{\hat{x}}_i$, we calculate the distances between the corresponding interest points to find out the features OBH (outer eye-brow height), IBH (inner eye-brow height), OLH (outer lip height), and ILH (inner lip height), eye opening, and LipCDT (lip corner distance), as illustrated in Figure~\ref{fig:face_annot}.
By disregarding the global transformation parameters, the extracted facial features are invariant to global translations, rotations, and scaling variations.
In addition to the features shown in Figure~\ref{fig:face_annot}, we separately incorporated three head pose features (Pitch, Yaw and Roll), based on the corresponding elements of the rotation matrix $\mathbf{R}$.
\begin{figure}
\centering
\includegraphics[width=0.35\textwidth]{./face_Annotated.eps}
\caption{Illustration of facial features: OBH (outer eye-brow height), IBH (inner eye-brow height), OLH (outer lip height), ILH (inner lip height), eye opening, and LipCDT (lip corner distance).}
\label{fig:face_annot}
\end{figure}
\subsubsection{Feature Normalization}
We concatenate the three types of features described above and obtain one combined feature vector. To remove any possible bias related to the range of values associated with a feature, we normalized each feature to have zero mean and unit variance, which allows treating all the features uniformly.
\subsection{Ground Truth Ratings and Turker Quality Estimation}
\label{section:turkerquality}
We aim to automatically estimate the reliability of each Turker, and the ground truth ratings based on the Turkers' ratings. We adapt a simplified version of the existing latent variable models~\cite{raykar2010learning} that treat each Turker's reliability and ground truth ratings as latent parameters, estimate their values using an EM-style iterative optimization technique.
Let us assume an input training dataset $\mathcal{D} = \{\mathbf{x}_i, y_i\}_{i=1}^N$
containing $N$ feature vectors $\mathbf{x}_i$ (one for each interview video), for which the ground truth label $y_i$ is not known.
Instead we acquire subjective labels $\{y_i^1, \ldots, y_i^K \}$ from $K$ Turkers on seven point Likert scale, i.e., $y_i^j \in \{1,2,\ldots, 7\}$.
Given this dataset $\mathcal{D}$, our goal is to learn the true rating ($y_i$) and the reliability of each worker ($\lambda_j$).
To simplify the estimation problem, we assume the Turkers' ratings to be real numbers, i.e., $y_i^j \in \mathbb{R}$. We also assume that each Turker's rating is a noisy version of the true rating $y_i \in \mathbb{R}$, perturbed via additive Gaussian noise. Therefore, the probability distribution for the $y_i^j$:
\begin{equation}
Pr[y_i^j | y_i, \lambda_j] = \mathcal{N} (y_i^j | y_i, 1/\lambda_j)
\end{equation}
where $\lambda_j$ is the unknown inverse-variance and the measure of reliability for the $j^{th}$ Turker. By taking logarithm on both side and ignoring constant terms, we get the log-likelihood function:\
\begin{equation}
L = \sum_{i=1}^N \sum_{j=1}^K \left[ \frac{1}{2} \log{\lambda_j} - \frac{\lambda_j}{2} (y_i^j - y_i)^2 \right]
\end{equation}
The log-likelihood function is non-convex in $y_i$ and $\lambda_j$ variables. However, if we fix $y_i$, the log-likelihood function becomes convex with respect to $\lambda_j$, and vice-versa. Assuming $\lambda_j$ fixed, and setting $\frac{\partial L}{\partial y_i} = 0$, we obtain the update rule:
\begin{equation}
y_i = \frac{ \sum_{j=1}^K \lambda_j y_i^j} { \sum_{j=1}^K \lambda_j }
\end{equation}
Similarly, assuming $y_i$ fixed, and setting $\frac{\partial L}{\partial \lambda_j} = 0$, we obtain the update rule:
\begin{equation}
\lambda_j = \frac{ \sum_{i=1}^N (y_i^j - y_i)^2 } {N}
\end{equation}
We alternately apply the two update rules for $y_i$ and $\lambda_j$ for $i = 1, \ldots, N$ and $j = 1, \ldots, K$ until convergence. After convergence, the estimated $y_i$ values are treated as ground truth ratings and used for training our prediction models.
\subsection{Score Prediction from Extracted Features}
Using the features described in the previous section, we train regression models to predict the interview scores. We also train models to predict other interview-specific traits such as excitement, friendliness, engagement, awkwardness, etc. We experimented with many different regression models: Support Vector Machine Regression (SVR)~\cite{Smola-04}, Lasso~\cite{Tibshirani-Lasso96}, L$_1$ Regularized Logistic Regression, Gaussian Process Regression, etc. We will only discuss SVR and Lasso, which achieved the best results with our dataset.
\subsubsection{Support Vector Regression (SVR)}
The Support Vector Machine (SVM) is a widely used supervised learning method. In this paper, we focus on the SVMs for regression, in order to predict the performance ratings from interview features. Suppose we are given a training data $\{ (\mathbf{x}_1, y_1), \ldots ,(\mathbf{x}_N, y_N))\}$, where $\mathbf{x}_i \in \mathbb{R}^d$ is a $d$-dimensional feature vector for the $i^{th}$ interview in the training set. For each feature vector $\mathbf{x}_i$, we have an associated value $y_i \in \mathbb{R}_+$ denoting the interview rating. Our goal is to learn the optimal weight vector $\mathbf{w} \in \mathbb{R}^d$ and a scalar bias term $b \in \mathbb{R}$ such that the predicted value for the feature vector $\mathbf{x}$ is: $\hat{y} = \mathbf{w}^T \mathbf{x} + b$. We minimize the following objective function:
\begin{equation}
\begin{aligned}
& \underset{\mathbf{w}, \xi_i, \hat{\xi}_i, b}{\text{minimize}}
& & \frac{1}{2} \| \mathbf{w} \|^2 + C \sum_{i = 1}^N (\xi_i + \hat{\xi}_i)\\
& \text{subject to}
& & y_i - \mathbf{w}^T \mathbf{x}_i - b \leq \epsilon + \xi_i, \ \forall i \\
&&& \mathbf{w}^T \mathbf{x}_i + b - y_i \leq \epsilon + \hat{\xi}_i, \ \forall i \\
&&& \xi_i, \hat{\xi}_i \geq 0, \ \forall i \\
\end{aligned}
\end{equation}
The $\epsilon \geq 0$ is the precision parameter specifying the amount of deviation from the true value that is allowed, and $(\xi_i, \hat{\xi}_i)$ are the slack variables to allow deviations larger than $\epsilon$. The tunable parameter $C > 0$ controls the tradeoff between goodness of fit and generalization to new data. The convex optimization problem is often solved by maximizing the corresponding dual problem. In order to analyze the relative weights of different features, we transform it back to the primal problem and obtain the optimal weight vector $\mathbf{w}^*$ and bias term $b^*$. The relative importance of the $j^{th}$ feature can be interpreted by the associated weight magnitude $|w_j^*|$.
\subsubsection{Lasso}
The Lasso regression method aims to minimize the residual prediction error in the presence of an $L_1$ regularization function. Using the same notation as the previous section, let the training data be $\{ (\mathbf{x}_1, y_1), \ldots ,(\mathbf{x}_N, y_N))\}$. Let our linear predictor be of the form: $\hat{y} = \mathbf{w}^T \mathbf{x} + b$. The Lasso method estimates the optimal $\mathbf{w}$ and $b$ by minimizing the following objective function:
\begin{equation}
\begin{aligned}
& \underset{\mathbf{w},b}{\text{minimize}}
& &\sum_{i=1}^N \left( y_i - \mathbf{w}^T \mathbf{x}_i - b \right)^2 \\
& \text{subject to}
& & \| \mathbf{w} \|_1 \leq \lambda \\
\end{aligned}
\end{equation}
where $\lambda > 0$ is the regularization constant, and $\| \mathbf{w} \|_1 = \sum_{j=1}^d |w_j|$ is the $L_1$ norm of $\mathbf{w}$. The $L_1$ regularization is known to push the coefficients of the irrelevant features down to zero, thus reducing the predictor variance. We control the amount of sparsity in the weight
vector $\mathbf{w}$ by tuning the regularization constant $\lambda$.
\subsection{Prediction using Automated Features}
\label{section:prediction}
\subsubsection{Prediction Accuracy using Trained Models}
\label{section:accuracy}
Given the feature vectors associated with each interview video, we would like to provide feedback to users about their overall performance in the interview, the likelihood of getting an offer, and insights into other personality traits that are relevant for job interviews. We train regression models for predicting ratings for a total of 16 traits or rating categories (as shown in Table~\ref{table:traits}).
The entire dataset has a total of 138 interview videos (for the 69 participants, 2 interviews for each participant). We used 80\% of the videos for training, and the remaining 20\% for testing. To avoid any artifacts related to how we split the data, we performed 1000 random trials. In each trial, we randomly select 80\% videos for training, and use the rest for testing.
We report our results averaged over these 1000 independent trials.
In each trial, we trained 16 different regression models for all 16 traits.
For each of the traits, we used exactly the same set of features.
The model automatically learned the weights for individual features for each trait.
We measure prediction accuracy by the correlation coefficients between the true ratings and predicted ratings in the test set.
Figure~\ref{fig:results} displays the correlation coefficients for different traits, both with SVM and Lasso.
The traits are shown in the order of their correlation coefficients.
We observe that we can predict several traits with 0.75 or higher correlation coefficients: engagement, excitement, and friendliness.
Furthermore, we performed well in predicting overall performance and hiring recommendation scores ($r ~ 0.70$ for SVM), which are the two most important scores for interview decision.
\begin{figure}[h]
\centering
\includegraphics[width=0.52\textwidth]{./correlation_LASSO_SVM.eps}
\caption{Regression coefficients using two different methods: Support Vector Machine (SVM) and Lasso.}
\label{fig:results}
\end{figure}
We also evaluate the learned regression models for a two-class classification task.
For each trait, we split the interviews into two groups by the median value for that trait.
Any interview with a score higher than the median value for a particular trait is considered to be in the positive class (for that trait), and the rest are placed in the negative class.
We then vary the threshold on the predicted scores by our regression models in the range $[1,7]$, and estimate the area under the Receiver Operator Curve (ROC).
The baseline area under the curve (AUC) value is 0.50, as we split the classes by the median value.
The AUC values for the learned models are presented in Table~\ref{table:auc}.
Again, we observe high accuracies for engagement, excitement, friendliness, hiring recommendation, and the overall score ($AUC > 0.80$ for SVM).
\begin{table}
\centering
\caption{The average area under the ROC curve.}
\label{table:auc}
\begin{tabular}{|| l || l || l ||} \hline \hline
Trait & SVM & Lasso \\ \hline \hline
Excited& 0.904 & 0.885 \\
Engagement & 0.858 & 0.850 \\
Smiled & 0.845 & 0.845\\
Friendly & 0.824 & 0.793\\
Recommend Hiring & 0.815 & 0.796 \\
Structured Answers & 0.812 & 0.799 \\
Not Awkward & 0.808 & 0.787 \\
Overall & 0.805 & 0.777\\
No Fillers & 0.803 & 0.855\\
Focused & 0.791 & 0.677 \\
Paused & 0.749 & 0.749 \\
Authentic & 0.688 & 0.642\\
Eye Contact & 0.676 & 0.622\\
Calm & 0.651 & 0.669\\
Speaking Rate & 0.608 & 0.546 \\
Not Stressed & 0.604 & 0.572\\ \hline
\end{tabular}
\end{table}
When we examine the traits with lower prediction accuracy, we observe: (1) either we have low interrater agreement for these traits, which indicates unreliable ground truth data (e.g., calm, stressed, structured answer, pause, etc.), or (2) we lack key features necessary to predict these traits (e.g., eye contact). In the absence of eye tracking information (which is very difficult to obtain automatically), we do not have enough informative features to predict eye contact.
\subsubsection{Feature Analysis}
\label{section:featureanalysis}
The relative weights of individual features in our regression model can provide valuable insights on essential constituents of a job interview.
To analyze this, we observed the features with highest weights for the SVM and the Lasso model.
We considered five traits with high accuracy: overall score, recommend hiring, excitement, engagement, and friendliness.
We considered the top twenty features in the order of descending weight magnitude, and estimate the summation of the weight magnitudes of the features in each of the three categories: prosodic, lexical, and facial features.
The relative proportion of prosodic, lexical and facial features are illustrated in Figure~\ref{fig:relWeights}, which shows that both SVM and Lasso assign higher weights to prosodic features while predicting engagement and excitement.
This indicates that engagement and excitement are expressed through prosodic features, which agrees with our intuition.
For both models, the relative weights of features for predicting the ``overall rating" and ``recommend hiring" are similar, which is expected, as these two traits are highly correlated (Figure~\ref{fig:relativeImportanceOfTraits}).
Since we had smaller number of facial features, the relative weights for facial features is much lower.
However, facial features, particularly the smile, were found significant for predicting friendliness. This result provides a solid ground for claiming that smile is very important in order to appear friendly.
\begin{figure*}
\centering
\subfigure[Relative proportion of the top twenty prosodic, lexical and facial (smile) features as learned by SVM and LASSO classifiers. The weights semantically match our perceptions on the traits]{
\includegraphics[width=0.80\textwidth]{./relative_features.eps}
\label{fig:relWeights}}
\subfigure[Correlation Coefficients for SVM, for different combinations of facial (F), prosodic (P), and lexical (L) features]{
\includegraphics[width=0.95\textwidth]{./feature_combination.eps}
\label{fig:featurecomb}}
\caption{Analysis of relative importance of facial, prosodic, and lexical features. }
\end{figure*}
Figure~\ref{fig:featurecomb} shows the importance of using multimodal features for predicting social traits in job interviews.
In most cases, the best correlation coefficient was obtained when we incorporated all three modalities.
Although lexical features were critical for predicting overall ratings and likelihood of getting hired, they were not strong predictors of excitement, engagement, and friendliness. Prosodic features played important role for predicting all the five traits, indicating that our speaking style plays a critical role in job interviews.
\subsubsection{Recommendation from our Framework}
\label{section:recommendation}
To better understand the recommended behavior in job interviews, we analyze the feature weights in our regression model. The weights with positive signs and higher magnitudes can potentially indicate elements of a successful job interview.
The negative weights, on the other hand, indicates behaviors we should avoid.
We sort the features by the magnitude of their weights and list the top twenty features (excluding the topic features) in Table~\ref{table:features:svm}.
We see from this table that people having higher speaking rate (higher words per second (\emph{wpsec}), total number of words (\emph{wc}), and total number of unique words (\emph{uc}), etc.) are perceived as better candidates in a job interview. People who speak more fluently and use less filler words (lower number of filler words per second (\emph{fpsec}), total number of filler words (\emph{Fillers}), total number non-fluency words (\emph{Non-fluencies}), less unvoiced region in speech (\emph{\%Unvoiced}), and fewer breaks in speech (\emph{\%Breaks})) are perceived as better candidates. We also find that higher interview score correlates with higher usage of words in LIWC category \emph{They} (e.g. they, they'll, them, etc.) and lower usage of words related to \emph{I}.
The overall interview performance and likelihood of hiring correlate positively with proportion of positive words, and negatively with proportions of negative words, which agrees with our experience.
Individuals who smiled more performed better in job interviews. Finally, those speaking with a higher proportion of quantifiers (e.g., best, every, all, few), perceptual words (e.g. see, observe, know), and other functional word classes (articles, prepositions, conjunctions) obtained higher scores in interview.
As we saw earlier, features related to prosody and speaking style are more important to appear excited and engaged.
Particularly the amplitude, variations in the voice intensity, and the first 3 formants had high positive weights in our prediction model. Finally, besides smiling, people who spoke more words related to \emph{``We''} than \emph{``I''} were perceived as friendlier.
\input{feature_table_svm}
\section{Results}
\label{section:results}
We organize our results in two sections. First, we analyze the ratings by Mechanical Turk workers (Section~\ref{section:turkresults}). The quality and reliability of Turkers' ratings are assessed by observing how well the Turkers agree with each other (Section~\ref{section:interrater}).
In addition, we identify which traits are important to succeed in job interviews by measuring the correlations of the ratings for individual traits with the overall ratings (Section~\ref{section:correlation:traits}).
Furthermore, we examine the correlations between the ratings for individual video segments with that for the entire videos. This allowed us to evaluate the temporal patterns in job interviews (Section~\ref{section:correlation:temporal}).
In Section~\ref{section:prediction}, we present the prediction accuracies for the trained regression models (SVR and Lasso) based on automatically extracted features, and analyze the relative influence of different modalities and features on prediction accuracy.
\subsection{Analysis of Mechanical Turk Dataset}
\label{section:turkresults}
\subsubsection{Inter-Rater Agreement}
\label{section:interrater}
\begin{figure}
\centering
\epsfig{file=./interRater.eps, width = 0.5\textwidth}
\caption{The inter-rater agreement among the turkers, measured by the Krippendorff's Alpha (varies in the range $[-1,1]$) and the average one-vs-rest correlation of their ratings (range $[-1,1]$).}\label{fig:InterRaterAgreement}
\end{figure}
To assess the quality of the ratings, we calculate Krippendorff's Alpha \cite{Krippendorff1970} for each trait.
In this case, Krippendorff's Alpha is more meaningful than the frequently used Fleiss' Kappa~\cite{Fleiss98}, as the ratings are ordinal values (on a 7-point Likert scale).
The value of Krippendorff's Alpha can be any real number in the range $[-1,1]$, with 1 being the perfect agreement and -1 being absolute disagreement among the raters.
We also estimate the correlation of each Turker's rating with the mean rating by the other Turkers for each trait.
Figure \ref{fig:InterRaterAgreement} shows that some traits have relatively good inter-rater agreement among the Turkers (e.g., ``engagement'', ``excitement'', ``friendliness''). Some other traits such as: ``stress", ``authenticity'', ``speaking rate'', and ``pauses'' have low inter-rater agreement. This may be because the Turkers were not in a position to judge those categories with the video data only.
\subsubsection{Correlation among the Behavioral Traits}
\label{section:correlation:traits}
We are interested in identifying the traits that correlate highly with overall ratings.
This knowledge can help interviewees understand the most important behavioral traits in job interviews.
We plot the mutual information and correlation between various ratings given by the Mechanical Turk workers and the overall rating of the interviewee performance in Figure~\ref{fig:relativeImportanceOfTraits}.
\begin{figure}
\centering
\epsfig{file=./relationAmongTraits.eps, angle = -90, width = 0.5\textwidth}
\caption{Correlation and Mutual information between overall rating and ratings on other traits.}
\label{fig:relativeImportanceOfTraits}
\end{figure}
The first bar in Figure~\ref{fig:relativeImportanceOfTraits} represents whether the rater will recommend hiring the interviewee.
It is another form of the overall rating and shows high correlation and mutual information with the overall rating.
It is evident from the plot that the most important trait in an interview is to stay focused.
This trait shows a 73\% correlation with the overall rating.
Some other top traits include possessing an engaging tone, not appearing awkward, being excited, and displaying an appropriate smile.
The mutual information and correlation coefficient closely follow the patterns.
This plot gives us an insight into what constitutes a good interview.
\subsubsection{First (and Last) Impression Matters}
\label{section:correlation:temporal}
We would like to understand how the performance in different interview questions during an interview affects the overall rating.
To understand this temporal relationship, we calculated the correlation and mutual information between the ratings for each individual interview question and the ratings for the entire videos.
In Figure \ref{fig:first_impression}, we plot this relationship.
It is evident from Figure~\ref{fig:firstImpressionMatters_overall} that performance on the first question correlates most with the overall performance.
After the first question, the correlation gradually decays.
We can interpret this result as follows: If an interviewee performs well for the first question, it is more likely that he/she will end up receiving an above average rating.
It is true in the opposite case as well; if an interviewee performs poorly in the first question, he/she is more likely to receive a poor overall rating. This finding is also supported by existing evidence from psychological point of view~\cite{Dougherty94,Curhan07}.
\begin{figure*}
\centering
\subfigure[Correlation and Mutual Information for overall performance]{
\includegraphics[width=0.3\textwidth]{./firstImpressionMatters_overall}
\label{fig:firstImpressionMatters_overall}
}
\subfigure[Traits following patterns similar to the overall performance]{
\includegraphics[width=0.29\textwidth]{./firstImpressionMatters_good}
\label{fig:firstImpressionMatters_good}
}
\subfigure[Traits not following patterns]{
\includegraphics[width=0.32\textwidth]{./firstImpressionMatters_bad}
\label{fig:firstImpressionMatters_bad}
}
\caption{Correlation between ratings of different segments and the rating on the whole interview.}
\label{fig:first_impression}
\end{figure*}
A similar pattern of \emph{first impression matters} holds for ratings on various other traits of the interviewee's behavior, such as whether he/she was excited, smiled, maintained eye contact, talked in engaging tone, or even appeared friendly.
Figure \ref{fig:firstImpressionMatters_good} illustrates this.
We notice from this figure that there is a sudden spike in correlation for the last question.
This indicates the fact that, although the first question matters the most, the interviewee can significantly change the interviewer's perception during the response to the final question.
Figure \ref{fig:firstImpressionMatters_bad} shows some traits (e.g., pause, calmness, stress) do not follow the pattern discussed above.
However, they have very low correlation values to begin with.
We believe it is difficult for Mechanical Turk workers to accurately judge these traits as these judgments demand considerable concentration.
We need to be cautious while interpreting this result.
Although the ratings for the first question had maximum correlation with the overall ratings for the entire interview, we can not say whether it is due to the temporal order or the verbal content of the question itself.
However, we would like to emphasize that our mock interviews start with a question about interviewee's background, which is consistent with many real-world job interviews.
\section{Appendix heading}
\ifCLASSOPTIONcompsoc
\section*{Acknowledgments}
\else
\section*{Acknowledgment}
\fi
The authors would like to thank Leon Weingard for helping with transcribing the audio, and Michaela Kerem for her extensive feedback.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2015-04-15T02:04:48",
"yymm": "1504",
"arxiv_id": "1504.03425",
"language": "en",
"url": "https://arxiv.org/abs/1504.03425"
}
|
\section{Introduction}
Systems such as Pig or Hive that implement SQL or relational algebra over MapReduce have mechanisms to deal with joins where there is significant skew; i.e.,
certain values of the join attribute(s) appear very frequently
(see, e.g.,~\cite{hive,pig,
Gates09}.
These systems use a two-round algorithm, where the first round identifies the {\em heavy hitters (HH)}, those values of the join attribute(s) that occur in at least some given fraction of the tuples. In the second round, tuples that do not have a heavy-hitter for the join attribute(s) are handled normally. That is, there is one reducer\footnote{In this paper, we use the term {\em reducer} to mean the application of the Reduce function to a key and its associated list of values. It should not be confused with a Reduce task, which typically executes the Reduce function on many key and their associated values.}
for each key, which is associated with a value of the join attribute(s).
Since the key is not a heavy hitter, this reducer handles only a small fraction of the tuples, and thus will not cause a problem of skew.
For tuples with heavy hitters, new keys are created that are handled along with the other keys (normal or those for other heavy hitters) in a single MR job.
The new keys in
these systems are created with a simple technique as in the following example:
\vspace*{-.3cm}
\begin{example}
\label{1-ex}
We have to compute the join $R(A,B) \bowtie S(B,C)$ using a given number, $k$, of reducers.
Suppose value $b$ for attribute $B$ is identified as a heavy hitter.
Suppose there are $r$ tuples of $R$ with $B=b$ and there are $s$ tuples of $S$ with $B=b$. Suppose also for convenience that $r>s$.
The distribution to $k$ buckets/reducers is done in earlier approaches by partitioning the data of one of the relations in $k$ buckets (one bucket for each reducer) and sending the data of the other relation to all reducers. Of course since $r>s$, it makes sense to choose relation $R$ to partition. Thus
values of attribute $A$ are hashed to $k$ buckets, using a hash function $h$, and each tuple of relation $R$ with $B=b$ is sent to one reducer -- the one that corresponds to the bucket that the value of the first argument of the tuple was hashed. The tuples of $S$ are sent to all the $k$ reducers.
Thus the number of tuples transferred from mappers to reducers is $ r + ks$.
\end{example}
\vspace*{-.3cm}
The approach described above appears not only in Pig and Hive, but dates back to~\cite{WolfDY93}. The latter work, which looked at a conventional parallel implementation of join rather than a MapReduce implementation, uses the same (non-optimal) strategy of choosing one side to partition and the other side to replicate.
In particular, these techniques are not optimal with respect to {\em communication cost} (i.e., the number of inputs
transferred from the mappers to the reducers~\cite{AfratiSSU13,SOCC}).
{\bf Our contribution:}
In Example~\ref{2-way-ex} we show how we can do significantly better than the standard technique of Example~\ref{1-ex}. In the rest of the paper we show how the idea
in Example~\ref{2-way-ex} can be
extended to apply on any multiway join and for any number of heavy hitters.
In particular, we show how to adapt Shares algorithm \cite{AfUl} to find a solution
that minimizes communication cost in the case there are heavy hitters.
\vspace*{-.5cm}
\begin{example}
\label{2-way-ex}
We take again the join $R(A,B) \bowtie S(B,C)$.
We partition the tuples of $R$ with $B=b$ into $x$ groups and we also partition the tuples of $S$ with $B=b$ into $y$ groups, where $xy = k$.
We use one of the $k$ reducers for each pair $(i, j)$ for a group $i$ from R and for a
group $j$ from $S$. Now we are going to
partition tuples from R and S and we use hash functions $h_r$ and $ h_s$ to do the partitioning.
We send each tuple $(a,b)$ of $R$ to all reducers of the form $(i, q)$,
where $i=h_r(a)$
is the group in which tuple $(a,b)$ belongs and $q$ ranges over all $y$ groups. Similarly, we send each tuple $(b,a)$ of $R$ to all reducers of the form $(q, i)$,
where $i=h_s(a)$
is the group in which tuple $(b,a)$ belongs and $q$ ranges over all $x$ groups.
Thus each tuple with $B=b$ from $R$ is sent to $y$ reducers and each tuple with
$B=b$ from $S$ is sent to $x$ reducers. Hence the
communication cost is $ry + sx$. We can show (see \cite{AfUl}) that by minimizing $ry + sx$ under
the constraint $xy = k$ we achieve communication cost equal to $\sqrt{2krs}$, which is always
less than what we found in Example~\ref{1-ex} which was $ r + ks$. The proof is easy: $\sqrt{2krs} \leq r + ks$
or $0 \leq \sqrt{r/s} - \sqrt{2k} + k\sqrt{s/r}$, which is a second order polynomial wrto $\sqrt{k}$ as unknown and it is positive for any $k$. Moreover observe that the improvement is significant:
The optimal communication cost grows as $\sqrt{k}$, while $ r + ks$ grows linearly with $k$.
\end{example}
\vspace*{-.4cm}
{\bf Related Work}
There is a lot of work over the decades about how to handle skew when we process queries.
We will limit ourselves here to recent work that considers joins in MapReduce or
discusses the Shares algorithm.
In \cite{BeameKS13} it is proven that with high probability the Shares algorithm distributes tuples evenly
on uniform databases (these are defined precisely in \cite{BeameKS13} to be databases which resemble the case of
random data). Then,
\cite{BeameKS14} generalizes and enhances results in \cite{BeameKS13} and
\cite{AfratiSSU13}.
\cite{BeameKS14}
describes how the Shares algorithm behaves on skewed data: it shows that the algorithm is resilient
to skew, and gives an upper bound even on skewed databases. However this resilience applies
to ordinary joins that use many of the attributes in one relation allowing thus the tuples with a heavy hitter
to be distributed in many reducers. However this is not the case in the 2-way join example we gave -- and many others.
\section{Shares Algorithm}
The algorithm is based on
a schema according to which we distribute the data to a given number of $k$
reducers. Each reducer is defined by a vector, where each component of the vector corresponds to an attribute.
The algorithm uses a number of independently chosen random hash functions $h_i$
one for
each attribute $X_i$. Each tuple is sent to a number of reducers depending on the value of $h_i$ for the
specific attribute $X_i$ in this tuple. If $X_i$ is not present in the tuple, then the tuple is sent to all
reducers for all $h_i$ values.
For an example, suppose we have the 3-way join $R_1(X_1,X_2) \bowtie
R_2(X_2,X_3) \bowtie R_3(X_3,X_1)$.
In this example each reducer is defined by a vector $(x,y,z)$.
A tuple $(a,b)$ of $R_1$ is sent to a number of reducers
and specifically to reducers
$(h_1(a), h_2(b), i)$ for all $i$. I.e., this tuple needs to be replicated a number of times, and specifically
in as many reducers as is the number of buckets into which $h_3$ hashes the values of attribute $X_3$.
\vspace*{-.1cm}
When the hash function $h_i$ hashes the values of attribute $X_i$ to $x_i$ buckets, we say that the {\em share} of $X_i$ is $x_i$.
The communication cost is calculated to be, for each relation, the size of the relation times the replication
that is needed for each tuple of this relation. This replication can be calculated to be the product
of the shares of all the attributes that do not appear in the relation.
In order to keep the number of reducers equal to $k$, we need to calculate the shares so that their product is equal to $k$.
\vspace*{-.1cm}
Thus, in our example, the communication cost is $r_1x_3+r_2x_1+r_3x_2$ and we must have $x_1x_2x_3=k$. (We denote the size of a relation $R_i$ by $r_i$.)
In \cite{AfUl}, it is explained how to use the Lagrangean method to find the shares that minimize the communication cost, for any multiway join.
\vspace*{-.1cm}
We are going to need an important observation that was proven in \cite{AfUl}.
An attribute $A$ is {\em dominated} by attribute $B$ in the join if $B$ appears in all relations where $A$ appears.
It is shown that if an attribute is dominated, then it does not get a share, or, in other words, its share is equal to 1.
\subsection{Our Setting}
We saw how to compute the 2-way join in Example~\ref{2-way-ex} for the tuples which have one HH.
For this join, we took two sets of keys:
\squishlisttwob
\item The set of keys as presented in Example~\ref{2-way-ex}
which send tuples with HH to a number of reducers in order to compute the join of tuples with HH.
\item The set of keys which send tuples without HH to a number of reducers in order to compute the join of tuples without HH. This second set is formed exactly as in the Shares algorithm.
\squishend
\vspace*{-.2cm}
It is convenient to see these two sets of keys as corresponding to two joins which we call
{\em residual joins}, and which actually differ only on the subset of the data they are applied. One applies the original join on the data with HH and the
other applies the original join on the data without HH.
The method we presented in Example~\ref{2-way-ex} is actually based on the Shares algorithm. To see this, we can be
equivalently thought as: We replace
each tuple of relation $R$ with a tuple where $B$ has distinct fresh values
$b_1,b_2,\ldots $ and the same for the tuples of relation $S$ with $B$ having values $b'_1,b'_2,\ldots$.
Now we can apply the Shares algorithm to find the shares and distribute the tuples to reducers normally. The only
problem with this plan is that the output will be empty because we have chosen $b_i$s and $b'_i$s
to be all distinct.
This problem however has an easy solution, because, we can keep this replacement
to the conceptual level, just so to create a HH-free join and be able to apply the Shares algorithm
and compute the shares optimally. When we transfer the tuples to the reducers, however, we transfer the
original tuples and thus, we produce the desired output. We explain this conceptual structure in Section~\ref{mod-shares-ex-sec}.
Our setting is as follows: We have $k$ reducers to use for computing all residual joins.
We assume each residual join $J_i$ uses $k_i$ of those reducers, thus one
constraint is $k_1+k_2+\cdots =k$.
For each residual join, we need to compute the communication cost expression. The objective function to minimize is the sum of the cost expressions over all residual joins, under the
constraint: for each residual join $J_i$, the product of the attribute shares must be equal to $k_i$.
The aim of this paper is to show how to systematically apply the idea explained for the 2-way join on
any multiway join with any number of HH.
The structure of the rest of the paper is the following:
\vspace*{-.2cm}
\squishlisttwo
\item We decompose into residual joins, i.e., we partition the data into subsets and we view a residual join as the
original join applied~on~one~of~the~subsets~(Section 3).
\item We explain how to form a {\em HH-free residual join} and how to compute the communication cost expression
for each residual join (Section 4).
\item We show how the cost expression for~each residual~join~is written in~a~simple~and~effective way~(Section 5).
\squishend
\section{Decomposition wrto HH}
First we need some definitions.
\vspace*{-.3cm}
For each attribute $X_i$ we define a set $L_{X_i}$ of {\em types}:
\vspace*{-.3cm}
\squishlisttwob
\item
If $X_i$ has no values that are heavy hitters, then $L_{X_i}$ comprises of only one type, $T_{-}$, called the {\em ordinary type}.\footnote{Ordinary type represents all other values of attribute $X_i$, the ones that are not heavy hitters.}
\item If $X_i$ has $p_i$ values that are heavy hitters, then $L_{X_i}$ comprises of $1+p_i$ types:
one type $T_b$ for each heavy hitter, $b$, of $X_i$, and one ordinary type $T_{-}$.
\squishend
\vspace*{-.1cm}
A {\em combination of types}, $C_T$, is an element of the Cartesian product of the sets $L_{X_i}, i=1,2\ldots$ and defines a {\em residual join}.
\vspace*{-.2cm}
E.g., for the query in in Example~\ref{2-way-ex}, we consider two residual joins, one for type combination $C_T=\{ A:T_{-}, B:T_{-}, C:T_{-} \}$ (without HH) and one for type combination $C_T=\{ A:T_{-}, B:T_{b}, C:T_{-} \}$ (with HH).
\vspace*{-.15cm}
Each $C_T$ defines a {\em residual join} which is the join computed only on a subset of the data.
Specifically, if an attribute $X$ has ordinary type in the current $C_T$ we exclude the tuples for which $X=HH$. E.g., if there are two HH $X=b_1$ and $X=b_2$, then we exclude (from all relations) all tuples with
$X=b_1$ and $X=b_2$. If attribute $X$ is of type $T_b$ then we exclude (from all relations) the
tuples with value $X\neq b$.
\vspace*{-.3cm}
\begin{example}
\label{run1-ex}
We take as our running example the 3-way join:~~~ $J=R(A,B)\bowtie S(B,E,C) \bowtie T(C,D)$
\vspace*{-.15cm}
Suppose attribute $B$ has two HHs, $B=b_1$ and $B=b_2$ and attribute $C$ has one HH,
and $C=c_1$. Thus attribute $B$ has three types, $T_{-}$, $T_{b_1}$ and $T_{b_2}$, attribute
$C$ has two types, $T_{-}$ and $T_{c_1}$ and the rest of the attributes have a single type, $T_{-}$.
Thus we have $3\times 2=6 $ residual joins, one for each combination.
By $r,s,t$ we denote the sizes of the relations that are {\em relevant} in each residual join, i.e., the number of tuples from each relation that
contribute in the particular residual join. We list the residual joins:
\vspace*{-.2cm}
\squishlisttwo
\item All attributes of type $T_{-}$. Here $r$ is the number of only those tuples of relation $R$ for which
$B\neq b_1$ and $B\neq b_2$, $s$ is the number of only those tuples of relation $S$ for which
$B\neq b_1$ and $B\neq b_2$ and $C\neq c_1$, and $t$ is the number of those tuples in relation $T$ for which $C\neq c_1$.
\item All attributes of type $T_{-}$, except $B$ of typle $T_{b_1}$.
\item All attributes of type $T_{-}$, except $B$ of typle $T_{b_2}$.
\item All attributes of type $T_{-}$, except $C$ of typle $T_{c_1}$.
\item All attributes a of type $T_{-}$, except $B$ of type $T_{b_1}$ and $C$ of typle $T_{c_1}$.
\item All attributes of type $T_{-}$, except $B$ of type $T_{b_2}$ and $C$ of typle $T_{c_1}$.
\squishend
\end{example}
\vspace*{-.3cm}
Each residual join is treated by the Shares algorithm as a separate join and a set of keys are defined that hash
each tuple as follows: A tuple $t$ of relation $R_j$ is sent to reducers of combinatrion $C_T$ only if the values of the tuple satisfy the constraints of $C_T$ as concerns values of HH.
\vspace*{-.3cm}
\begin{example}
\vspace*{-.2cm}
We continue from Example~\ref{run1-ex}.
Each tuple is sent to a number of reducers according to the keys created for each
residual join(we will provide more details later in the paper). E.g., a tuple $t$ from relation $R$ is sent to reducers as follows:
\vspace*{-.2cm}
\squishlisttwo
\item If $t$ has $B=b_1$ then it is sent to reducers created in items (2) and (5) in Example~\ref{run1-ex}.
\item If $t$ has $B\neq b_1$ and $B\neq b_2$ then it is sent to reducers created in items (1) and (4).
\item If $t$ has $B=b_2$ then it is sent to reducers created in items (3) and (6).
\squishend
\end{example}
\vspace*{-.5cm}
\section{Writing the Cost Expression}
\label{mod-shares-ex-sec}
In this section we will explain how to form a {\em HH-free residual join} and how to compute the communication cost expression
for each residual join.
The structure we use in this section is conceptual, for the sake of showing how to write the cost expression. In practice, we do not materialize $R(A,B_R)$ and $ S(B_S,C)$ or the auxiliary relation (definitions of these will be given shortly) -- as we
will explain in the next section. We begin with an example:
\vspace*{-.3cm}
\begin{example}
\label{res-2way-ex}
We consider the residual join with HH $B=b$ for the join of Example~\ref{2-way-ex} which we rewrite here: $R(A,B) \bowtie S(B,C)$.
In order to do that,
we will equivalently imagine that we have to compute:
\squishlisttwo
\item A join $R(A,B_R) \bowtie R_{aux} (B_R , B_S)
\bowtie S(B_S,C)$
\item On new database $D'$ which comes from $D$: We populate $R(A,B_R) $ with the same number of tuples as the original $R(A,B)$ has:
For each tuple $t$ with $B=b$, we add in $R(A,B_R) $ a tuple where we have replaced the
$B=b$ with $B_R=b.t.R$. We do similarly for $S(B_S,C)$ replacing $B=b$ in tuple $t$ by
$B_S=b.t.S$.
The relation $R_{aux}(B_R,B_S)$ is the Cartesian product of $B_R$ and $B_S$.
\squishend
\squishlisttwob
\item {\em Observation 1:} Database $D'$ now has no heavy hitters. So, we can apply the original
Shares algorithm. The introduction of auxiliary attributes and relations may seem now as if complicates things significantly, but, as we shall show in Section~\ref{dominance-sec}, it does not.
\squishend
\vspace*{-.3cm}
Thus if, in the database $D$, relation $R$ is $\{ (1,2),(3,2),(4,2)\}$ and $S$ is $\{ (2,5),(2,6)\}$
then, in the database $D'$ we have (assuming $B=2$ qualifies for HH):\\\\
$R(A,B_R) $ is $\{ (1,2.1.R),(3,2.3.R),(4,2.4.R)\}$.\\
$S(B_S,C)$ is $\{ (2.5.S,5),(2.6.S,6)\}$.\\
(I.e., we conveniently identify the tuple of $R$ with the value of its first argument and the tuple of $S$ with the value of its second argument.)\\
The auxiliary relation $R_{aux}(B_R,B_S)$ is :\\ $\{ (2.1.R,2.5.S),(2.3.R,2.5.S),(2.4.R,2.5.S),$\\ $(2.1.R,2.6.S),(2.3.R,2.6.S),(2.4.R,2.6.S)\}$
The residual join computation has no heavy hitters, thus, we apply the original Shares algorithm, only that, when we compute the cost expression
we ignore the communication cost for the auxiliary relation.\footnote{We can igonore it because we know what are the tuples in the
auxiliary relation and we can imagine that we can recreate them in the reducers. } Thus the communication cost of the
residual join is again $ry + sx$, which is the same expression as in Example~\ref{2-way-ex}.
\end{example}
\vspace*{-.3cm}
The conceptual structure in the general case is as follows:
For each combination of types, $C_T$, we compute a HH-free residual join whose cost expression is written as follows:
\squishlisttwo
\item If attribute $X_i$ has non-ordinary type in $C_T$ then:\\
--We introduce a number of
auxiliary attributes, one auxiliary attribute for each relation $R_j$ where attribute $X_i$ appears. We denote the auxiliary
attribute for relation $R_j$ by $X_{i-R_j}$.\\
--In the schema of each relation $R_j$ where $X_i$ appears, we replace $X_i$ with attribute $X_{i-R_j}$.
\item We form the residual join $J'$ for $C_T$ by adding to original join new relations as follows: one relation, $R^{X_i}_{aux}$,
for each attribute $X_i$ which is not
of ordinary type. The schema of~ that~ relation~ consists of the attributes
$X_{i-R_j}$ for each $j$ such that $X_i$ is an attribute of $R_j$.
\item Now we write the communication cost expression for $J'$ as in the Shares algorithm taking care that:
\squishlisttwo
\item[a.] The communication cost expression does not include a term for auxiliary relations.
\item[b.] The size of each relation in $J'$ that we use in the cost expression is the number of tuples that
have as values in the arguments with heavy hitters the specific value for this combination of types.
\squishend
\squishend
Now we will discuss in the next subsection how (and why) to simplify the cost expression not to inclue
share variables for the auxiliary attributes.
\section{Dominance Relation: Its Role in Simplifying the Cost Expression}
\label{dominance-sec}
The property of the dominance relation allows us to write the cost expression for each residual join
in a simple manner. We use the theorem:
\vspace*{-.3cm}
\begin{theorem}
The share of each auxiliary attribute is equal to 1 in the optimum solution.\footnote{Sometimes, we have a tie where in a relation all attributes appear only once; in this case we break ties declaring always the auxiliary
attribute as dominated.}
\end{theorem}
\begin{proof}
Each auxiliary attribute appears in one relation of the original join and in one auxiliary relation.
Since we do not add a term in the cost expression for the auxiliary
relation, we imagine that we write the cost expression for a join which is the residual join without the auxiliary relations. Hence,
an auxiliary attribute appears only in one relation,
hence it is dominated by a ordinary (non-HH in this residual join) attribute. There is the exception: when all attributes in a relation are auxiliary attributes. In this case
there is only one tuple in the relation in this particular residual join, so all attributes in the relation get share =1.
\end{proof}
\vspace*{-.3cm}
Thus we established that:
\squishlisttwob
\item The cost expression for each residual join can be derived from the cost expression of the original join (before dominance rule simplification) by making the shares of auxiliary attributes equal to 1.
\item Each tuple is hashed to reducers according to the values of the non-HH attributes in this tuple.
\squishend
\vspace*{-.5cm}
\begin{example}
We continue from Example~\ref{run1-ex} for the same HH as there.
Remember by $a,b,c,d,e$ we denote the shares for each attribute $A,B,C,D,E$ respectively and by $r,s,t$ we denote the sizes of the relations that are {\em relevant} in each residual join, i.e., the number of tuples from each relation that
contribute in the particular residual join.
We always start with the cost expression for the original join, $r cde + s ad + t abe $, and then simplify accordingly.
We list the cost expression for every residual join (and in the same order as) in Example~\ref{run1-ex}:
\squishlisttwo
\item Here all attributes are ordinary, so we simplifly the relation by observing that
$A$ is dominated by $B$ and $D$ is dominated by $C$, hence $a=1$ and $d=1$ and the expression is:
$r c + s + t b $.
\item Here $B$ is a non-ordinary attribute, hence $b=1$ and then, from the remaining attributes
only $D$ is dominated by $C$, hence $d=1$ and the expression is:
$r c + s a + t a $
\item $r c + s a + t a $, i.e., same as above, only the sizes of the relations will be different.
\item $r d + s d + t b $
\item Here we set both $b=1$ and $c=1$ and this gives us $r de + s ad + t ae $.
\item $r de + s ad + t ae $, i.e., same as above, only the sizes of the relations will be different.
\squishend
\end{example}
\bibliographystyle{abbrv}
|
{
"timestamp": "2015-04-14T02:16:22",
"yymm": "1504",
"arxiv_id": "1504.03247",
"language": "en",
"url": "https://arxiv.org/abs/1504.03247"
}
|
\section{Introduction}
The positron fraction spectrum $e{}^{+}$/($e{}^{+}$+$e{}^{-}$)
in the cosmic ray (CR) contains two components: secondary $e{}^{\pm}$
produced by nuclei collision and primary $e^{-}$.
It is currently believed that these two components, each of which
will produce a diffused power low spectrum, predict a positron fraction
which goes down with energy. However, the latest results measured
by the Alpha Magnetic Spectrometer (AMS-02) with high accuracy indicate
that the positron fraction increases with energy above $\sim$8
GeV and does not increase with energy above $\sim$200 GeV \cite{Accardo:2014lma,Aguilar:2014mma}.
This {}``increasing'' behavior, which is also observed by the payload for antimatter
matter exploration and light-nuclei astrophysics (PAMELA) \cite{Adriani:2008zr,Adriani:2011cu,Adriani:2014xoa}
and the Fermi Large Area Telescope (Fermi-LAT) \cite{Grasso:2009ma,FermiLAT:2011ab}, is
not compatible with only diffused power low components. The {}``cutoff''
behavior above 200 GeV , which can be well described by a common source term with an exponential cutoff parameter in the Eq.(1) of \cite{Accardo:2014lma}, indicates that potential sources produce the
exceed of electron and positron pairs.
AMS-02 \cite{Accardo:2014lma,Aguilar:2014mma} is a state-of-the-art astroparticle detector installed on the
International Space Station (ISS). It carries a Transition Radiation
Detector (TRD) and a Electromagnetic Calorimeter (ECAL). These two
sub-detectors provide independent proton/lepton identification, which
will achieve a much larger proton rejection power of AMS-02 compared
with PAMELA which has only one Electromagnetic Calorimeter for proton/lepton
identification using the 3D shower shape and Energy-Momentum match (E/P). Compared with Fermi-LAT, AMS-02 has a large magnet
which can identify charge sign of the particle. Thus, the contamination
of electrons (also called ``charge confusion'' in \cite{Accardo:2014lma})
in the positron sample of AMS-02 is much smaller that that of Fermi-LAT.
For the reasons given above, there is much less proton or charge confusion
contamination in AMS-02 measurement than that in PAMELA or Fermi-LAT.
Here, we only interpret AMS-02 result due to the lack of knowledge
of the contamination control in PAMELA and Fermi-LAT measurements.
The AMS-02's recent measurements of positron fraction \cite{Accardo:2014lma},
$e{}^{+}$ flux, $e{}^{-}$ flux \cite{Aguilar:2014mma} and ($e{}^{-}$+$e{}^{+}$) flux \cite{Aguilar:2014fea}were published. The
$e{}^{-}$ flux contains three components: primary $e{}^{-}$, secondary
$e{}^{-}$ and $e{}^{-}$ from unknown sources. The $e{}^{+}$ flux
contains only two components: $e{}^{+}$ from secondary production
and primary $e{}^{+}$ from sources to be indentified. To avoid the unnecessary uncertainty
of primary $e{}^{-}$, the $e{}^{+}$ flux seems to be an ideal spectrum
to study extra sources. However, there is an acceptance uncertainty
from the detector itself in the $e{}^{+}$ and $e{}^{-}$ fluxes.
This uncertainty in $e{}^{+}$ flux is strongly correlated with that
in $e{}^{-}$ flux \cite{Aguilar:2014mma}, especially at high energies. The
positron fraction can avoid this systematic uncertainty \cite{Accardo:2014lma}.
For example, one can clearly see a drop at the last point (350 GeV
$\sim$ 500 GeV) in the positron fraction but cannot tell
a drop at the last point (370 GeV $\sim$ 500 GeV) in the
$e{}^{+}$ flux due to its larger error bars. Therefore, positron
fraction is used to study extra sources while $e{}^{-}$ flux is used
to estimate the primary $e{}^{-}$ which will affect the denominator
of $e{}^{+}$/($e{}^{+}$+$e{}^{-}$).
Recent studies have proposed some interpretations, such as dark matter annihilation
or decay \cite{Bergstrom:2008gr,Cirelli:2008jk,Barger:2008su,Cholis:2008hb,Cirelli:2008pk,
Yuan:2013eja,Yin:2008bs,Lin:2014vja,Boudaud:2014dta}, supernova remnants (SNRs) \cite{Blasi:2009hv,Blasi:2009bd,Mertsch:2009ph,Kachelriess:2012ag,
Cholis:2013lwa,DiMauro:2014iia},
secondary production in the interstellar medium (ISM) \cite{Blum:2013zsa}
and pulsars \cite{Boulares1989,Atoian:1995ux,Aharonian:1995zz,Chi:1995id,Hooper:2008kg,Yuksel:2008rf,
Profumo:2008ms,Malyshev:2009tw,Delahaye:2010ji,Pochon2010,Kashiyama:2010ui,Mertsch:2010fn,
Linden:2013mqa,Yin:2013vaa,Delahaye:2014osa,Lin:2014vja,Boudaud:2014dta}.
Cosmic ray flux data can also be together with other observations (like the dark matter relic density and the direct detection experimental results etc.) to give a combined constraint on dark matter models \cite{Zheng:2010js,Yu:2011by}.
Besides dark matter scenario,
the others can provide astrophysical explanations which do not require
the existence of new particles. SNRs model, for instance in \cite{Cholis:2013lwa} and \cite{Tomassetti:2015cva}, introduce some new mechanisms for the propagation model or special distributions of the primary sources. The {}``model-independent''
approach from \cite{Blum:2013zsa}, sets an upper limit of the positron
fraction by neglecting radiative losses of electrons and positron
but does not indicate any obvious cutoff in the spectrum. Among them,
the pulsar interpretation is one of the scenarios which predict a
cutoff at a few hundred GeV in the positron fraction spectrum and do not contridict other cosmic ray spectrums (eg. boron-to-carbon). The pioneering works on pulsar interpretaion of positron fraction have been performed by \cite{Hooper:2008kg,Profumo:2008ms,Yin:2013vaa} a few years ago. Combined analyses of the recent AMS-02 lepton data have been performed by \cite{DiMauro:2014iia} and \cite{Lin:2014vja}, with a global fit on positron fraction \cite{Accardo:2014lma}, $e{}^{+}$ flux, $e{}^{-}$ flux and ($e{}^{-}$+$e{}^{+}$) flux. To avoid the over-estimation of the $\chi{}^{2}$, however, only two out of four spectrums should be used in the fit. As the reasons given by the previous paragraph, we only study positron fraction and $e{}^{-}$ flux in this paper.
A pulsar is widely regarded as a rotating neutron star with a strong magnetosphere, which
can accelerate electrons, which will induce an electromagnetic cascade
through the emission of curvature radiation
\cite{Ruderman:1975ju,Shapiro:1983du,Cheng:1986qt,Contopoulos:2009vm}. This leads
to the production of high energy photons which eventually induces $e{}^{+}$$e{}^{-}$ pair production. This process produces the same amount of high energy $e{}^{+}$ and
$e{}^{-}$, which can escape from the magnetosphere and propagate
to the earth. There is a cutoff energy of the photons produced in
a pulsar, which leads to a cutoff in the positron fraction.
In this paper, DRAGON \cite{DragonWebsite,Evoli:2008dv,Gaggero:2013rya,Gaggero:2013nfa, Bernardo:2013036D} is used as a numerical tool
to model the propagation environment, to tune the related parameters
and to estimate the $e{}^{\pm}$ background. The authors
of \cite{Gaggero:2013rya,Gaggero:2013nfa, Bernardo:2013036D, Kissmann:2015kaa}
did a very complete work on three-dimensional cosmic-ray modeling.
In the 3-D models, they pointed out the spiral arms have
an effect on the propagation parameters. A 2-D model is
used in this paper because we focus on the lepton spectra
implication. Due to the energy loss of leptons, the effect
of spiral arms on the high energy leptons is less important than
that of the additional nearby sources contribution. ROOT is used to minimize
$\chi{}^{2}$ to get the best fit results. We consider six nearby
pulsars from ATNF catalogue \cite{ATNFwebsite,Manchester:2004bp} as the possible extra single
sources of the high energy positrons. We find only four, which are
Geminga, J1741-2054, Monogem and J0942-5552, can survive from all
considered physical requirements. We then discuss the possibility
that these high energy $e{}^{\pm}$ are from multiple pulsars. The
multiple pulsars contribution predicts a positron fraction with some
structures at higher energies.
The paper is organized as follows. Section \ref{sec-background} shows the way where $e{}^{\pm}$
background is estimated. In Section \ref{sec-single}, the properties of
pulsars are described and the profile of $e{}^{\pm}$ fluxes produced by
a pulsar is derived. The interpretation of positron fraction with one single pulsar is discussed in Section \ref{sec-single-2} and the hypothesis about multiple pulsars interpretion is tested in Section \ref{sec-multiple}. The conclusions are drawn
in Section \ref{sec-con}. In Appendix \ref{app-A}, the diffusion energy-loss equation for a burst-like source is solved with the spherically symmetric approximation.
\section{Propagation parameters and $e^{\pm}$ background \label{sec-background}}
The Galacitc background of the lepton fluxes are considered as three
main components, which are primary electrons from CR sources,
secondary electrons and positrons from the interactions between the
CR and the interstellar medium (ISM). The propagation of $e^{\pm}$ in the Galaxy obeys the following Ginzburg and Syrovatskii's equation \cite{Ginzburg2013origin}, also in \cite{Evoli:2008dv,Strong:2007nh}
\begin{eqnarray}
&&\frac{\partial f_i}{\partial t}-\nabla\cdot[(D\nabla-\vec{v}_c)f_i]
-\frac{\partial}{\partial p}p^2D_{pp}\frac{\partial}{\partial p}\frac{f_i}{p^2}
+\frac{\partial}{\partial p}\left[(\dot{p}-\frac{p}{3}\nabla\cdot\vec{v}_c)f_i\right]\nonumber\\
&&=Q_i(\vec{x},t,p)+\sum_{j>i}c\beta n_{\mathrm{gas}}\sigma_{ji}f_j
-c\beta n_{\mathrm{gas}}\sigma_{\mathrm{in}}f_i
\end{eqnarray}
where $p\equiv|\vec{p}|$ is the particle momentum; $f_i(\vec{x},t,p)$ is the particle number density of a species $i$ per unit momentum interval; $\vec{v}_c$ is the convection velocity, $\beta\equiv v/c$ is the ratio of velocity to the speed of light; $\sigma_{\mathrm{in}}$ is the total inelastic cross section onto the ISM gas, whose density is $n_{\mathrm{gas}}$; $\sigma_{ji}$ is the production cross section of the species $i$ by the fragmentation of the species $j$ (with $j>i$); and $Q_i(\vec{x},t,p)$ is the source term of species $i$, which can be thought to be steady $Q_i=Q_i(\vec{x},p)$ for background CR particles.
The spatial diffusion coefficient $D$ in the cylindrical coordinate system $(r,z)$ may be parameterized as \cite{Evoli:2008dv,Strong:2007nh,DiBernardo:2010is}
\begin{equation}
\left\{
\begin{array}{r@{\;=\;}l}
D(\rho,r,z) & D(\rho,r) e^{-|z|/z_t}\quad\text{or}\quad D(\rho,r,z)=D(\rho,r) (-L<z<L)\\
D(\rho,r) & D_0f(r)\beta\left(\frac{\rho}{\rho_0}\right)^\delta \label{spatial-diffusion-coe}
\end{array}
\right.
\end{equation}
where $\rho\equiv pc/(Ze)$ is defined as the particle magnetic rigidity, $z_t$ is the scale height of the diffusion coefficient, $L$ is the halo size, and $\delta$ is the index of the power-law dependence of the diffusion coefficient on the rigidity. $D_0$ is the normalization of the diffusion coefficient at the reference rigidity $\rho_0=4$ GV. Previous DRAGON papers \cite{Gaggero:2013rya,Gaggero:2013nfa, Bernardo:2013036D} tested a few models with the exponential profile, i.e., the left formula in \eqref{spatial-diffusion-coe}, which is more physical than the constant one, i.e., the right one. The effect of choosing different profiles on the electron and positron background is small if the parameters are properly set. In this paper, the constant profile is used in order to compute the pulsar profile in an analytical way, i.e., eq. \eqref{eq:e_flux} in Section \ref{sec-single}. The function $f(r)$ describes a possible radial dependence of $D$, and it can be taken to be unity for simplicity.
The diffusion coefficient in momentum space $D_{pp}$ is related to the spatial diffusion coefficient $D$ by \cite{Yuan:2013eja,DiBernardo:2010is,Zhang:2008tb,Seo1994}
\begin{eqnarray}
D_{pp}D=\frac{4p^2v_A^2}{3\delta(4-\delta^2)(4-\delta)w}
\end{eqnarray}
where $v_A$ is the Alfven velocity, and $\delta$ is the power-law index as given in \eqref{spatial-diffusion-coe}. $w$ is the ratio of magnetohydrodynamic wave energy density to the magnetic field energy density, and it is usually taken to be 1.
DRAGON \cite{DragonWebsite,Evoli:2008dv,Gaggero:2013rya} is used to tune the propagation parameters according
to the B/C ratio, which is sensitive to the parameters. The Markov Chain Monte Carlo algorithm (MCMC, \cite{Lewis:2002ah}) is used to determine $D_{0}$ and $\delta$. The priors are shown in Table.~\ref{Tab:priors}. The posterior distributions can be shown in a contour in the $D_{0}$ and $\delta$ plane in Fig.~\ref{Fig:D_delta}.
\begin{table}[!htbp]
\centering
\caption{The priors of $D_{0}$ and $\delta$ }
\begin{tabular}{|c|c|c|c|}
\hline
& start value & minimum value & maximum value \tabularnewline
\hline
\hline
$ D_{0}(\times10^{28}~ \mathrm{cm}^{2}\mathrm{s}^{-1}) $ & 3 & 2.5 & 7.5 \tabularnewline
\hline
$\delta$ & 0.40 & 0.20 & 0.65 \tabularnewline
\hline
\end{tabular}
\label{Tab:priors}%
\end{table}
\begin{figure}[!htbp]
\includegraphics[width=1\textwidth]{pdf/D_delta}\caption{
Contour in the $D_{0}$ and $\delta$ plane. The cross shows the best fit value while the three closed curves from inside to outside show the 68.3\% C.L., 95.4\% C.L. and 99.7\% C.L. respectively.}
\label{Fig:D_delta}%
\end{figure}
The parameters with their 68\% C.L. uncertainties from the fit are as follows:
\begin{equation}
\left\{
\begin{array}{r@{\;=\;}l}
D_{0}\big|_{\rho_{0} = 4~\mathrm{GV}} & ( 6.20 \pm 0.31 ) \times10^{28}~ \mathrm{cm}^{2}\mathrm{s}^{-1}\\
\delta & 0.31 \pm 0.03\\
L & 4 ~\mathrm{kpc}\\
v_{A} & 40~\mathrm{km}/\mathrm{s}
\end{array}
\right.
\end{equation}
where the halo size $L$ is taken from the MED model of \cite{Donato:2003xg}, and the $v_{A}$ is fixed. These parameters are consistent with what the authors of Ref.\cite{Lin:2014vja} has got in the reaccelaration propagation model.
To avoid the uncertainty of solar modulation, AMS-02 proton flux \cite{Aguilar:2015ooa}
above 45 GV is fitted to get the injection spectra using MCMC \cite{Lewis:2002ah}.
Three breaks, which are 6.7 GV, 11 GV and 316($\pm$148) GV, are introduced
in the injection spectrum of nuclei. The proton spectral indice below and
above the breaks are 2.25, 2.35, 2.501($\pm$0.010) and 2.501-0.084($\pm$0.050), respectively.
The high energy spectral indices of helium, carbon and oxygen are shifted by -0.1 w.r.t those of proton
according to proton-to-helium ratio \cite{Adriani:2011cu}. The Ferriere
model \cite{Ferriere:2001rg} is used as the source distribution for the primary
components, e.g. SNRs for SNe type II.
To assure that the propagation parameters are correct, we need to compare
the model prediction with the boron-to-carbon ratio \cite{Adriani:2014xoa,Panov:2007fe, Ahn:2008my, Obermeier:2012vg} and the proton flux \cite{Aguilar:2015ooa}.
As shown in Fig.~\ref{Fig:BC_proton}, the set of
parameters used can reproduce the boron-to-carbon ratio and the proton flux well.
According to this set of parameters, we can obtain the fluxes of the secondary positrons and electrons.
\begin{figure}[!htbp]
\subfloat[]{\includegraphics[width=0.5\textwidth]{pdf/BC.pdf}}
\subfloat[]{\includegraphics[width=0.5\textwidth]{pdf/protons.pdf}}
\caption{(a) model prediction of B/C ratio compared with measurements from
PAMELA \cite{Adriani:2014xoa}, ATIC02 \cite{Panov:2007fe},
CREAM-I \cite{Ahn:2008my} and TRACER06 \cite{Obermeier:2012vg}.
(b) model prediction of proton flux compared with measurements from AMS-02 \cite{Aguilar:2015ooa}. The solar modulation
is taken as 500 MeV here. The red band in (a) shows
the variation of the propagation parameters $D_{0}$ and $\delta$ within 95\% C.L.
}
\label{Fig:BC_proton}%
\end{figure}
A power-law spectrum with two breaks is introduced to parameterize
the injection spectrum of the primary electrons as a function of rigidity,
\begin{equation}
Q(\rho)\propto\begin{cases}
\begin{array}{ll}
(\rho/\rho_{\mathrm{br1}}^{e})^{-\gamma_{1}} &\quad (\rho<\rho_{\mathrm{br1}}^{e})\\
(\rho/\rho_{\mathrm{br1}}^{e})^{-\gamma_{2}} &\quad
(\rho_{\mathrm{br1}}^{e}\leq \rho\leq \rho_{\mathrm{br2}}^{e})\\
(\rho_{\mathrm{br2}}^{e}/\rho_{\mathrm{br1}}^{e})^{-\gamma_{2}}\cdot
(\rho/\rho_{\mathrm{br2}}^{e})^{-\gamma_{3}} &\quad
(\rho>\rho_{\mathrm{br2}}^{e})
\end{array}
\end{cases}\label{eq:1}
\end{equation}
The parameters are adjusted according to the electron flux from AMS-02 \cite{Aguilar:2014mma}.
The agreement between the model and the data is shown in Section \ref{sec-multiple}.
These spectral indices are $\gamma_{1}=1.95$, $\gamma_{2}=2.75$
and $\gamma_{3}=2.5$ respectively. The breaks are $\rho_{\mathrm{br1}}^{e}=8.6$ GV
and $\rho_{\mathrm{br2}}^{e}=110$ GV. Since the high energy breaks of primary particles,
such as protons and helium, are found by PAMELA \cite{Adriani:2011cu} and recently
confirmed by AMS-02 \cite{Aguilar:2015ooa}, it is reasonable to assume that there is
also a high energy break in primary electron flux. More detailed discussion on the necessity of
the high energy break $\rho_{\mathrm{br2}}^{e}$ can be found in \cite{Lin:2014vja}
and \cite{Li:2014csu}, where the high energy break hypothesizes are in favor compared to the
no-break ones. Ref.~\cite{Li:2014csu} gave us an estimation by taking the primary electron
flux as $\Phi_{e-}-\Phi_{e+}$ and could roughly determine the break.
\section{e$^{\pm}$ from a single pulsar \label{sec-single}}
The pulsars are potential sources which could produce primary e$^{\pm}$
at high energy \cite{Hooper:2008kg,Yuksel:2008rf,Profumo:2008ms,Malyshev:2009tw,Yin:2013vaa}. Electrons can be accelerated by the strong
magnetosphere of the pulsars, and this acceleration produces photons. When those photons
annihilate with each other, they can produce e$^{\pm}$ pairs. Thus, the e$^{\pm}$ energies
are related to the pulsar magnetosphere. Assuming the pulsar magnetosphere
as a magnetic dipole, this magnetic dipole radiation energy is proportional
to the spin down luminosity. Due to this spin down (\textit{i.e.} slowing of rotation), the rotational
frequency of a pulsar $\Omega\equiv 2\pi/P$ (with $P$ being the period) is
a function of time as follows \cite{Hooper:2008kg,Profumo:2008ms,Yin:2013vaa}
\begin{equation}
\Omega(t)=\frac{\Omega_{0}}{\sqrt{1+t/\tau_{0}}},\label{eq:rotational_fre}
\end{equation}
where $\Omega_{0}$ is the initial spin frequency of the pulsar and
$\tau_{0}$ is a time scale which describes the spin-down luminosity
decays. $\tau_{0}$ cannot be directly obtained from pulsar timing observations, and it is assumed to be \cite{Profumo:2008ms,Yin:2013vaa}
\begin{equation}
\tau_{0}\simeq 10^{4}~\mathrm{yr}
\end{equation}
The rotational energy of the pulsar is $E(t)=(1/2)I\Omega^{2}(t)$.
Here $I$ is the moment of inertia, which is related to the mass
and the radius of the pulsar and can be regarded as a time independent
value. The magnetic dipole radiation energy is equal to the energy
loss rate,
\begin{equation}
|\dot{E}(t)|=I\Omega(t)\dot{|\Omega}(t)|
=\frac{I\Omega_{0}^{2}}{2}\frac{1}{\tau_{0}(1+t/\tau_{0})^{2}}\label{eq:Edot}
\end{equation}
The total energy loss of a pulsar is \cite{Profumo:2008ms,Malyshev:2009tw,Yin:2013vaa}
\begin{equation}
E_{\mathrm{tot}}(t)=\int_0^t dt'|\dot{E}(t')|
=\frac{I\Omega_{0}^{2}}{2}\frac{t/\tau_{0}}{1+t/\tau_{0}}
=|\dot{E}(t)|t\left(1+\frac{t}{\tau_{0}}\right)\label{eq:E_total}
\end{equation}
The total energy injection of $e^{\pm}$ out of a pulsar should be
proportional to the total energy loss
\begin{equation}
E_{\mathrm{out}}(t)=\eta E_{\mathrm{tot}}(t)=\eta|\dot{E}(t)|t\left(1+\frac{t}{\tau_{0}}\right)\label{eq:E_out}
\end{equation}
where $\eta$ is the efficiency of the injected $e^{\pm}$ energy
converted from the magnetic dipole radiation energy.
The pulsar characteristic age is defined as \cite{Manchester:2004bp}
\begin{equation}
T\equiv\frac{P}{2\dot{P}}=\frac{\Omega}{2|\dot{\Omega}|}=t+\tau_0
\end{equation}
For a mature pulsar with $t\gg \tau_0$, we have $T\simeq t$. In this condition, eqs.~\eqref{eq:Edot}, \eqref{eq:E_total} and \eqref{eq:E_out} become
\begin{eqnarray}
&&|\dot{E}(T)|\simeq \frac{I\Omega_{0}^{2}}{2}\frac{\tau_0}{T^2}\\
&&E_{\mathrm{tot}}(T)\simeq |\dot{E}(T)|\frac{T^2}{\tau_0}\label{E-tot-2}\\
&&E_{\mathrm{out}}(T)\simeq \eta |\dot{E}(T)|\frac{T^2}{\tau_0}\label{E-out-2}
\end{eqnarray}
The propagation equation for
the $e^{\pm}$ can be described as \cite{Hooper:2008kg,Yin:2013vaa}
\begin{equation}
\frac{\partial f}{\partial t}=D(E)\nabla^2f+\frac{\partial}{\partial E}[b(E)f]+Q(\vec{x},t,E),\label{eq:propagation_eq}
\end{equation}
where $f(\vec{x},t,E)$ is the number density per unit energy interval of $e^{\pm}$; $D(E)=(v/c)D_{0}(E/4~\mathrm{GeV})^{\delta}$
is the diffusion coefficient with the velosity $v$ of the particle,
the speed $c$ of light, $D_{0}$ and $\delta$ the same as the parameters
used to calculate the background in Section~\ref{sec-background}; and $b(E)\equiv-dE/dt=b_0E^{2}$
with $b_{0}=1.4\times10^{-16}~\mathrm{GeV}^{-1}\mathrm{s}^{-1}$ is the rate of energy
loss due to inverse Compton scattering and synchrotron \cite{Grasso:2009ma,Profumo:2008ms,Yin:2013vaa}.
The source term $Q(\vec{x},t,E)$ of a pulsar can be described by
a burst-like source with a power-law energy spectrum and an exponential
cutoff
\begin{equation}
Q(\vec{x},t,E)=Q_0E^{-\alpha}\exp\left(-\frac{E}{E_{\mathrm{cut}}}\right)
\delta^3(\vec{x}-\vec{x}_0)\delta(t-t_0),
\label{eq:Q_source}
\end{equation}
where $Q_{0}$ is the normalization factor related to the total injected energy $E_{\mathrm{out}}$,
$\alpha$ is the spectral index, and $E_{\mathrm{cut}}$ is the cutoff energy.
In Appendix~\ref{app-A},
we briefly review how to solve the equation \eqref{eq:propagation_eq} with the source \eqref{eq:Q_source}. The method is equivalent to many previous works (for example, Refs.~\cite{Atoian:1995ux,Yin:2013vaa}).
Using the results, \textit{i.e.}
eqs.~\eqref{solution-final} and \eqref{diff-distance-final}, in Appendix~\ref{app-A}, we obtain the electron or positron flux observed at the earth as follows:
\begin{equation}
\Phi_{e}(r,t_{\mathrm{dif}},E)=\frac{c}{4\pi}f
=\frac{c}{4\pi}\frac{Q_0E^{-\alpha}}{\pi^{\frac{3}{2}}r_{\mathrm{dif}}^3}
\left(1-\frac{E}{E_{\mathrm{max}}}\right)^{\alpha-2}
\exp\left[-\frac{E/E_{\mathrm{cut}}}{(1-E/E_{\mathrm{max}})}
-\frac{d^2}{r_{\mathrm{dif}}^2}\right],\label{eq:e_flux}
\end{equation}
where $d$ is the distance between the earth and
the source, the diffusion distance $r_{\mathrm{dif}}$ is given by
\begin{equation}
r_{\mathrm{dif}}(t_{\mathrm{dif}},E)=
2\sqrt{\frac{D(E)t_{\mathrm{dif}}}{(1-\delta)}
\frac{E_{\mathrm{max}}}{E}
\left[1-\left(1-\frac{E}{E_{\mathrm{max}}}\right)^{1-\delta}\right]}\label{eq:r_dif}
\end{equation}
and the diffusion time $t_{\mathrm{dif}}$ is the time a charged particle travels
in the ISM before it reaches the earth. The electrons and positrons may be trapped in
the pulsar wind nebula (PWN) for some time before they escape. The age of a pulsar is
$T = t_{\mathrm{escape}} + t_{\mathrm{dif}}$, where $t_{\mathrm{escape}}$ is the time
before the leptons escape from the PWN. In some case, $t_{\mathrm{escape}}$ and
$t_{\mathrm{dif}}$ can be of the same order of magnitude, and then the discussion will
be complicated. In some other case, $t_{\mathrm{escape}}$ could be negligible. For instance,
when the SNR is evolving into the "Sedov-Taylor" phase, the leptons in it are trapped
(See Ref.~\cite{Gaensler:2006ua} and references there in). In that case,
the time $t_{\mathrm{escape}}$, during which the SNR reverse shock collides with the PWN
forward shock, is typically a few $10^{3}$ yr~\cite{Gaensler:2006ua}, which is small comparing to the
ages of the pulsars we studied here, which are around $10^{5}$ yr. In this work, we consider
the latter case and neglect $t_{\mathrm{escape}}$ for simplicity. We leave the case of
large $t_{\mathrm{escape}}$ to a further specific study. Thus, we
assume that $t_{\mathrm{dif}}\simeq T$. The maxium energy $E_{\mathrm{max}}$ is defined as
\begin{equation}
E_{\mathrm{max}}=1/(b_{0}T).\label{eq:E_max}
\end{equation}
The positron fraction from AMS-02 implies a primary positron source with a cut-off
energy $1/E_{s} = 1.84 \pm 0.58\:TeV^{-1}$ in their "minimal" model \cite{Accardo:2014lma},
which corresponds to $E_{s} \in \left[490,790\right]$. Due to the limitation of statistics
of high energy e- and e+ measured by AMS-02, the upper bound 790 GeV is not a strict limit.
Thus, we consider a primary e+ and e- source contribution with a cut-off energy
$E_{\mathrm{cut-off}}\simeq$ (500 $\sim$ 5000) GeV, which corresponds
to a pulsar with an age $T\simeq (0.45\sim 4.5)\times10^{5}$ yr according to \eqref{eq:E_max}.
The term $\exp\left[-\frac{d^2}{r_{\mathrm{dif}}^2}\right]$ in \eqref{eq:e_flux} tells us that
a pulsar with $d>r_{\mathrm{dif}}$ requires a larger
normalization $Q_0$, which hints a larger $E_{\mathrm{out}}$, a larger $\eta$ in \eqref{eq:E_out}, or both.
$r_{\mathrm{dif}}>d$ is required in our study, whose physical interpretation is that the distance
a particle travels in the ISM should be larger
than the distance between the earth and the source.
Eq.~\eqref{eq:r_dif} tells us that $r_{\mathrm{dif}}$ is as a function of diffusion time $t_{\mathrm{dif}}$
and lepton energy $E$, as is shown by Fig.~\ref{Fig:r_d} where the color scale indicades $r_{\mathrm{dif}}$.
For $T\simeq (0.45\sim 4.5)\times10^{5}$ yr and the lepton energy $E=1000 $ GeV,
$r_{\mathrm{dif}}$ is always greater than 0.5 kpc.
Selecting pulsars with $d<0.5 $ kpc and $T\simeq (0.45\sim 4.5)\times10^{5}$ yr, high engery leptons
they produced can reach the earth.
\begin{figure}[!htbp]
\includegraphics[width=1\textwidth]{pdf/r_d}\caption{$r_{\mathrm{dif}}$ as a function of $t_{\mathrm{dif}}$ and $E$, which is from eq.~\eqref{eq:r_dif}. The lepton energy $E$
is the $e{}^{+}$ (or $e^{-}$) energy detected at location away from the
pulsar with the diffusion distance $r_{\mathrm{dif}}$. $r_{\mathrm{dif}}$ increases
with $E$.}
\label{Fig:r_d}%
\end{figure}
Thus, the pulsars with ages $T\simeq (0.45\sim 4.5)\times10^{5}$ yr
and distance $d<0.5$ kpc can explain the behavior of positron fraction
of AMS-02 at high energy range.
\section{Single pulsar interpretation \label{sec-single-2}}
A few simple examples using a single pulsar are given to explain high energy positron fraction
of AMS-02 \cite{Accardo:2014lma}. The background electrons
and positrons are described in Section~\ref{sec-background}. The primary electron flux
is scaled by a normalization factor $A{}_{prim,e^{-}}$ since it is not possible to constrain the electron flux contribution from SNRs. The
age $T$ and the distance $d$ are taken from the ATNF catalogue and the
positron fraction is fitted to obtain the free parameters in \eqref{eq:e_flux}, the spectral index $\alpha$, and the normalization $Q_0$. $Q_0$ is fixed by the relation \cite{DiMauro:2014iia,Malyshev:2009tw,Linden:2013mqa} $E_{\mathrm{out}}=\int_{E_{\mathrm{min}}}^{E_{\mathrm{max}}} dEEQ(E)\simeq\int_0^\infty dEEQ(E)$, which approximately yields $Q_{0}\simeq E_{\mathrm{out}}$ for $\alpha\simeq 2$. The cutoff energy $E_{\mathrm{cut}}$ is set to be 5000~GeV, which is large enough, as it does not change the shape
of pulsar contribution. Since we are interested in the positron excess at high energies, the fit is started from 10 GeV where the effect of solar modulation is negligible.
Six nearby single pulsars, whose ages $T\simeq (0.45\sim 4.5)\times10^{5}$ yr and distance $d<0.5$ kpc, are used to fit the positron fraction. Minuit
package in ROOT is used to determine the parameters to minimize $\chi^{2}$. The best results of the single pulsars are listed in Table \ref{Tab:fit_parameters_of_single_pulsars}.
\begin{table}[!htbp]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Pulsar name & $d$(kpc) & $T$($10{}^{5}$~yr) & $log{}_{10}(\frac{Q_{0}}{\mathrm{GeV}})$ & $\alpha$ & $A_{prim,e-}$ & $\chi^{2}/ndf$\tabularnewline
\hline
\hline
Geminga & 0.25 & 3.42 & 50.5 & 2.04 & 0.50 & 26.8/40\tabularnewline
\hline
J1741-2054 & 0.25 & 3.86 & 50.6 & 2.03 & 0.50 & 26.8/40\tabularnewline
\hline
Monogem & 0.28 & 1.11 & 50.1 & 2.15 & 0.50 & 27.3/40\tabularnewline
\hline
J0942-5552 & 0.30 & 4.61 & 50.6 & 2.01 & 0.49 & 27.7/40\tabularnewline
\hline
J1001-5507 & 0.30 & 4.43 & 50.1 & 2.34 & 0.47 & 27.6/40\tabularnewline
\hline
J1825-0935 & 0.30 & 2.32 & 50.5 & 2.61 & 0.44 & 28.8/40\tabularnewline
\hline
\end{tabular}
\caption{Parameters of six nearby single pulsars from the best fit results.
The $\chi^{2}/ndf$ from the fits of Geminga, J1741-2054, Monogem,
J0942-5552 and J1001-5507 are smaller than 1, which show a good agreement
between those single pulsar models and the experiment. }
\label{Tab:fit_parameters_of_single_pulsars}%
\end{table}
The results are also shown in Fig.~\ref{fig3}.
\begin{figure}[!htbp]
\subfloat[]{\includegraphics[width=0.5\textwidth]{pdf/Geminga}}
\subfloat[]{\includegraphics[width=0.5\textwidth]{pdf/J1741-2054}}
\subfloat[]{\includegraphics[width=0.5\textwidth]{pdf/Monogem}}
\subfloat[]{\includegraphics[width=0.5\textwidth]{pdf/J0942-5552}}
\subfloat[]{\includegraphics[width=0.5\textwidth]{pdf/J1001-5507}}
\subfloat[]{\includegraphics[width=0.5\textwidth]{pdf/J1825-0935}}
\caption{Single pulsar model can explain the positron fraction very well. According
to the fitting result, the spectral indices are almost the same. }
\label{fig3}
\end{figure}
Using the parameters of the best fit results, the positron fraction
can be well reproduced by these single pulsar's contributions. Table
\ref{Tab:fit_parameters_of_single_pulsars} tells us that the normalization
$A{}_{prim,e^{-}}$ are around 0.5 and the spectral
indices $\alpha$ of different pulsars are around 2.
We can estimate the injection efficiency $\eta$ from the pulsar.
Take Geminga as an example, the spin-down energy loss rate of Geminga
$|\dot{E}(T)|=3.2\times10^{34}~\mathrm{erg}/\mathrm{s}$. The total radiation energy of the
magnetic dipole can be derived from eq.~\eqref{E-tot-2} as $E_{\mathrm{tot}}(T)\simeq |\dot{E}(T)|T^2/\tau_0=1.2\times10^{49}~\mathrm{erg}$.
From the fit, we get the injection energy $E_{\mathrm{out}}/2=10{}^{50.5}~\mathrm{GeV}\simeq 5.19\times10^{47}$ erg.
From \eqref{E-out-2}, we get $\eta\sim8.7\%$. This efficiency is
consistent with the previous studies by \cite{Hooper:2008kg} and \cite{Yin:2013vaa}.
We can perform similar studies on the other five pulsars, whose results
are listed in Table \ref{Tab:injection_efficiency}.
\begin{table}[!htbp]
\centering
\begin{tabular}{|c|c|c|c|c|}
\hline
Pulsar name & $|\dot{E}|(10^{33}~\mathrm{erg})$ & $E{}_{\mathrm{tot}}(10^{49}~\mathrm{erg})$ & $E{}_{\mathrm{out}}(10^{47}~\mathrm{erg})$ & $\eta(\%)$\tabularnewline
\hline
\hline
Geminga & 32 & 1.2 & 5.19 & 8.7\tabularnewline
\hline
J1741-2054 & 9.5 & 0.46 & 5.83 & 25\tabularnewline
\hline
Monogem & 38 & 0.16 & 1.90 & 24\tabularnewline
\hline
J0942-5552 & 3.1 & 0.21 & 7.02 & 67\tabularnewline
\hline
J1001-5507 & 0.68 & 0.043 & 1.89 & 88\tabularnewline
\hline
J1825-0935 & 4.6 & 0.082 & 5.09 & 120\tabularnewline
\hline
\end{tabular}\caption{Electron injection efficiency $\eta$ of the six nearby pulsars. For
single pulsar interpretation of positron fraction, the results of Geminga,
J1741-2054, Monogem and J0942-5552 are thought to be reasonable while
the posibilities of J1001-5507 and J1825-0935 as the high energy positron
sources can be excluded. }
\label{Tab:injection_efficiency}%
\end{table}
A smaller $\eta$
means it is easier for this pulsar to produce the same amount of positrons
and electrons. The efficiency required by J1001-5507 or J1825-0935
is too large to satisfy the physics condition for single pulsar interpretation.
Geminga, J1741-2054, Monogem and J0942-5552 are the only candidates
which survive from our selection so far.\footnote{Considering that
the uncertainty of $log{}_{10}(\frac{Q_{0}}{\mathrm{GeV}})$ from the fit is $\pm0.1$,
the $E{}_{\mathrm{out}}$ for J1001-5507 is $1.89^{+0.44}_{-0.36}\times 10^{47}erg$ .
Thus, $\eta = 88^{+21}_{-17} $ \% for J1001-5507. There is no enough strong evidence that this $\eta$ is smaller than 1.
One should also note that $\eta = 67^{+14}_{-12} $ \% for J0942-5552, which is 2$\sigma$ smaller than 1.}
\section{Multiple pulsars interpretation \label{sec-multiple}}
The extra high energy positrons may come from serveral pulsars. We
perform similar study for multiple pulsars as we do for a single pulsar.
Benefiting from the
study in Section \ref{sec-single-2}, we can assume that the spectral indices $\alpha$
of all the pulsars are the same. Considering the physical models of the pulsars are similar,
we make another assumption that the electron injection efficiencies $\eta$ are the same.
These two assumptions help us reduce the number of
free parameters. The discussion on $\eta$ from single pulsar in
Section \ref{sec-single-2} tells us that Geminga, J1741-2054 and Monogem will give
a much larger contribution to the high energy positron than J0942-5552.
In other words, the $\eta$ of J0942-5552 in Table \ref{Tab:injection_efficiency}
is much larger than that of Geminga, which implies that the contribution from J0942-5552
in the multiple pulsars interpretation can be negligible compared
with that from Geminga.
We choose three from the four ``surviving'' pulsars in the multiple
pulsars discussion. The input parameters are the age $T$, the distance
$d$ and the energy loss rate $\dot{E}$ of each pulsar while the
parameters we get from the fit is the normalization factor of primary
electron $A_{prim,e^{-}}$, the spectral index $\alpha$ and the
electron injection efficiency $\eta$. As shown in Fig.~\ref{Fig:multi_electron_flux}
(a), we obtain a good result from the multiple pulsar fit where $\chi{}^{2}/ndf=26.9/40$.
The parameters we get are $A{}_{prim,e^{-}}=0.50$, $\alpha=2.07$
and $\eta=2.58\%$.
\begin{figure}[!htbp]
\subfloat[]{\includegraphics[width=0.5\textwidth]{pdf/multi}}
\subfloat[]{\includegraphics[width=0.5\textwidth]{pdf/Electron_flux}}\caption{Three pulsars fit to the positron fraction is presented here as an
example of multiple pulsars fit, as show in (a). Using the same parameters,
(b) shows the fitted parameters from positron fraction reproduce the
electron flux when the solar modulation potential 550 MV is applied
here. The error band in (a) shows the variation of propagation parameters within 95\% C.L., while that in (b) shows combined effect of propagation parameters and variation of solar modulation potential from 400 MeV to 800 MeV.}
\label{Fig:multi_electron_flux}%
\end{figure}
The multiple pulsars interpretaion predicts a positron
fraction with a decrease up to 600 GeV and after that a bump up to
2000 GeV, which is possible to be observed with more accumulating AMS-02
data.
Using the parameters from the fit, we can reproduce the electron flux
measured by AMS-02 \cite{Aguilar:2014mma} in Fig.~\ref{Fig:multi_electron_flux}
(b). It shows that our electron background estimation in Section 2 $+$ pulsar contribution
matches the experimental data especially at high energies. Fig.~\ref{Fig:multi_electron_flux}
also shows that the effect of the uncertainty due to the propagation model is small at high energy. The solar
modulation potential is taken as 550 MV in the best fit result. The
solar modulation potential is varied between 400 MV and 800 MV to
show that its effect on low energy is quite large. To
reproduce the low energy electron flux more accurately, we need a
monthly low energy electron fluxes, which may be published by AMS collaboration to model solar modulation.
\section{Discussion and conclusion \label{sec-con}}
In this work, we investigate the possibility that the rise of the
positron fraction measured by AMS-02 can be explained by pulsars.
The propagation parameters and the injection spectrums of nuclei and
electrons are tuned according to the Boron-to-Carbon ratio and the
proton flux. It will be better to tune those parameters with Boron-to-Carbon
ratio and proton flux measured by AMS-02 since they are in the same
data taking period as the lepton fluxes. We find both the single
pulsar model and the multiple pulsar model can explain the AMS-02 data
very well. Six nearby pulsars are investigated as the single pulsar
sources of the high energy positrons and finally four survive from
all the conditions. The $\chi{}^{2}$s of these single pulsars in this work
are much smaller that those in \cite{Boudaud:2014dta}, mainly because we
set the cut-off energy equals to 5000 GeV while the authors of \cite{Boudaud:2014dta}
set it to 1000 GeV. With three mostly contributing pulsars, the multiple
pulsars model predicts a positron fraction with a decrease up to 600 GeV
and a bump up to 2000 GeV. For the low energy, a simple solar modulation
potential potential can not explain the measurement well. Thus, we
need the monthly electron fluxes which can describe solar activity
during the whole period.
It is shown that the positron excess measured by AMS-02 can be explained
by the pulsar scenario. Since the multiple pulsars can explain the experimental
data well, it will be difficult to exclude pulsar scenario by isotropy.
With accumulating AMS-02 data and future experiments, we can see the
positron fraction behavior up to higher energy which will either confirm or reject the
multiple pulsars scenario. If we consider other scenarios such as
Dark Matter, we have to look into other productions, antiproton for instance,
which have no contribution from pulsars.
\vspace{0.6cm}
|
{
"timestamp": "2016-04-28T02:14:19",
"yymm": "1504",
"arxiv_id": "1504.03312",
"language": "en",
"url": "https://arxiv.org/abs/1504.03312"
}
|
\section{Introduction}
In 1969 Fillmore characterized the (positive) finite rank operators that are sums of projections (\cite{Fp69}). In 1994 Wu and Choi announced that positive operators with essential norm strictly larger than 1 are sums of projections (\cite{Wpy94} and \cite{ChoiWu2014}). See also \cite{KRS02} and \cite{KRS03} for the special case of scalar multiples of the identity and \cite {DFKLOW} and then \cite {AMRS} for a different approach motivated by frame theory.
The complete characterization of the positive operators that are infinite sums of projections converging in the strong topology, was obtained by Kaftal, Ng, and Zhang in \cite[Theorem 1.1]{SSP}. Their method did apply also to $\sigma$-finite von Neumann factors and using other methods their results were partially extended to some C*-algebras and their multiplier algebras in a series of articles (\cite {KNZPISpan}-\cite {KNZCompJOT}. In particular, they obtained that for $A\in \mathcal{M}^+$ to be a sum of projections (converging in the SOT) it is sufficient that
\begin{itemize}
\item $\tau\big((A-I)_+\big)=\infty$, ($\tau$ a faithful normal semifinite trace) when $\mathcal{M}$ is a type I$_\infty$ or a type II$_\infty$ factor;
\item $\|A\|> 1$, when $\mathcal{M}$ is a type III factor.
\end{itemize}
Notice that $$\|A\|_e>1 ~ \Rightarrow ~ \tau\big((A-I)_+\big)=\infty~\Rightarrow ~\|A\|_e\ge1,$$ where $\|\cdot\|_e$ is the essential norm relative to the ideal $\mathcal J$ of compact operators of $\mathcal{M}$ (the norm closed ideal generated by the finite projections of $\mathcal{M}$, also called the Breuer ideal \cite{Bm68}, \cite{Bm69}, see also \cite {Sm71}, \cite{Kv77} among others.) Denote also by $\sigma_e(\cdot)$ the essential spectrum relative to this ideal, i.e., the spectrum of the canonical image in the (generalized) Calkin algebra $\mathcal M/\mathcal J$.
In $B(\mathcal{H})$, sums of projections can be further decomposed into sums of rank-one projections, which are all Murray-von Neumann equivalent. Thus an extension of \cite {SSP} is the study by
Bourin and Lee in \cite {SMV} and \cite {UE} of decompositions of positive operators $A\in B(\mathcal{H})$ into sums of positive operators {\it equivalent} to a given positive operator $B\ne 0$.
The first notion of equivalence they considered is the {\it Murray-von Neumann equivalence} (also called the {\it Pedersen equivalence} see \cite[Definition 6.1.2] {Bla88} and for more background \cite {OrtegaRordamThiel}): $B\sim C$ if $B=XX^*$ and $C=X^*X$ for some $X$ or equivalently $B=VCV^*$ for some partial isometry $V$ such that $V^*V=R_C$, $VV^*= R_B$ where $R_B$, $R_C$ denote the range projections of $B$ and $C$ respectively (in a von Neumann algebra $\mathcal{M}$ we require that $X$, $V\in \mathcal{M}$). In this paper we will refer to $\sim$ simply as equivalence and we will denote by $\cong$ the relation of unitary equivalence.
Bourin and Lee proved the following results:
\begin{theorem}\label{SVM}
\cite [Theorem 1.2] {SMV}
If $A, B\in B(\mathcal{H})^+$, $\tau((A-I)_+ )=\infty$, and $0\ne B$ is a contraction, then $A=\sum_{j=1}^\infty B_j$ for some $B_j\sim B$.
\end{theorem}
\begin{theorem}\label{UEBL}
\cite [Theorem 1.1] {UE}
If $A, B\in B(\mathcal{H})^+$, $N_A=0$ ($N_A$ is the projection on the kernel of $A$), $\|A\|_e\ge 1$, $B$ is a strict contraction and $0\in \sigma_e(B)$, then $A=\sum_{j=1}^\infty B_j$ for some $B_j\cong B$.
\end{theorem}
By {\it strict contraction} Bourin and Lee mean that $\|B\xi\|<\|\xi\|$ for every non-zero vector $\xi$. If $B$ is a positive contractions this is equivalent to the condition $\chi_{\{1\}}(B)=0$. Indeed $\|B\xi\|=\|\xi\|$ if and only if $(I-B^2)\xi=0$, that is $ \xi\in \chi_{\{1\}}(B).$
They also conjectured in \cite {SMV} and \cite {UE} that the same theorems hold for von Neumann factors.
The goal of the present paper is to prove their conjecture.
Bourin and Lee based their proofs mostly on Bourin's ``pinching theorem" \cite {PT} from which they obtained:
\begin{lemma}\label{Lemma 2.4}\cite[Lemma 2.4]{SMV}
If $A\in B(\mathcal{H})^+$, $\|A\|_e>1$, $\{B_j\}_{j=1}^\infty$ is a family of non-zero positive contractions, $\beta>0$, and $B_j\ge \beta R_{B_j}$ for all $j$ then $A= \sum_{j=1}^\infty C_j$ for some $C_j\sim B_j$.
\end{lemma}
One part of their proof depended on the previous result \cite[Theorem 1.1]{SSP} on decompositions into sums of projections, but they announced an independent proof of that result.
Our approach is the opposite. We use decompositions into sums of projections from \cite{SSP}, which, as mentioned above, hold for $\sigma$-finite factors. From these we obtain decompositions into sums of positive operators (Theorem \ref {equiv infinite trace}). As a consequence we answer affirmatively the conjecture of Bourin and Lee and then we proceed to deduce a form of the ``pinching theorem" (Corollary \ref{pinching corollary}).
Our paper is organized as follows.
In section 2 we obtain some decomposition results for positive operators in a $\sigma$-finite factor $\mathcal M$ and we strengthen some results on decompositions into sums of projections for the type II$_\infty$ case.
In section 3 we use a series of reductions to sums of projections to obtain our main result, Theorem \ref {equiv infinite trace}, which proves that an $A\in \mathcal{M}^+$ with $\tau((A-I)_+)=\infty$ can be decomposed as a sum $A=\sum_{j=1}^\infty C_j$ with $C_j\sim B_j$ for a preset sequence of contractions $\{B_j\}_{j=1}^\infty$ provided there is a $\beta >0$ and a non-zero projection $P$ such that $P\prec \chi_{(\beta,\infty)}(B_{j})$ for infinitely many indices $j$.
Of course that condition is always satisfied in the case of a single operator $B\ne 0$, i.e., when $B_j=B$ for all $j$, which answers affirmatively the conjecture of Bourin and Lee.
The connection between decompositions into sums of projections of a positive $A\in \mathcal M$ and the (block) diagonals of $A$ used in \cite{SSP}, \cite{FSP} is easily extended in Proposition \ref {block diagonal decomp} to sums of positive elements. Thus Corollary \ref{pinching corollary} provides a form of the ``pinching theorem" of \cite {PT} for von Neumann factors.
In section 4 we find sufficient conditions for the decomposition to hold with $C_j$ unitarily equivalent to $B_j$ (Theorem \ref {unit equiv sequence}) under the additional hypothesis that $0\in \sigma_e(B_j)$ (resp., $0\in \sigma(B_j)$ if $\mathcal M$ is type III). Part of the proof of Theorem \ref {unit equiv sequence} is an adaptation to von Neumann factors of the methods used by Bourin and Lee in \cite{UE} for a single operator in $B(\mathcal{H})$.
In section 5 we specialize the previous results to the case where $B_j=B$ for all $j$ and we find necessary conditions for that case (Proposition \ref{nec equiv}). Theorem \ref {equiv} provides additional and independent sufficient conditions. These are also sufficient when $\mathcal M$ is type III, or when $\|A\|=\|A\|_e$ (Corollaries \ref {norm=essnorm} and \ref{N&S type III}). Similar results are obtained for unitary equivalence (Corollaries \ref {UE} and \ref{P:NecSuf}).
\section{Preliminary decompositions}
Throughout this paper, unless otherwise stated, $\mathcal{M}$ denotes a $\sigma$-finite, infinite factor acting on a Hilbert space $\mathcal{H}$ and all the operators considered will belong to $\mathcal{M}$. When $\mathcal{M}$ is type I$_{\infty}$, $\tau$ denotes the standard trace, normalized on rank one projections, when $\mathcal{M}$ is type II$_{\infty}$, $\tau$ denotes a faithful, normal, semifinite, tracial weight.
If $A\in \mathcal{M}$ we denote by $R_A$ and $N_A$ the range projection and the projection on the null space of $A$ respectively, and when $A=A^*$, we denote by $A_+$ the positive part of $A$.
$\mathcal{J}$ denotes the norm-closed two-sided ideal generated by the finite projections of $\mathcal{M}$. In particular, when $\mathcal{M}=B(\mathcal{H})$ then $\mathcal{J}= K(\mathcal{H})$, the ideal of compact operators, and when $\mathcal{M}$ is type III, $\mathcal{J}=\{0\}$.
When $\mathcal{M}$ is semifinite, denote by $\pi$ the canonical map from $\mathcal{M}$ onto $\mathcal{M}/\mathcal{J}$, which is a unital, surjective, *-homomorphism. Then the essential norm and essential spectrum of $A$ are defined as $\|A\|_e=\|\pi(A)\|$ and respectively $\sigma_e(A)=\sigma(\pi(A))$.
If $P$ is a non-zero projection in $\mathcal{M}$ we denote by $\mathcal{M}_P$ the compression of $\mathcal{M}$ by $P$ and by $\tau_P$ the restriction of $\tau$ on $\mathcal{M}_P$, i.e. $\mathcal{M}_P=P\mathcal{M}P|_{P\mathcal{H}}$ and $\tau_P(X_P)=\tau(PXP)$ for all $X\in\mathcal{M}^+$.
Notice that when $P$ is infinite and $AP=PA$ then $\|AP\|_e=\|A_P\|_e$ while $$\sigma_e(AP)=
\begin{cases}
\sigma_e(A_P), & \text{ if $I-P$ is finite},\\
\sigma_e(A_P)\cup\{0\}, & \text{ if $I-P$ is infinite},
\end{cases}
$$ where $\|A_P\|_e$ and $\sigma_e(A_P)$ are considered relative to $\mathcal{M}_P$ and $\tau_P$.
We also make the convention that whenever we write $A=\bigoplus_{i=1}^{N} A_i$, where $N$ is some positive integer or $\infty$, then $A_i=AE_i$ for some mutually orthogonal projections $E_i\in \mathcal{M}$ that commute with $A$. Similarly, if we write $A=\bigoplus_{i=1}^{N} AE_i$ we will mean that the projections $E_i$ are mutually orthogonal and commute with $A$.
We collect below two known properties of the essential norm and essential spectrum.
For $A\in \mathcal{M}^+$ and $\mathcal{M}$ semifinite we have that:
1) $\sigma_e(A)=\{\,\lambda\in\mathbb{R}\mid \tau\left(\chi_{(\lambda-\epsilon,\lambda+\epsilon)}(A)\right)=\infty,\forall\, \epsilon>0\,\},$
2) $\|A\|_e>\lambda$ if and only if there exists $\epsilon>0$ such that $\tau\left(\chi_{(\lambda+\epsilon,\infty)}(A)\right)=\infty.$
We will also need the following decompositions of positive operators.\\
\begin{lemma} \label{gen ess spect}
Let $\mathcal{M}$ be semifinite, $A \in \mathcal{M}^+$, and $N\geq 1$ an integer or $N=\infty$.
\item [ i)]
If $\lambda \in \sigma_e(A)$ then $A=\bigoplus_{i=1}^{N}AE_i$ for some mutually orthogonal infinite projections $E_i\in \{A\}'$ with $\sum_{i=1}^N E_i=I$ and with $\lambda \in \sigma_e(A_{E_i})$ for every $i$.
\item [ ii)] $A=\bigoplus_{i=1}^{N}A_i$ for some $A_i\in \mathcal M^+$ with $\|A_i\|_e=\|A\|_e$ for every $i$.
\item [ ii$\,'$)] If $N=\infty$ the positive operators $A_i$ in ii) can be chosen to be locally invertible, that is, $A_i\ge \alpha_iR_{A_i}$ for some $\alpha_i>0$ (we use the convention that the operator 0 is locally invertible).
\item [ iii)] If $\tau((A-tI)_+)=\infty$ for some $t>0$ then $A=\bigoplus_{i=1}^{N}A_i$ for some $A_i\in \mathcal M^+$ with $\tau\left((A_i-tI)_+\right)=\infty$ for all $i$.
\item [ iii$\,'$)] If $N=\infty$ the positive operators $A_i$ in iii) can be chosen to be locally invertible.\end{lemma}
\begin{proof}
\item [ i)] When $\tau\left(\chi_{\{\lambda\}}(A)\right)=\infty$, decompose $\chi_{\{\lambda\}}(A)=\sum_{k=1}^NQ_k$ into a sum of mutually orthogonal infinite projections $Q_k$. Take $E_k=Q_k$ for $k\ge2$ and $E_1=I-\sum_{k=2}^NQ_k$. It is immediate to verify that these projections satisfy the condition in i).
Assume therefore that $\tau\left(\chi_{\{\lambda\}}(A)\right)<\infty$. Choose sequences $\{c_k\}_{k=1}^{\infty}$ and $\{d_k\}_{k=1}^{\infty}$ such that $\lim_{k\to \infty}c_k=\lim_{k\to \infty}d_k=\lambda$, the intervals $[c_k,d_k)$ are disjoint and such that for every $L\ge 1$, $ \bigcup_{k=L}^{\infty}[c_k,d_k)= [\alpha_L, \beta_L)\setminus\{\lambda\}$ for some $\alpha_L< \lambda< \beta_L$. For instance one can take a strictly increasing sequence $\{a_k\}$ and a strictly decreasing sequence $\{b_k\}$, both convergent to $\lambda$, and then relabel the intervals $\{[a_k,a_{k+1})\}$ and $\{[b_{k+1},b_k)\}$ as $\{[c_k,d_k)\}$.
By 1) above,
$$
\tau\left(\chi_{\{\lambda\}}(A)\right)+\sum_{k=L}^{\infty}\tau\left(\chi_{[c_k,d_k)}(A)\right)= \tau\left(\chi_{[\alpha_L,\beta_L)}(A)\right)= \infty.
$$
Hence $\sum_{k=L}^{\infty}\tau\left(\chi_{[c_k,d_k)}(A)\right)=\infty$ for every $L\geq 1$. Now split the sequence of intervals $\{[c_k,d_k)\}_{k=1}^{\infty}$ into $N$ disjoint subsequences $\{[c_k^i,d_k^i)\}_{k=1}^{\infty}$, $1\leq i\leq N$, such that $\sum_{k=L}^{\infty}\tau\left(\chi_{[c_k^i,d_k^i)}(A)\right)=\infty$ for every $L\geq 1$ and every $1\leq i\leq N$.
Let $\Delta_i=\bigcup_{k=1}^{\infty}[c_k^i,d_k^i)$ and $E_i=\chi_{\Delta_i}(A)$. Furthermore, replace the projection $E_1$ with $I-\sum_{k=2}^N E_k$. From the construction and property 1) of the essential spectrum it is easy to verify that the projections $\{E_k\}_{k=1}^N$ have the desired properties.
\item [ ii)] follows from i) by noticing that $A\geq 0$ implies $\|A\|_e \in \sigma_e(A)$.
\item [ iii)] Because $\tau((A-tI)_+)=\infty$ we necessarily have $\tau\left(\chi_{(t,\infty)}(A)\right)=\infty$ which implies by property 2) of the essential norm that $\|A\|_e\geq t$.
If $\|A\|_e>t$ then by ii) we have $A=\bigoplus_{i=1}^{N}A_i$ for some $\|A_i\|_e=\|A\|_e>t$ for every $i$. We are done because $\|A_i\|_e>t$ implies $\tau((A_i-tI)_+)=\infty$.
Hence, assume that $\|A\|_e=t$. Then $\tau\left(\chi_{(s,\infty)}(A)\right)<\infty$ for every $s>t$, and in particular $\tau\left(\chi_{(u,v]}(A)\right)<\infty$ for every $(u,v]\subset(t,\infty)$. Write $(t,\infty)$ as $\bigcup_{i=1}^{\infty}(u_i,v_i]$ where the intervals are disjoint.
Then $$\infty=\tau\left((A-tI)\chi_{(t,\infty)}(A)\right)=\sum_{i=1}^{\infty}\tau\left((A-tI)\chi_{(u_i,v_i]}(A)\right)$$ and since $\tau\left((A-tI)\chi_{(u_i,v_i]}(A)\right)<\infty$ for every $i$ we can split the sequence $\{(u_i,v_i]\}_1^{\infty}$ into $N$ disjoint subsequences such that by denoting $\Delta_k$ the union of the intervals in the $k^{th}$ subsequence we obtain $\tau\left((A-tI)\chi_{\Delta_k}(A)\right)=\infty$ for every $k$. Take $E_1=\chi_{[0,t]\cup \Delta_1}(A)$, $E_k=\chi_{\Delta_k}(A)$ for $k\geq 2$, and $A_k:=AE_k$. Then $A=\bigoplus_{i=1}^{N}A_i$ and $\tau((A_i-tI)_+)=\infty$ for every $i$.
\item[ ii$\,'$) \& iii$\,'$)]
Choose $s>0$ such that $\|A\chi_{(s, \infty)}(A)\|_e= \|A\|_e$,
(resp., such that $\tau((A\chi_{(s, \infty)}(A)-tI)_+)=\infty$ in the case when $\tau((A-tI)_+)=\infty$ for some $t>0$).
By ii) (resp., iii)) applied to $A\chi_{(s, \infty)}(A)$, we can find a sequence of mutually orthogonal projections $F_j\le \chi_{(s, \infty)}(A)$ that commute with $A\chi_{(s, \infty)}(A)$ and hence with $A$ and such that $\|A\chi_{(s, \infty)}(A)F_j\|_e= \|A\chi_{(s, \infty)}(A)\|_e$ (resp., and such that $\tau((A\chi_{(s, \infty)}(A)F_j-tI)_+)=\infty)$. Thus $\|AF_j\|_e= \|A\|_e$ (resp., $\tau((AF_j-tI)_+)=\infty$) for all $j$. Let $s_n\downarrow 0$ with $s_1=s$ and let $E_j:=F_j\oplus \chi_{(s_{j+1}, s_j]}(A)$. Then it is immediate to verify that $A_j:=AE_j$ satisfy the required conditions.
\end{proof}
The next lemma is a natural analogue of Lemma \ref{gen ess spect} when dealing with type III factors. The proof is similar but simpler than the proof of Lemma \ref{gen ess spect} and hence we omit it.
\begin{lemma} \label{gen spect}
Let $\mathcal{M}$ be type III, $A \in \mathcal{M}^+$, and $N\geq 1$ an integer or $N=\infty$.
\item [ i)]
If $\lambda \in \sigma(A)$ then $A=\bigoplus_{i=1}^{N}AE_i$ for some mutually orthogonal non-zero projections $E_i\in\{A\}'$ with $\lambda \in \sigma(A_{E_i})$ for every $i$.
\item [ ii)] $A=\bigoplus_{i=1}^{N}A_i$ for some $A_i\in \mathcal M^+$ with $\|A_i\|=\|A\|$ for every $i$.
\item [ ii$\,'$)] If $N=\infty$ the positive operators $A_i$ in ii) can be chosen to be locally invertible.
\end{lemma}
When $\mathcal{M}$ is semifinite, $A \in \mathcal{M}^+$ and $\tau((A-I)_+)=\infty$, we know from \cite[Theorem 6.6]{SSP} that $A$ is a sum of projections. If $\mathcal{M}$ is of type I, by further decomposing these projections it follows that $A$ is a sum of equivalent projections. We need to show that the same conclusion holds also in the case when $\mathcal{M}$ is of type II$_\infty$. We will achieve this via embedding a type I$_\infty$ factor in the II$_\infty$ factor through the following reductions.
\begin{lemma} \label{dec diff}
Let $\mathcal{M}$ be semifinite and $A\in \mathcal{M}^+$ with $\tau(A)=\infty$, then there exist an infinite sequence of mutually orthogonal projections $E_n$ with $\tau(E_n)=1$ and a sequence of positive numbers $t_n$ with $\sum_{n=1}^{\infty}t_n=\infty$, such that $A\geq \sum_{n=1}^{\infty}t_nE_n$.
\end{lemma}
\begin{proof}
If $\|A\|_e>0$, then there is some $t>0$ for which $\tau\left(\chi_{(t,\infty)}(A)\right)=\infty$ and $A\geq t\chi_{(t,\infty)}(A)$.
The conclusion is now obvious by choosing mutually orthogonal projections $E_n$ such that $\tau(E_n)=1$ and $\chi_{(t,\infty)}(A)= \sum_{n=1}^{\infty}E_n$.
Assume now that $\|A\|_e=0$. The case when $\mathcal{M}$ is a type I$_{\infty}$ factor is immediate since $A$ is then diagonalizable.
So we can assume that $\mathcal{M}$ is type II$_{\infty}$. For $t\geq 0$ let $\mu_t(A)$ be the $t^{th}$ singular value of $A$, that is $$\mu_t(A)=\inf \{\,s\geq0\mid \tau\left(\chi_{(s,\infty)}(A)\right)\leq t\,\}.$$ For equivalent definitions and properties of the singular values see \cite{F}.
Let $t_1=\mu_1(A)$. Then $\|A\|\geq t_1>0$ because $\tau(A)=\infty$ implies that $\tau\left(\chi_{(0,\infty)}(A)\right)=\infty$ and $$\tau\left(\chi_{(t_1,\infty)}(A)\right)\leq 1\leq \tau\left(\chi_{[t_1,\infty)}(A)\right)<\infty,$$
the latter inequality holding because $\|A\|_e=0$.
Therefore $$0\leq1-\tau\left(\chi_{(t_1,\infty)}(A)\right)\leq\tau\left(\chi_{\{t_1\}}(A)\right)< \infty$$
and we can find a projection $P_1\leq \chi_{\{t_1\}}(A)$ such that $\tau(P_1)=1-\tau\left(\chi_{(t_1,\infty)}(A)\right)$.
Let $E_1=\chi_{(t_1,\infty)}(A)+P_1$. Then $E_1$ has trace 1 and commutes with $A$.
Apply the above step to $A_1=A(I-E_1)$ to find $0<\mu_1(A_1)=t_2\leq t_1$ and a trace 1 projection $E_2$ such that $E_2=\chi_{(t_2,\infty)}(A_1)+P_2$ for some projection $P_2\leq \chi_{\{t_2\}}(A_1)$. Since $I-E_1=\chi_{[0,t_1)}(A)+\left(\chi_{\{t_1\}}(A)-P_1\right)$ we see that $$E_2=\chi_{(t_2,\infty)}(A)(I-E_1)+P_2=\chi_{(t_2,t_1)}(A)+P_2+\left(\chi_{\{t_1\}}(A)-P_1\right).$$
Reasoning in the same way we construct by induction a non-increasing sequence of positive numbers $\{t_j\}_{j=1}^{\infty}$ and two sequences of projections $\{E_j\}_{j=1}^{\infty}$ and $\{P_j\}_{j=1}^{\infty}$ such that $\tau(E_j)=1$, $P_j \leq \chi_{\{t_j\}}(A)$ and $$E_j=\chi_{(t_j,t_{j-1})}(A)+P_j+\left(\chi_{\{t_{j-1}\}}(A)-P_{j-1}\right)$$ for every $j$, where $t_0=\|A\|$ and $P_0=0$.
We have that the projections $E_j$ are mutually orthogonal, and $$\chi_{(t_n,\infty)}(A)\leq E_1+\ldots+E_n\leq \chi_{[t_n,\infty)}(A)\quad\text{for every $n$.}$$ Together with $\tau(E_1+\ldots+E_n)=n$ and $\|A\|_e=0$ this implies that $\lim_{n\to {\infty}}t_n=0$ and therefore $\sum_{j=1}^{\infty}E_j=\chi_{(0,\infty)}(A)$.
From the construction we have that $E_j\in \{A\}'$ and $$\chi_{(t_j,t_{j-1})}(A)\le E_j\le \chi_{[t_j,t_{j-1}]}(A).$$ Hence $t_jE_j\leq AE_j\leq t_{j-1}E_j$ and hence $$\sum_{j=1}^{\infty}t_jE_j\le A\le \sum_{j=1}^{\infty}t_{j-1}E_j.$$ The conclusion is now obvious.
\end{proof}
\begin{lemma} \label{trace geq 1}
Let $\mathcal{M}$ be type II$_\infty$ and $A \in \mathcal{M}^+$ with $\tau((A-I)_+)=\infty$. Then $A=\sum_{i=1}^{\infty}R_i$ for some projections $R_i$ with $\tau(R_i)\geq1$.
\end{lemma}
\begin{proof}
We first show that $A\geq \sum_{n=1}^{\infty}P_n$ for some sequence $\{P_n\}$ of trace 1 projections.
Apply Lemma \ref{dec diff} to $(A-I)_+$ to find trace 1 mutually orthogonal projections $E_n$ and $t_n\ge 0$ such that $(A-I)_+\ge \sum_{n=1}^{\infty}t_nE_n$ and $\sum_{n=1}^{\infty}t_n=\infty$. Set $A_0:=(A-I)_+- \sum_{n=1}^{\infty}t_nE_n$.
Then
\begin{align}
A&=A\chi_{(0,1]}(A)+\chi_{(1,\infty)}(A)+(A-I)_+\notag \\
&=A\chi_{(0,1]}(A)+\chi_{(1,\infty)}(A)+A_0+\sum_{n=1}^{\infty}t_nE_n\notag \\
&=A\chi_{(0,1]}(A)+A_0+\big(\chi_{(1,\infty)}(A)-\sum_{n=1}^{\infty}E_n\big)+\sum_{n=1}^{\infty}(t_n+1)E_n\notag\\
&=A_1+A_2,\notag
\end{align}
where
\begin{align*} A_1&=A\chi_{(0,1]}(A)+A_0+\big(\chi_{(1,\infty)}(A)-\sum_{n=1}^{\infty}E_n\big)\geq 0\\mathcal{A}_2&=\sum_{n=1}^{\infty}(t_n+1)E_n.
\end{align*} Then $ \tau((A_2-I)_+)=\sum_{n=1}^{\infty}t_n=\infty.$
By considering a separable, infinite dimensional Hilbert space $\mathcal{H}$ and an injective, normal, *-homomorphism of $B(\mathcal{H})$ into $\mathcal{M}$ such that the projections $E_{n}$ correspond to rank one projections in $B(\mathcal{H})$ that sum to $I$, we can apply \cite [Theorem 1.1 i)]{SSP} to conclude that $A_2=\sum_{i=1}^{\infty}P_i$ where $\tau(P_i)=1$ for all $i$.
Hence $A\geq \sum_{i=1}^{\infty}P_i$ and $\tau(P_i)=1$ for all $i$.
Now by applying Lemma \ref{gen ess spect} $iii)$ we can write $A$ as $A=\bigoplus_{i=1}^4A_i$ such that $\tau((A_i-I)_+)=\infty$ for $1\le i\le4$. From the first part of the proof decompose $A_i$ for $1\le i\le 2$ as $A_i=\sum_{j=1}^{\infty}P_j^i+B_i$ with projections $\tau(P_j^i)=1$ for every $j\geq 1$ and with remainder $B_i\geq 0$. Then $$A=\Big(\sum_{j=1}^{\infty}P_j^1+(B_1\oplus A_3)\Big)\oplus \Big(\sum_{j=1}^{\infty}P_j^2+(B_2\oplus A_4)\Big).$$
Since $$\tau(((B_1\oplus A_3)-I)_+)=\tau(((B_2\oplus A_4)-I)_+)=\infty,$$
we can apply \cite[Theorem 6.6]{SSP} to conclude that $$B_1\oplus A_3=\sum_{j=1}^{\infty}Q_j^1\quad\text{and}\quad B_2\oplus A_4=\sum_{j=1}^{\infty}Q_j^2$$ for some projections $Q_j^1$ and $Q_j^2$.
Then
\begin{align*}
A&=\Big(\sum_{j=1}^{\infty}P_j^1+\sum_{j=1}^{\infty}Q_j^1\Big)\oplus \Big(\sum_{j=1}^{\infty}P_j^2+\sum_{j=1}^{\infty}Q_j^2\Big)\\ &=\sum_{j=1}^{\infty}\left(P_j^1\oplus Q_j^2\right)+\sum_{j=1}^{\infty}\left(P_j^2\oplus Q_j^1\right).
\end{align*}
Since $\tau\left(P_j^1\oplus Q_j^2\right)\geq 1$ and $\tau\left(P_j^2\oplus Q_j^1\right)\geq 1$ for every $j$, this concludes the proof.
\end{proof}
\begin{proposition}\label{proj}
Let $\mathcal{M}$ be semifinite, $0\ne P\in \mathcal{M}$ be a projection, and $A\in \mathcal{M}^+$ with
$\tau((A-I)_+)=\infty$. Then $A=\sum_{j=1}^{\infty}P_j$ for some projections $P_j\sim P$.
\end{proposition}
\begin{proof}
Assume first that $P$ is infinite.
By applying Lemma \ref{gen ess spect} $iii)$ we can write $A=\bigoplus_{i=1}^{\infty}A_i$ with $\tau((A_i-I)_+)=\infty$ for every $i$. Using Lemma \ref{trace geq 1} when $\mathcal{M}$ is type II$_{\infty}$ and \cite[Theorem 1.1 i)]{SSP} when $\mathcal{M}$ is type I$_{\infty}$ we obtain that $A_i=\sum_{j=1}^{\infty}P_{ij}$ with $\tau(P_{ij})\geq 1$ for every $j$ and every $i$.
So $$A=\bigoplus_{i=1}^{\infty}A_i=\bigoplus_{i=1}^{\infty}
\sum_{j=1}^{\infty}P_{ij}=\sum_{j=1}^{\infty}\bigoplus_{i=1}^{\infty}P_{ij}
=\sum_{j=1}^{\infty}P_{j}$$ where $P_j=\bigoplus_{i=1}^{\infty}P_{ij}$ is an infinite projection for every $j$. Since $\mathcal{M}$ is $\sigma$-finite we conclude that $P_j\sim P$ for every $j$.
If $P$ is finite decompose first $A$ as $A=\sum_{i=1}^{\infty}Q_i$ with $Q_i$ infinite and then write each $Q_i$ as $Q_i=\sum_{j=1}^{\infty}P_{ij}$ with $P_{ij}\sim P$ for every $j$ and $i$.
\end{proof}
\begin{corollary}\label{sequence of projections}
Let $\mathcal{M}$ be semifinite, $A\in \mathcal{M}^+$ with $\tau((A-I)_+)=\infty$, $\{P_j\}_{j=1}^{\infty}\subseteq \mathcal{M}$ be a sequence of projections where either infinitely many of the projections $P_j$ are infinite or finitely many are infinite and $\sum\{ \tau(P_j)\mid \tau(P_j)<\infty\}=\infty$. Then $A=\sum_{j=1}^{\infty}Q_j$ for some $Q_j\sim P_j$.
\end{corollary}
\begin{proof}
By passing if necessary to equivalent projections we can assume that the projections $P_j$ are mutually orthogonal. The assumption on the sequence $\{P_j\}$ guarantees that we can write $\mathbb{N}=\bigcup_{k=1}^\infty J_k$ for some infinite, disjoint sets $J_k$ such that $\sum_{j\in J_k}\tau(P_j)=\infty$ for every $k$.
By Proposition \ref {proj}, $A=\sum_{k=1}^{\infty}R_k$ with $R_k$ infinite projections for every $k$.
But then $R_k\sim \bigoplus_{j\in J_k}P_j$ and therefore $R_k=\sum_{j\in J_k}Q_j$ with $Q_j\sim P_j$ for $j\in J_k$, $k\ge1$.
The conclusion is now obvious.
\end{proof}
\begin{remark}
Assume that $A=\sum_{j=1}^{\infty}Q_j$ with $Q_j\sim P_j$ where the projections $P_j$ are infinite for $1\le j\le n$ and $\sum_{j=n+1}^{\infty}\tau(P_j)<\infty$. Then $\sum_{j=n+1}^{\infty}Q_j$ is trace class and consequently $\|A\|_e=\|Q_1+\dots+Q_n\|_e\le n$. Thus the conclusion of Corollary \ref {sequence of projections} fails if $\|A\|_e>n$.
\end{remark}
\section{Sums of equivalent copies of a sequence of operators}
It is clear that if we want to decompose a ``large" $A\in \mathcal{M}^+$ into $A=\sum_{j=1}^\infty C_j$ for some $C_j\sim B_j$, the sequence $B_j$ cannot be ``too small".
An obvious obstruction is that if every $B_j$ belongs to the ideal of compact operators $\mathcal J $, $\sum_{j=1}^\infty \|B_j\|< \infty$, and $A=\sum_{j=1}^\infty C_j$ for some $C_j\sim B_j$, then $A\in \mathcal J$. The same conclusion holds if $B_j=B_j'+ B''_j$ with $B_j', B''_j\in \mathcal J^+$,
$\sum_{j=1}^\infty \|B'_j\|< \infty$, and $\sum_{j=1}^\infty \tau(B''_j)< \infty$.
A natural condition on the sequence $\{B_j\}_{j=1}^{\infty}$ to avoid this obstruction is to ask that $\inf_{j\ge1}\tau(\chi_{(\beta, \infty)}(B_j))>0$ for some $\beta>0$. This is equivalent to the existence of a non-zero projection $P\prec \chi_{(\beta, \infty)}(B_j)$, that is, to the condition $B_j\ge \beta P_j$ with $P_j\sim P$.
Compare with \cite [Lemma 2.4]{SMV} where the condition on $B_j$ is that $B_j\ge \beta R_{B_j}$ for some $\beta >0$ and all $j$.
We start by showing that if $A\in \mathcal{M}^+$ is ``large" with respect to the norms of the sequence $\{B_j\}$, then we can absorb a sequence of ``copies" of these operators into $A$ and be left with a ``large" remainder.
\begin{lemma}\label{subsume a sequence}
Let $A\in \mathcal{M}^+$ and $\{B_j\}_{j=1}^\infty \subseteq \mathcal{M}^+$ with $\alpha:=\sup_{j\ge1}\|B_j\|<\infty$ and $\tau((A-\alpha I)_+)=\infty$ if $\mathcal{M}$ is semifinite (resp., $\|A\|>\alpha$ when $\mathcal{M}$ is type III). Then $A=\sum_{j=1}^\infty C_j+A'$ for some $C_j\sim B_j$, with the projection $N_{C_j}$ on the null space of $C_j$ being infinite for every $j$, and $A'\ge 0$ with $\tau((A'-\alpha I)_+)=\infty$ and $\|A'\|_e=\|A\|_e$ if $\mathcal{M}$ is semifinite and $\|A'\|=\|A\|$ when $\mathcal{M}$ is type III.
\end{lemma}
\begin{proof}
If $\alpha=0$ there is nothing to prove. So we can assume that $\alpha>0$.
From Lemma \ref{gen ess spect} ii) and iii) (resp., Lemma \ref{gen spect} ii) when $\mathcal{M}$ is type III) we can decompose $A$ as $A=\bigoplus_{j=0}^\infty A_j$ with $\tau((A_j-\alpha I)_+)=\infty$ and $\|A_j\|_e=\|A\|_e$ (resp., $\|A_j\|=\|A\|$ when $\mathcal{M}$ is type III) for all $j$. There are $s_j\ge \alpha$ such that $\chi_{(s_j,\infty)}(A_j)$ is infinite.
Since $R_{B_j}\prec I\sim \chi_{(s_j,\infty)}(A_j)$ for all $j\ge 1$, we can find a partial isometry $V_j$ with $V_j^*V_j=R_{B_j}$ and $V_jV_j^*\le \chi_{(s_j,\infty)}(A_j)$. Taking $C_j=V_jB_jV_j^*$ we see that $C_j\sim B_j$, $N_{C_j}$ is infinite and that $$C_j\le \|B_j\|V_jV_j^*\le \alpha\chi_{(s_j,\infty)}(A_j)\le s_j\chi_{(s_j,\infty)}(A_j)\le A_j.$$
Hence $A=(A_0\oplus (\sum_{j=1}^\infty A_j-C_j))+\sum_{j=1}^\infty C_j$ and by taking $$A'=A_0\oplus \Big(\sum_{j=1}^\infty (A_j-C_j)\Big)\ge 0$$ we have immediately the desired conclusion.
\end{proof}
The reason that we go to the additional step of choosing the operators $C_j$ with infinite null space in this lemma as well as in other results of this section, is that this will enable us to pass from the equivalence relation $\sim$ to the unitary equivalence relation $\cong$ in the proof of Lemma \ref {monotone sequence} below as well as in Section 4.
By using decomposition of positive elements into sums of projections, we can now eliminate the remainder in Lemma \ref {subsume a sequence} in the following special case.
\begin{lemma}\label{common proj direct summand}
Let $A\in \mathcal{M}^+$ and $\{B_j\}_{j=1}^\infty \subseteq \mathcal{M}^+$ with $\alpha:=\sup_{j\ge1}\|B_j\|<\infty$ and $\tau((A-\alpha I)_+)=\infty$ when $\mathcal{M}$ is semifinite (resp., $\|A\|>\alpha$ when $\mathcal{M}$ is type III). Assume furthermore that there is a number $t>0$ and a non-zero projection $P$ such that $B_j=tP_j\oplus B'_j$ for some $B'_j\ge 0$ and for projections $P_j\sim P$. Then $A=\sum_{j=1}^{\infty}C_j$ for some $C_j\sim B_j$ with $N_{C_j}$ infinite.
\end{lemma}
\begin{proof}
Assume first that $\mathcal{M}$ is semifinite.
By applying Lemma \ref{gen ess spect} iii) decompose $A$ as $A=\bigoplus_{i=1}^4A_i$ such that $\tau((A_i-\alpha I)_+)=\infty$ for $1\le i\le4$.
By Lemma \ref {subsume a sequence} we have that $A_i=\sum_{j=1}^\infty B''_{4j+1-i}+A_i'$ where $B''_{4j+1-i}\sim B'_{4j+1-i}$ for all $j$ and $\tau((A'_i-\alpha I)_+)=\infty$ for $1\le i\le4$.
Now $\tau\left((A_i'-\alpha I)_+\right)=\infty$ and $\alpha\ge t$ imply that $\tau\left(\left(\frac{1}{t}A_i'-I\right)_+\right)=\infty$ and hence
from Proposition \ref{proj}, we have that for each $i$, $\frac{1}{t}A_i'=\sum_{j=1}^{\infty}P'_{4j+i-4}$ with $P'_{4j+i-4}\sim P$.
Then,
\begin{align*}
A&=\Big(\sum_{ j=1}^\infty B''_{4j}+ t\sum_{j=1}^\infty P'_{4j-3}\Big)\oplus\Big(\sum_{ j=1}^\infty B''_{4j-1}+ t\sum_{ j=1}^\infty P'_{4j-2}\Big)\\&\oplus \Big(\sum_{j=1}^\infty B''_{4j-2}+ t\sum_{j=1}^\infty P'_{4j-1}\Big)\oplus \Big(\sum_{j=1}^\infty B''_{4j-3}+ t\sum_{j=1}^\infty P'_{4j}\Big)\\
&=\Big(\Big(\sum_{j=1}^\infty B''_{4j}\oplus \sum_{j=1}^\infty tP'_{4j}\Big)+\Big(\sum_{j=1}^\infty B''_{4j-1}\oplus \sum_{j=1}^\infty tP'_{4j-1}\Big)\Big)\\
&\oplus\Big(\Big(\sum_{j=1}^\infty B''_{4j-2}\oplus \sum_{j=1}^\infty tP'_{4j-2}\Big)+\Big(\sum_{j=1}^\infty B''_{4j-3}\oplus \sum_{j=1}^\infty tP'_{4j-3}\Big)\Big)\\
&=\Big(\sum_{j=1}^\infty C_{4j}+ \sum_{j=1}^\infty C_{4j-1}\Big)\oplus\Big(\sum_{j=1}^\infty C_{4j-2}+ \sum_{j=1}^\infty C_{4j-3}\Big)
\end{align*}
where $C_j=B''_j\oplus tP'_j\sim B_j$ for all $j$. Finally, notice that both $$\Big(\sum_{j=1}^\infty C_{4j}+ \sum_{j=1}^\infty C_{4j-1}\Big) \text{ and } \Big(\sum_{j=1}^\infty C_{4j-2}+ \sum_{j=1}^\infty C_{4j-3}\Big)$$ have infinite trace and hence infinite rank, which proves that $N_{C_j}$ is infinite for all $j$.
When $\mathcal{M}$ is type III the proof is similar, the only difference being that we must replace the essential norm with the operator norm, Lemma \ref{gen ess spect} with Lemma \ref{gen spect} and Proposition \ref{proj} with \cite[Theorem 1.1 (iii)]{SSP}.
\end{proof}
The condition in Lemma \ref {common proj direct summand} that each $B_j$ has a direct summand $tP_j$ with $P_j\sim P\ne 0$ is of course too limiting. A more natural condition and closer to the spirit of the one in Lemma \ref {Lemma 2.4} (i.e., \cite [Lemma 2.4]{SMV}) is that $B_j\ge tP_j$ with $P_j\sim P\ne 0$. In Theorem \ref{equiv infinite trace} we are going to prove that this condition is indeed sufficient and in fact, that it is enough to require it for infinitely many indices. The core of the argument is the following approximation lemma.
\begin{lemma}\label{monotone sequence}
Let $A\in \mathcal{M}^+$ and $\{B_j\}_{j=1}^\infty \subseteq \mathcal{M}^+$ with $\alpha:=\sup_{j\ge1}\|B_j\|<\infty$ and $\tau((A-\alpha I)_+)=\infty$ when $\mathcal{M}$ is semifinite (resp., $\|A\|>\alpha$ when $\mathcal{M}$ is type III). Assume that there is $\beta>0$ such that $\chi_{(\beta,\infty)}(B_j)$ is infinite for every $j$. Then for every $\epsilon >0$, $A=\sum_{j=1}^{\infty}C_j+R$ for some $C_j\sim B_j$ with $N_{C_j}$ infinite and $0\le R\le \epsilon A$.
\end{lemma}
\begin{proof}
By choosing a sequence of partial isometries $V_j$ with $V_j^*V_j=R_{B_j}$ and mutually orthogonal range projections $V_jV_j^*$, we can replace $B_j$ with the equivalent elements $B'_j:=V_jB_jV_j^*$ which have mutually orthogonal range projections $R_{B'_j}$. Since $R_{B'_j}\ge \chi_{(\beta,\infty)}(B'_j)\sim\chi_{(\beta,\infty)}(B_j)$ is infinite, we conclude that $N_{B'_j}$ is infinite for every $j$. Thus to simplify notations, assume directly that $N_{B_j}$ is infinite for every $j$.
First we show that there is a subsequence $\{j_k\}_{k=1}^{\infty}$ and a monotone sequence $\{\gamma_{j_k}\}_{k=1}^{\infty}\subseteq (\beta,\alpha]$ with $t:=\lim_{k\to \infty} \gamma_{j_k}$ and $\chi_{(\gamma_{j_k},t]}(B_{j_k})$ (resp., $\chi_{(t,\gamma_{j_k}]}(B_{j_k})$) is infinite for all $k$ if $\{\gamma_{j_k}\}$ is monotone increasing (resp., decreasing).
To see this, choose for every $j$ an interval $(\beta_j, \alpha_j]\subset (\beta, \alpha]$ of length $\le \frac{1}{2^{j}}$ and such that $ \chi_{(\beta_j, \alpha_j]}(B_j)$ is infinite. Then choose a subsequence $\{j_k\}$ such that both $\{\beta_{j_k}\}$ and $\{\alpha_{j_k}\}$ are monotone and set $ t:=\lim_k \beta_{j_k}=\lim_k\alpha_{j_k}$.
If both $\{\beta_{j_k}\}$ and $\{\alpha_{j_k}\}$ are increasing, then set $\gamma_{j_k}:=\beta_{j_k}$, if both are decreasing set $\gamma_{j_k}:=\alpha_{j_k}$. If $\{\beta_{j_k}\}$ is increasing and $\{\alpha_{j_k}\}$
is decreasing, then
$$\chi_{(\beta_{j_k},\alpha_{j_k}]}(B_{j_k})=\chi_{(\beta_{j_k},t]}(B_{j_k})+\chi_{(t, \alpha_{j_k}]}(B_{j_k})$$
hence for each $k$ at least one of the projections $\chi_{(\beta_{j_k},t]}(B_{j_k})$ and $\chi_{(t, \alpha_{j_k}]}(B_{j_k})$ must be infinite. Thus by passing if necessary to a subsequence we can assume that either the first projection is always infinite or the second projection is always infinite and define $\gamma_{j_k}$ accordingly.
To simplify notations, by invoking Lemma \ref{subsume a sequence} we can assume
that $j_k=k$ and that $0< \frac{|\gamma_1-t |}{\min\{t,\gamma_1\} } <\epsilon$.
Define
$
\begin{cases}
t_j=\gamma_j, s_j=t, s=t & \text{when $\{\gamma_j\}$ is increasing}\\
t_j=t, s_j=\gamma_j, s=\gamma_{1} & \text{when $\{\gamma_j\}$ is decreasing}
\end{cases}.
$\\
Then $0<\beta\le t_j<s_j\le s\le \alpha$, $\chi_{(t_{j},s_{j}]}(B_{j})$ is infinite for every $j$, and
$$
\sup_{j\ge1}\frac{s-t_{j}}{s}=
\begin{cases}
\frac{t-\gamma_{1}}{t} & \text{when $\{\gamma_j\}$ is increasing}\\
\frac{\gamma_1-t}{\gamma_1} & \text{when $\{\gamma_j\}$ is decreasing}\end{cases}
\le \frac{|\gamma_{1}-t|}{\min\{t,\gamma_{1}\}}<\epsilon.
$$
Now set $P_{j}:=\chi_{(t_{j},s_j]}(B_{j})$ and $D_{j}:=sP_{j}\oplus B_{j}(I-P_{j}).$
Then
$$0\le D_{j}-B_{j}=(sI-B_{j})P_{j}\le(s-t_{j})P_{j}\quad\forall\, j.$$
By construction, $R_{D_{j}}=R_{B_{j}}$ and hence $N_{D_{j}}= N_{B_{j}}$ is infinite for every $j$.
The hypotheses of Lemma \ref{common proj direct summand} are satisfied for $A$ and the sequence $\{D_{j}\}_{j=1}^{\infty}$, hence $A=\sum_{j=1}^{\infty} C'_{j}$ with $C'_{j}\sim D_{j}$ and $N_{C'_{j}}$ infinite for every $j$.
But then $C'_{j}\cong D_{j}$ for every $j$, i.e.
$C'_{j}=U_{j}D_{j}U_{j}^*$ for some unitaries $U_{j}$.
Taking $C_{j}:=U_{j}B_{j}U_{j}^*$, $N_{C_{j}}$ is infinite, and $C_{j} \sim B_j$.
Since $C'_{j}-C_{j}=U_{j}(D_{j}-B_{j})U_{j_i}^*$ we have
$$
0\le C'_{j}-C_{j}\le
U_{j}(s-t_j)P_{j}U_{j}^*
\le \sup_{j\ge1}\frac{s-t_{j}}{s}U_{j}sP_{j}U_{j}^*\le \epsilon U_jD_jU_j^* =\epsilon C'_j.
$$
Therefore $R:=\sum_{j=1}^{\infty}(C'_j-C_j)$
is strong operator convergent and hence so is $\sum_{j=1}^{\infty}C_{j}$. Then $A=\sum_{j=1}^{\infty}C_{j}+R$ and$$0\le R =\sum_{j=1}^{\infty}(C'_j-C_j)\le \epsilon \sum_{j=1}^{\infty}C'_j=\epsilon A.$$
\end{proof}
\begin{theorem}\label{equiv infinite trace}
Let $\mathcal{M}$ be a $\sigma$-finite infinite factor, $A\in \mathcal{M}^+$ and $\{B_j\}_{j=1}^\infty \subseteq \mathcal{M}^+$ with $\alpha:=\sup_{j\ge1}\|B_j\|<\infty$.
Each of the following two conditions is sufficient for $A=\sum_{j=1}^{\infty}C_j$ for some $C_j\sim B_j$ with $N_{C_j}$ infinite:
\item [ i)] $\tau((A-\alpha I)_+)=\infty$ when $\mathcal{M}$ is semifinite (resp., $\|A\|>\alpha$ when $\mathcal{M}$ is type III) and
there are a $\beta>0$ and a non-zero projection $P$ for which $P\prec \chi_{(\beta,\infty)}(B_{j})$ for infinitely many indices $j$.
\item [ ii)]
$\|A\|_e\ge\alpha$ when $\mathcal{M}$ is semifinite (resp., $\|A\|\ge\alpha$ when $\mathcal{M}$ is type III),
$\chi_{\{\alpha\}}(B_j)=0$ for all $j$ and
there are $0<\beta<\gamma<\alpha$ and a non-zero projection $P$ such that $ P\prec \chi_{(\beta,\gamma]}(B_j)$ for infinitely many indices $j$.
\end{theorem}
\begin{proof}
\item [ i)] Reasoning as in the proof of Lemma \ref {monotone sequence}, assume without loss of generality that the elements $B_j$ have mutually orthogonal projections and by invoking Lemma \ref {subsume a sequence} that the condition $P \prec \chi_{(\beta,\infty)}(B_{j})$ holds for all $j$.
Next, partition $\mathbb{N}=\bigcup_{i=1}^{\infty}N_i$ into a collection of disjoint infinite sets $N_i$ and set $B'_i=\bigoplus_{j\in N_i}B_j$. Then $\chi_{(\beta,\infty)}(B'_i)$ is infinite for every $i$. If we find a decomposition $A= \sum_{i=1}^\infty C'_i$ for some $C'_i\sim B'_i$ and $N_{C'_i}$ infinite for all $i$, we can then refine it into a decomposition $A=\sum_{j=1}^\infty C_j$ for some $C_j\sim B_j$ with $N_{C_j}$ infinite for every $j$. Thus to simplify notations assume directly that $\chi_{(\beta,\infty)}(B_{j})$ is infinite for every $j$.
Using Lemma \ref{gen ess spect} iii) (resp., Lemma \ref{gen spect} ii) when $\mathcal{M}$ is type III) decompose $A$ as $A=\bigoplus_{k=1}^{\infty}A_k$ with $\tau((A_k-\alpha I)_+)=\infty$ (resp., $\|A_k\|>\alpha$ when $\mathcal{M}$ is type III). Decompose $\mathbb{N}$ as $\mathbb{N}=\bigcup_{k=1}^{\infty}J_k$ for some disjoint, infinite subsets $J_k$.
Apply Lemma \ref{monotone sequence} to $A_1$ and $\{B_j\}_{j\in J_1}$ with $\epsilon=1$
to obtain that $$A_1=\sum_{j\in J_1}C_j+R_1$$ for some $C_j\sim B_j$ for $j\in J_1$ with infinite $N_{C_j}$ and $0\le R_1\le A_1$.
Then $R_1\perp A_2$ and $\tau((R_1+A_2-\alpha I)_+)=\infty$ if $M$ is semifinite (or $\|R_1+A_2\| > \alpha$ if $M$ is type III). Thus we can apply again Lemma \ref{monotone sequence} to $R_1+A_2$ and $\{B_j\}_{j\in J_2}$ with $\epsilon=\frac{1}{2}$ and obtain that
$R_1+A_2= \sum_{j\in J_2}C_j+R_2$ for some $C_j\sim B_j$ for $j\in J_2$ with infinite $N_{C_j}$ and $0\le R_2\le \frac{1}{2}A_2$. Then $$A_1+A_2= \sum_{k=1}^2\sum_{j\in J_2}C_j+R_2.$$
Iterating, we find $\{C_{j}\}_{j\in J_k}$ with $C_{j}\sim B_{j}$ and $N_{C_j}$ infinite, $0\le R_k\le \frac{1}{k}A_k\le \frac{1}{k}A$ such that for every $n$, $$\sum_{k=1}^nA_k= \sum_{k=1}^n \sum_{j\in J_{k+1}}C_{j}+R_{n+1}.$$
Since $\|R_n\|\to 0$ this concludes the proof of part i).
\item [ ii)]
Decompose $(\gamma,\alpha]$ as $(\gamma,\alpha]=\bigcup_{k=1}^{\infty}(t_k,t_{k+1}]$ for some sequence $\{t_k\}_{k=1}^{\infty}$ strictly increasing to $\alpha$.
Let $J:=\{j\mid P\prec \chi_{(\beta,\gamma]}(B_j)\}$ and decompose $\mathbb{N}$ as $\mathbb{N}=\bigcup_{k=1}^{\infty}J_k$ for some disjoint, infinite subsets $J_k$ with $J_k\cap J$ infinite for every $k$.
For each $k$, define $$B^j_k=\begin{cases}
B_j\chi_{(t_k,t_{k+1}]}(B_j)\oplus B_j\chi_{[0,\gamma]}(B_j) & \text{ if $j\in J_k$},\\
B_j\chi_{(t_k,t_{k+1}]}(B_j) &\text{ if $j\notin J_k$}.
\end{cases}
$$
Then for every $k\ge1$, $\sup_{j\ge1}\|B^j_k\|\le t_{k+1}<\alpha$ and $P\prec \chi_{(\beta,\gamma]}(B^j_k)$ for all $j\in J\cap J_k$.
Since $\chi_{\{\alpha\}}(B_j)=0$ for every $j$ we obtain that $$B_j=B_j\chi_{[0,\alpha)}(B_j)=\bigoplus_{k=1}^{\infty}B^j_k.$$
Using Lemma \ref{gen ess spect} ii) (resp., Lemma \ref{gen spect} ii) when $\mathcal{M}$ is type III,) decompose $A$ as $A=\bigoplus_{k=1}^{\infty}A_k$ with $\|A_k\|_e=\|A\|_e$ (resp., $\|A_k\|=\|A\|$ when $\mathcal{M}$ is type III.)
Then apply i) to $A_k$ and $\{B^j_k\}_{j\ge1}$. Hence $A_k=\sum_{j=1}^{\infty}C^j_k$ with $C^j_k\sim B^j_k$ for each $k$ and hence
$$
A=\bigoplus_{k=1}^{\infty}A_k=
\bigoplus_{k=1}^{\infty}\sum_{j=1}^{\infty}C^j_k
=\sum_{j=1}^{\infty}\bigoplus_{k=1}^{\infty}C^j_k=
\sum_{j=1}^{\infty}C_j,
$$
where $C_j=\bigoplus_{k=1}^{\infty}C^j_k\sim
\bigoplus_{k=1}^{\infty}B^j_k=B_j$.
The fact that $N_{C_j}$ can be chosen infinite follows from writing $A=A_1\oplus A_2$ with $\|A_1\|_e=\|A_2\|_e=\|A\|_e$ (resp., $\|A_1\|=\|A_2\|=\|A\|$ when $\mathcal{M}$ is type III) and splitting $\mathbb{N}$ as $\mathbb{N}=J'_1\cup J'_2$ with $J'_1\cap J'_2=\varnothing$ and $J'_1\cap J$ and $J'_2\cap J$ infinite. Then apply the first part of the proof to $A_1$ and $\{B_j\}_{j\in J'_1}$ obtaining $A_1= \sum_{j\in J'_1}C^1_j$ with $C^1_j\sim B_j$ for $j\in J'_1$. Then $C^1_j\le R_{A_1}$, hence $N_{C^1_j}\ge R_{A_2}$ is infinite. In the same way we obtain a decomposition of $A_2=\sum_{j\in J'_2}C^2_j$ with $C^2_j\sim B_j$ and $N_{C_j}$ infinite for $j\in J'_2$.
\end{proof}
Notice that if $\|A\|_e>\alpha$ then $\tau((A-\alpha I)_+)= \infty$ and therefore i) applies. But if $\|A\|_e=\alpha$ and $\tau((A-\alpha I)_+)< \infty$ the conditions on the sequence $\{B_j\}$ given in i) need indeed to be strengthened. For instance if $\mathcal{M}=B(\mathcal{H})$, $B_j$ are rank one projections (and hence satisfy i) but not ii)), and $A= I+K$ with $K$ a positive trace-class operator with $\tau (K)\not\in \mathbb N$, and hence $\|A\|_e=1$, then by \cite[Theorem 1.1 (i)]{SSP} $A$ cannot be a sum of projections.
\
We conclude this section by recalling the connection between decompositions of operators into sums of projections and block diagonal forms which was established in \cite[Proposition 3.1]{SSP} and \cite[Proposition 5.1] {FSP} and which can easily be extended as follows to decompositions of positive operators (see also \cite{KL} for the $B(\mathcal{H})$ rank-one projection case).
\begin{proposition}\label{block diagonal decomp}
Let $\mathcal{M}$ be a properly infinite von Neumann algebra, $A\in \mathcal{M}^+$ and $\{B_j\}_{j=1}^N\subseteq \mathcal{M}^+$, where $N\geq1$ is an integer or $N=\infty$.
The following are equivalent:
\item[ i)] $A=\sum_{j=1}^{N}C_j$ for some $C_j\sim B_j$.
\item[ ii)] There exist mutually orthogonal projections $\{E_j\}_{j=1}^N$ and a partial isometry $V$ such that $\sum_{j=1}^NE_j\ge VV^*$, $V^*V\ge R_A$ and $E_jVAV^*E_j\sim B_j$.
If in addition $N_A\sim I$ then in ii) we can take $V=I$ .
\end{proposition}
\begin{proof}
\item [ii)] $\Rightarrow$ i)
Let $C_j=A^{\frac{1}{2}}V^*E_jVA^{\frac{1}{2}}$.
Then $C_j\sim E_jVAV^*E_j\sim B_j$ and $$\sum_{j=1}^NC_j=A^{\frac{1}{2}}V^*\Big(\sum_{j=1}^NE_j\Big)VA^{\frac{1}{2}}=A^{\frac{1}{2}}V^*VA^{\frac{1}{2}}=A.$$
\item [i)] $\Rightarrow$ ii)
Since $\mathcal{M}$ is properly infinite we can find projections $\{E_j\}_{j=0}^N$ such that $E_j\sim I$ and $\sum_{j=0}^NE_j=I$. Choose isometries $\{W_j\}_{j=1}^N$ such that $W_jW_j^*=E_j$ and define $X_j=W_jC_j^{\frac{1}{2}}$, $j\ge1$.
Just as in the proof of \cite [Proposition 3.1]{SSP} where the $C_j$ are rank one projections, one can verify that if $N$ is infinite the series $\sum_{j=1}^NX_j$ is strong operator convergent to some operator $X$ and that $X^*X=A$. Then let $X=VA^{\frac{1}{2}}$ be the polar decomposition of $X$. In particular, $V^*V=R_A$. Since $E_jX=X_j$ and $VAV^*=XX^*$, it follows that $VV^*=R_X\le \sum_{j=1}^NE_j$ and $$E_jVAV^*E_j=E_jXX^*E_j=X_jX_j^*=W_jC_jW_j^*\sim C_j\sim B_j.$$
Finally, notice that
$$ I- VV^*\ge I-\sum_{j=1}^N E_j=E_0\sim I.$$ Thus if also $I-V^*V=N_A\sim I$, $V$ can be extended to a unitary $U$ and then the conclusion follows by taking the projections $U^*E_jU$ instead of $E_j$ for $j\ge2$ and $U^*E_1\oplus E_0U$ instead of $E_1$.
\end{proof}
We collect bellow a few remarks that are easy consequences of the proof.
\begin{remark}
\item[ 1)] We have actually proved that i) is equivalent to asking that $\sum_{j=1}^NE_j=I$, $E_j\sim I$, $V^*V=R_A$ and $E_jVAV^*E_j\sim B_j$.
We could equally well have found projections $\{E_j\}_{j=1}^N$ and a partial isometry $V$ such that $\sum_{j=1}^NE_j\ge VV^*$, $V^*V=R_A$, $R_{E_jVAV^*E_j}=E_j$ and $E_jVAV^*E_j\sim B_j$.
\item[ 2)] If $\mathcal{M}$ is an infinite factor and i) holds then for any sequence of mutually orthogonal projections $\{E_j\}_{j=1}^N$ such that $R_{B_j}\prec E_j$ we can find a partial isometry $V$ such that $\sum_{j=1}^NE_j\ge VV^*$, $V^*V=R_A$ and $E_jVAV^*E_j\sim B_j$.
\end{remark}
We further notice that in the case when $\mathcal M= B(\mathcal{H})$ and all the projections $E_j$ and operators $C_j$ have rank-one, then the partial isometry $V$ plays an important role in frame theory. Indeed the vectors $x_j\in \mathcal{H}$ for which $C_j=x_j\otimes x_j$ form a frame when $A$ is invertible (a Bessel sequence when it is not), and $V$ then coincides with the frame transform (also called analysis operator) of the Parseval frame associated with $\{x_j\}$.
The following example shows that we cannot expect to be able to choose $V$ unitary without some further hypotheses on $A$.
\begin{example}
Let $A= 2I$ and $B_j$ be a sequence of equivalent nonzero projections. Then $A=\sum_{j=1}^\infty C_j$ with $C_j\sim B_j$ but of course $E_jAE_j= 2E_j \not\sim B_j$. The same conclusion holds for every $A=\lambda I$ and $\lambda>1$ by \cite [Theorem 1.1]{SSP}.
\end{example}
Thus combining Theorem \ref {equiv infinite trace} and Proposition \ref {block diagonal decomp} we obtain the following form of the ``pinching theorem" of \cite {PT} for the case of positive operators in von Neumann factors.
\begin{corollary}\label{pinching corollary}
Let $\mathcal{M}$ be a $\sigma$-finite infinite factor, $A\in \mathcal{M}^+$ and $\{B_j\}_{j=1}^\infty \subseteq \mathcal{M}^+$ with $\alpha:=\sup_{j\ge1}\|B_j\|<\infty$.
Assume one of the following two conditions holds:
\item [ i)] $\tau((A-\alpha I)_+)=\infty$ when $\mathcal{M}$ is semifinite (resp., $\|A\|>\alpha$ when $\mathcal{M}$ is type III) and
there are a $\beta>0$ and a non-zero projection $P$ for which $P\prec \chi_{(\beta,\infty)}(B_{j})$ for infinitely many indices $j$.
\item [ ii)]
$\|A\|_e\ge\alpha$ when $\mathcal{M}$ is semifinite (resp., $\|A\|\ge\alpha$ when $\mathcal{M}$ is type III),
$\chi_{\{\alpha\}}(B_j)=0$ for all $j$ and
there are $0<\beta<\gamma<\alpha$ and a non-zero projection $P$ such that $ P\prec \chi_{(\beta,\gamma]}(B_j)$ for infinitely many indices $j$.
Then there exist mutually orthogonal projections $\{E_j\}_{j=1}^\infty$ with $\sum_{j=1}^\infty E_j=I$, $E_j\sim I$, and a partial isometry $V$ such that $V^*V\ge R_A$ and $E_jVAV^*E_j\sim B_j$.
If in addition $N_A\sim I$ then we can take $V=I$ .
\end{corollary}
\section{Sums of unitary equivalent conjugates of a sequence of operators}
To obtain decompositions into sums of unitary equivalent conjugates of a sequence $\{B_j\}_{j=1}^\infty$, we follow the approach of Bourin and Lee in \cite{UE} for the case when $\mathcal {M}= B(\mathcal{H})$ and $B_j=B$ for all $j$. In particular, in the first step here below (see \cite[Theorem 1.1 case I.(1)]{UE}) which deals with the case when $A$ is invertible, we can use their lemmas which apply without changes to the factor case. For completeness, we sketch the proof.
\begin{proposition}\label{unit equiv, invertible}
Let $A\in \mathcal{M}^+$ be invertible and $\{B_j\}_{j=1}^{\infty}\subseteq \mathcal{M}^+$
be a sequence with $\alpha:=\sup_{j\ge 1}\|B_j\|<\infty$ and such that there is a $\beta>0$ and a non-zero projection $P$ for which $P\prec \chi_{(\beta,\infty)}(B_{j})$ for infinitely many $j$. If
$$\begin{cases}\alpha< \|A\|_e,~ 0\in \sigma_e(B_j)&\text{when $\mathcal{M}$ is semifinite,}\\
\alpha< \|A\|, ~0\in \sigma(B_j)&\text{when $\mathcal{M}$ is type III,}\end{cases} $$
then $A=\sum_{j=1}^{\infty}C_j$ for some $C_j\cong B_j$.
\end{proposition}
\begin{proof}
\textbf{Step 1:} (see \cite[Lemma 2.4]{UE}) We first show that for every $\epsilon>0$, we can decompose $A=\sum_{j=1}^{\infty}C_j+R$ for some $C_j\cong B_j$ and $0\le R\le \epsilon I$.
Choose $0<\rho< \frac{\epsilon}{2}$ such that $A-\rho I\ge 0$ and $\|A-\rho I\|_e>\alpha$ when $\mathcal{M}$ is semifinite (resp., $\|A-\rho I\|>\alpha$ when $\mathcal{M}$ is type III).
Using Theorem \ref{equiv infinite trace}, decompose $A-\rho I$ as $A-\rho I=\sum_{j=1}^{\infty}C'_j$ with $C'_j\sim B_j$ and $N_{C'_j}$ infinite.
Since $N_{C'_j}$ is infinite, $0\in \sigma_e(C'_j)$ when $\mathcal{M}$ is semifinite (resp., $0\in \sigma(C'_j)$ when $\mathcal{M}$ is type III). The result of \cite[Lemma 2.3]{UE} holds also for factors with a similar proof, that is, there exist $ C_j\cong B_j$ such that $\| C'_j-C_j\| \le \frac{\rho}{2^{j}}$. Thus $\sum_{j=1}^{\infty}\|C'_j-C_j\|\le \rho.$ Let $R:= \rho I+\sum_{j=1}^{\infty}(C'_j-C_j)$. Then
$A=\sum_{j=1}^{\infty}C_j+ R$ and $0\le R \le \epsilon I.$
\textbf{Step 2:}
The result of \cite[Lemma 2.2]{UE} is true also in factors with a similar proof. Hence we can decompose $A$ as $A=\sum_{k=1}^{\infty}A_k$ with $A_k$ invertible and $\|A_k\|_e>\alpha$ or (resp., $\|A_k\|>\alpha$ if $\mathcal{M}$ is type III).
Write $\mathbb{N}=\bigcup_{k=1}^{\infty}J_k$ for some infinite, disjoint subsets such that the sequences $\{B_j\}_{j\in J_k}$ have the same property as $\{B_j\}_{j=1}^{\infty}$.
Apply Step 1 to conclude that $A_1=\sum_{j\in J_1}C_j+R_1$ with $C_j\cong B_j$ for $j \in J_1$ and $0\le R_1\le I$. Apply now Step 1 to $A_2+R_1$ to conclude that $A_2+R_1=\sum_{j\in J_2}C_j$ with $C_j\cong B_j$ for $j\in J_2$ and $0\le R_2\le \frac{1}{2}I$.
Continuing in the same way we construct $\{R_k\}_{k=1}^{\infty}$ such that $0\le R_k\le \frac{1}{k}I$ and $A_{k}+R_{k+1}=\sum_{j\in J_{k+1}}C_j+R_{k+1}$ with $C_j\cong B_j$ for $j\in J_k$. Just as in the last part of the proof of Theorem \ref{equiv infinite trace} it is easy to conclude that $A=\sum_{j=1}^{\infty}C_j$ with $C_j\cong B_j$ for every $j$.
\end{proof}
By Lemma \ref {gen ess spect} and \ref {gen spect} we know that we can decompose $A$ into a direct sum of locally invertible summands
``with the same properties". The next lemma shows that we can also decompose each element $B_j$ into a direct sum ``with the same properties".
\begin{lemma}\label{dec of sequence}
Let $\{B_j\}_{j=1}^{\infty}\subseteq \mathcal{M}^+$ with $\alpha:=\sup_{j\ge 1}\|B_j\|<\infty$ and assume that for every $j$, $0\in \sigma_e(B_j)$ when $\mathcal{M}$ is semifinite (resp., $0\in \sigma(B_j)$ when $\mathcal{M}$ is type III) and that there is a $\beta>0$ and a non-zero projection $P$ such that
$P\prec \chi_{(\beta,\infty)}(B_{j})$ for infinitely many integers $j$.
Then for every $j$ there is a sequence of mutually orthogonal infinite projections $\{F^j_k\}_{k=1}^{\infty}\subseteq \{B_j\}'$ such that $\sum_{k=1}^{\infty}F^j_k=I$, $0\in \sigma_e\big((B_j)_{F^j_k}\big)$ when $\mathcal{M}$ is semifinite (resp., $0\in \sigma\big((B_j)_{F^j_k}\big)$ when $\mathcal{M}$ is type III), and for every $k$, $P\prec \chi_{(\beta,\infty)}(B_jF^j_k)$ for infinitely many integers $j$.
If furthermore $\chi_{\{\alpha\}}(B_j)=0$ for every $j$ and for some $0<\beta<\gamma<\alpha$, $P\prec \chi_{(\beta,\gamma]}(B_{j})$ for infinitely many integers $j$, then the projections $\{F^j_k\}_{k=1}^{\infty}$ can be chosen so that for every $k$, $\sup_{j\ge1}\|B_jF^j_k\|<\alpha$ and
$P\prec \chi_{(\beta,\gamma]}(B_jF^j_k)$ for infinitely many integers $j$.
\end{lemma}
\begin{proof}
We prove the lemma when $\mathcal{M}$ is semifinite and leave to the reader the similar proof of the case when $\mathcal{M}$ is type III.
Let $J:=\{\, j\mid P\prec\chi_{(\beta, \infty)}(B_j)\,\}$. Decompose $\mathbb{N}=\bigcup_{k=1}^{\infty}J_k$ such that the sets $J_k$ are disjoint and $J_k\cap J$ is infinite for every $k$.
For every $j\ge 1$ let $F_j=\chi_{[0,\beta]}(B_j)$. From the hypothesis that $0\in \sigma_e(B_j)$ it follows that $F_j$ is infinite and $0\in \sigma_e\left((B_j)_{F_j}\right)$. Applying Lemma \ref{gen ess spect} i) to $\mathcal{M}_{F_j}$ and $(B_j)_{F_j}$, decompose each $B_jF_j$ as $$B_jF_j=\bigoplus_{k=1}^{\infty} B_j\tilde F^j_k$$ for some sequences of mutually orthogonal infinite projections $\{\tilde F_k^j\}_{k\ge1}\subseteq \{B_jF_j\}'$, with $\sum_{k=1}^{\infty}\tilde F_k^j=F_j$ and $0\in \sigma_e\left((B_j)_{\tilde F^j_k}\right)$.
For every $k$, define
$$F_k^j=\begin{cases}
\tilde F_k^j&\text{if $j\notin J_k$},\\
\tilde F_k^j \oplus (I-F_j)&\text{if $j\in J_k$}.
\end{cases}
$$
Then $\sum_{k=1}^{\infty}F_k^j=I$, $\{F_k^j\}_{k=1}^{\infty}\subseteq \{B_j\}'$, $F_k^j$ is infinite, and $0\in \sigma_e\left((B_j)_{F^j_k}\right)$ for every $k$ and $j$.
Finally, notice that $ \chi_{(\beta,\infty)}(B_j)=\chi_{(\beta,\infty)}(B_jF_k^j)$ for $j\in J_k$ and therefore $P\prec \chi_{(\beta,\infty)}(B_jF_k^j)$ for all $j\in J\cap J_k$ which concludes the first part of the proof.
Assume now that $\chi_{\{\alpha\}}(B_j)=0$ for every $j$ and $J:=\{\, j\mid P\prec\chi_{(\beta, \gamma]}(B_j)\,\}$ is infinite. Then define $J_k$, $F_j$ and $\{\tilde F_k^j\}_{k\ge1}$ just like above.
Let $\{t_k\}_{k=1}^{\infty}$ be a strictly increasing sequence to $\alpha$ with $t_1=\gamma$, and define for every $k$:
$$F_k^j=\begin{cases}
\tilde F_k^j\oplus\chi_{(t_k,t_{k+1}]}(B_j)&\text{if $j\notin J_k$},\\
\tilde F_k^j \oplus \chi_{(t_k,t_{k+1}]}(B_j)\oplus\chi_{(\beta,\gamma]}(B_j)&\text{if $j\in J_k$}.
\end{cases}
$$
Then $\sum_{k=1}^{\infty}F^j_k=I$ because $\chi_{\{\alpha\}}(B_j)=0$, $\sup_{j\ge1}\|B_jF^j_k\|\le t_{k+1}<\alpha$ for every $k$, and
$P\prec \chi_{(\beta,\gamma]}(B_jF_k^j)$ for all $j\in J\cap J_k$.
\end{proof}
\begin{theorem}\label{unit equiv sequence}
Let $\mathcal{M}$ be a $\sigma$-finite, infinite factor, $A$, $B\in \mathcal{M}^+$, and $\{B_j\}_{j=1}^{\infty}\subseteq \mathcal{M}^+$ with $\alpha:=\sup_{j\ge 1}\|B_j\|<\infty$.
Assume that for all $j$, $N_A\prec N_{B_j}$ and $0\in \sigma_e(B_j)$ when $\mathcal{M}$ is semifinite (resp., $0\in \sigma(B_j)$ when $\mathcal{M}$ is type III). Each of the following two conditions is sufficient for $A=\sum_{j=1}^{\infty}C_j$ for some $C_j\cong B_j$:
\item[ i)] $\|A\|_e>\alpha$ when $\mathcal{M}$ is semifinite (resp., $\|A\|>\alpha$ when $\mathcal{M}$ is type III) and there are a $\beta>0$ and a non-zero projection $P$ for which $P\prec \chi_{(\beta,\infty)}(B_{j})$ for infinitely many integers $j$.
\item[ ii)] $\|A\|_e\ge\alpha$ when $\mathcal{M}$ is semifinite (resp., $\|A\|\ge\alpha$ when $\mathcal{M}$ is type III), $\chi_{\{\alpha\}}(B_j)=0$ for every $j$, and there are $0<\beta<\gamma<\alpha$ and a non-zero projection $P$ for which $P\prec \chi_{(\beta,\gamma]}(B_{j})$ for infinitely many integers $j$.
\end{theorem}
\begin{proof}
By Theorem \ref{equiv infinite trace} it follows that $A=\sum_{j=1}^\infty C_j$ for some $C_j\sim B_j$ with $N_{C_j}$ infinite. If $N_A$ is infinite then $N_{B_j}$ is infinite for all $j$ and hence $N_{B_j}\sim N_{C_j}$ and thus $C_j\cong B_j$ for every $j$.
Assume henceforth that $N_A$ is finite. Then $N_A\sim R_j$ for some $R_j\le N_{B_j}$ and since in a factor finite equivalent projections are unitarily equivalent we obtain $N_A\cong R_j$. But then $R_A\cong R_j^{\perp}\ge R_{B_j}$ and hence $R_{B_j}\cong Q_j$ for some $Q_j\le R_A$.
Hence by replacing the $B_j$'s with unitary conjugates we can thus further assume that $R_{B_j}\le R_A$ for every $j$.
Then all the hypotheses of the theorem are satisfied by the compressions $A_{R_A}$ and $\{(B_j)_{R_A}\}_{j=1}^{\infty}$ belonging to the factor $\mathcal{M}_{R_A}$. To simplify notations, assume henceforth that $R_A=I$.
By using Lemma \ref{gen ess spect} ii$\,'$) (resp., Lemma \ref{gen spect} ii$\,'$) when $\mathcal{M}$ is type III) decompose $A$ as $A=\bigoplus_{k=1}^{\infty}AE_k$ with mutually orthogonal projections $E_k\in\{A\}'$ with $\|AE_k\|_e=\|A\|_e$ (resp., $\|AE_k\|=\|A\|$ when $\mathcal{M}$ is type III), and $AE_k$ locally invertible for all $k$. Notice that then $E_k$ must be infinite and because $R_A=I$, $R_{AE_k}=E_k$ and $\sum_{k=1}^{\infty}E_k=I$.
In case i), by using Lemma \ref{dec of sequence} decompose $B_j$ as $B_j=\bigoplus_{k=1}^{\infty}B_jF^j_k$ for some sequence of mutually orthogonal infinite projections $\{F^j_k\}_{k=1}^{\infty}\subseteq \{B_j\}'$ with $$\sum_{k=1}^{\infty}F^j_k=I, 0\in \sigma_e\big((B_j)_{F^j_k}\big) \text{ (resp., $0\in \sigma\big((B_j)_{F^j_k}\big)$ if $\mathcal{M}$ is type III), }$$ and for each $k$, $P\prec \chi_{(\beta,\infty)}(B_jF^j_k)$ for infinitely many indices $j$. Furthermore, $\sup_{j\ge1}\|B_jF^j_k\|_e\le \alpha < \|AE_k\|_e$ for all $k$ (resp., $\sup_{j\ge1}\|B_jF^j_k\|\le \alpha < \|AE_k\|$ for all $k$ when $\mathcal M$ is type III).
In case ii), and again by Lemma \ref{dec of sequence}, we obtain the same conclusion but with $\sup_{j\ge1}\|B_jF^j_k\|_e< \alpha \le \|AE_k\|_e$ for all $k$ (resp., $\sup_{j\ge1}\|B_jF^j_k\|< \alpha \le \|AE_k\|$ for all $k$ when $\mathcal M$ is type III).
In both cases, for every $j$, $\sum_{k=1}^\infty E_k=I$ and $\sum_{k=1}^\infty F^j_k=I$ and all the projections $E_k$ and $F^j_k$ are infinite and hence equivalent. Therefore there is a unitary $U_j\in \mathcal M$
such that $U_jF^j_kU_j^*=E_k$ for all $k$.
Let $B'_j=U_jB_jU_j^*$, hence $B'_j=\bigoplus_{k=1}^\infty B_j'E_k$. From the above constructions we see that for every $k$, $\mathcal{M}_{E_k}$, $A_{E_k}$ and $\{(B_j')_{E_k}\}_{j=1}^{\infty}$ satisfy the hypotheses of Proposition \ref{unit equiv, invertible}.
Thus $AE_k=\sum_{j=1}^{\infty}C_k^j$ for some $C_k^j\in \mathcal{M}_{E_k}$ and congruent to $B_j'E_k$ {\it within} $\mathcal{M}_{E_k}$, that is such that there are partial isometries $V_k^j\in \mathcal{M}$ with $V_k^j(V_k^j)^*=(V_k^j)^*V_k^j=E_k$ for which
$C_k^j= V_k^jB_j'E_k(V_k^j)^*$.
Then $V^j:=\bigoplus_{k=1}^\infty V_k^j$ is a unitary in $\mathcal{M}$ and
\begin{align*}
A&=\bigoplus_{k=1}^{\infty}AE_k
=\bigoplus_{k=1}^{\infty}\sum_{j=1}^{\infty}V_k^jB_j'E_k(V_k^j)^*\\
&=\sum_{j=1}^{\infty}\bigoplus_{k=1}^{\infty}V_k^jB_j'(V_k^j)^*=\sum_{j=1}^{\infty}V^jB_j'(V^j)^*.
\end{align*}
Taking $C_j:=V^jB_j'(V^j)^*\cong B_j'\cong B_j$ concludes the proof.
\end{proof}
\section{Equivalent and unitarily equivalent copies of a single operator}
In this section we apply the results of section 4 to the case when all the operators $B_j$ are equivalent or are unitarily equivalent to a given non-zero operator $B$. We are thus able to answer affirmatively the conjecture posed by Bourin and Lee in \cite{SMV} and \cite{UE}.
We start by considering necessary conditions:
\begin{proposition}\label{nec equiv}
Let $A$, $B$ $\in \mathcal{M}^+$.
If $A=\sum_{j=1}^{\infty}C_j$ for some $C_j\sim B$ for all $j$, then the following conditions hold:
\item [ i)]
$\|A\|\ge \|B\|$ and $R_B\prec R_A$. If $\mathcal{M}$ is semifinite, then $\|A\|_e\ge \|B\|$.
\item [ ii)] One of the following mutually exclusive conditions holds:
\begin{enumerate}
\item [ 1)] $\|A\|>\|B\|$,
\item [ 2)] $\|A\|=\|B\|$ and $\chi_{\{\|B\|\}}(B)=0$,
\item [ 3)]$\|A\|=\|B\|$,
$\chi_{\{\|B\|\}}(B)\ne 0$, $B\ne \|B\|\chi_{\{\|B\|\}}(B)$ and $\chi_{\{\|A\|\}}(A)$ is infinite,
\item [ 4)]$\|A\|=\|B\|$, $B=\|B\|\chi_{\{\|B\|\}}(B)$, $A=\|A\|\chi_{\{\|A\|\}}(A)$ and $\chi_{\{\|A\|\}}(A)$ is infinite.
\end{enumerate}
\noindent If $A=\sum_{j=1}^{\infty}C_j$ for some $C_j\cong B$ for all $j$, then in addition to the conditions i) and ii), the following conditions hold:
\item [ iii)] $N_A\prec N_B$,
\item [ iv)] $0\in \sigma(B)$ and if $\mathcal{M}$ is semifinite, then $0\in \sigma_e(B)$,
\item [ v)] If $\|A\|=\|B\|$, then $\chi_{\{\|B\|\}}(B) \prec N_B$.
\end{proposition}
\begin{proof}
\item [ i)] Since $A\ge C_j\sim B$ for all $j$, we have both $\|A\|\geq \|C_j\|=\|B\|$ and $R_A\ge R_{C_j}\sim R_B$. Assume now that $\mathcal{M}$ is semifinite. For every $0<t<\|B\|$ we have $B\geq tR$ for some non-zero projection $R$. Hence $C_j\geq tR_j$ for some projection $R_j\sim R$ and $A\geq t\sum_{j=1}^{\infty}R_j$. Thus $\|A\|_e\geq t\left\|\sum_{j=1}^{\infty}R_j\right\|_e$ and it is enough to show that $\left\|\sum_{j=1}^{\infty}R_j\right\|_e\geq 1$. Let $T:=\sum_{j=1}^{\infty}R_j$.
Since $\tau(T)=\infty$ we see that $R_T$ is infinite.
From \cite[Theorem 3.3]{SSP} we know that $\tau((T-I)_+)\geq \tau((R_T-T)_+)$. If $\tau((T-I)_+)=\infty$ then $\|T\|_e\geq 1$. If $\tau((T-I)_+)<\infty$ then both $(T-I)_+$ and $(R_T-T)_+$ belong to the ideal of compact operators relative to $\mathcal{M}$. Since $$T=(T-I)_+-(R_T-T)_++R_T,$$ it follows that $\|T\|_e=\|R_T\|_e=1$.
\item [ ii)]
Let $P=\chi_{\{\|B\|\}}(B)$ and $B'= \chi_{[0, \|B\|)}(B)$ so that $B=B'\oplus \|B\|P$. Let $V_j$ be partial isometries with $V_j^*V_j=R_B$ such that $C_j=V_jBV_j^*$ and hence $$C_j= V_jB'V_j^*\oplus \|B\|V_jPV_j^*.$$ Therefore $A=A'+A''$ where $A'=\sum_{j=1}^{\infty}V_jB'V_j^*$ and $A''=\|B\|\sum_{j=1}^{\infty}V_jPV_j^*$.
Assume that $P\ne 0$, then $\|A''\|\ge \|B\|$. If furthermore $\|A\|=\|B\|$, we have $\|B\|= \|A\|\ge \|A''\|\ge \|B\|$ whence $\|\sum_{j=1}^{\infty}V_jPV_j^*\|=1$. This implies that the projections $V_jPV_j^*$ must be mutually orthogonal and hence $Q:=\sum_{j=1}^{\infty}V_jPV_j^*$ is a projection, necessarily infinite. Then $A=A'+\|B\|Q$ with $A'\geq 0$ and $\|A\|=\|B\|$. Hence it follows that $A'\perp Q$. Thus $Q\leq \chi_{\{\|B\|\}}(A)=\chi_{\{\|A\|\}}(A)$, whence $\chi_{\{\|A\|\}}(A)$ is infinite.
If in addition we assume that $B=\|B\|\chi_{\{\|B\|\}}(B)$, i.e., $B'=0$, then $A'=0$, and $A=A''=\|B\|Q= \|A\|\chi_{\{\|A\|\}}(A)$.
Now consider the case when $C_j\cong B$ for all $j$.
\item [ iii)] Obvious since then $N_B\sim N_{C_1}\ge N_A$.
\item [ iv)]
Assume first that $\mathcal{M}$ is semifinite and let $\pi:
\mathcal M\to \mathcal M/\mathcal J$ be the quotient map. Then $\pi(A)\geq \sum_{j=1}^{N}\pi(C_j)$ for every $N\geq 1$. Assume by contradiction that $0\notin \sigma_e(B)$. Then $\pi(B)$ is invertible in $\mathcal{M}/\mathcal{J}$, hence $\pi(B)\geq t\pi(I)$ for some positive $t$. Since $\pi(C_j)\cong \pi(B)$ in $\mathcal{M}/\mathcal{J}$, we have $\pi(A)\geq Nt\pi(I)$ for every $N$, a contradiction.
Thus $0\in \sigma_e(B)$ and hence $0\in \sigma(B)$. When $\mathcal{M}$ is type III, the same argument, without the need to pass to the quotient algebra $\mathcal{M}/\mathcal{J}$, shows that $0\in \sigma(B)$.
\item [ v)] If $\chi_{\{\|B\|\}}(B)\ne 0$ and $\xi \in \chi_{\|B\|}(C_i)\mathcal{H}$ is a unit vector, then $ \langle C_i\xi,\xi\rangle= \|B\|$, hence
$$\|B\|= \|A\|\ge \langle A\xi,\xi\rangle=\|B\|+\sum_{j\ne i}^{\infty}\langle C_j\xi,\xi\rangle$$
which implies that $\xi\in N_{C_j}\mathcal{H}$ for every $j\ne i$, i.e., $\chi_{\{\|C_i\|\}}(B)\le N_{C_j}$. But since $C_j\cong B$ implies that $N_{C_j}\sim N_B$, it follows that $\chi_{\{\|B\|\}}(B)\prec N_B$ .
\end{proof}
Next we present sufficient conditions for the decomposition of $A$ into sums of positive operators equivalent to a fixed positive operator $B$. These are of course based on the decompositions obtained in section 3, but with the two additional cases iii) and iv).
\begin{theorem}\label{equiv}
Let $\mathcal{M}$ be a $\sigma$-finite, semifinite infinite factor, $A$, $B\in \mathcal{M}^+$, and $B\ne0$. Each of the following conditions is sufficient for $A=\sum_{j=1}^{\infty}C_j$ for some $C_j\sim B$ with $N_{C_j}$ infinite for all $j$:
\item [ i)] $\tau((A-\|B\|I)_+)=\infty$,
\item [ ii)] $\|A\|_e=\|B\|$ and $\chi_{\{\|B\|\}}(B)=0$,
\item [ iii)] $\|A\|_e=\|B\|$,
$\chi_{\{\|B\|\}}(B)\ne 0$, $B\ne \|B\|\chi_{\{\|B\|\}}(B)$ and $\chi_{\{\|A\|_e\}}(A)$ is infinite,
\item [ iv)] $\|A\|_e=\|B\|$, $B=\|B\|\chi_{\{\|B\|\}}(B)$, $A=\|A\|_e\chi_{\{\|A\|_e\}}(A)$ and $\chi_{\{\|A\|_e\}}(A)$ is infinite.
\end{theorem}
\begin{proof}
i) and ii) are direct consequences of Theorem \ref{equiv infinite trace}.
\item [ iii)]\& iv) Let $P:=\chi_{\{\|B\|\}}(B)\ne 0$. By decomposing $\chi_{\{\|A\|_e\}}(A)$ into the sum of two infinite projections, we can assume that $A=A'\oplus \|B\|Q$ where $\|A'\|_e=\|B\|$ and $Q$ is an infinite projection. Decompose further $Q=\sum_{j=1}^{\infty}Q_j$ into the sum of projections $Q_j\sim P$. Now decompose $B$ as $B=B'\oplus \|B\|P$ where $B'=B\chi_{(0, \|B\|)}(B)$.
\\
Assume that iii) holds, i.e., $B'\ne 0$. We can decompose $A'=\sum_{j=1}^{\infty}C'_j$ with $C'_j\sim B'$ by invoking i) in the case that $\|B'\|<\|B\|$, and hence $\|A'\|_e>\|B'\|$, whence $\tau((A'-\|B'\|)_+)=\infty$ or invoking ii) in the case that $\|B'\|=\|B\|$ because then $\chi_{\{\|B'\|\}}(B')=0$.
Thus
$$A= \Big(\sum_{j=1}^{\infty}C'_j\Big)\oplus \Big(\sum_{j=1}^{\infty}\|B\|Q_j\Big)=\sum_{j=1}^{\infty}\big(C'_j\oplus \|B\|Q_j\big)$$
and $C_j:=C'_j\oplus \|B\|Q_j\sim B'\oplus \|B\|P= B$.
Notice that by construction, $N_{C_j}\ge \sum_{k\ne j}Q_k$ is infinite for every $j$.
Finally, notice that $B'=A'=0$ in case iv) and hence the proof is a special case of iii).
\end{proof}
Thus combining these sufficient conditions with the necessary conditions of Proposition \ref {nec equiv} we obtain:
\begin{corollary} \label{norm=essnorm}
Let $\mathcal M$ be semifinite, $A, B\in \mathcal M^+$, $B\ne 0$ and $\|A\|_e=\|A\|$, then the condition ii) in Proposition \ref {nec equiv} is necessary and sufficient for $A=\sum_{j=1}^{\infty}C_j$ for some $C_j\sim B$. Furthermore if that condition is satisfied, the decomposition can be chosen so that $N_{C_j}$ is infinite for all $j$.
\end{corollary}
It is easy to see that condition ii) in Proposition \ref {nec equiv} is also sufficient in the type III case:
\begin{corollary}\label{N&S type III}
If $\mathcal M$ is type III, $A, B\in \mathcal M^+$ with $B\ne 0$, then the condition ii) in Proposition \ref {nec equiv} is necessary and sufficient for $A=\sum_{j=1}^{\infty}C_j$ for some $C_j\sim B$. Furthermore if that condition is satisfied, the decomposition can be chosen so that $N_{C_j}\ne 0$ for all $j$.
\end{corollary}
Decompositions into sums of operators unitarily equivalent to a given positive operator are a special case of Theorem \ref {unit equiv sequence}, but here too we can add the two additional conditions iii) and iv). These cases are easily obtained from the fact that equivalent positive operators with infinite null spaces are unitarily equivalent.
\begin{corollary}\label{UE}
Let $\mathcal{M}$ be semifinite, $A$, $B\in \mathcal{M}^+$, $B\neq 0$, $0\in \sigma_e(B)$, and that $N_A\prec N_B$. Then any of the following mutually exclusive conditions implies that $A=\sum_{j=1}^{\infty}C_j$ for some $C_j\cong B$:
\item [ i)] $\|A\|_e>\|B\|$,
\item [ ii)] $\|A\|_e=\|B\|$ and $\chi_{\{\|B\|\}}(B)=0$,
\item [ iii)] $\|A\|_e=\|B\|$, $\chi_{\{\|B\|\}}(B)\ne 0$, $B\ne \|B\|\chi_{\{\|B\|\}}(B)$, $\chi_{\{\|A\|_e\}}(A)$ is infinite and $N_B$ is infinite,
\item [ iv)] $\|A\|_e=\|B\|$, $B=\|B\|\chi_{\{\|B\|\}}(B)$, $A=\|A\|\chi_{\{\|A\|\}}(A)$, $\chi_{\{\|A\|\}}(A)$ is infinite and $N_B$ is infinite.
\end{corollary}
\begin{corollary}\label{P:NecSuf}
Let $\mathcal{M}$ be type III, $A$, $B\in \mathcal{M}^+$, $B\ne 0$.
Then $A=\sum_{j=1}^{\infty}C_j$ with $C_j\cong B$ if and only if
one of the following mutually exclusive conditions holds:
\item [i)] $\|A\|>\|B\|$, $0\in \sigma(B)$ and $N_A\prec N_B$,
\item [ii)] $\|A\|=\|B\|$, $\chi_{\{\|B\|\}}(B)=0$, $0\in \sigma(B)$ and $N_A\prec N_B$,
\item [iii)] $\|A\|=\|B\|$,
$\chi_{\{\|B\|\}}(B)\ne 0$, $B\ne \|B\|\chi_{\{\|B\|\}}(B)$, $\chi_{\{\|A\|\}}(A)\ne 0$ and $N_B\ne 0$,
\item [iv)] $\|A\|=\|B\|$, $B=\|B\|\chi_{\{\|B\|\}}(B)$, $A=\|A\|\chi_{\{\|A\|\}}(A)$, $\chi_{\{\|A\|\}}(A)\ne 0$ and $N_B\ne 0$.
\end{corollary}
|
{
"timestamp": "2015-12-31T02:11:40",
"yymm": "1504",
"arxiv_id": "1504.03193",
"language": "en",
"url": "https://arxiv.org/abs/1504.03193"
}
|
\chapter{Bayesian computational algorithms for social network analysis}
\label{BSNA}
\chapterauthor[Alberto Caimo, Isabella Gollini]
{Alberto Caimo\protect\footnote{Social Network Analysis Research Center, \\
Universit\`{a} della Svizzera italiana, Switzerland. \\
\tt{alberto.caimo@usi.ch}},
Isabella Gollini\protect\footnote{Department of Civil Engineering, \\
University of Bristol, United Kingdom. \\
\tt{isabella.gollini@bristol.ac.uk}}}
\section{Introduction}
\label{BSNA:intro}
Interest in statistical network analysis has grown massively in recent decades and its perspective and methods are now widely used in many scientific areas which involve the study of various types of networks for representing structure in many complex relational systems such as social relationships, information flows, protein interactions, etc.
Social network analysis is based on the study of social relations between actors so as to understand the formation of social structures by the analysis of basic local relations. Statistical models have started to play an increasingly important role because they give the possibility to explain the complexity of social behaviour and to investigate issues on how the global features of an observed network may be related to local network structures. The observed network is assumed to be generated by local social processes which depend on the self-organising dyadic relations between actors. The crucial challenge for statistical models in social network theory is to capture and describe the dependency giving rise to network global topology allowing inference about whether certain local structures are more common than expected.
Unfortunately the computational burden required to estimate these models is the main barrier to estimation. Recent theoretical developments and advances in approximate procedures have given the possibility to make important progress to overcome statistical inference problems.
In this chapter we review some of the most recent computational advances in the rapidly expanding field of statistical social network analysis (see \cite{tow:whi:gol:mur12} for a recent review) using the \verb+R+ open-source software.
In particular we will focus on Bayesian estimation for two important families of models: exponential random graph models (ERGMs) and latent space models (LSMs) and we will provide the \verb+R+ code used to produce the results obtained in this chapter.
The chapter is organised as follows: in Section~\ref{BSNA:randomgraphs}, we introduce the basic notation for social network analysis. In Section~\ref{BSNA:statmodels}, we highlight the basic statistical work on social networks citing recent references to enable interested readers to learn more. In particular, our interest lies on describing exponential random graph models and latent space models.
In Section 1.4, we discuss Bayesian analysis for these two families of models and computational methods on a well-known dataset using the \verb+R+ software. Predictive goodness-of-fit diagnostics are also described at the end of the section.
We conclude in Section~\ref{BSNA:conclusions} with a discussion of some future challenges.
\section{Social networks as random graphs}
\label{BSNA:randomgraphs}
Networks are relational data that can be defined as a collection of nodes interacting with each other and connected in a pairwise fashion. In typical applications, the nodes represent a set actors of various kind (people, organisations, countries, etc.) and the set edges represent a specific relationship between them (friendship, collaboration, etc.).
From a statistical point of view, networks are relational data represented as mathematical graphs. A graph consists of a set of $n$ nodes and a set of $m$ edges which define some sort of relationships between pair of nodes called dyads.
The connectivity pattern of a graph can be described by an $n \times n$ adjacency matrix $y$ encoding the presence or absence of an edge between node $i$ and $j$:
\begin{equation*}
y_{ij}
=\left\{\begin{matrix}
1, & \textrm{if }(i,j) \textrm{ are connected,}\\
0, & \textrm{otherwise.}\\
\end{matrix}
\right.
\end{equation*}
Two nodes are adjacent or neighbours if there is an edge between them. If $y_{ij} = y_{ji}, \forall i, j$ then the adjacency matrix is symmetric and the graph is undirected, otherwise the graph is directed and it is often called digraph. Edges connecting a node to itself (self-loops) are generally not allowed in many applications and will not be considered in this context. The nature of the edges between nodes can take a range of values indicating the strength, frequency, intensity, etc. of the relation between a dyad. In this paper we consider only binary networks. According to the generally used notation, $y$ will be used to indicate both a random graph and its adjacency matrix.
\section{Statistical modelling approaches to social network analysis}
\label{BSNA:statmodels}
Many probability models have been proposed in order to summarise the connectivity structure of social networks by utilising their network statistics.
The family of exponential random graph models (ERGMs) is a generalisation of various models which take different assumptions into account: the Bernoulli random graph model \cite{erd:ren59} in which edges are considered independent of each other; the $p_1$ model \cite{hol:lei81} where dyads are assumed independent, and its random effects variant the $p_2$ model \cite{van:sni:zij04}; and the Markov random graph model \cite{fra:str86} where each pair of edges is conditionally dependent given the rest of the graph.
The family of latent space models has been proposed by \cite{hof:raf:han02} under the assumption that each node of the graph has a unknown position in a latent space and the probability of the edges are functions of those positions and node covariates. The latent position cluster model of \cite{han:raf:tan07} represents a further extension of this approach that takes account of clustering.
Other latent variable modelling approaches are represented by stochastic blockmodels \cite{now:sni01} that involve block model structures whereby network nodes are partitioned into latent classes and the presence of any relationship between them depends on their block membership.
\subsection{Exponential random graph models (ERGMs)}
\label{BSNA:ERGMs}
Introduced by \cite{hol:lei81} to model individual heterogeneity of nodes and reciprocity of their edges, the family of exponential random graph models (ERGMs) was generalised by \cite{fra:str86}, \cite{was:pat96} and \cite{sni:pat:rob:han06}.
ERGMs constitute a broad class of network models (see \cite{rob:pat:kal:lus07} for an introduction) that assume that the topological structure of an observed
network $y$ can be explained in terms of the relative prevalence of a set of overlapping subgraph configurations $s(y)$ called network statistics:
\begin{equation}
p(y | \theta) = \frac{ \exp \{ \theta^t s(y)\} }
{z(\theta)}
\end{equation}
This equation states that the probability that an observed network $y$ given the set of parameters $\theta$ is equal to the exponent of an observed vector of network statistics $s(y)$ multiplied by its associated vector of unknown parameters $\theta$ divided by a normalising constant $z(\theta)$ to make all probabilities sum to one. The latter is calculated as the sum over all possible network configurations on the same set of $n$ nodes of the observed network. In practice $z(\theta)$ is computationally infeasible to calculate for non trivially-small networks.
\subsection{Latent space models (LSMs)}
\label{BSNA:LVMs}
Latent space models were introduced by \cite{hof:raf:han02} under the basic assumption that each node has an unknown position $z_i$ in a $d$-dimensional Euclidean latent space. Network edges are assumed to be conditionally independent given the latent positions, and the probability of an edge between nodes is modelled as a function of their positions.
Generally, in these models the smaller the distance between two nodes in the latent space, the greater their probability of being connected.
The likelihood function of latent space models can be written as follows:
\begin{equation*}
p(y|z,\alpha)=\prod_{i\neq j}^N \frac{\exp(\alpha-d_{ij})^{y_{ij}}}{1+\exp(\alpha-d_{ij})}
\end{equation*}
The standard metric is the Euclidean distance (ED in Table~\ref{tab:pack}) and is defined as: $d_{ij} = |z_i- z_j|$. As an alternative the squared Euclidean distance (SED in Table~\ref{tab:pack}) is defined as: $d_{ij} = |z_i- z_j|^2$ and has been proposed by \cite{gol:mur15} for computational reasons (see~\ref{BSNA:BLVMs}). The latent positions are assumed to be Normally distributed, or having a Gaussian mixture model structure in case of the latent position cluster models (LPCMs), a generalisation of latent space models where latent clusters are assumed to be useful to explain data heterogeneity.
For strongly asymmetric graph, it is suggested to use the bilinear latent model setting $d_{ij} = z_i'z_j$ so that the probability of a link depends on the angle between two actors. This model is available in the \verb+latentnet+ package through the \verb+bilinear+ argument included in the \verb+ergmm+ function.
All the presented latent space network models can be extended to incorporate covariate informations $x_{ij}$ introducing a parameter $\beta$, or the degree heterogeneity in sending or receiving links, these parameters are called sender and receiver if the network is directed, or sociality if the network is undirected \cite{Krivitsky2009}.
\section{Bayesian inference for social network models}
\label{BSNA:Bayes}
The Bayesian approach to statistical problems is probabilistic. Inference is based on the posterior distribution which is the conditional probability of the unknown quantities $\Omega$ given the data $y$. The posterior distribution extracts the information in the data and provide a complete summary of the uncertainty about the unknowns via Bayes' theorem:
\begin{equation}
p(\Omega | y) = \frac{p(y | \Omega) \; p(\Omega)}{p(y)}
\end{equation}
Bayesian analysis is able to give us a full probabilistic framework of uncertainty and this is something which is essential in the context of complex statistical modelling. Moreover recent research in social network analysis has demonstrated the advantages and effectiveness of probabilistic Bayesian approaches to relational data.
In this chapter we will focus on parameter inference so the uncertainties $\Omega$ will refer to the ERGM parameters $\theta$ or the LSM parameters $\alpha$ and latent positions $z$.
\subsection{R-based software tools}
\label{BSNA:Rtools}
Applied researchers interested in Bayesian statistics are increasingly attracted to \verb+R+ \cite{R} because of the ease of which one can code algorithms to sample from posterior distributions as well as the significant number of packages contributed to the Comprehensive R Archive Network (CRAN) that provide tools for Bayesian inference.
\verb+R+ represents a useful tool for social network analysis with many advantages over traditional software packages. With a little coding and patience, one can produce ad hoc analyses and visualisations for the problem under study.
Moreover \verb+R+ has a huge set of statistical libraries so that end users can complement their social network analysis research with any analysis of your choosing within \verb+R+ environment.
In this section we briefly review Bayesian tools for ERGMs and LSMs:
\begin{itemize}
\item The \verb+Bergm+ package (version 3.0.1) \cite{cai:fri14} implements Bayesian analysis for ERGMs using the methods proposed by \cite{cai:fri11,cai:fri13,cai:mir15}. The package provides a comprehensive framework for Bayesian inference and model selection using Markov chain Monte Carlo (MCMC) algorithms.
\item The \verb+latentnet+ package (version 2.5.1) \cite{latentnet, latentnet2}, which is part of the \verb+statnet+ suite of packages \cite{han:hun:but:goo:mor07}, provides comprehensive toolsets for Bayesian analysis for latent position and cluster network models using MCMC procedures.
\item The \verb+VBLPCM+ package (version 2.4.3) \cite{sal:mur13} contains a collection of functions implementing variational Bayesian Inference for the latent position cluster model.
\item The \verb+lvm4net+ package (version 0.2) \cite{gol15} contains a collection of functions implementing fast variational Bayesian inference for latent space models.
\end{itemize}
Other \verb+R+ implementations of Bayesian methods for statistical social network models include: \verb+RSiena+ \cite{rip:sni11} implementing stochastic actor-based models; \verb+hergm+ \cite{hergm} implementing hierarchical ERGMs with local dependence; \verb+sna+ (belonging to the \verb+statnet+ suite of packages) generating posterior samples from Butt's Bayesian network accuracy model using Gibbs sampling.
\section{Data}
\label{BSNA:data}
We demonstrate ideas and examples throughout the paper using the Dolphin network dataset, an undirected social network of frequent associations between 62 dolphins in a community living off Doubtful Sound, New Zealand (see Figure~\ref{fig:Dolphins_graph}), as compiled by \cite{lus:sch:boi:haa:slo:daw03}.
The results presented in this paper have been obtained using \verb+R+ version 3.1.3. To create, manipulate and visualise the observed network data $y$ we can use the function \verb+network+ and \verb+plot+ included in the \verb+statnet+ suite of packages.
\begin{verbatim}
y <- read.table("http://moreno.ss.uci.edu/dolphins.dat",
skip = 130)
y <- network(y, directed = FALSE)
plot(y, vertex.col = "blue")
\end{verbatim}
\begin{figure}[htp]
\centering
\includegraphics[scale=.8]{dolphins_graph}
\caption{Dolphin undirected network graph.}
\label{fig:Dolphins_graph}
\end{figure}
\subsection{Bayesian inference for exponential random graph models}
\label{BSNA:BERGMs}
Bayesian inference for ERGMs is based on the posterior distribution of $\theta$ given the data $y$:
\begin{equation}
p(\theta | y)
= \frac{p(y | \theta) \; p(\theta)}{p(y)}
= \frac{ \exp \{ \theta^t s(y)\} }
{z(\theta)} \frac{p(\theta)}{p(y)}
\end{equation}
where $p(y)$ is the evidence or marginal likelihood of $y$ which is typically intractable.
Standard MCMC methods such as the Metropolis-Hastings algorithm, can deal with posterior estimation as long as the target posterior density is known up to the model evidence $p(y)$. Unfortunately in the ERGM context the posterior density $p(\theta | y)$ of non-trivially small ERGMs includes two intractable normalising constants, the model evidence $p(y)$ and $z(\theta)$. For this reason, the ERGM posterior density is ``doubly intractable'' \cite{mur:gha:mac06}.
In order to carry out Bayesian inference for ERGMs, the \verb+Bergm+ package makes use of a combination of Bayesian algorithms and MCMC techniques \cite{cai:fri11,cai:fri13}. The approximate exchange algorithm circumvents the problem of computing the normalising constants of the ERGM likelihoods, while the use of multiple chains and efficient adaptive proposal strategies are able to speed up the computations and improve chain mixing quite significantly.
The approximate exchange algorithm implemented by the \verb+bergm+ function can be summarised in the following way:\\[.4cm]
For each chain, repeat until converge:\\[.3cm]
\begin{itemize}
\item[1)] generate $\theta'$ (using some proposal strategy)\\[.2cm]
\item[2)] simulate $s(y')$ from ERGM likelihood (using standard MCMC procedures such as \cite{mor:han:hun08})\\[.2cm]
\item[3)] update $\theta \rightarrow \theta'$ with the log of the probability:
\begin{equation*}
\min\left( 0,\; \left[ \theta - \theta'\right]^t \left[s(y') - s(y)\right]
+\log\left[
\frac{p(\theta')}
{p(\theta)}\right]\right)
\end{equation*}
\end{itemize}
Let us consider the following three dimensional model including the number of edges and two new specification statistics e.g.: geometrically weighted edgewise shared partners (gwesp) and geometrically weighted non-edgewise shared partners (gwesp) \cite{hun:han06}:
\begin{center}
\begin{tabular}{ll}
\verb+gwnsp+ =& $e^{\phi_v} \sum_{k=1}^{n-2}
\left \{ 1-\left( 1 - e^{-\phi_v} \right)^{k} \right \} NEP_k(y)$ \\
\verb+gwesp+ =& $e^{\phi_v} \sum_{k=1}^{n-2}
\left \{ 1-\left( 1 - e^{-\phi_v} \right)^{k} \right \} EP_k(y)$
\end{tabular}
\end{center}
where the scale parameters $\phi_v = \phi_u=0.6$, and $EP_k(y)$ and $NEP_k(y)$ are respectively the number of connected and non-connected pairs of nodes with exactly $k$ common neighbours.
We can use the \verb+bergm+ function to sample from the posterior distribution using the MCMC algorithm described above. In this example we use the parallel adaptive direction sampling (ADS) procedure described in \cite{cai:fri11} for step 1 and 1,200 iterations (\verb+main.iters+) for each chain. We set the number of MCMC chains to 9 by using the argument \verb+nchains+. The number of iterations used to simulate network statistics $s(y')$ at step 2 is defined by the argument \verb+aux.iters+ and it is set to $3,000$.
\begin{verbatim}
model <- y ~ edges +
gwnsp(.6, fixed = TRUE) +
gwesp(.6, fixed = TRUE)
post <- bergm(model,
main.iters = 1200,
aux.iters = 3000,
nchains = 9)
bergm.output(post, lag = 200)
\end{verbatim}
The \verb+bergm.output+ function produces MCMC diagnostic plots (Figure~\ref{fig:Bergm_post}) and the estimated posterior means, standard deviations, and acceptance rates for each of the 9 chains and for the aggregated overall chain.
{\footnotesize
\begin{verbatim}
MCMC results for Model:
y ~ edges + gwnsp(.6, fixed = TRUE) + gwesp(.6, fixed = TRUE)
Posterior mean:
theta1 (edges) theta2 (gwnsp.fixed.0.6) theta3 (gwesp.fixed.0.6)
Chain 1 -2.3512134 -0.1864153 0.7521076
Chain 2 -2.3889219 -0.1800818 0.7515701
Chain 3 -2.3362841 -0.1779192 0.7068975
Chain 4 -2.5628317 -0.1646549 0.7898211
Chain 5 -2.3709133 -0.1799828 0.7316151
Chain 6 -2.5407332 -0.1646283 0.7798916
Chain 7 -2.4301006 -0.1783869 0.7698418
Chain 8 -2.4523673 -0.1799679 0.8017089
Chain 9 -2.3681535 -0.1789971 0.7424549
Posterior sd:
theta1 (edges) theta2 (gwnsp.fixed.0.6) theta3 (gwesp.fixed.0.6)
Chain 1 0.33244522 0.03951240 0.11308154
Chain 2 0.43199257 0.04669447 0.11585996
Chain 3 0.37344505 0.04083082 0.10625747
Chain 4 0.41110025 0.04962470 0.11782669
Chain 5 0.48867437 0.05371030 0.14904932
Chain 6 0.36796911 0.04055858 0.13496058
Chain 7 0.42739511 0.04311092 0.14186529
Chain 8 0.48943818 0.05430663 0.12666345
Chain 9 0.38717484 0.04531716 0.12718701
Acceptance rate:
Chain 1 0.1316667
Chain 2 0.1375000
Chain 3 0.1158333
Chain 4 0.1550000
Chain 5 0.1475000
Chain 6 0.1566667
Chain 7 0.1700000
Chain 8 0.1525000
Chain 9 0.1500000
Overall posterior density estimate:
theta1 (edges) theta2 (gwnsp.fixed.0.6) theta3 (gwesp.fixed.0.6)
Post. mean -2.4223910 -0.17678157 0.7584343
Post. sd 0.4222507 0.04675703 0.1296237
Overall acceptance rate: 0.15
\end{verbatim}
}
In this example, we can observe a low density effect expressed by the negative value of the posterior mean of the edge effect parameter ($\theta_1$) combined with the negative value of multiple connectivity ($\theta_2$) and positive value of transitivity parameter ($\theta_3$).
\begin{figure}[htp]
\centering
\includegraphics[scale=.65]{out_bergm}\\[.5cm]
\caption{MCMC diagnostics for the overall chain. The 3 plot columns are: estimated marginal posterior densities (left), traces (center) and autocorrelation plots (right).}
\label{fig:Bergm_post}
\end{figure}
\subsection{Bayesian inference for latent space models}
\label{BSNA:BLVMs}
A fully Bayesian approach for latent space models allows the estimation of all the parameters and latent positions simultaneously e.g. via MCMC sampling or variational approximation.
In this paragraph, we perform an empirical Bayesian analysis in order to compare different computational approaches for the visualisation and prediction properties of LSMs with and without clustering. To carry out this type of analysis we can use the following \verb+R+ packages: \verb+latentnet+ \cite{latentnet}, \verb+VBLPCM+ \cite{sal:mur13} and \verb+lvm4net+ \cite{gol15}. Their main features of these packages are shown in Table~\ref{tab:pack}.
\begin{table}[ht]
\centering
\caption{Comparison of the main features of the packages for latent space modeling}\label{tab:pack}
\begin{tabular}{|l|cc|cc|c|}
\hline
& \multicolumn{2}{c|}{\textbf{Model}} &\multicolumn{2}{c|}{\textbf{Inference Method}} & \textbf{Clustering}\\
& ED & SED & MCMC & Variational & \\
\hline
\verb+latentnet+ & \ding{51} & \ding{55} & \ding{51} & \ding{55} & \ding{51} \\
\verb+VBLPCM+ & \ding{51} & \ding{55} & \ding{55} & \ding{51} & \ding{51} \\
\verb+lvm4net+ & \ding{55} & \ding{51} & \ding{55} & \ding{51} & \ding{55} \\
\hline
\end{tabular}
\end{table}
The \verb+latentnet+ package uses Bayesian MCMC algorithms so the model estimation is computationally very expensive, and the times to estimate the model can become extremely large even for networks of the order of hundreds of nodes. For this reason the variational Bayes approach to estimate the latent space model and the latent position cluster model in order to make feasible the modelling of larger networks \cite{sal:mur13,gol:mur15}.
The basic idea of this method is to find a lower bound of the log-likelihood by introducing a variational posterior distribution $q$ and maximize it \cite{jor:gha:jaa:sau99}.
The posterior probability of the unknown parameters $(z,\alpha)$ can be written in the following form:
\begin{equation*}
p(z,\alpha|y)=Cp(y|z,\alpha)p(\alpha)\prod_{i=1}^Np(z_i),
\end{equation*}
where $C$ is the unknown normalising constant. In the \verb+VBLPCM+ package, a hierarchical prior structure is taken into consideration.
In \cite{gol:mur15}, the variational posterior $q(z,\alpha|y)$ is defined in the following way:
\begin{equation*}\label{q.var}
q(z,\alpha|y)=q(\alpha)\prod_{i=1}^Nq(z_i),
\end{equation*}
where $ q(\alpha)=\mathcal{N}(\tilde{\xi},\tilde{\psi}^2)$ and $q(z_i)=\mathcal{N}(\tilde{z_i},\tilde{\Sigma})$.
The idea of using the squared Euclidean distance in the LSM was proposed by \cite{gol:mur15} in order to have less approximation to be made in the variational estimation procedure.
In the \verb+latentnet+ package, we use the function called \verb+ergmm+ to estimate the posterior distribution of the LSM parameters and latent positions. The argument \verb+d+ refers to the dimension of the latent space, which we set equal to 2 to make the visualisation of the latent positions of the nodes easier.
\begin{verbatim}
post.latentnet <- ergmm(y ~ euclidean(d = 2))
\end{verbatim}
In the \verb+VBLPCM+ package, we can use the function called \verb+vblpcmfit+ to estimate the posterior distribution of the LPCM by specifying the number of clusters. In order to estimate a LSM we consider one cluster by setting the argument \verb+G+ equal to 1.
\begin{verbatim}
post.vblpcm <- vblpcmfit(vblpcmstart(y, G = 1, d = 2))
\end{verbatim}
It is important to notice that the variational maximisation algorithm is subject of the risk of reaching local maximum.
The package \verb+VBLPCM+ provides a special function called \verb+vblpcmstart+ to generate initial latent positions. This algorithm is based on the Fruchterman-Reingold method by default (argument \verb+START+), but there is also the possibility of using random values, geodesic distances or Graph Laplacian methods. In this function other model features such as sociality effects, and node covariates can also be specified.
In \verb+lvm4net+ we use the function called \verb+lsm+ to estimate the posterior distribution of the LSM parameters and latent positions using a variational inferential approach. This function makes use of the Fruchterman-Reingold method to set the initial positions by default. Multi-start procedure can be implemented by changing the value associated to the argument \verb+nstart+ and only the values reaching the maximum are stored. It is also possible to start from random initial positions by setting the argument \verb+randomZ+ equal to \verb+TRUE+.
From Table~\ref{tab:compare}, we can see that the \verb+lsm+ function is much faster than the \verb+ergmm+ function. In this case, the squared Euclidean distance is used.
\begin{verbatim}
post.lvm4net <- lsm(y[,], D = 2)
\end{verbatim}
\begin{table}[ht]
\centering
\caption{Timings in seconds to fit LSMs (no clustering, G = 1).}\label{tab:compare}
\begin{tabular}{l|c}
\hline
& Time in sec. \\
\hline
\verb+latentnet+ & 111.20 \\
\verb+VBLPCM+ & 14.02 \\
\verb+lvm4net+ & 6.47 \\
\hline
\end{tabular}
\end{table}
The latent positions are invariant under rotation, reflection and translations. For this reason we can match the rotations using the \verb+rotXtoY+ function (included in \verb+lvm4net+) in order to visualise and compare the latent positions estimated by the three approaches using the \verb+plotY+ function (included in \verb+lvm4net+).
\begin{verbatim}
Z <- post.lvm4net$lsmEZ
Zm <- rotXtoY(post.latentnet$mkl$Z,Z)$X
Zv <- rotXtoY(post.vblpcm$V_z,Z)$X
plotY(y[,], EZ = Zm, main = "latentnet")
plotY(y[,], EZ = Zv, main = "VBLPCM")
plotY(y[,], EZ = Z, main = "lvm4net")
\end{verbatim}
\begin{figure}[htp]
\centering
\includegraphics[scale=.85]{z_latentnet_p}
\caption{Latent positions obtained by using the \texttt{latentnet} package.}
\label{fig:latentnet_plot}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[scale=.85]{z_vblpcm_p}
\caption{Latent positions obtained by using the \texttt{VBLPCM} package.}
\label{fig:vblpcm_plot}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[scale=.85]{z_lvm4net_p}
\caption{Latent positions obtained by using the \texttt{lvm4net} package.}
\label{fig:lvm4net_plot}
\end{figure}
In Figures~\ref{fig:latentnet_plot}, \ref{fig:vblpcm_plot}, \ref{fig:lvm4net_plot} we can see the estimated latent positions obtained using the three packages. In this example, the visualisation results obtained from \verb+latentnet+ and \verb+lvm4net+ are similar even though the distance model adopted is different.
Latent position cluster models (LPCMs) are latent space models which incorporate a Gaussian mixture model structure for the latent positions of nodes in the latent space in order to accommodate the clustering of nodes in the network. The \verb+latentnet+ and \verb+VBLPCM+ packages can be used to estimate latent position cluster models by fixing the number of clusters (\verb+G+). For our toy example, we choose 2 clusters.
\begin{verbatim}
post.latentnet.G2 <- ergmm(y ~ euclidean(d = 2, G = 2))
\end{verbatim}
\begin{figure}[htp]
\centering
\includegraphics[scale=.7]{z_latentnet_p2}
\caption{Estimated latent positions from LPCM with $2$ clusters obtained by using the \texttt{latentnet} package.}
\label{fig:latentnet2_plot}
\end{figure}
\begin{verbatim}
post.vblpcm.G2 <- vblpcmfit(vblpcmstart(y, G = 2, d = 2))
\end{verbatim}
\begin{figure}[htp]
\centering
\includegraphics[scale=.7]{z_vblpcm_p2}
\caption{Estimated latent positions from LPCM with $2$ clusters obtained by using the \texttt{VBLPCM} package.}
\label{fig:vblpcm2_plot}
\end{figure}
In Figures~\ref{fig:latentnet2_plot} and \ref{fig:vblpcm2_plot} we can see the latent positions and the clusters returned by the two packages. The two algorithms give very similar results as they find latent groups differing of just one node.
\begin{table}[ht]
\centering
\caption{Timings in seconds to fit LPCM with two clusters (G = 2).}\label{tab:compare2}
\begin{tabular}{l|c}
\hline
& Time in sec. \\
\hline
\verb+latentnet+ & 86.27 \\
\verb+VBLPCM+ & 11.95 \\
\hline
\end{tabular}
\end{table}
From Table~\ref{tab:compare2} it is possible to notice that the \verb+VBLPCM+ package is much faster than the \verb+latentnet+ package.
The \verb+latentnet+ package gives exact estimates as they are based on MCMC simulations from the posterior distribution. However it only allows to deal with small networks whereas the approximate approaches of the \verb+VBLPCM+ and \verb+lvm4net+ packages are able to handle networks on thousands of nodes.
\subsection{Predictive goodness-of-fit (GoF) diagnostics}
\label{BSNA:GOF}
An important feature of the Bayesian approach is to make available procedures to establish whether the estimated parameter posterior of the model achieves a good fit to the key topological features of the observed network.
The \verb++ function included in the \verb+Bergm+ package provides a useful tool for assessing Bayesian goodness-of-fit so as to examine the fit of the data to the posterior model obtained by the \verb+bergm+ function. The observed network data are compared with a set of networks simulated from independent parameter values of the posterior density estimate. This comparison is made in terms of high-level network statistics not explicitly included in the model \cite{hun:goo:han08}.
The \verb+R+ code below is used to compare some high level network statistics observed in the Dolphin network with a series of network statistics simulated from $100$ random realisations of the estimated posterior distribution \verb+post.est+ using $10,000$ auxiliary iterations for the network simulation step. The \verb+bgof+ function included in the \verb+Bergm+ package returns the plots shown in Figure~\ref{fig:Bergm_bgof}.
\begin{verbatim}
bgof(post,
n.deg = 20,
n.dist = 15,
n.esp = 15)
\end{verbatim}
\begin{figure}[htp]
\centering
\includegraphics[scale=.8]{gf_bergm}\\[.5cm]
\caption{GoF diagnostics for ERGM (\texttt{Bergm} package): The red line displays the goodness of fit statistics for the observed data together
with boxplots of GoF network statistics based on 100 simulated networks from the posterior distribution.}
\label{fig:Bergm_bgof}
\end{figure}
In Figure~\ref{fig:Bergm_bgof} we see, based on the various GoF statistics, that the networks simulated from the posterior distribution are in reasonable agreement with the observed network. We can therefore conclude that the model is a reasonable fit to the data, despite its simplicity.
In the LSM context, it is possible to use the \verb+gof+ function included in \verb+latentnet+ and \verb+VBLPCM+ and the \verb+goflsm+ function included in the \verb+lvm4net+ package to perform posterior GoF diagnostics. The \verb+GOF+ argument can be used to set the types of GoF statistics we want to analyse. Figures~\ref{fig:gf_latentnet}, \ref{fig:gf_vblpcm}, and \ref{fig:gf_lvm4net} display the GoF plots.
\begin{verbatim}
gf.latentnet <- gof(post.latentnet,
GOF = ~ degree + esp + distance)
plot(gf.latentnet)
gf.vblpcm <- gof(post.vblpcm,
GOF = ~ degree + esp + distance)
plot(gf.vblpcm)
gf.lvm4net <- goflsm(post.lvm4net,
Y = y[,],
stats = c("degree", "esp", "distance"),
doplot = FALSE)
plot(gf.lvm4net)
\end{verbatim}
The GoF analysis indicates that the LSM estimated by using the variational approximation with squared Euclidean distance implemented in the \verb+lvm4net+ package displays a better fit of the model to the data compared to the other two approaches.
\begin{figure}[htp]
\centering
\includegraphics[scale=.8]{gf_latentnet}\\[.5cm]
\caption{GoF diagnostics for LSM (\texttt{latentnet} package): The solid black line displays the goodness of fit statistics for the observed data together with boxplots of GoF network statistics based on 100 simulated networks from the posterior distribution.}
\label{fig:gf_latentnet}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[scale=.8]{gf_vblpcm}\\[.5cm]
\caption{GoF diagnostics for LSM (\texttt{VBLPCM} package): The solid black line displays the goodness of fit statistics for the observed data together with boxplots of GoF network statistics based on 100 simulated networks from the posterior distribution.}
\label{fig:gf_vblpcm}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[scale=.8]{gf_lvm4net}\\[.5cm]
\caption{GoF diagnostics for LSM (\texttt{lvm4net} package): The red line displays the goodness of fit statistics for the observed data together
with boxplots of GoF network statistics based on 100 simulated networks from the posterior distribution.}
\label{fig:gf_lvm4net}
\end{figure}
To display the GoF diagnostics for the LPCMs estimated above, we can use the same \verb+R+ functions.
\begin{verbatim}
gf.latentnet.G2 <- gof(post.latentnet.G2,
GOF = ~ degree + esp + distance)
plot(gf.latentnet.G2)
gf.vblpcm.G2 <- gof(post.vblpcm.G2,
GOF = ~ degree + esp + distance)
plot(gf.vblpcm.G2)
\end{verbatim}
\begin{figure}[htp]
\centering
\includegraphics[scale=.8]{gf_latentnet_G2}\\[.5cm]
\caption{GoF diagnostics for LPCM with 2 clusters (\texttt{latentnet} package): The solid black line displays the goodness of fit statistics for the observed data together with boxplots of GoF network statistics based on 100 simulated networks from the posterior distribution.}
\label{fig:gf_latentnet2}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[scale=.8]{gf_vblpcm_G2}\\[.5cm]
\caption{GoF diagnostics for LPCM with 2 clusters (\texttt{VBLPCM} package): The solid black line displays the goodness of fit statistics for the observed data together with boxplots of GoF network statistics based on 100 simulated networks from the posterior distribution.}
\label{fig:gf_vblpcm2}
\end{figure}
From Figures~\ref{fig:gf_latentnet2} and \ref{fig:gf_vblpcm2} we can see that the \verb+VBLPCM+ package has a better fit to the data in terms of edgewise shared partners distributions compared to the \verb+latetnet+ package. For this example, the inclusion of 2 clusters does not seem to produce a significant improvement in terms of GoF with respect to the LSM without clustering.
\section{Conclusions}
\label{BSNA:conclusions}
This chapter provided an overview of a number of social network models emphasising the computational perspective. In fact, the most important issue associated to statistical social network models is concerting their computational complexity which requires the development of efficient inferential algorithms and software able to deal with the increasing size of relational data available.
In particular, we have presented some recent advanced Bayesian approaches to parameter estimation of exponential random graph models and latent variable network models. We demonstrated that Bayesian inference is a very helpful and powerful approach allowing a formal treatment of uncertainty using the rules of probability.
We discussed how Bayesian parameter estimation for exponential random graph models and latent space models is a computationally intensive problem that can be tackled using advanced MCMC and variational techniques. We illustrated the main capabilities of the \verb+Bergm+, \verb+latentnet+, \verb+VBLPCM+ and \verb+lvm4net+ packages for the open-source \verb+R+ software through a tutorial analysis of a well-known social network dataset.
For each modelling approach we have also considered a Bayesian graphical test of goodness of fit to assess whether or not a given parametric model is compatible with the observed network data.
Advances in the Bayesian methodology and computing will prove crucial to effectively capture heterogeneity and organise different sources of information commonly available in social network data. For this reason, we believe statistical social network analysis will became fertile ground for interdisciplinary research in advanced statistics and social network analysis applications.
\newpage
|
{
"timestamp": "2015-04-14T02:14:04",
"yymm": "1504",
"arxiv_id": "1504.03152",
"language": "en",
"url": "https://arxiv.org/abs/1504.03152"
}
|
\section{Introduction}
Among higher gradient elasticity models \cite{Mindlin65,Mindlin68,aifantis2011gradient,maugin1980method,lazar2006note,MauginVirtualPowers} one of the very first models considered in the literature is the so called indeterminate couple stress model \cite{Grioli60,Mindlin62,Toupin64,Koiter64}
in which the higher gradient contributions only enter through gradients
of the continuum rotation, i.e. the total elastic energy can be written
as $$W(\nabla u,\nabla(\nabla u))=W_{\rm lin}({\rm sym}\nabla u)+W_{{\rm curv}}(\nabla{\rm curl}\,u).$$
In general, higher gradient elasticity models are used to
describe mechanical structures at the micro- and nano-scale or to
regularize certain ill-posed problems by means of these higher gradient
contributions.
In a series of papers which are either published \cite{hadjesfandiari2013fundamental,hadjesfandiari2011couple} or available as preprints \cite{Dargush,hadjesfandiari2013skew,hadjesfandiari2010polar,hadjesfandiari2014evo} Hadjesfandiari and Dargush
have reconsidered the linear indeterminate couple stress model. They are postulating a certain physically plausible split in the virtual work principle. Based on this postulate they claim that the second-order couple stress tensor must be skew-symmetric. Since their development has spread considerable confusion in the field of higher gradient elasticity, we were prompted to carefully re-examine their claim. In doing so we hope to contribute an important clarification in the field and to put an end to the above mentioned confusion.
In the course of our re-examination it turned out that the boundary conditions in the classical indeterminate couple stress theory have never been correctly derived. In \cite{MadeoGhibaNeffMunchCRM,MadeoGhibaNeffMunchKM,NeffGhibaMadeoMunch} we provide, for the first time the consistent and complete boundary conditions for the classical indeterminate couple stress model. In doing so, we find the underlying error in the argument by Hadjesfandiari and Dargush \cite{hadjesfandiari2011couple,hadjesfandiari2010polar,hadjesfandiari2011couple,hadjesfandiari2013fundamental,hadjesfandiari2013skew,hadjesfandiari2014evo}. While Hadjesfandiari and Dargush start with the linear general anisotropic couple stress response and only later specify to isotropy, for definiteness, we consider from the outset the linear isotropic indeterminate couple stress case. Nevertheless, our development is essentially independent of any isotropy assumption. It is clear that exhibiting the errors in their development for the simpler case of isotropy is sufficient for invalidating their claim.
Before discussing the papers by Hadjesfandiari and Dargush \cite{hadjesfandiari2011couple,hadjesfandiari2013fundamental,hadjesfandiari2014evo,hadjesfandiari2013skew} we will first recall the indeterminate couple stress model in its accepted format as far as kinematics and equilibrium equations are concerned. We also need to introduce the new set of traction boundary conditions which rectify the shortcomings in all previous papers. For comparison, the up to now accepted traction boundary conditions are also presented.
In the light of this new framework, we try to follow the argument given by Hadjesfandiari and Dargush \cite{hadjesfandiari2011couple,hadjesfandiari2013fundamental,hadjesfandiari2014evo,hadjesfandiari2013skew} as closely as possible. We will show that their implicitly formulated requirement, recast as a physically plausible postulate by us, leads to the skew-symmetry of the couple stress tensor if and only if the classical traction boundary conditions are assumed. However, within the corrected format of traction boundary conditions no similar conclusion is possible.
Despite the finally erroneous claim by Hadjesfandiari and Dargush we recognize their work in being the reason to reconsider the indeterminate couple stress model and to finally find the underlying error which occurred in the accepted boundary conditions and not directly in Hadjesfandiari and Dargush's work.
\section{Notational agreements}
In this paper, we denote by $\R^{3\times 3}$ the set of real $3\times 3$ second order tensors, written with
capital letters. For $a,b\in\R^3$ we let $\langle {a},{b}\rangle_{\R^3}$ denote the scalar product on $\R^3$ with
associated vector norm $\|{a}\|^2_{\R^3}=\langle {a},{a}\rangle_{\R^3}$.
The standard Euclidean scalar product on $\R^{3\times 3}$ is given by
$\langle{X},{Y}\rangle_{\R^{3\times3}}=\tr({X Y^T})$, and thus the Frobenius tensor norm is
$\|{X}\|^2=\langle{X},{X}\rangle_{\R^{3\times3}}$. In the following we omit the index
$\R^3,\R^{3\times3}$. The identity tensor on $\R^{3\times3}$ will be denoted by $\id$, so that
$\tr({X})=\langle{X},{\id}\rangle$. We adopt the usual abbreviations of Lie-algebra theory, i.e.,
$\so(3):=\{X\in\mathbb{R}^{3\times3}\;|X^T=-X\}$ is the Lie-algebra of skew symmetric tensors
and $\sL(3):=\{X\in\mathbb{R}^{3\times3}\;| \tr({X})=0\}$ is the Lie-algebra of traceless tensors.
For all $X\in\mathbb{R}^{3\times3}$ we set $\sym X=\frac{1}{2}(X^T+X)\in\Sym$, $\skw X=\frac{1}{2}(X-X^T)\in \so(3)$ and the deviatoric part $\dev X=X-\frac{1}{3}\;\tr(X)\id\in \sL(3)$ and we have
the \emph{orthogonal Cartan-decomposition of the Lie-algebra} $\gl(3)$
\begin{align}
\gl(3)&=\{\sL(3)\cap \Sym(3)\}\oplus\so(3) \oplus\mathbb{R}\!\cdot\! \id,\quad
X=\dev \sym X+ \skw X+\frac{1}{3}\tr(X) \id\,.
\end{align}
Throughout this paper (when we do not specify else) Latin subscripts take the values $1,2,3$. Typical conventions for differential
operations are implied such as comma followed
by a subscript to denote the partial derivative with respect to
the corresponding cartesian coordinate. We also use the Einstein notation of the sum over repeated indices if not differently specified. Here,
we consider the operators $\axl:\so(3)\rightarrow\mathbb{R}^3$ and $\anti:\mathbb{R}^3\rightarrow \so(3)$ through
\begin{align}
(\axl \overline{A})_k=-\frac{1}{2}\, \epsilon_{ijk}\overline{A}_{ij},\qquad \overline{A}.\, v=(\axl \overline{A})\times v, \quad \quad (\anti(v))_{ij}=-\varepsilon_{ijk}v_k, \quad\quad \overline{A}_{ij}=\anti(\axl \overline{A})_{ij},
\quad \quad
\end{align}
for all $ v\in\mathbb{R}^3$ and $\overline{A}\in \so(3)$, where $\epsilon_{ijk}$ is the totally antisymmetric third order permutation tensor. We recall that for a third order tensor $\mathbb{E}$ and $X\in \mathbb{R}^{3\times 3}$, $v\in \mathbb{R}^3$ we have the contraction operations $\mathbb{E}: X\in \mathbb{R}^{3}$, $\mathbb{E}. \, v\in \mathbb{R}^{3\times 3}$ and $X.\, v\in \mathbb{R}^3$, with the components
\begin{align}
(\mathbb{E}:\, X)_{i}=\mathbb{E}_{ijk}\,X_{kj}\, , \qquad (\mathbb{E}. \,v)_{ij}=\mathbb{E}_{ijk}\,v_{k}\,,\qquad (X.\, v)_{i}=X_{ij}\,v_j.
\end{align}
For multiplication of two matrices we will not use other specific notations, this means that for $A,B\in \mathbb{R}^{3\times 3}$ we are setting $(A B)_{ij}=A_{ik}B_{kj}$.
We consider a body which occupies a bounded open set $\Omega$ of the three-dimensional Euclidian space $\R^3$ and assume that its boundary
$\partial \Omega$ is a piecewise smooth surface. An elastic material fills the domain $\Omega\subseteq \R^3$ and we refer the motion of the body to rectangular axes $Ox_i$. Let us consider an open subset $\Gamma$ of $\partial \Omega$.
Here, $\nu$ is a vector tangential to the surface $\Gamma$ and which is orthogonal to its boundary $\partial \Gamma$, $\tau=n\times \nu$ is the tangent to the curve $\partial \Gamma$ with respect to the orientation on $\Gamma$.
We assume that $\partial \Omega$ is a smooth surface. Hence, there are no singularities of the boundary and the jump $\jump{a\cdot \nu}:=[a\cdot \nu]^{+}+[a\cdot \nu]^{-}=([a]^{+}-[a]^{-})\cdot \nu$ of $a$ across the joining
curve $\partial\Gamma$ arises only as consequence of possible discontinuities of the corresponding quantities which follows from the prescribed boundary conditions on $\Gamma$ and $\partial \Omega\setminus \overline{\Gamma}$, where
$$[\,\cdot\,]^-:=\hspace*{0cm}\dd\lim\limits_{\footnotesize{\begin{array}{c}x\in\partial \Omega\setminus \overline{\Gamma}\\
\ x\rightarrow \partial \Gamma\end{array}}}\hspace*{0cm} [\,\cdot\,], \qquad \qquad [\,\cdot\,]^+:=\hspace*{-0.2cm}\dd\lim\limits_{\footnotesize{\begin{array}{c}x\in \Gamma\\
\ x\rightarrow \partial \Gamma\end{array}}}\hspace*{-0.2cm} [\,\cdot\,].$$
The usual Lebesgue spaces of square integrable functions, vector or tensor fields on $\Omega$ with values in $\mathbb{R}$, $\mathbb{R}^3$ or $\mathbb{R}^{3\times 3}$, respectively will be denoted by $L^2(\Omega)$. Moreover, we introduce the standard Sobolev spaces \cite{Adams75,Raviart79,Leis86}
\begin{align}
\begin{array}{ll}
{\rm H}^1(\Omega)=\{u\in L^2(\Omega)\, |\, {\rm grad}\, u\in L^2(\Omega)\}, &\|u\|^2_{{\rm H}^1(\Omega)}:=\|u\|^2_{L^2(\Omega)}+\|{\rm grad}\, u\|^2_{L^2(\Omega)}\,,\vspace{1.5mm}\\
{\rm H}({\rm curl};\Omega)=\{v\in L^2(\Omega)\, |\, {\rm curl}\, v\in L^2(\Omega)\}, &\|v\|^2_{{\rm H}({\rm curl};\Omega)}:=\|v\|^2_{L^2(\Omega)}+\|{\rm curl}\, v\|^2_{L^2(\Omega)}\, ,\vspace{1.5mm}\\
{\rm H}({\rm div};\Omega)=\{v\in L^2(\Omega)\, |\, {\rm div}\, v\in L^2(\Omega)\}, &\|v\|^2_{{\rm H}({\rm div};\Omega)}:=\|v\|^2_{L^2(\Omega)}+\|{\rm div}\, v\|^2_{L^2(\Omega)}\, ,
\end{array}
\end{align}
of functions $u$ or vector fields $v$, respectively. Furthermore, we introduce their closed subspaces $H_0^1(\Omega)$, ${\rm H}_0({\rm curl};\Omega)$ as completion under the respective graph norms of the scalar valued space $C_0^\infty(\Omega)$ (the set of infinitely
differentiable functions with compact support in $\Omega$).
For vector fields $v$ with components in ${\rm H}^{1}(\Omega)$, i.e.
$
v=\left( v_1, v_2, v_3\right)^T\, , v_i\in {\rm H}^{1}(\Omega),
$
we define
$
\nabla \,v=\left(
(\nabla\, v_1)^T,
(\nabla\, v_2)^T,
(\nabla\, v_3)^T
\right)^T
$, while for tensor fields $P$ with rows in ${\rm H}({\rm div}\,; \Omega)$, i.e.
$
P=\left(
P_1^T,
P_2^T,
P_3^T
\right)$, $P_i\in {\rm H}({\rm div}\,; \Omega)$
we define
$ {\rm Div}\,P=\left(
{\rm div}\, P_1,
{\rm div}\,P_2,
{\rm div}\,P_3
\right)^T$.
The corresponding Sobolev-spaces will be denoted by
$
H^1(\Omega), H^1(\Div;\Omega).
$
\section{The classical indeterminate couple stress model}\label{sectaxlb}\setcounter{equation}{0}
We are now shortly re-deriving the classical equations based on the $\nabla [\axl (\skw \nabla u)]$-formulation of the indeterminate couple stress model. This part does not contain new results, see, e.g., \cite{MadeoGhibaNeffMunchKM} for further details, but is included for setting the stage for this contribution.
The linear isotropic indeterminate couple stress problem can be viewed as a minimization problem
\begin{align}
\int_\Omega\Big[ \mu\, \|\sym \nabla u\|^2+\frac{\lambda}{2}\, [\tr(\nabla u)]^2+W_{\rm curv}(\nabla \curl u)-\langle f, u\rangle \Big]dv \quad \rightarrow \text{min. w.r.t.} \quad u,
\end{align}
subjected to geometric and mechanical boundary conditions, in part depending on the form of $W_{\rm curv}(\nabla \curl u)$, which will be specified later on.
In the following, in order to place the subject in the literature, we outline some curvature energies proposed in different isotropic second gradient elasticity properly models:
\begin{itemize}
\item {\bf the indeterminate couple stress model} (Grioli-Koiter-Mindlin-Toupin model) \cite{Grioli60,Aero61,Koiter64,Mindlin62,Toupin64,Sokolowski72,grioli2003microstructures} in which the higher derivatives (apparently) appear only through derivatives of the infinitesimal continuum rotation $\curl u$. Hence, the curvature energy has the equivalent forms
\begin{align}\label{KMTe}
W_{\rm curv}( \nabla \curl\, u)&=\mu\, L_c^2\,\left[\frac{\alpha_1}{4}\, \|\sym \nabla \curl\, u\|^2+\frac{\alpha_2}{4}\,\| \skw \nabla \curl\, u\|^2\right]\notag\\
&=\mu\, L_c^2\,\left[{\alpha_1}\, \|\sym\nabla[\axl(\skw \nabla u)]\|^2+{\alpha_2}\,\| \skw \nabla[\axl(\skw \nabla u)]\|^2\right]\\
&=\mu\, L_c^2\,\left[\frac{\alpha_1}{4}\, \|\dev \sym \nabla \curl\, u\|^2+\frac{\alpha_2}{4}\,\| \skw \nabla \curl\, u\|^2\right].\notag
\end{align}
We remark that the spherical part of the couple stress tensor remains {\bf indeterminate} since $\tr(\nabla \curl u)={\rm div} (\curl u)=0$. In order to prove the pointwise uniform positive definiteness it is assumed following \cite{Koiter64}, that ${\alpha_1}>0, {\alpha_2}>0$. Note that pointwise uniform positivity is often assumed \cite{Koiter64} when deriving analytical solutions for simple boundary value problems because it allows to invert the couple stress-curvature relation. It is clear that pointwise positive definiteness is not necessary for well-posedness \cite{Neff_JeongMMS08}.
Mindlin \cite[p. 425]{Mindlin62} explained the relations between Toupin's constitutive equations \cite{Toupin62} and Grioli's \cite{Grioli60} constitutive equations and concluded that the obtained equations in the linearized theory are identical, since the extra constitutive parameter $\eta^\prime$ of Grioli's model does not explicitly appear in the equations of motion but enters only the boundary conditions. The same extra constitutive coefficient appears in Mindlin and Eshel's version \cite{Mindlin68}.
\item
{\bf the modified - symmetric couple stress model - the conformal model}. On the other hand, in the conformal case \cite{Neff_Jeong_IJSS09,Neff_Paris_Maugin09} one may consider that $\alpha_2=0$, which makes the couple stress tensor $\widetilde{m}$ symmetric and trace free. This conformal curvature case has been
considered by Neff in \cite{Neff_Jeong_IJSS09}, the curvature energy having the form
\begin{align}
W_{\rm curv}( \nabla \curl\, u)&=\mu\, L_c^2\,\frac{\alpha_1}{4}\, \|\sym \nabla \curl\, u\|^2=\mu\, L_c^2\,\alpha_1\, \|\dev \sym \nabla[\axl(\skw \nabla u)\|^2.
\end{align}
Indeed, there are two major reasons uncovered in \cite{Neff_Jeong_IJSS09} for using the modified couple stress model. First, in order to avoid singular stiffening behaviour for smaller and smaller samples in bending \cite{Neff_Jeong_bounded_stiffness09} one has to take $\alpha_2=0$. Second, based on a homogenization procedure invoking an intuitively appealing natural ``micro-randomness" assumption (a strong statement of microstructural isotropy) requires conformal invariance, which is again equivalent to $\alpha_2=0$. Such a model is still well-posed \cite{Neff_JeongMMS08} leading to existence and uniqueness results with only one additional material length scale parameter, while it is {\bf not} pointwise uniformly positive definite.
\item {\bf the skew-symmetric couple stress model - the non-conformal model}.
{Hadjesfandiari and Dargush} strongly advocate \cite{hadjesfandiari2011couple,hadjesfandiari2013fundamental,hadjesfandiari2013skew} the opposite extreme case, $\alpha_1=0$ and $\alpha_2>0$, i.e. they used the curvature energy
\begin{align}
W_{\rm curv}( \nabla \curl\, u)&=\mu\, L_c^2\,\frac{\alpha_2}{4}\, \|\skw \nabla({\rm curl}\, u)\|^2=\mu\, L_c^2\,\alpha_2\, \|\axl \skw \nabla({\rm curl}\, u)\|^2=4\, \mu\, L_c^2\, \alpha_2\, \|{\rm curl}\,({\rm curl}\, u)\|^2\notag\notag.
\end{align}
In that model the nonlocal force stresses and the couple stresses are both assumed to be skew-symmetric. Their reasoning, based in fact on an incomplete understanding of boundary conditions is critically discussed in this paper and generally refuted, while mathematically it is well-posed, which will also be shown.
\end{itemize}
\subsection{Equilibrium and constitutive equations}
Taking free variations $\delta u\in C^\infty(\Omega)$ in the energy $W(\sym \nabla u,\nabla \curl u)=W_{\rm lin}(\sym \nabla u)+W_{\rm curv}(\nabla \curl u)$, where
\begin{align}\label{gradeq11}
W_{\rm lin}(\sym \nabla u)=&\mu\, \|\sym \nabla u\|^2+\frac{\lambda}{2}\, [\tr(\nabla u)]^2=\mu\, \|\dev \sym \nabla u\|^2+\frac{2\, \mu+3\,\lambda}{6}\, [\tr(\nabla u)]^2,\notag\\
W_{\rm curv}(\nabla \curl u)=& \mu\,L_c^2\,[\alpha_1\, \|\dev\sym \nabla [\axl (\skw \nabla u)]\|^2+
\alpha_2\, \|\skw\nabla [\axl (\skw \nabla u)]\|^2],
\end{align}
we obtain the virtual work principle
\begin{align}\label{gradeq211}
\frac{\rm d}{\rm dt}\int_\Omega W(\nabla u+t\,\nabla \delta u)\,dv\Big|_{t=0}=&\int_\Omega\bigg[ 2\mu\,\langle\sym \nabla u, \sym \nabla \delta u \rangle+\lambda \tr(\nabla u)\,\tr( \nabla \delta u)\notag\\&+\mu\,L_c^2\,[2\, \alpha_1\, \langle \dev\sym \nabla [\axl (\skw \nabla u)],\dev\sym \nabla [\axl (\skw \nabla \delta u)]\rangle \\&+
2\, \alpha_2\, \langle \skw\nabla [\axl (\skw \nabla u)],\skw\nabla [\axl (\skw \nabla \delta u)]\rangle]+\langle f,\delta u\rangle \bigg]\, dv =0,\notag
\end{align}
where $f$ denotes the body force density.
The classical divergence theorem
leads to
\begin{align}\label{germaneq311}
\int_\Omega\langle \Div (\sigma-\widetilde{\tau})+f, \delta u \rangle \, dv\underbrace{-\int_{\partial \Omega}\langle (\sigma-\widetilde{\tau}).\, n, \delta u\rangle \,dv
-\int_{\partial\Omega}\langle \widetilde{ {m}}.\, n, \axl (\skw \nabla \delta u) \rangle \, da}_{\tiny \text{the virtual power work of the surface forces}}=0,
\end{align}
where
\begin{align}\label{consteq}
\widetilde{\sigma}_{\rm total}&=\sigma-\widetilde{\tau}\not\in{\rm Sym}(3)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\ \ \, \text{total force-stress tensor} \vspace{1.2mm}\notag\\
\sigma&=2\, \mu \, \sym \nabla u+\lambda \, \tr(\nabla u)\id \in {\rm Sym}(3)\qquad\qquad\qquad\qquad\quad\quad\ \,\text{local force-stress tensor} \vspace{1.2mm}\notag\\
\widetilde{\tau}&=\dd\frac{1}{2}\anti
{\rm Div}[\widetilde{ {m}}]\in \so(3),\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \text{nonlocal force-stress tensor} \vspace{1.2mm}\\
\widetilde{ {m}}&=\mu\,L_c^2\,[{\alpha_1}\sym \nabla (\curl u)+ {\alpha_2}\,\skw\nabla (\curl u)]\qquad\qquad\qquad\quad \text{couple stress tensor}\vspace{1.2mm}\notag\\
&=\mu\,L_c^2\,[{\alpha_1}\dev\sym \nabla (\curl u)+ {\alpha_2}\,\skw\nabla (\curl u)]\text{}\vspace{1.2mm}\notag\\
&=\mu\,L_c^2\,[2\,{\alpha_1}\dev\sym \nabla[\axl (\skw \nabla u)]+2\, {\alpha_2}\,\skw\nabla [\axl (\skw \nabla u)]],\notag
\end{align}
and $n$ is the unit outward normal vector at the surface $\partial \Omega$.
The equilibrium equation are therefore
\begin{align}\label{ec11}
\Div \,\widetilde{\sigma}_{\rm total}+f=0.
\end{align}
Note that the local force-stress tensor $\sigma$ is always symmetric, the nonlocal force-stress tensor $\widetilde{\tau}$ is automatically skew-symmetric, while the second order hyperstress tensor (the couple stress tensor) $\widetilde{m}$ may or may not be symmetric, depending on the material parameters. The asymmetry of force stress is a hidden constitutive assumption, compare to \cite{NeffGhibaMadeoMunch}.
\subsection{Boundary conditions}
\subsubsection{Classical (incomplete) Grioli-Koiter-Mindlin-Tiersten boundary conditions}
Since the variations at the boundary and the interior can be assigned independently, see \eqref{germaneq311}, we must have:
\begin{align}\label{integralasuprafata}
&-\int_{\partial \Omega}\langle (\sigma-\widetilde{\tau}).\, n, \delta u\rangle \,da
-\int_{\partial\Omega}\langle \widetilde{ {m}}.\, n, \axl(\skw \nabla \delta u) \rangle\, da=
0\qquad\text{or equivalently}\qquad\\
&-\int_{\partial \Omega}\langle (\sigma-\widetilde{\tau}).\, n, \delta u\rangle \,da
-2\,\int_{\partial\Omega}\langle \widetilde{ {m}}.\, n, \curl \delta u \rangle\, da=
0.\notag
\end{align}
This suggests 6 possible independent prescriptions of mechanical boundary conditions; three for the normal components of the total force stress $(\sigma-\widetilde{\tau}).n$ and three for the normal components of the couple stress tensor.
The possible Dirichlet boundary conditions on $\Gamma\subset \partial \Omega$ seem to be the 6 conditions\footnote{as indeed proposed by Grioli \cite{Grioli60} in concordance with the Cosserat kinematics for independent fields of displacements and microrotation.}
\begin{align}\label{bc11}
u=\widetilde{u}, \qquad \axl(\skw \nabla u)=\widetilde{w} \quad (\text{or equivalently} \ \ \curl u=2\,\widetilde{w}),
\end{align}
for two given functions $\widetilde{u}, \widetilde{w}:\mathbb{R}^3\rightarrow\mathbb{R}^3$ at the open subset $\Gamma\subset\partial \Omega$ of the boundary (3+3 boundary conditions).
However, following Koiter we note
\begin{remark}\label{remarkcc} {\rm [independent variations and curl] }Assume $u\in C^\infty(\overline{\Omega})$ and $u\Big|_{{\Gamma}}$ is known. Then ${\rm curl}\Big|_{{\Gamma}}$ exists and for any open subset ${\Gamma}\subset \partial \Omega$ the integral $\int_{{\Gamma}}\langle {\rm curl}
\,u, n\rangle\, da$ is already known by Stokes theorem, while $\int_{{\Gamma}}\langle {\rm curl}\, u,\tau\rangle \,da$ is still free, where $\tau$ is any tangential vector field on the open set ${\Gamma}\subset \partial \Omega$.
Only the two tangential components of ${\rm curl}\, u$ may be independently prescribed on an open subset of the boundary.
\end{remark}
Already, Mindlin and Tiersten \cite{Mindlin62} have rightly remarked that also in this formulation only 5 mechanical boundary conditions can be prescribed. They rewrote \eqref{integralasuprafata} in a further separated form
\begin{align}\label{integralasuprafata0}
-\int_{\partial \Omega}\langle (\sigma-\widetilde{\tau}).\, n -\underbrace{\frac{1}{2} n\times \nabla[\langle (\sym \widetilde{ {m}}).n,n\rangle]}_{\begin{array}{c}\text{\tiny{performs already}}\vspace{-1.5mm}\\ \text{\tiny{work only against \ $\delta u$} }\end{array}}, \delta u\rangle \,da
-\int_{\partial\Omega}\langle (\id-n\otimes n)\,\widetilde{ {m}}.n,\!\!\!\!\!\!\!\!\underbrace{(\id-n\otimes n)\,[\axl(\skw \nabla \delta u)]}_{\begin{array}{c}\text{\tiny{cannot be assigned arbitrarily independent of $\delta u$}}\vspace{-1.5mm}\\
\text{\tiny{since it still contains certain}}\vspace{-1.5mm}\\\text{\tiny{ "tangential" derivatives of $ \delta u\in C^\infty(\Omega)$}}
\end{array} }\!\!\!\!\!\!\!\!\!\!\!\rangle\,
da\notag\\
\\
=-\int_{\partial \Omega}\langle (\sigma-\widetilde{\tau}).\, n -\underbrace{\frac{1}{2} n\times \nabla[\langle (\sym \widetilde{ {m}}).n,n\rangle]}_{\begin{array}{c}\text{\tiny{performs already}}\vspace{-1.5mm}\\ \text{\tiny{work only against \ $\delta u$} }\end{array}}, \delta u\rangle \,da
-2\,\int_{\partial\Omega}\langle (\id-n\otimes n)\,\widetilde{ {m}}.n,(\id-n\otimes n)\,[\curl \delta u]\rangle\,
da=
0,\notag
\end{align}
which already shows that the second term in \eqref{integralasuprafata} still contains contributions which perform work against $\delta u$, namely
$\frac{1}{2} n\times \nabla[\langle (\sym \widetilde{ {m}}).n,n\rangle]$, while the remaining higher order term $(\id-n\otimes n)\,\widetilde{ {m}}.n$ performs work against combinations of second derivatives. Mindlin and Tiersten \cite{Mindlin62} concluded that 3 boundary conditions derive from the first integral (correctly) and two other from the second integral, since \cite[p.~432]{Mindlin62} ``the normal component of the couple stress vector [$\langle \widetilde{ {m}}.n,n\rangle=\langle \sym\widetilde{ {m}}.n,n\rangle$] on" $\partial \Omega$ ``enters only in the combination with the force-stress vector shown in the coefficient of" $\delta u$ ``in the surface integral (our first term on the right hand side of \eqref{integralasuprafata0}).
Therefore, Mindlin and Tiersten \cite{Mindlin62} concluded that the boundary conditions consists in:
\begin{itemize}
\item \textbf{Geometric boundary conditions} on $\Gamma\subset \partial \Omega$:
\begin{align}\label{bcme1}
&\hspace{4.52cm}u\ =\ \widetilde{u}^0, \qquad \qquad \quad\qquad\quad\qquad\qquad \qquad \qquad (3\ \text{bc})\notag\\
&\hspace{0.06cm}\left\{
\begin{array}{rcl}
(\id -n\otimes n).\,\axl(\skw \nabla u)&=&\ \ \!\!\!(\id -n\otimes n).\,\axl(\skw \nabla \widetilde{u}^0),\vspace{1.2mm}\\
\text{or}\qquad (\id -n\otimes n).\,\curl u&=&\!\!\!2\, (\id -n\otimes n).\,\curl \widetilde{u}^0, \qquad \qquad \qquad \qquad (2\ \text{bc})
\end{array}\right.
\end{align}
for a given function $\widetilde{u}^0:\mathbb{R}^3\rightarrow\mathbb{R}^3$ at the boundary.
The latter condition prescribes only the tangential component of $\axl(\skew \nabla u)$. Therefore, we may prescribe only 3+2 independent boundary conditions.
\item \textbf{Traction boundary conditions} on $\partial \Omega\setminus\overline{\Gamma}$:
\begin{align}\label{bcme2}
(\sigma-\widetilde{\tau}).\, n-\frac{1}{2} n\times \nabla[\langle (\sym \widetilde{ {m}}).n,n\rangle]&=\widetilde{t},\qquad\qquad \qquad\qquad\text{traction}\qquad\qquad \qquad\qquad\quad\text{ \ (3 bc)}\notag\\
(\id-n\otimes n)\,\widetilde{ {m}}.n&=(\id-n\otimes n)\,\widetilde{g}, \qquad \text{``double force normal traction" (2 bc)}
\end{align}
for prescribed functions $\widetilde{t},\widetilde{g}:\mathbb{R}^3\rightarrow\mathbb{R}^3$ at the boundary.
\end{itemize}
However, while \eqref{bcme1} and \eqref{bcme2} correctly describe the maximal number of independent boundary conditions in the indeterminate couple stress model and these conditions have been rederived again and again by Yang et al. \cite{Yang02}, Park and Gao \cite{Park07}, \cite{lubarda2003effects}, etc. among others they are not correct. This is explained in the following paragraphs.
\subsubsection{Complete (corrected) traction boundary conditions}\label{sectioncomplete}
The indeterminate couple stress model is not simply obtained as a constraint Cosserat model \cite{Sokolowski72}, i.e. assuming that $\overline{A}=\axl(\skew \nabla u)$. In the indeterminate couple stress model the only independent kinematical degree of freedom is $u$. We understand that the indeterminate couple stress model constructed as a constraint Cosserat model represents at most an approximation of the indeterminate couple stress model, in the sense that the boundary conditions are not correctly/completely considered. We remark that the quantity $\langle \widetilde{ {m}}.n,(\id-n\otimes n)[\axl(\skw \nabla \delta u)]\rangle$ does still contain contributions performing work against $\delta u$ alone (even though there is a projection $\id -n\otimes n$ involved), which can be assigned arbitrarily and are therefore somehow related to independent variation $\delta u$. This case is not similar to the Cosserat theory in which we assume a priori that displacement $u$ and microrotation $\overline{A}\in\so(3)$ are independent kinematical degrees of freedom.
At this point, it must also be considered that the tangential trace of
the gradient of virtual displacement can be integrated by parts once
again and that the surface divergence theorem can be applied to this
tangential part of $\nabla\delta{u}$. Let ${n}$ be the unit normal vector
at a considered
surface point.
As it is well known from differential geometry, the projectors $\id -{n}\otimes{n}$ and ${n}\otimes{n}$ allow to split a given vector or tensor field in one part projected
on the plane tangent to the considered surface and one projected on
the normal to such surface (see also \cite{NthGrad,TheseSeppecher,dell2012beyond} for details)). For the sake of simplicity we assume that $\partial\Omega$ is a smooth surface of class $C^2$. As in the notation section, we consider the curve $\partial \Gamma$ which joins the open subsets $\Gamma$ and $\partial \Omega\setminus \Gamma$ of the boundary. Therefore, in our case and in our abbreviations, the surface divergence \cite[p.~58, ex.~7]{gurtin2010mechanics} reads:
\begin{equation}
\int_{\partial \Omega}\mathrm{Div}^{S}\left(v\right)da:=\int_{\partial \Omega}\langle \id-n\otimes n,\nabla\left((\id-n\otimes n)\cdot v\right)]\rangle\, da=\int_{\partial \Gamma}\jump{\:\langle v,{\nu}\rangle\text{ }}\,ds.\label{eq:Surface_Div_Th}
\end{equation}
for any field $v\in \mathbb{R}^3$.
Indeed, using the surface divergence theorem, we have obtained in \cite{MadeoGhibaNeffMunchKM} that
\begin{align}\label{axldeltan}
-\int_{\partial \Omega}\langle (\sigma&-\widetilde{\tau}).\, n, \delta u\rangle \,da
-\int_{\partial \Omega}\langle \widetilde{ {m}}.\, n, \axl(\skw \nabla \delta u) \rangle\, da\notag\\
=&
-\int_{\partial \Omega}\langle (\sigma-\frac{1}{2}\anti
{\rm Div}[\widetilde{ {m}}]).\, n -\frac{1}{2} n\times \nabla[\langle (\sym \widetilde{ {m}}).n,n\rangle]\\
&\qquad\qquad\ -\frac{1}{2}\{\nabla[(\anti[(\id-n\otimes n)\widetilde{ {m}}.\, n])\, (\id -{n}\otimes{n})]:(\id -{n}\otimes{n}), \delta u\rangle \,da\notag\\& -\frac{1}{2}
\int_{\partial \Omega}\underbrace{\langle (\id -n\otimes n)\anti[(\id-n\otimes n)\widetilde{ {m}}.\, n].n, \nabla \delta u.n \rangle}_{\tiny\begin{array}{c}\text{\tiny{completely}}\ {\tiny \delta u}\text{\tiny{-independent second}}\vspace{0mm}\\
\text{\tiny{order normal variation of the gradient}}
\end{array} } da-\frac{1}{2}
\int_{\partial \Gamma}\langle \jump{\anti[(\id-n\otimes n)\widetilde{ {m}}.\, n]. \,\nu}, \delta u\rangle ds.\notag
\end{align}
Hence, there are indeed two terms
$$(\sigma-\frac{1}{2}\anti
{\rm Div}[\widetilde{ {m}}]).\, n -\frac{1}{2} n\times \nabla[\langle (\sym \widetilde{ {m}}).n,n\rangle]-\frac{1}{2}\nabla[(\anti[(\id-n\otimes n)\widetilde{ {m}}.\, n])(\id -{n}\otimes{n})]:(\id -{n}\otimes{n})$$ and $$\jump{\anti[(\id-n\otimes n)\widetilde{ {m}}.\, n]. \,\nu}$$
which perform work against $\delta u$, while only the term
$$(\id -n\otimes n)\anti[(\id-n\otimes n)\widetilde{ {m}}.\, n].n$$
is related solely to the independent second order normal variation of the gradient $ \nabla \delta u.n$.
This split of the boundary condition is not the one as obtained e.g. by Gao and Park \cite{park2008variational} and seems to be entirely new in the context of the indeterminate couple stress model.
Therefore, we need to adjoin on $\partial \Omega$ the following complete set of boundary conditions:
\begin{itemize}
\item Geometric (essential) boundary conditions on $\Gamma\subset\partial \Omega$:
\begin{align}\label{bc1110}
u&=\widetilde{u}^0, \qquad \qquad \quad\quad\quad\quad\quad \quad\ \ \qquad\qquad\qquad\qquad (3\ \text{bc})\\
(\id-n\otimes n)(\nabla u).n&= (\id-n\otimes n)(\nabla \widetilde{u}^0).n, \qquad \quad \qquad\qquad\qquad\qquad (2\ \text{bc})\notag
\end{align}
where $\widetilde{u}^0:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ is a prescribed function (i.e. 3+2=5 boundary conditions), as in \cite{Mindlin62}.
\item Traction boundary conditions on $\partial \Omega\setminus\overline{\Gamma}$:
\begin{align}\label{bc1001}\hspace{-1.3cm}
\begin{array}{rcl}
(\sigma-\widetilde{\tau}).\, n -\frac{1}{2} n\times \nabla[\langle (\sym \widetilde{ {m}}).n,n\rangle] \hspace{4cm}&&\vspace{1.2mm}\\
-\frac{1}{2}\{\nabla[(\anti[(\id-n\otimes n)\widetilde{ {m}}.\, n])(\id-n\otimes n)]:(\id-n\otimes n) &=&t,\vspace{1.2mm}\\
\dd(\id -n\otimes n)\anti[(\id-n\otimes n)\widetilde{ {m}}.\, n].n&=&(\id-n\otimes n)\,g
\end{array}
&
\begin{array}{r}
\\(3\ \text{bc})\vspace{1.2mm}\\
(2\ \text{bc})
\end{array}
\end{align}
where $t, g:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ are prescribed functions on $\partial \Omega\setminus \overline{\Gamma}$.
\item Jump boundary conditions on $\partial \Gamma\subset \partial \Omega$:
\begin{align}\begin{array}{rcl}
\jump{\anti[(\id-n\otimes n)\widetilde{ {m}}.\, n]. \,\nu}&=&\widetilde{\pi},\hspace{2.3cm}
\end{array}
\qquad
\qquad \qquad\qquad\qquad \begin{array}{r}
(3\ \text{bc})
\end{array}
\end{align}
where $\widetilde{\pi}$ is prescribed on $\partial \Gamma$ and leads to 3 boundary conditions.
\end{itemize}
\section{The Hadjesfandiari and Dargush's postulate}\setcounter{equation}{0}
Let us now turn to Hadjesfandiari and Dargush's far reaching claims.
In the abstract of their paper \cite{hadjesfandiari2011couple} Hadjesfandiari and Dargush write:
\begin{quote}
\textit{``By relying on the definition of admissible boundary conditions, the principle of virtual work and some
kinematical considerations, we establish the skew-symmetric character of the couple-stress tensor {\rm [}$\widetilde{m}${\rm ]} in
size-dependent continuum representations of matter. This fundamental result, which is independent of
the material behavior {\rm [e.g. isotropy]}, resolves all difficulties in developing a consistent couple stress theory."}
\end{quote}
\noindent
In their appendix of \cite[p. 1263-1264]{hadjesfandiari2013fundamental} they add:
\begin{quote}
\textit{``Therefore, the present determinate theory is not mathematically a special case of {\rm [the]} indeterminate {\rm [couple stress]} theory obtained by letting {\rm [}$\alpha_1=0${\rm ]}. This is just a coincidence for the linear isotropic case where equations in both theories have some similarities. It should be realized that the determinate theory is not simply about fixing the constitutive equations for {\rm [the]} linear isotropic couple stress theory of elasticity. The present size-dependent couple stress theory is the consistent couple stress theory in continuum mechanics. This has been achieved by discovering the skew-symmetric character of the couple-stress tensor {\rm [}$\widetilde{m}${\rm ]}. Mindlin, Tiersten and Koiter did not recognize the mean curvature tensor {\rm [}$\skw[\nabla{\rm curl}\, u]${\rm ]} as the consistent measure of deformation in continuum mechanics."}\footnote{In \cite[p. 1283, A.16]{hadjesfandiari2013fundamental} Hadjesfandiari and Dargush erroneously take the strict positivity of curvature parameters $\alpha_1,\alpha_2>0$, used initially by Koiter, Mindlin and others, as belonging to the definition of the indeterminate couple stress model. This is clearly not the case. It can be shown that both limit cases $\alpha_1>0, \alpha_2=0$ (modified couple stress theory) and $\alpha_1=0, \alpha_2>0$ (Hadjesfandiari and Dargush choice) are mathematically admitted with the provision of using the attendant correct boundary conditions being defined by the solution space depending on the values of $\alpha_1, \alpha_2$, see Section \ref{existHD}.}
\end{quote}
In our understanding this claim is completely unfounded. Assuming isotropic response, we treat their ``consistent and determinate" couple stress theory as the linear and isotropic accepted indeterminate couple stress model with constitutive parameters $\alpha_1=0$, $\alpha_2>0$ in \eqref{consteq}.
Turning to their most important claim regarding the skew-symmetry of the coule stress tensor $\widetilde{m}$ we will exhibit their line of thought. Their reasoning is based on their fundamental hypothesis that the normal component of the couple stress traction vector $\langle \widetilde{m}.n,n\rangle $ should vanish on any bounding surface of an arbitrary volume\footnote{They also wrote \cite[p.13]{hadjesfandiari2014evo}:
\textit{``...the corresponding generalized force must be zero and, for the normal component of the surface moment-traction vector {\rm [}$\widetilde{m}.n${\rm ]}, we must enforce the condition {\rm [}$\langle \widetilde{m}.n,n\rangle=0${\rm ]}."}.}. In the case $\alpha_1=0$ we have $\widetilde{m}\in\so(3)$ and $\langle \widetilde{m}.n,n\rangle =0$ is satisfied automatically.
\smallskip
\noindent In our notation the argument of Hadjesfandiari and Dargush is given as follows \cite[p. 12-13]{hadjesfandiari2010polar}:
\begin{quote}
\textit{``From kinematics, since {\rm [$\omega^{nn}:=\langle {\rm curl} u,n\rangle$]} is not an independent generalized degree of freedom, its apparent
corresponding generalized force must be zero. Thus, for the normal component of the surface
couple vector {\rm [$\widetilde{m}.n$]}, we must enforce the condition
\[
[\langle \widetilde{m}.n,n\rangle=0].
\]
{\rm [...]} we notice that
the energy equation can be written for any arbitrary volume with arbitrary surface within the
body. Therefore, for any point on any arbitrary surface with unit normal $n$, we must have
\[
[\langle \widetilde{m}.n,n\rangle=0].
\]
Since $n_in_j$ is symmetric and arbitrary in {\rm [..]}, {\rm [$\widetilde{m}$]} must be skew-symmetric. Thus,
{\rm [$\widetilde{m}^T=-\widetilde{m}$.]}
This is the fundamental property of the couple-stress tensor in polar continuum mechanics,
which has not been recognized previously."} (see also \cite[p. 2500]{hadjesfandiari2011couple}).
\end{quote}
\noindent Let us interpret this statement. We rephrase it for our purpose through the formulation of an implicit
\begin{quote}
\vspace{-0.5cm}
\begin{postulate}{\rm [Hadjesfandiari and Dargush as we understand it]}\label{postulate} In any extended continuum model only the total force stress traction vector $(\sigma-\widetilde{\tau}).\, n$ should perform work against the independent virtual displacement $\delta u$ at the part $\partial \Omega\setminus{\overline{\Gamma}}$ of the boundary $\partial \Omega$, where traction boundary conditions are applied.
\end{postulate}
\end{quote}
\begin{remark}
Incidentally, this postulate is automatically satisfied in classical elasticity, Cosserat and micromorphic models \cite{Neff_Forest_jel05}, since the possible variations of the field variables are independent anyway. Whether such a postulate can be satisfied in a higher gradient continuum is the concern of Hadjesfandiari and Dargush. We will see that this is not possible.
\end{remark}
Hadjesfandiari and Dargush apply this postulate to the classical (incomplete) Mindlin and Tiersten's format of the boundary conditions, namely \eqref{bcme1} and \eqref{bcme2}. Inspection of the indeterminate couple stress model within the framework of these erroneous classical boundary conditions, see e.g. \eqref{inspHD} and \eqref{integralasuprafata01}, shows that choosing $\sym \widetilde{m}=0$ is indeed sufficient for this postulate to be satisfied.
Hadjesfandiari and Dargush accept
\begin{align}
\sigma.n=\sigma_n,\qquad \widetilde{m}.n=m_n,
\end{align}
and they also realize correctly that the number of geometric or mechanic boundary conditions is 5 since the tangential component of the test function $\delta u$ cannot independently be varied, which is by now well established. To this aim, they spilt the term
\begin{align}\label{HDpw}
\int_{\partial V}\langle \widetilde{m}.n, \axl\skew \nabla \delta u\rangle \,da&=2\,\int_{\partial V}\langle \widetilde{m}.n, \curl \delta u\rangle\, da\\
&=2\,\int_{\partial V}\underbrace{\langle (n\otimes n)\, \widetilde{m}.n}_{\text{\tiny normal part}}, \curl \delta u\rangle\, da+2\,\int_{\partial V}\underbrace{\langle (\id-n\otimes n)\,\widetilde{m}.n}_{\text{\tiny tangential part}}, \curl \delta u\rangle\, da\notag
\\
&=2\,\int_{\partial V}\langle\widetilde{m}.n, \underbrace{(n\otimes n).\,\curl \delta u\rangle}_{\text{\tiny normal part}}\, da+2\,\int_{\partial V}\langle\widetilde{m}.n,\underbrace{(\id-n\otimes n).\,\curl \delta u\rangle}_{\text{\tiny tangential part}}\, da\notag
\end{align}
on any arbitrary subdomain $V\subset \Omega$, into its tangential and normal part. They observe that the normal part
\begin{align}\label{HDPw2}
\int_{\partial V}\langle (n\otimes n)\, \widetilde{m}.n, \curl \delta u\rangle\, da=\int_{\partial V}\underbrace{\langle (\sym \widetilde{ {m}}).n,n\rangle}_{\text{\tiny generalized force}} \, \!\!\!\underbrace{\langle \curl \delta u, n\rangle}_{\begin{array}{c}\vspace{-6mm}\\\text{\tiny no independent }\vspace{-2mm}\\
\text{\tiny degree of freedom $(\surd)$}\end{array}}\, da
\end{align}
cannot be prescribed independently of $\delta u$, since indeed their $\omega^{nn}:=\langle \curl \delta u, n\rangle$ cannot independently be prescribed due to Stokes theorem. To avoid a somehow felt inconsistency, they state that the corresponding generalized force must be zero and, for the normal component of the surface moment-traction vector, they enforce accordingly the (misguided) condition
\begin{align}
\underbrace{\langle (\sym \widetilde{ {m}}).n,n\rangle}_{\text{\tiny generalized force}} =0
\end{align}
on any arbitrary subdomain $V\subset \Omega$ having the boundary $\partial V$.
The equilibrium equations considered by Hadjesfandiari and Dargush ($\alpha_1=0$)\cite{hadjesfandiari2011couple} read therefore
\begin{align}\label{HD1}
\Div \widetilde{\sigma} _{\rm total}+f=0,
\end{align}
where the total force stress is given by
\begin{align}\label{HD2}
\widetilde{\sigma}_{\rm total}&=\sigma-\widetilde{\tau}\not\in{\rm Sym}(3)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\ \ \, \text{total force-stress tensor} \vspace{1.2mm}\notag\\
\sigma&=2\, \mu \, \sym \nabla u+\lambda \, \tr(\nabla u)\id \in {\rm Sym}(3)\qquad\qquad\qquad\qquad\quad\quad\qquad\qquad\ \,\text{local force-stress tensor} \vspace{1.2mm}\notag\\
\widetilde{\tau}&=\dd\frac{1}{2}\anti
{\rm Div}[\widetilde{ {m}}]\in \so(3),\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \text{nonlocal force-stress tensor} \vspace{1.2mm}\\
\widetilde{ {m}}&=\mu\,L_c^2\, {\alpha_2}\,\skw\nabla (\curl u)=2\,\mu\,L_c^2\, {\alpha_2}\,\skw\nabla [\axl (\skw \nabla u)]]\in \so(3)\quad\ \text{couple stress tensor}.\notag
\end{align}
To the equilibrium equation, Hadjesfandiari-Dargush adjoin on $\partial \Omega$ the following boundary conditions
\begin{itemize}
\item \textbf{Geometric boundary conditions} on $\Gamma\subset \partial \Omega$:
\begin{align}\label{HD3}
&\hspace{4.45cm}u\ =\ \widetilde{u}^0, \notag\\
&\hspace{0.06cm}\left\{
\begin{array}{rcl}
(\id -n\otimes n).\axl(\skw \nabla u)&=&\ \ \!\!\!(\id -n\otimes n).\axl(\skw \nabla \widetilde{u}^0),\\
\text{or}\qquad (\id -n\otimes n).\,\curl u&=&\!\!\!2\, (\id -n\otimes n).\,\curl \widetilde{u}^0,
\end{array}\right.
\end{align}
for a given function $\widetilde{u}^0:\mathbb{R}^3\rightarrow\mathbb{R}^3$ at the boundary.
The latter condition prescribes only the tangential component of $\axl(\skew \nabla u)$. Therefore, they prescribe(correctly) only 3+2 independent boundary conditions.
We may consider the equivalent geometric boundary conditions
\begin{align}\label{HD30}
\begin{array}{rclll}
u&=&\widetilde{u}_0 &\text{on}&\Gamma,\\
(\id-n\otimes n)\,\nabla u.\,n&=& (\id-n\otimes n)\,\nabla \widetilde{u}_0.\,n &\text{on} &\Gamma,
\end{array}
\end{align}
where $\widetilde{u}_0:\mathbb{R}^3\rightarrow\mathbb{R}^3$ is given, i.e. 3+2 boundary conditions.
\item \textbf{Traction boundary conditions} on $\partial \Omega\setminus\overline{\Gamma}$:
\begin{align}
\text{The Hadjesfandiari-Dargush-choice:}\qquad
\begin{array}{rclll}\label{HBBC}
(\sigma+\widetilde{\tau}).\, n&=&\widetilde{t}& \text{on}& \partial \Omega\setminus\overline{\Gamma},\\
(\id-n\otimes n)\widetilde{ {m}}.n&=&(\id-n\otimes n).\, \widetilde{h} & \text{on}&\partial \Omega\setminus\overline{\Gamma},
\end{array}
\end{align}
where $\widetilde{t},\widetilde{h}:\mathbb{R}^3\rightarrow\mathbb{R}^3$ are given.
\end{itemize}
\section{No reason for the Hadjesfandiari-Dargush formulation with skew-symmetric couple stress tensor}\label{HDF}\setcounter{equation}{0}
Let us try to follow the argument of Hadjesfandiari and Dargush (bona fide). It follows that the split considered by
Hadjesfandiari-Dargush is not complete, since they nowhere do use the surface divergence theorem and they do not prescribe fully independent geometrically boundary conditions. We see, on the contrary, that, when integrated over $\partial \Omega$, even upon the Hadjesfandiari and Dargush restriction the remaining tangential part from \eqref{HDpw}, namely
\begin{align}
\int_{\partial \Omega}\underbrace{\langle (\id-n\otimes n)\widetilde{m}.n, \curl \delta u\rangle}_{\text{\tiny tangential part}} da=&\quad\frac{1}{4} \int_{\partial \Omega}\Big\{\langle \nabla[(\anti(\widetilde{ {m}}.\, n))\,(\id-n\otimes n)]: (\id-n\otimes n), \delta u\rangle \\
&\qquad\qquad\qquad+
\underbrace{\langle (\id -n\otimes n)\anti(\widetilde{ {m}}.\, n).n, \nabla \delta u.n \rangle}_{\begin{array}{c}\vspace{-6mm}\\\text{\tiny completely} \,\mbox{\tiny $\delta u$}\text{\tiny-independent second order}\vspace{-2mm}\\
\text{\tiny normal variation of gradient}
\end{array} }\Big\} \,da\notag,
\end{align}
still contains some parts with independent degrees of freedom, performing work against the normal derivative $\nabla \delta u. n$.
Further on, we explain this in more detail. Looking at the Mindlin and Tiersten approach, see also \cite{MadeoGhibaNeffMunchKM}, in the framework of Hadjesfandiari-Dargush's assumption $\widetilde{m}\in\so(3)$, we get
\begin{align}\label{inspHD}
\langle\widetilde{ {m}}.\,n, \axl(\skw \nabla \delta u)\rangle&=
\langle (\id-n\otimes n)\,\widetilde{ {m}}.n,\axl(\skw \nabla \delta u)\rangle\notag\\&\quad+\frac{1}{2}\langle n,\curl[\!\!\!\!\!\!\!\!\!\!\!\!\underbrace{\langle (\sym \widetilde{ {m}}).n,n\rangle}_{\begin{array}{c}\vspace{-6mm}\\ \text{\tiny zero generalized force}\vspace{-2mm}\\
\text{\tiny according to}\vspace{-2mm}\\\text{\tiny Hadjesfandiari-Dargush's assumption}\end{array}}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\, \delta u]\rangle
-\frac{1}{2}\underbrace{\langle n\times \nabla[\langle (\sym \widetilde{ {m}}).n,n\rangle]}_{\begin{array}{c}\vspace{-6mm}\\
\text{\tiny performs work against} \ \mbox{\tiny $\delta u$}\vspace{-2mm}\\
\text{\tiny and should therefore vanish}\vspace{-2mm}\\
\text{\tiny corresponding to their Postulate \ref{postulate}}
\end{array}}, \delta u\rangle,\\
&=
\langle (\id-n\otimes n)\,\widetilde{ {m}}.n,\axl(\skw \nabla \delta u)\rangle=\frac{1}{2}
\langle (\id-n\otimes n)\,\widetilde{ {m}}.n,\curl \delta u\rangle.\notag
\end{align}
Therefore \eqref{integralasuprafata0} can immediately be written as
\begin{align}\label{integralasuprafata01}
-\int_{\partial \Omega}\langle \Big\{(\sigma-\widetilde{\tau}).\, n, \delta u\rangle\, da
-\int_{\partial\Omega}\langle(\id-n\otimes n)\, \widetilde{ {m}}.n,(\id-n\otimes n)\,[\axl(\skw \nabla \delta u)]\rangle\,
da=
0,
\end{align}
which is used and accepted by Hadjesfandiari and Dargush, see \eqref{HDpw}. We comprehend their curvature parameter choice (based on the incomplete format of the independent boundary conditions): normal tractions would be automatically completely separated into pure total force-stress tractions and pure couple stress tractions.
However, after integration, the second term from \eqref{integralasuprafata01} $\langle(\id-n\otimes n)\, \widetilde{ {m}}.n,(\id-n\otimes n)\,[\axl(\skw \nabla \delta u)]\rangle$ leads to two new quantities: one performing work against the normal derivative $\nabla \delta u. n$ and one still performing work against $\delta u$. This means that the assumption of Hadjesfandiari and Dargush does not remove all quantities which may also perform work against $\delta u$, besides the total force stress tensor $(\sigma-\widetilde{\tau}).n$. This fact follows as a direct consequence of \eqref{axldeltan}, since, also in the Hadjesfandiari-Dargush formulation similar to \cite{MadeoGhibaNeffMunchKM,MadeoGhibaNeffMunchCRM}, it is possible for us to use the surface divergence theorem and to obtain \begin{align}
-\int_{\partial \Omega}\langle (\sigma&-\widetilde{\tau}).\, n, \delta u\rangle \,da
-\int_{\partial\Omega}\langle \widetilde{ {m}}.\, n, \axl(\skw \nabla \delta u) \rangle\, da\notag\\
=&
-\int_{\partial \Omega}\langle (\sigma-\widetilde{\tau}).\, n -\frac{1}{2}\nabla[(\anti[(\id-n\otimes n)\widetilde{ {m}}.\, n])(\id-n\otimes n)]:(\id-n\otimes n), \delta u\rangle \,da\notag\\& -\frac{1}{2}
\int_{\partial \Omega}\langle (\id -n\otimes n)\anti[(\id-n\otimes n)\widetilde{ {m}}.\, n].n, \nabla \delta u.n \rangle \,da\\&-\frac{1}{2}
\int_{\partial\Gamma}\langle \jump{\anti((\id-n\otimes n)\widetilde{ {m}}.\, n)}.\, \nu, \delta u\rangle\, ds\notag,\notag
\\
=&
-\int_{\partial \Omega}\langle (\sigma-\widetilde{\tau}).\, n -\frac{1}{2}\nabla[(\anti[\widetilde{ {m}}.\, n])(\id-n\otimes n)]:(\id-n\otimes n), \delta u\rangle \,da\notag\\& -\frac{1}{2}
\int_{\partial \Omega}\langle (\id -n\otimes n)\anti[\widetilde{ {m}}.\, n].n, \nabla \delta u.n \rangle \,da\notag-\frac{1}{2}
\int_{\partial\Gamma}\langle \jump{\anti(\widetilde{ {m}}.\, n)}.\, \nu, \delta u\rangle \,ds\notag,\notag
\end{align}
for all variations $ \delta u\in C^\infty(\Omega)$, where we have used that $\langle \widetilde{m}.\, n,n\rangle=0$ implies
$$(\id-n\otimes n)\widetilde{ {m}}.\, n=\widetilde{ {m}}.\, n-n\, \langle \widetilde{m}.\, n,n\rangle=\widetilde{ {m}}.\, n.$$
We consider this last representation as the only correct form following the Hadjesfandiari-Dargush assumption $\langle \widetilde{m}.\, n,n\rangle=0$ made in \eqref{HDPw2}.
Similar to the correction to the Mindlin and Tiersten's approach with the specification that $\widetilde{m}\in \so(3)$ (which is now seen to be not necessary but possible), we arrive rather, upon the Hadjesfandiari-Dargush assumption $\langle \widetilde{m}.\, n,n\rangle=0$, at the traction boundary condition
\begin{align}\label{HBBC1}
\hspace{-0.5cm}
\begin{array}{rcl}
[(\sigma-\widetilde{\tau}).\, n -\frac{1}{2}\nabla[(\anti[(\id-n\otimes n)\widetilde{ {m}}.\, n])](\id -{n}\otimes{n})]:(\id -{n}\otimes{n})\,(x) &=&\widetilde{g}(x),\vspace{1.2mm}\\
\dd[(\id -n\otimes n)\anti[(\id-n\otimes n)\widetilde{ {m}}.\, n].n]\,(x)&=&[(\id-n\otimes n)\,\widetilde{s}]\,(x)
\end{array}
\
\begin{array}{r}
(3\ \text{bc})\vspace{1.2mm}\\
(2\ \text{bc})
\end{array}
\end{align}
on $\partial \Omega\setminus {\Gamma}$, while on $\partial \Gamma$ we have to prescribe the jump conditions
\begin{align}
\hspace{4.5cm}
\begin{array}{rcl}
\dd\{\jump{\anti[(\id-n\otimes n)\widetilde{ {m}}.\, n]}.\,\nu\}(x)&=&\widetilde{\pi}(x),
\end{array}
\quad
\quad\quad\qquad\
\begin{array}{r}
(3\ \text{bc})
\end{array}\notag
\end{align}
From \eqref{HBBC1}, we see finally that their intended response is not satisfied:
\begin{remark}
Assuming that $\langle (\sym \widetilde{ {m}}).n,n\rangle=0$ (the Hadjesfandiari-Dargush assumption) implies only that
\begin{align}
\int_{\partial \Omega}\langle \nabla[(\anti[(n\otimes n)\widetilde{ {m}}.\, n])(\id -{n}\otimes{n})]:(\id -{n}\otimes{n}), \delta u\rangle \,da=0.
\end{align}
However, from $$\nabla[(\anti[\widetilde{ {m}}.\, n])(\id -{n}\otimes{n})]:(\id -{n}\otimes{n}),$$ there remains another contribution $$\nabla[(\anti[(\id-n\otimes n)\widetilde{ {m}}.\, n])(\id -{n}\otimes{n})]:(\id -{n}\otimes{n})$$ which still performs work against $\delta u$. The raison d'\^{e}tre of the Hadjesfandiari-Dargush formulation, as we understand it, was to avoid that any parts related to ``couple stress normal traction" $\widetilde{m}.n$, other than the contributions to the total force-stresses, i.e. other than $\sigma-\widetilde{\tau}$, would perform work against $\delta u$. Our calculation shows that this requirement can never be satisfied in any higher gradient elasticity theory.
\end{remark}
Continuing, Hadjesfandiari tries to support the claim of a skew-symmetric couple stress tensor with some independent motivations in \cite{hadjesfandiari2013skew}. There, he defines the ``torsion-tensor" $\Chi$ and the ``mean curvature tensor" $\omega$
\begin{align}
\Chi(u)=\sym \nabla \curl u, \qquad \omega(u)=\skew\, \nabla \curl u,
\end{align}
respectively. In Section 3.2. of
\cite[eq.~32]{hadjesfandiari2013skew} Hadjesfandiari claims that an inhomogeneous state $u$ of constant torsional deformation $\Chi(u)=0$ cannot exist. However, the conformal mapping, see Appendix \ref{aapendixconf}
\begin{align}
\phi_c(x)=\frac{1}{2}\left(2\langle \axl\widehat{W},x\rangle \,x-\axl\widehat{W})\|x\|^2\right)+[\widehat{p}\, \id+\widehat{A}]. x+\widehat{b},
\end{align}
where $\widehat{W},\widehat{A}\in \so(3)$, $\widehat{b}\in \mathbb{R}^3$, $\widehat{p}\in \mathbb{R}$ are arbitrary but constant is on the one hand inhomogeneous in the displacement $u=\phi_c-x$ but on the other hand gives precisely
\begin{align}
\Chi(u)=\Chi(\phi_c(x)-x)=\sym \nabla \curl \phi_c(x)=0.
\end{align}
For instance, a simple conformal displacement field is given by
\begin{align}
\notag
\phi_c(x)&=\left(\begin{array}{c}
2\, x_1^2 -(x_1^2+x_2^2+x_3^2)\\
2\, x_1\, x_2 \\
2\, x_1\, x_3
\end{array}\right)=\left(\begin{array}{c}
x_1^2 -(x_2^2+x_3^2)\\
2\, x_1\, x_2 \\
2\, x_1\, x_3
\end{array}\right)\ \ \Rightarrow\ \ \nabla \phi_c(x)=\left(\begin{array}{ccc}
2\, x_1& -2\, x_2& -2\, x_3\\
2\, x_2&2\, x_1& 0 \\
2\, x_3& 0&2\, x_1
\end{array}\right)\\ &\Rightarrow\ \ \skew\nabla \phi_c(x)=\left(\begin{array}{ccc}
0& -2\, x_2& -2\, x_3\\
2\, x_2&0& 0 \\
2\, x_3& 0&0
\end{array}\right)\ \ \Rightarrow\ \ \axl\skew\nabla \phi_c(x)=\left(\begin{array}{c}
0\\
-2\, x_3 \\
2\, x_2
\end{array}\right) \\ &\Rightarrow\ \ \curl \phi_c(x)=\left(\begin{array}{c}
0\\
-\, x_3 \\
\, x_2
\end{array}\right)\ \ \Rightarrow\ \ \nabla \curl \phi_c(x)=\left(\begin{array}{ccc}
0& 0& 0\\
0&0& -1 \\
0& 1&0
\end{array}\right)\Rightarrow\ \ \sym\nabla \curl\phi_c(x)=0.\notag
\end{align}
\section{Existence and uniqueness of the solution in the Hadjesfandiari-Dargush formulation}\label{existHD}\setcounter{equation}{0}
In view of the previous discussion, we do not think that the Hadjesfandiari-Dargush formulation has a sound physical motivation. Nevertheless, mathematically it is possible to consider the parameter choice inherent in the Hadjesfandiari-Dargush formulation. Let us consider for simplicity null boundary conditions. Hence, in the following we study the existence of the solution in the space
\begin{equation}
{\widetilde{\mathcal{X}}_0}\,{=}\,\big\{ u\,{\in}\,{H}^1_0(\Omega)\,|\, \curl u\in \, H(\curl; \Omega), \ \ (\id-n\otimes n).\,\curl u\big|_{\Gamma}=0\big\}.
\end{equation}
On ${\widetilde{\mathcal{X}}_0}$ we define the norm
\begin{equation}
\| u \|_{\widetilde{\mathcal{X}}_0}=\left(\|\nabla u\|^2_{L^2(\Omega)}+\| \skew \nabla\axl(\skew \nabla u)\|^2_{L^2(\Omega)} \right)^{\frac{1}{2}}=\left(\|\nabla u\|^2_{L^2(\Omega)}+\frac{1}{4}\| \curl \curl u\|^2_{L^2(\Omega)} \right)^{\frac{1}{2}},
\end{equation}
and the bilinear form
\begin{align}\label{proscalar}
((u,v))=\int_\Omega\bigg[& 2\mu\,\langle\sym \nabla u, \sym \nabla v \rangle+\lambda \tr(\nabla u)\,\tr( \nabla v)\notag\\&+2\,\mu\,L_c^2\, \alpha_3\, \langle \skw[\nabla \axl(\skw \nabla u)],\skw[\nabla \axl(\skw \nabla v)]\rangle\bigg]\,dv \\
=\int_\Omega\bigg[& 2\mu\,\langle\sym \nabla u, \sym \nabla v \rangle+\lambda \tr(\nabla u)\,\tr( \nabla v)+2\,\mu\,L_c^2\, \alpha_3\, \langle \curl\curl u,\curl\curl v\rangle\bigg]\,dv,\notag
\end{align}
where
$u,v\in{\widetilde{\mathcal{X}}_0}$. Let us define the linear operator $l:{\widetilde{\mathcal{X}}_0}\rightarrow\mathbb{R}$, describing the influence of external loads,
$
l(v)=\int_\Omega \langle f, v \rangle \, dv$ {for all} $\widetilde{w}\in{\mathcal{X}_0}.
$ We say that $w$ is a weak solution of the problem $(\mathcal{P})$ if and only if
\begin{equation}\label{wfdh}
((u,v))=l(v) \ \ \text{ for all } \ \ v\in {\mathcal{X}_0}.
\end{equation}
A classical solution $u\in{\mathcal{X}_0}$
of the problem $(\mathcal{P})$ is also a weak solution.
\begin{theorem}\label{thexdh}
Assume that
\begin{itemize}
\item[i)] the constitutive coefficients satisfy $\mu>0, \quad 3\, \lambda+2\mu>0, \quad \alpha_3\geq 0$;
\item[ii)] the loads satisfy the regularity condition $f\in L^2(\Omega)$.
\end{itemize}
Then there exists one and only one solution of the problem {\rm (\ref{wfdh})}.
\end{theorem}
\begin{proof}
In the case $\alpha_3=0$ we have the boundary value problem from classical elasticity. Further, we consider the case $\boldsymbol{\alpha_3> 0}$.
The Cauchy-Schwarz inequality, the inequalities $(a\pm b)^2\leq 2(a^2+b^2)$ and the assumption upon the constitutive coefficients lead to
\begin{align}
((u,v))&\leq \dd \,C\, \|w\|_{{\widetilde{\mathcal{X}}_0}}\,\,\|\widetilde{w}\|_{{\widetilde{\mathcal{X}}_0}}\, ,
\end{align}
which means that $((\cdot,\cdot))$ is bounded. On the other hand, we have
\begin{align}
(({u},{u}))
=\int_\Omega\bigg[& 2\mu\,\|\sym \nabla u\|^2+\lambda\,[ \tr(\nabla u)]^2+2\,\mu\,L_c^2\, \alpha_3\, \|\skw[\nabla \axl(\skw \nabla u)]\|^2]\bigg]\, dv ,
\end{align}
for all
$u\in {\widetilde{\mathcal{X}}_0}$. Moreover, as a consequence of the properties i) of the constitutive coefficients
we have that there exist the positive constant $c$
\begin{align}
(({u},{u}))
&\geq \dd c\,\int _\Omega\biggl(\| \sym\nabla u\|^2+ \|\skw[\nabla \axl(\skw \nabla u)]\|^2\biggl)\, dv.
\end{align}
From linearized elasticity we have Korn's inequality \cite{Neff00b}, that is
\begin{align}\label{Korn}
\| \nabla u\|_{L^2(\Omega)}\leq C \|\sym {\nabla}\, u\|_{L^2(\Omega)}\, ,
\end{align}
for all functions $u\in H_0^1(\Omega;\Gamma)$ with some constants $C>0$, for bounding the deformation of an elastic medium in terms of the symmetric strains.
Hence, using the Korn's inequality \eqref{Korn}, it results that there is a positive constant $C$ such that
\begin{align}
({u},{u})
&\geq \dd c\,\int _\Omega\biggl(\|\nabla u\|^2+ \|\skw[\nabla \axl(\skw \nabla u)] \|^2\biggl)\, dv=c\,\|u\|^2_{{\widetilde{\mathcal{X}}_0}}.
\end{align}
Hence our bilinear form $((\cdot,\cdot))$ is coercive. The Cauchy-Schwarz inequality and the Poincar\'{e}-inequality imply that the linear operator $l(\cdot)$ is bounded. By the Lax-Milgram theorem it follows that (\ref{wfdh}) has one and only one solution and the proof is complete.
\end{proof}
\begin{remark} The Lax-Milgram theorem used in the proof of the previous theorem also offers a continuous dependence result on the load $f$. Moreover, the weak solution $u$ minimizes on ${\mathcal{X}_0}$ the energy functional
\begin{align}
I(u)=\int_\Omega\bigg[& 2\,\mu\,\|\sym \nabla u\|^2+\lambda\,[ \tr(\nabla u)]^2\notag+2\, \mu\,L_c^2\, \alpha_3\, \|\skw(\Curl (\sym \nabla u))\|^2-\langle f,u\rangle \bigg]\, dv.\notag
\end{align}
\end{remark}
\section{The constrained Cosserat formulation of the Hadjesfandiari-Dargush model: well posedness of a degenerate Cosserat model}\setcounter{equation}{0}
Similarly to the classical indeterminate couple stress model, also the Hadjesfandiari-Dargush formulation can be obtained as a constrained Cosserat model. We only need to adapt the curvature energy. We consider the replacement ${\skew \,\nabla u \mapsto \overline{A}}\in \so(3)$, and we obtain the energy
\begin{align}
\mathcal{W}&=2\,\mu\,\|{\rm sym} \nabla u\|^2+\lambda\,[ \tr(\nabla u)]^2+\mu_c\,\|\skw \nabla u-\overline{A}\|^2+2\,\mu\,L_c^2\,\|\skew\nabla \axl(\overline{A})\|^2\\
&=2\,\mu\,\|{\rm sym} \nabla u\|^2+\lambda\,[ \tr(\nabla u)]^2+\mu_c\,\|\skw \nabla u-\overline{A}\|^2+2\,\mu\,L_c^2\,\|\curl \axl(\overline{A})\|^2\notag
\end{align}
for the Cosserat model.
Using the usual procedure, it follows that there exists a unique solution $(u,\overline{A})$ of the corresponding minimization problem, i.e. to find the minimum of the energy
\begin{align}
I(u)=\int_\Omega\bigg[& 2\,\mu\,\|{\rm sym} \nabla u\|^2+\lambda\,[ \tr(\nabla u)]^2+\mu_c\|\skw \nabla u-\overline{A}\|^2+2\,\mu\,L_c^2\,\|\curl \axl(\overline{A})\|^2\\
&-\langle f,u\rangle-\langle \axl(\mathfrak{M}),\axl(\overline{A})\rangle \bigg]\, dv,\notag
\end{align}
where $f:\Omega\rightarrow\mathbb{R}^3$ and $\mathfrak{M}:\Omega\rightarrow\mathbb{R}^{3\times 3}$ are prescribed, such that
$u\in H_0^1(\Omega)$ and $\axl(\overline{A})\in H(\Curl;\Omega)$, $\axl(\overline{A})\times n\big|_{\Gamma}=0$.
\begin{figure}
\setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(10,30)
\thicklines
\put(-40,18){\oval(86,43)}
\put(-81,36){\footnotesize{\bf \ \ \ $\boldsymbol{\skew \nabla u \mapsto \overline{A}}\in \so(3)$, degenerate Cosserat model }}
\put(-81,32){\footnotesize{ ${\sigma\sim 2\,\mu\, \sym\nabla u+2\, \mu_c \skw(\nabla u-\overline{A})\not\in {\rm Sym}(3)}$}}
\put(-81,28){\footnotesize{ $m=2\,\mu\,L_c^2\, \skew\nabla \axl(\overline{A})\in \so(3)$}}
\put(-81,24){\footnotesize{ $\mathcal{E}\sim\mu\,\|{\rm sym} \nabla u\|^2+\mu_c\|\skw \nabla u-\overline{A}\|^2+\mu\,L_c^2\,\|\skew\nabla \axl(\overline{A})\|^2$}}
\put(-81,20){\footnotesize{ \bf invariant under: \!$u\mapsto u+\overline{W}.x+\overline{b}, \, \overline{b}\in \mathbb{R}^3$}}
\put(-81,16){\footnotesize{\bf \qquad\qquad\quad \qquad\ $A\mapsto \overline{A}+{\overline{W}}, \,\,\,\overline{W}\in \so(3)$}}
\put(-81,12){\footnotesize{ {\bf well-posed: } $u\in H^1(\Omega)$, $\axl(\overline{A})\in H({\rm curl};\Omega)$, }}
\put(-81,8){\footnotesize{ {\bf degenerate curvature energy}, }}
\put(-81,4){\footnotesize{ \bf $\boldsymbol{3+2=5}$ geometric bc: $u\big|_\Gamma, \ \ \langle \axl(\overline{A}),\tau_\alpha\rangle\big|_\Gamma$}}
\put(-81,0){\footnotesize{ \bf $\boldsymbol{3+2=5}$ traction bc: $\sigma.n\big|_{\partial \Omega\setminus \Gamma},\ \ \ \langle m.n,\tau_\alpha\rangle\big|_{\partial \Omega\setminus \Gamma}$ }}
\put(3,18){\vector(1,0){19}}
\put(8,28){\footnotesize{$\nabla u=\overline{A}$}}
\put(8,24){\footnotesize{constrain}}
\put(8,20){\footnotesize{$\mu_c\rightarrow\infty$}}
\put(57,19){\oval(69,44)}
\put(25,36){\footnotesize{ \bf Hadjesfandiari-Dargush model}}
\put(25,32){\footnotesize{ \bf $\sigma\sim 2\,\mu\, \sym\nabla u\in {\rm Sym}(3)$}}
\put(25,28){\footnotesize{ $\widetilde{ {m}}=2\,\mu\,L_c^2\, \skew \nabla \axl(\skew \nabla u)\in\so(3)$}}
\put(25,24){\footnotesize{ $\widetilde{\tau}=2\,\mu\,L_c^2\, \anti[\Div(\skew\nabla \axl(\skew \nabla u))]\in\so(3)$}}
\put(25,20){\footnotesize{ $\mathcal{E}\sim\mu\,\|{\rm sym} \nabla u\|^2+\mu\,L_c^2\,\|\skew \nabla \axl(\skew \nabla u)\|^2$}}
\put(25,16){\footnotesize{ \bf invariant under: $u\mapsto u+\overline{W}.x+\overline{b}, \overline{b}\in \mathbb{R}^3$}}
\put(25,12){\footnotesize{ \bf \ \quad\quad\quad\quad \qquad\ $\nabla u\mapsto \nabla u+\overline{W}, {\overline{W}}\in \so(3)$}}
\put(25,8){\footnotesize{{\bf well-posed:} $u\in H^1(\Omega)$, $\curl u\in H(\curl;\Omega)$ }}
\put(25,4){\footnotesize{\bf 5 geometric bc}: $u\big|_\Gamma$, $\langle \curl u,\tau_\alpha\rangle\big|_{\Gamma}$}
\put(25,0){\footnotesize{\bf 5 traction bc}}
\end{picture}
\end{center}
\caption{A possibility of lifting the 4th.-order indeterminate couple stress model to a 2nd.-order micromorphic or Cosserat-type formulation formulation. Here $\tau_\alpha$, $\alpha=1,2$ denote two independent tangential vectors on the boundary. }\label{limitmodel}
\end{figure}
\section{Conclusion}
First, Hadjesfandiari and Dargush reject the notion of additional independent degrees of freedom, which
is purely arbitrary \cite[p. 2496]{hadjesfandiari2011couple}.
The ``consistent" theory proposed by Hadjesfandiari and Dargush ``resolving all difficulties...", taking only the skew-symmetric
part of the curvature tensor $\nabla \curl \, u$,
is simply a special case corresponding to some vanishing moduli in the theory. As we have seen, this choice is a restriction, but not a necessity.
In summary, Hadjesfandiari and Dargush have raised \cite[p.~17]{hadjesfandiari2011couple} three major concerns in the indeterminate couple stress model (see also \cite{hadjesfandiari2013skew}):
\begin{quote}
\begin{itemize}
\item[1)] \textit{The body-couple is present in the constitutive relations for the [total] force-stress tensor in the {\rm [indeterminate couple stress]} theory}.
\item[2)] \textit{The spherical part of the couple-stress tensor is indeterminate, because the curvature
tensor {\rm [$\widetilde{k}=\nabla {\rm curl} \,u$]} is deviatoric}.
\item[3)] \textit{The boundary conditions are inconsistent, because the normal component of moment
traction {\rm [$\langle n,\widetilde{m}.n\rangle$]} appears in the formulation {\rm [of the force stress tensor]}}.
\end{itemize}
\end{quote}
Regarding the above three ``serious
inconsistencies" presented in \cite[p. 17]{hadjesfandiari2011couple} we may answer:
\begin{quote}
\begin{itemize}
\item[1)] This is of course not inconsistent.
It is well-known that the Cauchy-like total force stress tensor $\sigma-\widetilde{\tau}$ is not the constitutive stress. The
constitutive stresses and couples are those arising from the virtual work principle
\eqref{gradeq211} by fixing $\sym \nabla u$ and $\nabla \curl u$ as independent constitutive quantities\footnote{However, this does not imply that we may independently prescribe
$u$ and $\curl u$ on the boundary.}. More precisely, the
constitutively dependent quantities are the energetic conjugates of $\frac{\rm d}{\rm dt}\sym \nabla u$ and $\frac{\rm d}{\rm dt}\nabla \curl u$, respectively. We have to note that the constitutive dependent quantities are not the energetic conjugates of $\frac{\rm d}{\rm dt}u$ and $\frac{\rm d}{\rm dt}\curl u$, respectively.
The relation between the Cauchy-like stress and constitutive stress
indeed involve the volume simple double and triple forces (see \cite{Germain1} in
French, first part on the classical theory).
Note that in the Hadjesfandiari and Dargush-formulation \eqref{HBBC}, the boundary conditions would acquire the same form and meaning as in the classical format: the total force stress tensor $\sigma-\widetilde{\tau}$ would be the Cauchy stress tensor and the curvature of the surface normal would not intervene in the traction boundary conditions. However, the incorrect boundary conditions used by them \eqref{HBBC} are not any more in the completely independent decomposition form \eqref{HBBC1} of the boundary conditions;
\item[2)] The indeterminacy of the spherical part of the couple stress is not inconsistent.
Like the pressure in an incompressible body, it is indeterminate in the local
constitutive law but can be found from the boundary conditions after solving the equilibrium equations.
It is a reaction stress, as it is well-known in the theory of continua with
internal constraints.
Should we say that the theory of incompressible bodies is ``inconsistent"? Surely
not, it is mathematically clear even though it may induce
computational difficulties.
\item[3)] As we have shown in this paper, see also \cite{MadeoGhibaNeffMunchKM}, the boundary conditions used by Hadjesfandiari and Dargush \cite{hadjesfandiari2011couple} are incomplete, therefore no inconsistency occurs, see again \eqref{HBBC1} vs. \eqref{HBBC}.
\end{itemize}
\end{quote}
Finally, we like to mention that the extra constitutive parameter $\eta^\prime$ of Grioli's model does not intervene in the field
partial differential equations.
It will arise through the boundary conditions.
We do not see which mechanical principle is violated by that. In summary, there is a fully consistent version of the indeterminate-couple stress model with only 3 constitutive parameters. This is the modified couple stress model, see \cite{Neff_Jeong_IJSS09,NeffGhibaMadeoMunch} and a novel variant of it is recently discussed in \cite{NeffGhibaMadeoMunch}.
\section*{Acknowledgement} We are grateful to Ali Reza Hadjesfandiari (University at Buffalo) and Gary F. Dargush (University at Buffalo) for sending us the paper \cite{hadjesfandiari2014evo} prior to publication. We would like to thank Samuel Forest (CNRS Mines ParisTech) for detailed discussions. Ionel-Dumitrel Ghiba acknowledges support from the Romanian National Authority for Scientific Research (CNCS-UEFISCDI), Project No. PN-II-ID-PCE-2011-3-0521.
\bibliographystyle{plain}
\addcontentsline{toc}{section}{References}
\begin{footnotesize}
|
{
"timestamp": "2015-04-20T02:07:01",
"yymm": "1504",
"arxiv_id": "1504.03105",
"language": "en",
"url": "https://arxiv.org/abs/1504.03105"
}
|
\section{Introduction}
Tensor decomposition is an important research area, and it has found numerous applications in data mining \cite{Kolda2006,Kolda2008,Kolda2009}, computational neuroscience \cite{Field1991,Comon2014}, and statistical learning for latent variable models \cite{Anan14}.
An important class of tensor decomposition is sum-of-squares (SOS) tensor decomposition. It is known that to determine a given even order symmetric tensor is positive
semi-definite or not is an NP-hard problem in general. On the other hand, an interesting feature of SOS tensor decomposition is checking whether a given even order symmetric tensor has SOS decomposition or not can be verified by solving a semi-definite programming problem (see for example \cite{Hu14}), and hence, can be validated efficiently. SOS tensor decomposition has a close connection with SOS polynomials, and SOS polynomials are very
important in polynomial theory \cite{Ch77,Ch95,Habi39,H88,Power98,Rez00} and polynomial optimization \cite{KLYZ,JB01,M09,Le14,parrilo03,shor98}.
It is known that an even order symmetric tensor having SOS decomposition
is positive semi-definite, but the converse is not true in general. Recently, a few classes of structured tensors such as B tensors \cite{qi14} and diagonally dominated tensor \cite{Qi05},
have been shown to be positive semi-definite in the even order symmetric case. It then raises a natural and interesting question: Will these structured tensors admit an SOS decomposition? Providing an answer for this question is important because this will enrich the theory of SOS tensor decomposition, achieve
a better understanding for these structured tensors, and lead to efficient numerical methods for solving problems involving these structured tensors.
In this paper, we make the following contributions in answering the above theoretical question and providing applications on important numerical problems involving structured tensors:
\begin{itemize}
\item[{\rm (1)}] We first show that several classes of symmetric structured tensors available in the literature have SOS decomposition when the order is even. These classes include positive Cauchy tensors, weakly diagonally dominated tensors,
$B_0$-tensors, double $B$-tensors, quasi-double $B_0$-tensors, $MB_0$-tensors, $H$-tensors, absolute tensors
of positive semi-definite $Z$-tensors and extended $Z$-tensors.
\item[{\rm (2)}] Secondly, we examine the SOS-rank for tensors with SOS decomposition and the SOS-width for SOS tensor cones.
The SOS-rank of tensor $\mathcal{A}$ is defined to be the minimal number of the squares which appear in the sums-of-squares decomposition
of the associated homogeneous polynomial of $\mathcal{A}$, and, for a given SOS tensor cone, its SOS-width is the maximum possible SOS-rank for all the tensors in this cone. We deduce an upper bound for the SOS-rank of general SOS tensor decomposition and the SOS-width for the general SOS tensor cone using the known result in
polynomial theory \cite{Ch95}. We then provide a sharper explicit upper bound of the SOS-rank for tensors with bounded exponent and identify the exact SOS-width for the cone consists of all such tensors with SOS decomposition.
\item[{\rm (3)}] Finally, as applications, we show how the derived SOS tensor decomposition can be used to compute the minimum $H$-eigenvalue of an even order symmetric extended $Z$-tensor and test the positive definiteness of an associated multivariate form. Numerical experiments are also provided to show the efficiency of the proposed numerical method ranging from small size to large size numerical examples.
\end{itemize}
The rest of this paper is organized as follows. In Section 2, we recall some basic definitions and facts for tensors and polynomials.
We also present some properties of SOS tensor cone and its duality. In Section 3, we present SOS decomposition property for various classes of structured tensors. In Section 4, we study the SOS-rank of SOS tensor decomposition and SOS-width for a given SOS tensor cone.
In particular, we examine SOS tensor decomposition with bounded exponents and the SOS-width
of the cone constituted by all such tensors with SOS decomposition, and provide their sharper explicit estimate.
In Section 5, as applications for the derived SOS decomposition of structure tensors, we show that
the minimum $H$-eigenvalue of an even order extended $Z$-tensor can be computed via polynomial optimization technique.
Accordingly, this also leads to an efficient test for positive definiteness of an associated multivariate form.
Numerical experiments are also provided to illustrate the significance of the result. Final remarks and some questions are listed in Section 6.
Before we move on, we briefly mention the notation that will be used in the sequel. Let $\mathbb{R}^n$
be the $n$ dimensional real Euclidean space and the set consisting of all positive integers
is denoted by $\mathbb{N}$. Suppose $m, n\in \mathbb{N}$ are two natural numbers. Denote $[n]=\{1,2,\cdots,n\}$. Vectors are denoted by bold lowercase letters i.e. ${\bf x},~ {\bf y},\cdots$, matrices are denoted by capital letters i.e. $A, B, \cdots$, and tensors are written as calligraphic capitals such as
$\mathcal{A}, \mathcal{T}, \cdots.$ The $i$-th unit coordinate vector in $\mathbb{R}^n$ is denoted by ${\bf e_i}$.
If the symbol $|\cdot|$ is used on a tensor $\mathcal{A}=(a_{i_1 \cdots i_m})_{1\leq i_j\leq n}$, $j=1,\cdots,m$, it denotes another
tensor $|\mathcal{A}|=(|a_{i_1 \cdots i_m}|)_{1\leq i_j\leq n}$, $j=1,\cdots,m$.
\setcounter{equation}{0}
\section{Preliminaries}
A real $m$th order n-dimensional tensor $\mathcal{A}=(a_{i_1i_2\cdots i_m})$ is a multi-array
of real entries $a_{i_1i_2\cdots i_m}$, where $i_j \in [n]$ for $j\in [m]$.
If the entries $a_{i_1i_2\cdots i_m}$ are invariant under any permutation of
their indices, then tensor $\mathcal{A}$ is called a symmetric tensor. In this paper, we always consider
symmetric tensors defined in $\mathbb{R}^n$. The identity tensor $\mathcal{I}$ with order $m$ and dimension $n$ is given by $\mathcal{I}_{i_1\cdots i_m}=1$ if $i_1=\cdots=i_m$ and $\mathcal{I}_{i_1\cdots i_m}=0$ otherwise.
We first fix some symbols and recall some basic facts of tensors and polynomials. Let $m,n \in \mathbb{N}$. Consider
$S_{m,n}:=\{\mathcal{A}: \mathcal{A} \mbox{ is an } m\mbox{th-order } n\mbox{-dimensional} \mbox{ symmetric tensor}\}.$
Clearly, $S_{m,n}$ is a vector space under the addition and multiplication
defined as below: for any $t \in \mathbb{R}$,
$\mathcal{A}=(a_{i_1 \cdots i_m})_{1 \le i_1,\cdots,i_m
\le n}$ and $\mathcal{B}=(b_{i_1 \cdots i_m})_{1 \le
i_1,\cdots,i_m \le n},$
\[
\mathcal{A}+\mathcal{B}=(a_{i_1 \cdots i_m}+b_{i_1 \cdots i_m})_{1
\le i_1,\cdots,i_m \le n} \mbox{ and } t
\mathcal{A}=(ta_{i_1 \cdots i_m})_{1 \le i_1,\cdots,i_m
\le n}.
\]
For each $\mathcal{A}, \mathcal{B} \in S_{m,n}$, we define the inner
product by
\[
\langle
\mathcal{A},\mathcal{B}\rangle:=\sum_{i_1,\cdots,i_m=1}^{n}a_{i_1 \cdots
i_m}b_{i_1 \cdots i_m}.
\]
The corresponding norm is defined by $\displaystyle
\|\mathcal{A}\|=\left(\langle
\mathcal{A},\mathcal{A}\rangle\right)^{1/2}=\left(\sum_{i_1,\cdots,i_m=1}^{n}(a_{i_1
\cdots i_m})^2\right)^{1/2}$.
For a vector ${\bf x}\in
\mathbb{R}^n$, we use $x_i$ to denote its $i$th component.
Moreover, for a vector ${\bf x}\in
\mathbb{R}^n$, we use ${\bf x}^m$ to denote the $m$th-order $n$-dimensional symmetric rank one tensor induced by ${\bf x}$, i.e.,
\[
({\bf x}^m)_{i_1 i_2\cdots i_m}=x_{i_1}x_{i_2}\cdots x_{i_m}, \ \forall \, i_1,\cdots,i_m \in \{1,\cdots,n\}.
\]
We note that an $m$th order $n$-dimensional symmetric tensor uniquely defines an $m$th degree homogeneous
polynomial $f_{\mathcal{A}}$ on $\mathbb{R}^n$: for all ${\bf x}=(x_1,\cdots,x_n)^T
\in \mathbb{R}^n$,
\begin{equation}\label{e21}
f_{\mathcal{A}}({\bf x})= \mathcal{A}{\bf x}^m=\sum_{i_1,i_2,\cdots, i_m\in [n]}a_{i_1i_2\cdots i_m}x_{i_1}x_{i_2}\cdots x_{i_m}.
\end{equation}
Conversely, any $m$th degree homogeneous
polynomial function $f$ on $\mathbb{R}^n$ also uniquely corresponds a symmetric tensor. Furthermore,
a tensor $\mathcal{A}$ is called positive semi-definite (positive definite) if $f_{\mathcal{A}}({\bf x}) \geq 0$ ($f_{\mathcal{A}}({\bf x})> 0$)
for all ${\bf x}\in \mathbb{R}^n$ (${\bf x}\in \mathbb{R}^n \backslash \{\bf 0\}$).
{We now recall the following definitions on eigenvalues and eigenvectors for a tensor \cite{Lim05, Qi05}.
\bd\label{def21} Let $\mathbb{C}$ be the complex field. Let $\mathcal{A}=(a_{i_1i_2\cdots i_m})$ be an order $m$ dimension $n$ tensor. A pair $(\lambda, {\bf x})\in \mathbb{C}\times \mathbb{C}^n\setminus \{0\}$ is called an
eigenvalue-eigenvector pair of tensor $\mathcal{A}$, if they satisfy
$$
\mathcal{A}{\bf x}^{m-1}=\lambda {\bf x}^{[m-1]},
$$
where $\mathcal{A}{\bf x}^{m-1}$ and ${\bf x}^{[m-1]}$ are all n dimensional column vectors given by
$$\mathcal{A}{\bf x}^{m-1}=\left(\sum_{i_2,\cdots,i_m=1}^n a_{ii_2\cdots i_m}x_{i_2}\cdots x_{i_m} \right)_{1\leq i\leq n}$$ and ${\bf x}^{[m-1]}=(x_1^{m-1},\ldots,x_n^{m-1})^T \in \mathbb{R}^n$.
\ed
If the eigenvalue $\lambda$ and the eigenvector ${\bf x}$ are real, then
$\lambda$ is called an $H$-eigenvalue of $\mathcal{A}$ and ${\bf x}$ is its corresponding $H$-eigenvector \cite{Qi05}.
An important fact which will be used frequently later on is that
an even order symmetric tensor is positive semi-definite (definite) if and only if all $H$-eigenvalues of the tensor are nonnegative (positive).}
Suppose that $m$ is even. In (\ref{e21}),
if $f_{\mathcal{A}}({\bf x})$ is a sums-of-squares (SOS) polynomial, then we say $\mathcal{A}$ has an {\bf SOS tensor decomposition} (or an SOS decomposition, for simplicity). It is clear that a tensor with SOS decomposition and an SOS polynomial must have even degree.
If a given tensor has SOS decomposition, then
the tensor is positive semi-definite, but not vice versa. Next, we recall a useful lemma which provides a test for verifying whether a homogeneous
polynomial is a sums-of-squares polynomial or not. To do this, we introduce some basic notions.
For all ${\bf x}\in \mathbb{R}^n$, consider a homogeneous polynomial $f({\bf x})=\sum_{\alpha}f_{\alpha}{\bf x}^{\alpha}$ with degree $m$ ($m$ is an even number), where $\alpha=(\alpha_1,\cdots,\alpha_n) \in (\mathbb{N} \cup \{0\})^n$, ${\bf x}^{\alpha}=x_1^{\alpha_1}\cdots x_n^{\alpha_n}$ and $|\alpha|:=\sum_{i=1}^n \alpha_i=m$.
Let $f_{m,i}$ be the coefficient associated with $x_i^{m}$. Let ${\bf e_i}$ be the $i$th unit vector and let
\begin{equation}\label{e22} \Omega_f=\{\alpha=(\alpha_1,\cdots,\alpha_n) \in (\mathbb{N} \cup \{0\})^n: f_{\alpha} \neq 0 \mbox{ and } \alpha \neq m \, {\bf e_i}, \ i=1,\cdots,n\}.\end{equation}
Then, $f$ can be decomposed as
$f({\bf x})=\sum_{i=1}^n f_{m,i} x_i^{m}+\sum_{\alpha \in \Omega_f}f_{\alpha}{\bf x}^{\alpha}$. Recall that $2\mathbb{N}$ denotes the set consisting of all the even numbers. Define
$$
\hat{f}({\bf x})=\sum_{i=1}^n f_{m,i} x_i^{m}-\sum_{\alpha \in \Delta_f}|f_{\alpha}|{\bf x}^{\alpha},
$$
where
\begin{equation}\label{e23}
\Delta_f:=\{\alpha=(\alpha_1,\cdots,\alpha_n) \in \Omega_f: f_{\alpha} < 0 \mbox{ or } \alpha \notin (2\mathbb{N} \cup \{0\})^n\}.
\end{equation}
\begin{lemma}\cite[Corollary 2.8]{FK11}\label{lema21}
Let $f$ be a homogeneous polynomial of degree
$m$, where $m$ is an even number. If $\hat{f}$ is a polynomial which always takes nonnegative values, then $f$ is a sums-of-squares polynomial.
\end{lemma}
\subsection*{SOS tensor cone and its dual cone}
In this part, we study the cone consisting of all tensors that have SOS decomposition, and its dual cone \cite{Luo}. We use ${\rm SOS}_{m,n}$ to denote the cone consisting of all order $m$ and dimension $n$ tensors, which have SOS decomposition. The following simple lemma from \cite{Hu14} gives some basic properties of ${\rm SOS}_{m,n}$.
\begin{lemma}\label{lema31} {\rm (cf. \cite{Hu14})}
Let $m,n \in \mathbb{N}$ and $m$ be an even number. Then, ${\rm SOS}_{m,n}$ is a closed convex cone with dimension at most $I(m,n)=\binom {n+m-1} {m}.$
\end{lemma}
For a closed convex cone $C$, we recall that the dual cone of $C$ in
$S_{m,n}$ is denoted by $C^{\oplus}$ and defined by
$C^{\oplus}=\{\mathcal{A} \in S_{m,n}: \langle
\mathcal{A},\mathcal{C}\rangle \ge 0$ for all $\mathcal{C} \in C\}$.
Let $\mathcal{M}=(m_{i_1,i_2,\cdots,i_m}) \in S_{m,n}$. We also define the symmetric tensor
${\rm sym}(\mathcal{M} \otimes \mathcal{M}) \in S_{2m,n}$ by $${\rm
sym}(\mathcal{M} \otimes \mathcal{M}){\bf x}^{2m}=
(\mathcal{M}{\bf x}^ m)^2=\sum_{1 \le i_1,\cdots, i_m,
j_1,\cdots,j_m \le n}m_{i_1, \cdots ,i_m}m_{j_1,
\cdots ,j_m}x_{i_1}\cdots x_{i_m}x_{j_1}\cdots x_{j_m}.$$
Moreover, in the case where the degree $m=2$,
${\rm SOS}_{2,n}$ and its dual cone are equal, and both reduce to the cone of positive semidefinite $(n \times n)$ matrices. Therefore, to avoid triviality,
we consider the duality of the SOS tensor cone ${\rm SOS}_{m,n}$ in the case where $m$ is an even number with $m \ge 4$.
\begin{proposition}{\bf (Duality between tensor cones)}
Let $n \in \mathbb{N}$ and $m$ be an even number with $m \ge 4$. Then, we have
${\rm SOS}_{m,n}^{\oplus} = \{\mathcal{A} \in S_{m,n}: \langle \mathcal{A}, {\rm sym}(\mathcal{M} \otimes \mathcal{M})\rangle \ge 0, \,~ \forall~\mathcal{M} \in S_{\frac{m}{2},n}\}$ and ${\rm SOS}_{m,n} \nsubseteqq {\rm SOS}_{m,n}^{\oplus}$.
\end{proposition}
\proof We define ${\rm SOS}_{m,n}^h$ to be the cone consisting of all $m$th-order $n$-dimensional symmetric tensors such that
$f_{\mathcal{A}}({\bf x}):=\langle \mathcal{A}, {\bf x}^ m \rangle$ is a polynomial which can be written as sums of finitely many homogeneous polynomials. We now see that indeed ${\rm SOS}_{m,n}^{h} = {\rm SOS}_{m,n}$. Clearly, ${\rm SOS}_{m,n}^{h} \subseteq {\rm SOS}_{m,n}$. To see the reverse inclusion, we let $\mathcal{A} \in {\rm SOS}_{m,n}$. Then, there exists $l \in \mathbb{N}$ and $f_1,\cdots,f_l$ are real polynomials with degree at most $\frac{m}{2}$ such that
$\langle \mathcal{A}, {\bf x}^m \rangle=\sum_{i=1}^l f_i({\bf x})^2$. In particular, for all $t \ge 0$, we have
\[
t^m \, \langle \mathcal{A}, {\bf x}^m\rangle =\langle \mathcal{A}, (t{\bf x})^m \rangle=\sum_{i=1}^l f_i(t{\bf x})^2
\]
Dividing $t^m$ on both sides and letting $t \rightarrow +\infty$, we see that
$\langle \mathcal{A}, {\bf x}^m\rangle =\sum_{i=1}^l f_{i,\frac{m}{2}} ({\bf x})^2$,
where $f_{i,\frac{m}{2}}$ is the $\frac{m}{2}$th-power term of $f_i$, $i=1,\cdots,l$. This shows that $\mathcal{A} \in {\rm SOS}_{m,n}^{h}$. Thus, we have ${\rm SOS}_{m,n}^{h} = {\rm SOS}_{m,n}$. It then follows that
\begin{eqnarray*}
\big({\rm SOS}_{m,n} \big)^{\oplus} = \big({\rm SOS}_{m,n}^h \big)^{\oplus}
&=&\{\mathcal{A} \in S_{m,n}: \langle \mathcal{A}, \mathcal{C}\rangle \ge 0 \mbox{ for all } \mathcal{C} \in {\rm SOS}^h_{m,n}\}\\
& = & \{\mathcal{A} \in S_{m,n}: \langle \mathcal{A}, \mathcal{C}\rangle \ge 0 \mbox{ for all } \mathcal{C}=\sum_{i=1}^l {\rm sym}(\mathcal{M}_i \otimes \mathcal{M}_i), \\
& & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathcal{M}_i \in S_{\frac{m}{2},n}, i=1,\cdots,l\} \\
& = & \{\mathcal{A} \in S_{m,n}: \langle \mathcal{A}, {\rm sym}(\mathcal{M} \otimes \mathcal{M})\rangle \ge 0 \mbox{ for all } \mathcal{M} \in S_{\frac{m}{2},n}\}.
\end{eqnarray*}
We now show that ${\rm SOS}_{m,n} \nsubseteqq {\rm
SOS}_{m,n}^{\oplus}$ if $m \ge 4$. Let
$f({\bf x})=x_1^4+x_2^4+\frac{1}{4}x_3^4+6x_1^2x_2^2+6x_1^2x_3^2+6x_2^2x_3^2$
and let $\mathcal{A} \in S_{4,3}$ be such that
$\mathcal{A}{\bf x}^4=f({\bf x})$. Then, $\mathcal{A}$ has an SOS
decomposition and $\mathcal{A}_{1,1,1,1}=\mathcal{A}_{2,2,2,2}=1$,
$\mathcal{A}_{3,3,3,3}=\frac{1}{4}$,
$\mathcal{A}_{1,1,3,3}=\mathcal{A}_{1,1,2,2}=\mathcal{A}_{2,2,3,3}=1$.
We now see that $\mathcal{A} \notin {\rm SOS}_{m,n}^{\oplus}$. To
see this, we only need to find $M \in S_{2,3}$ such that
$\langle \mathcal{A},{\rm sym}(M \otimes M)\rangle<0.$
To see this, let $M={\rm diag}(1,1,-4)$. Then, ${\rm sym}(M \otimes M){\bf x}^4=({\bf x}^TM{\bf x})^2=(x_1^2+x_2^2-4x_3^2)^2$. Direct verification shows that
${\rm sym}(M \otimes M){\bf x}^4=x_1^4+x_2^4+16x_3^4+2x_1^2x_2^2-8x_1^2x_3^2-8x_2^2x_3^2.$
So, ${\rm sym}(M \otimes M)_{1,1,1,1}={\rm sym}(M \otimes M)_{2,2,2,2}=1$, ${\rm sym}(M \otimes M)_{3,3,3,3}=16$, ${\rm sym}(M \otimes M)_{1,1,2,2}=\frac{1}{3}$, ${\rm sym}(M \otimes M)_{1,1,3,3}={\rm sym}(M \otimes M)_{2,2,3,3}=-\frac{4}{3}$. Therefore,
\[
\langle \mathcal{A},{\rm sym}(M \otimes M)\rangle=1+1+\frac{1}{4}\cdot 16+ 6\left(1 \cdot \frac{1}{3}\right)+6 \left(1 \cdot \left(-\frac{4}{3}\right)\right) +6 \left(1 \cdot \left(-\frac{4}{3}\right)\right)=-8 <0,
\]
and the desired results hold.\qed
{\bf Question:}
It is known from polynomial optimization (see \cite[ Proposition 4.9]{M09} or \cite{JB01}) that the dual cone of the cone consisting of all sums-of-squares polynomials (possibly nonhomogeneous) is the moment cone (that is, all the sequence whose associated moment matrix is positive semi-definite). Can we link the dual cone of ${\rm SOS}_{m,n}$ to the moment matrix? Can the membership problem of ${\rm SOS}_{m,n}^{\oplus}$ be solvable in polynomial time?
\setcounter{equation}{0}
\section{SOS Decomposition of Several Classes of Structured Tensors}
In this section, we examine the SOS decomposition of several classes of symmetric even order structured tensors,
such as positive Cauchy tensor, weakly diagonally dominated tensors, $B_0$-tensors, double $B$-tensors, quasi-double $B_0$-tensors, $MB_0$-tensors, $H$-tensors, absolute tensors of positive semi-definite $Z$-tensors and extended $Z$-tensors.
\subsection{Characterizing SOS decomposition for even order Cauchy tensors}
Symmetric Cauchy tensors was first studied in \cite{chen14}. Some checkable sufficient and necessary conditions for
an even order symmetric Cauchy tensor to be positive semi-definite or positive definite were provided in \cite{chen14}, which extends the matrix cases established in \cite{Fied10}.
Let ${\bf c}=(c_1,c_2,\cdots,c_n)^T\in \mathbb{R}^n$ with $c_{i_1}+c_{i_2}+\cdots+c_{i_m} \neq 0$ for all $i_j \in \{1,\ldots,n\}$, $j=1,\ldots,m$. Let the real tensor $\mathcal{C}=(c_{i_1i_2\cdots i_m})$ be defined by
$$
c_{i_1i_2\cdots i_m}=\frac{1}{c_{i_1}+c_{i_2}+\cdots+c_{i_m}},\quad j\in [m],~i_j \in [n].
$$
Then, we say that $\mathcal{C}$ is a symmetric {\bf Cauchy tensor} with order $m$ and dimension $n$ or simply a Cauchy tensor. The corresponding vector ${\bf c}\in \mathbb{R}^n$ is called the generating vector of $\mathcal{C}$.
To establish the SOS decomposition of Cauchy tensors, we will also need another class of tensors called completely positive tensor, which has an SOS tensor
decomposition in the even order case.
Tensor $\mathcal{A}$ is called a {\bf completely decomposable tensor} if there are
vectors ${\bf x}_j\in \mathbb{R}^n$,~$j\in [r]$ such that $\mathcal{A}$ can be written as sums of rank-one tensors generated by the vector ${\bf x}_j$, that is,
$$\mathcal{A}=\sum\limits_{j\in [r]}{\bf x}_j^m.$$
If ${\bf x}_j\in \mathbb{R}^n_+$ for all $j\in [r]$, then $\mathcal{A}$ is called a {\bf completely positive tensor} \cite{QXX}. It was shown that a strongly
symmetric, hierarchically dominated nonnegative tensor is a completely positive tensor \cite{QXX}.
We now characterize the SOS decomposition and completely positivity for even order Cauchy tensors.
\bt\label{them36}
Let ${\bf c}=(c_1,c_2,\cdots,c_n)^T\in \mathbb{R}^n$ with $c_{i_1}+c_{i_2}+\cdots+c_{i_m} \neq 0$ for all $i_j \in \{1,\ldots,n\}$, $j=1,\ldots,m$. Let $\mathcal{C}$ be a Cauchy tensor generated by ${\bf c}$ with even order $m$ and dimension $n$.
Then, the following statements are equivalent.
\begin{itemize}
\item[{\rm (i)}] the Cauchy tensor $\mathcal{C}$ has an SOS tensor decomposition;
\item[{\rm (ii)}] the Cauchy tensor $\mathcal{C}$ is positive semi-definite;
\item[{\rm (iii)}] the generating vector of the Cauchy tensor $c_i$, $i \in [n]$, are all positive;
\item[{\rm (iv)}] the Cauchy tensor $\mathcal{C}$ is a completely positive tensor.
\end{itemize}
\et
\proof Since $m$ is even, by definitions of completely positive tensor, SOS tensor decomposition and positive
semi-definite tensor, we can easily obtain ${\rm (i)}\Rightarrow{\rm (ii)}$ and ${\rm (iv)}\Rightarrow{\rm (i)}$. By Theorem 2.1 of \cite{chen14}, we know that $\mathcal{C}$ is positive semi-definite if
and only if $c_i>0, i\in [n]$, and hence, ${\rm (ii)} \Leftrightarrow {\rm (iii)}$ holds.
So, we only need to prove ${\rm (iii)}\Rightarrow{\rm (iv)}$, that is, any Cauchy tensors with positive generating vector is completely positive.
Assume ${\rm (iii)}$ holds. Then, for any ${\bf x}\in \mathbb{R}^n$,
$$
\begin{aligned}
\mathcal{C} {\bf x}^m =& \sum_{i_1,i_2,\cdots,i_m=1}^n \frac{x_{i_1}x_{i_2} \cdots x_{i_m}}{c_{i_1}+c_{i_2}+\cdots+c_{i_m}} \\
= & \sum_{i_1,i_2,\cdots,i_m=1}^n \left(\int_0^1 t^{c_{i_1}+c_{i_2}+\cdots+c_{i_m}-1}x_{i_1}x_{i_2} \cdots x_{i_m} dt \right) \\
=&\int_0^1 \left(\sum_{i_1,i_2,\cdots,i_m=1}^n t^{c_{i_1}+c_{i_2}+\cdots+c_{i_m}-1} x_{i_1}x_{i_2} \cdots x_{i_m} \right)dt \\
=& \int_0^1 \left(\sum_{i=1}^n t^{c_i-\frac{1}{m}}x_i
\right)^m dt.
\end{aligned}
$$
By the definition of Riemann integral, we have
$$
\mathcal{C} {\bf x}^m = \lim_{k \rightarrow \infty}\sum_{j=1}^k
\frac{\left(\sum_{i=1}^n (\frac{j}{k})^{c_i-\frac{1}{m}} x_i
\right)^m}{k}.
$$
Let $\mathcal{C}_k$ be the symmetric tensor such that
$$
\begin{aligned}
\mathcal{C}_k {\bf x}^m=&\sum_{j=1}^k \frac{\left(\sum_{i=1}^n (\frac{j}{k})^{c_i-\frac{1}{m}} x_i \right)^m}{k} \\
= & \sum_{j=1}^k \left(\sum_{i=1}^n \frac{(\frac{j}{k})^{c_i-\frac{1}{m}} }{k^{\frac{1}{m}}}x_i \right)^m \\
=& \sum_{j=1}^k \left(\langle u^j, {\bf x} \rangle \right)^m,
\end{aligned}
$$
where
$$
u^j=\left(\frac{(\frac{j}{k})^{c_1-\frac{1}{m}}
}{k^{\frac{1}{m}}},\cdots, \frac{(\frac{j}{k})^{c_n-\frac{1}{m}}
}{k^{\frac{1}{m}}}\right) \in \mathbb{R}^n, \ j=1,\cdots,k.
$$
Let ${\rm CD}_{m,n}$ denote the set consisting of all completely decomposable tensor with order $m$ and dimension $n$. From \cite[Theorem 1]{LQX}, ${\rm CD}_{m,n}$ is a closed convex cone when $m$ is even. It then follows that
$\mathcal{C}=\lim_{k \rightarrow \infty}\mathcal{C}_k$ is also a completely positive tensor.
\qed
\subsection{Even order symmetric weakly diagonally dominated tensors have SOS decompositions}
In this section, we establish that even order symmetric weakly diagonally dominated tensors have SOS decompositions.
Firstly, we give the definition of weakly diagonally dominated tensors. To do this, we introduce an index set $\Delta_{\mathcal{A}}$
associated with a tensor $\mathcal{A}$.
Now, let $\mathcal{A}$ be a tensor with order $m$ and dimension $n$, and let $f_{\mathcal{A}}$ be its associated homogeneous polynomial such that
$f_{\mathcal{A}}({\bf x})=\mathcal{A} {\bf x}^m$. We then define the index set $\Delta_{\mathcal{A}}$ as $\Delta_{f}$ with $f=f_{\mathcal{A}}$, as given as in
(\ref{e23}).
\begin{definition}\label{def31}
We say $\mathcal{A}$ is a {\bf diagonally dominated tensor} if, for each $i=1,\cdots,n$,
\[
a_{i i \cdots i} \ge \sum_{(i_2, \cdots, i_m) \neq (i \cdots i)} |a_{i i_2 \cdots i_m}|.
\]
We say $\mathcal{A}$ is a {\bf weakly diagonally dominated tensor} if, for each $i=1,\cdots,n$,
\[
a_{i i \cdots i} \ge \sum_{(i_2 \cdots i_m) \neq (i \cdots i), \atop {(i,i_2 \cdots,i_m) \in \Delta_{\mathcal{A}}}} |a_{i i_2 \cdots i_m}|.
\]
\end{definition}
Clearly, any diagonally dominated tensor is a weakly diagonally dominated tensor. However, the converse is, in general, not true.
\begin{theorem}\label{them31}
Let $\mathcal{A}$ be a symmetric weakly diagonally dominated tensor with order $m$ and dimension $n$. Suppose that $m$ is even. Then, $\mathcal{A}$ has an
SOS tensor decomposition.
\end{theorem}
\proof
Denote $I=\{(i,\cdots,i)\mid 1 \le i \le n\}$.
Let ${\bf x} \in \mathbb{R}^n$. Then,
\begin{eqnarray*}
\mathcal{A}{\bf x}^m & = & \sum_{i=1}^n a_{i i \cdots i} x_i^m + \sum_{(i_1,\cdots,i_m) \notin I} a_{i_1 i_2 \cdots i_m} x_{i_1} x_{i_2} \cdots x_{i_m} \\
& = & \sum_{i=1}^n \left(a_{i i \cdots i}-\sum_{(i_2 \cdots i_m) \neq (i \cdots i)\atop { (i,i_2 \cdots,i_m) \in \Delta_{\mathcal{A}}}} |a_{i i_2 \cdots i_m}| \right) x_i^m + \\
& & \sum_{i=1}^n \sum_{(i_2 \cdots i_m) \neq (i \cdots i) \atop { (i,i_2 \cdots,i_m) \in \Delta_{\mathcal{A}}}} |a_{i i_2 \cdots i_m}| x_i^m + \sum_{(i_1,\cdots,i_m) \notin I} a_{i_1 i_2 \cdots i_m} x_{i_1} x_{i_2} \cdots x_{i_m} \\
& = & \sum_{i=1}^n \left(a_{i i \cdots i}-\sum_{(i_2 \cdots i_m) \neq (i \cdots i) \atop (i,i_2 \cdots,i_m) \in \Delta_{\mathcal{A}}} |a_{i i_2 \cdots i_m}| \right) x_i^m \\
& & +\sum_{i=1}^n \sum_{(i_2 \cdots i_m) \neq (i \cdots i) \atop { (i,i_2 \cdots,i_m) \in \Delta_{\mathcal{A}}}} |a_{i i_2 \cdots i_m}| x_i^m + \sum_{i=1}^n \sum_{(i_2 \cdots i_m) \neq (i \cdots i) \atop { (i,i_2 \cdots,i_m) \in \Delta_{\mathcal{A}}}}a_{i i_2 \cdots i_m} x_{i} x_{i_2} \cdots x_{i_m} \\
& & + \sum_{i=1}^n \sum_{(i_2 \cdots i_m) \neq (i \cdots i) \atop { (i,i_2 \cdots,i_m) \notin \Delta_{\mathcal{A}}}}a_{i i_2 \cdots i_m} x_{i} x_{i_2} \cdots x_{i_m}
\end{eqnarray*}
Define
\[
h({\bf x})= \sum_{i=1}^n \sum_{(i_2 \cdots i_m) \neq (i \cdots i) \atop { (i,i_2 \cdots,i_m) \in \Delta_{\mathcal{A}}}} |a_{i i_2 \cdots i_m}| x_i^m + \sum_{i=1}^n \sum_{(i_2 \cdots i_m) \neq (i \cdots i) \atop { (i,i_2 \cdots,i_m) \in \Delta_{\mathcal{A}}}}a_{i i_2 \cdots i_m} x_{i} x_{i_2} \cdots x_{i_m}.
\]
We now show that $h$ is a sums-of-squares polynomial.
To see $h$ is indeed sums-of-squares, from Lemma \ref{lema21}, it suffices to show that
\[
\hat{h}({\bf x}):= \sum_{i=1}^n \sum_{(i_2 \cdots i_m) \neq (i \cdots i) \atop (i,i_2 \cdots,i_m) \in \Delta_{\mathcal{A}}} |a_{i i_2 \cdots i_m}| x_i^m - \sum_{i=1}^n \sum_{(i_2 \cdots i_m) \neq (i \cdots i) \atop (i,i_2 \cdots,i_m) \in \Delta_{\mathcal{A}}}|a_{i i_2 \cdots i_m}| x_{i} x_{i_2} \cdots x_{i_m}
\]
is a polynomial which always takes nonnegative values. As $\hat{h}$ is a homogeneous polynomial with degree $m$ on $\mathbb{R}^n$, let $\hat{\mathcal{H}}$ be a
symmetric tensor with order $m$ and dimension $n$ such that
$\hat{h}({\bf x})=\hat{\mathcal{H}}{\bf x}^m$.
Since $\mathcal{A}$ is symmetric, the nonzero entries of $\hat{\mathcal{H}}$ are the same as the corresponding entries of $\mathcal{A}$.
Now, let $\lambda$ be an arbitrary $H$-eigenvalue of $\hat{\mathcal{H}}$, from the Gershgorin Theorem for eigenvalues of tensors \cite{Qi05}, we have
\[
\left|\lambda - \sum_{(i_2 \cdots i_m) \neq (i \cdots i) \atop (i,i_2 \cdots,i_m) \in \Delta_{\mathcal{A}}} |a_{i i_2 \cdots i_m}| \right| \le \sum_{(i_2 \cdots i_m) \neq (i \cdots i) \atop (i,i_2 \cdots,i_m) \in \Delta_{\mathcal{A}}} |a_{i i_2 \cdots i_m}|.
\]
So, we must have $\lambda \ge 0$. This shows that all $H$-eigenvalues of $\hat{\mathcal{H}}$ must be nonnegative, and so, $\hat{\mathcal{H}}$ is positive semi-definite \cite{Qi05}. Thus, $\hat{h}$ is
a polynomial which always takes nonnegative values.
Now, as $\mathcal{A}$ is a weakly diagonally dominated tensor and $m$ is even,
\[
\sum_{i=1}^n \left(a_{i i \cdots i}-\sum_{(i_2 \cdots i_m) \neq (i \cdots i) \atop (i,i_2 \cdots,i_m) \in \Delta_{\mathcal{A}}} |a_{i i_2 \cdots i_m}| \right) x_i^m
\]
is an SOS polynomial. Moreover, from the definition of $\Delta_{\mathcal{A}}$, for each $(i_1 \cdots i_m) \notin \Delta_{\mathcal{A}}$, $a_{i_1 \cdots i_m} \ge 0$ and
$x_{i_1} \cdots x_{i_m}$ is a squares term. Then,
\[
\sum_{i=1}^n \sum_{(i_2 \cdots i_m) \neq (i \cdots i) \atop { (i,i_2 \cdots,i_m) \notin \Delta_{\mathcal{A}}}}a_{i i_2 \cdots i_m} x_{i} x_{i_2} \cdots x_{i_m}
\]
is also a sums-of-square polynomial.
Thus, $\mathcal{A}$ has an SOS tensor decomposition. \qed
As a diagonally dominated tensor is weakly diagonally dominated, the following corollary follows immediately.
\begin{corollary}
Let $\mathcal{A}$ be a symmetric diagonally dominated tensor with even order $m$ and dimension $n$. Then, $\mathcal{A}$ has an SOS tensor decomposition.
\end{corollary}
\subsection{The absolute tensor of an even order symmetric positive semi-definite $Z$-tensor has an SOS decomposition}
Let $\mathcal{A}$ be an order $m$ dimension $n$ tensor. If all off-diagonal elements of $\mathcal{A}$ are non-positive,
then $\mathcal{A}$ is called a $Z$-tensor \cite{Zhang12}. A $Z$-tensor $\mathcal{A}=(a_{i_1,\ldots,i_m})$ can be written as
\begin{equation}\label{e31} \mathcal{A}=\mathcal{D}-\mathcal{C}, \end{equation}
where $\mathcal{D}$ is a diagonal tensor where its $i$th diagonal elements equals $a_{i i \ldots i}$, $i=1,\ldots,n$, and $\mathcal{C}$ is a nonnegative tensor (or a tensor with nonnegative entries) such that diagonal entries all equal to zero.
We now define the absolute tensor of $\mathcal{A}$ by
$$|\mathcal{A}|=|\mathcal{D}|+\mathcal{C}.$$
Note that all even order symmetric positive semi-definite $Z$-tensors have SOS decompositions \cite{Hu14,HLQS},
a natural interesting question would be: do all absolute tensors of even order symmetric positive semi-definite $Z$-tensors have SOS decompositions?
Below, we provide an answer for this question.
\bt\label{them34} Let $\mathcal{A}$ be a symmetric $Z$-tensor with even order $m$ and dimension $n$ defined as in (\ref{e31}).
If $\mathcal{A}$ is positive semi-definite, then $|\mathcal{A}|$ has an SOS tensor decomposition.
\et
\proof Let $\mathcal{A}=(a_{i_1 \ldots i_m})$ be a symmetric positive semi-definite $Z$-tensor. From (\ref{e31}), we have $\mathcal{A}= \mathcal{D}-\mathcal{C}$, where $\mathcal{D}$ is a diagonal tensor where the diagonal entries
of $\mathcal{D}$ is $d_i:=a_{i \ldots i}, i\in [n]$ and
$\mathcal{C}=(c_{i_1i_2\cdots i_m})$ is a nonnegative tensor with zero diagonal entries. Define three index sets as follows:
$$
\begin{aligned} I=&\{(i_1, i_2,\cdots,i_m)\in [n]^m~|~i_1=i_2=\cdots=i_m\};\\
\Omega=&\{(i_1, i_2,\cdots,i_m)\in [n]^m~|~ c_{i_1i_2\cdots i_m}\neq0~~ and ~~(i_1, i_2,\cdots,i_m) \notin I\}; \\
\Delta=&\{(i_1, i_2,\cdots,i_m)\in \Omega~|~ c_{i_1i_2\cdots i_m}>0~~{\bf or}~~{\rm at~~least~~one~~index~~in}~(i_1, i_2,\cdots,i_m)~~{\rm exists~~odd~~times}\}.
\end{aligned}
$$
Let $f({\bf x})=|\mathcal{A}|{\bf x}^m$ and define
a polynomial $\hat{f}$ by
$$\hat{f}({\bf x})=\sum_{i=1}^nd_ix_i^m-\sum_{(i_1,i_2,\cdots,i_m)\in \Delta}|c_{i_1i_2\cdots i_m}|x_{i_1}x_{i_2}\cdots x_{i_m}.$$
From Lemma \ref{lema21}, to see polynomial $f({\bf x})=|\mathcal{A}|{\bf x}^m$ is a sums-of-squares polynomial, we only need to show that
$\hat{f}$ always takes nonnegative value. To see this,
as $\mathcal{A}$ is positive semi-definite, we have $d_i\geq 0$.
Since $c_{i_1i_2\cdots i_m}\geq0$, $i_j\in [n],~j\in [m]$, it follows that
$$
\begin{aligned}
\hat{f}({\bf x})=&\sum_{i=1}^nd_ix_i^m-\sum_{(i_1,i_2,\cdots,i_m)\in \Delta}c_{i_1i_2\cdots i_m}x_{i_1}x_{i_2}\cdots x_{i_m}\\
=&\sum_{i=1}^nd_ix_i^m-\sum_{(i_1,i_2,\cdots,i_m)\in \Omega}c_{i_1i_2\cdots i_m}x_{i_1}x_{i_2}\cdots x_{i_m}+\sum_{(i_1,i_2,\cdots,i_m)\in \Omega \backslash \Delta}c_{i_1i_2\cdots i_m}x_{i_1}x_{i_2}\cdots x_{i_m}\\
=&\mathcal{A}{\bf x}^m+\sum_{(i_1,i_2,\cdots,i_m)\in \Omega \backslash \Delta}c_{i_1i_2\cdots i_m}x_{i_1}x_{i_2}\cdots x_{i_m} \\
\geq&0.
\end{aligned}
$$
Here, the last inequality follows from the fact that $m$ is even, $\mathcal{A}$ is positive semi-definite and
$x_{i_1}x_{i_2}\cdots x_{i_m}$ is a square term if $(i_1,i_2,\cdots,i_m)\in \Omega \backslash \Delta$.
Thus, the desired result follows.
\qed
\subsection{SOS tensor decomposition for even order symmetric extended $Z$-tensors}
In this subsection, we introduce a new class of symmetric tensor which extends symmetric $Z$-tensors to the cases where the off-diagonal elements can be positive, and examine its SOS tensor decomposition.
Let $f$ be a polynomial on $\mathbb{R}^n$ with degree $m$. Let $f_{m,i}$ be the coefficient of $f$ associated with $x_i^{m}$, $i\in [n]$. We say $f$ is an extended $Z$-polynomial if there exist $s \in \mathbb{N}$ with $s \le n$ and index sets $\Gamma_l \subseteq \{1,\cdots,n\}$, $l=1,\cdots,s$ with $\bigcup_{l=1}^s \Gamma_l=\{1,\cdots,n\}$ and $\Gamma_{l_1} \cap \Gamma_{l_2} =\emptyset$ for all $l_1 \neq l_2$ such that
\[
f({\bf x})=\sum_{i=1}^n f_{m,i} x_i^{m}+\sum_{l=1}^s \sum_{\alpha_l \in \Omega_l }f_{\alpha_l} {\bf x}^{\alpha_l},\]
where
\[
\Omega_l=\left\{\alpha\in ([n]\cup \{0\})^n:~|\alpha|=m, {\bf x}^{\alpha}=x_{i_1}x_{i_2}\cdots x_{i_m}, \{i_1,\cdots,i_m\}\subseteq \Gamma_l,~
\mbox{and} ~~ \alpha \neq m {\bf e_i}, \ i=1,\cdots,n \right \}
\]
for each $l=1,\cdots,s$ and either one of the following two conditions holds:
\begin{itemize}
\item[{\rm (1)}] $f_{\alpha_l}=0$ for all but one $\alpha_l \in \Omega_l$;
\item[{\rm (2)}] $f_{\alpha_l} \le 0$ for all $\alpha_l \in \Omega_l$.
\end{itemize}
We now say a symmetric tensor $\mathcal{A}$ is an {\bf extended $Z$-tensor} if its associated polynomial $f_{\mathcal{A}}({\bf x})=\mathcal{A}{\bf x}^m$ is a
an extended $Z$-polynomial.
From the definition, it is clear that any $Z$-tensor is an extended $Z$-tensor with $s=1$ and $\Gamma_1=\{1,\cdots,n\}$. On the other hand, an extended $Z$-tensor allows a few elements of the off-diagonal elements to be positive, and so, an extended $Z$-tensor need not to be a $Z$-tensor. For example, consider a symmetric tensor $\mathcal{A}$ where its associated polynomial $f_{\mathcal{A}}({\bf x})=\mathcal{A}{\bf x}^m=x_1^6+x_2^6+x_3^6+x_4^6+4x_1^3x_2^3+6x_3^2x_4^4$. It can be easily see that $\mathcal{A}$ is an extended $Z$-tensor but not a $Z$-tensor (as there are positive off-diagonal elements). In \cite{LQX15}, partially $Z$-tensors are introduced. There is no direct relation between these two concepts, except that both of them contain $Z$-tensors. But they do have intersection which is larger than the set of all $Z$-tensors. Actually, the example just discussed is not a $Z$-tensor, but it is an extended $Z$-tensor and a partially $Z$-tensor as well.
We now see that any positive semi-definite extended $Z$-tensor has an SOS tensor decomposition. To achieve this, we recall the following useful lemma, which provides us a
simple criterion for determining whether a homogeneous polynomial with only one mixed term is a sum of
squares polynomial or not.
\bl\label{lema52}$^{\cite{FK11}}$ Let $b_1,b_2,\cdots,b_n\geq 0$ and $d\in \mathbb{N}$.
Let $a_1,a_2,\cdots,a_n\in \mathbb{N}$ be such that $\sum_{i=1}^n=2d$. Consider the homogeneous
polynomial $f({\bf x})$ defined by
$$f({\bf x})=b_1x_1^{2d}+\cdots+b_nx_n^{2d}-\mu x_1^{a_1}\cdots x_n^{a_n}.$$
Let $\mu_0=2d\prod _{a_i\neq 0, 1\leq i\leq n}(\frac{b_i}{a_i})^{\frac{a_i}{2d}}.$
Then, the following statements are equivalent:
\begin{itemize}
\item[{\rm (i)}] $f$ is a nonnegative polynomial i.e. $f({\bf x})\geq 0$ for all ${\bf x}\in \mathbb{R}^n$;
\item[{\rm (ii)}] either $|\mu|\leq \mu_0$ or $\mu<\mu_0$ and all $a_i$ are even;
\item[{\rm (iii)}] $f$ is an SOS polynomial.
\end{itemize}
\bt \label{th:5.2}
Let $\mathcal{A}$ be an even order positive semi-definite extended $Z$-tensor. Then, $\mathcal{A}$ has an SOS tensor decomposition.
\et
\proof
Let $f_{\mathcal{A}}({\bf x})=\mathcal{A}{\bf x}^m$. As $\mathcal{A}$ is a positive semi-definite symmetric extended $Z$-tensor, there exist $s \in \mathbb{N}$ and index sets $\Gamma_l \subseteq \{1,\cdots,n\}$, $l=1,\cdots,s$ with $\bigcup_{l=1}^s \Gamma_l=\{1,\cdots,n\}$ and $\Gamma_{l_1} \cap \Gamma_{l_2} =\emptyset$ for all $l_1 \neq l_2$ such that
for all ${\bf x} \in \mathbb{R}^n$
\[
f({\bf x})=\sum_{i=1}^n f_{m,i} x_i^{m}+\sum_{l=1}^s \sum_{\alpha_l \in \Omega_l}f_{\alpha_l} {\bf x}^{\alpha_l}
\]
such that, for each $l=1,\cdots,s$, either one of the following two condition holds:
(1) $f_{\alpha_l}=0$ for all but one $\alpha_l \in \Omega_l$; (2) $f_{\alpha_l} \le 0$ for all $\alpha_l \in \Omega_l$.
Define, for each $l=1,\cdots,s$,
\[
h_l({\bf x}):=\sum_{i \in \Gamma_l} f_{m,i} x_i^{m}+ \sum_{\alpha_l \in \Omega_l}f_{\alpha_l} {\bf x}^{\alpha_l}.
\]
It follows that each $h_l$ is an extended $Z$-polynomial. Moreover, from the construction, $\sum_{l=1}^s h_l=f_{\mathcal{A}}$ and so, $\inf_{{\bf x} \in \mathbb{R}^n} \sum_{l=1}^s h_l({\bf x})=0$. Note that each $h_l$ is also a homogeneous polynomial, and hence
$\inf_{{\bf x} \in \mathbb{R}^n} h_l({\bf x}) \le 0$. Noting that each $h_l$ is indeed a polynomial on $(x_i)_{i \in \Gamma_l}$, $\bigcup_{l=1}^s \Gamma_l=\{1,\cdots,n\}$ and $\Gamma_{l_1} \cap \Gamma_{l_2} =\emptyset$ for all $l_1 \neq l_2$, we have $\inf_{{\bf x} \in \mathbb{R}^n} \sum_{l=1}^s h_l({\bf x}) = \sum_{l=1}^s \inf_{{\bf x} \in \mathbb{R}^n} h_l({\bf x})$. This enforces that $\inf_{{\bf x} \in \mathbb{R}^n} h_l({\bf x})=0$. In particular,
each $h_l$ is a polynomial which takes nonnegative values. We now see that $h_l$, $1 \le l \le s$, are SOS polynomial. Indeed, if $f_{\alpha_l}=0$ for all but one $\alpha_l \in \Omega_l$, then $h_l$ is a homogeneous polynomial with only a mixed term, and so,
Lemma \ref{lema52} implies that $h_l$ is a SOS polynomial. On the other hand, if $f_{\alpha_l} \le 0$ for all $\alpha_l \in \Omega_l$, $h_l$ corresponds to a $Z$-tensor, and so, $h_l$ is also a SOS polynomial in this case because any positive semi-definite $Z$-tensor has an SOS tensor decomposition \cite{Hu14}.
Thus, $f_{\mathcal{A}}=\sum_{l=1}^sh_l$ is also a SOS polynomial, and hence the conclusion follows. \qed
\begin{Remark}\label{remark:3.3}
A close inspection of the above proof indicates that we indeed shows that the associated polynomial $f_{\mathcal{A}}({\bf x})=\mathcal{A}{\bf x}^m$ satisfies
$f_{\mathcal{A}}=\sum_{l=1}^s h_l$ where each $h_l$ is an SOS polynomial in $(x_i)_{i \in \Gamma_l}$.
\end{Remark}
\subsection{Even order symmetric $B_0$-tensors have SOS decompositions}
In this part, we show that even order symmetric $B_0$ tensors have SOS tensor decompositions.
Recall that a tensor $\mathcal{A}=(a_{i_1i_2\cdots i_m})$ with order $m$ and dimension $n$ is called a $B_0$-tensor \cite{qi14} if
\[
\sum_{i_2, \cdots, i_m=1}^n a_{i i_2 \cdots i_m} \ge 0
\]
and
\[
\frac{1}{n^{m-1}} \sum_{i_2, \cdots, i_m=1}^n a_{i i_2 \cdots i_m} \ge a_{i j_2 \cdots j_m} \mbox{ for all } (j_2,\cdots, j_m) \neq (i, \cdots, i).
\]
To establish that a $B_0$-tensor has an SOS tensor decomposition, we first present the SOS tensor decomposition of the all-one-tensor. We say $\mathcal{E}$ is an all-one-tensor if
with each of its elements of $\mathcal{E}$ is equal to one.
\bl\label{lema32} Let $\mathcal{E}$ be an even order all-one-tensor. Then, $\mathcal{E}$ has an SOS tensor decomposition.
\el
\proof Let $\mathcal{E}=(e_{i_1i_2\cdots i_m})$ be an all-one-tensor with even order $m$ and dimension $n$.
For all ${\bf x}\in \mathbb{R}^n$, one has
$$
\begin{aligned}
\mathcal{E}{\bf x}^m=&\sum_{i_1,i_2,\cdots,i_m \in [n]}e_{i_1i_2\cdots i_m}x_{i_1}x_{i_2}\cdots x_{i_m}\\
=&\sum_{i_1,i_2,\cdots,i_m \in [n]}x_{i_1}x_{i_2}\cdots x_{i_m} \\
=&(x_1+x_2+\cdots+x_n)^m \\
\geq & 0,
\end{aligned}
$$
which implies that $\mathcal{E}$ has an SOS tensor decomposition. \qed
Let $J\subset [n]$. $\mathcal{E}^J$ is called a partially all-one-tensor if its elements are defined such that
$e_{i_1i_2\cdots i_m}=1$, $i_1, i_2, \cdots, i_m\in J$ and $e_{i_1i_2\cdots i_m}=0$ for the others.
{Similar to Lemma 3.2}, it is easy to check that all even order partially all-one-tensors
have SOS decompositions.
We also need the following characterization of $B_0$-tensors established in \cite{qi14}.
\begin{lemma}\label{lema33}
Suppose that $\mathcal{A}$ is a $B_0$-tensor with order $m$ and dimension $n$.
Then either $\mathcal{A}$ is a diagonally dominated symmetric $M$-tensor itself, or we have
\[
\mathcal{A}=\mathcal{M}+ \sum_{k=1}^s h_k \mathcal{E}^{J_k},
\]
where $\mathcal{M}$ is a diagonally dominated symmetric $M$-tensor, $s$ is a positive integer, $h_k>0$ and $J_k \subseteq \{1,\cdots,n\}$, for $k=1,\cdots,s$, and $J_k \cap J_l=\emptyset$,
for $k \neq l$.
\end{lemma}
From Theorem \ref{them31}, Lemma \ref{lema32} and Lemma \ref{lema33}, we have the following result.
\bt\label{them32} All even order symmetric $B_0$-tensors have SOS tensor decompositions.
\et
{
{ Before we move on to the next part,} we note that, stimulated by $B_0$-tensors in \cite{qi14},
symmetric double $B$-tensors, symmetric quasi-double $B_0$-tensors and symmetric $MB_0$-tensors {have been} studied
in \cite{LCL,LCQL}. Below, we briefly explain that, using a similar method of proof as above, these three classes of tensors all have SOS decompositions. To do this, let us
recall the definitions of these three classes of tensors. }
For a real symmetric tensor $\mathcal{B}=(b_{i_1i_2\cdots i_m})$ with order $m$ and dimension $n$,
denote $$\beta_i(\mathcal{B})=\max_{j_2,\cdots,j_m\in [n], (i,j_2,\cdots,j_m)\notin I}\{0, b_{ij_2\cdots j_m}\};$$
$$ \Delta_i(\mathcal{B})=\sum_{j_2,\cdots,j_m\in [n],(i,j_2,\cdots,j_m)\notin I}(\beta_i(\mathcal{B})-b_{ij_2\cdots j_m});$$
$$\Delta^i_j(\mathcal{B})=\Delta_j(\mathcal{B})-(\beta_j(\mathcal{B})-b_{jii\cdots i}),~i\neq j.$$
As defined in \cite[Definition 3]{LCL}, $\mathcal{B}$ is called a double $B$-tensor if, $b_{ii\cdots i}> \beta_i(\mathcal{B}),$ for all $i\in [n]$ and for all
$i, j\in [n], i\neq j$ such that
$$b_{ii\cdots i}-\beta_i(\mathcal{B})\geq \Delta_i(\mathcal{B})$$
and
$$(b_{ii\cdots i}-\beta_i(\mathcal{B}))(b_{jj\cdots j}-\beta_j(\mathcal{B}))>\Delta_i(\mathcal{B})\Delta_j(\mathcal{B}).$$
If $b_{ii\cdots i}> \beta_i(\mathcal{B}),$ for all $i\in [n]$ and
$$(b_{ii\cdots i}-\beta_i(\mathcal{B}))(b_{jj\cdots j}-\beta_j(\mathcal{B})-\Delta^i_j(\mathcal{B}))\geq (\beta_j(\mathcal{B})-b_{ji\cdots i})\Delta_i(\mathcal{B}),$$
then tensor $\mathcal{B}$ is called a quasi-double $B_0$-tensor (see Definition 2 of \cite{LCQL}).
Let $\mathcal{A}=(a_{i_1i_2\cdots i_m})$ such that
$$a_{i_1i_2\cdots i_m}=b_{i_1i_2\cdots i_m}-\beta_{i_1}(\mathcal{B}),~{\rm for~~ all}~~ i_1\in [n].$$
If $\mathcal{A}$ is an $M$-tensor, then $B$ is called an $MB_0$-tensor (see Definition 3 of \cite{LCQL}).
It was shown in \cite{LCQL} that all quasi-double $B_0$-tensors are $MB_0$-tensors.
In \cite{LCL}, Li et al. proved that, for any symmetric double $B$-tensor $\mathcal{B}$, either
$\mathcal{B}$ is a doubly strictly diagonally dominated (DSDD) $Z$-tensor, or $\mathcal{B}$ can be decomposed to
the sum of a DSDD $Z$-tensor and several positive multiples of partially all-one-tensors (see Theorem 6 of \cite{LCL}).
From Theorem 4 of \cite{LCL}, we know that an even order symmetric DSDD $Z$-tensor is positive definite. This together with
the fact that any positive semi-definite $Z$-tensor has an SOS tensor decomposition \cite{Hu14} implies that any even order symmetric double $B$-tensor $\mathcal{B}$
has an SOS tensor decomposition. Moreover, from Theorem 7 of \cite{LCQL}, we know that, for any symmetric $MB_0$-tensor, it is either an $M$-tensor itself or it can be decomposed
as the sum of an $M$-tensor and several positive multiples of partially {all-one-tensors}. As even order symmetric $M$-tensors are positive semi-definite $Z$-tensors \cite{Zhang12} which have, in particular, SOS decomposition, we see that any even order symmetric $MB_0$ tensor also has an SOS tensor decomposition. Combining these and noting that any quasi-double $B_0$-tensor is an $MB_0$-tensor, we arrive at the following conclusion.
\bt\label{them33} Even order symmetric double $B$-tensors, even order symmetric quasi double $B_0$-tensors
and even order symmetric $MB_0$-tensors all have SOS tensor decompositions.
\et
{ \subsection{Even order symmetric $H$-tensors with nonnegative diagonal elements have SOS decompositions}
In this part, we show that any even order symmetric $H$-tensor with nonnegative diagonal elements has an SOS tensor decomposition.
{Recall that an $m$th order $n$ dimensional tensor $\mathcal{A}=(a_{i_1i_2\cdots i_m})$, it's comparison tensor is
defined by $\mathcal{M}(\mathcal{A})=(m_{i_1i_2\cdots i_m})$ such that
$$m_{ii\cdots i}=|a_{ii\cdots i}|,~~{\rm and}~~ m_{i_1i_2\cdots i_m}=-|a_{i_1i_2\cdots i_m}|,
$$
for all $i,i_1,\cdots,i_m\in [n], (i_1,i_2,\cdots,i_m)\notin I$.
Then, tensor $\mathcal{A}$ is called an $H$-tensor \cite{Ding13} if there exists a tensor $\mathcal{Z}$ with nonnegative entries such that
$\mathcal{M}(\mathcal{A})=s \mathcal{I}- \mathcal{Z}$ and $s \ge \rho(\mathcal{Z})$, where $\mathcal{I}$ is the identity tensor and $\rho(\mathcal{Z})$ is the spectral
radius of $\mathcal{Z}$ defined as the maximum of modulus of all eigenvalues of $\mathcal{Z}$. If $s > \rho(\mathcal{Z})$, then $\mathcal{A}$ is called
a nonsingular $H$-tensor.} A characterization for nonsingular $H$-tensors was given in \cite{Ding13} which states
$\mathcal{A}$ is a nonsingular
$H$-tensor if and only if there exists an enteritis positive vector ${\bf y}=(y_1,y_2,\cdots,y_n)\in \mathbb{R}^n$ such that
$$|a_{ii\cdots i}|y_i^{m-1}>\sum_{(i,i_2,\cdots,i_m)\notin I}|a_{ii_2\cdots i_m}|y_{i_2}y_{i_3}\cdots y_{i_m},~~\forall~i\in[n].$$
We note that the above definitions were first introduced in \cite{Ding13}. These were further
examined in \cite{Kannan15,LWZ} where the authors in \cite{LWZ} { referred} nonsingular $H$-tensors simply as $H$-tensors and the authors in \cite{Kannan15} { referred} nonsingular $H$-tensors as strong $H$-tensors.
\bt\label{them35} Let $\mathcal{A}=(a_{i_1i_2\cdots i_m})$ be a symmetric $H$-tensor with even order $m$ dimension $n$. Suppose that all the diagonal elements of $\mathcal{A}$ are nonnegative. Then, $\mathcal{A}$ has an SOS tensor decomposition.
\et
\proof We first show that any nonsingular $H$-tensor with positive diagonal elements has an SOS tensor decomposition. Let $\mathcal{A}=(a_{i_1i_2\cdots i_m})$ be a nonsingular $H$-tensor with even order $m$ dimension $n$ such that $a_{i i \cdots i} > 0$, $i \in [n]$.
Then, there exists a vector ${\bf y}=(y_1,\cdots,y_n)^T \in \mathbb{R}^n$ with $y_i>0$, $i=1,\cdots,n$, such that
\begin{equation}\label{e34}
a_{ii\cdots i}y_i^{m-1} > \sum_{(i,i_2,\cdots,i_m)\notin I}|a_{ii_2\cdots i_m}|y_{i_2}y_{i_3}\cdots y_{i_m},~\forall~i\in [n].
\end{equation}
To prove the conclusion, by Lemma \ref{lema21}, we only need to prove
$$\hat{f}_{\mathcal{A}}({\bf x})=\sum_{i\in [n]}a_{ii\cdots i}x_i^m-\sum_{(i_1,i_2,\cdots,i_m)\in \Delta_{\mathcal{A}}}|a_{i_1i_2\cdots i_m}|x_{i_1}x_{i_2}\cdots x_{i_m}\geq 0,~\forall~{\bf x}\in \mathbb{R}^n.$$
From (\ref{e34}), we know that
\begin{equation}\label{e35}
\begin{aligned}
\hat{f}_{\mathcal{A}}({\bf x})\geq &\sum_{i\in [n]} \left(\sum_{(i,i_2,\cdots,i_m)\notin I}|a_{ii_2\cdots i_m}|y_i^{1-m}y_{i_2}y_{i_3}\cdots y_{i_m} x_i^m\right)
-\sum_{(i_1,i_2,\cdots,i_m)\in \Delta_{\mathcal{A}}}|a_{i_1i_2\cdots i_m}|x_{i_1}x_{i_2}\cdots x_{i_m}.
\end{aligned}
\end{equation}
Here, for any fixed tuple $(i_1^0,i_2^0,\cdots,i_m^0)\in \Delta_{\mathcal{A}}$,
assume $(i_1^0,i_2^0,\cdots,i_m^0)$ is constituted by $k$ distinct indices $j^0_1,j^0_2,\cdots, j^0_k$, $k\leq m$,
which appear $s_1,s_2,\cdots,s_k$ times in $(i_1^0,i_2^0,\cdots,i_m^0)$ respectively, $s_l\in[m], l\in [k]$. Then, one has $s_1+s_2+\cdots+s_k=m$.
Without loss of generality, we denote $a=|a_{i^0_1i^0_2\cdots i^0_m}|>0$. { Let $\pi(i_1^0,i_2^0,\cdots,i_m^0)$ be the set consisting of all permutations of $(i_1^0,i_2^0,\cdots,i_m^0).$}
So, on the right side of (\ref{e35}), there are some terms corresponding to the fixed tuple $(i^0_1,i^0_2,\cdots,i^0_m)$ such that{
\begin{eqnarray*}
& & \sum_{(j^0_1,i_2,\cdots,i_m)\in \pi(i_1^0,i_2^0,\cdots,i_m^0)}|a_{j^0_1i_2\cdots i_m}|y_{j_1^0}^{1-m}y_{i_2}y_{i_3}\cdots y_{i_m} x_{j_1^0}^m\\
& & +\sum_{(j^0_2,i_2,\cdots,i_m)\in \pi(i_1^0,i_2^0,\cdots,i_m^0)}|a_{j^0_2i_2\cdots i_m}|y_{j_2^0}^{1-m}y_{i_2}y_{i_3}\cdots y_{i_m} x_{j_2^0}^m\\
& & +\cdots\\
& & +\sum_{(j^0_k,i_2,\cdots,i_m)\in \pi(i_1^0,i_2^0,\cdots,i_m^0)}|a_{j^0_ki_2\cdots i_m}|y_{j_k^0}^{1-m}y_{i_2}y_{i_3}\cdots y_{i_m} x_{j_k^0}^m\\
& & -\sum_{(i_1,i_2,\cdots,i_m)\in \pi(i_1^0,i_2^0,\cdots,i_m^0)}|a_{i_1i_2\cdots i_m}|x_{i_1}x_{i_2}\cdots x_{i_m}\\
&= & \binom {m-1} {s_1-1} \binom {m-s_1} {s_2} \binom {m-s_1-s_2}{s_3}\cdots \binom {m-s_1-s_2\cdots -s_{k-1}} {s_k}ay_{j^0_1}^{s_1-m}y_{j^0_2}^{s_2}\cdots y_{j^0_k}^{s_k}x_{j^0_1}^m \\
& & +\binom {m-1} {s_2-1} \binom {m-s_2} {s_1} \binom {m-s_1-s_2}{s_3}\cdots \binom {m-s_1-s_2\cdots -s_{k-1}} {s_k}ay_{j^0_2}^{s_2-m}y_{j^0_1}^{s_1}y_{j^0_3}^{s_3}\cdots y_{j^0_k}^{s_k}x_{j^0_2}^m \\
& & +\cdots \cdots \\
& & +\binom {m-1} {s_k-1} \binom {m-s_k} {s_1} \binom {m-s_k-s_1}{s_2}\cdots \binom {m-s_k-s_1\cdots -s_{k-2}} {s_{k-1}}ay_{j^0_k}^{s_k-m}y_{j^0_1}^{s_1}\cdots y_{j^0_{k-1}}^{s_{k-1}}x_{j^0_k}^m \\
& & -\binom {m} {s_1} \binom {m-s_1} {s_2} \binom {m-s_1-s_2}{s_3}\cdots \binom {m-s_1-s_2\cdots -s_{k-1}} {s_k}ax_{j^0_1}^{s_1}x_{j^0_2}^{s_2} \cdots x_{j^0_k}^{s_k}\\
& = & \frac{(m-1)!ay_{j^0_1}^{s_1}y_{j^0_2}^{s_2}\cdots y_{j^0_k}^{s_k}}{s_1!s_2!\cdots s_k!} \left[s_1\left(\frac{x_{j^0_1}}{y_{j^0_1}}\right)^m +s_2\left(\frac{x_{j^0_2}}{y_{j^0_2}}\right)^m +\cdots+s_k\left(\frac{x_{j^0_k}}{y_{j^0_k}}\right)^m -m\left(\frac{x_{j^0_1}}{y_{j^0_1}}\right)^{s_1} \left(\frac{x_{j^0_2}}{y_{j^0_2}}\right)^{s_2}\cdots \left(\frac{x_{j^0_k}}{y_{j^0_k}}\right)^{s_k}\right] \\
& \geq & 0,
\end{eqnarray*}}
where the last inequality follows the arithmetic-geometric inequality and the fact ${\bf y}>{\bf 0}$.
Thus, each tuple $(i_1,i_2,\cdots,i_m)\in \Delta_{\mathcal{A}}$ corresponds to a nonnegative value on the right side of (\ref{e35}), which implies that
$\hat{f}({\bf x})\geq 0$ for all ${\bf x}\in \mathbb{R}^n$. Hence, by Lemma \ref{lema21}, $\mathcal{A}$ has an SOS tensor decomposition.
Now, let $\mathcal{A}$ be a general $H$-tensor with nonnegative diagonal elements. Then, for each $\epsilon>0$, $\mathcal{A}_{\epsilon}:=\mathcal{A}+\epsilon \mathcal{I}$ is a
nonsingular $H$-tensor with positive diagonal elements. Thus, $\mathcal{A}_{\epsilon} \rightarrow \mathcal{A}$, and for each $\epsilon>0$, $\mathcal{A}_{\epsilon}$ has an SOS tensor decomposition. As ${\rm SOS}_{m,n}$ is a closed convex cone, we see that $\mathcal{A}$ also has an SOS tensor
decomposition and the desired results follows. \qed}
In \cite{DQW1, LQX}, SOS decomposition of some classes of Hankel tensors was given.
\setcounter{equation}{0}
\section{ The SOS-Rank of SOS tensor Decomposition}
In this section, we study the SOS-rank of SOS tensor decomposition. Let us formally define the SOS-rank of SOS tensor decomposition as follows.
Let $\mathcal{A}$ be a tensor with even order $m$ and dimension $n$. Suppose $\mathcal{A}$ has a SOS tensor decomposition.
As shown in Proposition 3.1, ${\rm SOS}_{m,n}={\rm SOS}_{m,n}^h$ where ${\rm SOS}_{m,n}$ is the SOS tensor cone and
${\rm SOS}_{m,n}^h$ is the cone consisting of all $m$th-order $n$-dimensional symmetric tensors such that
$f_{\mathcal{A}}({\bf x}):=\langle \mathcal{A}, {\bf x}^ m \rangle$ is a polynomial which can be written as sums of finitely many \emph{homogeneous polynomials}. Thus, there exists
$r\in \mathbb{N}$ such that the homogeneous polynomial
$f_{\mathcal{A}}({\bf x})=\mathcal{A}{\bf x}^m$ can be decomposed by
$$ f_{\mathcal{A}}({\bf x})=f_1^2({\bf x})+f_2^2({\bf x})+\cdots+f_r^2({\bf x}),~\forall ~{\bf x}\in \mathbb{R}^n,$$
where $f_i({\bf x}),~i\in [r]$ are homogeneous polynomials with degree $\frac{m}{2}$.
The minimum value $r$ is called the {\bf SOS-rank} of $\mathcal{A}$, and is denoted by
${\rm SOSrank}(\mathcal{A}).$
Let $C$ be a convex cone in the SOS tensor cone, that is, $C \subseteq {\rm SOS}_{m,n}$. We define the {\bf SOS-width} of the convex cone $C$ by
\[
\mbox {SOS-width}(C)=\sup\{ {\rm SOSrank}(\mathcal{A}): \mathcal{A} \in C\}.
\]
Here, we do not care about the minimum of the SOS-rank of all the possible tensors in the cone $C$ as it will be always zero.
Recall that it was shown by Choi et al. in \cite[Theorem 4.4]{Ch95} that, an SOS homogeneous polynomial can be decomposed as sums of at most $\Lambda$
many squares of homogeneous polynomials where \begin{equation}\label{eq:lambda}\Lambda=\frac{\sqrt{1+8a}-1}{2} \mbox{ and } a=\binom {n+m-1} {m}.\end{equation} This immediately gives us that
\bp \label{proposition} Let $\mathcal{A}$ be a tensor with even order $m$ and dimension $n$, $m, n \in \mathbb{N}$.
Suppose $\mathcal{A}$ has an SOS tensor decomposition.
Then, its SOS-rank satisfies
${\rm SOSrank(\mathcal{A})}\leq \Lambda$,
where $\Lambda$ is given in (\ref{eq:lambda}).
In particular, {\rm SOS-width}$({\rm SOS}_{m,n}) \leq \Lambda.$
\ep
In the matrix case, that is, $m=2$, the upper bound $\Lambda$ equals the dimension $n$ of the symmetric tensor which is tight in this case.
On the other hand, in general, the upper bound is of the order $n^{m/2}$ and need not to be tight.
However, for a class of structured tensors with bounded exponent (BD-tensors) that have SOS decompositions, we show that their SOS-rank is less or
equal to the dimension $n$ which is significantly smaller than the
upper bound in the above proposition. Moreover, in this case,
the SOS-width of the associated BD-tensor cone can be determined explicitly. To do this, let us recall the definition of polynomials with bounded exponent and define
the BD-tensors. Let $e \in \mathbb{N}$. Recall that $f$ is said to be a degree $m$ homogeneous polynomials on $\mathbb{R}^n$ with bounded exponent $e$ if
\[
f({\bf x})=\sum_{\alpha} f_{\alpha} {\bf x}^{\alpha}= \sum_{\alpha} f_{\alpha} x_1^{\alpha_1}\cdots x_n^{\alpha_n},
\]
where $0 \le \alpha_j \le e$ and $\sum_{j=1}^n \alpha_j=m$. We note that degree $4$ homogeneous polynomials on $\mathbb{R}^n$ with bounded exponent $2$ is nothing but the bi-quadratic
forms in dimension $n$. Let us denote ${\rm BD}_{m,n}^{e}$ to be the set consists of all degree $m$ homogeneous polynomials on $\mathbb{R}^n$ with bounded exponent $e$.
An interesting result for characterizing when a positive semi-definite (PDF) homogeneous polynomial with bounded exponent has SOS tensor decomposition was established in \cite{Ch77} and can be stated as follows.
\begin{lemma}
Let $n \in \mathbb{N}$ with $n \ge 3$. Suppose $e,m$ are even numbers and $m \ge 4$.
\begin{itemize}
\item[{\rm (1)}] If $n \ge 4$, then ${\rm BD}_{m,n}^{e} \cap {\rm PSD}_{m,n} \subseteq {\rm SOS}_{m,n}$ if and only if $m \ge en-2$;
\item[{\rm (2)}] If $n =3$, then ${\rm BD}_{m,n}^{e} \cap {\rm PSD}_{m,n} \subseteq {\rm SOS}_{m,n}$ if and only if $m=4$ or $m \ge 3e-4$.
\end{itemize}
\end{lemma}
Now, we say a symmetric tensor $\mathcal{A}$ is a BD-tensor with order $m$, dimension $n$ and exponent $e$ if $f({\bf x})=\mathcal{A}{\bf x}^m$ is a degree $m$ homogeneous polynomial on $\mathbb{R}^n$ with bounded exponent $e$.
We also define ${\rm BD}_{m,n}$ to be the set consisting of all symmetric BD-tensors with order $m$, dimension $n$ and exponent $e$. It is clear that ${\rm BD}_{m,n}^e$ is a convex cone.
\bt
Let $n \in \mathbb{N}$ with $n \ge 3$. Suppose $e,m$ are even numbers and $m \ge 4$. Let $\mathcal{A}$ be a BD-tensor with order $m$, dimension $n$ and exponent $e$. Suppose that $\mathcal{A}$
has an SOS tensor decomposition. Then, we have ${\rm SOSrank} (\mathcal{A}) \le n$. Moreover, we have
$$\mbox{\rm SOS-width}({\rm BD}_{m,n}^{e} \cap {\rm SOS}_{m,n})=\left\{\begin{array}{ll}
1 & \mbox{ if } m=en \\
n & \mbox{ otherwise}.
\end{array}
\right.$$
\et
\begin{proof} As $\mathcal{A}$ is a BD-tensor and it has SOS decomposition, the preceding lemma implies that either {\rm (i)} $n \ge 4$ and $m \ge en-2$ {\rm (ii)} $n=3$ and $m=4$
and {\rm (iii)} $n=3$ and $m \ge 3e-4$. We now divide the discussion into these three cases.
Suppose that Case {\rm (i)} holds, i.e., $n \ge 4$ and $m \ge en-2$. From the construction, we have $m \le en$. If $m=en$, then $\mathcal{A}$ has the form $a x_1^{e}\cdots x_n^{e}$. Here, $a \ge 0$ because
$\mathcal{A}$ has SOS decomposition and $e$ is an even number. In this case, ${\rm SOSrank}(\mathcal{A})=1$. Now, let $m=en-2$. Then,
\[
\mathcal{A} {\bf x}^m= x_1^{e} \cdots x_n^{e} \left(\sum_{(i,j) \in F} a_{ij} x_i^{-1} x_j^{-1}\right),
\]
for some $a_{ij} \in \mathbb{R}$, $(i,j)\in F$ and for some $F \subseteq \{1,\cdots,n\} \times \{1,\cdots,n\}$. As $e$ is an even number and $\mathcal{A}$ has SOS decomposition, we have
\[
\sum_{(i,j) \in F} a_{ij} x_i^{-1} x_j^{-1} \ge 0 \mbox{ for all } x_i \neq 0 \mbox{ and } x_j \neq 0.
\]
Thus, by continuity, $Q(t_1,\cdots,t_n)=\sum_{(i,j) \in F} a_{ij} t_i t_j$ is a positive semi-definite quadratic form, and so, is at most sums of $n$ many squares of linear functions in $t_1,\cdots,t_n$.
Let $Q(t_1,\cdots,t_n)=\sum_{k=1}^n \big[q_k(t_1,\cdots,t_n)\big]^2$ where $q_k$ are linear functions. Then,
\[
\mathcal{A} {\bf x}^m= x_1^{e} \cdots x_n^{e} \left(\sum_{i=1}^n \left[q_k(x_1^{-1},\cdots,x_n^{-1})\right]^2\right)=\sum_{i=1}^n\left(x_1^{e} \cdots x_n^{e}\left[q_k(x_1^{-1},\cdots,x_n^{-1})\right]^2\right),
\]
Note that $$x_1^{e} \cdots x_n^{e}\left[q_k(x_1^{-1},\cdots,x_n^{-1})\right]^2=\left[x_1^{\frac{e}{2}} \cdots x_n^{\frac{e}{2}}q_k(x_1^{-1},\cdots,x_n^{-1})\right]^2$$
is a square. Thus, {${\rm SOSrank}(\mathcal{A}) \le n$ in this case}.
Suppose that Case {\rm (ii)} holds, i.e., $n = 3$ and $m=4$. Then by Hilbert's theorem \cite{H88}, ${\rm SOSrank} (\mathcal{A}) \le 3=n$.
Suppose that Case {\rm (iii)} holds, i.e., $n=3$ and $m \ge 3e-4$. In the case of $m=en-2=3e-2$ and $m=en=3e$, using similar argument as in the Case {\rm (i)}, we see that the conclusion follows. The only remaining case is when $m=3e-4$. In this case, as $\mathcal{A}$ is a BD-tensor with order $m$, dimension $3$ and exponent $e$ and $\mathcal{A}$ has SOS decomposition, we have
\[
\mathcal{A}{\bf x}^m=x_1^{e}x_2^ex_3^e G(x_1^{-1},x_2^{-1},x_3^{-1}),
\]
where $G$ is a positive semi-definite form and is of 3 dimension and degree $4$. It then from Hilbert's theorem \cite{H88} that $G(t_1,t_2,t_3)$ can be expressed as at most the sum of 3 squares of $3$-dimensional quadratic forms. Thus, using similar line of argument as in Case {\rm (i)} and noting that $e \ge 4$ (as $m=3e-4$ and $m \ge 4$), we have ${\rm SOSrank}(\mathcal{A}) \le n=3$.
Combining these three cases, we see that ${\rm SOSrank}(\mathcal{A}) \le n$, and ${\rm SOSrank}(\mathcal{A})=1$ if $m=en$. In particular, we have
${\rm \mbox {SOS-width}}({\rm BD}_{m,n}^{e} \cap {\rm SOS}_{m,n}) \le n$, and ${\rm \mbox {SOS-width}}({\rm BD}_{m,n}^{e} \cap {\rm SOS}_{m,n}) =1$ if $m=en$.
To see the conclusion, we consider the homogeneous polynomial $$
f_0({\bf x})=\left\{\begin{array}{ll}
x_1^e\cdots x_n^e (\sum_{i=1}^n x_i^{-2}) & \mbox{ if } n \ge 3 \mbox{ and } m = en-2 \\
x_1^2x_2^2+x_2^2x_3^2+x_3^2x_1^2 & \mbox{ if } n=3 \mbox{ and } m=4 \\
x_1^ex_2^ex_3^e(x_1^{-2}x_2^{-2}+x_2^{-2}x_3^{-2}+x_3^{-2}x_1^{-2}) & \mbox{ if } n=3 \mbox{ and } m = 3e-4 \\
\end{array}
\right.$$
and its associated BD-tensor $\mathcal{A}_0$ such that $f_0({\bf x})=\mathcal{A}_0{\bf x}^m$. It can be directly verified that
$${\rm SOSrank}(\mathcal{A}_0)=\left\{\begin{array}{ll}
n & \mbox{ if } n \ge 3 \mbox{ and } m = en-2, \\
3 & \mbox{ if } n=3 \mbox{ and } m=4, \\
3 & \mbox{ if } n=3 \mbox{ and } m = 3e-4. \\
\end{array}
\right.$$
For example, in the case $n \ge 3$ and $m = en-2$, to see ${\rm SOSrank}(\mathcal{A}_0)=n$, we only need to show ${\rm SOSrank}(\mathcal{A}_0) \ge n$. Suppose on the contrary that ${\rm SOSrank}(\mathcal{A}_0) \le n-1$. Then, there exists $r \le n-1$ and homogeneous polynomial $f_i$ with degree $m/2=\frac{e}{2}n-1$ such that
\[
x_1^e\cdots x_n^e \left(\sum_{i=1}^n x_i^{-2}\right)=\sum_{i=1}^r f_i({\bf x})^2.
\]
This implies that for each ${\bf x}=(x_1,\cdots,x_n)$ with $x_i \neq 0$, $i=1,\cdots,n$
\[
\sum_{i=1}^n x_i^{-2} = \sum_{i=1}^r \left[\frac{f_i({\bf x})}{x_1^{\frac{e}{2}} \cdots x_n^{\frac{e}{2}}}\right]^2
\]
Letting $t_i=x_i^{-1}$, by continuity, we see that the quadratic form $\sum_{i=1}^n t_i^2$ can be written as a sum of at most $r$ many squares of rational functions in $(t_1,,\cdots,t_n)$. Then, the Cassels-Pfister's Theorem \cite[Theorem 17.3]{book0} (see also \cite[Corollary 17.6]{book0}), implies that the quadratic form $\sum_{i=1}^n t_i^2$ can be written as a sum of at most $r$ many sums of squares of polynomial functions in $(t_1,,\cdots,t_n)$, which is impossible.
In the case $n =3$ and $m = 4$, we only need to show ${\rm SOSrank}(\mathcal{A}_0) \ge 3$. Suppose on the contrary that ${\rm SOSrank}(\mathcal{A}_0) \le 2$. Then, there exist $a_i,b_i,c_i,d_i,e_i,f_i \in \mathbb{R}$, $i=1,2$, such that \begin{eqnarray*}
x_1^2x_2^2+x_2^2x_3^2+x_3^2x_1^2&= & (a_1 x_1^2+b_1 x_2^2 + c_1x_3^2 +d_1 x_1x_2+e_1 x_1x_3+f_1x_2x_3)^2 \\
& & +(a_2 x_1^2+b_2 x_2^2 + c_2x_3^2 +d_2 x_1x_2+e_2 x_1x_3+f_2x_2x_3)^2.
\end{eqnarray*}
Comparing with the coefficients gives us that $a_1=a_2=b_1=b_2=c_1=c_2=0$ and
\[
\left\{\begin{array}{l}
d_1^2+d_2^2=1 \\
e_1^2+e_2^2=1 \\
f_1^2+f_2^2=1 \\
d_1e_1+d_2e_2=0 \\
d_1f_1+d_2f_2=0 \\
e_1f_1+e_2f_2=0.
\end{array} \right.
\]
From the last three equations, we see that one of $d_1,d_2,e_1,e_2,f_1,f_2$ must be zero. Let us assume say $d_1=0$. Then, the first equation shows $d_2=\pm 1$ and hence, $e_2=0$ (by the fourth equation). This implies that $e_1= \pm 1$ and $f_2=0$. Again, we have $f_1=\pm 1$ and hence
\[
e_1f_1+e_2f_2=(\pm 1)(\pm 1)+0=\pm 1 \neq 0.
\]
This leads to a contradiction.
For the last case, suppose again by contradiction that ${\rm SOSrank}(\mathcal{A}_0) \le 2$. Then, there exist two homogeneous polynomial $f_i$ with degree $m/2=\frac{3e}{2}-2$ such that
\[
x_1^ex_2^ex_3^e(x_1^{-2}x_2^{-2}+x_2^{-2}x_3^{-2}+ x_3^{-2}x_1^{-2})=\sum_{i=1}^2 f_i({\bf x})^2.
\]
This implies that for each $x=(x_1,\cdots,x_n)$ with $x_i \neq 0$, $i=1,\cdots,n$
\[
x_1^{-2}x_2^{-2}+x_2^{-2}x_3^{-2}+x_3^{-2}x_1^{-2} = \sum_{i=1}^2 \left[\frac{f_i({\bf x})}{x_1^{\frac{e}{2}} \cdots x_3^{\frac{e}{2}}}\right]^2
\]
Letting $t_i=x_i^{-1}$, using a similar line argument in the case $m=en-2$, we see that the polynomial $t_1^2t_2^2+t_2^2t_3^2+t_3^2t_1^2$ can be written
as sums of $2$ squares of polynomials in $(t_1,t_2,t_3)$. This is impossible by the preceding case. Therefore, the conclusion follows. \qed
\end{proof}
Below, let us mention that calculating the exact SOS-rank of SOS tensor decomposition is not a trivial task even for the identity tensor, and this relates to
some open question in algebraic geometry in the literature. To explain this, we recall that
the identity tensor $\mathcal{I}$ with order $m$ and dimension $n$ is given by $\mathcal{I}_{i_1\cdots i_m}=1$ if $i_1=\cdots=i_m$ and $\mathcal{I}_{i_1\cdots i_m}=0$ otherwise. The identity tensor $\mathcal{I}$ induces the polynomial $f_{\mathcal{I}}({\bf x})=\mathcal{I}{\bf x}^m=x_1^m+\cdots +x_n^m$. It is clear that, $\mathcal{I}$ has an SOS tensor decomposition when $m$ is even and the corresponding SOS-rank of $\mathcal{I}$ is less than or equal to $n$. It was conjectured by Reznick \cite{Rez0} that $f_{\mathcal{I}}({\bf x})$ cannot be written as sums of $(n-1)$ many squares, that is, ${\rm SOSrank}(\mathcal{I})=n$. The positive answer for this conjecture in the special case
of $m=n=4$ was provided in \cite{JPAA1,JPAA2}. On the other hand, the answer for this conjecture in the general case is still open to the best of our knowledge. Moreover, this conjecture relates to another conjecture of Reznick \cite{Rez0} in the same paper where he showed that the polynomial $f_R({\bf x})=x_1^n+\cdots +x_n^n- n x_1 \cdots x_n$ can be written as sums of $(n-1)$ many squares whenever $n=2^k$ for some $k \in \mathbb{N}$, and he
conjectured that the estimate of the numbers of squares is sharp. Indeed, he also showed that this conjecture is true whenever the previous conjecture of ``$f_{\mathcal{I}}({\bf x})$ cannot be written as sums of $(n-1)$ many squares" is true.
\setcounter{equation}{0}
\section{Applications}
In this section, we provide some applications for the SOS tensor decomposition of the structure tensors such as finding the minimum $H$-eigenvalue of an
even order extended $Z$-tensor and testing the positive definiteness of a multivariate form. We also provide some numerical examples/experiments to support the theoretical findings.
Throughout this section, all numerical experiments are performed on a desktop, with 3.47 GHz quad-core Intel E5620 Xeon 64-bit CPUs and 4 GB RAM,
equipped with Matlab 2015.
\subsection{Finding the minimum $H$-eigenvalue of an even order symmetric extended $Z$-tensor}
Finding the minimum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra, and has found numerous applications including automatic control and image processing \cite{Qi05}.
Recently, it was shown that the minimum $H$-eigenvalue of an even order symmetric $Z$-tensor \cite{HLQS,Hu14} can be found by solving a sums-of-squares optimization problem, which can be
equivalently reformulated as a semi-definite programming problem, and so, can be solved efficiently. In \cite{HLQS}, some upper and lower estimates for the minimum $H$-eigenvalue of general symmetric tensors with even order are
provided via sums-of-squares programming problems. Examples show that the estimate can be sharp in some cases.
On the other hand, it was unknown in \cite{Hu14,HLQS} that whether similar results can continue to hold for some classes of symmetric tensors which are not $Z$-tensors, that is, for symmetric tensors
with possible positive entries on the off-diagonal elements. In this section, as applications of the derived SOS decomposition of structured tensors, we show that the class of even order symmetric extended $Z$-tensor serves as one such class. To present the conclusion, the following Lemma plays an important role in our later analysis.
\bl\label{lema51}$^{\cite{Qi05}}$ \label{lemma:5.1} Let $\mathcal{A}$ be a symmetric tensor with even order $m$ and dimension $n$. Denote the
minimum $H$-eigenvalue of $\mathcal{A}$ by
$\lambda_{min}(\mathcal{A})$. Then, we have
\begin{equation}\label{e51}\lambda_{min}(\mathcal{A})=\min_{{\bf x\neq {\bf 0}}}\frac{\mathcal{A}{\bf x}^m}{\|{\bf x}\|_m^m}=\min_{\|{\bf x}\|_m=1}\mathcal{A}{\bf x}^m,
\end{equation}
where $\|{\bf x}\|_m=(\sum_{i=1}^n|x_i|^m)^{\frac{1}{m}}$.
\el
\begin{theorem}{\bf (Finding the minimum $H$-eigenvalue of an even order symmetric extended $Z$-tensor)} \label{th:5.1}
Let $m$ be an even number. Let $\mathcal{A}$ be a symmetric extended $Z$-tensor with order $m$ and dimension $n$. Then, we have
\begin{eqnarray*}
\lambda_{{\rm min}}(\mathcal{A})&=& \max_{\mu, r \in \mathbb{R}}\{\mu: f_{\mathcal{A}}({\bf x})-r (\|{\bf x}\|_m^m-1)-\mu \in \Sigma^2_m[{\bf x}]\},
\end{eqnarray*}
where $f_{\mathcal{A}}({\bf x})=\mathcal{A}{\bf x}^m$ and $\Sigma_m^2[{\bf x}]$ is the set of all SOS polynomials with degree at most $m$.
\end{theorem}
\proof
Consider the following problem
\[
(P) \ \ \ \min \{\mathcal{A}{\bf x}^m: \|{\bf x}\|_m^m=1\}
\]
and denote its global minimizer by ${\bf a}=(a_1,\cdots,a_n)^T \in \mathbb{R}^n$. Clearly, $\sum_{i=1}^na_i^m=1$. Then, $\lambda_{\rm min}(\mathcal{A})=f_{\mathcal{A}}({\bf a})=\mathcal{A}{\bf a}^m$.
It follows that for all ${\bf x} \in \mathbb{R}^n\backslash\{0\}$
\begin{eqnarray*}
f_{\mathcal{A}}({\bf x})-\lambda_{\rm min}(\mathcal{A}) \sum_{i=1}^nx_i^m & = & f_{\mathcal{A}}({\bf x})-f_{\mathcal{A}}({\bf a})\sum_{i=1}^nx_i^m \\
&= & \sum_{i=1}^nx_i^m\bigg(f_{\mathcal{A}}(\frac{{\bf x}}{(\sum_{i=1}^nx_i^m)^{\frac{1}{m}}})-f_{\mathcal{A}}({\bf a})\bigg) \ge 0,
\end{eqnarray*}
where the last inequality holds as $m$ is even and $\overline{{\bf x}}=\frac{{\bf x}}{(\sum_{i=1}^nx_i^m)^{\frac{1}{m}}}$ belongs to the feasible set of (P).
This shows that
$g({\bf x}):=f_{\mathcal{A}}({\bf x})-\lambda_{\rm min}(\mathcal{A}) \sum_{i=1}^nx_i^m$ is a homogeneous polynomial which always take nonnegative values. As $\mathcal{A}$ is an extended $Z$-tensor,
there exist $s \in \mathbb{N}$ and index sets $\Gamma_l \subseteq \{1,\cdots,n\}$, $l=1,\cdots,s$ with $\bigcup_{l=1}^s \Gamma_l=\{1,\cdots,n\}$ and $\Gamma_{l_1} \cap \Gamma_{l_2} =\emptyset$ such that
for all ${\bf x} \in \mathbb{R}^n$
\begin{equation}\label{eq:96}
f_{\mathcal{A}}({\bf x})=\sum_{i=1}^n f_{m,i} x_i^{m}+\sum_{l=1}^s \sum_{\alpha_l \in \Omega_l}f_{\alpha_l} {\bf x}^{\alpha_l}
\end{equation}
such that, for each $l=1,\cdots,s$, either one of the following two condition holds:
(1) $f_{\alpha_l}=0$ for all but one $\alpha_l \in \Omega_l$; (2) $f_{\alpha_l} \le 0$ for all $\alpha_l \in \Omega_l$.
Thus,
\[
g({\bf x})=\sum_{i=1}^n (f_{m,i}-\lambda_{\rm min}(\mathcal{A})) x_i^{m}+\sum_{l=1}^s \sum_{\alpha_l \in \Omega_l} f_{\alpha_l} {\bf x}^{\alpha_l},
\]
is an extended $Z$-polynomial which always takes nonnegative values. Let $\mathcal{B}$ be a symmetric tensor such that $g({\bf x})=\mathcal{B}{\bf x}^m$. Then,
$\mathcal{B}$ is a positive semi-definite extended $Z$-tensor and so is SOS by Theorem \ref{th:5.2}. Thus, $g({\bf x})$ is an SOS polynomial with degree $m$. Note that $g({\bf x})=
f_{\mathcal{A}}({\bf x})-\lambda_{\rm min}(\mathcal{A}) \sum_{i=1}^nx_i^m=
f_{\mathcal{A}}({\bf x})-\lambda_{\rm min}(\mathcal{A}) (\sum_{i=1}^nx_i^m-1)-\lambda_{\rm min}(\mathcal{A})$. This shows that
\[
\lambda_{\rm min}(\mathcal{A}) \le \max_{\mu, r \in \mathbb{R}}\{\mu: f_{\mathcal{A}}({\bf x})-r (\|{\bf x}\|_m^m-1)-\mu \in \Sigma^2_m[{\bf x}]\}.
\]
To see the reverse inequality, take any $(\mu,r)$ with $f_{\mathcal{A}}({\bf x})-r (\|{\bf x}\|_m^m-1)-\mu \in \Sigma^2_m[{\bf x}]$. Then, for all ${\bf x} \in \mathbb{R}^n$,
\[
f_{\mathcal{A}}({\bf x})-r (\|{\bf x}\|_m^m-1)-\mu \ge 0.
\]
This shows that $r \ge \mu$ and $f_{\mathcal{A}}({\bf x}) \ge r \|{\bf x}\|_m^m$ for all ${\bf x} \in \mathbb{R}^n$. This shows that $\lambda_{\rm min}(\mathcal{A}) \ge r \ge \mu$, and so, the conclusion follows. \qed
\begin{Remark}\label{remark:5.1}
Let $\mathcal{A}$ be an extended $Z$-tensor. As in (\ref{eq:96}), its associated polynomial $f_{\mathcal{A}}$ can be written as $f_{\mathcal{A}}({\bf x})=\sum_{i=1}^n f_{m,i} x_i^{m}+\sum_{l=1}^s \sum_{\alpha_l \in \Omega_l}f_{\alpha_l} {\bf x}^{\alpha_l}$
. Then, Remark \ref{remark:3.3} implies that
\begin{eqnarray*}
\lambda_{{\rm min}}(\mathcal{A}) & = & \max_{\mu, r \in \mathbb{R}}\{\mu: f_{\mathcal{A}}({\bf x})-r (\|{\bf x}\|_m^m-1)-\mu \in \Sigma^2_m[{\bf x}]\} \\
& = & \max_{\mu, r \in \mathbb{R}}\{\mu: f_{\mathcal{A}}({\bf x})-r \|{\bf x}\|_m^m \in \Sigma^2_m[{\bf x}], r-\mu \ge 0\} \\
& = & \max_{\mu, r \in \mathbb{R}}\{\mu: \sum_{i \in \Gamma_l} f_{m,i} x_i^{m}+ \sum_{\alpha_l \in \Omega_l}f_{\alpha_l} {\bf x}^{\alpha_l}-r \|{\bf x}^{(l)}\|_m^m \in \Sigma^2_m[{\bf x}^{(l)}], l=1,\cdots,s \\
& & \ \ \ \ \ \ \ \ \ \ \ \ \ r-\mu \ge 0\}
\end{eqnarray*}
where, for each $l=1,\cdots,s$, ${\bf x}^{(l)}=(x_i)_{i \in \Gamma_l}$ and $\Sigma^2_m[{\bf x}^{(l)}]$ is the set of all
SOS polynomials in ${\bf x}^{(l)}$.
\end{Remark}
As explained in \cite{Hu14,HLQS}, the sums-of-squares problem $$\max_{\mu, r \in \mathbb{R}}\{\mu: f_{\mathcal{A}}({\bf x})-r (\|{\bf x}\|_m^m-1)-\mu \in \Sigma^2_m[{\bf x}]\}$$ can be equivalently rewritten as a semi-definite programming problem (SDP), and so, can be solved efficiently. Indeed, this conversion can be done by using the commonly used
Matlab Toolbox YALMIP \cite{Yalmip1,Yalmip2}.
On the other hand, the size of the equivalent SDP problem of the relaxation problem increase dramatically when the dimension/order of the tensor increases.
For example, as illustrate in Table 1, for a $4$th-order $50$-dimensional tensor, the equivalent SDP problem has $1326$ variables and $316251$ constraints. Fortunately, a robust SDP software (SDPNAL \cite{zhao}) has been established recently which enables us to solve large-scale SDP (dimension up to 5000 and number of constraint of the SDP up to 1 million). This
enables us to find the minimum $H$-eigenvalue for medium-size tensor. Later on, we will explain how to use SDPNAL together with the observation in Remark \ref{remark:5.1}
to find the minimum $H$-eigenvalue for large-size tensor.
We first illustrate how to compute the minimum $H$-eigenvalue of an extended $Z$-tensor $\mathcal{A}$ using the above sums-of-squares problem via Matlab Toolbox YALMIP \cite{Yalmip1,Yalmip2} via
two small-size problems. We will show the performance of the method for various larger-size problem later.
\begin{example}
Consider the symmetric tensor $\mathcal{A}$ with order $6$ and dimension $4$ where
\[
\mathcal{A}_{111111}=\mathcal{A}_{222222}=\mathcal{A}_{333333}=\mathcal{A}_{444444}=1,
\]
\[
\mathcal{A}_{i_1 \cdots i_6}=\frac{1}{5}, \mbox{ for all } (i_1,\cdots,i_6)=\sigma(1,1,1,2,2,2),
\]
\[
\mathcal{A}_{i_1 \cdots i_6}=\frac{2}{5}, \mbox{ for all } (i_1,\cdots,i_6)=\sigma(3,3,4,4,4,4),
\]
and $\mathcal{A}_{i_1 \cdots i_6}=0$ otherwise. Here $\sigma(i_1,\cdots,i_6)$ denotes all the possible permutation of $(i_1,\cdots,i_6)$. The associated polynomial
$$f_{\mathcal{A}}({\bf x})=\mathcal{A}{\bf x}^m=x_1^6+x_2^6+x_3^6+x_4^6+4x_1^3x_2^3+6x_3^2x_4^4$$ is an extended $Z$-polynomial. So, $\mathcal{A}$ is an extended $Z$- tensor. It can be easily verified that $\mathcal{A}$ is not a $Z$-tensor.
To compute its minimum $H$-eigenvalue, we note that the corresponding sums-of-squares optimization problem reads
\[
\max_{\mu, r \in \mathbb{R}}\{\mu: f_{\mathcal{A}}({\bf x})-r (\|{\bf x}\|_6^6-1)-\mu \in \Sigma^2_6[{\bf x}]\}.
\]
Convert this sums-of-squares optimization problem into a semi-definite programming problem using the Matlab Toolbox {\rm YALMIP} \cite{Yalmip1,Yalmip2}, and solve it by using the
SDP software {\rm SDPNAL} we obtain that $\lambda_{\rm min}(\mathcal{A})=-1$.
The simple code using {\rm YALMIP} is appended as follows:
\begin{verbatim}
sdpsettings('solver','sdpnal')
sdpvar x1 x2 x3 x4 r mu
f = x1^6+x2^6+x3^6+x4^6+4*x1^3*x2^3+6*x3^2*x4^4;
g = [(x1^6+x2^6+x3^6+x4^6)-1];
F = [sos(f-mu-r*g)];
solvesos(F,-mu,[],[r;mu])
\end{verbatim}
Moreover, note from the geometric mean inequality that
$|x_1^3x_2^3|=(x_1^6)^{\frac{1}{2}} (x_2^6)^{\frac{1}{2}}\le \frac{1}{2} x_1^6+\frac{1}{2} x_1^6$.
It follows that
\[
f_{\mathcal{A}}({\bf x})+\|{\bf x}\|_6^6=2x_1^6+2x_2^6+2x_3^6+2x_4^6+4x_1^3x_2^3+6x_3^2x_4^4 \ge 0 \mbox{ for all } {\bf x} \in \mathbb{R}^n.
\]
On the other hand, consider $\bar {\bf x}=(\sqrt[6]{\frac{1}{2}},-\sqrt[6]{\frac{1}{2}},0,0)$. We see that $f_{\mathcal{A}}(\bar {\bf x})+\|\bar {\bf x}\|_6^6=0$. This shows that $\lambda_{\rm min}(\mathcal{A})=\min\{f_{\mathcal{A}}({\bf x}): \|{\bf x}\|_6=1\}=-1$. This verifies the correctness of our computed minimum $H$-eigenvalue.
\end{example}
\begin{example}
Let $\alpha,\beta \in \mathbb{R}$ and consider the symmetric tensor $\mathcal{A}$ with order $6$ and dimension $4$ where
\[
\mathcal{A}_{111111}=\mathcal{A}_{222222}=\mathcal{A}_{333333}=\mathcal{A}_{444444}=1,
\]
\[
\mathcal{A}_{i_1 \cdots i_6}=\alpha, \mbox{ for all } (i_1,\cdots,i_6)=\sigma(1,1,1,2,2,2),
\]
\[
\mathcal{A}_{i_1 \cdots i_6}=\beta, \mbox{ for all } (i_1,\cdots,i_6)=\sigma(3,3,3,4,4,4),
\]
and $\mathcal{A}_{i_1 \cdots i_6}=0$ otherwise. Here $\sigma(i_1,\cdots,i_6)$ denotes all the possible permutation of $(i_1,\cdots,i_6)$. The associated polynomial
$$f_{\mathcal{A}}({\bf x})=\mathcal{A}{\bf x}^m=x_1^6+x_2^6+x_3^6+x_4^6+20 \alpha \,x_1^3x_2^3+20 \beta\, x_3^3x_4^3$$ is an extended $Z$-polynomial. So, $\mathcal{A}$ is an extended $Z$- tensor. It can be easily verified that if either $\alpha>0$ or $\beta>0$, then $\mathcal{A}$ is not a $Z$-tensor.
To compute its minimum $H$-eigenvalue, we randomly generate 100 instance of $(\alpha,\beta) \in [-5,5] \times [-5,5]$. For each $(\alpha,\beta)$, we convert the corresponding sums-of-squares optimization problem
\[
\max_{\mu, r \in \mathbb{R}}\{\mu: f_{\mathcal{A}}({\bf x})-r (\|{\bf x}\|_6^6-1)-\mu \in \Sigma^2_6[{\bf x}]\}
\]
into a semi-definite programming problem using the Matlab Toolbox {\rm YALMIP} \cite{Yalmip1,Yalmip2}, and solve it by using the
SDP software {\rm SDPNAL}. We then compare the computed minimum $H$-eigenvalue with the true minimum $H$-eigenvalue of $\mathcal{A}$. Indeed, similar to the preceding example, we can verify that $\lambda_{\rm min}(\mathcal{A})=m(\alpha,\beta)$ where
\[
m(\alpha,\beta):=\left\{
\begin{array}{cll}
1-10|\alpha| & \mbox{ if } & |\alpha| \ge |\beta|,\\
1-10 |\beta| & \mbox{ if } & |\alpha| < |\beta|.
\end{array}
\right.
\]
For all the $100$ generated $(\alpha,\beta)$, the maximum difference of the computed $H$-minimum eigenvalue and the true $H$-minimum eigenvalue is
$6.2039e-05$.
\end{example}
\subsection*{Medium-size examples}
We now consider a few medium-size examples which involves symmetric extended $Z$-tensor with order up to $30$ or dimension up to $60$ .
\begin{example}
Let $m=10k$ with $k \in \mathbb{N}$. Consider the symmetric tensor $\mathcal{A}$ with order $m$ and dimension $4$ where
\[
\mathcal{A}_{1 \cdots 1}=\mathcal{A}_{2 \cdots 2}=\mathcal{A}_{3 \cdots 3}=\mathcal{A}_{4 \cdots 4}=1,
\]
\[
\mathcal{A}_{i_1 \cdots i_{m}}=\alpha, \mbox{ for all } (i_1,\cdots,i_{m})=\sigma(\underbrace{1,\cdots,1}_{m/2},\underbrace{2,\cdots,2}_{m/2}),
\]
\[
\mathcal{A}_{i_1 \cdots i_{m}}= \beta, \mbox{ for all } { (i_1,\cdots,i_{m})=\sigma(\underbrace{3,\cdots,3}_{m/5},\underbrace{4,\cdots,4}_{4m/5}),}
\]
\[
\mathcal{A}_{i_1 \cdots i_{m}}= \beta, \mbox{ for all } { (i_1,\cdots,i_{m})=\sigma(\underbrace{3,\cdots,3}_{4m/5},\underbrace{4,\cdots,4}_{m/5}),}
\]
with $\alpha=2 {m \choose m/2} ^{-1}$ and $\beta=- {m \choose m/5} ^{-1}$,
and $\mathcal{A}_{i_1 \cdots i_m}=0$ otherwise. Here $\sigma(i_1,\cdots,i_m)$ denotes all the possible permutation of $(i_1,\cdots,i_m)$. The associated polynomial
$${ f_{\mathcal{A}}({\bf x})=\mathcal{A}{\bf x}^m=x_1^{m}+x_2^{m}+x_3^{m}+x_4^{m}+ 2x_1^{\frac{m}{2}}x_2^{\frac{m}{2}}-x_3^\frac{m}{5}x_4^{\frac{4m}{5}}-x_3^{\frac{4m}{5}}x_4^{\frac{m}{5}},}$$
is an extended $Z$-polynomial. So, $\mathcal{A}$ is an extended $Z$-tensor. It can be easily verified that $\mathcal{A}$ is not a $Z$-tensor.
Moreover, using geometric mean inequality, we can directly verify that the true minimum $H$-eigenvalue is $0$.
We compute the minimum
$H$-eigenvalue by solving the corresponding sums-of-squares problem for the case $m=20, 30$, and compare with the true minimum $H$-eigenvalue. The results are
summarized in Table 1.
\end{example}
\begin{example}
Let $n=4k$ with $k \in \mathbb{N}$. Consider the symmetric tensor $\mathcal{A}$ with order $4$ and dimension $n$ where
\[
\mathcal{A}_{1111}=\mathcal{A}_{2222}=\cdots=\mathcal{A}_{nnnn}=n,
\]
\[
\mathcal{A}_{i_1i_2i_3i_4}=\frac{1}{6}, \mbox{ for all } (i_1,i_2,,i_3,i_4)=\sigma(4i-3,4i-2,4i-1,4i), i=1,\cdots,\frac{n}{4},
\]
and $\mathcal{A}_{i_1i_2i_3i_4}=0$ otherwise. Here $\sigma(i_1,\cdots,i_4)$ denotes all the possible permutation of $(i_1,\cdots,i_4)$. The associated polynomial
$$f_{\mathcal{A}}({\bf x})=\mathcal{A}{\bf x}^m= n(x_1^{4}+\cdots+x_n^4) + 4\sum_{i=1}^{n/4}x_{4i-3}\, x_{4i-2}\, x_{4i-1}\, x_{4i}$$ is an extended $Z$-polynomial. So, $\mathcal{A}$ is an extended $Z$-tensor. It can be easily verified that $\mathcal{A}$ is not a $Z$-tensor.
Moreover, using geometric mean inequality, we can directly verify that the true minimum $H$-eigenvalue is $n-1$.
We compute the minimum
$H$-eigenvalue by solving the corresponding sums-of-squares problem for the case $n=20,40,50,60$, and compare with the true minimum $H$-eigenvalue. The results are
summarized in Table 1.
\end{example}
The following table summarizes the numerical results of Example 5.3 and Example 5.4 where we compute the minimum $H$-eigenvalue by first converting the corresponding sums-of-squares problem to
an SDP problem using YALMIP and solving this SDP problem using SDPNAL. In particular, the data of the following table are explained as follows. \begin{itemize}
\item $m$: the order of the symmetric tensor,
\item $n$: the dimension of the symmetric tensor,
\item $NV$: the number of variables of the equivalent SDP problem,
\item $NC$: the number of constraints in the equivalent SDP problem,
\item Computed eigenvalue: the calculated minimum $H$-eigenvalue,
\item True eigenvalue: the true minimum $H$-eigenvalue
\item Time (YALMIP): the CPU-time for converting the sums-of-squares problem to SDP (measured in seconds).
\item Time (SDPNAL): the CPU-time for solving SDP via SDPNAL (measured in seconds).
\end{itemize}
\begin{table}[h!]
\begin{center}
\caption{Test results for medium size tensors}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|}\hline
Problem & m & n & NV & NC & Computed & True & Time (YALMIP) & Time (SDPNAL) \\
& & & & & eigenvalue & eigenvalue & & \\ \hline
Example 5.3 & 20 & 4 & 1001 & 1001 & -1.7634e-09 & 0 & 11.9487 & 0.5700 \\ \hline
Example 5.3 & 30 & 4 & 3876 & 6936 & 1.1382e-12 & 0 & 198.8141 & 8.2700 \\ \hline
Example 5.4 & 4 & 20 & 231 & 10626 & 19.0000 & 19 & 4.6951 & 0.4763 \\ \hline
Example 5.4 & 4 & 40 & 861 & 135751 & 39.0000 & 39 & 440.8231 & 1.7727 \\ \hline
Example 5.4 & 4 & 50 & 1326 & 316251 & 49.0000 & 49 & 2365.9043& 5.1109 \\ \hline
Example 5.4 & 4 & 60 & 1891 & 635376 & 59.0000 & 59 & 9322.0631 & 50.2934 \\ \hline
\end{tabular}
\end{center}
\end{table}
We observe that, for all the above numerical examples, the minimum $H$-eigenvalues can be found successfully
for medium-size tensors.
\subsection{Large size examples}
Finally, we illustrate with an example that using SDPNAL together with the observation in Remark \ref{remark:5.1} enables us to solve
some large size tensors (dimension up to 2000).
As one can observed in Table 1, most of the time are occupied in YALMIP in converting the sums-of-squares problem into an SDP problem. This process involves matching up
the coefficients of all the involved ${m+n-1 \choose m}$ many monomials, and so, can be time-consuming. On the other hand, by using
the sums-of-squares problem discussed in Remark \ref{remark:5.1} and letting $k=\max_{1 \le l \le s}|\Gamma_l|$, the corresponding process only involves $s {m+k-1 \choose m}$ many monomials which is much smaller than ${m+n-1 \choose m}$ when
$s$ is large and $k$ is small. For example, as in Example 5.4, we can set $s=n/4$, $k=4$ and $m=4$, and so, $s {m+k-1 \choose m}$ is of the order $n$; while ${m+n-1 \choose m}={n+3 \choose 4}$ which
is of the order $n^4$.
The following table summarizes the numerical results of Example 5.4 with dimension from 500 to 2000, where we compute the minimum $H$-eigenvalue by first converting the corresponding sums-of-squares problem
discussed in Remark \ref{remark:5.1} to
an SDP problem using YALMIP and solving this SDP problem using SDPNAL. We observe that, for all the instances, the minimum $H$-eigenvalues can be found successfully.
The meaning of the data are the same as in Table 1.
\begin{table}[h!]
\begin{center}
\caption{Test results for large size tensors}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|}\hline
Problem & m & n & NV & NC & Computed & True & Time (YALMIP) & Time (SDPNAL) \\
& & & & & eigenvalue & eigenvalue & & \\ \hline
Example 5.4 & 4 & 500 & 1250 & 1375 & 499.0000 & 499 & 4.6299 & 6.8295 \\ \hline
Example 5.4 & 4 & 1000 & 2500 & 2750 & 999.0000 & 999 & 8.8298 & 66.5566 \\ \hline
Example 5.4 & 4 & 2000 & 5000 & 5500 & 1999.0000 & 1999 & 20.9729 & 563.6903 \\ \hline
\end{tabular}
\end{center}
\end{table}
\newpage
\subsection{Testing positive definiteness of a multivariate form}
For a multivariate form $\mathcal{A}{\bf x}^m$, we say it is positive definite if $\mathcal{A}{\bf x}^m>0$ for all ${\bf x} \neq {\bf 0}$.
Testing positive definiteness of a multivariate form $\mathcal{A}{\bf x}^m$ is an important
problem in the stability study of nonlinear autonomous systems via Lyapunov's direct
method in automatic control \cite{Qi05}. Researchers in automatic control have studied the
conditions of such positive definiteness intensively. However, for $n \ge 3$
and $m \ge 4$, this is, in general, a hard problem in mathematics. Recently, some efficient methods based on eigenvalues
of tensors were proposed to solve the problem in the case where $m=4$ \cite{Ni}.
In this part, we show that testing positive definiteness of a multivariate form $\mathcal{A}{\bf x}^m$ where $\mathcal{A}$ is an extended $Z$-tensor
can be computed by sums-of-squares problem via Theorem \ref{th:5.1}. Indeed, a direct consequence of Theorem \ref{th:5.1} and Lemma \ref{lemma:5.1} give us the following useful test:
\begin{corollary}
Let $\mathcal{A}$ be an extended $Z$-tensor. Then, the associated multivariate form $\mathcal{A}{\bf x}^m$ is positive definite if and only if
\[
\max_{\mu, r \in \mathbb{R}}\{\mu: f_{\mathcal{A}}({\bf x})-r (\|{\bf x}\|_m^m-1)-\mu \in \Sigma^2_m[{\bf x}]\}>0,
\]
where $f_{\mathcal{A}}({\bf x})=\mathcal{A}{\bf x}^m$ and $\Sigma_m^2[{\bf x}]$ is the set of all SOS polynomials with degree at most $m$.
\end{corollary}
We now use the above corollary to test the positive definiteness of extended $Z$-tensors. To do this, we first generate 100 extended $Z$-tensors as numerical examples. These extended $Z$-tensors are randomly generated by the
following procedure.
\bigskip
{\bf Procedure 1}
\begin{itemize}
\item[(i)] Given $(m, n,s,k,M)$ with $m$ is an even number and $n=sk$, where $n$ and $m$ are the dimension and the order of the
randomly generated tensor, respectively, and $M$ is a large positive constant.
\item[(ii)] Randomly generate a random positive integer $L$ and a partition of the index set $\{1,\cdots,n\}$, $\{\Gamma_1,\cdots,\Gamma_s\}$, such that $|\Gamma_i|=k$, $i=1,\cdots,s$ and $\Gamma_{i} \cap \Gamma_{i'}=\emptyset$ for all $i \neq i'$.
For each $i=1,\cdots,s-1$, generate a random multi-index $(l_1^i,\cdots,l_m^i)$ with $l_j^i \in \Gamma_i$, $j=1,\cdots,m$ and a random number $\bar{a}_{l_1^i \cdots l_m^i} \in [0,1]$. { Generate one randomly $m$th-order $k$-dimensional symmetric tensor $\mathcal{B}$, such that all elements
of $\mathcal{B}$ are in the interval $[0, 1]$.}
\item[(iii)] We define extended $Z$-tensor $\mathcal{A}=(a_{i_1i_2\cdots i_m})$ such that
\[
a_{i_1 \cdots i_m}= \left\{\begin{array}{cll}
(-1)^L M & \mbox{ if } & i_1=\cdots=i_m=i \mbox{ for all } i=1,\cdots,n, \\
\bar{a}_{l_1^i \cdots l_m^i} & \mbox{ if } & (i_1,\cdots,i_m)=\sigma(l_1^i,\cdots,l_m^i) \mbox{ with } l_1^i,\cdots,l_m^i \in \Gamma_i, i=1,\cdots,s-1, \\
-\mathcal{B}_{i_1 \cdots i_m} & \mbox{ if } & { i_1,\cdots,i_m \in \Gamma_s,}\\
0 & \mbox{ othewise. } &
\end{array}
\right.
\]
Here $\sigma(i_1,\cdots,i_m)$ denotes all the possible permutation of $(i_1,\cdots,i_m)$.
\end{itemize}
From the construction of $\mathcal{A}$, it can be verified that $\mathcal{A}$ is an extended $Z$-tensor.
Let $f_{\mathcal{A}}({\bf x})=\mathcal{A}{\bf x}^m$. We then solve the sums-of-squares problem $$\max_{\mu, r \in \mathbb{R}}\{\mu: f_{\mathcal{A}}({\bf x})-r (\|{\bf x}\|_m^m-1)-\mu \in \Sigma^2_m[{\bf x}]\}$$
and use the preceding corollary to determine whether $\mathcal{A}{\bf x}^m$ is a positive definite multivariate form or not. Here, to speed up the algorithm, as we did for the large size
tensors, we first convert the sums-of-squares problem into an SDP by using Remark \ref{remark:5.1} and {\rm YALMIP}. Then, we solve the equivalent
SDP by using the software {\rm SDPNAL}. The correctness can be verified by looking at the randomly generated positive number $L$. Indeed, from the construction, if $L$ is an even number and $M$ is a large positive number, the diagonal elements will strictly dominate the sum of the off-diagonal elements, and so, $\mathcal{A}{\bf x}^m$ is a positive definite multivariate form. On the other hand, if $L$ is an odd number, then the diagonal elements will be negative, and so, $\mathcal{A}{\bf x}^m$ is not a positive definite multivariate form in this case.
The following table summarize the results for the correctness of testing the positive definiteness of a multivariate form generated by an extended $Z$-tensor.
As we can see the results, in our numerical experiment, all the $100$ randomly generated instance has been correctly identified.
\el
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
m & n & s & k & M & PD & NPD & Correctness \\ \hline
4 & 20 & 4 & 5 & 100 & 48 & 52 & 100\% \\
4 & 25 & 5 & 5 & 100 & 46 & 54 &100\% \\
4 & 40 & 4 & 10 & 100 & 52 & 48 & 100\% \\
4 & 60 & 4 & 15 & 100 & 45 & 55 & 100\% \\
4 & 100 & 4 & 25 & 100 & 44 & 56 & 100\% \\
\hline
\end{tabular}
\end{center}
\section{Conclusions and Remarks}
In this paper, we establish SOS tensor decomposition of various even order symmetric structured tensors available in the current literature.
These include positive Cauchy tensors, weakly diagonally dominated tensors,
$B_0$-tensors, { double $B$-tensors, quasi-double $B_0$-tensors, $MB_0$-tensors,} $H$-tensors, absolute tensors
of positive semi-definite $Z$-tensors and extended $Z$-tensors.
We also examine the SOS-rank of SOS tensor decomposition and the SOS-width for SOS tensor cones. In particular, we provide an explicit sharp
estimate for SOS-rank of tensors with bounded exponent and SOS-width for the tensor cone consisting of all such tensors with bounded exponent that have SOS decomposition.
It is shown that the SOS-rank of SOS tensor decomposition is equal to the
optimal value of a related rank optimization problem over positive semi-definite matrix constraints. Finally, applications for the SOS decomposition
of extended $Z$-tensors are provided and several numerical experiments illustrate the significance.
Below, we raise some open questions which might be interesting for future work:
{\bf Question 1}: Can we evaluate the SOS-rank of symmetric $B_0$-tensors?
{\bf Question 2}: Can we evaluate the SOS-rank of symmetric $Z$-tensors?
{\bf Question 3}: Can we evaluate the SOS-rank of symmetric diagonally dominated tensors?
{ {\bf Question 4}: Can we use the techniques in Section 5 to find the minimum $H$-eigenvalue
of an even order symmetric structured tensors other than the extended $Z$-tensors? }
\bigskip
\noindent
{\bf Acknowledgment} We are thankful to Prof. Man-Duen Choi, Prof. Changqing Xu, Dr. Xin Liu and Dr. Ziyan Luo for their comments, which improved our paper.
|
{
"timestamp": "2015-10-13T02:07:06",
"yymm": "1504",
"arxiv_id": "1504.03414",
"language": "en",
"url": "https://arxiv.org/abs/1504.03414"
}
|
\section*{Results and discussion}
\subsection*{Experiment}
\begin{figure}[]
\centering
\includegraphics[width = 8.5 cm]{Fig1}
\caption{\textbf{Comparison of the critical current density and the defect density.}
The critical current density (solid line) and the defect density (symbols) of
the vortex lattice of the irradiated (F$_\text{n}$ = 3 $\times$ 10$^{21}$\,m$^{-2}$)
NbSe$_2$ single crystal are almost directly proportional from 0.14 to 0.9\,T at 4.2\,K.
\label{fig1}}
\end{figure}
The experiments were carried out on NbSe$_2$ single crystals (see
methods). The critical current density of the as-grown (unirradiated) samples had just one maximum at a low magnetic field and declined rapidly with
increasing field. Introducing material defects, capable of pinning vortices, by
exposing the samples to neutron irradiation made the current density increase
and, in addition to the low-field maximum, a second current peak appear at high fields, called the second peak effect (see figure~\ref{fig1} and Supplementary section I). It should be noted that the samples have a finite thickness of some 100\,$\mu$m, along which vortices can be bent, making the vortex matter system more three-dimensional. Double-sided decoration measurements revealed rather stiff vortices in NbSe$_2$ \cite{Mar98a}.
Mapping the density of states of the sample surface, scanning tunneling
spectroscopy reveals the locations of the vortices (see methods).
To get the same history-dependent vortex matter state as with the macroscopic current measurements, it was necessary to first set the temperature to 4.2\,K and then apply the magnetic field. This procedure leads to a small gradient in the macroscopic vortex density proportional to the critical current, establishing a corresponding driving force on the vortices
\cite{Bra95a}.
\begin{figure*}[]
\centering
\includegraphics[width = 0.95\textwidth]{Fig2}
\caption{\textbf{The order-disorder transition of the vortex lattice in NbSe$_2$ at
4.2\,K.} The vortex lattice of NbSe$_2$, shown at 0.16, 0.18, 0.19, 0.20, 0.25,
and 0.50\,T (cp. with figure~\ref{fig1}), becomes more disordered with increasing field. Each thin line indicates
a bond between two adjacent vortices, the large (colored) symbols indicate
lattice defects, namely vortices with four ({\color{violet} $\circ$}), five
({\color{red} $\circ$}), seven ({\color{blue} $\bullet$}), or eight
({\color{green} $\bullet$}) nearest neighbors. The horizontal bars correspond to a length of 1\,$\mu$m. The image boundaries were not used for the evaluation. \label{fig2}}
\end{figure*}
\subsection*{The vortex lattice and the critical current density}
Figure~\ref{fig2} shows the vortex lattice of the sample of figure~\ref{fig1} at
several magnetic fields from 0.16\,T, which is close to the minimum of the
current, to 0.50\,T, which is deep in the second peak region. Here
each line connects two adjacent vortices, hence each intersection marks the
position of a vortex. As each vortex has six nearest neighbors in a
hexagonal lattice, a lattice defect is defined as a vortex having more or
less than six neighbors and is highlighted by a large colored symbol in the figures.
Figure~\ref{fig2} makes it manifest that the disorder, reflected
by the lattice defect density, increases with field. Figure~\ref{fig1} shows that
the lattice defect density is almost directly proportional
to the macroscopic critical current density from about 0.14 to 0.9\,T.
Such a relation is not a-priori expected and was indeed not observed at higher fields. The
microscopic vortex lattice and the macroscopic currents were determined in the same sample.
\subsection*{Defects in the vortex lattice and the order-disorder transition}
Different kinds of defect \cite{Gru07a} appear in the two-dimensional
vortex lattice images. A single defect, having no defect as a direct neighbor, is called
a disclination. Two adjacent defects from which one has five and the other seven neighbors (shown by a red-blue pair in the figures) indicate a dislocation. Two connected dislocations form a dislocation pair. Finally, also larger defect clusters show up.
The most established theoretical explanation of how a two-dimensional particle
system undergoes an order-disorder transition was worked out by Kosterlitz,
Thouless, Halperin, Nelson, and Young (KTHNY \cite{Kos73a,Nel79a,Hal78a,You79a}). In the ordered state, characterized by a quasi-long-range translational and a long-range orientational order, defects appear only in the form of dislocation pairs. Pushing the
system towards the disordered state first generates the so-called hexatic state by dissolving the pairs into single dislocations, making the translational order short-range. The final disordered state is dominated by disclinations, making also the orientational order short-range. Alternatively, Chui \cite{Chu83a} proposed that
the dislocations form grain boundaries instead of dissolving into disclinations. Both scenarios were backed by experiments and simulations.
Previous measurements \cite{Hec14a} revealed a rather well ordered defect-free vortex lattice in an unirradiated clean NbSe$_2$ single crystal at fields where the critical current vanished. Magnetic decoration images by Kim et al.\cite{Kim99a} showed that defect-free regions may contain more than 10$^5$ vortices in clean superconductors. In contrast, several dislocation pairs and a small number of single dislocations showed up in a sample irradiated to a low neutron fluence (F$_\text{n}$ = 10$^{21}$\,m$^{-2}$) at fields between the two peaks where the critical current was also close to zero (Supplementary section II).
\subsection*{The different stages of the order-disorder transition}
\begin{figure}[]
\centering
\includegraphics[width = 8.5 cm]{Fig3}
\caption{\textbf{Histograms of the defect cluster size.} The panels show the number of
defect clusters per 1000 vortices as a function of the cluster size of all
lattices of figure~\ref{fig2}. \label{fig3}}
\end{figure}
Now I come to the details of figures~\ref{fig2} and ~\ref{fig3}, which refer to
a higher irradiated (F$_\text{n}$ = 3 $\times$ 10$^{21}$\,m$^{-2}$) sample, whose
current, shown in figure~\ref{fig1}, does not vanish between the peaks. At
0.16\,T, a field close to the minimum of the critical current, several dislocation pairs and a few single dislocations appear. To compare lattices with
different sizes, I normalized the number of defects to a lattice with 1000
vortices. This gives 13.5 dislocation pairs and 4.5 single dislocations
(figure~\ref{fig3}). No other defects are visible. Note that all dislocation
pairs form loops with alternating coordination numbers between adjacent
defects, thus minimizing the Burgers vector \cite{Gru07a} and the corresponding defect energy, and destroying the order merely locally. It is reasonable that dislocation pairs dominate at low
disorder, for they can easily be created just by locally displacing the involved
vortices. In contrast, single dislocations are topological defects, which cannot
be created by such simple transformations \cite{Gru07a}, but two are obtained by splitting a
dislocation pair, or they are introduced at the sample edges. Likewise, dissolving a dislocation leads to two disclinations. The pairs also dominate in the lower irradiated sample (F$_\text{n}$ = 10$^{21}$\,m$^{-2}$) at fields between the two current peaks and also form loops with alternating coordination number (Supplementary section II), but even here single dislocations exist. Some of these single dislocations may be remnants of the more disordered state at low fields, where the first current peak appears. Note that some vortices may be pinned strongly enough to maintain their position during the magnetization process.
Increasing the field to 0.18\,T leads to merely minor modifications. There are still more
pairs (10.4) than single dislocations (3.2), but, for the first time, also a
larger defect cluster, consisting of six vortices, emerges. Moreover, some of
the pairs no longer exist as loops but form lines that are not closed, leading to rather
large Burgers vectors and defect energies. This may be the first stage of
a process that dissolves the pairs into single dislocations. It also becomes apparent that the defects are not homogeneously distributed in the lattice but rather form chains.
At 0.19\,T the changes are more fundamental. In contrast to the low-field
results, the number of single dislocations (13) exceeds that of the pairs (5) (figure~\ref{fig3}). Most single dislocations are located close to another one,
indicating that they have just been separated. Additionally, larger clusters
with up to twelve vortices and one disclination, located, however, at the image
boundary and hence presumably being part of a larger cluster, show up. Apart from this
disclination, all clusters consist of an even number of vortices.
With increasing field, the basic trends, observed at 0.19\,T, carry on.
The number of single dislocations and pairs does not
change considerably, while the size of the larger clusters grows. The number of
clusters decays roughly exponentially with their size (figure~\ref{fig3}).
Disclinations emerge more frequently though do not become dominant.
Going to 0.2\,T slightly increases the critical current, though the field is still close to the onset of the second peak regime. Here, the formation of grain boundaries, consisting of defect clusters, becomes manifest. At 0.25\,T the critical current has increased by more than a factor of two and the disorder has grown accordingly. The defect cluster chains become more bulky though defect-free grains are still visible.
Finally, at 0.5\,T, which is deep in the second peak region, the lattice is quite disordered. The defect density is about one third, and for the first time, vortices with more than seven or less than five nearest neighbors emerge. The defect amalgamations are no longer line-shaped but
rather bulky, yet defect-free vortex grains are still present. A further increase of the magnetic field to 1.6\,T basically raises the disorder but does not alter the nature of the vortex lattice structure.
Although the finite sample thickness leads to a more three-dimensional character of the vortex lines, some similarities with KTHNY theory are striking. In particular I observed in accordance with KTHNY theory that in the most ordered lattice, dislocation pairs dominate, which first form closed loops, so that the order is only locally destroyed, and later also become elongated. Then, single dislocations, destroying the quasi-long-range translational order, appear more frequently. But finally, in contrast to KTHNY theory, which predicts disclocations to dissolve into disclinations, though in agreement with Chui's proposal \cite{Chu83a}, the dislocations become clustered and form a grain boundary structure getting more bulky with increasing disorder.
\subsection*{Correlation functions}
\begin{figure}[]
\centering
\includegraphics[width = 8.5 cm]{Fig4}
\caption{\textbf{The correlation functions of the vortex lattices.} The symbols show
the (upper) envelope of the orientational (left panel) and of the translational
(right panel) correlation functions of the unirradiated sample at 1\,T
and of the irradiated sample of figure~\ref{fig2} at different
magnetic fields. The fit functions (solid lines) follow either a power law (red)
or an exponential (blue) decay behavior. \label{fig4}}
\end{figure}
Correlation functions offer additional insight into the ordering state of our
systems (see methods). The translational correlation function measures the
translational symmetry of the system and the bond-orientational correlation
function the relative orientation of two bonds between nearest neighbors as a
function of distance. Both functions change from one for a perfectly ordered to
zero for a completely disordered state.
Figure~\ref{fig4} shows these functions for our systems. The highest order is
found in the defect-free lattice of the unirradiated sample. Its slightly
decaying though quasi-long-range translational order and its virtually constant
orientational order agree with the properties of a Bragg glass \cite{Gia95a,Kle01a}.
In the disordered lattices, that is, at and above 0.5\,T, both functions decay
exponentially for not too large distances and have similar correlation lengths
of 1-2 lattice parameters. This would match the vortex glass behavior, though
the lattice at 0.5\,T is not amorphous. The flat long-range behavior
reflects the fact that the mean orientation of different vortex grains does not
change significantly, presumably an aftermath of the ordered state that had been run through during applying the field at a constant temperature. Additionally, the finite sample thickness may also contribute to the unusual long-range behavior. At higher fields the translational correlation does not change significantly, while the orientational correlation further declines.
Near the onset of the second peak, for instance when the field changes from 0.18 to 0.20\,T,
the orientational order remains almost unchanged and is quasi long-range, whereas the translational order decreases appreciably and has a much shorter correlation length in both cases. This is what one expects of the hexatic state \cite{Nel79a}. The observation of the hexatic state, associated with the appearance of free dislocations, was also reported for the melting process of the vortex lattice in a superconducting tungsten-based thin film \cite{Gui09a}. The hexatic phase in a three-dimensional vortex lattice was discussed in Ref.~\onlinecite{Mar90a}.
\subsection*{Further discussion}
In conclusion, the order-disorder transition of a superconductor's flux-line lattice follows reasonably well the theoretical predictions of the grain boundary scenario. While in most cases, the systems are cooled down from a disordered state, our system reached the final state by increasing the magnetic field at a constant temperature and thus always first passed the low-field disordered state and then the most ordered state at the onset of the second peak. But only this procedure makes a comparison of the microscopic vortex lattice with the macroscopic critical current useful. Moreover, it is a change in temperature that evokes the transition in most experiments, whereas it is a change in magnetic field in our case. Increasing the magnetic field increases the number of vortices and thus reduces the distances between them, as it is the case when pressure is applied to a conventional system.
The experiments confirm that the transition is induced mainly by pinning and not by thermal effects, since no high-field disordered state, at least below 1.7\,T, which is 85 per cent of the upper critical field, was observed in the unirradiated sample, in which vortex pinning is weak. Moreover, the disordered vortex lattice configurations did not change significantly with time, as was found from measuring the same sample area several times in immediate succession. This agrees with the common assumption that the order-disorder transition is a solid-to-solid transition \cite{Mik01a,Kie04a}, at least in low-temperature superconductors. It was shown theoretically \cite{Mik01a,Kie04a} that in samples with a very small Ginzburg number, such as in NbSe$_2$ (10$^{-5}$ - 10$^{-6}$), thermal effects are not expected to become relevant at fields lower than some 90 per cent of the upper critical field, which is far from our transition field. Pinning effects extend the region where thermal energy matters, but the predicted shift of the melting line was found to be small for NbSe$_2$ even when pinning effects are strong \cite{Mik03a}.
The assumption of a two-stage nature of the order-disorder transition in vortex matter was recently confirmed by two other scanning tunneling spectroscopy studies \cite{Gui14a,Gan14a}. In the first, Guillamon et al.\cite{Gui14a} investigated a tungsten-based superconducting thin film, in which the order-disorder transition was generated by a modulation of the sample thickness. No lattice defects were found at low magnetic fields. In the first stage of the transition, pair and single dislocations showed up, and in the second stage, the dislocation density grew strongly and free disclinations were created. The translational and orientational correlation functions revealed good agreement with KTHNY theory.
Ganguli et al. \cite{Gan14a} analyzed a cobalt-doped NbSe$_2$ single crystal having a second peak effect at high magnetic fields. They also observed a proliferation of dislocations in the first stage of the transition and the appearance of single disclinations, driving the lattice into an amorphous glass, in the second stage. They showed that the increasing and the decreasing field branch led to similar vortex structures, but differences were observed in the long-range behavior of the correlation functions at fields in the transition region, which confirms the assumption suggested above that the deviations of the correlation functions from KTHNY theory for long distances may be explained by the specific history of the magnetization process. Cooling the sample at a constant field led to the expected higher disorder in the lattices.
Though all three studies confirm the two-step nature of the transition, a detailed comparison reveals some remarkable differences, which may be caused by different material properties and sample thicknesses, leading to a more three- (Ref.~\onlinecite{Gan14a} and this work) or a more two-dimensional (Ref.~\onlinecite{Gui14a}) character of the vortex matter. First, in the present work, I found the defect structure of the rather ordered lattice to be dominated by dislocation pairs, which agrees with the prediction of KTHNY theory. In contrast, merely single dislocations showed up in Ref.~\onlinecite{Gan14a} at this stage. Second, I found the defects to form a grain boundary structure, getting more bulky with increasing field. Such a structure was not reported by the other groups, though several vortex lattice images of Ref.~\onlinecite{Gan14a} might indicate an inhomogeneous defect distribution.
Vortex lattice defects concentrated mainly in grain boundaries were reported by Fasano et al. \cite{Fas02a,Fas08a} in NbSe$_2$. These results were, however, obtained at very low magnetic fields, namely at 3.6\,mT, which was not in the second peak regime but close to the first critical current peak. Employing magnetic decoration, they were able to assess large sample areas and thus to detect vortex grains including several hundred vortices in both field-cooled and zero-field-cooled experiments. Based on the assumption that the field-cooled lattices had been frozen in close to the upper critical field in the second peak region, they concluded that the vortex lattice disorder does not correlate with the macroscopic current in the second peak region, which is in contrast to the proportionality found in this work. However, the proportionality between current and lattice disorder was only shown for fields between the onset and the maximum of this region in this work, while the frozen lattice of Ref.~\onlinecite{Fas02a} would refer to the high-field or high-temperature end of this region, where the correlation can be quite different. A grain boundary structure was also found in simulations \cite{Mig03a,Das03a,Cha04a,Mor05a,Mor09a} of the two-dimensional vortex matter. Other two- or quasi-two-dimensional systems such as electron sheets \cite{Gri79a}, dusty plasmas \cite{Tho96a,Nos09a}, colloidal \cite{Mur87a,Mar96a,Deu13a} and other systems \cite{Str88a,Gru07a} were found to become disordered by the same or a similar two-stage process, though only some of these systems developed a grain-boundary structure.
\section*{Methods}
\subsection*{Samples} The material used for this study is NbSe$_2$, which becomes
superconducting below some 7.2\,K. All experiments were carried
out at 4.2\,K with the magnetic field parallel to the uniaxial axis of the single crystal's hexagonal unit cell, for which the upper critical field is some 2\,T.
Recently, NbSe$_2$ was shown to be a two-band superconductor similar to
MgB$_2$ \cite{Zeh10a}. Our samples had a size of 1-3\,mm in lateral direction and 100 -
200\,$\mu$m parallel to the uniaxial axis.
\subsection*{Neutron irradiation} Neutron irradiation of the samples took place
in a TRIGA MARK II research fission reactor \cite{Web86a}. This procedure introduces crystal defects that are efficient for vortex pinning
in many superconductors. For instance, in MgB$_2$ \cite{Mar08a} and YBa$_2$Cu$_3$O$_{7-\delta}$ \cite{Fri94a}, transmission electron microscopy revealed spherical defects with a diameter close to the superconducting coherence length. Neutron irradiation tremendously strengthens vortex pinning also in NbSe$_2$, as shown by
its effects on the critical current (Supplementary section I). One sample was irradiated to a fast neutron ($E > 0.1$\,MeV) fluence of 1 $\times$ 10$^{21}$\,m$^{-2}$ and the sample to which most results of this article refer to 3 $\times$ 10$^{21}$\,m$^{-2}$.
\subsection*{Critical current density} The critical current density was obtained
from SQUID magnetometry by measuring the magnetic moment as a function of
magnetic field at 4.2\,K. Assuming the current to be constant in the sample, one can derive the
critical current from the hysteresis of the magnetic moment in
increasing and decreasing field \cite{Zeh09a}. The unirradiated samples showed
a finite though small critical current only at low fields. A low neutron fluence
enhanced the current significantly at low fields and additionally produced a
second peak close to the upper critical field. With increasing fluence the
low-field current became larger and the second peak more prominent.
\subsection*{Scanning tunneling spectroscopy}
Scanning tunneling spectroscopy reveals the electronic density of states at the
sample surface. In a superconductor the procedure produces
the well-known energy gap of the quasi-particles near the Fermi level. This gap,
however, does not exist in the vortex core, and hence the tunneling signal changes,
thus identifying the vortex core positions, which makes scanning tunneling spectroscopy the sole known method for measuring the vortex lattice in real-space at arbitrary magnetic fields \cite{Hes89a,Iav08a,Han12a,Hec14a,Fis07a,Sud14a}.
For the measurements, a sample was cleaved in air and then put into the sample tube, which was first evacuated and then flooded with pure helium
gas. The measurements were carried out with a PtIr tip. Having placed the sample in a helium cryostat, I cooled it to 4.2\,K and then applied a magnetic
field. Then, after waiting for several hours, the surface topography was mapped by a constant-current mode measurement and, at the same time, the density of states at an energy level slightly
above the Fermi level but below the energy gap width of some 2\,meV by a lock-in
technique (Supplementary section III). The densities of state maps usually showed a clear
contrast for the vortex core positions (Supplementary section III). Applying simple filters such as a
Gaussian filter to the raw data reduced the noise, so that the vortex core
positions were easily acquired by seeking all local maxima. The nearest
neighbors of each vortex were determined by a Delaunay triangulation.
\subsection*{Correlation functions}
The translational correlation function \cite{Nel79a} is defined as
\begin{equation}
G_\text{T} (|\vec{r} - \vec{r}'|) = \langle e^{i \vec{G} \vec{u}} e^{-i
\vec{G} \vec{u}'} \rangle
\end{equation}
where $\vec{G}$ is a reciprocal lattice vector, and $\vec{u}$ is the
displacement of a vortex at $\vec{r}$ from its lattice position $\vec{R}$, that
is, $\vec{u}$ = $\vec{r} - \vec{R}$ and $e^{i \vec{G} \vec{R}} = 1$. The brackets
indicate an averaging over all vortex pairs. The reciprocal lattice vector was
determined by seeking the vector with the largest value near the boundary of the first Brillouin zone after a Fourier transformation of the lattice. The
distances were discretized into interval lengths of $0.1\,a_0$.
For the bond orientational correlation function \cite{Nel79a}, one has to calculate
\begin{equation}
G_6 (|\vec{r} - \vec{r}'|) = \langle b_6(\vec{r})~ b_6^\star
(\vec{r}')\rangle
\end{equation} with
\begin{equation}
b_6(\vec{r}) = \frac{1}{n}\sum_{j=1}^n e^{i6\phi(\vec{r},\vec{r}_\text{j})}.
\end{equation}
The sum runs over all nearest neighbors j of the vortex at $\vec{r}$, and
$\phi(\vec{r},\vec{r}_\text{j})$ is the angle of the bond connecting the two
vortices with respect to the $x$-axis.
|
{
"timestamp": "2015-04-14T02:13:27",
"yymm": "1504",
"arxiv_id": "1504.03123",
"language": "en",
"url": "https://arxiv.org/abs/1504.03123"
}
|
\section{Introduction}
In the last decades, far-field fluorescence microscopy has experienced great advances with applications in medicine and biology, among other fields, offering the possibility to obtain high resolution images in a non-invasive manner. In lens-based light microscopes, the image of a point object obtained from the fluorescence emitted by a fluorophore placed in the sample is, in principle, limited by diffraction \cite{Abbe'73}.
This image becomes a finite-size spot in the focal plane, mathematically described by the point-spread function (PSF), whose full width at half maximum (FWHM) is $\lambda/(2\,\rm NA)$, being $\lambda$ the wavelength of the addressing light, and $\rm NA$ the numerical aperture of the objective. Due to the reduced dimensions of the samples to investigate, frequently around few nanometers, high resolution images overcoming the diffraction limit are necessary.
To this aim, one of the most extended group of techniques is based on the general concept of reversible saturable optical fluorescence transition (RESOLFT) \cite{Hell'03} between two distinguishable molecular states, such as stimulated-emission-depletion (STED) \cite{Hell'94} and ground-state-depletion (GSD) \cite{Hell'95}. RESOLFT exploits the spatial inhibition of the fluorescence signal from a light-excited fluorophore in order to engineer the effective PSF, reducing its FWHM.
Parallely, in recent years, there has been an intense activity in the spatial localization of atomic population in $\Lambda$-type systems coherently interacting with two electromagnetic fields (see e.g., \cite{Agarwal'06,Gorshkov'08}). All these methods are based on the so-called coherent population trapping (CPT) technique or close variations, and have also been considered for microscopy \cite{Yavuz'07, Kapale'10,Li'08}).
In the context of atomic population localization, some of us have recently proposed the subwavelength localization via adiabatic passage (SLAP) technique~\cite{Mom'09}.
In SLAP, position-dependent stimulated Raman adiabatic passage (STIRAP) \cite{Bergmann'98} is performed using spatially dependent fields. This approach provides population peaks narrower than using other coherent localization techniques \cite{Agarwal'06,Yavuz'07} and its application in nanolitography and patterning of BEC's \cite{Mom'09}, and single-site addressing of ultracold atoms \cite{Vis'12}, has already been discussed.
In this work, we propose a SLAP-based nanoscale resolution technique for fluorescence microscopy.
At variance with respect to \cite{Mom'09, Vis'12}, two driving laser pulses are applied here to a continuous distribution of emitters, and and we take into account the effect of the objective lens in our model.
In section \ref{sec:Model}, we describe the physical system under consideration, derive an expression for the resolution, and compare it with the one obtained for the CPT case. In section \ref{sec:NumericalResults}, we show numerical results of the SLAP proposal and compare it with the STED technique. Finally, we present the conclusions and point out a possible implementation of the proposed technique using quantum dots.
\section{Model}
\label{sec:Model}
We consider a $\Lambda$ scheme, where the population is initially in \ensuremath{\left|1\right\rangle}, see top of Fig. \ref{f:fig1}(a).
Two laser pulses, Stokes (S) and pump (P), couple to transitions \ensuremath{\left|3\right\rangle}$\leftrightarrow$\ensuremath{\left|2\right\rangle} and \ensuremath{\left|1\right\rangle}$\leftrightarrow$\ensuremath{\left|2\right\rangle} with Rabi frequencies $\Omega_S(x,t)\equiv \ensuremath{\bm{\mu}}_{32} \cdot{\ensuremath{\bm{E}}_{\rm S}}(x,t)/\hbar$ and $\Omega_P(x,t)\equiv\ensuremath{\bm{\mu}}_{12}\cdot{\ensuremath{\bm{E}}_{\rm P}}(x,t)/\hbar$, respectively, where ${\ensuremath{\bm{E}}_{\rm S}}$ (${\ensuremath{\bm{E}}_{\rm P}}$) is the electric field amplitude of the Stokes (pump), $\ensuremath{\bm{\mu}}_{32}$ ($\ensuremath{\bm{\mu}}_{12}$) is the electric dipole moment of the \ensuremath{\left|3\right\rangle}$\leftrightarrow$\ensuremath{\left|2\right\rangle} (\ensuremath{\left|1\right\rangle}$\leftrightarrow$\ensuremath{\left|2\right\rangle}) transition, and $\hbar$ is the reduced Planck constant. The excited level \ensuremath{\left|2\right\rangle} has a decay rate $\gamma_{21}$ ($\gamma_{23}$) to the ground state \ensuremath{\left|1\right\rangle} (\ensuremath{\left|3\right\rangle}). If the two-photon resonance condition is fulfilled, one of the eigenstates of the Hamiltonian, the so-called dark-state, takes the form:
\begin{align}\label{eq:darkstate}
\ket{D(x,t)}=[\Omega_{\rm S}^{*}(x,t)\ket{1}-\Omega_{\rm P}^{*}(x,t)\ket{3}]/\Omega(x,t),
\end{align}
where $\Omega(x,t)=(|\Omega_{\rm P}(x,t)|^2+|\Omega_{\rm S}(x,t)|^2)^{1/2}$.
Note that $\ket{D(x,t)}$ does not involve the excited state \ensuremath{\left|2\right\rangle}.
In our scheme, and in order to adiabatically follow the dark state, both pulses are sent in a counterintuitive temporal sequence, applying first the Stokes and, with a temporal delay $T$, the pump [see bottom of Fig. \ref{f:fig1}(a)]. In addition, to ensure no coupling between the different energy eigenstates, a global adiabatic condition must be fulfilled, which imposes $\Omega(x)\,T\geq A$, where $A$ is a dimensionless constant that for optimal Gaussian profiles and delay times takes values around 10 \cite{Bergmann'98}. In the SLAP technique, the pump pulse has a central node in its spatial profile such that the described temporal sequence of the pulses produces an adiabatic population transfer from \ensuremath{\left|1\right\rangle} to \ensuremath{\left|3\right\rangle}, except at the position of the pump node. Therefore, we obtain a narrow peak of population remaining in \ensuremath{\left|1\right\rangle}, whose profile can be considered as the PSF function referred to an image object in a lens-based microscope. Note that the FWHM of the population peak determines the resolution of the technique. Finally, an exciting E pulse is used to pump the population remaining in \ensuremath{\left|1\right\rangle} either to state \ensuremath{\left|2\right\rangle} or to an auxiliary excited state, allowing for the subsequent registration of the fluorescence.
If state \ensuremath{\left|2\right\rangle} decays radiatively, the pump (P) could be used also as the exciting (E) pulse, reducing the experimental requirements. Figure \ref{f:fig1}(b) shows a possible setup for the proposal.
Right to left, the Stokes, pump and exciting pulses are sent, and we assume that due to the Stokes's shift, the fluorescence (left arrow) can be separated from the light of the different pulses by, e.g., dichroic mirrors (DM).
\begin{figure}
\includegraphics[width=0.8\columnwidth]{fig1a.pdf}
\includegraphics[width=0.8\columnwidth]{fig1b.pdf}
\caption{
(a) SLAP technique. Top: $\Lambda$-system with pump (P) and Stokes (S) pulses coupling the \ensuremath{\left|1\right\rangle}$\leftrightarrow$\ensuremath{\left|2\right\rangle} and \ensuremath{\left|3\right\rangle}$\leftrightarrow$\ensuremath{\left|2\right\rangle} transitions, respectively.
The decay rate from \ensuremath{\left|2\right\rangle} to \ensuremath{\left|1\right\rangle} (\ensuremath{\left|3\right\rangle}) is $\gamma_{21}$ ($\gamma_{23}$). Bottom: Pulse temporal sequence, being E the exciting pulse. (b) Schematic setup of the SLAP-based fluorescence microscopy. Note the central node of the pump pulse. DM accounts for dichroic mirrors.}
\label{f:fig1}
\end{figure}
\begin{figure}[htbp]
\centering\includegraphics[width=1\columnwidth]{fig2.pdf}
\caption{
Ratio between the FWHM using SLAP and CPT techniques as a function of $R$, for $k=0.1$ (blue solid line), $k=0.4$ (green dashed line), and $k=0.9$ (red dotted line).}
\label{f:fig2}
\end{figure}
To take into account the effect of the objective lens in our model, we consider the Rabi frequency of the Stokes pulse having a Bessel beam spatial profile, and the one of the pump being the result of superimposing two Bessel beams focused with a lateral offset, producing a node in its center. The spatio-temporal profiles of the Stokes and pump fields read:
\begin{align}
\Omega_{\rm S}(\upsilon,t)&=\Omega_{\rm S0}F(\upsilon,0)\,e^{-(t-t_{\rm S})^2/\sigma^2}, \label{eq:fields1} \\
\Omega_{\rm P}(\upsilon,t)&=\Omega_{\rm P0}[F(\upsilon,\delta)+F(\upsilon,-\delta)]\,e^{-(t-t_{\rm P})^2/\sigma^2},\label{eq:fields2}
\end{align}
where $\Omega_{\rm S0}$ ($\Omega_{\rm P0}$) is the peak Rabi frequency of the Stokes (pump) pulse, $F(\upsilon,r)\equiv\frac{2J_{1}(\upsilon + r)}{\upsilon+r}$, where $J_{1}(\upsilon)$ is the first order Bessel function and $\upsilon=(2\pi x \rm NA)/\lambda$ is the optical unit corresponding to the Cartesian coordinate $x$ in the focal plane, $\sigma$ is the temporal width of the pulses, $T=t_{\rm P}-t_{\rm S}$ is the temporal delay between the pulses, which is proporcional to $\sigma$,
and $\delta=1.22\pi$ is the offset with respect to $\upsilon=0$ corresponding to the first cutoff of $F(\upsilon,0)$.
It is possible to obtain an analytical expression for the FWHM of the final population peak in \ensuremath{\left|1\right\rangle}, $p_1(x)$, by considering that the global adiabaticity condition is reached for $\left|\upsilon\right|\approx\rm FWHM$, assuming a Gaussian population peak profile and considering $\left|\upsilon\right|\ll\delta$. Thus, from the spatial profiles in Eqs. (\ref{eq:fields1})-(\ref{eq:fields2}), the FWHM of the population distribution is
\begin{align}\label{eq:FWHM-SLAP}
{\rm FWHM_{SLAP}}=\frac{\lambda}{2\rm NA}\frac{\delta}{\pi}\left(\sqrt{\frac{4R}{k^{-2}-1}}+1\right)^{-1/2},
\end{align}
where $R\equiv(\Omega_{\rm P0}/\Omega_{\rm S0})^2$ is the intensity ratio between the pump and the Stokes pulses, and $k\equiv\Omega_{\rm S0}T/A$ must fulfill $0<k<1$.
In such a SLAP-based fluorescence microscope, the effective PSF is given by the product
$h_{\rm exc}(\upsilon)\,p_1(\upsilon)$ normalized to 1, where $h_{\rm exc}(\upsilon)$ is the PSF of the E pulse. If we assume that all the localized population leads to fluorescence, the lateral resolution is determined by Eq. (\ref{eq:FWHM-SLAP}), where the first factor accounts for diffraction, the second one is 1.22, and the third one can be less than 1 depending on the adiabaticity of the process, and tends to zero in the limit $R\rightarrow\infty$, i.e., when $\Omega_{\rm P0}\gg\Omega_{\rm S0}$.
Other dark-state techniques proposed to obtain atomic localization \cite{Agarwal'06,Yavuz'07} are based on the so-called coherent population trapping (CPT) or close variations, in which the fields can be either two continuos waves or two perfectly overlapping long pulses. In this case, to obtain the analytical expression for the FWHM of the final population peak in state \ensuremath{\left|1\right\rangle}, we have considered, as in~\cite{Agarwal'06}, that $|\left\langle 1|D(\upsilon) \right\rangle|^2=1/2$ for $\upsilon=\rm FWHM/2$ .
Using the profiles given in (\ref{eq:fields1}) and (\ref{eq:fields2}), the FWHM for CPT is
\begin{align}\label{eq:FWHM-CPT}
{\rm FWHM_{CPT}}=\frac{\lambda}{2\rm NA}\frac{2\delta}{\pi}\left(\sqrt{2\sqrt{R}+1}\right)^{-1/2}.
\end{align}
Figure~\ref{f:fig2} shows the ratio between the analytical FWHM obtained for SLAP [Eq. (\ref{eq:FWHM-SLAP})] and CPT [Eq. (\ref{eq:FWHM-CPT})] as a function of $R$ for different values of $k$. From the figure, we can see that in the whole range of parameters considered, the peak obtained with SLAP is significantly narrower than the one with CPT. Note that the adiabatic nature of the SLAP technique allows to increase the final resolution by increasing the time delay $T$, while fixing the intensities.
\section{Numerical results}
\label{sec:NumericalResults}
\begin{figure}
{
\includegraphics[width=0.6\columnwidth]{fig3a.pdf}
\includegraphics[width=1\columnwidth]{fig3bc.pdf}
}
\caption{(a) STED technique. Top: Fluorophore energy levels interacting with the exciting (E) and the depletion (D) fields, and their decay rates. Bottom: Pulse temporal sequence. (b) Analytical (solid line) and numerical (crosses) values for the FWHM of the population peak in \ensuremath{\left|1\right\rangle}, using SLAP, as a function of $R$. Inset: Spatial profiles of pump (P) and Stokes (S) Rabi frequencies for $R=150$. (c) Numerical results of the population $p_1(x)$ in \ensuremath{\left|1\right\rangle} for SLAP using different values of $R$, and the population $n_1(x)$ in the first excited vibrational level of Rhodamine B for STED typical values (see text).}
\label{f:fig3}
\end{figure}
In the following, we are interested in comparing our scheme with the STED microscopy technique {[see Fig.~\ref{f:fig3}(a)]. In the STED technique, two beams, the exciting (E) and the depletion (D) lasers, interact with an organic fluorophore, with a time delay $\Delta t$. Typical values for $\Delta t$ are some hundreds of ps, larger than the vibrational relaxation time $\tau$ ($\sim$ps) but much shorter than the fluorescence time $\tau_{\rm fl}$ ($\sim$ns) of the transition $n_1\rightarrow n_0$. First, the E field excites all the population to state $n^{\rm vib}_1$, which rapidly decays to $n_1$. Next, the D field, which has a doughnut-like spatial profile, produces a spatial depletion of the population in $n_1$ by stimulated emission. Out of the node, the excited population is removed resulting in fluorescence inhibition, and reducing the width of the effective PSF. Note here that the main distinctive feature of SLAP with respect to general RESOLFT techniques, e.g., STED, is the adiabatic nature of the state transfer process, which as discussed below, confers robustness and flexibility on our method.
In Fig.~\ref{f:fig3}(b), values for the FWHM of the final population peak obtained from numerical simulations using SLAP (crosses) and the analytical curve (solid line) given by Eq. (\ref{eq:FWHM-SLAP}) using $A=20$ are represented.
For the simulations, we have used the density-matrix formalism for a $\Lambda$-system with degenerated ground states with the following parameter setting: $\gamma_{21}=\gamma_{23}=2\pi\times 6.36$ GHz, $\Omega_{\rm S0}/\gamma_{21}=1.5$, $\sigma=100$ ps, $T=1.5\sigma$, $\rm NA=1.4$, and $\lambda=490$ nm.
Figure~\ref{f:fig3}(c) shows the final population $p_1(x)$ in \ensuremath{\left|1\right\rangle} using the SLAP technique for $R=10$ (dashed line), $R=50$ (dotted line) and $R=300$ (dotted-dashed line), marked with vertical arrows in Fig.~\ref{f:fig3}(b). In addition, the population $n_1(x)$ using the STED technique (black solid line) is also shown. For the STED simulation, we have used rate equations for the Rhodamine B dye with intensity profiles of the exciting and depletion pulses corresponding to Eq.~(\ref{eq:fields1}) and Eq.~(\ref{eq:fields2}), respectively, and typical values \cite{Hell'94} for the absorption cross sections $\sigma_{\rm cs}=10^{-17}$ cm$^2$, peak intensity of the depletion laser $h_{\rm D}^{\rm peak}=1300$ MW/cm$^2$, $\tau=1$ ps, $\tau_{\rm fl}=2$ ns, $\Delta$t $=90$ ps, $\sigma=100$ ps, $\rm NA=1.4$, $\lambda_{\rm E}=490$ nm, and $\lambda_{\rm D}=600$ nm, obtaining $\rm FWHM=65.2$ nm.
Note that the final peak in STED does not reach unity due to the loss of population by fluorescence while depletion acts. This does not occur in SLAP, since the final peak corresponds to the population in a ground state.
\section{Conclusions and perspectives}
In conclusion, we have presented a proposal to implement nanoscale resolution microscopy using the SLAP technique.
We have derived an analytical expression to estimate the lateral resolution and we have compared it with the corresponding one for the CPT case, showing that SLAP yields a better resolution.
In both cases, the resolution can be improved by increasing the pump intensity $\propto\Omega_{\rm P0}^2$, provided $\Omega_{\rm S0}$ is kept constant. This behavior is similar in STED microscopy, whose resolution improves by increasing the intensity of the depletion laser.
Then, we have performed a numerical comparison between the STED technique using typical parameter values and the SLAP technique with Rabi frequencies of the order of GHz, obtaining a similar resolution in both cases.
All the previous results suggest that the localization via adiabatic passage may offer interesting features for fluorescence microscopy, for which coherent interaction between a $\Lambda$-type system and pump and Stokes fields would be required. In this sense, fluorescent semiconductor nano-crystals, also known as quantum-dots \cite{Alivisatos'96} (q-dots), could be good candidates to act as a diluted medium over the sample. Q-dots have excellent photostability, exhibit efficient fluorescence, can be selectively attached onto the item to study, and have been used effectively for imaging cells and tissues \cite{Michalet'05}. Further, coherent population transfer via adiabatic passage in two \cite{Hohenester'00} and three \cite{Fabian'05} coupled q-dots has been recently proposed. Q-dots have recombination times of the order of some hundreds of ps, corresponding to Rabi frequencies of GHz, and intensity of the pump beam around $10^6$ W/cm$^2$ for $\left|\ensuremath{\bm{\mu}}\right|\simeq10^{-28}$ C m \cite{Eliseev'00}. This intensity is similar to that used recently in RESOLFT-based microscopy using q-dots \cite{Irvine'07}, and up to three orders of magnitude below the one of the depletion beam used in STED microscopy with conventional fluorophores.
Nevertheless, due to the adiabatic nature of SLAP: (i) it is possible to enhance the resolution by increasing the temporal duration of the pulses, without increasing the fields intensity, thus reducing the possibility of damaging the sample, (ii) the spontaneous decay rate from \ensuremath{\left|2\right\rangle} does not play any role in the localization process, and (iii) there is no need of using resonant lasers in our localization method provided the two photon resonance condition is fulfilled. This fact should permit to use the same experimental arrangement with different types of emitters.
\section*{Acknowledgments}
We acknowledge David Artigas for fruitful comments, and funding from the Spanish Ministry of Economy and Competitiveness under Contract No. FIS2011-23719, and from the Catalan Government under Contract No. SGR2009-00347.
|
{
"timestamp": "2015-04-14T02:14:29",
"yymm": "1504",
"arxiv_id": "1504.03171",
"language": "en",
"url": "https://arxiv.org/abs/1504.03171"
}
|
\section{Introduction}
\noindent
With the motivation to provide a common framework for studying
the ultragraph $C^*$-algebras
(\cite{To1, To2}) and
the shift space $C^*$-algebras
(see \cite{Ca, CM, Ma} among others), Bates and Pask \cite{BP1}
introduced the $C^*$-algebras associated to labeled graphs (more
precisely, labeled spaces).
Graph $C^*$-algebras
(see \cite{BHRS, BPRS, KPR, KPRR, R} among many others)
and Exel-Laca algebras \cite{EL} are
ultragraph $C^*$-algebras and all these
algebras are defined as universal objects
generated by partial isometries
and projections
satisfying certain relations
determined by graphs (for graph $C^*$-algebras), ultragraphs
(for ultragraph $C^*$-algebras),
and infinite matrices (for Exel-Laca algebras).
In a similar but more complicated manner,
a labeled graph $C^*$-algebra
$C^*(E,{\mathcal{L}}, {\mathcal{B}})$ is also defined as a
$C^*$-algebra generated by partial isometries
$\{s_a:a\in {\mathcal{A}}\}$ and projections $\{p_A:A\in {\mathcal{B}}\}$,
where ${\mathcal{A}}$ is an alphabet
onto which a {\it labeling map} ${\mathcal{L}}:E^1\to {\mathcal{A}}$ is
given from the edge set $E^1$ of the directed graph $E$,
and ${\mathcal{B}}$, an {\it accommodating set},
is a set of vertex subsets $A\subset E^0$ satisfying certain conditions.
The family of these generators is assumed to obey a set of rules
regulated by the triple $(E,{\mathcal{L}},{\mathcal{B}})$ called a {\it labeled space}
and moreover it should be universal in the sense
that any $C^*$-algebra generated by
a family of partial isometries and
projections satisfying the same rules must be a quotient
algebra of
$C^*(E, {\mathcal{L}},{\mathcal{B}})$.
The universal property allows the group $\mathbb T$
to act on $C^*(E,{\mathcal{L}},{\mathcal{B}})$ in a canonical way, and
this action $\gamma$ (called the {\it gauge action})
plays an important role throughout the study of generalizations of
the Cuntz-Krieger algebras.
The Cuntz-Krieger algebras \cite{CK}
(and the Cuntz algebras \cite{C81})
are the $C^*$-algebras of finite graphs from which
many generalizations have emerged in various ways including
the $C^*$-algebras of higher-rank graphs whose study started in
\cite{KP}.
Simplicity and pure infiniteness results for
labeled graph $C^*$-algebras are obtained in \cite{BP2},
and particularly
it is shown that there exists a purely infinite simple
labeled graph $C^*$-algebra which is not
stably isomorphic to any graph $C^*$-algebras.
Thus we can say that the class of simple labeled graph $C^*$-algebras
is strictly larger than that of simple graph $C^*$-algebras.
As is shown in \cite{To2}, every simple ultragraph $C^*$-algebra
is either AF or purely infinite, whereas
we know from \cite{PRRS} that
among higher rank graph $C^*$-algebras
there exist simple $C^*$-algebras which are
neither AF nor purely infinite,
more specifically there exist such simple $C^*$-algebras which are
stably isomorphic to irrational rotation algebras or Bunce-Deddens algebras.
These examples of finite (but non-AF) simple $C^*$-algebras
associated to higher rank graphs raise a natural question
of whether
there exist labeled graph $C^*$-algebras that
are simple finite but non-AF.
The purpose of this paper is to answer this question positively
by providing a family of such simple
labeled graph $C^*$-algebras.
The $C^*$-algebras in this family are
$A\mathbb T$-algebras (limit circle algebras) with traces
that are isomorphic to
crossed products $C(X)\times_T \mathbb Z$
of Cantor minimal systems $(X,T)$,
where the compact metric spaces $X$ are subshifts
over finite alphabets.
A dynamical system $(X,T)$ consists of
a compact metrizable space $X$ and a
transformation $T:X\to X$ which is a homeomorphism.
This determines a $C^*$-dynamical system
$(C(X), \mathbb Z, T)$ where
$T(f):=f\circ T^{-1}$, $f\in C(X)$ and thus
gives rise to the crossed product
$C(X)\times_T \mathbb Z$.
If two dynamical systems $(X_i,T_i)$, $i=1,2$, are topologically
conjugate, namely if there is an homeomorphism $\phi: X_1\to X_2$
satisfying
$T_2(\phi(x))=\phi(T_1(x))$ for all $x\in X$, then
it is rather obvious that the crossed products are
isomorphic.
As a consequence of the Markov-Kakutani fixed point theorem,
one can show that there exists a Borel probability measure $m$ on $X$
which is $T$-invariant in the sense that
$m\circ T^{-1}=m$ (for example, see \cite[Theorem VIII. 3.1]{Da}).
If there exists a unique $T$-invariant measure, we call
$(X,T)$ {\it uniquely ergodic}.
If $X$ is the only non-empty closed $T$-invariant subspace of $X$, the system
$(X,T)$ is said to be {\it minimal}, and
as is well known, a dynamical system $(X,T)$ is minimal
if and only if
each $T$-orbit $\{T^ix: i\in \mathbb Z\}$, $x\in X$,
is dense in $X$.
A Cantor space is characterized as a compact metrizable
totally disconnected space with no isolated points,
and a dynamical system $(X,T)$ on a Cantor space $X$
is called a {\it Cantor system}.
The family of Cantor minimal systems is important for the study
of whole minimal dynamical systems in view of the fact that
every minimal system is a factor of a Cantor minimal system
(see \cite[Section 1]{GPS}).
For a dynamical system $(X,T)$ on an infinite space $X$,
the crossed product $C(X)\times_{T} \mathbb Z$
is well known to be simple exactly when
the system $(X,T)$ is minimal.
In particular, if $(X,T)$ is a minimal dynamical system on a Cantor space
$X$, this simple crossed product turns out to be an $A\mathbb T$-algebra,
an inductive limit of finite direct sums of
matrix algebras over $\mathbb C$ or $C(\mathbb T)$
(for example, see \cite[Chapter VIII]{Da}).
It should be noted here
that these simple crossed products
$C(X)\times_{T} \mathbb Z$ of Cantor minimal systems are never
AF since their $K_1$ groups are
all equal to $\mathbb Z$, hence nonzero (\cite[Theorem 1.4]{HPS}).
For a finite alphabet ${\mathcal{A}}$ ($|{\mathcal{A}} |\geq 2$),
the set ${\mathcal{A}}^{\mathbb Z}$ of all two-sided infinite sequences
becomes a compact metrizable space in the product topology
and forms a dynamical system $({\mathcal{A}}^{\mathbb Z}, T)$
together with the shift transformation $T$ given by
$T(\omega)_i:=\omega_{i+1}$,
$\omega\in {\mathcal{A}}^{\mathbb Z}$, $i\in \mathbb Z$.
If $X\subset {\mathcal{A}}^{\mathbb Z}$ is a $T$-invariant closed subspace,
we call the dynamical system $(X,T)$ a {\it subshift} of $({\mathcal{A}}^{\mathbb Z}, T)$.
For a sequence $\omega\in {\mathcal{A}}^{\mathbb Z}$, let ${\mathcal{O}}_\omega$ denote
the the closure of the $T$-orbit of $\omega$.
Then, as is well known, the subshift $({\mathcal{O}}_\omega, T)$
becomes a Cantor minimal system
whenever $\omega$ is an almost periodic and aperiodic sequence.
In order to explain how we form a labeled space
from a Cantor minimal subshift $({\mathcal{O}}_\omega, T)$,
let $E_{\mathbb Z}$ be the directed graph
with the vertex set $\{v_n:n\in \mathbb Z\}$ and
the edge set $\{ e_n: n\in \mathbb Z\}$, where
each $e_n$ is an arrow from $v_n$ to $v_{n+1}$, $n\in \mathbb Z$.
Then we consider a labeling map
${\mathcal{L}}_\omega$ on the graph $E_{\mathbb Z}$ which assigns
to an edge $e_n$ a letter $\omega_n$ for each $n\in \mathbb Z$.
In this way we obtain a labeled graph $C^*$-algebra
$C^*(E_{\mathbb Z}, {\mathcal{L}}_\omega, {\overline{\mathcal{E}}}_{\mathbb Z})$, where
${\overline{\mathcal{E}}}_{\mathbb Z}$ is the smallest set amongst the normal
accommodating sets.
Then we first show that these unital labeled graph algebras
$C^*(E_{\mathbb Z}, {\mathcal{L}}_\omega, {\overline{\mathcal{E}}}_{\mathbb Z})$ are all simple
and have traces.
In the simple crossed product $C({\mathcal{O}}_\omega)\times_T \mathbb Z$,
we then find a family of partial isometries
and projections satisfying the same relations
required for the canonical generators of
$C^*(E_{\mathbb Z}, {\mathcal{L}}_\omega, {\overline{\mathcal{E}}}_{\mathbb Z})$,
which proves from universal property of
$C^*(E_{\mathbb Z}, {\mathcal{L}}_\omega, {\overline{\mathcal{E}}}_{\mathbb Z})$
that there exists an isomorphism
of $C^*(E_{\mathbb Z}, {\mathcal{L}}_\omega, {\overline{\mathcal{E}}}_{\mathbb Z})$ to
the crossed product $C({\mathcal{O}}_\omega)\times_T \mathbb Z$.
Our results can be summarized as follows:
\vskip 1pc
\begin{thm} (Theorem~\ref{main thm 1} and Theorem~\ref{main thm 2})
Let ${\mathcal{A}}$ be a finite alphabet with $|{\mathcal{A}}|\geq 2$,
and let $\omega\in {\mathcal{A}}^{\mathbb Z}$ be a sequence such that
the subshift $({\mathcal{O}}_\omega,T)$ is a Cantor minimal system.
If ${\mathcal{L}}_\omega$ is a labeling map on the graph $E_{\mathbb Z}$
by the sequence $\omega$, the labeled graph $C^*$-algebra
$C^*(E_{\mathbb Z}, {\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})$ is a non-AF simple unital $C^*$-algebra.
Moreover there is an isomorphism
$$\pi: C^*(E_{\mathbb Z}, {\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})\to
C({\mathcal{O}}_\omega) \times_{T} \mathbb Z$$
such that the restriction of $\pi$ onto
the fixed point algebra $C^*(E_{\mathbb Z}, {\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})^\gamma$
of the gauge action $\gamma$ is an isomorphism onto
$C({\mathcal{O}}_\omega)$.
\end{thm}
\vskip 1pc\noindent
The crossed products $C(X)\times_T \mathbb Z$ of
Cantor minimal systems have been studied intensively
(especially in \cite{GPS, HPS}).
Perhaps one important result from the works, in our viewpoint,
would be the fact that the crossed products
$C({\mathcal{O}}_\omega) \times_{T} \mathbb Z$ can be completely classified
by their ordered $K_0$-groups with distinguished oder units
(\cite[Theorem 2.1]{GPS}).
Also from the above theorem and \cite[Theorem 1.4]{HPS}
we know that
the labeled graph $C^*$-algebras $C^*(E_{\mathbb Z}, {\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})$
are $A\mathbb T$-algebras with
$K_1(C^*(E_{\mathbb Z}, {\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z}))=\mathbb Z$, hence
they are not AF.
Finally,
regarding the question of abundance of those Cantor minimal subshift systems,
we notice a well known fact that
$(X,T)$ is topologically conjugate to a two-sided subshift
if and only if it is expansive, and also from \cite[Theorem 1]{DM} that
this is the case if a Cantor system $(X,T)$
has a finite rank $K$ and $K\geq 2$ while
odometer systems are the systems of rank one
(we refer the reader to \cite{DM} for definitions and
properties of this sort of systems).
\vskip 1pc
\section{Preliminaries}
\subsection{\bf Labeled spaces}
We will follow notational conventions of \cite{KPR} for
graph $C^*$-algebras and of \cite{BP2, BCP} for
labeled spaces and their $C^*$-algebras.
A {\it directed graph} $E=(E^0,E^1,r,s)$
consists of a countable vertex set $E^0$, a countable edge set $E^1$,
and the range, source maps $r$, $s: E^1\to E^0$.
If $v\in E^0$ emits (receives, respectively)
no edges it is called a {\it sink} ({\it source}, respectively).
Throughout this paper, we assume that
{\it graphs have no sinks and no sources}.
$E^n$ denotes the set of all finite paths $\lambda=\lambda_1\cdots \lambda_n$
of {\it length} $n$ ($|\lambda|=n$),
($\lambda_{i}\in E^1,\ r(\lambda_{i})=s(\lambda_{i+1}), 1\leq i\leq n-1$).
We write $E^{\leq n}$ and $E^{\geq n}$
for the sets $\cup_{i=1}^n E^i$ and $\cup_{i=n}^\infty E^i$, respectively.
The range and source maps, $r$ and $s$, naturally extend to
all finite paths $E^{\geq 0}$, where
$r(v)=s(v)=v$ for $v\in E^0$.
If a sequence of edges $\lambda_i\in E^1(i\geq 1)$ satisfies $r(\lambda_{i})=s(\lambda_{i+1})$,
one has an infinite path $\lambda_1\lambda_2\lambda_3\cdots $
with the source vertex $s(\lambda_1\lambda_2\lambda_3\cdots):=s(\lambda_1)$.
By $E^\infty$ we denote the set of all infinite paths.
A {\it labeled graph} $(E,{\mathcal{L}})$ over a countable alphabet ${\mathcal{A}}$
consists of a directed graph $E$ and
a {\it labeling map} ${\mathcal{L}}:E^1\to {\mathcal{A}}$.
For $\lambda=\lambda_1\cdots\lambda_n\in E^{\geq 1}$, we call
${\mathcal{L}}(\lambda):={\mathcal{L}}(\lambda_1)\cdots{\mathcal{L}}(\lambda_n)$ a ({\it labeled}) {\it path}.
Similarly one can define an infinite labeled path
${\mathcal{L}}(\lambda)$ for $\lambda\in E^\infty$.
A labeled graph $(E,{\mathcal{L}})$ is said to have a {\it repeatable path} $\beta$
if $\beta^n:=\beta\cdots\beta(\text{repeated $n$-times})\in{\mathcal{L}}(E^{\geq 1})$
for all $n\geq 1$.
The {\it range} $r(\alpha)$ of a labeled path
$\alpha\in {\mathcal{L}}(E^{\geq 1})$ is defined to be a vertex subset of $E^0$:
$$
r(\alpha) =\{r(\lambda) \,:\, \lambda\in E^{\geq 1},\,{\mathcal{L}}(\lambda)=\alpha\},
$$
and the {\it source} $s(\alpha)$ of $\alpha$ is defined similarly.
The {\it relative range of $\alpha\in {\mathcal{L}}(E^{\geq 1})$
with respect to $A\subset 2^{E^0}$} is defined to be
$$
r(A,\alpha)=\{r(\lambda)\,:\, \lambda\in E^{\geq 1},\ {\mathcal{L}}(\lambda)=\alpha,\ s(\lambda)\in A\}.
$$
For notational convenience, we use a symbol $\epsilon$ such that
$r(\epsilon) =E^0$, $r(A, \epsilon) = A$ for all $A \subset E^0$,
and $\alpha=\epsilon\alpha=\alpha\epsilon$ for all $\alpha\in {\mathcal{L}}(E^{\geq 1})$, and
write
$${\mathcal{L}}^\#(E):={\mathcal{L}}(E^{\geq 1})\cup \{\epsilon \}.$$
We denote the subpath $\alpha_i\cdots \alpha_j$ of
$\alpha=\alpha_1\alpha_2\cdots\alpha_{|\alpha|}\in {\mathcal{L}}(E^{\geq 1})$
by $\alpha_{[i,j]}$ for $1\leq i\leq j\leq |\alpha|$.
A subpath of the form $\alpha_{[1,j]}$ is called
an {\it initial path} of $\alpha$.
The symbol $\epsilon$ is regarded as an initial (and terminal)
path of every path.
Let ${\mathcal{B}}\subset 2^{E^0}$ be a collection of subsets of $E^0$.
If $r(A,\alpha)\in {\mathcal{B}}$ for all $A\in {\mathcal{B}}$ and $\alpha\in {\mathcal{L}}(E^{\geq 1})$,
${\mathcal{B}}$ is said to be
{\it closed under relative ranges} for $(E,{\mathcal{L}})$.
We call ${\mathcal{B}}$ an {\it accommodating set} for $(E,{\mathcal{L}})$
if it is closed under relative ranges,
finite intersections and unions and contains $r(\alpha)$ for all $\alpha\in {\mathcal{L}}(E^{\geq 1})$.
The triple $(E,{\mathcal{L}},{\mathcal{B}})$ is called a {\it labeled space} when
${\mathcal{B}}$ is accommodating for $(E,{\mathcal{L}})$.
For $A,B\in 2^{E^0}$ and $n\geq 1$, let
$$ AE^n =\{\lambda\in E^n\,:\, s(\lambda)\in A\},\ \
E^nB=\{\lambda\in E^n\,:\, r(\lambda)\in B\}.$$
We write $E^n v$ for $E^n\{v\}$ and $vE^n$ for $\{v\}E^n$,
and will use notations like $AE^{\geq k}$ and $vE^\infty$
which should have obvious meaning.
A labeled space $(E,{\mathcal{L}},{\mathcal{B}})$ is said to be {\it set-finite}
({\it receiver set-finite}, respectively) if for every $A\in {\mathcal{B}}$ and $l\geq 1$
the set ${\mathcal{L}}(AE^l)$ (${\mathcal{L}}(E^lA)$, respectively) is finite.
A labeled space $(E,{\mathcal{L}},{\mathcal{B}})$ is {\it finite} if
there are only finitely many labels.
In this paper, we will always assume that
labeled spaces $(E,{\mathcal{L}},{\mathcal{B}})$ are
{\it weakly left-resolving}, namely
$$r(A,\alpha)\cap r(B,\alpha)=r(A\cap B,\alpha)$$
for all $A,B\in {\mathcal{B}}$ and $\alpha\in {\mathcal{L}}(E^{\geq 1})$.
$(E,{\mathcal{L}},{\mathcal{B}})$ is {\it left-resolving} if
${\mathcal{L}} : r^{-1}(v) \rightarrow \mathbf{{\mathcal{A}}}$ is injective
for each $v \in E^0$.
Left-resolving labeled spaces are weakly left-resolving.
For each $l\geq 1$, the following relation on $E^0$,
$$ v\sim_l w \ \text{ if and only if \ } {\mathcal{L}}(E^{\leq l} v)={\mathcal{L}}(E^{\leq l} w)$$
is actually an equivalence relation, and
the equivalence class $[v]_l$ of $v\in E^0$ is called a
{\it generalized vertex}.
If $k>l$, $[v]_k\subseteq [v]_l$ is obvious and
$[v]_l=\cup_{i=1}^m [v_i]_{l+1}$
for some vertices $v_1, \dots, v_m\in [v]_l$ (\cite[Proposition 2.4]{BP2}).
\vskip 1pc
\begin{notation}
Given a labeled graph $(E,{\mathcal{L}})$,
${\overline{\mathcal{E}}}$ denotes the smallest {\it normal} accommodating set, that is
the smallest one among the accommodating sets which are
closed under relative complements.
\end{notation}
\vskip 1pc
\begin{prop}{\rm (\cite[Remark 2.1 and Proposition 2.4.(ii)]{BP2},
\cite[Proposition 2.3]{JKK})}
\label{prop-barE}
Let $(E,{\mathcal{L}})$ be a labeled graph ($E$ has no sinks or sources). Then
$${\overline{\mathcal{E}}}=\{ \cup_{i=1}^{n} [v_i]_l \,:\,
v_i \in E^0,\ l,n\geq 1 \}.$$
\end{prop}
\vskip 1pc
\subsection{\bf Labeled graph $C^*$-algebras}
Here we review the labeled graph $C^*$-algebras
which are associated to set-finite, receiver set-finite,
and weakly left-resolving labeled spaces
(whose underlying graphs have no sinks or sources)
although our results are concerning only about
finite left-resolving spaces.
Let $(E,{\mathcal{L}},{\mathcal{B}})$ be a labeled space such that
${\overline{\mathcal{E}}}\subset {\mathcal{B}}$.
Recall from \cite[Definition 2.1]{BCP} that
a {\it representation} of $(E,{\mathcal{L}},{\mathcal{B}})$ is a collection of
projections $\{p_A : A\in {\mathcal{B}}\}$ and
partial isometries
$\{s_a : a\in {\mathcal{A}}\}$ such that for $A, B\in {\mathcal{B}}$ and $a, b\in {\mathcal{A}}$,
\begin{enumerate}
\item[(i)] $p_{\emptyset}=0$, $p_Ap_B=p_{A\cap B}$, and
$p_{A\cup B}=p_A+p_B-p_{A\cap B}$,
\item[(ii)] $p_A s_a=s_a p_{r(A,a)}$,
\item[(iii)] $s_a^*s_a=p_{r(a)}$ and $s_a^* s_b=0$ unless $a=b$,
\item[(iv)] for each $A\in {\mathcal{B}}$,
\begin{eqnarray}\label{representation} p_A=\sum_{a\in {\mathcal{L}}(AE^1)} s_a p_{r(A,a)}s_a^*.
\end{eqnarray}
\end{enumerate}
It follows from (iv) that $p_A=\sum_{\alpha\in{\mathcal{L}}(AE^n)}s_\alpha p_{r(A,\alpha)}s_\alpha^*$
for $n\geq 1$.
By $C^*(p_A,s_a)$ we denote the $C^*$-algebra generated by
$\{s_a,p_A: a\in {\mathcal{A}},\, A\in {\mathcal{B}} \}$.
\vskip 1pc
\begin{remark}\label{review remarks}
Let $(E,{\mathcal{L}},{\mathcal{B}})$ be a labeled space such that
${\overline{\mathcal{E}}}\subset {\mathcal{B}}$.
\begin{enumerate}
\item[(i)]
There exists a $C^*$-algebra
generated by a universal representation
$\{s_a,p_A\}$ of $(E,{\mathcal{L}},{\mathcal{B}})$ (see the proof of \cite[Theorem 4.5]{BP1}).
If $\{s_a,p_A\}$ is a universal representation of $(E,{\mathcal{L}},{\mathcal{B}})$,
we call $C^*(s_a,p_A)$, denoted $C^*(E,{\mathcal{L}},{\mathcal{B}})$, the {\it labeled graph $C^*$-algebra} of
$(E,{\mathcal{L}},{\mathcal{B}})$.
Note that $s_a\neq 0$ and $p_A\neq 0$ for $a\in {\mathcal{A}}$
and $A\in {\mathcal{B}}$, $A\neq \emptyset$, and that
$s_\alpha p_A s_\beta^*\neq 0$ if and only if $A\cap r(\alpha)\cap r(\beta)\neq \emptyset$.
By definition of representation
and \cite[Lemma 4.4]{BP1},
it follows that
\begin{eqnarray}\label{eqn-elements}
C^*(E,{\mathcal{L}},{\mathcal{B}})=\overline{span}\{s_\alpha p_A s_\beta^*\,:\,
\alpha,\,\beta\in {\mathcal{L}}^{\#}(E),\ A\in {\mathcal{B}}\},
\end{eqnarray}
where $s_\epsilon$ is regarded as the unit of the multiplier algebra of
$C^*(E,{\mathcal{L}},\mathcal B)$.
\item[(ii)] Universal property of $C^*(E,{\mathcal{L}},{\mathcal{B}})=C^*(s_a, p_A)$
defines the {\it gauge action}
$\gamma:\mathbb T\to Aut(C^*(E,{\mathcal{L}},{\mathcal{B}}))$ such that
for $a\in {\mathcal{L}}(E^1)$, $A\in {\mathcal{B}}$, and $z\in \mathbb T$,
$$\gamma_z(s_a)=z s_a \ \text{ and } \ \gamma_z(p_A)=p_A.$$
\item[(iii)]
The fixed point algebra of $\gamma$ is an AF algebra such that
\begin{eqnarray}\label{fixed point}
C^*(E,{\mathcal{L}},{\mathcal{B}})^\gamma=\overline{\rm span}\{s_\alpha p_A s_\beta^*: |\alpha|=|\beta|,\ A\in {\mathcal{B}}\}
\end{eqnarray}
Moreover, since $\mathbb T$ is a compact group,
there exists a faithful conditional expectation
$$\Psi: C^*(E,{\mathcal{L}},{\mathcal{B}})\to C^*(E,{\mathcal{L}},{\mathcal{B}})^\gamma.$$
\end{enumerate}
\end{remark}
\vskip 1pc
Recall \cite{BP2, JK} that for a labeled space $(E,{\mathcal{L}},{\overline{\mathcal{E}}})$,
a path $\alpha\in {\mathcal{L}}([v]_l E^{\geq 1})$ is {\it agreeable} for
a generalized vertex $[v]_l$ if
$\alpha=\beta^k\beta'$ for some $\beta\in {\mathcal{L}}([v]_l E^{\leq l})$ and its initial path $\beta'$, and $k\geq 1$.
A labeled space $(E,{\mathcal{L}},{\overline{\mathcal{E}}})$ is said to be {\it disagreeable}
if every $[v]_l$, $l\geq 1$, $v\in E^0$, is disagreeable
in the sense that there is an $N\geq 1$ such that for all $n\geq N$
there is a path $\alpha\in {\mathcal{L}}([v]_l E^{\geq n})$ which is not {\it agreeable}.
\vskip 1pc
\begin{remark}\label{repeatable}
If $(E,{\mathcal{L}},{\overline{\mathcal{E}}})$ is disagreeable, every representation $\{s_a,p_A\}$ such that
$p_A\neq 0$ for all non-empty set $A\in {\overline{\mathcal{E}}}$ gives rise to a $C^*$-algebra
$C^*(s_a,p_A)$ isomorphic to $C^*(E,{\mathcal{L}},{\overline{\mathcal{E}}})$ (\cite[Theorem 5.5]{BP2} and
\cite[Corollary 2.5]{JKP}).
A labeled space $(E,{\mathcal{L}},{\overline{\mathcal{E}}})$ is disagreeable if
there is no repeatable paths in $(E, {\mathcal{L}})$ (\cite[Proposition 4.12]{JKK}).
\end{remark}
\vskip 1pc
$K$-theory of labeled graph $C^*$-algebras was obtained
in \cite{BCP}.
Let $(E,{\mathcal{L}},{\mathcal{B}})$ be a normal labeled space.
Since we assume that $E$ has no sink vertices ($E^0_{\rm sink} =\emptyset$),
the set ${\mathcal{B}}_J$ given in (2) of \cite{BCP}
coincides with ${\mathcal{B}}$, and by \cite[Theorem 4.4]{BCP}
the linear map
$(1-\Phi) :{\rm span}_{\mathbb Z}\{\chi_A: A\in {\mathcal{B}}\} \to
{\rm span}_{\mathbb Z}\{\chi_A: A\in{\mathcal{B}}\}$ given by
\begin{equation}\label{Phi}
(1-\Phi)(\chi_A)=\chi_A-\sum_{a\in {\mathcal{L}}(AE^1)} \chi_{r(A,a)},\ \ A\in {\mathcal{B}}
\end{equation}
determines the $K$-groups of $C^*(E,{\mathcal{L}},{\mathcal{B}})$ as follows:
\begin{align}
K_0(C^*(E,{\mathcal{L}},{\mathcal{B}}))& \cong {\rm span}_{\mathbb Z}\{\chi_A : A\in {\mathcal{B}} \}/{\rm Im}(1-\Phi)\label{K0}\\
K_1(C^*(E,{\mathcal{L}},{\mathcal{B}}))& \cong \ker(1-\Phi)\label{K1}.
\end{align}
In (\ref{K0}), the isomorphism is given by $[p_A]_0\mapsto \chi_A+{\rm Im}(1-\Phi)$ for
$A\in {\mathcal{B}}$.
\vskip 1pc
\subsection{\bf Cantor minimal systems that are subshifts}
A (topological) dynamical system $(X,T)$ consists of
a compact metrizable space $X$
and a homeomorphism $T$ on $X$.
By Krylov-Bogolyubov Theorem, a dynamical system $(X,T)$
admits a Borel probability measure $m$ which is $T$-invariant,
that is $m(T^{-1}(E))=m(E)$ for all Borel sets $E$.
If there exists exactly one $T$-invariant probability measure,
we say that the system $(X,T)$ is {\it uniquely ergodic}.
We will focus on the Cantor systems $(X,T)$ that are
subshifts, and here we briefly review definitions and
basic properties of such Cantor systems.
For an alphabet ${\mathcal{A}}$ ($|{\mathcal{A}}|\geq 2)$,
a {\it word} (or {\it block}) over ${\mathcal{A}}$
is a finite sequence $b=b_1\cdots b_k$ of symbols (or letters) $b_i$'s in ${\mathcal{A}}$
of length $|b|:=k\geq 1$.
By ${\mathcal{A}}^+$, we denote the set of all {\it words}.
Let $\epsilon$ be the empty word of length zero and let
${\mathcal{A}}^*:={\mathcal{A}}^+\cup \{\epsilon\}$.
The set
$${\mathcal{A}}^{\mathbb Z}:=\{\omega=\cdots \omega_{-1}\omega_0\omega_1\cdots : \omega_i\in {\mathcal{A}}\}$$
of all two-sided infinite sequences on ${\mathcal{A}}$,
endowed with the product topology of the discrete topology on
${\mathcal{A}}$, is a totally disconnected compact metrizable space. Actually
the {\it cylinder sets}
$${}_t[b]:=\{\omega \in {\mathcal{A}}^{\mathbb Z} : \omega_{[t, t+|b|-1]}=b\},$$
$b\in {\mathcal{A}}^+$, $t\in \mathbb Z$,
are clopen and form a base for the topology, where
$\omega_{[t_1,t_2]}$ denotes the block $\omega_{t_1}\cdots \omega_{t_2}$ ($t_1\leq t_2$).
Thus the characteristic functions $\chi_{{}_t[b]}$
are continuous for all $b\in {\mathcal{A}}^+$, $t\in \mathbb Z$.
If $b=\omega_{[t_1,t_2]}$ holds for $b\in {\mathcal{A}}^+$ and
$\omega\in {\mathcal{A}}^{\mathbb Z}\cup {\mathcal{A}}^+$,
$b$ is called a {\it factor} of $\omega$.
For $\omega\in {\mathcal{A}}^{\mathbb Z}$ (or ${\mathcal{A}}^{\mathbb N}$),
the set of all factors of $\omega$ is denoted by
$$L_\omega=\{\omega_{[t_1,t_2]} : t_1\leq t_2\}.$$
For convenience, we will use the following notation:
$$[.b]:={}_0[b],\ \ [b.]:={}_{-|b|}[b],\ \ [b.c]:={}_{-|b|}[bc]$$
for words $b,c\in {\mathcal{A}}^+$.
The {\it shift} transform $T:{\mathcal{A}}^{\mathbb Z}\to {\mathcal{A}}^{\mathbb Z}$ given by
$$(Tx)_k=x_{k+1}, \ k\in \mathbb Z,$$
is a homeomorphism.
A {\it subshift} on ${\mathcal{A}}$ is a (topological) dynamical system $(X,T)$
which consists of a $T$-invariant closed subset $X\subset {\mathcal{A}}^{\mathbb Z}$ and
the restriction $T|_X$ which we denote by $T$ again.
If we consider the shift transform $T$ on the space ${\mathcal{A}}^{\mathbb N}$ of one-sided
infinite sequences, it is a
continuous transform (but not a homeomorphism).
For $\omega\in{\mathcal{A}}^{\mathbb Z}$, the closure of the orbit of $\omega$
is denoted by
$${\mathcal{O}}_\omega:=\overline{\{ T^i(\omega): i\in \mathbb Z\}}\ \subset {\mathcal{A}}^{\mathbb Z}.$$
A dynamical system $(X, T)$ is {\it minimal} if
every orbit is dense in $X$, namely
${\mathcal{O}}_x=X$ for all $x\in X$.
It is well known that a subshift $({\mathcal{O}}_\omega,T)$
is minimal if and only if $\omega$ is {\it almost periodic}
(or {\it uniformly recurrent}) in the sense that
each factor of $\omega$ occurs with bounded gaps.
\vskip 1pc
We provide examples of subshifts that are Cantor minimal systems:
\vskip 1pc
\begin{ex}(\textbf{Generalized-Morse sequences})
(\cite{Ke}) \label{generalized Morse}
Let ${\mathcal{A}}=\{0,1\}$.
For a one-sided sequence $x\in {\mathcal{A}}^{\mathbb N}$, let
$ \mathscr{O}_x:=\{\omega\in {\mathcal{A}}^{\mathbb Z} : L_\omega\subset L_x\}$.
Note that each block $b\in {\mathcal{A}}^+$
defines a block $\tilde{b}$,
called the {\it mirror image} of $b$,
such that $\tilde{b}_i=b_i+1$ (mod 2).
For $c=c_0\cdots c_n\in {\mathcal{A}}^+$, the product
$b\times c$ of $b$ and $c$ denotes the block
(of length $|b|\times |c|$) obtained by
putting $n+1$ copies of either $b$ or $\tilde b$ next to each other
according to the rule of choosing the $i$th copy as $b$ if $c_i=0$ and
$\tilde b$ if $c_i=1$. For example, if
$b=01$ and $c=011$, then the product block $b\times c$ is equal to
$b\tilde b \tilde b=011010$.
Let $\{b^i:=b^i_0\cdots b^i_{|b^i|-1}\}_{i\geq 1}\subset {\mathcal{A}}^+$
be a sequence of blocks with length $|b^i|\geq 2$ such that
$b^i_0=0$ for all $i\geq 0$.
Since the product operation $\times$ is associative, one can consider
a sequence of the form
$$x=b^0\times b^1\times b^2\times \cdots \in {\mathcal{A}}^{\mathbb N}$$
which is called a (one-sided) {\it recurrent} sequence
(see \cite[Definition 7]{Ke}).
We call $x=b^0\times b^1\times b^2\times \cdots\in {\mathcal{A}}^{\mathbb N}$ a
{\it (generalized) one-sided Morse sequence} if it is non-periodic and
$$ \sum_{i=0}^\infty \min (r_0(b^i),r_1(b^i))=\infty,$$
where $r_a(b)$ is the {\it relative frequency of occurrence} of
$a\in {\mathcal{A}}$ in $b\in {\mathcal{A}}^+$
(see \cite[p.338]{Ke}).
If $x \in {\mathcal{A}}^{\mathbb N}$ is a non-periodic recurrent sequence,
it is almost periodic, and
there exists $\omega\in \mathscr{O}_x$ with $x= \omega_{[0,\infty)}$.
Moreover, $x$ is a one-sided Morse sequence
if and only if ${\mathcal{O}}_\omega$ is minimal and uniquely ergodic,
and if this is the case, then
${\mathcal{O}}_\omega=\mathscr{O}_x$.
By a {\it generalized Morse sequence},
we mean a two-sided sequence $\omega\in {\mathcal{A}}^{\mathbb Z}$ such that
$x:=\omega_{[0,\infty)}$ is a one-sided Morse sequence and
$L_\omega=L_x$.
(Note that the term a {\it two-sided generalized
Morse sequence} used in \cite{Ke} means a sequence
$\omega\in \mathscr{O}_x$ for some one-sided Morse sequence $x$.)
The subshifts $({\mathcal{O}}_\omega,T)$ for generalized Morse sequences
$\omega$ are uniquely ergodic Cantor minimal systems.
\end{ex}
\vskip 1pc
\begin{ex} (\textbf{Substitution subshifts}) (\cite{Ho})
Let ${\mathcal{A}}$ be a finite alphabet with $|{\mathcal{A}}|\geq 2$.
A {\it substitution} on ${\mathcal{A}}$ is a map $\sigma: {\mathcal{A}}\to {\mathcal{A}}^+$.
$\sigma$ can be iterated to define maps $\sigma^k: {\mathcal{A}}\to {\mathcal{A}}^+$
for all positive integer $k$,
and is called {\it primitive} if there exists $k\geq 1$ such that
$b$ appears in $\sigma^k(a)$ for all $a,b\in {\mathcal{A}}$.
By the {\it language} $L_\sigma$ of a substitution $\sigma$ we mean
the set of words that are factors of $\sigma^k(a)$
for some $k\geq 1$ and $a\in {\mathcal{A}}$.
The subshift
$$X_\sigma:=\{x\in {\mathcal{A}}^{\mathbb Z}\mid L_x\subset L_\sigma\},$$
associated to this language $L_\sigma$ is called
the {\it substitution subshift} defined by $\sigma$.
If $\sigma$ is primitive, it is known that the system $(X_\sigma,T)$ is minimal
and thus a Cantor minimal system.
A sequence $\omega\in {\mathcal{A}}^{\mathbb Z}$ is called a
{\it fixed point} of $\sigma$ if
$\sigma(\omega)=\omega$, and it is known that for any primitive substitution $\sigma$,
there is an $n\geq 1$ such that $\sigma^n$ admits a fixed point $\omega$ in $X_\sigma$.
Since $\sigma^n$ and $\sigma$ define the same dynamical system, we can
only consider primitive substitutions $\sigma$ with a fixed point $\omega\in X_\sigma$,
and in this case, $X_\sigma={\mathcal{O}}_\omega$ follows.
To avoid the case where $X_\sigma$ is finite, or equivalently $\omega$ is
shift periodic, we also assume that $\sigma$ is an {\it aperiodic} substitution
(giving rise to the infinite system $X_\sigma$).
Then the substitution subshifts $(X_\sigma, T)=({\mathcal{O}}_\omega,T)$ are
uniquely ergodic minimal Cantor systems.
\end{ex}
\vskip 1pc
\begin{ex}(\textbf{Thue-Morse sequence})\label{Thue Morse}
Let ${\mathcal{A}}=\{0,1\}$ and $b^i:=01\in {\mathcal{A}}^+$ for all $i\geq 0$.
Then the recurrent sequence
$$x:=b^0\times b^1\times b^2\times \cdots\ =\, 01\times b^1\times\cdots
=0110\times b^2\times \cdots= 01101001\times b^3\times\cdots$$
is a one-sided Morse sequence called the {\it Thue-Morse sequence} and
$$\omega:=x^{-1}.x
=\cdots 10010110.011010011001 \cdots\in \mathscr{O}_x$$
is a generalized Morse sequence,
where
$x^{-1}:= \cdots x_{2} x_{1} x_{0}$ is the sequence obtained
by writing $x= x_0 x_1 \cdots$ in reverse order.
In fact, $\omega$ is the sequence constructed from $x$ in the proof of
\cite[Lemma 4]{Ke}, and
it is well known \cite{GH} that $\omega$ is characterized as a sequence with
no blocks of the form
$b b b_0$ for any $b=b_0\cdots b_{|b|-1}\in {\mathcal{A}}^+$.
By Example~\ref{generalized Morse}, the subshift
$({\mathcal{O}}_\omega, T)$ is a uniquely ergodic Cantor minimal system.
On the other hand, this Thue Morse sequence $\omega$ is
the fixed point of the primitive
aperiodic substitution $\sigma:{\mathcal{A}}\to {\mathcal{A}}^+$ given by
$$\sigma(0)=01 \ \text{ and }\ \sigma(1)=10,$$
so that the subshift $({\mathcal{O}}_\omega,T)$ can also be viewed
as a substitution subshift $(X_\sigma,T)$.
\end{ex}
\vskip 1pc
\section{Main results}
\noindent
Throughout this section,
$E_{\mathbb Z}$ will denote the following graph:
\vskip 1.5pc
{} \hskip 1.1cm
\xy /r0.3pc/:(-44.2,0)*+{\cdots};(44.3,0)*+{\cdots .};
(-40,0)*+{\bullet}="V-4";
(-30,0)*+{\bullet}="V-3";
(-20,0)*+{\bullet}="V-2";
(-10,0)*+{\bullet}="V-1"; (0,0)*+{\bullet}="V0";
(10,0)*+{\bullet}="V1"; (20,0)*+{\bullet}="V2";
(30,0)*+{\bullet}="V3";
(40,0)*+{\bullet}="V4";
"V-4";"V-3"**\crv{(-40,0)&(-30,0)};
?>*\dir{>}\POS?(.5)*+!D{};
"V-3";"V-2"**\crv{(-30,0)&(-20,0)};
?>*\dir{>}\POS?(.5)*+!D{};
"V-2";"V-1"**\crv{(-20,0)&(-10,0)};
?>*\dir{>}\POS?(.5)*+!D{};
"V-1";"V0"**\crv{(-10,0)&(0,0)};
?>*\dir{>}\POS?(.5)*+!D{};
"V0";"V1"**\crv{(0,0)&(10,0)};
?>*\dir{>}\POS?(.5)*+!D{};
"V1";"V2"**\crv{(10,0)&(20,0)};
?>*\dir{>}\POS?(.5)*+!D{};
"V2";"V3"**\crv{(20,0)&(30,0)};
?>*\dir{>}\POS?(.5)*+!D{};
"V3";"V4"**\crv{(30,0)&(40,0)};
?>*\dir{>}\POS?(.5)*+!D{};
(-35,1.5)*+{{e_{-4}}};(-25,1.5)*+{{e_{-3}}};
(-15,1.5)*+{{e_{-2}}};(-5,1.5)*+{{e_{-1}}};(5,1.5)*+{e_{0}};
(15,1.5)*+{e_{1}};(25,1.5)*+{{e_{2}}};(35,1.5)*+{{e_{3}}};
(0.1,-2.5)*+{v_0};(10.1,-2.5)*+{v_1};
(-9.9,-2.5)*+{v_{-1}};
(-19.9,-2.5)*+{v_{-2}};
(-29.9,-2.5)*+{v_{-3}};
(-39.9,-2.5)*+{v_{-4}};
(20.1,-2.5)*+{v_{2}};
(30.1,-2.5)*+{v_{3}};
(40.1,-2.5)*+{v_{4}};
\endxy
\vskip 1.5pc
\noindent
Given a two-sided sequence
$\omega=\cdots \omega_{-1}\omega_0\omega_1 \cdots \in {\mathcal{A}}^{\mathbb Z}$,
we obtain a labeled graph $(E_{\mathbb Z},{\mathcal{L}}_\omega)$ shown below
\vskip 1.5pc
\xy /r0.3pc/:(-44.2,0)*+{\cdots};(44.3,0)*+{\cdots,};
(-40,0)*+{\bullet}="V-4";
(-30,0)*+{\bullet}="V-3";
(-20,0)*+{\bullet}="V-2";
(-10,0)*+{\bullet}="V-1"; (0,0)*+{\bullet}="V0";
(10,0)*+{\bullet}="V1"; (20,0)*+{\bullet}="V2";
(30,0)*+{\bullet}="V3";
(40,0)*+{\bullet}="V4";
"V-4";"V-3"**\crv{(-40,0)&(-30,0)};
?>*\dir{>}\POS?(.5)*+!D{};
"V-3";"V-2"**\crv{(-30,0)&(-20,0)};
?>*\dir{>}\POS?(.5)*+!D{};
"V-2";"V-1"**\crv{(-20,0)&(-10,0)};
?>*\dir{>}\POS?(.5)*+!D{};
"V-1";"V0"**\crv{(-10,0)&(0,0)};
?>*\dir{>}\POS?(.5)*+!D{};
"V0";"V1"**\crv{(0,0)&(10,0)};
?>*\dir{>}\POS?(.5)*+!D{};
"V1";"V2"**\crv{(10,0)&(20,0)};
?>*\dir{>}\POS?(.5)*+!D{};
"V2";"V3"**\crv{(20,0)&(30,0)};
?>*\dir{>}\POS?(.5)*+!D{};
"V3";"V4"**\crv{(30,0)&(40,0)};
?>*\dir{>}\POS?(.5)*+!D{};
(-35,1.5)*+{\omega_{-4}};(-25,1.5)*+{\omega_{-3}};
(-15,1.5)*+{\omega_{-2}};(-5,1.5)*+{\omega_{-1}};(5,1.5)*+{\omega_0};
(15,1.5)*+{\omega_1};(25,1.5)*+{\omega_{2}};(35,1.5)*+{\omega_{3}};
(0.1,-2.5)*+{v_0};(10.1,-2.5)*+{v_1};
(-9.9,-2.5)*+{v_{-1}};
(-19.9,-2.5)*+{v_{-2}};
(-29.9,-2.5)*+{v_{-3}};
(-39.9,-2.5)*+{v_{-4}};
(20.1,-2.5)*+{v_{2}};
(30.1,-2.5)*+{v_{3}};
(40.1,-2.5)*+{v_{4}};
(-53,0)*+{(E_{\mathbb Z},{\mathcal{L}}_\omega)};
\endxy
\vskip 1.5pc
\noindent
where the labeling map ${\mathcal{L}}_\omega: E_{\mathbb Z}^1\to {\mathcal{A}}$ is given by
${\mathcal{L}}_\omega (e_n)=\omega_n$ for $e_n\in E_{\mathbb Z}^1$.
Then we also have a labeled space $(E_{\mathbb Z},{\mathcal{L}}_\omega, {\overline{\mathcal{E}}}_{\mathbb Z})$
with the smallest accommodating set ${\overline{\mathcal{E}}}_{\mathbb Z}$
which is closed under relative complements.
\vskip 1pc
\noindent
{\bf Assumption}. In this section, unless stated otherwise,
${\mathcal{A}}$ is a finite alphabet with $|{\mathcal{A}}|\geq 2$ and
$\omega \in{\mathcal{A}}^{\mathbb Z}$ denotes an almost periodic sequence
such that the subshift $({\mathcal{O}}_\omega, T)$ is a
Cantor minimal system.
\vskip 1pc
\subsection{The fixed point algebra
$C^*(E_{\mathbb Z},{\mathcal{L}}_\omega, {\overline{\mathcal{E}}}_{\mathbb Z})^\gamma$ of the gauge action $\gamma$}
Let $C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})=C^*(s_a, p_A)$ be
the labeled graph $C^*$-algebra
associated with the labeled space $(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})$.
Since the labeled paths ${\mathcal{L}}_\omega(E_{\mathbb Z}^{\geq 1})$
are exactly the factors of the sequence $\omega$,
from now on we briefly denote
the whole labeled paths by $L_\omega$.
By (\ref{fixed point}), we know that
the fixed point algebra of the gauge action $\gamma$
is generated by elements of the form $s_\alpha p_A s_\beta^*$ ($|\alpha|=|\beta|$).
But, in the case $(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})$,
it is rather obvious that
$s_\alpha p_A s_\beta^*\neq 0$, $|\alpha|=|\beta|$, only if
$\alpha=\beta$ and $A\cap r(\alpha)\neq \emptyset$.
Since ${\mathcal{L}}_\omega(E^l v)$ consists of
a single path for each vertex $v$ and $l\geq 1$,
every generalized vertex $[v]_l$ is equal to
the range $r(\alpha)$ for a path $\alpha$ with ${\mathcal{L}}_\omega(E^l v) =\{\alpha\}$.
Hence, by Proposition~\ref{prop-barE},
we have
\begin{eqnarray}\label{fixed point}
C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})^\gamma
=\overline{\rm span}\{s_\alpha p_{r(\beta\alpha)} s_\alpha^*:
\alpha, \beta \in L_\omega \}.
\end{eqnarray}
Moreover $C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})^\gamma$
is easily seen to be a commutative $C^*$-algebra.
For each $k\geq 1$, let
$$F_k:={\rm span}\{ s_\alpha p_{r(\alpha'\alpha)} s_\alpha^*: \alpha,\alpha'\in L_\omega,
|\alpha|=|\alpha'|=k \}.$$
The (finitely many) elements $s_\alpha p_{r(\alpha'\alpha)} s_\alpha^*$ in
$F_k$ are linearly independent and actually orthogonal to each other so that
$F_k$ is a finite dimensional subalgebra of $C^*(E_{\mathbb Z},{\mathcal{L}}_\omega, {\overline{\mathcal{E}}}_{\mathbb Z})^\gamma$.
Moreover $F_k$ is a subalgebra of $F_{k+1}$ because
$$s_\alpha p_{r(\alpha'\alpha)} s_\alpha^*
=\sum_{b\in{\mathcal{A}}} s_{\alpha b} p_{r(\alpha'\alpha b)} s_{\alpha b}^*
=\sum_{a,b\in{\mathcal{A}}} s_{\alpha b} p_{r(a\alpha'\alpha b)} s_{\alpha b}^*.$$
This gives rise to an inductive sequence
$\displaystyle F_1 \xrightarrow{\iota_1} F_2 \xrightarrow{\iota_2} \cdots$
of finite dimensional $C^*$-algebras, where
the connecting maps $\iota_k:F_k\to F_{k+1}$ are the inclusions for all $k\geq 1$,
from which
we obtain an AF algebra $\varinjlim F_k$.
\vskip 1pc
\begin{prop}\label{AF structure}
For the labeled space $(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})$, we have
$$C^*(E_{\mathbb Z},{\mathcal{L}}_\omega, {\overline{\mathcal{E}}}_{\mathbb Z})^\gamma= \varinjlim F_k.$$
\end{prop}
\begin{proof} Since
$F_k\subset C^*(E_{\mathbb Z},{\mathcal{L}}_\omega, {\overline{\mathcal{E}}}_{\mathbb Z})^\gamma$ for all $k\geq 1$
and $\overline{\cup_k F_k}=\varinjlim F_k$,
it is clear that $\varinjlim F_k\subset C^*(E_{\mathbb Z},{\mathcal{L}}_\omega, {\overline{\mathcal{E}}}_{\mathbb Z})^\gamma$.
Thus it suffices to know that the algebra $\cup_k F_k$ is dense in
$C^*(E_{\mathbb Z},{\mathcal{L}}_\omega, {\overline{\mathcal{E}}}_{\mathbb Z})^\gamma$
and then by (\ref{fixed point})
we only need to show that for $y:=s_\alpha p_{r(\beta\alpha)} s_\alpha^*$,
there is $k\geq 1$ such that $y\in F_k$.
If $|\beta\alpha|=2|\alpha|$, then $y\in F_k$ for $k=|\alpha|$.
If $|\beta\alpha|> 2|\alpha|$, then
$$y=s_\alpha p_{r(\beta\alpha)} s_\alpha^*=
\sum_{\nu\in {\mathcal{L}}_\omega(E^{|\beta|-|\alpha|})}
s_{\alpha\nu}p_{r(\beta\alpha\nu)}s_{\alpha\nu}^*\in F_k$$
for $k=|\beta|$.
If $|\beta\alpha|< 2|\alpha|$, we also have
$$y=s_\alpha p_{r(\beta\alpha)} s_\alpha^*=
\sum_{\nu\in {\mathcal{L}}_\omega(E^{|\alpha|-|\beta|})} s_{\alpha}p_{r(\nu\beta\alpha)}s_{\alpha}^*
\in F_k$$ for $k=|\alpha|$.
\end{proof}
\vskip 1pc
\begin{prop}\label{AF isomorphism} There is a surjective isomorphism
\begin{eqnarray}\label{eqn AF iso}
\rho: C^*(E_{\mathbb Z},{\mathcal{L}}_\omega, {\overline{\mathcal{E}}}_{\mathbb Z})^\gamma\to C({\mathcal{O}}_\omega)
\end{eqnarray} such that
$\rho(s_\alpha p_{r(\alpha'\alpha)} s_\alpha^*)=\chi_{[\alpha'.\alpha]}$
for $s_\alpha p_{r(\alpha'\alpha)} s_\alpha^*\in F_k$, $k\geq 1$.
\end{prop}
\begin{proof} Note that for each $k\geq 1$,
the map $\rho_k: F_k\to C({\mathcal{O}}_\omega)$ given by
$$\rho_k(s_\alpha p_{r(\alpha'\alpha)} s_\alpha^*)=\chi_{[\alpha'.\alpha]}$$
is a $*$-homomorphism (we omit the proof) such that
for $y=s_\alpha p_{r(\alpha'\alpha)} s_\alpha^*\in F_k$,
$$\rho_k(y)=\rho_{k+1}(\iota_k(y)),$$
where $\iota_k:F_k\to F_{k+1}$ is the inclusion map.
In fact,
$\iota_k(y)= \sum_{a,b\in{\mathcal{A}}} s_{\alpha b} p_{r(a\alpha'\alpha b)} s_{\alpha b}^*$,
so that
$$\rho_{k+1}(\iota_k(y))=
\rho_{k+1}\big( \sum_{a,b\in{\mathcal{A}}} s_{\alpha b} p_{r(a\alpha'\alpha b)} s_{\alpha b}^*\big)
=\sum_{a,b\in{\mathcal{A}}} \chi_{[a\alpha'.\alpha b]}.$$
But $\sum_{a,b\in{\mathcal{A}}} \chi_{[a\alpha'.\alpha b]}=\chi_{[\alpha'.\alpha]}$ is obvious from
$\cup_{a,b\in{\mathcal{A}}}[a\alpha'.\alpha b]=[\alpha'.\alpha]$.
Hence,
there exists a $*$-homomorphism
$\rho: \varinjlim F_k \to C({\mathcal{O}}_\omega)$ satisfying
$\rho(y)=\rho_k(y)$ for all $y\in F_k$, $k\geq 1$.
Since each $\rho_k$ is injective, so is $\rho$.
Now we show that
$\rho$ is surjective to complete the proof.
Let $\chi_{{}_t[\beta]}\in C({\mathcal{O}}_\omega)$ for $t\in \mathbb Z$ and
$\beta\in L_\omega$.
Assuming $t>0$, we can write
$\displaystyle \chi_{{}_t[\beta]} =\sum_{\alpha,\nu} \chi_{ [\alpha.\nu\beta]}$, where
the sum is taken over all $\alpha$, $\nu$ with $|\nu|=t$ and $|\alpha|=|\nu\beta|$.
Then for $k:=|\beta|+t$, we have
$$\chi_{{}_t[\beta]}=\rho_k\big(\,\sum_{\alpha,\nu} s_\alpha p_{r(\alpha\nu\beta)}s_\alpha^*\big)\in \rho(F_k).
$$
In the case $t\leq 0$, a similar argument shows
that $\chi_{{}_t[\beta]}\in \rho(F_k)$ for some $k$.
Thus $\rho$ is surjective since
${\rm span}\{\chi_{{}_t[\beta]} : t\in \mathbb Z,\,\beta\in L_\omega \}$
is a dense subalgebra of $C({\mathcal{O}}_\omega)$.
\end{proof}
\vskip 1pc
\begin{remark}\label{measure}
It follows from general theory for dynamical systems that
the systems $({\mathcal{O}}_\omega,T)$ considered in this paper have always
$T$-invariant ergodic probability measure
(for example, see \cite[Chapter VIII]{Da}).
If $m_\omega$ is such a $T$-invariant ergodic measure,
the unital commutative AF algebra $C({\mathcal{O}}_\omega)$ of all
continuous functions on ${\mathcal{O}}_\omega$
admits a (tracial) state
$$
f\mapsto \int_{{\mathcal{O}}_\omega} f {\rm d} m_\omega: C({\mathcal{O}}_\omega)\to \mathbb C
$$
which we also write $m_\omega$.
Since $m_\omega$ is $T$-invariant, it easily follows that
$m_\omega (\chi_{{}_t[b]})=m_\omega (\chi_{{}_t[b]}\circ T)=m_\omega (\chi_{{}_{t+1}[b]})$,
and hence
\begin{eqnarray}\label{T-invariance}
m_\omega (\chi_{{}_t[b]})=m_\omega (\chi_{[.b]})
\end{eqnarray}
holds for all $t\in \mathbb Z$ and $b\in L_\omega$.
\end{remark}
\vskip 1pc
\begin{lem}\label{trace on AF}
Let
$\rho: C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})^\gamma\to C({\mathcal{O}}_\omega)$ be
the isomorphism given in {\rm (\ref{eqn AF iso})}.
Then a $T$-invariant ergodic measure $m_\omega$ on ${\mathcal{O}}_\omega$
defines a tracial state
$$\tau_0:=m_\omega\circ\rho: C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})^\gamma\to \mathbb C$$
on the fixed point algebra
$ C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})^\gamma$
such that
for $\alpha, \beta\in L_\omega $,
$$\tau_0(s_\alpha p_{r(\beta\alpha)} s_\alpha^*)=\tau_0 (p_{r(\beta\alpha)} ).$$
\end{lem}
\begin{proof}
Note that $p_{r(\beta\alpha)} = \sum_{\nu} s_\nu p_{r(\beta\alpha\nu)}s_\nu^*$, where
the sum is taken over the paths $\nu$ with $|\nu|=|\beta\alpha|$.
We then have
$$\rho(p_{r(\beta\alpha)})
= \rho(\, \sum_{|\nu|=|\beta\alpha|} s_\nu p_{r(\beta\alpha\nu)}s_\nu^*)
= \sum_{|\nu|=|\beta\alpha|} \chi_{[\beta\alpha.\nu]}
= \chi_{\cup_\nu [\beta\alpha.\nu]}=\chi_{[\beta\alpha.]}.
$$
Thus $$\tau_0 (p_{r(\beta\alpha)})= m_\omega(\chi_{[\beta\alpha.]}).$$
On the other hand,
if $|\beta\alpha|> 2|\alpha|$,
$s_\alpha p_{r(\beta\alpha)} s_\alpha^*=
\sum_{|\nu|= |\beta|-|\alpha|} s_{\alpha\nu}p_{r(\beta\alpha\nu)}s_{\alpha\nu}^*$
so that
$$\tau_0 (s_\alpha p_{r(\beta\alpha)} s_\alpha^*)
=m_\omega (\sum_{|\nu|= |\beta|-|\alpha| } \chi_{[\beta.\alpha\nu]})
=m_\omega(\chi_{[\beta.\alpha]}).$$
But the equality $m_\omega(\chi_{[\beta\alpha.]})=m_\omega(\chi_{[\beta.\alpha]})$
follows from the fact that
$m_\omega$ is $T$-invariant (see (\ref{T-invariance})).
The case where $|\beta\alpha|\leq 2|\alpha|$ can be done in a similar way.
\end{proof}
\vskip 1pc
\begin{lem}\label{lemma for trace}
The labeled graph $C^*$-algebra $C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})$ admits a tracial state
$$\tau_0\circ\Psi: C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})\to \mathbb C,$$
where
$\Psi: C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})\to
C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})^\gamma$ is the conditional expectation onto
the fixed point algebra of the gauge action.
\end{lem}
\begin{proof}
To see that $\tau_0\circ\Psi$ is a trace,
we claim
\begin{eqnarray} \label{tracial}
\tau_0(\Psi(XY))= \tau_0(\Psi(YX))
\end{eqnarray}
for
$X,Y\in {\rm span}\{s_\alpha p_A s_\beta^* : \alpha,\beta\in L_\omega,\,
A\in {\overline{\mathcal{E}}}_{\mathbb Z}, \, A\subset r(\alpha)\cap r(\beta)\}$.
Since the map $\tau_0\circ \Psi$ is linear,
we only need to show (\ref{tracial}) for
$X=s_\alpha p_A s_\beta^*$ and $Y=s_\mu p_B s_\nu^*$.
But also by (\ref{representation}),
it suffices to consider the case of $|\beta|=|\mu|$, so that
$XY=\delta_{\beta,\mu} s_\alpha p_{A\cap B} s_\nu^*$.
In this case if $|\alpha|\neq |\nu|$, then $\Psi(XY)=\Psi(YX)=0$ follows immediately.
Hence now let $|\alpha|=|\nu|$.
If $\alpha\neq \nu$, it is easy to see that $XY=YX=0$ and (\ref{tracial}) holds.
If $\alpha=\nu$, then
$YX= s_\beta p_{B\cap A} s_\beta^*$ and $XY=s_\alpha p_{A\cap B} s_\alpha^*$,
and by Lemma~\ref{trace on AF} we have
$$\tau_0(\Psi(XY))=\tau_0(XY)=\tau_0(s_\alpha p_{A\cap B} s_\alpha^*)
=\tau_0(p_{A\cap B})= \tau_0(\Psi(YX)).$$
The fact that $\tau_0\circ \Psi$ is a state comes from
$$ (\tau_0\circ \Psi)(1)=\tau_0 \big(\sum_{a,b\in{\mathcal{A}}} s_b p_{r(ab)}s_b^*\big)
= m_\omega (\sum_{a,b\in{\mathcal{A}} } \chi_{[a.b]})= m_\omega (\chi_{{\mathcal{O}}_\omega})
=1.$$
\end{proof}
\vskip 1pc
To prove the simplicity of the labeled graph $C^*$-algebra
$C^*(E_{\mathbb Z}, {\mathcal{L}}_\omega, {\overline{\mathcal{E}}}_{\mathbb Z})$,
we need the following lemma which might be well known in the theory of
dynamical systems, but we provide a proof here for the reader's
convenience.
\vskip 1pc
\begin{lem}\label{disagreeable}
Let $\omega\in {\mathcal{A}}^{\mathbb Z}$ be a sequence which is
almost periodic but not periodic.
Then the labeled space $(E_{\mathbb Z}, {\mathcal{L}}_\omega, {\overline{\mathcal{E}}}_{\mathbb Z})$ is
disagreeable.
\end{lem}
\begin{proof}
It is enough to show that the labeled space has no repeatable paths
(see Remark~\ref{repeatable}).
For this, suppose there is a repeatable path $\alpha$.
We may assume that $\alpha$ has the smallest length.
If $\beta\in L_\omega$, by the assumption that $\omega$ is almost periodic,
there exists a $d\geq 1$ such that
every block $\omega_{[t,t+d]}$, $t\in \mathbb Z$,
has $\beta$ as its factor.
Thus any path $\alpha^k\in L_\omega$, $k$ large enough, has
$\beta$ as a factor, so that $\alpha^k=\mu\beta\nu$ for some
$\mu, \nu\in L_\omega\cup\{\epsilon\}$.
In other words, every $\beta\in L_\omega$ must be of the form
$\beta=\alpha''\alpha^l\alpha'$ for an initial path $\alpha'$ and
terminal path $\alpha''$ of $\alpha$ and $l\geq 0$.
Now we can apply this fact to the paths $\beta=\omega_{[0,n]}$, $n\geq 1$,
to obtain that $\omega_{[0,\infty)}$ is of the form $\alpha''\alpha^\infty$.
But then, considering the blocks of the form $\omega_{[-n,n]}\in L_\omega$
($n\to \infty$)
we can easily see that
$\omega=(\alpha)^\infty\alpha'. \alpha''(\alpha)^\infty$,
where $\alpha=\alpha'\alpha''$. Thus $\omega$ is periodic, which is a contradiction.
\end{proof}
\vskip 1pc
Since we assume that a Cantor system $({\mathcal{O}}_\omega,T)$ is minimal, or
equivalently $\omega$ is almost periodic (and not periodic),
the labeled space $(E_{\mathbb Z}, {\mathcal{L}}_\omega, {\overline{\mathcal{E}}}_{\mathbb Z})$ considered in
this section is
always disagreeable by Lemma~\ref{disagreeable}.
The following theorem shows that there exist
simple labeled graph $C^*$-algebras that are not
stably isomorphic to simple graph $C^*$-algebras.
\vskip 1pc
\begin{thm}\label{main thm 1}
Let ${\mathcal{A}}$ be a finite alphabet with $|{\mathcal{A}}|\geq 2$,
and let $\omega\in {\mathcal{A}}^{\mathbb Z}$ be a sequence such that
the subshift $({\mathcal{O}}_\omega,T)$ is a Cantor minimal system.
Then the labeled graph $C^*$-algebra
$C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})$ is a
non-AF simple unital $C^*$-algebra with a tracial state
$\tau$
which satisfies
$$\tau(s_\alpha p_{r(\nu\alpha)}s_\beta^*)
=\tau\circ \Psi(s_\alpha p_{r(\nu\alpha)}s_\beta^*)=\delta_{\alpha,\beta}\tau(p_{r(\nu\alpha)})$$
for labeled paths $\alpha, \beta, \nu\in {\mathcal{L}}_\omega (E_{\mathbb Z}^{\geq 1})$.
Moreover if the system $({\mathcal{O}}_\omega, T)$ is uniquely ergodic,
$C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})$ has a unique tracial state.
\end{thm}
\begin{proof}
For the simplicity of
$C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})=C^*(p_A,s_a)$,
we show that any nonzero homomorphism
$\pi:C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})\to C^*(q_A, t_a)$
onto a $C^*$-algebra generated by
$q_A:=\pi(p_A),\ t_a:=\pi(s_a)$ for $ A\in {\overline{\mathcal{E}}}_{\mathbb Z}$,
$a\in {\mathcal{A}}$, is faithful.
Since the labeled space
$(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})$ is disagreeable
by Lemma~\ref{disagreeable},
we see from \cite[Theorem 5.5]{BP2} that
$\pi$ is faithful whenever
$\pi(p_{[v]_l})\neq 0$ for all $v\in E^0$ and $l\geq 1$.
Suppose on the contrary that
$$q_{[v]_m}=\pi(p_{[v]_m})=0$$
for some $[v]_m=r(\alpha)$ with $|\alpha|=m$.
Since $\alpha\in L_\omega$ and $\omega$ is almost periodic,
one finds a $d\geq 1$ such that for all $s\geq 0$,
$$T^{s+j}\omega \in [.\alpha]$$
for some $0\leq j\leq d$.
This means that if $\beta\in L_\omega$ is a block
with length $|\beta|\geq d$, it must have $\alpha$ as a factor.
Thus $\beta$ must be of the form $\beta=\beta'\alpha\beta''$
for some $\beta',\beta''\in {\mathcal{L}}_\omega^\sharp (E)\,(=L_\omega \cup\{\epsilon\})$.
For $\beta$ with $|\beta|\geq d$ we have $q_{r(\beta)}=0$.
In fact,
\begin{align*}
q_{r(\beta)} & \ = q_{r(\beta'\alpha\beta'')}=q_{r(r(\beta'\alpha),\beta'')}\\
& \ = q_{r(r(\beta'\alpha),\beta''))}t_{\beta''}^*t_{\beta''}q_{r(r(\beta'\alpha),\beta'')}\\
& \ \sim t_{\beta''}q_{r(r(\beta'\alpha),\beta'')}t_{\beta''}^*\\
& \ \leq q_{r(\beta'\alpha)}\leq q_{r(\alpha)}\\
& \ = q_{[v]_m}=0.
\end{align*}
On the other hand, since $\pi$ is a nonzero homomorphism,
there exists a
$\delta\in L_\omega$ with $q_{r(\delta)}=\pi(p_{r(\delta)})\neq 0$.
But then, with an $n>\max\{ |\delta|,d\}$, we have
$$q_{r(\delta)}=\pi(p_{r(\delta)})
=\pi\big(\sum_{|\delta\mu_i|=n} s_{\mu_i}p_{r(\delta\mu_i)}s_{\mu_i}^* \big)
= \sum_{|\delta\mu_i|=n} t_{\mu_i}q_{r(\delta\mu_i)}t_{\mu_i}^* =0,$$
a contradiction, and
$C^*(E_{\mathbb Z}, {\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})$ is simple.
With ${\overline{\mathcal{E}}}_{\mathbb Z}$ in place of ${\mathcal{B}}$ in (\ref{K1})
it is rather obvious that $\mathcal N=\emptyset$ and $\hat{\mathcal{B}}=\hat {\mathcal{B}}_J={\overline{\mathcal{E}}}_{\mathbb Z}$.
Since
$\chi_A\in \ker(1-\Phi)$ if and only if
$\chi_A=\sum_{a\in {\mathcal{A}}} \chi_{r(A,a)}$ (see (\ref{Phi})) which
actually holds for $A=E_{\mathbb Z}^0$, we have
$K_1(C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z}))
= {\rm ker}(1-\Phi)\neq 0$.
Thus $C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})$ is not AF.
(We will see later from
Theorem~\ref{main thm 2} that
$K_1(C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z}))=\mathbb Z$.)
If $\tau_0$ is the tracial state of
$C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})^\gamma$ induced by
an ergodic measure of $({\mathcal{O}}_\omega, T)$,
the tracial state
$\tau:=\tau_0\circ \Psi:C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})\to \mathbb C$
of Lemma~\ref{lemma for trace} satisfies
\begin{eqnarray}\label{trace condition}
\tau(s_\alpha p_{r(\nu\alpha)} s_\beta^*)=\delta_{\alpha,\beta}\tau(p_{r(\nu\alpha)})
\end{eqnarray}
for
$s_\alpha p_{r(\nu\alpha)}s_\beta^* \in C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})$.
Now let $({\mathcal{O}}_\omega, T)$ be uniquely ergodic and
again let $\tau_0$ be the tracial state on
the fixed point algebra
$C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})^\gamma$ and
$\tau:=\tau_0\circ \Psi$ the extension of $\tau_0$ to the
whole labeled graph $C^*$-algebra as before.
To show that $\tau$ is the unique tracial state on
$C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})$,
we claim that if $\tau'$ is a tracial state on
$C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})$, then $\tau'\circ\Psi=\tau'$ holds, and that
the state $\tau'\circ \rho^{-1}$ on $C({\mathcal{O}}_\omega)$ is $T$-invariant.
For the first claim, suppose $\tau'\circ\Psi\neq\tau'$. Then
there exists an element $s_\alpha p_{r(\alpha)} s_\beta^*$ ($|\beta|<|\alpha|$) such that
$\tau'(s_\alpha p_{r(\alpha)} s_\beta^*)\neq 0$.
Since $\tau'$ is tracial, we have
$0\ne \tau'(s_\alpha p_{r(\alpha)} s_\beta^*)= \tau'(s_\beta^*s_\alpha p_{r(\alpha)})$.
Thus $\alpha$ must be of the form $\alpha=\beta\alpha'$ for some path $\alpha'$, and then
$0\neq \tau'(s_\beta^*s_\alpha p_{r(\alpha)})=\tau'(s_{\alpha'} p_{r(\alpha)})$.
Again the tracial property of $\tau'$ gives
$$0\neq \tau'(s_{\alpha'} p_{r(\alpha)})=\tau'( p_{r(\alpha)} s_{\alpha'})=
\tau'(s_{\alpha'} p_{r(\alpha\af')})=\cdots = \tau'(s_{\alpha'} p_{(r(\alpha), (\alpha')^n)})$$
for all $n\geq 1$.
But this means that the generalized vertex $[v]_l:=r(\alpha)$, $l=|\alpha|$,
is not disagreeable emitting only agreeable paths,
which is a contradiction to Lemma~\ref{disagreeable}.
To see that the state $\tau'\circ \rho^{-1}:C({\mathcal{O}}_\omega)\to \mathbb C$ is $T$-invariant,
let $\chi_{{}_t[\beta]}\in C({\mathcal{O}}_\omega)$. We assume $t>0$.
Since $$\rho^{-1}(\chi_{{}_t[\beta]})
=\rho^{-1}\big(\sum_{\substack{\alpha,\beta\\ |\alpha|=|\sigma\beta|=t+|\beta|}} \chi_{[\alpha.\sigma\beta]}\,\big)
=\sum_{\substack{\alpha,\beta\\ |\alpha|=|\sigma\beta|=t+|\beta|}} s_{\sigma\beta} p_{r(\alpha\sigma\beta)}s_{\sigma\beta}^*,$$
we have
$\displaystyle\tau'(\rho^{-1}(\chi_{{}_t[\beta]}))
=\tau'\big( \sum_{\substack{\alpha,\beta\\ |\alpha|=|\sigma\beta|
=t+|\beta|}} p_{r(\alpha\sigma\beta)}\,\big)=\tau'(p_{r(\beta)}).$
This implies that
$$\tau'\circ\rho^{-1}(\chi_{{}_t[\beta]})=\tau'\circ\rho^{-1}(\chi_{{}_{t+1}[\beta]})
=\tau'\circ\rho^{-1}(\chi_{{}_t[\beta]}\circ T),$$
which can also be shown for $t\leq 0$ in a similar way.
Thus $\tau'\circ\rho^{-1}$ is $T$-invariant because the span of functions
$\chi_{{}_t[\beta]}$ is dense in $C({\mathcal{O}}_\omega)$.
\end{proof}
\vskip 1pc
\begin{remarks}
(1) Simplicity of $C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})$
can also be shown by analyzing the path structure of the labeled space
$(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})$.
For a labeled graph $(E,{\mathcal{L}})$, set
$$\overline{{\mathcal{L}}(E^\infty)}:=\{x\in {\mathcal{A}}^{\mathbb N}\mid
x_{[1,n]}\in {\mathcal{L}}(E^n) \ \text{for all } n\geq 1\}.$$
Then ${\mathcal{L}}(E^\infty)\subset \overline{{\mathcal{L}}(E^\infty)}$ is obvious, but
it is possible to have ${\mathcal{L}}(E^\infty)\subsetneq \overline{{\mathcal{L}}(E^\infty)}$.
For example, if $\omega= 0^\infty. 0101^201^301^4\cdots\in \{0,1\}^{\mathbb Z}$,
then
the path $1^\infty\in \overline{{\mathcal{L}}_\omega(E_{\mathbb Z})}$ does not appear
as an infinite labeled path in ${\mathcal{L}}_{\omega}(E_{\mathbb Z}^\infty)$.
We say that a labeled space $(E,{\mathcal{L}},{\overline{\mathcal{E}}})$ is {\it strongly cofinal} if
for each $x\in \overline{{\mathcal{L}}(E^\infty)}$ and $[v]_l\in {\overline{\mathcal{E}}}$, there exist
an $N\geq 1$ and a finite number of paths $\lambda_1, \dots, \lambda_m\in {\mathcal{L}}(E^{\geq 1})$ such that
$$r(x_{[1,N]})\subset \cup_{i=1}^m r([v]_l,\lambda_i).$$
This definition of strong cofinality is a modification
of the definitions given in \cite{BP2, JK} and the proof of \cite[Theorem 6.4]{BP2}
can be slightly modified to prove that if
$(E,{\mathcal{L}},{\overline{\mathcal{E}}})$ is strongly cofinal and disagreeable, the $C^*$-algebra
$C^*(E,{\mathcal{L}},{\overline{\mathcal{E}}})$ is simple.
If $\omega$ is a sequence satisfying the assumption of this section,
it is not hard to see that the labeled space
$(E_{\mathbb Z}, {\mathcal{L}}_\omega, {\overline{\mathcal{E}}}_{\mathbb Z})$ is
strongly cofinal.
Then by Lemma~\ref{disagreeable}, we know that
$C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})$ is simple.
(2) In case $\omega$ is the Thue Morse sequence given in
Example~\ref{Thue Morse},
one can directly show that the simple unital $C^*$-algebra
$C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})$
admits a unique tracial state.
Moreover, its exact values on typical elements of the form $s_\alpha p_A s_\beta^*$
can be obtained explicitly, which will be done in \cite{Ki}.
\end{remarks}
\vskip 1pc
\begin{remark}
If $(X,T)$ is a Cantor minimal system,
$T$ induces an automorphism $T$ of $C(X)$,
$$T(f)=f\circ T^{-1},\ \ f\in C(X),$$
and
it is well known that the crossed product $C(X)\times_{T} \mathbb Z$
is always simple (for example, see \cite{Da}).
It is also known \cite{GPS} that
the crossed products $C(X)\times_{T} \mathbb Z$ are not AF
because $K_1(C(X)\times_{T} \mathbb Z)=\mathbb Z$.
But they are all $A\mathbb T$ algebras, hence
finite algebras of stable rank one, and have real rank zero by \cite{BBEK}.
Moreover their isomorphism classes are determined by the ordered $K_0$-groups
$$(K_0(C(X)\times_{T} \mathbb Z), K_0^+(C(X)\times_{T} \mathbb Z), [1]_0)$$
together with the distinguished order units $[1]_0$, where $1$ is the unit projection
of the crossed product.
\end{remark}
\vskip 1pc
If a Cantor minimal system
$({\mathcal{O}}_\omega, T)$ is uniquely ergodic,
the following theorem implies together with Theorem~\ref{main thm 1}
that the crossed product $C({\mathcal{O}}_\omega) \times_{T} \mathbb Z$
has a unique tracial state, which is well known for
uniquely ergodic minimal systems $(X, T)$ of infinite
spaces $X$ (see \cite[Corollary VIII.3.8]{Da}).
\vskip 1pc
\begin{thm}\label{main thm 2}
Let ${\mathcal{A}}$ be a finite alphabet with $|{\mathcal{A}}|\geq 2$,
and let $\omega\in {\mathcal{A}}^{\mathbb Z}$ be a sequence such that
the subshift $({\mathcal{O}}_\omega,T)$ is a Cantor minimal system.
Then there is an isomorphism
$$\pi: C^*(E_{\mathbb Z}, {\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})\to
C({\mathcal{O}}_\omega) \times_{T} \mathbb Z$$
such that the restriction of $\pi$ onto
the fixed point algebra $C^*(E_{\mathbb Z}, {\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})^\gamma$
of the gauge action $\gamma$ is an isomorphism onto
$C({\mathcal{O}}_\omega)$.
\end{thm}
\begin{proof}
Proposition~\ref{AF isomorphism} (and its proof) says that
the fixed point algebra $C^*(E_{\mathbb Z}, {\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})^\gamma$ is
isomorphic to $C({\mathcal{O}}_\omega)$ via the map $\rho$
given by
$$\rho(s_\alpha p_{r(\beta\alpha)}s_\alpha^*)=\chi_{[\beta.\alpha]},\ \
\alpha,\beta\in {\mathcal{L}}_\omega^\sharp (E_{\mathbb Z}).$$
We show that there exists an isomorphism
of $C^*(E_{\mathbb Z}, {\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})$ onto
the crossed product
$C^*(E_{\mathbb Z}, {\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})^\gamma\rtimes_{T'} \mathbb Z$,
where
$T':=\rho^{-1}\circ T\circ \rho$ is the automorphism of
$C^*(E_{\mathbb Z}, {\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})^\gamma$.
Note first that $T'$ satisfies the following
\begin{eqnarray}\label{T'}
T'(s_\alpha p_{r(\beta\alpha)}s_\alpha^*)=s_{\alpha_2\cdots\alpha_n}p_{r(\beta\alpha)}s_{\alpha_2\cdots\alpha_n}^*
\end{eqnarray}
for $\alpha,\beta\in {\mathcal{L}}_\omega^\sharp (E_{\mathbb Z})$.
In fact,
$
\rho\big(T'(s_\alpha p_{r(\beta\alpha)}s_\alpha^*)\big) =T(\rho(s_\alpha p_{r(\beta\alpha)}s_\alpha^*))
= T(\chi_{[\beta.\alpha]})
=\chi_{T([\beta.\alpha])}
=\chi_{[\beta\alpha_1.\alpha_2\cdots \alpha_n]}
=\rho\big(s_{\alpha_2\cdots\alpha_n}p_{r(\beta\alpha)}s_{\alpha_2\cdots\alpha_n}^*\big)
$ where $n:=|\alpha|$.
With the unitary $u$ implementing the automorphism $T'$
(namely, $T'=Ad u$),
this can be written as
$$ T'(s_\alpha p_{r(\beta\alpha)}s_\alpha^*)
= u(s_\alpha p_{r(\beta\alpha)}s_\alpha^*)u^*
= s_{\alpha_2\cdots\alpha_n}p_{r(\beta\alpha)}s_{\alpha_2\cdots\alpha_n}^*.$$
Particularly,
\begin{eqnarray} \label{eqn-t_a}
up_{r(\beta)}u^*=u\big(\sum_{a\in {\mathcal{A}}}s_ap_{r(\beta a)} s_a^*\big)u^*=
\sum_{a\in {\mathcal{A}}} p_{r(\beta a)}
\end{eqnarray}
holds.
To find a desired isomorphism, we will find a representation of the labeled
space $(E_{\mathbb Z}, {\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})$ in the crossed product, and then
apply the universal property of the $C^*$-algebra
$C^*(E_{\mathbb Z}, {\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})$.
We actually show that the following partial isometries
$$t_a:=u^*p_{r(a)},\ \ a\in {\mathcal{A}}$$
in the crossed products
$C^*(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})^\gamma\times_{T'} \mathbb Z$
form a representation of $(E_{\mathbb Z},{\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})$
together with the family of projections $\{p_A : A\in {\overline{\mathcal{E}}}_{\mathbb Z})$.
By (\ref{eqn-t_a}),
$t_a^* t_a=p_{r(a)}$ and $t_a^* t_b=\delta_{a,b} p_{r(a)}$
are immediate for $a,b\in {\mathcal{A}}$.
We also have
\begin{align*}
p_{r(\beta)}t_a & =p_{r(\beta)}u^*p_{r(a)}
= u^*\big(\sum_{b\in {\mathcal{A}}} p_{r(\beta b)}\big) p_{r(a)}\\
& = u^* p_{r(\beta a)} =u^* p_{r(a)} p_{r(\beta a)}
= t_a p_{r(\beta a)} \\
& =t_a p_{r(r(\beta), a)}.
\end{align*}
Since every $A\in {\overline{\mathcal{E}}}_{\mathbb Z}$ can be written as
a finite union of generalized vertices (by Proposition~\ref{prop-barE})
and a generalized vertex $[v]_l$ is clearly equal to
a range $r(\beta)$ of $\beta\in {\mathcal{L}}_\omega(E^lv)$,
we know that
the above equalities hold for any $A\in{\overline{\mathcal{E}}}_{\mathbb Z}$.
Finally we have to check
$$p_{r(\beta)}= \sum_{a\in {\mathcal{A}}} t_a p_{r(\beta a)} t_a^*,$$
but this follows directly from the definition of $t_a$ and (\ref{eqn-t_a}).
Thus $\{t_a, p_A\}$ forms a representation of the labeled space
$(E_{\mathbb Z}, {\mathcal{L}}_\omega, {\overline{\mathcal{E}}}_{\mathbb Z})$ in the $C^*$-algebra
$C^*(E_{\mathbb Z}, {\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})^\gamma\rtimes_{T'} \mathbb Z$,
and hence
there exists a homomorphism
$$\pi: C^*(E_{\mathbb Z}, {\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})\to
C^*(E_{\mathbb Z}, {\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})^\gamma\rtimes_{T'} \mathbb Z$$
such that
$\pi(s_a)=t_a$ and $\pi(p_A)=p_A$ ($a\in {\mathcal{A}}$, $A\in {\overline{\mathcal{E}}}_{\mathbb Z}$).
The homomorphism $\pi$ is injective since
$C^*(E_{\mathbb Z}, {\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})$ is simple
by Theorem~\ref{main thm 1},
and is surjective since $u^*=u^*(\sum_{a\in {\mathcal{A}}}p_{r(a)})=
\sum_{a\in {\mathcal{A}}} t_a \in {\rm Im}(\pi)$ and
$s_{\alpha}p_{r(\beta\alpha)}s_{\alpha}^*=(u^*)^{|\alpha|}p_{r(\beta\alpha)}u^{|\alpha|}\in {\rm Im}(\pi)$
for all generators $s_{\alpha}p_{r(\beta\alpha)}s_{\alpha}^*$ of
$C^*(E_{\mathbb Z}, {\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})^\gamma$.
For the last assertion, it is enough to see that
for $\alpha,\beta\in {\mathcal{L}}_\omega^\sharp (E_\mathbb Z)$,
$$\pi(s_\alpha p_{r(\beta\alpha)} s_\alpha^*)=s_\alpha p_{r(\beta\alpha)} s_\alpha^*.$$
If $a\in {\mathcal{A}}$, then $\pi(p_{r(a)})=\pi(s_a^* s_a)=t_a^* t_a
=p_{r(a)}u u^* p_{r(a)}=p_{r(a)}$, and hence
$\pi(p_{r(\alpha)})= p_{r(\alpha)}$ holds for all $\alpha\in {\mathcal{L}}_\omega^\sharp (E_\mathbb Z)$.
The equality (\ref{T'}) shows that the inverse $(T')^{-1}$ of
the automorphism $T'$ on $C^*(E_{\mathbb Z}, {\mathcal{L}}_\omega,{\overline{\mathcal{E}}}_{\mathbb Z})^\gamma$
maps a projections $p_{r(\alpha)}$ to the projection
$s_a p_{r(\alpha)} s_a^*$,
where
$a\in {\mathcal{A}}$ is the last letter of $\alpha$.
(If $\alpha=\epsilon$ is the empty word,
$p_{r(\epsilon)}=s_\epsilon$ is the unit of
the labeled graph $C^*$-algebra, hence
$(T')^{-1}(p_{r(\epsilon)})=p_{r(\epsilon)}=
s_{\epsilon}p_{r(\epsilon)}s_\epsilon^*$ also holds.)
Then for $\alpha=\alpha'a$ with
$\alpha'\in {\mathcal{L}}_\omega^\sharp (E_\mathbb Z)$, $a\in {\mathcal{A}}$,
we have
$$
\pi(s_a p_{r(\alpha)} s_a^*) =t_a p_{r(\alpha)} t_a^* = u^* p_{r(\alpha)} u
= (T')^{-1} (p_{r(\alpha)}) = s_a p_{r(\alpha)} s_a^*
$$
as desired.
\end{proof}
\vskip 1pc
|
{
"timestamp": "2016-03-01T02:11:02",
"yymm": "1504",
"arxiv_id": "1504.03455",
"language": "en",
"url": "https://arxiv.org/abs/1504.03455"
}
|
\section*{Methods}
The fabrication process, cryogenic setup, gate biasing and data acquisition is identical to that described by Muhonen~\textit{et~al.}\cite{Muhonen2014nn}. We combine the signals of three different generators (Agilent E8257D/E8267D PSG for ESR and Agilent N5182 MXG for NMR) to apply pulses to each of the transitions described in the main text. We can achieve $\pi_{\textsc{mw}\textmwone{1},\textmwtwo{2}}$ rotations in $\sim3$~$\mu$s and $\pi_{\textrf{\textsc{rf}}}$ rotations in $\sim30$~$\mu$s. The generators are pulse modulated using a SpinCore PulseBlaster ESR TTL pulse generator. For phase control in the density matrix tomography experiments, we use the internal baseband arbitrary waveform generator (AWG) in the E8267D vector source, and we use an Agilent 81180A AWG to gate the I/Q inputs of the N5182.
|
{
"timestamp": "2015-04-14T02:12:58",
"yymm": "1504",
"arxiv_id": "1504.03112",
"language": "en",
"url": "https://arxiv.org/abs/1504.03112"
}
|
\section{Background on \texorpdfstring{$p$}{Lg}-local finite groups}\label{2}
We give here a very short introduction to $p$-local finite groups.
We refer the reader interested in more details to Aschbacher, Kessar and Oliver \cite{AKO}.
Roughly speaking, fusion systems encode the conjugation data of a finite group with respect to a choice of a Sylow $p$-subgroup.
For $G$ a finite group and $g\in G$, we will denote by $c_g$ the homomorphism $x\in G\mapsto gxg^{-1}\in G$. Given subgroups $H,K\leq G$, we shall denote by $\text{Hom}_G(H,K)$ the set of all group homomorphisms $c_g$ for $g\in G$ such that $c_g(H)\leq K$.
\begin{defi}\label{defF}
Let $S$ be a finite $p$-group.
A \emph{fusion system} over $S$ is a small category $\mathcal{F}$, where $\text{Ob}(\mathcal{F})$ is the set of all subgroups of $S$ and which satisfies the following two
properties for all $P,Q\leq S$:
\begin{enumerate}[(a)]
\item $\text{Hom}_S(P,Q)\subseteq \text{Mor}_\mathcal{F}(P,Q)\subseteq \text{Inj}(P,Q)$;
\item each $\varphi\in\text{Mor}_\mathcal{F}(P,Q)$ is the composite of an $\mathcal{F}$-isomorphism followed by an inclusion.
\end{enumerate}
A fusion system is \emph{saturated} if it satisfies two more technical conditions called the saturation axioms (we refer the reader to \cite{AKO}, Definition I.2.1 for a proper definition).
\end{defi}
The composition in a fusion system is given by composition of homomorphisms. We usually write $\text{Hom}_\mathcal{F}(P,Q)=\text{Mor}_\mathcal{F}(P,Q)$ to emphasize the fact that the
morphims in $\mathcal{F}$ are actual group homomorphisms.
The typical example of saturated fusion system is the fusion system $\mathcal{F}_S(G)$ of a finite group $G$ over $S\in\text{Syl}_p(G)$.
\begin{defi}\label{pcentr}
Let $\mathcal{F}$ be a saturated fusion system over a $p$-group $S$.
A subgroup $P\leq S$ is \emph{$\mathcal{F}$-centric} if $C_S(Q)=Z(Q)$ for every $Q\in P^\mathcal{F}$.
We will denote by $\mathcal{F}^c$ the full subcategory of $\mathcal{F}$ with set of objects all the $\mathcal{F}$-centric subgroups of $S$.
\end{defi}
If $\mathcal{F}$ is the saturated fusion system associated to a finite group $G$ with $S$ as Sylow $p$-subgroup, then a subgroup $P\leq S$ is $\mathcal{F}$-centric if and only if $P$ is $p$-centric, i.e. $Z(P)$ is a Sylow $p$-subgroup of $C_G(P)$.
Before defining the notion of centric linking system let us first recall a well-known result about saturated fusion system.
\begin{thm}[Alperin's Fusion Theorem]\label{AFT}
Let $\mathcal{F}$ be a saturated fusion system over a $p$-group $S$.
Then, every morphism is a composite of restrictions of automorphisms of $\mathcal{F}$-centric subgroups.
\end{thm}
In other words, a saturated fusion system $\mathcal{F}$ is generated by $\mathcal{F}^c$.
In fact, Alperin's Fusion Theorem is more precise and says that we just need automorphisms of $S$ and $\mathcal{F}$-essential subgroups of $S$.
For more details, we refer to Section I.3 of Aschbacher, Kessar and Oliver \cite{AKO}.
For $\mathcal{F}$ a fusion system over a $p$-group, $\mathcal{T}_S^c(S)$ will denote the usual transporter category of $S$ with set of objects $\text{Ob}(\mathcal{F}^c)$.
\begin{defi}\label{linkdef}
Let $\mathcal{F}$ be a fusion system over a $p$-group $S$.
A \emph{centric linking system} associated to $\mathcal{F}$ is a finite category $\mathcal{L}$ together with a pair of functors
\[
\xymatrix{\mathcal{T}_S^c(S)\ar[r]^-{\delta} &\mathcal{L} \ar[r]^-{\pi}& \mathcal{F}}
\]
satisfying the following conditions:
\begin{itemize}
\item[(A)] $\delta$ is the identity on objects, and $\pi$ is the inclusion on objects. For each $P,Q\in\text{Ob}(\mathcal{L})$ such that $P$ is
fully centralized in $\mathcal{F}$, $C_S(P)$ acts freely on $\text{Mor}_\mathcal{L}(P,Q)$ via $\delta_{P,P}$ and right composition, and
\[
\xymatrix{\pi_{P,Q}:\text{Mor}_\mathcal{L}(P,Q)\ar[r] & \text{Hom}_\mathcal{F}(P,Q)}
\]
is the orbit map for this action.
\item[(B)] For each $P,Q\in\text{Ob}(\mathcal{L})$ and each $g\in T_S(P,Q)$, the application $\pi_{P,Q}$ sends $\delta_{P,Q}(g)\in\text{Mor}_\mathcal{L}(P,Q)$ to $c_g\in\text{Hom}_\mathcal{F}(P,Q)$.
\item[(C)] For each $P,Q\in\text{Ob}(\mathcal{L})$, all $\psi\in\text{Mor}_\mathcal{L}(P,Q)$ and all $g\in P$, the diagram
\[
\xymatrix{ P\ar[d]_ {\delta_P(g)} \ar[r]^\psi & Q \ar[d]^{\delta_Q(\psi(g))} \\
P \ar[r]^\psi & Q}
\]
commutes in $\mathcal{L}$.
\end{itemize}
A \emph{$p$-local finite group} is a triple $(S,\mathcal{F},\mathcal{L})$ where $S$ is a $p$-group, $\mathcal{F}$ a saturated fusion system over $S$,
and $\mathcal{L}$ is a linking system associated to $\mathcal{F}$.
The \emph{classifying space} of $(S,\mathcal{F},\mathcal{L})$ is then given by $\pcompl{|\mathcal{L}|}$.
\end{defi}
\section{Cohomology and \texorpdfstring{$\mathcal{F}^c$}{Lg}-stable elements}\label{3}
In this section we introduce our cohomology functor with twisted coefficients defined on $\mathcal{F}^c$ and we define the notion of $\mathcal{F}^c$-stable elements. We refer the reader to \cite{We} for all the necessary results on homological algebra as well as the classicial notion of $\delta$-functor. We also refer the reader to \cite{We}, \cite{Br}, \cite{CE} or \cite{AM} for details on group cohomology.
\vspace{10pt}
Before introducing the notion of $\mathcal{F}^c$-stable elements, we need to understand the action of $S$ on a $\mathbb{Z}_{(p)}[\pi_1(|\mathcal{L}|)]$-module.
For each pair of $\mathcal{F}$-centric subgroups $P\leq Q$, set
\[\iota_P^Q=\delta_P^Q(1).\]
We denote by
\[\pi_\mathcal{L}=\pi_1(|\mathcal{L}|,S)
\]
the fundamental group of the geometric realization $|\mathcal{L}|$ with base point at the vertex $S$.
For $G$ a discrete group, let $\mathcal{B}(G)$ be the category with
a unique object, and morphism set equals to $G$ (hence, $|\mathcal{B}(G)|=BG$).
Consider the functor
\[\xymatrix{\omega:\mathcal{L}\ar[r] & \mathcal{B}(\pi_\mathcal{L})}\]
which maps each object to the unique object in the target and sends each morphism $\varphi\in\text{Mor}_\mathcal{L}(P,Q)$ to the class of the loop
$\iota_Q.\varphi. \overline{\iota_P}$ where $\overline{\iota_P}$ is the edge $\iota_P$ followed in the opposite direction.
In particular, every $\mathbb{Z}_{(p)}[\pi_\mathcal{L}]$-module is naturally a $\mathbb{Z}_{(p)}[S]$-module where the action is given by the following composition:
\[\xymatrix{\mathcal{B}(S)=\mathcal{B}(\text{Mor}_{\mathcal{T}^c_S(S)}(S,S))\ar[r]^-{\delta_S} & \mathcal{L} \ar[r]^-\omega & \mathcal{B}(\pi_\mathcal{L})}.\]
\vspace{10pt}
Let $(S,\mathcal{F},\mathcal{L})$ be a $p$-local finite group and let $M$ be a $\mathbb{Z}_{(p)}[\pi_\mathcal{L}]$-module. As we work with an action of $\pi_1(|\mathcal{L}|)$,
we can define a functor on $\mathcal{L}$ using the bi-functoriality of group cohomology.
Recall that group cohomology defines a contravariant functor
\[\xymatrix{H^*(-,-):\mathcal{D}\ar[r] & \mathbb{Z}_{(p)}\text{-Mod}}\]
where $\mathcal{D}$ is the category of pairs $(G,M)$ with $G$ a group and $M$ a $\mathbb{Z}_{(p)}[G]$-module. A morphism in $\mathcal{D}$ from $(G,M)$ to $(H,N)$ is a pair, $(\varphi,\rho)$ where $\xymatrix{\varphi:G\ar[r]& H}$ is a group homomorphism and $\xymatrix{\rho: N\ar[r] & M}$ is a linear map such that, for every $n\in N$ and
every $g\in G$, $g\rho(n)=\rho\left(\varphi(g)n\right)$.
Given $\varphi\in\text{Mor}(\mathcal{L})$, we have $(\pi(\varphi),\omega(\mathcal{L}))\in\text{Mor}(\mathcal{D})$ and we define our cohomology functor as the following.
\[
\xymatrix@R=1mm @C=1cm{**[r]H^*(-,M):& \mathcal{L} \ar[r] & **[r]\mathbb{Z}_{(p)}\text{-Mod} \\
& P\in\text{Ob}(\mathcal{L}) \ar@{|->}[r] & **[r]H^*(P,M)\\
& \varphi\in\text{Mor}_\mathcal{L}(P,Q)\ar@{|->}[r] & **[r] H^*(\varphi,M)=\varphi^*:=H^*(\pi(\varphi),\omega(\varphi)^{-1}).}
\]
For $P,Q$ two subgroups of $S$ and $\varphi\in\text{Mor}_\mathcal{L}(P,Q)$, $H^*(\varphi,M)$ can also be defined on the chain level as follows:
\[
\xymatrix@R=1mm{\text{Hom}_{\mathbb{Z}_{(p)}[Q]}\left(R_\bullet,M\right) \ar[r] & \text{Hom}_{\mathbb{Z}_{(p)}[P]}\left(R_\bullet,M\right)\\
f \ar@{|->}[r] & \left( \omega(\varphi)^{-1}\circ f\circ\pi(\varphi)_*\right)}
\]
where $(R_\bullet)$ is a projective resolution of the trivial $\mathbb{Z}_{(p)}[S]$-module $\mathbb{Z}_{(p)}$.
Finally, it can also be defined as the morphism between the two derived functors of $(-)^Q$ and $(-)^P$ induced by
\[
\xymatrix{ x\in M^Q \ar@{|->}[r] & \omega(\varphi)^{-1}x \in M^P.}
\]
By construction, it defines a morphism of $\delta$-functors.
\begin{prop}\label{phimorhdelta}
Let $(S,\mathcal{F},\mathcal{L})$ be a $p$-local finite group.
If $P,Q\leq S$ are $\mathcal{F}$-centric and $\varphi\in\text{Mor}_\mathcal{L}(P,Q)$, then
\[
\xymatrix{H^*(\varphi,-): H^*(Q,-)\ar[r] & H^*(P,-)}
\]
is a morphism of $\delta$-functors from $\left(H^*(Q,-),\delta_{H^*(Q,-)}\right)$ to $\left(H^*(P,-),\delta_{H^*(P,-)}\right)$.
\end{prop}
By construction, this functor naturally extends the group cohomology functor defined on $\mathcal{T}_S^c(S)$.
\[
\xymatrix{ \mathcal{T}_S^c(S)\ar[rr]^-{H^*(-,M)}\ar[rd]_\delta& & \mathbb{Z}_{(p)}\text{-Mod} \\
& \mathcal{L} \ar[ru]_-{H^*(-,M)}& }
\]
In particular, for every $P\leq S$ and $g\in P$, $H^*(\delta_P(g),M)=c_g^*$.
Moreover, it also factors through $\mathcal{F}^c$ along $\xymatrix{\pi:\mathcal{L}\ar[r]& \mathcal{F}^c}$.
\begin{prop}
Let $\varphi,\beta\in\text{Mor}_\mathcal{L}(P,Q)$ with $P,Q\in\mathcal{L}$.
If $\pi(\varphi)=\pi(\beta)$ then $H^*(\varphi,M)=H^*(\beta,M)$.
\end{prop}
\begin{proof}
If $\pi(\varphi)=\pi(\beta)$, then there exists $u\in Z(P)$ such that $\varphi=\beta\circ \delta_P(u)$ and thus
\[H^*(\varphi,M)=H^*(\delta_P(u),M)\circ H^*(\beta,M).\]
However $H^*(\delta_P(u),M)= H^*(\pi(\delta_P(u)),\omega(u)^{-1})=H^*(c_u,\omega(u)^{-1})=c_u^*$
is the automorphism of $H^*(P,M)$ induced by the conjugation by $u$, and, as $u\in Z(P)\leq P$,
this is the identity.
\end{proof}
In particular, if $\pi(\varphi)=\text{incl}^Q_P$, then $H^*(\varphi,M)=H^*(\iota^Q_P,M)=H^*(\text{incl}^Q_P,\text{Id}_M)=\text{Res}^Q_P$. Hence, $H^*(-,M)$ factors naturally through $\mathcal{F}^c$ along $\pi$.
For $M$ a $\mathbb{Z}_{(p)}[\pi_\mathcal{L}]$-module, $P,Q\leq S$ two $\mathcal{F}$-centric subgroups and $\varphi\in\text{Hom}_\mathcal{F}(P,Q)$ we write $\varphi^*:=H^*(\psi,M)$
where $\psi\in\text{Mor}_\mathcal{L}(P,Q)$ is such that $\pi(\psi)=\varphi$.
\begin{defi}
An element $x\in H^*(S,M)$ is called \emph{$\mathcal{F}$-centric stable}, or just \emph{$\mathcal{F}^c$-stable}, if for all $P\in\text{Ob}(\mathcal{F}^c)$ and all $\varphi\in\text{Hom}_\mathcal{F}(P,S)$,
\[\varphi^*(x)=\text{Res}_P^S(x).\]
We denote by $H^*(\mathcal{F}^c,M)\subseteq H^*(S,M)$ the submodule of all $\mathcal{F}^c$-stable elements.
\end{defi}
This submodule of $\mathcal{F}^c$-stable elements corresponds to the inverse limit of $H^*(-,M)$ on the category $\mathcal{F}^c$,
\[H^*(\mathcal{F}^c,M)\cong\limproj{\mathcal{F}^c} H^*(-,M).\]
Moreover, if $M$ is a $\mathbb{Z}_{(p)}$-module with a trivial action of $\pi_\mathcal{L}$, then, by Alperin's Fusion Theorem (Theorem \ref{AFT}),
$H^*(\mathcal{F}^c,M)\cong H^*(\mathcal{F},M)$ and the notion of $\mathcal{F}^c$-stable elements naturally extends the notion of $\mathcal{F}$-stable elements.
In general, we cannot expect to define a cohomology functor on all $\mathcal{F}$. For example, every morphism of $\mathcal{L}$ induces the identity on the trivial subgroup $\{e\}$. Thus, if the cohomology functor was defined on all $\mathcal{F}$, every morphism in $\mathcal{L}$ should act trivially on $M=H^0(\{e\},M)$, which is absurd.
Instead we consider the following construction.
Let $(S,\mathcal{F},\mathcal{L})$ be a $p$-local finite group.
Let $Q\leq S$ be a $\mathcal{F}$-centric subgroup of $S$, $P_0\leq Q$, $\psi\in\text{Aut}_\mathcal{L}(Q)$ and denote $P_1=\pi(\psi)(P_0)\leq Q$.
For $M$ a $\mathbb{Z}_{(p)}[\pi_\mathcal{L}]$-module, even if $P_0$ is not $\mathcal{F}$-centric, we can consider the morphism $\xymatrix{H^*(P_1,M)\ar[r]& H^*(P_0,M)}$
given by $H^*(\pi(\psi)|^{P_1}_{P_0},\omega(\psi))$. This can also be defined on the chain level by,
\[
\xymatrix@R=1mm{\text{Hom}_{\mathbb{Z}_{(p)}[P_1]}\left(R_\bullet,M\right) \ar[r] & \text{Hom}_{\mathbb{Z}_{(p)}[P_0]}\left(R_\bullet,M\right)\\
f \ar@{|->}[r] & \left( \omega(\psi)^{-1}\circ f\circ(\pi(\psi)|^{P_1}_{P_0})_*\right)}
\]
where $(R_\bullet)$ is a projective resolution of the trivial $\mathbb{Z}_{(p)}[S]$-module $\mathbb{Z}_{(p)}$.
This is well-defined because, by Definition \ref{linkdef} $(C)$,
\[\xymatrix @R=1mm { M \ar[r] & M \\
x\ar@{|->}[r] & \omega(\psi)^{-1}x}\]
defines a linear map such that, for every $p\in P_0$ and $x\in M$,
\[
\omega(\psi)^{-1} \omega\left(\delta_Q(p)\right) x=\omega\left(\delta_Q(\psi(p))\right) \omega(\psi)^{-1}x
\]
and thus $(\pi(\psi)|^{P_1}_{P_0},\omega(\psi))\in\text{Mor}(\mathcal{D})$.
Notice that if $P$ is $\mathcal{F}$-centric, then this is just $H^*(\psi,M)$.
Note also that, if the action of $\omega(\psi)$ on $M$ is trivial, this is just the usual morphism induced in cohomology by $\pi(\psi)|^{P_1}_{P_0}$.
Let $P,Q\leq S$ and $\varphi\in\text{Hom}_\mathcal{F}(P,Q)$. By Alperin's Fusion Theorem \ref{AFT}, there exist $P=P_0, P_1,\dots,P_r=\varphi(P)$ subgroups of $S$, $Q_1,\dots,Q_r$ $\mathcal{F}$-centric subgroups of $S$ and $\psi_i\in\text{Aut}_\mathcal{L}(Q_i)$ for every $i$ such that $\pi(\psi_i)(P_{i-1})=P_{i}$ and
\[\varphi=\pi(\psi_r)|^{P_r}_{P_{r-1}}\circ\pi(\psi_{r-1})|^{P_{r-1}}_{P_{r-2}}\circ\dots\circ\pi(\psi_1)|^{P_1}_{P_0}.\]
We then consider the following composite
\[
\xymatrix@C=3cm{ H^*(P_r,M) \ar[r]^-{H^*(\pi(\psi_1)|^{P_1}_{P_0},\omega(\psi_1)^{-1})} & \cdots \ar[r]^-{H^*(\pi(\psi_r)|^{P_r}_{P_{r-1}},\omega(\psi_r)^{-1})} & H^*(P,M)}
\]
composed on the right by $Res^Q_{P_r}$ which gives us a morphism
\[\xymatrix{H^*(Q,M)\ar[r] & H^*(P,M).}\]
Note that this morphism depends on the choice of the decomposition of $\varphi$ into restrictions of automorphisms of $\mathcal{F}$-centric subgroups and
not only on $\varphi$. As an example, we can again look at the trivial subgroup $\{e\}$ in a given fusion system $\mathcal{F}$:
each morphism in $\mathcal{F}^c$ restricts to the identity on $\{e\}$, but, if $M$ is not a trivial $\mathbb{Z}_{(p)}[\pi_\mathcal{L}]$-module,
not every $\varphi\in\text{Mor}(\mathcal{F}^c)$ acts trivially on $M=M^{\{e\}}=H^0(\{e\},M)$.
\begin{rem}\label{phimorhdelta2}
By construction, a morphism $\xymatrix{H^*(Q,M)\ar[r] & H^*(P,M)}$ obtained from $\varphi$ by this process defines a morphism of $\delta$-functors.
\end{rem}
\section{Bisets and idempotents}\label{4}
An important result in Broto, Levi and Oliver \cite{BLO2}, and a crucial tool in the proof of Theorem \ref{trivial},
is the existence of an $\mathcal{F}$-characteristic $(S,S)$-biset which leads to an idempotent of $H^*(S,\mathbb{F}_p)$ whose image is $H^*(\mathcal{F},\mathbb{F}_p)$.
This section is divided into several subsections. We start by recalling some results about left-free bisets. Next, we describe the interaction of bisets with cohomology with trivial coefficients. The third subsection analyzes the relation of bisets with nontrivial coefficients. After that, we present the construction of an idempotent from an $\mathcal{F}$-characteristic biset. As an example, we consider in a fifth subsection the particular case of constrained fusion systems. Finally, the last section gives the link with $\delta$-functors.
\subsection{Background on bisets}\label{4.1}
Let $G,H$ be two finite groups.
Transitive $(G,H)$-bisets (here, $G$ acts on the left and $H$ on the right) are isomorphic to bisets of the form $(G\times H)/K$ for $K$ a subgroup of $G\times H$.
We can then use the Goursat Lemma to describe all these subgroups.
Here, we are just interested in isomorphism classes of $(G,H)$-bisets where the action of $G$ is free.
In this setting, the classes of transitive left-free $(G,H)$-bisets are given by pairs $(K,\varphi)$,
where $K$ is a subgroup of $G$ and $\varphi\in\text{Hom}(K,H)$ is a group homomorphism.
\begin{nota}
For all $(K,\varphi)$, with $K$ a subgroup of $G$ and $\varphi\in\text{Hom}(K,H)$ a group homomorphism, we write
\[\Delta(K,\varphi)=\left\{(k,\varphi(k))\;;\; k\in K\right\}\leq G\times H.\]
For a $(G,H)$-pair $(K,\varphi)$, the set $\lbrace K,\varphi\rbrace:=(G\times H)/\Delta(K,\varphi)$ defines a $(G,H)$-biset.
Moreover, its isomorphic class is determined by the conjugacy class of $\Delta(K,\varphi)$ and we denote by $[K,\varphi]$ this class.
\end{nota}
We can also define a category $\mathcal{B}$, often called the \textit{Burnside category}, where the objects are the finite groups and,
for all finite groups $G$ and $H$, $\mathcal{B}(G,H)$ is the set of isomorphism classes of $(G,H)$-bisets.
The composition is given by the following construction.
\begin{defi} \label{compdef}
Let $G,H$ and $K$ be finite groups, $\Omega$ a $(G,H)$-biset and $\Lambda$ a $(H,K)$-biset.
We define,
\[\Omega\circ \Lambda=\Omega\times_H \Lambda=\Omega\times \Lambda/\sim\]
where, for all $x\in \Omega$, $y\in \Lambda$ and $h\in H$, $(x,h y)\sim (x h,y)$.
\end{defi}
This construction is compatible with isomorphisms, and, endowed with the induced composition law, $\mathcal{B}$ defines a category.
As we work with left-free bisets, we consider the subcategory $\mathcal{A}\subseteq \mathcal{B}$ where
the objects are the same but we restrict the morphisms to isomorphism classes of left-free bisets.
This gives us a category and the composition follows from the next lemma.
\begin{lem}\label{comp}
Let $G,H$ and $K$ be finite groups.
Let $[K,\varphi]\in\mathcal{A}(G,H)$ and $[L,\psi]\in \mathcal{A}(H,K)$.
Then, \[[K,\varphi]\circ[L,\psi]=\coprod_{x\in \varphi(K)\backslash H/L} [\varphi^{-1}(\varphi(K)\cap xLx^{-1}),\psi\circ c_{x^{-1}}\circ\varphi].\]
\end{lem}
\begin{proof}
We refer to that identity as the double coset formula and it is a direct consequence of \cite{Bo} Proposition 1.
\end{proof}
\subsection{\texorpdfstring{$\mathcal{F}$}{Lg}-characteristic bisets and trivial coefficients}\label{4.2}
Let $(S,\mathcal{F},\mathcal{L})$ be a $p$-local finite group.
When we work with trivial coefficients, the idea is to consider the category $\mathcal{A}_\mathcal{F}$ defined as follows. $\text{Ob}(\mathcal{A}_\mathcal{F})$ is the set of subgroups of $S$ and, for $P,Q\leq S$, $\mathcal{A}_\mathcal{F}(P,Q)$ is the set of isomorphism classes of $\mathcal{F}$-generated left-free $(P,Q)$-bisets, i.e. the $(P,Q)$-bisets union of transitive bisets of the form $[R,\varphi]$ with $R\leq P$ and $\varphi\in\text{Hom}_\mathcal{F}(R,Q)$.
Then, for $M$ a $\mathbb{Z}_{(p)}$-module, we construct a functor
\[\xymatrix{M:\mathcal{A}_\mathcal{F}\ar[r] & \mathbb{Z}_{(p)}\text{-Mod}}\]
defined on objects by $M(P)=H^*(P,M)$ for every $P\leq S$ and on morphisms as follows.
For every $P,Q\leq S$, $R\leq P$ and $\varphi\in\text{Hom}_\mathcal{F}(P,Q)$, $({}_P[R,\varphi]_Q)_*=\text{tr}_R^P\circ\varphi^*$. More generally, for every $\mathcal{F}$-generated left-free $(P,Q)$-biset $\Omega$ we define $\Omega_*$ by sum of its transitive components.
The existence of this functor will help us to construct an idempotent of $H^*(S,M)$ with image $H^*(\mathcal{F}^c,M)$.
For that, we also need the notion of $\mathcal{F}$-characteristic $(S,S)$-biset.
\begin{defi}\label{fcstable}
Let $\Omega$ be a left-free $(S,S)$-biset.
\begin{enumerate}[(a)]
\item We say that $\Omega$ is \emph{$\mathcal{F}$-generated} if it is the union of $(S,S)$-bisets of the form $[P,\varphi]$ with $P\in\text{Ob}(\mathcal{F})$ and $\varphi\in \text{Hom}_\mathcal{F}(P,S)$.
\item We say that $\Omega$ is \emph{left-$\mathcal{F}$-stable} if for all $P\in \text{Ob}(\mathcal{F})$ and $\varphi\in\text{Hom}_\mathcal{F}(P,S)$, we have
$_\varphi\Omega_S\cong {}_P\Omega_S$, i.e.
\[\left({}_P[P,\varphi]_S\right)\circ [\Omega]=\left({}_P[P,\text{incl}^S_P]_S\right)\circ [\Omega].\]
\item We say that $\Omega$ is \emph{right-$\mathcal{F}$-stable} if for all $P\in \text{Ob}(\mathcal{F})$ and $\varphi\in\text{Hom}_\mathcal{F}(P,S)$, we have
$_S\Omega_\varphi\cong {}_S\Omega_P$, i.e.
\[[\Omega]\circ \left({}_S[\varphi(P),\varphi^{-1}]_P\right) = [\Omega]\circ \left({}_S[P,\text{Id}_P^S]_P\right).\]
\item We say that $\Omega$ is \emph{non degenerate} if $|\Omega|/|S|\neq 0$ modulo $p$.
\end{enumerate}
If $\Omega$ satisfies all this four properties, we say that $\Omega$ is an $\mathcal{F}$-characteristic $(S,S)$-biset.
\end{defi}
The notion of $\mathcal{F}$-characteristic biset was first motivated by unpublished work of Linckelmann and Webb.
They are the ones who first formulated these conditions and
recognized the importance of finding a biset with these properties. Broto, Levi and Oliver proved that such a biset always exists if the fusion system is saturated.
\begin{prop}[\cite{BLO2}, Proposition 5.5]
Let $\mathcal{F}$ be a fusion system over a $p$-group $S$.
If $\mathcal{F}$ is saturated, then there exists an $\mathcal{F}$-characteristic $(S,S)$-biset.
\end{prop}
In fact, Ragnarsson and Stancu (\cite{RS}, Theorem A), and independently Puig (\cite{P7}, Proposition 21.9), proved that a fusion system $\mathcal{F}$ is saturated if, and only if,
there exists a $\mathcal{F}$-characteristic $(S,S)$-biset.
Let $M$ be a $\mathbb{Z}_{(p)}$-module, any $\mathcal{F}$-characteristic biset induces
an idempotent of $H^*(S,M)$ with image $H^*(\mathcal{F}^c,M)$.
\begin{prop}[cf. \cite{BLO2}, Proposition 5.5]\label{triv}
Let $(S,\mathcal{F},\mathcal{L})$ be a $p$-local finite group and $M$ be a $\mathbb{Z}_{(p)}$-module (with a trivial action of $\pi_\mathcal{L}$).
If $\Omega$ is an $\mathcal{F}$-characteristic biset, then $\frac{|S|}{|\Omega|}\Omega_*\in\text{End}(H^*(S,M))$ defines an idempotent with image $H^*(\mathcal{F}^c,M)$.
\end{prop}
\begin{proof}
In \cite{BLO2}, Proposition 5.5, this is proved for $M=\mathbb{F}_p$ but the general case works the same way.
\end{proof}
\subsection{Bisets and twisted coefficients}\label{4.3}
Let $(S,\mathcal{F},\mathcal{L})$ be a $p$-local finite group. When we work with twisted coefficients, one has to be more careful.
Unlike the case of trivial coefficients, defining a functor from $\mathcal{A}_\mathcal{F}$ to $\mathbb{Z}_{(p)}\text{-Mod}$ does not work in general.
In fact, for $M$ a $\mathbb{Z}_{(p)}[\pi_\mathcal{L}]$-module, our cohomological functor $H^*(-,M)$ cannot be defined on $\mathcal{F}$ but only on $\mathcal{F}^c$ and thus, we can only consider $\mathcal{F}^c$-generated bisets.
\begin{defi}
Let $P,Q$ be two $\mathcal{F}$-centric subgroups of $S$.
A left-free $(P,Q)$-biset is \emph{$\mathcal{F}^c$-generated} if it is an union of transitive bisets of the form $[R,\varphi]$ with $R\in\text{Ob}(\mathcal{F}^c)$ and $\varphi\in\text{Hom}_\mathcal{F}(R,Q)$.
\end{defi}
Unfortunately, we can see from Lemma \ref{comp} that the set of isomorphism classes of $\mathcal{F}^c$-generated bisets is not stable with respect to composition.
Hence, we can not, by analogy with $\mathcal{A}_\mathcal{F}$, define a category $\mathcal{A}_{\mathcal{F}^c}$
where the objects are the $\mathcal{F}$-centric subgroups of $S$
and for $P$ and $Q$ two $\mathcal{F}$-centric subgroups of $S$, $\mathcal{A}_{\mathcal{F}^c}(P,Q)$
is the set of isomorphism classes of $\mathcal{F}^c$-generated left-free $(P,Q)$-bisets.
Nevertheless, for all $\mathbb{Z}_{(p)}[\pi_\mathcal{L}]$-module $M$ and $P,Q\leq S$, we have a map from the set $A_{\mathcal{F}^c}(P,Q)$ of isomorphism classes of $\mathcal{F}^c$-generated left-free $(P,Q)$-bisets to $\text{Hom}(H^*(P,M),H^*(Q,M))$.
For $P,Q,R\in\mathcal{F}^c$ with $R\leq Q$ and $\varphi\in\text{Hom}_\mathcal{F}(R,P)$,
we can associate to the $(P,Q)$-pair $\lbrace R,\varphi\rbrace$ a morphism
\[\xymatrix@R=1mm{ \lbrace R,\varphi\rbrace_*=\text{tr}_R^Q\circ\varphi^*: H^*(P,M)\ar[r] & H^*(Q,M).}\]
If we consider another $(P,Q)$-biset $\lbrace R',\varphi'\rbrace$ isomorphic to $\lbrace R,\varphi\rbrace$
(this implies that $R'$ is also $\mathcal{F}$-centric), we obtain the same morphism.
Then we can set $[R,\varphi]_*$ as the composite $\text{tr}_R^Q\circ\varphi^*$ and it is well-defined.
Finally, for $\Omega$ a left-free $\mathcal{F}^c$-generated $(P,Q)$-biset, we define $\Omega_*$ by the sum of its transitive components.
\begin{rem}\label{Omegadeltafunctor}
By Proposition \ref{phimorhdelta}, for $\varphi\in\text{Mor}(\mathcal{F}^c)$, $\varphi^*=H^*(\varphi,-)$ is a morphism of $\delta$-functors.
Hence, as $\Omega_*$ is a sum of composites of transfers, restrictions and $\varphi^*$, for $\varphi\in\text{Mor}(\mathcal{F}^c)$,
which are all morphisms of $\delta$-functors, it is an endomorphism of the $\delta$-functor $\left(H^*(S,-),\delta_{H^*(S,-)}\right)$.
\end{rem}
\subsection{Idempotents and twisted coefficients}\label{4.4}
In general, an $\mathcal{F}$-characteristic biset is not $\mathcal{F}^c$-generated.
Hence, when we are working with twisted coefficients, we cannot use directly an $\mathcal{F}$-characteristic biset as in the trivial case.
However, we can define, from an $\mathcal{F}$-generated $(S,S)$-biset, an endomorphism of $H^*(S,M)$ but not in a unique way.
That is why we introduce the notion of $\Omega$-endomorphism.
\begin{defi}
Let $(S,\mathcal{F},\mathcal{L})$ be a $p$-local finite group.
Let $P\leq S$ and $\varphi\in\text{Hom}_\mathcal{F}(P,S)$. Let
\[
\varphi= \text{incl}_{\varphi(P)}^S\circ\pi(\psi_1)|^{\varphi(P)}_{P_{r-1}}\circ\dots\circ\pi(\psi_1)|^{P_1}_{P}
\]
be a decomposition of $\varphi$ into automorphisms of $\mathcal{F}$-centric subgroups.
Given a $\mathbb{Z}_{(p)}[\pi_\mathcal{L}]$-module $M$, a \emph{${}_S[P,\varphi]_S$-endomorphism} is an endomorphism of $H^*(S,M)$ given by the following composition
\begin{align*}
\text{tr}_P^S \circ \left( H^*(\pi(\psi_1)|^{P_1}_{P},\omega(\psi_1)^{-1}) \right.&\circ H^*(\pi(\psi_1)|^{P_2}_{P_1},\omega(\psi_2)^{-1})&\\
&\circ\left. \cdots \circ H^*(\pi(\psi_1)|^{\varphi(P)}_{P_{r-1}},\omega(\psi_r)^{-1})\circ\text{Res}_{\varphi(P)}^S \right).&
\end{align*}
More generally, we define an \emph{$\Omega$-endomorphism}, for $\Omega$ an $\mathcal{F}$-generated $(S,S)$-biset as a
sum of the previous morphisms given by the transitive components of $\Omega$.
\end{defi}
\begin{rem}\label{omegatrivial}
If the action of $\pi_\mathcal{L}$ on $M$ is trivial, every $\Omega$-endomorphism corresponds to $\Omega_*$ and,
if $\Omega$ is $\mathcal{F}$-characteristic, by Proposition \ref{triv}, $\frac{|S|}{|\Omega|}\Omega_*$ is an idempotent with image $H^*(\mathcal{F}^c,M)$.
\end{rem}
Let us look at the behavior of $\Omega$-endomorphisms induced by $\mathcal{F}$-characteristic $(S,S)$-bisets with $\mathcal{F}^c$-stable elements.
\begin{lem}\label{nontriv}
Let $(S,\mathcal{F},\mathcal{L})$ be a $p$-local finite group and $M$ be a $\mathbb{Z}_{(p)}[\pi_\mathcal{L}]$-module.
If $\Omega$ is an $\mathcal{F}$-characteristic $(S,S)$-biset and $\omega_*$ is an $\Omega$-endomorphism,
then $\frac{|S|}{|\Omega|}\omega_*\in\text{End}(H^*(S,M))$ restricted to $H^*(\mathcal{F}^c,M)$ is the identity.
\end{lem}
\begin{proof}
Let $P$ be a subgroup of $S$, $\varphi\in\text{Hom}_\mathcal{F}(P,S)$ and $\lambda_*$ a ${}_S[P,\varphi]_S$-endomorphism.
For every $x\in H^*(\mathcal{F}^c,M)$,
\[
\lambda_*(x)=\text{tr}_P^S\circ \text{Res}_P^S(x)=[S:P]x=\frac{|[P,\varphi]|}{|S|}x.
\]
Hence, for every $x\in H^*(\mathcal{F}^c,M)$,
\[\omega_*(x)= \frac{|\Omega|}{|S|}x.\]
\end{proof}
\begin{rem}\label{omegadeltafunctor}
Notice also that for $\Omega$ an $\mathcal{F}$-characteristic $(S,S)$-biset,
by construction and Remark \ref{phimorhdelta2}, an $\Omega$-endomorphism defines a morphism of $\delta$-functors.
\end{rem}
We remind the reader that in this article, every $p$-group is finite. This turn out to be a crucial property in the following result.
\begin{prop}\label{idem}
Let $(S,\mathcal{F},\mathcal{L})$ be a $p$-local finite group and let $M$ be an abelian $p$-group with an action of $\pi_\mathcal{L}$.
Let $\Omega$ be an $\mathcal{F}$-characteristic $(S,S)$-biset and $\omega_*$ an $\Omega$-endomorphism.
For every $k\geq 0$, there is a natural number $N_{k,M}>0$ such that
$\left(\frac{|S|}{|\Omega|}\omega_k\right)^{N_{k,M}}$ defines an idempotent $\overline{\omega}_{k,M}$ of $H^k(S,M)$ and we have
\[H^k(\mathcal{F}^c,M)\subseteq \text{Im}\,(\overline{\omega}_{k,M}).\]
\end{prop}
\begin{proof}
To simplify the notation, we write $\omega=\frac{|S|}{|\Omega|}\omega_k$.
For any $k\geq 0$, we have the following decreasing family of subgroups of $H^k(S,M)$.
\[
H^k(\mathcal{F}^c,M)\subseteq \dots \subseteq \text{Im}\,(\omega^r)\subseteq\text{Im}\,(\omega^{r-1})\subseteq\dots \subseteq \text{Im}\,(\omega^1)\subseteq\text{Im}\,(\omega^0)=H^k(S,M).
\]
As $H^k(S,M)$ is a finite abelian $p$-group, this sequence stabilizes.
Thus there is an $n_0\geq 1$ such that for all $n\geq n_0$ $\text{Im}\,(\omega^n)=\text{Im}\,(\omega^{n_0})$.
In particular, $\omega^{n_0}|_{\text{Im}\,(\omega^{n_0})}$ is a permutation of the finite set $\text{Im}\,(\omega^{n_0})$ and there is an $l$
such that $(\omega^{n_0}|_{\text{Im}\,(\omega^{n_0})})^l=\text{Id}_{\text{Im}\,(\omega^{n_0})}$.
Thus, for $N_{k,M}=l\times n_0$, the endomorphism $\overline{\omega}_{k,M}=\omega^{N_{k,M}}\in\text{End}(H^k(S,M))$ is an idempotent with image $\text{Im}\,(\omega^{n_0})\supseteq H^k(\mathcal{F}^c,M)$.
\end{proof}
Hence, we can define an idempotent of $H^*(S,M)$ as follows. For every $k\geq 0$ and every $x\in H^k(S,M)$,
\[
\overline{\omega}_{k,M}(x)=\left(\frac{|S|}{|\Omega|}\omega_k\right)^{\prod_{i=0}^k N_{i,M}} (x).
\]
Moreover, this definition only depends on the $\Omega$-endomorphism $\omega$.
\begin{defi}
For $\Omega$ an $\mathcal{F}$-characteristic $(S,S)$-biset and $\omega_*$ an $\Omega$-endomorphism,
the idempotent $\overline{\omega}_{*,-}$ of $H^*(S,-)$ obtained by the previous process is called the \emph{$\mathcal{F}^c$-characteristic idempotent} associated to $\omega$.
Let $M$ be an abelian $p$-group with an action of $\pi_\mathcal{L}$, we denote by $I_\omega^*(M)\subseteq H^*(S,M)$ the image of $\overline{\omega}_{*,M}$.
\end{defi}
\begin{rem}\label{rk}
Notice that, by Remark \ref{omegatrivial}, if the action on $M$ is trivial, then $I_\omega^*(M)=H^*(\mathcal{F}^c,M)$.
\end{rem}
\begin{prop}\label{Omegadelta}
Let $(S,\mathcal{F},\mathcal{L})$ be a $p$-local finite group.
If $\Omega$ is an $\mathcal{F}$-characteristic $(S,S)$-biset and $\omega_*$ an $\Omega$-endomorphism,
then $\overline{\omega}_{*,-}$, the $\mathcal{F}^c$-characteristic idempotent induced by $\omega_*$,
defines an endomorphism of the $\delta$-functor $\left(H^*(S,-),\delta_{H^*(S,-)}\right)$.
\end{prop}
\begin{proof}
For $M$ an abelian $p$-group with an action of $\pi_\mathcal{L}$ and $k\geq 0$, we denote by $N_{k,M}$ a natural number as in Proposition \ref{idem}.
We have first to show that $\overline{\omega}_{*,-}$, the $\mathcal{F}^c$-characteristic idempotent associated to $\omega_*$,
defines a natural transformation from the functor $H^*(S,-)$ to itself.
For every pair of abelian $p$-groups $(M,N)$ with an action of $\pi_\mathcal{L}$ and every $\varphi\in\text{Hom}_{\mathbb{Z}_{(p)}[\pi_\mathcal{L}]}(M,N)$,
let us consider, for $k\geq 0$, the following diagram,
\[
\xymatrix{
H^k(S,M) \ar[r]^\text{Id} \ar[d]_{\overline{\omega}_{k,M}} & H^k(S,M) \ar[r]^{\varphi_k} \ar[d]_{\tilde{\omega}_{k,M,N}} & H^k(S,N) \ar[d]_{\tilde{\omega}_{k,M,N}} \ar[r]^\text{Id} & H^k(S,N) \ar[d]^{\overline{\omega}_{k,N}} \\
H^k(S,M) \ar[r]_\text{Id} & H^k(S,M) \ar[r]_{\varphi_k} & H^k(S,N) \ar[r]_\text{Id} & H^k(S,N)
}
\]
where $\tilde{\omega}_{k,M,N}=\left(\omega_k\right)^{\prod_{i=0}^k N_{i,M}\times \prod_{i=0}^k N_{i,N} }$.
The middle square commutes as $\tilde{\omega}_{k,M,N}$ is a finite iteration of $\frac{|S|}{|\Omega|}\omega_k$
and $\omega_*$ is an endomorphism of $\delta$-functors by Remark \ref{omegadeltafunctor}.
The leftmost square commutes because, as $\overline{\omega}_{k,M}$ is an idempotent of $H^k(S,M)$,
$\tilde{\omega}_{k,M,N}=\overline{\omega}_{k,M}^{\prod_{i=0}^k N_{i,N}}=\overline{\omega}_{k,M}$.
Finally, the rightmost one commutes because, as $\overline{\omega}_{k,N}$ is an idempotent of $H^k(S,N)$,
$\tilde{\omega}_{k,M,N}=\overline{\omega}_{k,N}^{\prod_{i=0}^k N_{i,M}}=\overline{\omega}_{k,N}$.
Hence, the exterior diagram commutes.
Now, to show that it defines a morphism of $\delta$-functor,
let us consider a short exact sequence of abelian $p$-groups with an action of $\pi_\mathcal{L}$, $\xymatrix{0\ar[r] & L\ar[r] & M\ar[r] & N\ar[r] & 0}$.
By the previous argument we just have to show that, for $k\geq 0$, the following diagram commutes,
\[
\xymatrix{
H^k(S,N) \ar[r]^{\delta}\ar[d]_{\overline{\omega}_{k,N}} & H^{k+1}(S,L) \ar[d]^{\overline{\omega}_{k+1,L}} \\
H^k(S,N) \ar[r]_\delta & H^{k+1}(S,L)
}
\]
where $\delta=\delta_{H^*(S,-)}$ corresponds to the connecting homomorphism.
Consider then the following diagram,
\[
\xymatrix{
H^k(S,N) \ar[r]^\text{Id} \ar[d]_{\overline{\omega}_{k,N}} & H^k(S,N) \ar[r]^{\delta} \ar[d]_{\tilde{\omega}_{k,L,N}} & H^{k+1}(S,L) \ar[d]_{\tilde{\omega}_{k+1,L,N}} \ar[r]^\text{Id} & H^{k+1}(S,L) \ar[d]^{\overline{\omega}_{k+1,L}} \\
H^k(S,N) \ar[r]_\text{Id} & H^k(S,N) \ar[r]_{\delta} & H^{k+1}(S,L) \ar[r]_\text{Id} & H^{k+1}(S,L)
}
\]
where
\[\tilde{\omega}_{k,L,N}=\left(\overline{\omega}_k\right)^{\prod_{i=0}^{k+1} N_{i,L}\times \prod_{i=0}^{k+1} N_{i,N} }\]
and
\[\tilde{\omega}_{k+1,L,N}=\left(\overline{\omega}_{k+1}\right)^{\prod_{i=0}^{k+1} N_{i,L}\times \prod_{i=0}^{k+1} N_{i,N} }.\]
The middle square commutes as $\tilde{\omega}_{k,L,N}$ and $\tilde{\omega}_{k+1,L,N}$
are finite iterations of $\overline{\omega}_k$ and $\overline{\omega}_{k+1}$, and $\overline{\omega}_*$
is an endomorphism of $\delta$-functors by Remark \ref{omegadeltafunctor}.
The leftmost square commutes because, as $\overline{\omega}_{k,N}$ is an idempotent of $H^k(S,N)$,
$\tilde{\omega}_{k,L,N}=\overline{\omega}_{k,N}^{N_{k+1,N}\times\prod_{i=0}^{k+1} N_{i,L}}=\overline{\omega}_{k,N}$.
The rightmost one commutes because, as $\overline{\omega}_{k,L}$ is an idempotent of $H^k(S,L)$,
$\tilde{\omega}_{k+1,L,N}=\overline{\omega}_{k+1,L}^{\prod_{i=0}^{k+1} N_{i,N}}=\overline{\omega}_{k+1,L}$.
Thus, the exterior diagram commutes.
\end{proof}
\subsection{The idempotent for a constrained fusion system}\label{4.5}
When we work with a constrained fusion system, the $(S,S)$-characteristic biset is $\mathcal{F}^c$-generated and, working
with a suitable category, it induces, for every $\mathbb{Z}_{(p)}[\pi_\mathcal{L}]$-module $M$, an idempotent of $H^*(S,M)$ with image $H^*(\mathcal{F}^c,M)$.
Let us first recall the notion of constrained fusion system.
\begin{defi}\label{constrained}
Let $\mathcal{F}$ be a fusion system over a $p$-group $S$.
A subgroup $Q\leq S$ is \emph{normal in $\mathcal{F}$} if $Q\trianglelefteq S$, and for all $P,R\leq S$ and every $\varphi\in\text{Hom}_\mathcal{F}(P,R)$, $\varphi$ extends to a morphism
$\overline{\varphi}\in\text{Hom}_\mathcal{F}(PQ,RQ)$ such that $\overline{\varphi}(Q)=Q$.
We write $O_p(\mathcal{F})$ for the maximal subgroup of $S$ which is normal in $\mathcal{F}$.
We say that $\mathcal{F}$ is \emph{constrained} if $O_p(\mathcal{F})$ is $\mathcal{F}$-centric.
\end{defi}
Define, for $\mathcal{F}$ a fusion system over a $p$-group $S$ and $P_0$ a subgroup of $S$, $\mathcal{A}_{\mathcal{F}\geq P_0}$
as follow.
\[\text{Ob}(\mathcal{A}_{\mathcal{F}\geq P_0})=\lbrace P_0\leq P\leq S\rbrace\text{ is the set of all subgroups of $S$ containing $P_0$}\]
and for all $P,Q\in\text{Ob}(\mathcal{A}_{\mathcal{F}\geq P_0})$,
\[\mathcal{A}_{\mathcal{F}\geq P_0}(P,Q)=\lbrace\text{$\mathcal{F}$-generated left-free $(P,Q)$-bisets union of $[R,\varphi]$ with $R\geq P_0$}\rbrace.\]
$A_{\mathcal{F}\geq P_0}$ is not in general a subcategory of $\mathcal{A}_\mathcal{F}$.
The problem comes from Lemma \ref{comp}:
the set \[\text{Mor}(\mathcal{A}_{\mathcal{F}\geq P_0})=\bigsqcup_{P,Q\in\text{Ob}(\mathcal{A}_{\mathcal{F}\geq P_0})}\mathcal{A}_{\mathcal{F}\geq P_0}(P,Q)\] is not stable with respect to composition.
But it is stable when the subgroup $P_0\leq S$ is \textit{weakly closed} in $\mathcal{F}$, i.e.
$P^\mathcal{F}=\{P\}$.
\begin{lem}
Let $\mathcal{F}$ be a fusion system over a $p$-group $S$.
If $P_0\trianglelefteq S$ is weakly closed in $\mathcal{F}$, then $\mathcal{A}_{\mathcal{F}\geq P_0}$, with the composition defined in \ref{compdef}, is a subcategory of $\mathcal{A}_\mathcal{F}$.
\end{lem}
\begin{proof}
As $P_0$ is weakly closed in $\mathcal{F}$, for every $R,P\geq P_0$, $s\in S$ and $\varphi\in\text{Hom}_{\mathcal{F}}(R,S)$,
\[\varphi^{-1}(\varphi(R)\cap sPs^{-1})\geq \varphi^{-1}(\varphi(P_0)\cap sP_0s^{-1})=P_0.\]
Thus, by Lemma \ref{comp}, $\text{Mor}(\mathcal{A}_{\mathcal{F}\geq P_0})$ is stable with respect to composition
and $\mathcal{A}_{\mathcal{F}\geq P_0}$ defines a subcategory of $\mathcal{A}_\mathcal{F}$.
\end{proof}
For example, the subgroup $O_p(\mathcal{F})$ is normal in $\mathcal{F}$. Thus it is weakly closed in $\mathcal{F}$ and $\mathcal{A}_{\mathcal{F}\geq O_p(\mathcal{F})}$ is a subcategory of $\mathcal{A}_\mathcal{F}$.
\vspace{10pt}
When $\mathcal{F}$ is constrained, $O_p(\mathcal{F})$ is $\mathcal{F}$-centric. Thus, every biset $\Omega\in\text{Mor}(\mathcal{A}_{\mathcal{F}\geq O_p(\mathcal{F})})$ is $\mathcal{F}^c$-generated.
Hence, if $\mathcal{F}$ is constrained, for every $\mathbb{Z}_{(p)}[\pi_\mathcal{L}]$-module $M$, we have, as in the trivial case, a functor
\[
\xymatrix@R=1mm{\mathcal{A}_{\mathcal{F}\geq O_p(\mathcal{F})}\ar[r] & \mathbb{Z}_{(p)}\text{-Mod} \\
P\ar@{|->}[r] & H^*(P,M) \\
{}_P[R,\varphi]_Q\ar@{|->}[r]& \text{tr}_R^P\circ\varphi^*.}
\]
Moreover, if we look at the \emph{minimal} $\mathcal{F}$-characteristic $(S,S)$-biset i.e. the smallest $\mathcal{F}$-characteristic $(S,S)$-biset, we have the following.
\begin{prop}\label{constrained biset}
Let $\mathcal{F}$ be a constrained fusion system over a $p$-group $S$.
If $\Omega$ is the minimal $\mathcal{F}$-characteristic biset, then $\Omega\in\mathcal{A}_{\mathcal{F}\geq O_p(\mathcal{F})}$.
\end{prop}
\begin{proof}
This is a direct corollary of \cite{GRh}, Proposition 9.11.
Indeed, by \cite{GRh}, Proposition 9.11, every $[P,\varphi]$ which appears in the decomposition of $\Omega$ satisfies $P\geq O_p(\mathcal{F})$.
\end{proof}
Hence, using the same argument as for Proposition \ref{triv}, we have the following theorem.
\begin{thm}
Let $(S,\mathcal{F},\mathcal{L})$ be a $p$-local finite group and $M$ be a $\mathbb{Z}_{(p)}[\pi_\mathcal{L}]$-module.
Let $\Omega$ be the minimal $\mathcal{F}$-characteristic $(S,S)$-biset.
If $\mathcal{F}$ is a constrained fusion system, then $\frac{|S|}{|\Omega|}\Omega_*\in\text{End}(H^*(S,M))$ is an idempotent with image the $\mathcal{F}^c$-stable elements $H^*(\mathcal{F}^c,M)$.
\end{thm}
\begin{proof}
By Proposition \ref{constrained biset}, $\Omega\in\mathcal{A}_{\mathcal{F}\geq O_p(\mathcal{F})}$, and the proof is the same as the proof of Proposition \ref{triv}.
\end{proof}
\subsection{A \texorpdfstring{$\delta$}{Lg}-functor}\label{4.6}
Let $(S,\mathcal{F},\mathcal{L})$ be a $p$-local finite group, $\Omega$ be an $\mathcal{F}$-characteristic $(S,S)$-biset,
and $\omega_*$ an $\Omega$-endomorphism.
For $M$ an abelian $p$-group with an action of $\pi_\mathcal{L}$,
let $\overline{\omega}_{*,-}\in\text{End}(H^*(S,M))$ be the associated $\mathcal{F}^c$-characteristic idempotent.
Let us start with the behavior of $\delta$-functors with idempotents.
We recall that a $\delta$-functor can be seen as a functor from the category $\mathcal{S}_\mathcal{A}$ of short exact sequences in $\mathcal{A}$ to $\text{Ch}(\mathcal{B})$, the category of $\mathbb{Z}$-graded chain complexes in $B$, which sends any short exact sequence to an acyclic chain complex. We refer the reader to \cite{We} for more details and properties.
\begin{lem}\label{imexact}
Let
$(M_*,f_*)=\left(\cdots\xrightarrow{f_{l-2}}M_{l-1}\xrightarrow{f_{l-1}} M_l\xrightarrow{f_l} M_{l+1}\xrightarrow{f_{l+1}} \cdots\right)_{l\in\mathbb{Z}}$ be
a long exact sequence in an abelian category $\mathcal{A}$.
Let $i_*:(M_*,f_*)\rightarrow (M_*,f_*)$ be a morphism of long exact sequences such that, for all $l\in\mathbb{Z}$, $i_l$ is an idempotent of $M_l$.
Then the sequence \[\cdots\xrightarrow{f_{l-2}}\text{Im}\,(i_{l-1})\xrightarrow{f_{l-1}}\text{Im}\,(i_l)\xrightarrow{f_l} \text{Im}\,(i_{l+1})\xrightarrow{f_{l+1}} \cdots\]
is exact.
\end{lem}
\begin{proof}
Let $l\in \mathbb{Z}$ and $x\in \text{Im}\,(i_l)$ such that $f_l(x)=0$. By exactness of $(M_*,f_*)$ in $l$,
there is a $y\in M_ {l-1}$ such that $f_{l-1}(y)=x$.
Thus $x=i_l(x)=i_l\circ f_{l-1}(y)=f_{l-1}\circ i_{l-1}(y)$ and hence we obtain the exactness of $(\text{Im}\,(i_*),f_*)$ in degree $l$.
\end{proof}
\begin{prop}\label{deltaidem}
Let $\mathcal{A},\mathcal{B}$ be two abelian categories and let $\left(F^*,\delta_F\right):\mathcal{A}\rightarrow\mathcal{B}$ be a $\delta$-functor.
If $\xymatrix{i^*:\left(F^*,\delta_F\right)\rightarrow \left(F^*,\delta_F\right)}$
is an idempotent of $\delta$-functors, then $\left(\text{Im}\,(i^*),\delta_F\right)$ defines a $\delta$-functor.
\end{prop}
\begin{proof}
A $\delta$-functor can be seen as a functor from the category $\mathcal{S}_\mathcal{A}$ of short exact sequences in $\mathcal{A}$ to $\text{Ch}(\mathcal{B})$
which sends any short exact sequence to an acyclic chain complex.
A morphism of $\delta$-functors is then a natural transformation in that setting.
With this point of view, this is just a corollary of Lemma \ref{imexact}.
\end{proof}
\begin{thm}\label{deltafunctor}
Let $(S,\mathcal{F},\mathcal{L})$ be a $p$-local finite group, $\Omega$ an $\mathcal{F}$-characteristic $(S,S)$-biset
and $\omega_*$ an $\Omega$-endomorphism.
Then, the functor $I_\omega^*(-)$, with the connecting homomorphism $\delta_{H^*(S,-)}$,
defines a $\delta$-functor from the category of finite $\mathbb{Z}_{(p)}[\pi_\mathcal{L}]$-modules to $\mathbb{Z}_{(p)}\text{-Mod}$.
\end{thm}
\begin{proof}
This is a direct corollary of Proposition \ref{Omegadelta} and Proposition \ref{deltaidem}.
\end{proof}
In the next section, we will show that if the action on $M$ is nilpotent, then $I_\omega^*(M)=H^*(\mathcal{F}^c,M)$.
But this is not clear at all in general. This raised the following question.
\begin{conj}
Let $(S,\mathcal{F},\mathcal{L})$ be a $p$-local finite group, $\Omega$ an $\mathcal{F}$-characteristic $(S,S)$-biset and $\omega$ an $\Omega$-endomorphism. If $M$ is an abelian $p$-group with an action of $\pi_1(\pcompl{|\mathcal{L}|})$, then
\[H^*(\mathcal{F}^c,M)\cong I_\omega^*(M).\]
\end{conj}
We insist that, in view of the counterexamples given by Levi and Ragnarsson (\cite{LR} Proposition 3.1), we cannot expect $I_\omega^*(M)$ to be isomorphic to $H^*(|\mathcal{L}|,M)$ in general. But it is natural to ask if $I_\omega(M)$ always corresponds to the $\mathcal{F}^c$-stable elements.
\section{The cohomology of the geometric realization of a linking system with nilpotent coefficients}\label{5}
We give here a proof of the main theorem.
\begin{lem}\label{lem2}
Let $(S,\mathcal{F},\mathcal{L})$ be a $p$-local finite group.
The natural inclusion $\delta_S$ of $\mathcal{B}(S)$ in $\mathcal{L}$ induces, for any $\mathbb{Z}_{(p)}[\pi_\mathcal{L}]$-module $M$, a natural morphism in cohomology
\[ \xymatrix{ H^*(|\mathcal{L}|,M)\ar[r] & H^*(\mathcal{F}^c,M)\subseteq H^*(S,M).}\]
\end{lem}
\begin{proof}
This follows easily from the functoriality of the geometric realization.
\end{proof}
\begin{lem}\label{lem}
Let $(S,\mathcal{F},\mathcal{L})$ be a $p$-local finite group and let $\Omega$ be an $\mathcal{F}$-characteristic $(S,S)$-biset.
Let also $0\rightarrow L \rightarrow M \rightarrow N \rightarrow 0$ be a short exact sequence of finite $\mathbb{Z}_{(p)}[\pi_\mathcal{L}]$-modules.
If $\delta_S$ induces isomorphisms $ H^*(|\mathcal{L}|,L)\cong I_\omega^*(L)$ and $ H^*(|\mathcal{L}|,N)\cong I^*_\omega(N)$,
then $\delta_S$ induces an isomorphism \[ H^*(|\mathcal{L}|,M)\cong I_\omega^*(M).\]
\end{lem}
\begin{proof}
Consider the exact sequences in cohomology induced by the short exact sequence
\[\xymatrix{0 \ar[r] & L \ar[r] & M\ar[r] & N \ar[r] & 0}\]
and look at the following diagram (where $\overline{\omega}_{*,-}$ denote the $\mathcal{F}^c$-characteristic idempotent associated to $\Omega$).
\[
\xymatrix{\cdots\ar[r] & H^{n-1}(|\mathcal{L}|,N)\ar[r]\ar[d]_{\overline{\omega}_{n-1,N}\circ\delta_S^*} & H^{n}(|\mathcal{L}|,L)\ar[r]\ar[d]_{\overline{\omega}_{n,L}\circ\delta_S^*} & H^{n}(|\mathcal{L}|,M)\ar[r]\ar[d]_{\overline{\omega}_{n,M}\circ\delta_S^*} & H^{n}(|\mathcal{L}|,N)\ar[r]\ar[d]_{\overline{\omega}_{n,N}\circ\delta_S^*} & \cdots \\
\cdots\ar[r] & I^{n-1}_\Omega (N)\ar[r] & I^{n}_\Omega (L)\ar[r] & I^{n}_\Omega (M)\ar[r] & I^{n}_\Omega (N)\ar[r] & \cdots}
\]
As $H^*(|\mathcal{L}|,-)$ is a $\delta$-functor and, by Theorem \ref{deltafunctor}, $I^{*}_\Omega$ is also a $\delta$-functor,
the two lines are exact and, as by Proposition \ref{Omegadelta} the $\mathcal{F}$-characteristic idempotent associated to $\Omega$
defines a morphism of $\delta$-functors, this diagram is commutative.
An application of the Five Lemma then finishes the proof.
\end{proof}
We can now state the main theorem.
\begin{thm}\label{main}
Let $(S,\mathcal{F},\mathcal{L})$ be a $p$-local finite group.
If $M$ is an abelian $p$-group with a nilpotent action of $\pi_1(|\mathcal{L}|)$, then $\delta_S$ induces a natural isomorphism
\[ H^*(|\mathcal{L}|,M)\cong H^*(\mathcal{F}^c,M).\]
\end{thm}
\begin{proof}
As the action of $\pi_\mathcal{L}$ is nilpotent, there is a sequence
\[0=M_0\subseteq M_1\subseteq\dots\subseteq M_n=M\]
such that, for every $1\leq i \leq n$, the action of $\pi_\mathcal{L}$ on $M_{i}/M_{i-1}$ is trivial.
We know, by Theorem \ref{trivial} and Remark \ref{rk}, that for $1\leq i \leq n$, $\delta_S$ induces an isomorphism
\[H^*(|\mathcal{L}|,M_{i}/M_{i-1})\cong H^*(\mathcal{F}^c,M_{i}/M_{i-1})=I^*_\Omega(M_{i}/M_{i-1}).\]
By induction on $n$, and by Lemma \ref{lem}, we get that $H^*(|\mathcal{L}|,M)\cong I_\omega^*(M)$.
Finally we also have by Lemma \ref{lem2} that \[\delta_S(H^*(|\mathcal{L}|,M))\subseteq H^*(\mathcal{F}^c,M)\subseteq I_\omega^*(M).\]
Then $H^*(|\mathcal{L}|,M)\cong H^*(\mathcal{F}^c,M)= I^*_\Omega(M)$.
\end{proof}
\section{The cohomology with twisted coefficients of \texorpdfstring{$p$}{Lg}-good spaces}\label{6}
We finish with a result on the cohomology with twisted coefficients of the $p$-completion of a $p$-good space
and we apply it, with Theorem \ref{main}, to compare the
cohomology with twisted coefficients of $|\mathcal{L}|$ and the $\mathcal{F}^c$-stable elements.
We refer the reader to Bousfield and Kan \cite{BK} for more details about $p$-completion.
There is also a brief introduction in Aschbacher, Kessar and Oliver \cite{AKO}.
Here, for $X$ a space,
\[\xymatrix{\lambda_X:X\rightarrow \pcompl{X}}\]
denote the structural natural transformation of the $p$-completion and we recall that if $X$ is $p$-good, it induces an isomorphism
\[
H^*(\pcompl{X},\mathbb{F}_p)\cong H^*(X,\mathbb{F}_p).
\]
\begin{lem}\label{5lemmapcomp}
Let $X$ be a space and let $0\rightarrow L \rightarrow M \rightarrow N \rightarrow 0$ be a short exact sequence of $\mathbb{Z}_{(p)}[\pi_1(\pcompl{X})]$-modules.
If $\lambda_X$ induces isomorphisms $ H^*(\pcompl{X},L)\cong H^*({X},L)$ and $ H^*(\pcompl{X},N)\cong H^*({X},N)$,
then $\lambda_X$ induces an isomorphism \[ H^*(\pcompl{X},M)\cong H^*({X},M).\]
\end{lem}
\begin{proof}
This is a straightforward application of the Five Lemma.
\end{proof}
\begin{prop}\label{pgoodspace}
Let $X$ be a space and $M$ be an abelian $p$-group with an action of $\pi_1(\pcompl{X})$.
If $X$ is $p$-good and $\pi_1(\pcompl{X})$ is a finite $p$-group, then $\lambda_X$ induces a natural isomorphism
\[
H^*(\pcompl{X},M)\cong H^*(X,M).
\]
\end{prop}
\begin{proof}
As $X$ is $p$-good, $\lambda_X$ induces an isomorphism,
$H^*(\pcompl{X},\mathbb{F}_p)\cong H^*(X,\mathbb{F}_p).$
Moreover, as $\pi_1(\pcompl{X})$ is a $p$-group quotient of $\pi_1(X)$, the action of $\pi_1(\pcompl{X})$ on $M$ is nilpotent:
there is a sequence
\[\{0\}=M_0\subseteq M_1\subseteq\dots\subseteq M_n=M\]
such that, for any $1\leq i \leq n$, $M_{i}/M_{i-1}\cong \mathbb{F}_p$ is the trivial module.
We conclude by induction on $n$ using Lemma \ref{5lemmapcomp}.
\end{proof}
\begin{cor}\label{pgood}
Let $(S,\mathcal{F},\mathcal{L})$ be a $p$-local finite group.
If $M$ is an abelian $p$-group with an action of $\pi_1(\pcompl{|\mathcal{L}|})$, $\lambda_{|\mathcal{L}|}$ induces an isomorphism
\[H^*(\pcompl{|\mathcal{L}|},M)\cong H^*(|\mathcal{L}|,M).\]
\end{cor}
\begin{proof}
As $|\mathcal{L}|$ is a $p$-good space and $\pi_1(\pcompl{|\mathcal{L}|})$ is a finite $p$-group (\cite{AKO}, Theorem III.4.17), we can apply Proposition \ref{pgoodspace}.
\end{proof}
\begin{cor}\label{main2}
Let $(S,\mathcal{F},\mathcal{L})$ be a $p$-local finite group.
If $M$ is an abelian $p$-group with an action of $\pi_1(\pcompl{|\mathcal{L}|})$, then $\lambda_{|\mathcal{L}|}\circ\delta_S^*$ induces a natural isomorphism
\[
H^*(\pcompl{|\mathcal{L}|},M)\cong H^*(\mathcal{F}^c,M).
\]
\end{cor}
\begin{proof}
By \cite{AKO}, Theorem III.4.17, $|\mathcal{L}|$ is $p$-good and $\pi_1(\pcompl{|\mathcal{L}|})$ is a $p$-group. In particular, the action of $\pi_1(\pcompl{|\mathcal{L}|})$ on $M$ is nilpotent.
Hence, this is just a corollary of Theorem \ref{main} and Corollary \ref{pgood}.
\end{proof}
|
{
"timestamp": "2016-09-14T02:00:52",
"yymm": "1504",
"arxiv_id": "1504.03191",
"language": "en",
"url": "https://arxiv.org/abs/1504.03191"
}
|
\section{Introduction}
Brown dwarfs (BDs) are the bridge between late-type stars and planetary-mass objects.
T dwarfs (with effective temperatures of $T_{\rm eff}\sim$ 1500-500\,K) and Y-dwarfs ($T_{\rm eff} \leq$ 500\,K )
are among the coolest and least luminous objects detected,
and their properties can help us to understand giant extrasolar planets.
In recent years, a significant number of BDs have
been identified in wide-field optical and infrared imaging surveys, e.g.,
the Two Micron All Sky Survey \citep[2MASS,][]{Skrutskie2006}, the DEep Near Infrared Survey \citep[DENIS,][]{Epchtein1997},
or the UKIRT Infrared Deep Sky Survey \citep[UKIDSS,][]{Lawrence2007}.
For the coolest objects, deep near-IR and mid-IR surveys such as the Canada-France Brown Dwarf Survey \citep[CFBDS,][]{Delorme2008}
or the Wide-Field Infrared Survey Explorer \citep[WISE,][]{Wright2010} have contributed to significantly increasing the
number of known T and Y-dwarfs \citep[e.g., ][]{Burningham2010,Albert2011,Kirk2011,Cushing2011,Kirk2012,Tinney2012,Pinfield2014,Pinfield2014b}.
Currently, there are $\sim$355 and 15 identified T and Y dwarfs, respectively.\footnote{www.dwarfarchives.org}
Multiplicity is directly related to the stellar and substellar formation process, and it has been
studied for different mass regimes. In the case of T dwarfs, several works have focused on studying the
companions around these sources \citep[e.g., ][]{Burg2003,Bouy2003,Burg2006,Bouy2008,Gelino2011,Liu2011,Liu2012}.
Most of them have benefited from laser guide star (LGS) adaptive optics (AO) assisted observations at large
ground-based telescopes or/and the {\em Hubble Space Telescope} (HST) data in order to achieve the best angular resolution and sensitivity.
As a result there are now $\sim$14 identified T-dwarf binaries. Most of them show
very small separations \citep[$\sim$2-5\,au,][]{Burg2003,Burg2006,Liu2011}, although there are also wide binaries that have been detected
\citep[e.g., WISEJ1711+3500, $\rho=15\pm2$\,au, ][]{Liu2012}. In a recent work, \citet{aberasturi2014} have derived a
binary fraction of $<$16 or $<$25\%, depending on the underlying mass ratio distribution.
Motivated by the large number of T and Y dwarfs identified by WISE, we started a program to search for companions to newly
discovered BDs in 2012 using high angular resolution LGS AO-assisted observations in the near-IR. Most of the selected targets were T and Y dwarfs identified
with WISE and listed in \citet{Kirk2011}. In this paper we present the results from our AO
imaging campaigns. In Sections~\ref{Samp} and \ref{Observations} we described the sample selection,
the observations, and the data reduction. In Section~\ref{Results} we discuss the main results, including the
sensitivity limits. The main conclusions are summarized in Section~\ref{conclusions}.
\section{The sample of brown dwarfs.\label{Samp}}
\begin{table*}[th!]
\caption{Sample of observed brown dwarfs}\label{sample}
\begin{tabular}{lccllll}\hline
Target & RA & DEC & Sp. Type & $H^*$ & $d$ & Reference\\
& (J2000) & (J2000) & & (mag) & (pc) & \\ \hline
WISE J0612-3036 & 06 12 13.8 & -30 36 12.2 & T6 & 17.06$\pm$0.11 & 17.0 & 1\\
2MASS J1114-2618 & 11 14 51.3 & -26 18 23.5 & T7.5 & 15.73$\pm$0.12 & 10$\pm$2 & 2,3 \\
WISE J1436-1814 & 14 36 02.2 & -18 14 21.9 & T8 pec & $>$17.62 & 16.8 & 1,4 \\
WISE J1612-3420 & 16 12 15.9 & -34 20 27.1 & T6.5 & 16.96$\pm$0.03$^{**}$ & 16.2 & 1,4\\
WISE J1959-3338 & 19 59 05.7 & -33 38 33.7 & T8 & 17.18$\pm$0.05$^{**}$ & 11.6 & 1 \\
WISE J2255-3118 & 22 55 40.7 & -31 18 42.0 & T8 & 17.70$\pm$0.11$^{**}$ & 13.0 & 1\\
WISE J2327-2730 & 23 27 28.7 & -27 30 56.5 & L9 & 15.481$\pm$0.103 & 21.7 & 1 \\ \hline
\end{tabular}
{\bf Notes:} $^*$\,2MASS photometry except for $^{**}$ with MKO filter system photometry; $^1$\,\citet{Kirk2011};
$^2$\,\citet{Tinney2005};
$^3$\,\citet{aberasturi2014};
$^4$\,\citet{Kirk2012}
\end{table*}
\begin{table*}
\caption{Observing log.}\label{obslog}
\begin{tabular}{lllllllllr}\hline
Date & Target & Filter & Exp. time & Seeing$^1$ &$t_0$$^1$ & Airmass & LGS mode$^2$ & FWHM$^3$ & Strehl$^4$ \\
& & & (sec) & (arcsec) & (ms) & & & (arcsec) & (\%) \\ \hline
2012-09-08 & WISE 0612-3036 & $J$ & 300 & 0.8 & 2.5 & 1.1 & non-AO & -- & -- \\
& & $H$ & 180 & 0.7 & 2.5 & 1.3 & $se$ & -- & -- \\
& & $K_{\rm s}$ & 420 & 0.7 & 3.0 & 1.2 & non-AO & -- & -- \\
& WISE 2255-3118 & $H$ & 780 & 1.3 & 1.1 & 1.0 & $se$ & 0.50 & 2.1 \\
& & $K_{\rm s}$ & 840 & 1.0 & 1.5& 1.0 & $se$ & 0.40 & 3.3 \\
& WISE 2327-2730 & $H$ & 660 & 0.9 & 1.7& 1.3& $se$ & 0.37 & 3.2 \\
\hline
2013-05-19 & 2MASS J1114-2618 & $J$ & 480 & 0.9& 1.5 & 1.2 & $tt$ & 0.40 & 1.8 \\
& & $H$ & 480 & 0.9 & 2.0 & 1.1 & $tt$ & 0.22 & 4.8 \\
& & $K_{\rm s}$ & 480 & 0.9 & 2.0 & 1.1 & $tt$ & 0.14 & 11.0 \\
& WISE 1959-3338 & $J$ & 600 & 0.9 & 2.0& 1.0 & $tt$ & 0.39 & 2.7\\
& & $H$ & 400 & 0.9 & 1.7 & 1.0 & $tt$ & 0.27 & 5.0 \\
& & $K_{\rm s}$ & 300 & 0.8 & 2.0 & 1.0 & $tt$ & 0.25 & 8.0\\ \hline
2013-07-01 & WISE 1436-1814 & $ H$ & 240 & 1.1 & 4.0 & 1.0 & $se$ & 0.53& 2.3\\
& & $K_{\rm s}$ & 240 & 1.1 & 4.0 & 1.0 & $se$ & 0.29& 4.6\\
& WISE 1612-3420 & $H$ & 900 & 1.0 & 4.0 & 1.0 & non-AO & 0.41& 2.4\\
& & $K_{\rm s}$ & 800 & 1.0 & 4.0 & 1.0 & non-AO & 0.30 & 4.0\\ \hlin
\end{tabular}
{\bf Notes:} $^1$ Average values from the Differential Image Motion Monitor (DIMM); $^2$ $se$: seeing enhancer, $tt$: tip-tilt
(see text for details); $^3$ Full width at half maximum (FWHM) measured on the target; $^4$ Strehl ratio measured on the target, except in the case
of WISE\,0612-3036, which is resolved into a tight binary system.
\end{table*}
Our target list was mainly selected from the newly found BDs with {\it WISE},
which is a mid-IR satellite launched in 2009 that surveyed the entire sky in
four bands: 3.4, 4.6, 12, and 22\,$\mu$m.
Finding BDs was a primary science goal of WISE, and the strategy was based on
a particular color cut: the methane absorption feature near 3.3\,$\mu$m made the [3.6]-[4.6]
color extremely red ($\geq$2\,mag), unique among substellar objects.
\citet{Kirk2011} summarize the first WISE BDs discoveries that
included a list of a hundred of detected substellar objects.
For our study, we initially selected WISE T and Y dwarfs from this work. We kept only those
observable from Paranal Observatory and never studied for multiplicity before
\citep{Gelino2011,Liu2012}.
Additional dwarfs were discovered and studied afterwards \citep[][]{Cushing2011,Gizis2011,Kirk2012}, and we
included them in our target list.
Finally, we also considered several bright targets (mainly previously known L-type and early-T dwarfs) as backups to be observed under bad conditions.
One of these targets, 2MASS J1114-2618, was observed by the $HST$ in the meantime, and the
results were recently presented by \citet{aberasturi2014}. We include it here for comparison.
In total, we selected $\sim$50 targets for our study.
\section{Observations and data reduction \label{Observations}}
The observations presented here were obtained with NAOS-CONICA (NACO),
the AO system and near-IR camera \citep{Lenzen2003,Rousset2003} at the
Very Large Telescope (VLT). Since most of the targets are too faint to be used as
natural guide stars (NGS), we made use of the laser guide star (LGS)
facility at the VLT.
To obtain a full AO correction, the LGS facility should be used in combination with
a tip-tilt ($tt$) reference star brighter than $V$\,=\,17\,mag, located at a maximum separation of 40$\arcsec$ from the target.
In the absence of a $tt$ star, the LGS can be used in the so-called `seeing
enhancer' ($se$) mode \citep{Girard2010}, that is, LGS-assisted AO observation but without correcting for tip/tilt.
Because the low orders of the wavefront corrugations are not corrected, the SE
mode provides an image quality intermediate between simple (non-AO) imaging
and full LGS closed loop observations. (We refer to the NACO manual for a full description
of this mode. \footnote{http://www.eso.org/sci/facilities/paranal/instruments/naco/doc)}
For our observations, we used a $tt$ reference star whenever available,
and switched to $se$ mode otherwise.
The observations were allocated in seven different nights over 2012 and 2013.
We could not collect any data in four of the nights
(2012-06-10, 2012-07-12, 2012-12-08, and 2013-03-07) because of inadequate atmospheric
conditions for AO and the LGS and/or technical issues associated to the laser itself.
The atmospheric conditions in the three other nights were moderate to poor for the LGS,
with average coherence times ($t_0$) $\le$ 3\,ms and variable seeing.
In these conditions we observed a total of 20 targets, but we could only obtain good quality datasets
for seven of them. They are included in Table~\ref{sample}. As shown in the table,
some of the targets were even observed without AO owing to the unsuitable atmospheric conditions and
poor LGS performance.
All the targets were observed with the S54 objective, providing a field of
view of 54$\arcsec$$\times$54$\arcsec$ and a nominal plate scale of
0\farcs054. The data was obtained in one or several near-IR filters
($JHK_s$), depending on the atmospheric conditions and
brightness. To estimate and remove the sky background, we observed the targets
applying random offsets between the individual exposures (`Autojitter' mode).
The data was reduced using {\em eclipse} \citep{Devi1997} and
following the standard steps for near-IR imaging data: the images were dark-subtracted and
flat-field-divided, and bad pixels were masked out. The sky background was estimated
and subtracted from all the individual images. Finally, the frames were shifted
and co-added to generate the final stacked image. Since the atmospheric conditions
were variable during the observations, some of the individual frames showed a low image quality.
To improve the final spatial resolution of the combined data, we
removed the worst individual images, i.e. those with the lowest Strehl ratio and the largest FWHM, and kept only those with the best AO performance.
The observing log is given in Table~\ref{obslog}. As can be seen, the seeing and coherence time were
not optimal for AO+LGS observations, and this is reflected in the final FWHM and Strehl ratios measured on the combined images.
\section{Results~\label{Results}}
\begin{figure}[!t]
\includegraphics[angle=0,scale=0.5]{sensitMULTI3.eps}
\caption{Sensitivity limit of the NACO observations. We show the curves for the
five apparently single sources, one target per row, in the different
NACO filters. They were computed as the 3-$\sigma$ standard deviation of the
PSF radial profile. }\label{sensit}
\end{figure}
Five of the seven BDs observed with NACO+LGS are apparently single.
We estimated the sensitivity limit as a function of the separation for all of them (see Fig.~\ref{sensit})
in the different NACO filters.
The curves have been estimated from the point spread function (PSF) radial profiles of
the targets, while computing the standard deviation of the
noise in bins of 20 pixels and one pixel
step. We derived the difference in magnitudes at a
3-$\sigma$ level normalizing by the peak intensity of the PSF.
Usually, the best combination of sensitivity and angular resolution is achieved in the H band.
To derive the sensitivity limits in this band, we used the $H$ mag listed in Table~\ref{sample}.
For the targets with MKO filter system photometry, we converted the magnitudes into the
2MASS system using the relations from \citet{Stephens2004}. We obtain $H_{\rm 2MASS}\sim H_{\rm MKO}-0.04$\,mag for
the given spectral types. We reach 3-$\sigma$ magnitude limits between $H$$\sim$19-21\,mag for separations of 0\farcs5 (see Table~\ref{limits}).
For an average distance of $\sim$15\,pc and ages of 1\,Gyr and 5\,Gyr, this magnitude interval corresponds to masses of $\sim$11-7\,$M_{\rm Jup}$,
and $\sim$30-18\,$M_{\rm Jup}$, respectively, according to COND
evolutionary tracks \citep{Chabrier2000}.
For each individual target, we estimated the approximate spectral type (Sp. Type $_{\rm lim}$) that corresponds to the
$H$-band limiting magnitude (3-$\sigma$) using the $M_{\rm H}$ versus spectral type relation derived by \citet{Kirk2012}, and the
individual distances from Table~\ref{sample}. This value is included
in the last column of Table~\ref{limits}.
In the case of WISE\,1114-2618, we achieve a $\Delta H$ value of 4.7\,mag at 0\farcs5. This is deeper than the HST/F164N observations presented by \citet{aberasturi2014}, with a contrast of 1.5\,mag at a separation of 0\farcs6.
\begin{table}\label{limits}
\caption{Contrast and sensitivity limits (3-$\sigma$) in the $H$-band at a separation of 0\farcs5. The magnitudes are provided in the 2MASS system.}
\begin{tabular}{llll}
\hline
Target & $\Delta H$ & $H_{\rm lim}$@0.5\arcsec & Sp. type $_{\rm lim}$$^1$ \\
& (mag) & (mag) & \\ \hline
2MASS\,1114- 2618 & 4.7 & 20.4 & Y0\\
WISE\,1436-1814 & 2.0 & $>$19.6 & $>$T9\\
WISE\,1612-3420 & 3.2 & 20.2 & T9\\
WISE\,1959-3338 & 3.5 & 20.7 & Y0\\
WISE\,2327-2730 & 3.7 & 19.2 & T8\\ \hline
\end{tabular}
$^1$ Spectral type that corresponds to the limiting $H$-mag according to the $M_{H}$-Sp.~Type relation from
\citet{Kirk2012}.
\end{table}
\subsection{WISE\,0612-3036}
\begin{figure}[!t]
\includegraphics[angle=0,scale=0.54]{figure1bis3.eps}
\caption{NACO/LGS observations of WISE\,0612-3036. The top panel shows the $\sim$54\arcsec$\times$54\arcsec
NACO FOV. The target, WISE-0612-3036, is in the center of the black square. The star
2MASSJ06121227-3036100, used to calibrate the JHKs photometry of the binary system, is the brightest
source in the west. The bottom panels show a zoom on the newly discovered binary in $JHK_s$.}\label{fig1}
\end{figure}
We have resolved WISE 0612-3036 (WISE0612, hereafter) into a subarcsecond binary
(Fig.~\ref{fig1}). The two objects are detected in the three bands, $JHK_s$
with an estimated separation of $\rho$\,=\,350$\pm$5\,mas and
position angle of PA\,=\,235$\pm$1$^{\circ}$. The values are derived computing the average
and standard deviation of the measurements in the three bands. We note that only the H-band data was
obtained with LGS assisted AO, while the subsequent $JK_{\rm s}$
data was obtained in open loop owing to technical problems with the laser.
We performed PSF stellar photometry of the binary system with ALLSTAR in
DAOPHOT~II \citep{Stetson1992}. For the photometry we used the final stacked images in each
filter. The PSF was constructed using three or four stars in the field of view for the $J$ and $HK_{\rm s}$ bands, respectively. The typical PSF
errors of both components are in the range of 0.012--0.034\,mag. For the
transformation to the standard 2MASS $JHK_{\rm s}$ system, we used the
only star in the field with 2MASS photometry: 2MASS\,J06121227-3036100
with $\alpha$(2000)=6$^{\rm h}$ 06$^{\rm m}$ 12\fs27 and $\delta(2000)=-30\degr 36\arcmin 10\farcs0$ and
apparent magnitudes of $J$=15.915$\pm$0.080\,mag, $H$=15.141$\pm$0.066\,mag, and $K_{\rm s}$=15.024$\pm$0.139\,mag.
The final errors of the photometry are weighted means of the PSF errors and errors of the standard star.
To double check the PSF photometric calibration with the bright star
2MASS\,J06121227-3036100, we estimated its $K_{\rm s}$-band magnitude using a standard
star observed at the end of the night. We find that the difference between the 2MASS tabulated value
and our calibrated photometry is only 0.04\,mag.
The estimated magnitudes for the two components in the 2MASS system are
provided in Table~\ref{bin0612}. \citet{Kirk2011} report magnitudes of $J$=17.00$\pm$0.09, $H$=17.06$\pm$0.11, and
$K_{\rm s}$= 17.34$\pm$0.21 for the unresolved system. If we correct them for binarity, the
difference between our PSF photometry and these values are -0.12, 0.14, and 0.16 in $JHK_{\rm s}$, respectively, which are consistent
within the given uncertainties.
The primary (A) is the brightest object located to the NE (see Fig.~\ref{fig1}).
\citet{Kirk2011} classify the unresolved object as a T6 dwarf based on near-IR spectroscopy.
The difference in magnitudes between the two components are $\Delta J$= --0.05$\pm$0.14\,mag,
$\Delta H$= --0.10$\pm$0.14\,mag, and $\Delta K_{\rm s}$=--0.13$\pm$0.24\,mag, so we can assume similar spectral types for the two objects.
We have plotted the binary components in a near-IR color-color diagram (see Fig.~\ref{ccd}) and compared them
with well known T-type dwarfs from \citet[][Table 10]{Dupuy2012} with available 2MASS photometry. We have only
included objects with photometric uncertainties smaller than 0.2\,mag in the three bands. The WISE0612 system falls
within the region of the late-T dwarfs with a very blue $H-K_{\rm s}$ color. As explained in \citet{Burg2002}, this is expected in late-T dwarfs since
the 2.1$\mu$m peak starts to flatten until it is suppressed for the latest type objects. This is indeed the case of WISE0612, which shows
no emission peak at 2.1 $\mu$m in the near-IR spectrum obtained by \citet{Kirk2011}.
\begin{table}
\caption{Properties of the binary system WISE\,0612-3036\,AB.}\label{bin0612}
\begin{tabular}{lrr}
& NACO photometry & \\ \hline
Parameter & WISE 0612A & WISE 0612B \\ \hline
$J$ & 17.85$\pm$0.10 & 17.89$\pm$0.10 \\
$H$ & 17.62$\pm$0.10 & 17.73$\pm$0.10 \\
$K_{\rm s}$ & 18.18$\pm$0.14 & 18.33$\pm$0.20\\ \hline
$J-H$ & 0.25$\pm$0.14 & 0.20$\pm$0.14 \\
$H-K_{\rm s}$ & --0.60$\pm$0.17 & --0.60$\pm$0.22 \\
$J-K_{\rm s}$ & --0.40$\pm$0.17 & --0.42$\pm$0.22 \\ \hline
& Binary parameters & \\ \hline
$\rho$ (mas) & 350$\pm$ 5& \\
$\rho$ (au) & 11$\pm$2 & \\
PA (deg) & 235$\pm$1 & \\
Distance (pc) & 31$\pm$6 & \\ \hline
\end{tabular}
\end{table}
The spectrophotometric distance to the unresolved source, based on its WISE W2 magnitude and
its spectral type, is $\sim$17\,pc \citep{Kirk2011}.
To derive the distance to the new binary system we have
the mean 2MASS absolute magnitudes as a function of the spectral type from \citet{Dupuy2012}.
The values derived for a T6 dwarf are
$M_{J}$=15.35$\pm$0.02, M$_{H}$=15.32$\pm$0.02\,mag, and $M_{K_{\rm s}}$=15.68$\pm$0.01\,mag, with typical dispersions of
0.11, 0.17, and 0.33\,mag, respectively. Using these values, we derived distances of $\sim$ 31$\pm$2\,pc ($J$-band), 29$\pm$3\,pc ($H$-band) and 32$\pm$5\,pc ($K_{\rm s}$-band) and adopted an average value of $d$ = 31$\pm$6\,pc for the rest of the paper.
Assuming that the two objects are bound, we represent the
binary components in several color-absolute magnitude diagrams (see
Fig.~\ref{cmd}), together with a sample of MLT dwarfs with the measured
parallaxes included in \citet{Dupuy2012}. We excluded objects
with 2MASS photometric errors larger than 0.5\,mag, and relative
errors in their parallaxes ($\delta\pi/\pi$) larger than 0.6. As
expected, the two binary components of WISE0612 lie in the region
between T5-T7 dwarfs, with very blue $H-K_{\rm s}$ and $J-K_{\rm s}$
colors.
For the assumed distance of 31$\pm$6\,pc, the projected separation of the binary is $\rho \sim$ 11$\pm$2\,au. Since the age of the system is unknown, we estimated the mass of the objects assuming two ages of $\sim$ 1\,Gyr and 5\,Gyr.
According to COND evolutionary tracks \citep[2MASS system,][]{Chabrier2000}, we derived
masses of $\sim$30\,$M_{\rm Jup}$ (1\,Gyr) and $\sim$60\,$M_{\rm Jup}$ (5\,Gyr) for the individual components. For a circular orbit, we derived orbital periods of $\sim$150\,yr for 1\,Gyr and $\sim$105\,yr for 5\,Gyr.
Finally, we note that a single-epoch observation is not enough to classify the binary as bound.
Although additional data is needed to confirm it as a common proper motion pair, the small angular separation
makes a random coincidence unlikely.
We have estimated the probability of a background object lying close to the target using the number
sources detected in the NACO $H$-band image. We detect eight sources in the $\sim$56\arcsec$\times$56\arcsec field of view.
Since the companion candidate is detected at a separation of 0\farcs35, the probability of detecting an object
in an area of $\pi$$\times$$0\farcs35^2$= 0.3848 arcsec$^2$ around the target is 8$\times$(0.3848/56$^2$), that is, 0.09\%.
If we only consider the detected sources with a similar brightness to the companion (2), we derive a probability of 0.02\%.
We have also checked that there is no object at the given position in the Palomar Observatory Sky Survey (POSS) II images.
On the other hand, the proper motion of the primary estimated by
\citet{Kirk2011} is $\mu_\alpha.cos\delta$ = -137$\pm$22\,mas/yr and
$\mu_\delta=$-256$\pm$21\,mas/yr, that is, $\sim$0.3\arcsec/yr.
Therefore, the binary can be confirmed with second-epoch non-AO
observations obtained after 3-4\,yrs, to allow the BD enough time to
move far enough from the companion, if it turns out that the latter
is not co-moving.
\begin{figure}[!t]
\includegraphics[angle=0,scale=0.55]{cc_diag_2m.ps}
\caption{Color-color near-IR diagram. We have included a sample of ultracool dwarfs from \citet{Dupuy2012}:
the small gray diamonds represent M- and L-dwarfs, while the
green diamonds are T dwarfs. We have only plotted objects with photometric uncertainties
smaller or equal to 0.2\,mag. The binary components of WISE0612 are represented by blue triangles.}\label{ccd}
\end{figure}
\begin{figure}[!t]
\includegraphics[angle=0,scale=0.46]{dupuy2012_diag_2m.ps}
\caption{Near-IR color-magnitude diagrams. We represent a sample of ultracool dwarfs with measured parallaxes and 2MASS photometry from
\citet{Dupuy2012}: gray diamonds represent ML-dwarfs and green diamonds the T dwarfs.
We have highlighted in red the T-dwarfs with spectral types between T5 and T7. The binary components of WISE0612 are represented
by blue open triangles. In the lower central panel, the faint companion candidate to WISE2255 is represented by a filled square and
WISE\,1711+3500B \citep[T9.5,][]{Liu2012} by a filled circle. A sample
of 4 Y-dwarfs \citep{Liu2012,Leggett2013} are indicated by asterisks.}\label{cmd}
\end{figure}
\subsection{WISE 2255-3118}
In the case of the T8 dwarf WISE\,J2255-3118 (WISE2255, hereafter),
the final image quality was not good enough to detect very close
companions. However, we have detected a very faint source in the
$K_{\rm s}$ band at a separation of $\sim$1\farcs3 (see
Fig.~\ref{fig2}). This source is marginally detected in the $H$ band.
The estimated distance to the source, 13\,pc \citep{Kirk2011}, implies
a projected separation of $\sim$17\,au.
We performed aperture photometry on the $HK_s$-band images and calibrated it using standard stars observed during the night. We
derive $H$=17.64$\pm$0.18\,mag and $K_{\rm s}$= 17.47$\pm$0.25\,mag
for the primary. For the faint object we
estimate $K_{\rm s}$=20.5$\pm$0.3\,mag and $H >$20.6\,mag and a color of
$H-K_{\rm s}$ > 0.1\,mag.
If bound, the companion candidate would be at a distance of 13\,pc, which implies a $M_{\rm Ks}$$\sim$19.9\,mag. When we compare its properties
to those of the T dwarfs represented in the central lower panel of Fig.~\ref{cmd}, we see that the object does not follow the sequence of the
coldest T dwarfs, displaying a redder $H-K_{\rm s}$ color.
In this panel we also compared WISE2255 with WISE\,1711+3500, a physical binary system with very similar properties:
it consists of a T8 primary and a T9.5 companion with $\Delta K$=3\,mag
\citep{Liu2012}. The difference in the $H$ band between the two components is $\Delta H$ of -2.83, and its
$H-K_{\rm s}$$_{\rm , MKO}$ color is -0.42$\pm$0.17. Using the relations by \citet{Stephens2004} for a T9 dwarf (the coldest object in their sample),
we can convert its magnitudes into the 2MASS system ($K_{\rm s,2MASS}$ = $K_{\rm s,MKO}$ +0.16, and $H_{\rm 2MASS}$ = $H_{\rm MKO}$ +0.04), obtaining $H-K{\rm s}$$_{\rm , 2MASS}$ = -0.54, significantly bluer than the upper limit reported for the
WISE2255 companion candidate.
Finally, we also compared the photometry of the companion candidate with five Y dwarfs included in \citet{Liu2012} and \citet{Leggett2013}.
Since they all have $HK_s$ photometry in the MKO system, we have converted them to the 2MASS system following the same procedure described above.
The result is displayed in Fig.~\ref{cmd} (low central panel). The companion is redder than the Y dwarfs, but it lies close enough to their locus, preventing us from drawing a firm conclusion about the nature of this object.
We searched in public archives for additional data on this source and
found $HST$/WFC3 data obtained in two near-IR filters, F110W and
F160W (Program 12972, P.I. Gelino). The observations were obtained on
2012 October 17 in MULTIACCUM mode and following a four dithering
pattern. The total exposure time was $\sim$1200\,sec in the two
filters. We retrieved the final calibrated images ({\em flt} name
extension) from the Hubble Legacy Archive (HLA) and reprocessed them
using the routines within the DrizzlePac\footnote{http://drizzlepac.stsci.edu/} package. In
particular, we used {\sc multidrizzle} to combine the exposures into
a master image, resampling onto a finer pixel scale and correcting
for geometric distortions in the camera. (We selected a final\_scale
of 0.09 "/pixel and final\_pixfrac of 0.8.)
The T dwarf and the companion are clearly detected in the final $HST$ images (see Fig.~\ref{hst}), and the analysis of the two images with {\sc SEXtractor} \citep{Bertin1996} and {\sc PSFEx} \citep{Bertin2011} reveals that the faint companion candidate is an extended source. To visualize it, we represented the HST PSF-subtracted images in Figure~\ref{hst}, where the companion shows strong residuals in comparison to the T dwarf. This is consistent with the concentration index (CI) derived by the $HST$ team and included in the HLA {\sc SEXtractor} catalog for the F160W band (Data Release 8). This index is defined as the difference in magnitude from two different apertures (0\farcs09 and 0\farcs27), and it is used to separate extended from point-like sources. They measure a CI\,=1.4 for the companion candidate, while point-like sources in the WFC3 show 0.5 $<$ CI $<$ 1.0.
We note that the $HST$ image is relatively crowded, with a large number of spatially resolved galaxies. In fact, the HLA catalog contains
275 detected sources, with 235 displaying CI $>$ 1.0.
An extended source is consistent with a background galaxy or with a binary. In the latter case, and if bound to the primary,
the binary components would be cooler than T8, and probably in the Y-dwarf regime.
However, with the current data we cannot distinguish between the two scenarios.
That the detected source is extended, together with the red $H-K_{\rm s}$ color, suggests that the faint object is most probably
a background galaxy, although additional imaging is needed to test whether the objects are co-moving or not.
\begin{figure}[!t]
\centering
\includegraphics[angle=0,scale=0.45]{test2.ps}
\caption{NACO/LGS $K_{\rm s} \& H$ observations of WISE\,2255-3118. We display a field of 8\arcsec$\times$8\arcsec. A faint source (encircled) is detected at $\sim$1\farcs3 SE from the T dwarf.}\label{fig2}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[angle=0,scale=0.44]{ds9.ps}
\caption{HST/WFC3 F110W and F160W reduced images of WISE\,2255-3118. We display a field of 8\arcsec$\times$8\arcsec. North is up and east to the left. The companion candidate is clearly resolved in the two filters. The PSF-subtracted images are displayed on the right panels, where the
companion candidate shows large residuals consistent with an extended source.}\label{hst}
\end{figure}
\begin{table*}
\caption{Summary of known T-dwarf binary systems}\label{allbinT}
{\tiny
\begin{tabular}{lllllll} \hline
Name & SpT A & SpT B & Distance & $\rho$ & $\rho$ & Reference \\
& & & [pc] & [mas] & [au] & \\ \hline
SDSS\,102109.69-030420.1 & T1 & T4 & 29$\pm$4 & 172$\pm$5 & 5.0$\pm$0.7 & \citet{Burg2006} \\
2MASS J14044941-3159329 & T1 &T5 & $\sim$23 & 133.6$\pm$0.6 & $\sim$3.1 & \citet{Looper2008} \\
$\epsilon$ Indi~B & T1 & T6 & 3.626$\pm$0.013 & 732$\pm$2 & 2.654$\pm$0.012& \citet{McCaughrean2004} \\
SDSS\,153417.05+161546.1 & T1.5 &T5.5 & $\sim$36 & 110$\pm$5 & $\sim$4 & \citet{Liu2006} \\
SIMP J1619275+031350AB & T2.5 & T4 & 22$\pm$3 & 691$\pm$2 & 15.4$\pm$2.1 & \citet{Artigau2011} \\
SDSS\,092615.38+584720.9 & T4 & T4 & 38$\pm$7 & 70$\pm$6 & 2.6$\pm$0.5 & \citet{Burg2006}\\
WISEPA\,184124.73+700038.0 & T5 & T5 & 40.2$\pm$4.9 & 70$\pm$14 & 2.8$\pm$0.7 & \citet{Gelino2011} \\
2MASS J15344984-2952274 & T5 & T5.5 & 16$\pm$5 & 140.3$\pm$0.57 & 2.3$\pm$0.5 & \citet{Burg2003,Liu2008} \\
2MASS J12255432-2739466 & T6 &T8 & 11.2$\pm$0.5 & 282$\pm$5 & 3.17$\pm$0.14 & \citet{Burg2003} \\
2MASS J15530228+1532369 & T6.5 & T7 & 12$\pm$2 & 349$\pm$5 & 4.2$\pm$0.7 & \citet{Burg2006} \\
WISEPA\,J171104.60+35000036.8 & T8 &T9.5 & 19$\pm$3 & 780$\pm$2 & 15.0$\pm$2.0 & \citet{Liu2012} \\
WISEPA\,J045853.90+643452.6 & T8.5 & T9 & 10.5$\pm$1.4& 510$\pm$20 & 5.0$\pm$0.4 & \citet{Gelino2011} \\
WISE J014656.66+423410.0 & T9 & Y0 & 10.6$\pm$1.5 & 87.5$\pm$2.0 & 0.93$\pm$0.14 & \citet{Dupuy2015} \\
WISEPC\,J121756.91+162640.2 & T9 & Y0 & 10.5$\pm$1.7 & 759.2$\pm$3.3 & 8.0$\pm$1.3 & \citet{Liu2012}\\%Liu2012 \\
CFBDSIR J1458+1013 & T9.5 &T10 & 23.1$\pm$2.4 & 110$\pm$5 & 2.6$\pm$0.3 & \citet{Liu2011} \\
\hline
\end{tabular}
}
\end{table*}
\subsection{Comparison with T-dwarf binaries}
To put our study into context, we compared the properties of WISE\,0612-3036 with other T-dwarf binaries from
different AO and HST studies (see Table~\ref{allbinT}). We have represented the spectral types of the secondaries
versus the spectral types of the primaries in Figure~\ref{tbinaries}. Their projected separation in the sky (in au)
have been color-coded. As seen in Figure~\ref{tbinaries}, all the early-T primaries with SpT $<$ T3 have secondaries later
or equal than T4, while most of the late-T primaries (SpT $\geq$ T4) show nearly equal mass companions
(typical errors in spectral typing are between 0.5-1 type). The separations range between $\sim$2-16\,au, with
only four binaries showing separations larger than or equal to 8\,au. WISE\,0612-3036 is among these four binaries with a projected
separation of 11\,au.
\begin{figure}[!t]
\includegraphics[scale=0.56]{fig_tbinaries4.eps}
\caption{Known T-dwarf binaries. We have represented the spectral types of the secondaries versus the primaries for known T-dwarf binaries (circles).
0 stands for T0, 10 for T10, and 11 for Y0.
The color code is related with the projected separation (in au) of the binaries. Most of them show separations smaller than 5\,au, while only four
show $\rho \ge$ 8\,au. WISE\,0612-3036 (filled square) is among these four binaries with a projected separation of 11\,au.}\label{tbinaries}
\end{figure}
\section{Conclusions}\label{conclusions}
We have observed seven BDs with the NACO+LGS system to look for close companions.
In general, the observing conditions were not optimal for the LGS, resulting in poor
corrections that did not allow the innermost regions around the targets to be explored.
Our main findings can be summarized as follows:
\begin{itemize}
\item Five targets are apparently single with 3-$\sigma$ limiting magnitudes of $H$ between 19--21 mag at 0\farcs5.
\item The T-dwarf WISE\,0612-3036 is clearly resolved into a subarsecond binary. PSF-photometry of the binary
has allowed us to derive the individual $JHK_{\rm s}$ magnitudes of the two components, which are consistent with a very
similar spectral type of T6. Using absolute magnitude-spectral type relationships in $JHKs$, we estimate a distance
to the binary of $d\sim31\pm6$\,pc, which implies a projected separation
of $\rho=11\pm2$\,au. Additional observations are needed to confirm the system as bound.
\item The target WISE\,2255-3031 shows a very faint source at a separation of
1\farcs3 SE from the primary. Its $H-K_{\rm s}$ color ($>$\,0.1\,mag) is redder than the colors of the latest T dwarfs and Y dwarfs.
Since $HST$ public archival data in two near-IR filters shows that the source is extended,
we suggest that it is most probably a background galaxy.
\item{Similar to other known late T-dwarf binaries, WISE\,0612-3036 is comprised of two nearly equal-mass objects.
Its projected separation, 11\,au, is larger than the average value for T dwarf binaries ($\sim$5\,au).}
\end{itemize}
\begin{acknowledgements}
We thank the referee, C. Reyle, for her useful comments. This research has been funded by Spanish grants AYA2010-21161-C02-02 and AYA2012-38897-C02-01. Support for NH, JB,RK, and DM is provided by the Ministry of Economy, Development, and Tourism's Millennium Science Initiative
through grant IC120009, awarded to The Millennium Institute of Astrophysics, MAS. JB is supported by FONDECYT No.1120601,
RK is supported by Fondecyt Reg. No. 1130140. DM is also supported by FONDECYT No.1130196 and by the Center for Astrophysics
and Associated Technologies CATA PFB-06. CC acknowledges the support from project CONICYT FONDECYT Postdoctorado 3140592.
We are indebted to the Paranal staff for their support during the runs. NH thanks H. Bouy for useful discussions. This research has benefited from the M, L, T,
and Y dwarf compendium housed at DwarfArchives.org.
\end{acknowledgements}
\bibliographystyle{aa}
|
{
"timestamp": "2015-05-25T02:07:00",
"yymm": "1504",
"arxiv_id": "1504.03150",
"language": "en",
"url": "https://arxiv.org/abs/1504.03150"
}
|
\section{Introduction}
\subsection{Overview of studies of scattering in one-dimensional channel with emitter}
With current advances in experimental nanooptics, the problem of light scattering in quasi-one dimensional waveguides becomes an important cornerstone for understanding physics behind the light-matter interaction in a confined geometry. A number of recent experimental studies have been devoted to photon scattering when a single emitter is coupled to a one-dimensional (1D) scattering channel \cite{Akimov}, \cite{Astafiev}, \cite{Dayan}, \cite{coherent1}, \cite{coherent2}, \cite{chal1}, \cite{chal2}, \cite{Eichler1}, \cite{Eichler2}. The focus of these studies is made on a possibility of making few-photon devices (transistors, mirrors, switchers, transducers, etc.) as building blocks for either all-photonic or hybrid quantum devices. While a number of few-photon emitters based on single molecules, diamond color centers and quantum dots are available nowadays \cite{Yamamoto},\cite{1-photon-rev}, an understanding of the extreme quantum regime of a few-photon scattering in a 1D fiber or transmission line \cite{Claudon},\cite{Imamoglu} should be supplemented by microscopic studies of scattering of a coherent light (e.g., generated by a laser driving) off an emitter in a confined 1D geometry. This is the main motivation of the present work. In addition, it is worth mentioning that the model studied here can be derived as an effective model in a 3D scattering geometry when scattering channels are restricted to the photonic states with the lowest angular momentum values ($s$-wave scattering).
Theoretical studies of quantum models describing light propagation in 1D geometry have been pioneered in 1980's by Rupasov and Yudson \cite{R1}, \cite{R2}, \cite{RY1}, \cite{RY2}, \cite{Y1}, \cite{Y2}, \cite{Y-entropy}. They introduced and solved a broad class of the Bethe ansatz integrable one-dimensional models, and even managed to determine exactly time evolution of the certain initial states \cite{Y1}. In the next decade these studies have extended by the other authors \cite{Le1}, \cite{KoLe}, \cite{LLLS}, \cite{Le2}. The exactly solvable class of models includes linearly dispersed photons interacting with a single qubit, a Dicke cluster, and distributed emitters. However, the integrability imposes a rather strict constraint -- it requires the absence of backscattering thus limiting this class to the chiral, or unidirectional, models. This constraint, however, is not restrictive if a scatterer is local: transforming left- and right-propagating states of photons to the basis of their even and odd combinations, one can observe that the odd modes decouple from the scatterer and thus a model with backscattering is mapped onto an effective chiral model for even modes. In turn, to realize the physical chiral model with distributed emitters it has been recently proposed \cite{RPG} to employ scattering of edge states in topological photonic insulators. Experimentally a quantum nondemolition measurement of a single
unidirectionally propagating microwave photon has been achieved in Ref.~\cite{QNDChalmers} using a chain of transmons cascaded through circulators which suppress photon backscattering.
A revival of interest to problems of photonic transport in 1D geometries has been triggered in 2000's by the progress in quantum information science, which resulted in series of publications from various groups \cite{Kojima1}, \cite{Kojima2}, \cite{Kojima3}, \cite{Shen-Fan1}, \cite{Shen-Fan2}, \cite{Shen-Fan3}, \cite{Shen-Fan4}, \cite{Chang}, \cite{Nori1}, \cite{YR}, \cite{Chang2}, \cite{Busch}, \cite{Sorensen}, \cite{Nori2}, \cite{Baranger1}, \cite{Roy}, \cite{Shi-Fan-Sun}, \cite{Hafezi1}, \cite{Baranger2}, \cite{Hafezi2}, \cite{Liao1}, \cite{Liao2}, \cite{PGnjp}, \cite{Oehri}, \cite{sanchez}, \cite{werra}, \cite{martens}, \cite{auf1}, \cite{auf2}. In these works, a variety of different setups have been carefully analyzed, comprising three- and four-level emitters, the nonlinear photon dispersion, effects of driving and dissipation. In addition, the recent experimental achievements \cite{Wallraff-2} motivate a theoretical consideration of models containing both distributed emitters and the backscattering \cite{Baranger3}, \cite{LP}, \cite{auf3}, \cite{2qubit-plasmon}.
\begin{figure*}[ht]
\includegraphics[width=\textwidth]{scat1.eps}
\caption{(Color online) Our system consists of a two-level emitter coupled to a waveguide (transmission line) at $x=0$. The sketch shows spatial snapshots of the wavepacket propagation. The coherent initial pulse $|\alpha_{0} \rangle$ of the length $L$ (shown in pink with dashed contour) is injected at time $t=-t_0$ and the point $x=-t_0$. At time $t=0$ its front hits the scatterer. The scattered pulse (shown in blue with solid contour) leaves the scattering region, and after time $t_0$ its front reaches a detector located at $x=t_0$. At time $t=t_0 +T$ the detector starts counting photons which lasts during the time interval $\tau$. It is assumed that $t_0 \gg L > \tau$. } \label{fig:fcs}
\end{figure*}
\subsection{Scattering approach, role of detector}
In this paper we focus on a basic model consisting of a two-level qubit coupled to a 1D channel and driven by a coherent field. We develop a complete and exact quantum description of all physical properties of this system using the scattering formalism.
A characterization of different scattering regimes in this model can be obtained by introducing parameters quantifying (i) an initial state, (ii) a qubit (iii), a waveguide, and (iv) a detector. Throughout the paper we assume that the initial state is a pulse of the spatial length $L$. In the units $\hbar = v_g=1$, where $v_g$ is the group velocity of linearly dispersed photons in a waveguide, the parameter $1/L$ defines one of the important energy scales -- the wavepacket width in the $k$-space. Another energy scale is given by the qubit relaxation rate $\Gamma=\pi g^{2}$, where $g$ is a photon-qubit interaction strength. In addition, we have a dimensionless parameter $\bar{N}$ characterizing the mean number of photons in the initial pulse. In terms of these parameters, one can distinguish three different regimes in this problem: (a) $\bar{N} \gg\Gamma L\gg 1$; (b) $\Gamma L\gg \bar{N} \gg 1$; (c) $\Gamma L\gg 1\gg \bar{N}$. Note that in all cases we assume $\Gamma L \gg 1$ meaning the long-$L$ (or narrow bandwidth) pulse's limit.
The regime (a) was studied back in 1970's in connection with the resonance fluorescence phenomenon \cite{Mollow} (see also the review \cite{KM-rev}). In this regime, a semiclassical description of the laser beam is sufficient.
In contrast to the regime (a), in the regime (c) single-photon (elastic) processes dominate, while a contribution of many-photon (inelastic) processes to the scattering outcome is weak; the most remarkable inelastic effect in this regime is, perhaps, a formation of the two-photon bound state. Various aspects of the regime (c) have been recently studied in numerous publications cited above.
To complement the previous studies, we wish to achieve a comprehensive understanding of the crossover regime (b), where the mean number of scattered photons is already large, but the Rabi frequency $\propto \sqrt{\bar{N} \Gamma/L}$ is still much smaller than $\Gamma$. To this end, we apply the quantum scattering approach developed in our earlier paper \cite{PGnjp}. It will be shown in the following that it is eventually capable to cover all three regimes, thereby establishing a theoretical platform for studying the classical-to-quantum crossover in this model.
In addition to the system related parameters, our approach can accommodate information contained in the detection protocol as shown in Fig.~\ref{fig:fcs}. A pulse of the spatial length $L$ (shown in pink) is injected into the waveguide at time $t=-t_0$ and the coordinate $x=-t_0$ (recall that $v_g=1$). Due to the linear dispersion, it moves without changing its shape toward the qubit coupled to the waveguide at the point $x=0$. From the time instant $t=0$ the photons in the pulse start interacting with the qubit, and this interaction lasts approximately for a time $\sim L$. Subsequently the scattered pulse leaves the interaction region around $x=0$ (only the transmitted part is shown in blue in the figure), its shape is being modified, and at $t=t_0$ its front reaches the detector (the ``eye'') at the position $x=t_0$. To make the scattering formalism applicable we must assume that $t_0 \gg L$, meaning that the detector is far away from the scattering region. After the front of the scattered pulse reaches the detector at $t=t_0$, the latter remains switched off during the {\it waiting} time $T$, and at $t=t_0+T$ it switches on and starts to count transmitted photons during the subsequent time interval $\tau$ ({\it counting} time). Thus, the setup in Fig.~\ref{fig:fcs} is characterized by the two important time scales $T$ and $\tau$. In the following we will assume $L > \tau$ (which, in particular, allows us to take the limit $L \to \infty$ {\it before} the limit $\tau \to \infty$). This assumption is based on an instrumental possibility to create an initial wavepacket of a sufficiently narrow width in the $k$-space; from the conceptual viewpoint we impose this condition to enable an open-system description of the scattered pulse, for which the aforementioned order of limits is essential.
The detector-related time scales together with the system-related energy scales exhaust the most relevant parameters of this problem.
\subsection{Physical observables and their generating function}
Major statistical properties of a scattered light field are characterized by a collection of $m$-point ($m=2,4, \ldots$) correlation functions \cite{Glauber}.
Another basic observable quantity is a transmission (reflection) spectrum. It quantifies the amplitude of the transmitted (reflected) field, and it has been measured in various experiments with microwave transmission lines \cite{Astafiev}, \cite{Dayan}, \cite{coherent1}, \cite{coherent2}. A more involved quantity of interest is the resonance fluorescence power spectrum, which is a Fourier transform of the field-field ($g^{(1)}$) correlation function. This (Mollow) spectrum is know to feature a three-peak structure \cite{Mollow}. Further experimental practice is to collect information about the density-density ($g^{(2)}$) correlation function using the Hanbury-Brown-Twiss setup \cite{HBT}. Higher order correlators can also (at least, in principle) be accessed in an experiment, and this motivates us to study the generating function of all moments of the density operator, being related to the $m$-point functions. In mesoscopic physics, such a generating function is known as the Full Counting Statistics (FCS), though in the
historical retrospective this concept has been originally introduced in
the quantum-optical context \cite{Glauber}, \cite{MaWo} in order to characterize statistical
properties of a {\it non-interacting} quantized electromagnetic field. Its fully quantum derivation has
been presented in Ref.~\cite{KK}. For a coherent light field driving a two-level system (the resonance fluorescence model) the FCS has been studied in \cite{Mandel}, \cite{Cook}, \cite{Lenstra}, \cite{Smirnov-Troshin}. It has been observed \cite{Mandel} that the FCS distribution of the fluorescent photons is narrower than the Poissonian distribution. An important measure of this effect is the Mandel's $Q$-parameter, $Q= [\langle N^{(2)} \rangle - \langle N^{(1)}\rangle^2 ]/\langle N^{(1)}\rangle$ which is obtained from the first and the second factorial moments of the FCS.
Later on, the concept of the FCS has been borrowed and actively developed in
the field of mesoscopic physics \cite{LL}, \cite{LLL} for studying statistical properties of electronic currents
in meso- and nanoscopic devices for both non-interacting and interacting electrons
\cite{NB}, \cite{Belzig}, \cite{BN}, \cite{BUGS}, \cite{GK}, \cite{Schonh}, \cite{KS}, \cite{BSS}, \cite{BBSS}, \cite{CBS}. It has been recently
shown that the FCS can be very useful for characterizing classical-quantum crossover \cite{SB}, quantum
entanglement \cite{TF}, \cite{KL}, \cite{LeHur}, and phase transitions \cite{IA}, \cite{FG}. The FCS
of nonlocal observables can be used to quantify correlations \cite{GADP}, \cite{IGD}, \cite{Hoffer}, \cite{LF}, and
prethermalization in many-body systems \cite{Kitagawa}, \cite{prethermalization}, as well as to define some kind of a topological order parameter \cite{IA2}.
The studies of the FCS in mesoscopic physics have generated a back flow of ideas to the quantum optics community. Inspired by the recent experiments in the context of the 1D resonance fluorescence, the subject of the FCS has received the renewed attention, see \cite{deflection}, \cite{BB}, \cite{Vogl}, \cite{LJ}.
\subsection{This paper: content and results}
Motivated by previous developments we revisit the original problem of computing the FCS for photons
interacting with an emitter. This is the first goal of the present work. We give it a detailed quantum consideration, treating the interaction
nonperturbatively. We present an {\it exact} calculation of the FCS in the basic model of light-matter
interaction (see Fig.~\ref{fig:fcs}): a multimode propagating photonic field in 1D interacting with a
two-level emitter. Since one of the objectives in nanophotonics research is to
obtain strong photon nonlinearities as well as a strong photon-emitter interaction for the purpose
of an efficient control over individual atoms and photons, knowledge of
statistical properties of an {\it interacting} photon-emitter device becomes essential.
The geometry of our system (see Fig.~\ref{fig:fcs}) suggests to define the three types of the counting statistics: the FCS of transmitted photons, the FCS of reflected photons, and the FCS of the chiral model. Below we discuss this classification in more detail. The multimode nature of the waveguide implies an emergence of many-body correlations. As such, interactions modify the statistics of photons in forward
and backward scattering channels in comparison with the Poissonian one of the incident coherent beam. As a by-product of our FCS computation, we revisit the transmission properties and evaluate the Mandel's $Q$-factor exhibiting super-Poissonian, sub-Poissonian, and Poissonian statistics for the three counting models in question.
We also provide a new derivation of the Mollow spectrum based on knowledge of the exact scattering wavefunction that avoids the usage of the quantum regression theorem, and reproduces the original expression \cite{Mollow} derived for the 3D scattering geometry. It helps to understand how the resonance fluorescence can be decomposed into elementary scattering processes.
Yet another quantity that has recently received a great deal of attention due to developments in quantum information science is the {\it entanglement entropy}, see, e.g., the recent review \cite{Eisert}. While several measures of entanglement exist, the entropy of entanglement has
several nice properties like additivity and convexity.
In quantum information theory, the entanglement entropy gives the efficiency of conversion of partially entangled to maximally entangled states by local operations \cite{Bennett1}, \cite{Bennett2}. In other terms, it gives the amount of classical information required to
specify the reduced density matrix. A large degree of entanglement is what makes quantum information exponentially more powerful than classical information,
so states with lower entanglement entropy are less complex. For extended systems of condensed matter physics it is customary to distinguish between {\it area} and {\it volume} law \cite{Eisert} behavior of the entanglement entropy. Here the notions of area or volume refer to a typical geometric measure of a region bounded by a subsystem $A$ with respect to the rest of a system. Thus in 1D case, relevant for our discussion here, the area of an interval $A$ consists of just two end points, while the volume is a length of the interval of the subsystem $A$.
Systems with volume law behavior entanglement possess much higher potential for applications in quantum simulations and computing. It was shown \cite{Eisert} that in most cases a quantum ground state wave function of gapped systems exhibits the area law, while typical excited states mostly follow the volume law. An intermediate logarithmic behavior of the entanglement entropy is related to gapless systems. These features should be understood as asymptotic properties of a system, when the area and the volume of a subsystem entangled with the rest part of a system become large. Our complete knowledge of the scattering state allows us to calculate explicitly the entanglement entropy of the scattered pulse's interval of the length $\tau$ (see Fig.~\ref{fig:fcs}) for different values of $\tau$, $T$, and system parameters. The (dimensionless) duration $\Gamma\tau$ of the observation interval plays the role of the volume of the subsystem $A$ in this context. One of our central results in that section is a demonstration of the existence of the {\it absolute limit} for the entanglement entropy in our system: it is bounded by $\log 4$, the entropy of four-level system. Another important observation is that while the entanglement entropy at large $\tau$ asymptotically approaches the {\it area law} bounded by $\log 4$, it can behave very differently for small and intermediate values of $\tau$ -- we even observe its nonmonotonous oscillatory behavior for some intermediate regime of parameters.
\section{Definitions and approximations}
We start our analysis defining the model and approximations involved in its derivation, the bosonic operators creating the initial pulse, and the FCS.
\subsection{Theoretical model.}
\subsubsection{Approximations and the effective Hamiltonian.}
Our model is described by the Hamiltonian $H=H_{ph}+H_{em}+H_{ph-em}$, where $H_{ph}$ is the Hamiltonian of the free propagating photonic field, $H_{em}$ is the Hamiltonian of an emitter, and $H_{ph-em}$ describes the field-emitter coupling. We involve approximations which are customary in quantum optics: (i) the dipole approximation for the interacting Hamiltonian; (ii) the two-level approximation for the emitter Hamiltonian; (iii) the rotating wave approximation (RWA); and (iv) the Born-Markov approximation (energy independence) for the coupling constant. In addition, we linearize the photonic spectrum around some appropriately chosen frequency $\Omega_{0}$ which is commensurate with the emitter's transition frequency $\Omega$, and extend the linearized dispersion to infinities. With these assumptions [except for (iii)] we obtain an effective low-energy Hamiltonian
\beq
H&=&\sum_{\xi=r,l}\int dk(\Omega_{0}+\xi k)a^{\dag}_{\xi k}a_{\xi k}+\frac{\Omega}{2}\sigma^{z}\nonumber\\
&+&g_0 \sum_{\xi=r,l}\int dk (a_{\xi k}^{\dag}+a_{\xi k})(\sigma^{+}+\sigma^{-}),
\eeq
featuring the two-branch linear dispersion with right- ($\xi=r=+$) and left- ($\xi=l=-$) propagating modes. Here
$\sigma^{\pm}=(\sigma^{x}\pm i\sigma^{y})/2$ are expressed in terms of the Pauli matrices $\sigma^{a}$, $a=x,y,z$. The states of the emitter are separated by the transition frequency $\Omega$. To implement the RWA in a systematic way, we first perform the gauge transformation $H \to U^{\dag} H U + i (d U^{\dag}/ dt) U$ with
\beq
U=\exp\left[-i\Omega_{0}t\left(\sum_{\xi=r,l}\int dk \, a^{\dag}_{\xi k} a_{\xi k}+\frac{\sigma^z}{2} \right)\right],
\eeq
which leads us to the Hamiltonian
\beq
H&=&\sum_{\xi=r,l}\int dk \, \xi \, k \, a^{\dag}_{\xi k}a_{\xi k}+\frac{\Delta}{2}\sigma^{z}\nonumber\\
&+&g_0 \sum_{\xi=r,l}\int dk (a_{\xi k}^{\dag}\sigma^{-}+a_{\xi k}\sigma^{+}) \label{start-Ham} \nonumber\\
&+& g_0 \sum_{\xi=r,l}\int dk (a_{\xi k}^{\dag}\sigma^{+} e^{2i\Omega_{0}t}+a_{\xi k}\sigma^{-} e^{-2i\Omega_{0}t}),
\label{osc-terms}
\eeq
where $\Delta=\Omega-\Omega_{0}$. As soon as $g_0^2/\Omega_0 \ll 1$, the time-oscillating terms in (\ref{osc-terms}) can be treated as a time-dependent perturbation. In zeroth order it is simply neglected, which is equivalent to the RWA. Note that this approximation is consistent with the assumption about the absence of lower and upper bounds in the linearized dispersion.
\subsubsection{Transformation to the ``even-odd'' basis.}
Due to energy independence of the coupling constant $g_0$, one can decouple the Hilbert space of the model defined in (\ref{start-Ham}) into two sectors. To this end, one introduces \emph{even} (symmetric) and \emph{odd} (antisymmetric) combinations of fields corresponding to the same energy $|k|$
\be
a_{ek} = \frac{a_{rk} + a_{l, -k}}{\sqrt{2}}, \quad a_{ok} = \frac{a_{rk}- a_{l, -k}}{\sqrt{2}} .
\ee
By virtue of this canonical transformation the Hamiltonian (\ref{start-Ham}) turns into a sum of the two terms, $H=H_e + H_o$, defined by
\beq
H_e &=& \int dk \left[ k \, a^\dag_{e k} a_{e k}
+ g \left( a^\dag_{e k}\sigma^- + a_{e k}\sigma^+ \right) \right]
+ \frac{\Delta}{2} \sigma^z , \nonumber\\
H_o &=& \int dk \, k \, a^\dag_{ok} a_{ok},
\eeq
where $g=g_0 \sqrt{2}$. Note that the odd Hamiltonian $H_{o}$ is noninteracting, and therefore odd modes do not scatter off a local emitter ($S_o \equiv 1$). The even Hamiltonian $H_{e}$ can be interpreted in terms of a chiral model with a single branch of the linear dispersion.
A similar decomposition can be applied to an initial state. Both even and odd photons are labeled by a momentum value $k$ lying on a single branch of the linear dispersion.
\subsection{Definitions of wave packet field operators and the initial state.}
In order to define the incident coherent state we need field operators annihilating/creating states which are normalized by unity. The field operators $a_{k}$ and $a^{\dag}_{k}$, which are the Fourier transforms of $a (x)$ and $a^{\dag} (x)$, do not fulfill this requirement, as they obey the commutation relation $[a_k, a^{\dagger}_{k'}] = \delta (k-k')$ (in other words, they annihilate and create {\it unnormalizable} states). To circumvent this difficulty, we construct wavepacket field operators
\beq
b_k^{\dagger} &=& \frac{1}{\sqrt{L}} \int_{-L/2}^{L/2} d x a^{\dagger} (x ) e^{i k x},
\label{op_c}
\eeq
which do satisfy the desired commutation relation $[b_{k}, b_{k}^{\dagger}] =1$. For example, the operators
$b_{r,k_0}^{\dagger} = \frac{1}{\sqrt{L}} \int_{-L/2}^{L/2} d x a^{\dagger}_{r} (x) e^{i k_0 x} $ and $b_{l,-k_0}^{\dagger} = \frac{1}{\sqrt{L}} \int_{-L/2}^{L/2} d x a^{\dagger}_{l} (x) e^{-i k_0 x}$ create wavepackets which are centered around $+k_0 (-k_0)$ of the right (left) branch of the spectrum and broadened over the width $\sim 2 \pi/L$. In the coordinate representation, they create states which are spatially localized on a finite interval of the length $L$. We note the identity $[a_{\xi} (x), b_{\xi',k_0}^{\dagger}] = \delta_{\xi \xi'} \frac{e^{i k_0 x}}{\sqrt{L}} \Theta (L/2-|x|)$. In the following, $k_0$ denotes the laser driving frequency (measured from the linearization point).
Having introduced $b_{r/l,k0}^{\dagger}$, we define the initial (incoming)
state $|\mathrm{in} \rangle =|\alpha_0 \rangle_{r} \otimes |\downarrow \rangle$, where the
incident right-moving photons are prepared in the coherent state $|\alpha_0 \rangle_{r} = D_{r} (\alpha_0) |0 \rangle$, and the two-level emitter is initially in the
ground state~$|\downarrow\rangle$. Here $|0\rangle$~denotes the photonic vacuum, and $D_{r} (\alpha_0) = \exp (\alpha_0 b^{\dagger}_{r, k_0} - \alpha_0^* b_{r, k_0}) $ is the coherent state displacement operator.
The mean number of photons
in the state $|\alpha_0 \rangle_{r}$ is given by $\bar{N}_0 = |\alpha_0 |^2$. We also quote a useful relation
\beq
D_{r}^{\dagger} (\alpha_0) a_r (x ) D_{r} (\alpha_0) = a_r (x) + \frac{\alpha_0 e^{i k_0 x}}{\sqrt{L}} \Theta (L/2 - |x|),
\label{til_a}
\eeq
which follows from the commutation relation between $a$ and $b^{\dagger}$ operators.
The initial coherent state defined in the original right-left basis admits a decomposition into the product state in the even-odd basis
\beq
|\alpha_0 \rangle_{r}&=&e^{\alpha_0 b^{\dag}_{r, k_0} -|\alpha_0|^2/2} | 0 \rangle \nonumber\\
&=&
e^{(\alpha_0/\sqrt{2}) b^{\dag}_{e,k_0} -|\alpha_0|^2/4} e^{(\alpha_0/\sqrt{2}) b^{\dag}_{o,k_0} - |\alpha_0|^2/4} | 0 \rangle \nonumber\\
&\equiv& D_e (\alpha )D_{o} (\alpha ) |0\rangle\equiv | \alpha \rangle_e \otimes | \alpha \rangle_o ,\nonumber
\label{init-state}
\eeq
where $\alpha = \alpha_0/\sqrt{2}$, and the displacement operators $D_{e,o}$ are defined using the mutually commuting operators $b_{e, k_{0}}^{\dagger}$ and $b_{o,k_{0}}^{\dagger}$, respectively. Importantly, $\, _r\langle \alpha_0 | \alpha_0 \rangle_r =1$, i.e. the incoming state is properly normalized.
The major consequence of the even-odd decoupling is a factorization of the scattering operator $S$
into the product $S= S_e S_o$, where $S_o$ is the identity operator, and $S_e$ can be studied in the context of the effective one-channel chiral model described by $H_e$. Scattering in chiral models has been studied by us in Ref.~\cite{PGnjp} for arbitrary initial states, including the coherent state. In particular, we have established the explicit form of the operator $S_e$ in the latter case, which provides the full information about the scattering wavefunction. Here we take over this result and use it for calculation of observables announced in the Introduction. All expressions necessary for this purpose are quoted below for readers' convenience.
\subsection{Definition of the full counting statistics}
The statistics of the initial field, defined by the probability $p_{\alpha_0} (n)$ to find $n$ photons in
the mode $k_0$, is given by the Poissonian distribution $p_{\alpha_0} (n)= e^{-\bar{N}_0}
\frac{\bar{N}_0^{n}}{n!}$ for the coherent field, with the mean value $\bar{N}_0 =|\alpha_0|^2$. Due to the photonic dispersion and inelastic scattering processes photons can leak from the right-moving
mode $k_0$ to other modes on both branches of the spectrum by virtue of scattering processes, thus modifying the photon statistics.
A fraction of photons is reflected, and their statistics is also of great
interest. We propose a calculation of the FCS in both forward and backward scattering channels,
which is {\it exact} and {\it nonperturbative} in both $g_0$ and $\bar{N}_0$.
Generally speaking, the FCS can be defined as a generating function $F(\chi)
=\sum_{n=0}^{\infty} e^{i \chi n} p (n)$ associated with a probability distribution $p(n)$ to detect $n$ photons in some given state. The function $F (\chi) $
generates $m$-th order moments of the distribution $p (n)$ which are determined by evaluating the $m$-th derivative with respect to $i \chi$ at $\chi=0$.
The Fourier expansion of the $2\pi$-periodic function $F(\chi)$ yields power series in terms of
the ``fugacity'' $z=e^{i\chi}$, $F(\chi) = \sum_{n=0}^{\infty} z^{n} p(n) \equiv {\hat
F}(z)$. The normalization of a probability distribution implies $\hat F(1) = 1$. The expansion $\hat{F} (z)= \sum_{r=0}^{\infty} \frac{(z-1)^r}{r!} \langle N^{(r)} \rangle$ around $z=1$ gives the factorial moments of a distribution $\langle N^{(r)} \rangle = \langle N (N-1) \ldots (N-r+1)\rangle$.
To define the photon FCS we need the two main constituents: (i) the scattering (outgoing) state $|\mathrm{out}\rangle = (S_e | \alpha \rangle_e ) \otimes |\alpha \rangle_o $ necessary to perform the average, and (ii) a meaningful and experimentally measurable counting operator $N$. In particular, as such we can choose the number of transmitted photons which pass through the detector during the time $\tau$ (see the Fig.~\ref{fig:fcs}). Because of the linear dispersion, the same operator characterizes the number of photons in the spatial interval of the length $v_g \tau$ viewed in the frame co-moving in the right direction with the velocity $v_g$. Introducing the coordinate system in this co-moving frame such that the pulse's front has the coordinate value $+L/2$, we define the photon number operator
\beq
N_{r, \tau} = \int_{z_1}^{z_2} d x a_r^{\dagger} (x) a_r (x)
\label{Ndef}
\eeq
of transferred photons which appear in the spatial interval $[z_1,z_2]$. Here $-L/2<z_1<z_2 <L/2$ (we note that the tail of the scattered pulse extends to $-\infty$, see below; however, we will focus on counting intervals $\tau \equiv z_2 -z_1 < L$). The corresponding FCS reads
\beq
F_{r, \tau} (\chi_0) = \langle e^{i \chi_0 N_{r,\tau}} \rangle .
\label{Fdef}
\eeq
In the chosen coordinate system, the waiting time is expressed by $T =L/2 -z_2$.
In the same setting one can consider the FCS of the chiral model.
To define the FCS for reflected photons, one can put the second detector at $x=-t_0$ and consider the co-moving frame with the velocity $-v_g$, defining in it the counting operator $N_{l,\tau}$ via $a_l (x)$.
\section{Method of computing observables based on exact scattering matrix}
\subsection{Scattering of the coherent state in the chiral model}
As we discussed above, we need to know an expression $S_e | \alpha \rangle_e$ for the scattering state in the effective chiral model for the even modes. In this subsection we quote the result of Section 5 in Ref.~\cite{PGnjp} for the coherent light scattering in the chiral model. In particular, we copy the Eq.~(134) from this reference, adapting notations therein to the present paper.
The subscript $e$ is also omitted in the following.
Thus, for the incoming coherent state $| \alpha \rangle$ in the mode $k_0$, the outgoing scattering state amounts to $S_0 | \alpha \rangle$, where
\beq
S_0 = S_0^a [L/2,-L/2] + S_0^b [L/2,-L/2] \frac{\lambda}{\sqrt{2 \Gamma}} A_0^{\dagger}.
\label{S0-factor}
\eeq
The operator
\be
A_0^{\dagger} = \sqrt{2 \Gamma} \int_{-\infty}^{-L/2} d x_0 e^{i k_0 x_0} e^{- i (\delta + i \Gamma) (L/2 +x_0)} a^{\dagger} (x_0),
\ee
describing the states in the tail of the scattered pulse, is normalized by $\langle 0 | A_0 A_0^{\dagger} | 0 \rangle =1$.
In turn, the states within the initial pulse's size $L$ are expressed via the operators
\beq
S_0^a [y,x] &=& 1 + \sum_{n=1}^{\infty} \lambda^n \int {\cal D}x_{n}\label{Sa0}\\
& \times & d_0 (y - x_n) a^{\dagger} (x_n) e^{i k_0 x_n}\nonumber\\
&\times& d_0 (x_n - x_{n-1}) a^{\dagger} (x_{n-1})e^{i k_0 x_{n-1}} \nonumber\\
&\times & \ldots d_0 (x_2 -x_1) a^{\dagger} (x_1 ) e^{i k_0 x_1} , \nonumber\\
S_0^b [y,x] &=& d_0 (y-x) + \sum_{n=1}^{\infty} \lambda^n \int {\cal D}x_{n}\label{Sb0}\\
& \times & d_0 (y - x_n) a^{\dagger} (x_n) e^{i k_0 x_n}\nonumber\\
&\times & d_0 (x_n - x_{n-1})
a^{\dagger} (x_{n-1}) e^{i k_0 x_{n-1}} \nonumber \\
&\times& \ldots d_0 (x_2 - x_1) a^{\dagger} (x_1 ) e^{i k_0 x_1} d_0 (x_1 -x). \nonumber
\eeq
Here the parameter
\beq
\lambda = - \frac{2 i \Gamma \alpha}{(\delta + i \Gamma) \sqrt{L}},
\eeq
is expressed in terms of the detuning $\delta = k_0 - \Delta$ and the relaxation rate $\Gamma = \pi g^2$; $d_0 (x)=1-\exp [i (\delta + i \Gamma) x]$ is the bare single-photon propagator, and the short-hand notation has been used for the integration measure ${\cal D}x_{n} = \Theta (y > x_n > \ldots > x_1 > x) d x_n \ldots d x_1 $. We also consider $L/2 \geq y >x \geq -L/2$.
For the later use we also define the following operators
\beq
S_0^c [y,x] &=& 1+ \sum_{n=1}^{\infty} \lambda^n \int {\cal D}x_{n}\label{Sc0}\\
& \times & a^{\dagger} (x_n) e^{i k_0 x_n} d_0 (x_n - x_{n-1}) \nonumber\\
&\times& a^{\dagger} (x_{n-1})e^{i k_0 x_{n-1}} d_0 (x_{n-1} -x_{n-2})\ldots \nonumber\\
&\times& d_0 (x_2 -x_1) a^{\dagger} (x_1 ) e^{i k_0 x_1} , \nonumber
\eeq
\beq
S_0^{\bar{a}} [y,x] &=& 1 + \sum_{n=1}^{\infty} \lambda^n \int {\cal D}x_{n} \label{Sat0} \\
& \times & a^{\dagger} (x_n) e^{i k_0 x_n} d_0 (x_n - x_{n-1})\nonumber\\
&\times& a^{\dagger} (x_{n-1}) e^{i k_0 x_{n-1}}d_0 (x_{n-1} - x_{n-2}) \ldots \nonumber \\
&\times& a^{\dagger} (x_1 ) e^{i k_0 x_1} d_0 (x_1 -x).
\nonumber
\eeq
The set of operators $S^{a}_{0},S^{b}_{0},S^{c}_{0},S^{\bar{a}}_{0}$ is complete in that sense that they exhaust all possible arrangements of the bare propagators $d_{0}(x)$.
\subsection{Algebra of scattering operators}
The scattering state (134) of Ref.~\cite{PGnjp} (equivalent of $S_0 | \alpha \rangle$) can be used for a computation of observable quantities. In particular, we will be interested in correlation functions $\langle \alpha |S_0^{\dagger} a^{\dagger} (z_1) \ldots a^{\dagger} (z_m) a(z_m) \ldots a(z_1) S_0 | \alpha \rangle$, and therefore we need to know how the local annihilation operators $a (z_k)$ commute with the many-body scattering operator $S_0$ defined on a finite spatial interval. For a systematic treatment, we observe the following algebraic properties of the scattering operators $S^{a}_{0},S^{b}_{0},S^{c}_{0},S^{\bar{a}}_{0}$.
Let us choose an arbitrary point $z \in [x, y]$. Using the obvious identity
\beq
& &\Theta (y > x_n > \ldots > x_1 > x) \\
&=& \sum_{j=0}^n \Theta (y > x_n > \!\ldots >\! x_{j+1} > z > x_j > \!\ldots > \!x_1 > x),\nonumber
\eeq
where $x_{n+1}=y$ and $x_0=x$, one can show by rewriting Eqs.~\eq{Sa0}, \eq{Sb0}, \eq{Sc0}, and \eq{Sat0}, that the operators $S^{a}_{0},S^{b}_{0},S^{c}_{0},S^{\bar{a}}_{0}$ satisfy the following closed algebra with respect to the interval splitting operation
\beq
S_0^a [y,x] \!=\! S_0^a [y,z] S_0^a [z,x]\! + S_0^b [y,z] \! \{ S_0^c [z,x]\!- \!S_0^a [z,x] \},\nonumber\\
S_0^b [y,x] \!=\! S_0^a [y,z] S_0^b [z,x]\! + S_0^b [y,z] \! \{ S_0^{\bar{a}} [z,x]\! -\! S_0^b [z,x] \}, \nonumber\\
S_0^c [y,x] \!= \! S_0^c [y,z] S_0^a [z,x]\! + S_0^{\bar{a}} [y,z] \! \{S_0^c [z,x] \!-\!S_0^a [z,x]\}, \nonumber\\
S_0^{\bar{a}} [y,x]\! =\! S_0^c [y,z] S_0^b [z,x] \! + S_0^{\bar{a}} [y,z] \! \{ S^{\bar{a}} [z,x]\! - \! S_0^b [z,x]\}.\nonumber\\
\label{split_alg}
\eeq
If one divides the interval $[x,y]$ into three parts $y>z_2>z_1>x$ by arbitrary points $z_{1}$ and $z_{2}$, one can prove by a direct calculation that the algebra \eq{split_alg} is associative, as expected.
The algebra \eq{split_alg} also allows us to express the action of annihilation operators on the scattered state in a simple form
\beq
& &a (z) S_0^a [y,x] |0 \rangle = S_0^b [y,z] \lambda e^{i k_0 z} S_0^a [z,x] |0 \rangle,\label{aS0a}\\
& &a (z) S_0^b [y,x] | 0 \rangle = S_0^b [y,z] \lambda e^{i k_0 z} S_0^b [z,x] |0 \rangle, \label{aS0b}\\
& &a (z) S_0^c [y,x] |0 \rangle= S_0^{\bar{a}} [y,z] \lambda e^{i k_0 z} S_0^a [z,x] |0 \rangle, \label{aS0c} \\
& &a (z) S_0^{\bar{a}} [y,x] |0 \rangle = S_0^{\bar{a}} [y,z] \lambda e^{i k_0 z} S_0^{b} [z,x] | 0 \rangle .
\label{aS0abar}
\eeq
For the proof of \eq{aS0a}-\eq{aS0abar} we used the property $d_0 (0) =0$.
Similarly, for an action of the ordered product of two annihilation operators $a(z_{2})a(z_{1})$, with $y>z_{2}>z_{1}>x$, we obtain
\beq
& & a (z_2) a (z_1) S_0^a [y,x] |0 \rangle \label{aaS0a} \\
&=& S_0^b [y, z_2] \lambda e^{i k_0 z_2} S_0^b [z_2, z_1] \lambda e^{i k_0 z_1} S_0^a [z_1, x] |0 \rangle ,
\nonumber \\
& & a (z_2 ) a (z_1) S_0^b [y,x] |0 \rangle \label{aaS0b} \\
&=& S_0^b [y, z_2] \lambda e^{i k_0 z_2}S_0^b [z_2, z_1] \lambda e^{i k_0 z_1} S_0^b [z_1, x] |0 \rangle,
\nonumber \\
& & a (z_2) a (z_1) S_0^c [y,x] |0 \rangle \label{aaS0c} \\
&=& S_0^{\bar{a}} [y, z_2] \lambda e^{i k_0 z_2} S_0^b [z_2 , z_1] \lambda e^{i k_0 z_1} S_0^a [z_1, x] |0 \rangle,
\nonumber \\
& & a (z_2) a (z_1) S_0^{\bar{a}} [y,x] |0 \rangle \label{aaS0abar} \\
&=& S_0^{\bar{a}} [y, z_2] \lambda e^{i k_0 z_2} S_0^b [z_2,z_1] \lambda e^{i k_0 z_1} S_0^b [z_1, x] |0 \rangle. \nonumber
\eeq
Iterating this procedure, one can find an action of the ordered product of $m$ annihilation operators $a (z_m) \ldots a (z_1)$, $y>z_m>\ldots z_1>x$. It produces the product of $m+1$ $S$-operators, what can be symbolically written as
\beq
a^m S_0^a & \to & S_0^b \ldots S_0^b \ldots S_0^a , \label{ama} \\
a^m S_0^b & \to & S_0^b \ldots S_0^b \ldots S_0^b , \label{amb} \\
a^m S_0^c & \to & S_0^{\bar{a}} \ldots S_0^b \ldots S_0^a , \label{amc} \\
a^m S_0^{\bar{a}} & \to & S_0^{\bar{a}} \ldots S_0^b \ldots S_0^b . \label{ambara}
\eeq
Note that in all intermediate positions appears only $S_0^b$. To classify leftmost and rightmost operators in these expressions, we introduce the mappings $\sigma$ and $\mu$ according to the table
\beq
\sigma (a) =& \! b , \quad \mu (a) =& \!a , \\
\sigma (b) =& \! b , \quad \mu (b) =& \! b , \\
\sigma (c) =& \! \bar{a}, \quad \mu (c) =& \! a , \\
\sigma (\bar{a}) =& \! \bar{a} , \quad \mu (\bar{a}) =& \! b .
\eeq
In their terms, the relations \eq{ama}-\eq{ambara} acquire the compact form
\beq
a^m S_0^{\beta} \to S_0^{\sigma (\beta)} \left( S_0^b \right)^{m-1} S_0^{\mu (\beta)}.
\label{comp_am}
\eeq
\subsection{Dressing $S$-operators}
In the following we will also need {\it shifted} scattering operators
\beq
S_v^{\beta} [y, x] &=& D^{\dagger} (v) S_0^{\beta} [y,x] D (v) \nonumber\\
&=& e^{v^* b_{k_0}} S^{\beta} [y,x] e^{-v^* b_{k_0}} ,
\label{disp}
\eeq
where $\beta=a, b, c,\bar{a}$, and $D(v)= \exp [v b_{k_0}^{\dagger} - v^* b_{k_0}]$ is the displacement operator of the fields, such that $D^{\dagger} (v) a^{\dagger} (x) D (v) = a^{\dagger} (x) + v^* e^{-i k_0 x}/\sqrt{L}$. Our next goal is to establish explicit expressions for the operators $S_v^{\beta} [y, x]$ for arbitrary complex-valued parameter $v$.
Performing the displacement \eq{disp}, we obtain the new series in field operators defining $S_v^{a, b,c,\bar{a}}$. Appropriately reorganizing (re-summing) them, we find the following expressions
\beq
S_v^a [y,x] &=& \tilde{d}_v (y-x) + \sum_{n=1}^{\infty} \lambda^n \int {\cal D}x_{n} \label{Sa} \\
& \times & d_v (y - x_n) a^{\dagger} (x_n) e^{i k_0 x_n} \nonumber\\
&\times& d_v (x_n - x_{n-1}) a^{\dagger} (x_{n-1})e^{i k_0 x_{n-1}} \nonumber\\
&\times & \ldots d_v (x_2 -x_1) a^{\dagger} (x_1 ) e^{i k_0 x_1} \tilde{d}_v (x_1 -x),
\nonumber
\eeq
\beq
S_v^b [y,x] &=& d_v (y-x) + \sum_{n=1}^{\infty} \lambda^n \int {\cal D} x_{n} \label{Sb}
\\
& \times & d_v (y - x_n) a^{\dagger} (x_n) e^{i k_0 x_n} \nonumber\\
&\times& d_v (x_n - x_{n-1}) a^{\dagger} (x_{n-1}) e^{i k_0 x_{n-1}} \nonumber\\
&\times& \ldots d_v (x_2 -x_1) a^{\dagger} (x_1 ) e^{i k_0 x_1} d_v (x_1 -x),
\nonumber
\eeq
\beq
S_v^c [y,x] &=& \tilde{\tilde{d}}_v (y-x) + \sum_{n=1}^{\infty} \lambda^n \int {\cal D} x_{n} \label{Stc} \\
& \times & \tilde{d}_v (y - x_n) a^{\dagger} (x_n) e^{i k_0 x_n} \nonumber\\
&\times& d_v (x_n - x_{n-1}) a^{\dagger} (x_{n-1})e^{i k_0 x_{n-1}} \nonumber\\
&\times & \ldots d_v (x_2 -x_1) a^{\dagger} (x_1 ) e^{i k_0 x_1} \tilde{d}_v (x_1 -x), \nonumber
\eeq
\beq
S_v^{\bar{a}} [y,x] &=& \tilde{d}_v (y-x) + \sum_{n=1}^{\infty} \lambda^n \int {\cal D} x_{n} \label{Sta} \\
& \times & \tilde{d}_v (y - x_n) a^{\dagger} (x_n) e^{i k_0 x_n} \nonumber\\
&\times& d_v (x_n - x_{n-1}) a^{\dagger} (x_{n-1})e^{i k_0 x_{n-1}} \nonumber\\
&\times & \ldots d_v (x_2 -x_1) a^{\dagger} (x_1 ) e^{i k_0 x_1} d_v (x_1 -x),
\nonumber
\eeq
as well as
\beq
S_v &=& D^{\dagger} (v) S_0 D(v) \nonumber \\
&=& S_v^a [L/2, -L/2] + S^b_v [L/2, -L/2] \frac{\lambda}{\sqrt{2 \Gamma}} A_0^{\dagger},
\label{S-factor}
\eeq
where
\beq
d_v (x) &=& - \frac{p_+ + p_-}{p_+ - p_-} \left[ e^{- i p_+ x} - e^{- i p_- x}\right] , \label{dv}\\
\tilde{d}_v (x) &=& - \frac{p_-}{p_+ - p_-} e^{- i p_+ x} + \frac{p_+ }{p_+ - p_-} e^{- i p_- x},
\label{tdv}\\
\tilde{\tilde{d}}_v (x) &=& - \frac{p_-^2}{p_+^2 - p_-^2} e^{- i p_+ x} + \frac{p_+^2}{p_+^2 -p_-^2} e^{- i p_- x} \label{ttdv} \nonumber \\
& \equiv & \tilde{d}_v (x) + \frac{i \lambda v^*}{(\delta +i \Gamma) \sqrt{L}} d_v (x)
\eeq
are the dressed single-photon propagators, and $p_{\pm} \equiv p_{\pm} (v^*)= \frac{-(\delta+i \Gamma) \pm \sqrt{(\delta+i \Gamma)^2 + 8 \Gamma \alpha v^*/L}}{2} $.
Additional details on evaluation of \eq{Sa}-\eq{Sta} are presented in the Appendix \ref{dress_det}.
\subsection{Factorization property}
To evaluate the FCS we prove the following key property of {\it generalized} $m$-point correlation functions: their {\it factorization} into the $(m+1)$-fold product of the two-point functions.
Generalized correlation functions are defined by
\beq
& &G_{\beta' \beta}^{(m)} (\{z_l \}; y,x) \label{corr_func} \\
&=& \langle 0 | S_{u}^{\beta' \, \dagger} [y,x] \left( : \prod_{l=1}^m a^{\dagger} (z_l) a (z_l) : \right) S_{v}^{\beta} [y,x] |0 \rangle , \nonumber
\eeq
where the ``time-forward'' $S_v^{\beta}$ and the ``time-backward'' $S_u^{\beta' \, \dagger}$ scattering operators depend on {\it different} and {\it arbitrary} displacement parameters $v$ and $u$, respectively. We also assume here that $y>z_m > \ldots > z_1 > x$. Note that an additional symbol for the path-ordering in this expression is not required: operators $a (z_l)$ and $a (z_{l'})$ commute with each other, since (i) they are bosonic, and (ii) they are written in the interaction picture (which is equivalent to the Schr\"{o}dinger picture in the co-moving frame).
We observe that the relations \eq{split_alg} are also fulfilled by the dressed operators $S_v^{\beta}$: one has just to dress \eq{split_alg} with $D(v)$. Along with the property $d_v (0)=0$, this implies that all relations \eq{aS0a}-\eq{comp_am} also remain valid under the replacement $S_0^{\beta} \to S_v^{\beta}$. These properties allow us to split both $S_v^{\beta}$ and $S_u^{\beta' \, \dagger}$ into $m+1$ operators defined on the ordered intervals, see Eq.~\eq{comp_am}. Applying the Wick's theorem, we contract operators from $S_v^{\beta}$ and $S_u^{\beta' \, \dagger}$ belonging to the same interval; in total we have $m+1$ pairwise contractions of intervals. This procedure leads to a factorization of \eq{corr_func}
\beq
& & G_{\beta' \beta}^{(m)} (\{z_l\};y,x)
= |\lambda|^{2m} \mathcal{G}_{\sigma (\beta') \sigma (\beta)} (y-z_m) \nonumber\\
& & \qquad \times\left[ \prod_{l=1}^{m-1} \mathcal{G}_{bb} (z_{l+1} -z_l) \right] \mathcal{G}_{\mu (\beta') \mu (\beta)} (z_1 - x)
\label{fact_prop}
\eeq
into the product of $m+1$ two-point functions defined by
\beq
\mathcal{G}_{\beta' \beta} (z_{l+1}-z_l) &=& \langle 0 | S_u^{\beta' \, \dagger} [z_{l+1},z_l] S_v^{\beta} [z_{l+1},z_l]| 0\rangle .
\label{G_def}
\eeq
In the following we will also use the special case of \eq{G_def} with $u=v=\alpha$
\beq
\bar{\mathcal{G}}_{\beta' \beta} (z_{l+1}-z_l) &=& \langle 0 | S_{\alpha}^{\beta' \, \dagger} [z_{l+1},z_l] S_{\alpha}^{\beta} [z_{l+1},z_l]| 0\rangle ,
\label{G_def0}
\eeq
corresponding to the standard definition of the correlation functions. The factorization property \eq{fact_prop} in this case is well-known (see, e.g., Ref.~\cite{MaWo}).
\section{Computation of the FCS}
\subsection{Detailing the definition}
In our setup shown in Fig.~\ref{fig:fcs} we have the right-moving photons in the incoming state. Therefore, in the outgoing state the right-moving photons correspond to the transmitted particles, while the left-moving photons are those which are reflected. Extending the definition \eq{Fdef} we represent
\beq
F_{r/l,\tau} (\chi_0 ) &=& \langle 0 | D_o^{\dagger} (\alpha) D_e^{\dagger} (\alpha) S_0^{(e) \, \dagger} \nonumber \\
& & \times e^{i \chi_0 N_{r/l, \tau} } S_0^{(e)} D_e (\alpha) D_o (\alpha)| 0\rangle ,
\label{Fdef2}
\eeq
where
\beq
N_{r/l,\tau} = \int_{z_1}^{z_2} d x \frac{a_e^{\dagger} (x) \pm a_o^{\dagger} (x)}{\sqrt{2}} \frac{a_e (x) \pm a_o (x)}{\sqrt{2}} .
\eeq
Since $D_o (\alpha)$ commutes with $D_e (\alpha)$ and $S_0^{(e)}$,
and it holds
\beq
D_o (\alpha)^{\dagger} N_{r/l,\tau} D_o (\alpha) = D_e (\pm \alpha)^{\dagger} N_{r/l,\tau} D_e (\pm\alpha),
\eeq
we can cast
\eq{Fdef2} to
\beq
F_{r/l,\tau} (\chi ) &=& \langle 0 | D_e^{\dagger} (\alpha) S_0^{(e) \, \dagger} D_e^{\dagger} (\pm \alpha) \nonumber \\
& & \times e^{i \chi_0 N_{r/l, \tau} } D_e (\pm \alpha) S_0^{(e)} D_e (\alpha) | 0\rangle .
\label{Fdef3}
\eeq
Using the identity
\beq
& & e^{i \chi_0 N_{r/l,\tau}} = : e^{(z_0-1) N_{r/l,\tau}} : \label{id_norm1} \\
&=& 1 + \sum_{m=1}^{\infty} (z_0-1)^m \int {\cal D}z'_{m}\left( : \prod_{k=1}^m a_{r/l}^{\dagger} (z'_k) a_{r/l} (z'_k) : \right),\nonumber
\eeq
where $z_0 = e^{i \chi_0}$, and taking into account that in \eq{Fdef3} there are only even operators from both sides of $e^{i \chi_0 N_{r/l,\tau}}$ as well as the vacuum average is performed, we can integrate out the odd modes. It can be effectively done by replacing $a_{r/l}^{\dagger} (z'_k) \to \frac{a_e^{\dagger} (z'_k)}{\sqrt{2}}$ and $a_{r/l} (z'_k) \to \frac{a_e (z'_k)}{\sqrt{2}}$ in \eq{id_norm1}. This eventually results in the replacement
\beq
e^{i \chi_0 N_{r/l,\tau}} \to e^{i \chi N_{e, \tau}},
\eeq
where
\beq
N_{e,\tau} = \int_{z_1}^{z_2} d x a_e^{\dagger} (x) a_e (x),
\eeq
and $\chi$ is defined by $z-1 = \frac{z_0 -1}{2}$ and $z=e^{i \chi}$. Thus,
\beq
F_{r/l,\tau} (\chi ) &=& \langle 0 | D_e^{\dagger} (\alpha) S_0^{(e) \, \dagger} D_e^{\dagger} (\pm \alpha) \nonumber \\
& & \times e^{i \chi N_{e, \tau} } D_e (\pm \alpha) S_0^{(e)} D_e (\alpha) | 0\rangle .
\label{Fdef3a}
\eeq
For the chiral model, we define the FCS by
\beq
F_{e,\tau} (\chi) = \langle 0 | D_e^{\dagger} (\alpha) S_0^{(e)\, \dagger} e^{i \chi N_{e,\tau}} S_0^{(e)} D_e (\alpha) | 0\rangle .
\label{Fdef4}
\eeq
This expression is very similar to \eq{Fdef3a}, so we can combine them together
\beq
F_{\tau}^{(\kappa)} (\chi) &=& \langle 0 | D^{\dagger} (\alpha) S_0^{\dagger} D^{\dagger} ((\kappa -1) \alpha) \nonumber \\
& & \times e^{i \chi N_{\tau} } D ((\kappa -1) \alpha) S_0 D (\alpha) | 0\rangle ,
\label{Fdef5}
\eeq
where the parameter $\kappa =0,1,2$ distinguishes between the reflected, chiral, and transmitted modes, respectively. We have also suppressed the label $e$, since all subsequent calculations will be performed in the effective chiral basis using the results of the previous section.
After a simple transformation of \eq{Fdef5} we find
\beq
F_{\tau}^{(\kappa)} (\chi) &=& e^{-\kappa^2 |\alpha |^2} \langle 0 | S_0^{\dagger} e^{- (\kappa -1) \alpha b_{k_0}^{\dagger}} e^{\kappa \alpha^* b_{k_0}}
\nonumber\\
& & \times e^{i \chi N_{\tau}} e^{\kappa \alpha b_{k_0}^{\dagger}} e^{- (\kappa-1) \alpha^* b_{k_0}} S_0 |0 \rangle ,
\label{fcs_def}
\eeq
which is our starting expression for a computation of the FCS.
\subsection{Integrating out the ``future'' and the ``past''}
\label{int_out}
The operator $N_{\tau} = \int_{z_1}^{z_2} dx a^{\dagger} (x) a (x)$ in the definition \eq{fcs_def} contains only fields belonging to the counting interval $[z_1,z_2]$. In turn, the operator
\beq
b_{k_0} = \frac{1}{\sqrt{L}} \int_{-L/2}^{L/2} d x \, a (x) \, e^{-i k_0 x} = \bar{b}_{k_0} + \tilde{b}_{k_0}
\eeq
contains the counting part
\beq
\bar{b}_{k_0} &=& \frac{1}{\sqrt{L}} \int_{z_1}^{z_2} d x \, a (x) \, e^{-i k_0 x} ,
\eeq
and its complement $\tilde{b}_{k_0} = b_{k_0}^f + b_{k_0}^p$. The latter consists of the ``future'' ($z_1 >x>-L/2$) and ``past'' ($L/2 > x >z_2$) parts, see Fig.~\ref{fig:fcs}, which are defined by
\beq
b^f_{k_0} &=& \frac{1}{\sqrt{L}} \int_{-L/2}^{z_1} d x \, a (x) \, e^{-i k_0 x} ,\\
b^p_{k_0} &=& \frac{1}{\sqrt{L}} \int_{z_2}^{L/2} d x \, a (x) \, e^{-i k_0 x}.
\eeq
Integrating out the states lying outside of the counting interval (see the Appendix \ref{deriv_fcs}), we obtain
\beq
& & F_{\tau}^{(\kappa)} (\chi) e^{-(z-1) \kappa^2 |\alpha |^2 \frac{\tau}{L}} \nonumber \\
&=& \left[ 1 + R (T) \left(1 - \frac{|\lambda |^2}{2 \Gamma} \right) - 2\, \mathrm{Re} \, C (T) \right] \nonumber \\
& & \times \left( \Lambda_{aa} (\tau)+ \frac{|\lambda|^2}{2 \Gamma} \Lambda_{bb} (\tau) \right) \nonumber \\
&+& R (T) \left( \Lambda_{cc} (\tau) + \frac{|\lambda|^2}{2 \Gamma} \Lambda_{\bar{a} \bar{a}} (\tau) \right) \nonumber \\
&+& [C (T)-R(T)] \left( \Lambda_{ac} (\tau) + \frac{|\lambda|^2}{2 \Gamma} \Lambda_{b \bar{a}} (\tau) \right) \nonumber \\
&+& [C^* (T)- R (T)] \left( \Lambda_{ca} (\tau) + \frac{|\lambda|^2}{2 \Gamma} \Lambda_{\bar{a} b} (\tau) \right) ,
\label{fcs:c}
\eeq
where
\beq
\Lambda_{\beta' \beta} (\tau) = \langle 0| S_{u}^{\beta' \, \dagger} [z_2,z_1] e^{i \chi N_{\tau}} S_{v}^{\beta} [z_2,z_1] | 0 \rangle
\label{Lam_term}
\eeq
evaluated at $u = z_{\kappa} \alpha$ and $v^* = z_{\kappa} \alpha^*$, where $z_{\kappa} = \kappa (z - 1) +1$, and
\beq
R (T) &=& \bar{\mathcal{G}}_{bb} (L/2 -z_2) , \\
C (T) &=& \bar{\mathcal{G}}_{ab} (L/2-z_2) .
\eeq
The properties of the functions $R (T)$ and $C (T)$ are summarized in the Appendix \ref{app:rcmn}.
The last remaining step is to compute the terms \eq{Lam_term}.
Using the identity
\beq
& & e^{i \chi N_{\tau}} = : e^{(z-1) N_{\tau}} : \\
& & =1 + \sum_{m=1}^{\infty} (z-1)^m \int {\cal D}z'_{m}\left( : \prod_{l=1}^m a^{\dagger} (z'_l) a (z'_l) : \right),\nonumber
\eeq
the definition of the generalized correlation functions \eq{corr_func}, and their factorization property \eq{fact_prop}, we express
\beq
& & \Lambda_{\beta' \beta} (\tau) = \mathcal{G}_{\beta' \beta} (z_2 -z_1) + \sum_{m=1}^{\infty} \left [(z-1) |\lambda|^{2} \right]^m \nonumber \\
& & \qquad \times \int {\cal D}z'_{m} \nonumber
\mathcal{G}_{\sigma (\beta') \sigma (\beta)} (z_2-z'_m) \nonumber\\
& & \qquad \times\left[ \prod_{l=1}^{m-1} \mathcal{G}_{bb} (z'_{l+1} -z'_l) \right] \mathcal{G}_{\mu (\beta') \mu (\beta)} (z'_1 - z_1).
\eeq
Defining the Laplace transform for $x>0$ and its inverse
\beq
\mathcal{G}_{\beta' \beta} (p) &=& \int_0^{\infty} d x \, \mathcal{G}_{\beta' \beta} (x) e^{i p x}, \label{LT} \\
\mathcal{G}_{\beta' \beta} (x) &=& \int_{-\infty +i 0}^{\infty +i 0} \frac{dp}{2 \pi} e^{-i p x} \mathcal{G}_{\beta' \beta} (p) ,
\eeq
we establish
\beq
& & \Lambda_{\beta' \beta} (\tau) = \int \frac{d p}{2 \pi} e^{- i p \tau} \left[ \mathcal{G}_{\beta' \beta} (p) \right. \nonumber \\
& & \left. + (z-1) |\lambda |^2 \frac{\mathcal{G}_{\sigma (\beta') \sigma (\beta)} (p) \mathcal{G}_{\mu (\beta') \mu (\beta)} (p)}{1 - (z-1) |\lambda|^2 \mathcal{G}_{bb} (p)} \right].
\label{lam_fin}
\eeq
The Laplace transforms \eq{LT} for all components $\mathcal{G}_{\beta' \beta} (p)$ are listed in the Appendix \ref{LT_app}. With their help we find (see the Appendix \ref{app:norm}) the following expressions
\beq
& & \Lambda_{aa} (\tau) + \frac{|\lambda|^2}{2 \Gamma} \Lambda_{bb} (\tau) \label{lam_aa_res} \\
&=& \int \frac{d p}{2 \pi} \frac{ i e^{-i p \tau} }
{p + \frac{\Omega_r^2}{R_0 (p)} (z-1) (p+i \Gamma) [ i \Gamma - \kappa (p+i \Gamma)]} \nonumber \\
& & \times \left[1 - \frac{\Omega_r^2 \kappa (z-1) (p+i \Gamma)}{2 R_0 (p)} \right], \nonumber \\
& & \Lambda_{cc} (\tau) + \frac{|\lambda|^2}{2 \Gamma} \Lambda_{\bar{a} \bar{a}} (\tau) \label{lam_cc_res}\\
&=& \int \frac{d p}{2 \pi} \frac{i e^{-i p \tau} }{p + \frac{\Omega_r^2}{R_0 (p)} (z-1) (p+i \Gamma) [ i \Gamma - \kappa (p+i \Gamma)]}\nonumber \\
& & \times \left[ 1+\frac{|\lambda|^2}{2 \Gamma} +\frac{i \Gamma \Omega_r^2 (z-1) (1-\kappa)}{2 R_0 (p)} \left( 1 + \frac{(p + 2 i \Gamma)^2}{\delta^2 + \Gamma^2} \right) \right. \nonumber \\
& & \left. - \frac{\Omega^2_r \kappa (z -1)(p+i \Gamma) }{R_0 (p)} \left( 1 + \frac{ \Omega_r^2 (\kappa (z -1) +3)}{ 8 (\delta^2 +\Gamma^2) } \right) \right] , \nonumber \\
& & \Lambda_{ca} (\tau) + \frac{|\lambda|^2}{2 \Gamma} \Lambda_{\bar{a} b} (\tau)
\label{lam_ca_res} \\
&=& \int \frac{d p}{2 \pi} \frac{ i e^{-i p \tau}}{p + \frac{\Omega_r^2}{R_0 (p)} (z-1) (p+i \Gamma) [ i \Gamma - \kappa (p+i \Gamma)]} \nonumber \\
& & \times \left[ 1 + \frac{ i \Gamma \Omega_r^2 (z-1) (1-\kappa) (p + \delta + i \Gamma )}{2 R_0 (p) (\delta -i \Gamma) } \right. \nonumber \\
& & \left. \quad - \frac{\Omega_r^2 \kappa (z - 1)(p+i \Gamma)}{2 R_0 (p)} \left( 1 + \frac{p(p + \delta+i \Gamma ) }{ 2 (p+i \Gamma) (\delta -i \Gamma) } \right) \right], \nonumber
\eeq
where $\Omega_r = \sqrt{8 \Gamma |\alpha|^2/L}$ denotes the Rabi frequency (note also the relation $\frac{|\lambda|^2}{2 \Gamma} = \frac{\Omega_r^2}{4 (\delta^2+ \Gamma^2)}$), and
\beq
R_0 (p) &=& p^3 + 4 i \Gamma p^2 - p \left( \Omega_r^2 + \delta^2 + 5 \Gamma^2\right) \nonumber \\
& & - i \Gamma \left( \Omega_r^2 + 2 \delta^2 + 2 \Gamma^2 \right)
\label{R0}
\eeq
is the third-order polynomial. An expression for $ \Lambda_{ac} (\tau) + \frac{|\lambda|^2}{2 \Gamma} \Lambda_{b\bar{a}} (\tau)$ is obtained from \eq{lam_ca_res} by flipping the sign $\delta \to - \delta$.
The expressions \eq{fcs:c} and \eq{lam_aa_res}-\eq{R0} completely define the FCS for reflected, chiral, and transmitted photons in the model under consideration. One should also keep in mind that for $\kappa =0,2$ it is necessary to replace in the end of calculation $z-1 \to \frac{z_0 -1}{2}$, since the counting parameter for the reflected and transmitted photons is $\chi_0$, and it is related to $z_0 = e^{i \chi_0}$ (see also discussion before \eq{Fdef3a}).
\subsection{Normalization of probability distribution}
We must check normalization of the probability distribution generated by \eq{fcs:c}, which is expressed by the condition $F_{\tau}^{(\kappa)} (0)=1$. Noticing that at $\chi=0$
the functions $\Lambda_{\beta' \beta} (\tau)$, $\mathcal{G}_{\beta' \beta} (\tau)$, and $ \bar{\mathcal{G}}_{\beta' \beta} (\tau)$ coincide with each other, we check the following identities
\beq
\bar{\mathcal{G}}_{aa} (\tau) + \frac{|\lambda |^2}{2 \Gamma} \bar{\mathcal{G}}_{bb} (\tau) &=& 1 , \label{norm_id1} \\
\bar{\mathcal{G}}_{cc} (\tau) + \frac{|\lambda |^2}{2 \Gamma} \bar{\mathcal{G}}_{\bar{a}\bar{a}} (\tau) &=& 1 + \frac{|\lambda |^2}{2 \Gamma} , \label{norm_id2} \\
\bar{\mathcal{G}}_{ca} (\tau) + \frac{|\lambda |^2}{2 \Gamma} \bar{\mathcal{G}}_{\bar{a}b} (\tau) &=& 1 , \label{norm_id3}
\eeq
which hold for arbitrary $\tau$, by setting $z=1$ in \eq{lam_aa_res}-\eq{lam_ca_res}, and insert them into \eq{fcs:c}. We see that $F_{\tau}^{(\kappa)} (0) =1$ is indeed fulfilled for all $\tau$ and $T$. This also means that the scattering wavefunction is properly normalized.
\subsection{Limiting cases of the waiting time $T$}
\subsubsection{Waiting regime $T \to \infty$}
One of the important detection regimes is when the detector's waiting time is long enough, $T \to \infty$, what physically means that $T$ is much larger than all system's time scales, but still smaller than $L$. Depending on the context we will also call this regime {\it stationary} (for the resonance fluorescence) and {\it bulk} (for the entanglement entropy, see below).
The stationary values $R (\infty) = C (\infty) =\frac{ \Gamma }{\Gamma + |\lambda|^2}$ (see the Appendix \ref{app:rcmn}) allow us to express \eq{fcs:c} as
\beq
& & F_{\tau}^{(\kappa)} (\chi) e^{-(z-1) \kappa^2 |\alpha |^2 \frac{\tau}{L}} \label{fcs:inf} \\
&=& \frac{|\lambda |^2}{2 (\Gamma+|\lambda |^2)} \left( \Lambda_{aa} (\tau)+ \frac{|\lambda|^2}{2 \Gamma} \Lambda_{bb} (\tau) \right) \nonumber \\
& & + \frac{\Gamma}{\Gamma+|\lambda |^2} \left( \Lambda_{cc} (\tau) + \frac{|\lambda|^2}{2 \Gamma} \Lambda_{\bar{a} \bar{a}} (\tau) \right) \nonumber \\
&=& \int \frac{d p}{2 \pi} \frac{ i e^{-i p \tau} W^{(\kappa)} (p)}{p + \frac{\Omega_r^2}{R_0 (p)} (z-1) (p+i \Gamma) [ i \Gamma - \kappa (p+i \Gamma)]} , \nonumber
\eeq
where
\beq
W^{(\kappa)} (p) &=& 1+ \frac{\Gamma \Omega_r^2 (z-1) }{(|\lambda |^2 + \Gamma ) R_0 (p)} \label{Wk} \\
& \times& \left[ \frac{i \Gamma (1-\kappa)}{2} \left( 1 + \frac{(p + 2 i \Gamma)^2}{\delta^2 +\Gamma^2} \right) \right. \nonumber \\
&-& \left. \kappa (p+i \Gamma) \left( 1 + \frac{ \Omega_r^2 (\kappa (z -1) +4)}{ 8 (\delta^2 +\Gamma^2) } \right) \right].\nonumber
\eeq
To find the relation of this result to the photon number statistics in the stationary resonance fluorescence \cite{Mandel},\cite{Smirnov-Troshin}, we must consider the case $\kappa=0$ corresponding to the reflected photons. We note that the fluorescent photons do not interfere with the driving field, and this is precisely the case for the left-moving photons in the presence of the right-propagating driving field. In fact, we find the full agreement of \eq{fcs:inf}, \eq{Wk} at $\kappa=0$ with the result of Mandel \cite{Mandel} and especially with that of Smirnov and Troshin \cite{Smirnov-Troshin}, who have also expressed it in terms of the Laplace transform.
\subsubsection{Waiting regime $T =0$}
In this regime (which we will also call {\it boundary} in the context of the entanglement entropy below) the detection starts from the forefront of the pulse. Using the initial values $R(0) =C (0)=0$ (see the Appendix \ref{app:rcmn}) we cast \eq{fcs:c} to
\beq
F_{\tau}^{(\kappa)} (\chi) e^{-(z-1) \kappa^2 |\alpha |^2 \frac{\tau}{L}} = \Lambda_{aa} (\tau) + \frac{|\lambda|^2}{2 \Gamma} \Lambda_{bb} (\tau) .
\eeq
In the case $\kappa =0$ our result
\beq
F_{\tau}^{(0)} (\chi_0)
&=& \int \frac{d p}{2 \pi} \frac{ i e^{-i p \tau} }
{p + \frac{i \Gamma \Omega_r^2}{2 R_0 (p)} (z_0 -1) (p+i \Gamma) }
\eeq
identically coincides with that of Lenstra \cite{Lenstra} which was derived for the corresponding regime of the resonance fluorescence.
\subsection{Limit of long counting time $\tau$}
When the counting time $\tau$ is much larger than all system's time scales, the main contribution to $F_{\tau}^{(\kappa)}$ comes from the pole in the vicinity of zero, the contributions from other poles being exponentially suppressed. The dependence on $(z-1)$ in the numerators of Eqs.~\eq{lam_aa_res}-\eq{lam_ca_res} can be also neglected in the large-$\tau$ limit, and we obtain
\beq
& & F_{\tau}^{(\kappa)} (\chi) e^{-(z-1)\kappa^2 |\alpha |^2 \frac{\tau}{L}} \label{pre_pois} \label{fcs:tau} \\
& & \approx \int \frac{d p}{2 \pi} \frac{ i e^{-i p \tau} }
{p + \frac{\Omega_r^2}{R_0 (p)} (z-1) (p+i \Gamma) [ i \Gamma - \kappa (p+i \Gamma)]} ,
\nonumber
\eeq
independently of the waiting time $T$.
Setting $p=0$ in the remaining term $\propto (z-1)$ in the denominator of \eq{fcs:tau}, we arrive at the Poissonian distribution $F_{\tau}^{(\kappa)} (\chi) \approx e^{(z-1) \langle N \rangle}$ characterized by the mean value
\beq
\langle N \rangle = \tau \left[ \kappa^2 \frac{|\alpha |^2}{L} + \frac{\Omega_r^2 \Gamma ( 1- \kappa )}{\Omega_r^2 + 2 \delta^2 + 2 \Gamma^2} \right].
\label{pois_mean}
\eeq
In particular, in the chiral model the mean number of photons remains the same, $\tau |\alpha|^2/L$, as in the incident beam, while the mean numbers of reflected and transmitted photons are
\beq
\langle N_l \rangle &=& \frac12 \langle N \rangle_{\kappa=0} = \frac{\tau |\alpha_0|^2}{ L} \frac{ \Gamma^2}{\frac12 \Omega_r^2 + \delta^2 + \Gamma^2} , \label{meanNl} \\
\langle N_r \rangle &=& \frac12 \langle N \rangle_{\kappa=2} = \frac{\tau |\alpha_0|^2}{L} \frac{\frac12 \Omega_r^2 + \delta^2 }{\frac12 \Omega_r^2 + \delta^2 + \Gamma^2}. \label{meanNr}
\eeq
Recall the necessary replacement $z-1 \to \frac{z_0 -1}{2}$, giving the additional factor $1/2$ in the two last expressions.
To find corrections to the Poissonian distribution, we expand the denominator in \eq{pre_pois} to
the linear order in $p$
\beq
& \approx & [1+ (z-1) Z] \left( p - i \frac{(z-1)}{1+ (z-1) Z} \frac{\Omega_r^2 \Gamma (1-\kappa)}{\Omega_r^2 + 2 \delta^2 +2 \Gamma^2} \right) \nonumber \\
& \approx & p - i (z-1) (1-(z-1) Z) \frac{\Omega_r^2 \Gamma (1-\kappa)}{\Omega_r^2 + 2 \delta^2 +2 \Gamma^2},
\eeq
where
\beq
Z &=& \frac{d}{dp} \left[ \frac{\Omega_r^2}{R_0 (p)} (p+i \Gamma) ( i \Gamma - \kappa (p+i \Gamma))\right]_{p=0} \label{Zfactor} \\
&=& \Omega_r^2 \frac{\kappa (\Omega_r^2 + 3 \delta^2 - \Gamma^2) +3 \Gamma^2- \delta^2 }{(\Omega_r^2 + 2 \delta^2 + 2 \Gamma^2)^2} .\nonumber
\eeq
Thereby we achieved the $O ((z-1)^2)$ correction to the pole position in the vicinity of zero, which leads to a modification of the Poissonian form of the generating function
\beq
F_{\tau}^{(\kappa)} (\chi) \approx e^{(z-1) \langle N \rangle} e^{-\tau (z-1)^2 Z \frac{\Omega_r^2 \Gamma (1-\kappa)}{\Omega_r^2 + 2 \delta^2 +2 \Gamma^2} }.
\label{post_pois}
\eeq
\begin{figure}[t]
\includegraphics[width=0.48\textwidth]{pn.eps}
\caption{(Color online) Probability distributions $p_r (n)$ (dark yellow diamonds) and $p_l (n)$ (magenta squares) calculated on the basis of Eq.~\eq{post_pois} for $\delta=0$, $\Omega_r = \sqrt{2}\Gamma$, and $\langle N_r \rangle = \langle N_l \rangle = \frac{\Gamma \tau}{4} =50$. Blue circles indicate the Poissonian distribution with the same mean value. Solid lines correspond to the Gaussian approximations $\frac{e^{-\frac{(n -\langle N_r \rangle )^2}{2\langle N_r \rangle (1+Q_r) }}}{\sqrt{2 \pi \langle N_r \rangle (1+Q_r)}}$ (dark yellow), $\frac{e^{-\frac{(n -\langle N_r \rangle )^2}{2\langle N_r \rangle }}}{\sqrt{2 \pi \langle N_r \rangle}} $ (blue), and $\frac{e^{-\frac{(n -\langle N_r \rangle )^2}{2\langle N_r \rangle (1+Q_l) }}}{\sqrt{2 \pi \langle N_r \rangle (1+Q_l)}} $ (magenta), which become rather accurate at large mean values. The Mandel's $Q$-factors, which are $Q_r = \frac58$ and $Q_l =- \frac38$ for the chosen parameters, quantify deviations of the variances from that of the Poissonian distribution.} \label{fig:pn}
\end{figure}
Deviations of \eq{post_pois} from the Poissonian statistics can be quantified in terms of the Mandel's $Q$-factor \cite{Mandel}
\beq
Q = \lim_{\tau \to \infty} \frac{\langle N^{(2)} \rangle -\langle N \rangle^2}{\langle N \rangle}, \label{Qfactor}
\eeq
where $\langle N^{(2)} \rangle$ is the second factorial moment. For $Q<0$ a distribution is narrower than the Poissonian, and it is called sub-Poissonian; for $Q>0$ a distribution is broader than the Poissonian, and it is called super-Poissonian. If $Q=0$, a distribution is almost indistinguishable from the Poissonian. Performing an expansion of \eq{post_pois} in $(z-1)$ up to the quadratic term we establish
\beq
Q & =& -2 \frac{ Z (1-\kappa) \Gamma^2}{\frac{\kappa^2}{8} (\Omega_r^2 + 2 \delta^2 +2 \Gamma^2)+ \Gamma^2 (1-\kappa)} .
\eeq
We note that in the chiral model ($\kappa=1$) the Mandel's $Q$-factor identically vanishes, rendering it Poissonian. Considering
$\kappa=0$ and $\kappa=2$, we find the factors $Q_l$ and $Q_r$ for the reflected and transmitted photons (recall again the necessary replacement $z-1 \to \frac{z_0 -1}{2}$)
\beq
Q_l &=& \frac12 Q_{\kappa=0}= - \Omega_r^2 \frac{3 \Gamma^2- \delta^2 }{(\Omega_r^2 + 2 \delta^2 + 2 \Gamma^2)^2} ,
\label{Ql} \\
Q_r &=& \frac12 Q_{\kappa=2} = \frac{ \Omega_r^2 \Gamma^2}{\frac{1}{2} \Omega_r^2 + \delta^2 } \frac{2 \Omega_r^2 + 5 \delta^2 +\Gamma^2}{(\Omega_r^2 + 2 \delta^2 + 2 \Gamma^2)^2}.
\label{Qr}
\eeq
We see that for $\delta < \Gamma \sqrt{3}$ the statistics of the reflected photons is sub-Poissonian, and it turns into super-Poissonian for $\delta > \Gamma \sqrt{3}$, while the statistics of the transmitted photons is always super-Poissonian. At $\delta=0$ the expression \eq{Ql} coincides with the original result of Mandel \cite{Mandel} for the photon number statistics in the stationary resonance fluorescence.
In Fig.~\ref{fig:pn} we show the probability distributions $p_r (n)$ and $p_l (n)$ of transmitted and reflected photons for the long counting time $\tau = \frac{200}{\Gamma}$, which are generated by \eq{post_pois}. We also choose $\delta=0$ and $\Omega_r = \sqrt{2} \Gamma$, which provide $\langle N_r \rangle =\langle N_l \rangle$, to ease a comparison of these two distributions. This plot elucidates the physical meaning of the Mandel's $Q$-factor.
We also mention that these theoretical results on photon statistics are supported by the recent measurements of the second-order correlation function \cite{chal1} showing photon antibunching in the reflected field and superbunching in the transmitted field.
\section{Transmission, reflection, and the Mollow triplet}
The mean field and the resonance fluorescence power spectrum, also known as the Mollow triplet \cite{Mollow}, are usually computed using the equation of motion method and the quantum regression theorem. In this section we demonstrate how to obtain these expressions in the framework of our scattering approach.
To compute the mean field $\langle a (z_1) \rangle$ and the first order correlation function $g^{(1)} (z'_1, z_1) = \langle a^{\dagger} (z'_1) a (z_1) \rangle$ we employ the expression \eq{Fdef5}, in which we replace $e^{i \chi N_{\tau}}$ by $a (z_1)$ and $a^{\dagger} (z'_1) a (z_1)$. Thus,
\beq
\langle a (z_1) \rangle &=& \langle 0 | S_{\alpha}^{\dagger} D^{\dagger} (\kappa \alpha ) a (z_1)D (\kappa \alpha) S_{\alpha} | 0\rangle \nonumber \\
&=& \frac{\kappa \alpha}{\sqrt{L}} e^{i k_0 z_1} + \langle 0 | S_{\alpha}^{\dagger} a (z_1) S_{\alpha} | 0\rangle , \label{mf1} \\
\langle a^{\dagger} (z'_1) a (z_1) \rangle &=& \langle 0 | S_{\alpha}^{\dagger} D^{\dagger} (\kappa \alpha ) a^{\dagger} (z'_1) a (z_1)D (\kappa \alpha) S_{\alpha} | 0\rangle \nonumber \\
&=& \frac{\kappa^2 |\alpha |^2}{L} e^{-i k_0 (z'_1 - z_1)}
\nonumber \\
&+& \frac{\kappa \alpha^*}{\sqrt{L}} e^{-i k_0 z'_1} \langle a (z_1) \rangle + \langle a (z'_1) \rangle^* \frac{\kappa \alpha}{\sqrt{L}} e^{i k_0 z_1}
\nonumber \\
&+& \langle 0 | S_{\alpha}^{\dagger} a^{\dagger} (z'_1) a (z_1) S_{\alpha} | 0\rangle .
\label{g1_1}
\eeq
Note that to find the reflected and transmitted ($\kappa=0,2$) fields and their $g^{(1)}$ functions, it is necessary to multiply \eq{mf1} and \eq{g1_1} additionally by the factors $1/\sqrt{2}$ and $1/2$.
So we see that it suffices to compute
\beq
& & \langle 0 | S_{\alpha}^{\dagger} a (z_1) S_{\alpha} |0 \rangle
\nonumber \\
&=& \langle 0 | S_{\alpha}^{a \, \dagger} [L/2,-L/2] a (z_1) S_{\alpha}^a [L/2,-L/2] |0 \rangle \nonumber\\
&+& \frac{|\lambda|^2}{2 \Gamma} \langle 0 | S_{\alpha}^{b \, \dagger} [L/2,-L/2] a (z_1) S_{\alpha}^b [L/2,-L/2] |0 \rangle \nonumber
\eeq
and
\beq
& &\tilde{g}^{(1)} (z'_1 , z_1) \equiv \langle 0 | S_{\alpha}^{\dagger} a^{\dagger} (z'_1) a (z_1) S_{\alpha} |0 \rangle
\nonumber \\
&=& \langle 0 | S_{\alpha}^{a \, \dagger} [L/2,-L/2] a^{\dagger} (z'_1) a (z_1) S_{\alpha}^a [L/2,-L/2] |0 \rangle \nonumber\\
&+& \frac{|\lambda|^2}{2 \Gamma} \langle 0 | S_{\alpha}^{b \, \dagger} [L/2,-L/2] a^{\dagger} (z'_1) a (z_1) S_{\alpha}^b [z_1,-L/2] |0 \rangle .\nonumber
\eeq
Applying the $\alpha$-shifted versions of \eq{aS0a} and \eq{aS0b} we obtain
\beq
\langle 0 | S_{\alpha}^{\dagger} a (z_1) S_{\alpha} |0 \rangle &=& \lambda e^{i k_0 z_1}
\nonumber \\
& & \times \left(\langle 0 | S_{\alpha}^{a \, \dagger} [L/2,-L/2] \right. \nonumber \\
& & \qquad \times S_{\alpha}^b [L/2,z_1] S_{\alpha}^a [z_1,-L/2] |0 \rangle
\nonumber\\
&+& \frac{|\lambda|^2}{2 \Gamma} \langle 0 | S_{\alpha}^{b \, \dagger} [L/2,-L/2] \nonumber \\
& & \left. \qquad \times S_{\alpha}^b [L/2,z_1] S_{\alpha}^b [z_1,-L/2] |0 \rangle \right) \nonumber
\eeq
and
\beq
\tilde{g}^{(1)} (z'_1 , z_1) &=& |\lambda|^2 e^{-i k_0 (z'_1 -z_1)}
\nonumber \\
& & \times \left(\langle 0 | S_{\alpha}^{a \, \dagger} [z'_1,-L/2] S_{\alpha}^{b \, \dagger} [L/2,z'_1] \right. \nonumber \\
& & \qquad \times S_{\alpha}^b [L/2,z_1] S_{\alpha}^a [z_1,-L/2] |0 \rangle \nonumber\\
&+& \frac{|\lambda|^2}{2 \Gamma} \langle 0 | S_{\alpha}^{b \, \dagger} [z'_1,-L/2] S_{\alpha}^{b \, \dagger} [L/2,z'_1] \nonumber \\
& & \left. \qquad \times S_{\alpha}^b [L/2,z_1] S_{\alpha}^b [z_1,-L/2] |0 \rangle \right).\nonumber
\eeq
Next, using \eq{split_alg} we split in the first case the operators $S_{\alpha}^{a,b \, \dagger} [L/2,-L/2]$ into the subintervals $[-L/2, z_1]$ and $[z_1, L/2]$. Assuming $z'_1 > z_1$ in the second case, we split the operators $S_{\alpha}^{a,b \, \dagger} [z'_1,-L/2]$ into the subintervals $[-L/2, z_1]$ and $[z_1, z'_1]$ and the operator $S_{\alpha}^b [L/2,z_1]$ into the subintervals $[z_1, z'_1]$ and $[z'_1,L/2]$. After that, we apply the Wick's theorem, use the identities \eq{norm_id1}-\eq{norm_id3}, and obtain
\beq
& & \langle 0 | S_{\alpha}^{\dagger} a (z_1) S_{\alpha} |0 \rangle = \lambda e^{i k_0 z_1} C (L/2 -z_1), \\
& & \tilde{g}^{(1)} (z'_1 , z_1) = |\lambda|^2 e^{-i k_0 (z'_1 -z_1)} \label{til_g1} \\
& \times & \left(R (T) M (z'_1 -z_1) + [C^* (T) -R (T)] C (z'_1 -z_1) \right),
\nonumber
\eeq
where $T=L/2 -z'_1$ and $M (\tau) = \bar{\mathcal{G}}_{a \bar{a}} (\tau)$. The properties of the latter function are studied in the Appendix \ref{app:rcmn}.
In the stationary regime $L/2 - z_1 \to \infty$ the mean field equals
\beq
\langle a (z_1) \rangle \approx e^{i k_0 z_1} \left( \frac{\kappa \alpha}{\sqrt{L}} + \frac{\lambda \Gamma}{\Gamma+ |\lambda |^2}\right).
\eeq
In particular, we find the mean reflected ($\kappa=0$) and transmitted ($\kappa=2$) fields (recall also about the additional factor $1/\sqrt{2}$)
\beq
\langle a_l (z_1) \rangle & \approx & - \frac{\alpha_0 e^{i k_0 z_1}}{\sqrt{L}} \frac{i \Gamma (\delta - i \Gamma)}{\frac12 \Omega_r^2 + \delta^2 + \Gamma^2} , \\
\langle a_r (z_1) \rangle & \approx & \frac{\alpha_0 e^{i k_0 z_1}}{\sqrt{L}} \left( 1 - \frac{ i \Gamma (\delta - i \Gamma)}{\frac12 \Omega_r^2 +\delta^2 + \Gamma^2 } \right).
\eeq
Dividing these expressions by $\frac{\alpha_0 e^{i k_0 z_1}}{\sqrt{L}}$, we obtain the reflection and transmission amplitudes, in full agreement with \cite{coherent1} and \cite{coherent2}.
We note that for the strong drive the quantities $ |\langle a_{l/r} (z_1) \rangle |^2$ differ from the mean numbers of photons per unit time $N_{l/r}/\tau$, defined by \eq{meanNl}, \eq{meanNr}. These observables begin to coincide in the limit of the weak driving field $\alpha_0 \to 0$ and $\Omega_r \to 0$, both converging to the single-photon reflection $\frac{\Gamma^2}{\delta^2 + \Gamma^2} \frac{|\alpha_0 |^2}{L}$ and transmission $\frac{\delta^2}{\delta^2 + \Gamma^2} \frac{|\alpha_0 |^2}{L}$ probabilities (times the incident photon density), what indicates the suppression of the inelastic scattering processes.
The information about the inelastic -- Mollow -- part of the power spectrum is contained in $\tilde{g}^{(1)} (z'_1, z_1)$ expressed by \eq{til_g1}, and, more precisely, in the functions
\beq
M (\tau) &=& \int \frac{d p}{2 \pi} e^{-i p \tau} \left[ \tilde{r} (p) + \frac{|\lambda |^2 \bar{c} (p) c (p)}{1 -|\lambda |^2 r (p)} \right] \\
&=& \frac{\Gamma}{\Gamma + |\lambda|^2} \left(1 +\frac{|\lambda|^2}{\Gamma} \int \frac{d p}{2 \pi} e^{-i p \tau} \frac{i M_0 (p)}{R_0 (p)} \right), \nonumber \\
C (\tau) &=& \int \frac{dp}{2 \pi} e^{-i p \tau} \frac{c (p)}{1-|\lambda|^2 r (p)} \\
&=& \frac{\Gamma}{\Gamma + |\lambda|^2} \left(1 - \int \frac{dp }{2 \pi} e^{-i p \tau} \frac{i C_0 (p)}{R_0 (p)} \right), \nonumber
\eeq
in the form of terms containing the third-order polynomial $R_0 (p)$ which is defined in \eq{R0}. Here $r (p) = r (p; \alpha, \alpha^*)$, $\tilde{r} (p) = \tilde{r} (p; \alpha, \alpha^*)$, $c (p) = c (p; \alpha, \alpha^*)$ are the special cases of the functions \eq{r_expl}, \eq{rt_expl}, \eq{c_expl}, and
\beq
& & M_0 (p) = (p+2 i \Gamma)^2 - \frac{\Omega_r^2}{2}, \label{M0} \\
& & C_0 (p) = M_0 (p) - \frac{ \Gamma+ |\lambda|^2}{\Gamma} (\delta + i \Gamma) (p+2 i \Gamma) .
\label{C0}
\eeq
We note the inelastic power spectrum is the same (up to the factor $1/2$) for the chiral and reflected/transmitted photons, therefore it is sufficient to consider only the chiral case.
We analyze first the stationary regime $T \to \infty$ of \eq{til_g1}. Here we have $R (\infty) = C (\infty) =\frac{\Gamma}{\Gamma + |\lambda|^2}$, and the Mollow spectrum is completely defined by $\frac{\Gamma}{\Gamma + |\lambda|^2}\mathrm{Re} \frac{i M_0 (p)}{R_0 (p)}$, in the full agreement with the known results \cite{Mollow}, \cite{KM-rev}. The roots of $R_0 (p)$ define the positions and widths of all three Mollow peaks, while the functions $M_0 (p)$ [Eq.~\eq{M0}] contribute to their weights.
The expression \eq{til_g1} shows how the shape of the Mollow spectrum evolves in time $T$ toward its stationary value discussed above. At $T=0$ we have $R(0)=C(0)=0$, and the inelastic power spectrum is absent. For finite $T>0$ its weight starts to grow, it acquires the three-peak form, however its transient shape differs from the stationary one due to the presence of the additional contribution in $C_0 (p)$ [Eq.~\eq{C0}] which is not proportional to $M_0 (p)$.
\begin{figure}[t]
\includegraphics[width=0.48\textwidth]{mollow.eps}
\caption{(Color online) Evolution of the Mollow triplet (from bottom to top) with increasing time $T=\frac{0.01}{\Gamma},\frac{0.05}{\Gamma},\frac{0.1}{\Gamma},\frac{1}{\Gamma},\frac{10}{\Gamma}$ for the parameters $\Omega_{r}=10 \Gamma$ and $\delta=0$. For $T=\frac{10}{\Gamma}$ (top green curve) it is already indistinguishable from the stationary shape given by Eq.~\eq{mollow_stat}.} \label{fig:mollow}
\end{figure}
In the formal expression the power spectrum amounts to
\beq
P (\omega) &=& \frac{k_0}{2 \pi T_0} \int d z'_1 d z_1 \langle a^{\dagger} (z'_1) a (z_1 ) \rangle e^{i \omega (z'_1 -z_1)} \nonumber \\
&=& \frac{k_0}{2 \pi T_0} \mathrm{Re} \int_0^{T_0} d T \int_0^{\infty} d \tau e^{i \omega \tau} \nonumber \\
& & \times \langle a^{\dagger} (L/2-T) a (L/2-T - \tau)\rangle , \label{Pom}
\eeq
where $T_0$ is the maximal waiting time. The inelastic part of \eq{Pom} equals
\beq
P_{\mathrm{inel}} (\omega) &=& \frac{k_0 |\lambda |^2}{2 \pi T_0} \mathrm{Re} \int_0^{T_0} d T \int_0^{\infty} d \tau e^{i (\omega - k_0) \tau} \label{Pom_in} \\
& & \times (R (T) M_{\mathrm{inel}} (\tau) + [C^* (T) - R(T)] C_{\mathrm{inel}} (\tau)) , \nonumber
\eeq
where
\beq
M_{\mathrm{inel}} (\tau) &=& \frac{|\lambda|^2}{\Gamma +|\lambda|^2} \int \frac{d p}{2 \pi} e^{-i p \tau} \frac{i M_0 (p)}{R_0 (p)}, \\
C _{\mathrm{inel}} (\tau) &=& - \frac{\Gamma}{\Gamma +|\lambda|^2} \int \frac{d p}{2 \pi} e^{-i p \tau} \frac{i C_0 (p)}{R_0 (p)}.
\eeq
Computing \eq{Pom_in} we obtain
\beq
P_{\mathrm{inel}} (\omega) &=& \frac{k_0}{T_0} \int_0^{T_0} d T p_{\mathrm{inel}} (\omega, T) , \\
p_{\mathrm{inel}} (\omega, T) &=& \frac{|\lambda |^4 R (T)}{2 \pi (\Gamma +|\lambda|^2)} \mathrm{Re} \left\{ \frac{i M_{0} (\omega -k_0)}{R_0 (\omega -k_0)}
\right\} \label{p_in} \\
& & - \mathrm{Re}\frac{|\lambda |^2 \Gamma [C^* (T) - R(T)] i C_{0} (\omega-k_0)}{2 \pi (\Gamma +|\lambda|^2) R_0 (\omega -k_0)} . \nonumber
\eeq
For $T_0 \to \infty$ we recover the stationary Mollow spectrum
\beq
P_{\mathrm{inel}}^{\mathrm{stat}} (\omega) &=& k_0 \, p_{\mathrm{inel}} (\omega, \infty) \nonumber \\
&=& \frac{k_0 |\lambda |^4 \Gamma}{2 \pi (\Gamma +|\lambda|^2)^2} \mathrm{Re} \left\{ \frac{i M_{0} (\omega -k_0)}{R_0 (\omega -k_0)}
\right\} .
\label{mollow_stat}
\eeq
At large Rabi frequency $\Omega_r \gg \Gamma, \delta$ it acquires the most familiar form
\beq
& & P_{\mathrm{inel}}^{\mathrm{stat}} (\omega) = \frac{k_0 \Gamma}{4 \pi} \\
& \times & \left\{ \frac{\Gamma}{(\omega- k_0)^2 +\Gamma^2} + \frac12 \sum_{s =\pm} \frac{\frac{3 \Gamma}{2}}{(\omega -k_0 -s \Omega_r)^2+ \frac{9 \Gamma^2}{4}} \right\}. \nonumber
\eeq
For finite $T_0$ we show in Fig.~\ref{fig:mollow} the augmentation of $p_{\mathrm{inel}} (\omega, T)$ with increasing time $T$.
\section{Reduced density matrix and entanglement entropy}
Explicit knowledge of the scattering state allows us to determine a reduced density matrix of some spatial interval, which we continue to call a counting interval. It suffices to trace out ``past'' and ``future'' states of the full density matrix $(S_0 |\alpha \rangle) (\langle \alpha |S_0^{\dagger})$ by a procedure similar to that described in the section \ref{int_out}. Knowing the reduced density matrix, whose computation by other methods is questionable, we can study the entanglement entropy in our model.
Let us consider an arbitrary many-body operator $\hat{A}$ in the chiral model which is defined on the counting interval $[z_1 , z_2]$. Its average value in the scattering state reads
\beq
\langle \hat{A} \rangle &=& \langle 0 | D^{\dagger} (\alpha) S_0^{\dagger} \hat{A} S_0 D (\alpha) | 0 \rangle \\
&=& \langle 0 | S_{\alpha}^{\dagger} D^{\dagger} (\alpha) \hat{A} D (\alpha) S_{\alpha} | 0 \rangle \nonumber \\
&=& \langle 0 | S_{\alpha}^{a \, \dagger} [L/2,-L/2] \hat{A}_{\alpha} S_{\alpha}^a [L/2,-L/2] | 0 \rangle
\nonumber \\
&+& \frac{|\lambda |^2}{2 \Gamma} \langle 0 | S_{\alpha}^{b \, \dagger} [L/2,-L/2] \hat{A}_{\alpha} S_{\alpha}^b [L/2,-L/2] | 0 \rangle , \nonumber
\eeq
where $\hat{A}_{\alpha} = D^{\dagger} (\alpha) \hat{A} D (\alpha) $. Repeating the same steps following Eq.~\eq{fcs:fcp}, we obtain an expression for $\langle \hat{A}_{\alpha} \rangle$ analogous to \eq{fcs:c} -- it is only necessary to set $\kappa=1$ and $z=1$, and to replace $e^{i \chi N_{\tau}}$ by $\hat{A}_{\alpha}$. This implies that $\hat{A}$ can be expressed as a trace $\mathrm{Tr}_{\tau} (\hat{A} \hat{\rho}_{\tau})$ over the states in the spatial interval $\tau=z_2-z_1$ with the reduced density matrix of this interval
\beq
\hat{\rho}_{\tau} &=& \left[ 1 + \left( 1- \frac{|\lambda |^2}{2 \Gamma} \right) R (T) - 2 \mathrm{Re} \, C (T) \right] \nonumber\\
&\times& \left( |\psi^a \rangle \langle \psi^a| + \frac{|\lambda |^2}{2 \Gamma} |\psi^b \rangle \langle \psi^b| \right) \nonumber \\
&+& R (T) \left( |\psi^c \rangle \langle \psi^c | + \frac{|\lambda |^2}{2 \Gamma} |\psi^{\bar{a}} \rangle \langle \psi^{\bar{a}} | \right)
\nonumber \\
&+& \left[ C (T) - R(T) \right] \left( |\psi^a \rangle \langle \psi^{c} | + \frac{|\lambda |^2}{2 \Gamma} |\psi^b \rangle \langle \psi^{\bar{a}} | \right) \nonumber \\
&+& \left[ C^* (T) - R(T) \right] \left( |\psi^{c} \rangle \langle \psi^a| + \frac{|\lambda |^2}{2 \Gamma} |\psi^{\bar{a}} \rangle \langle \psi^b| \right) \nonumber \\
&=& \sum_{\beta,\beta'} \rho_{\beta \beta'} (T) | \psi^{\beta} \rangle \langle \psi^{\beta'} |,
\label{red_matr}
\eeq
where
\beq
| \psi^{\beta} \rangle = D (\alpha) S^{\beta}_{\alpha} [z_2 , z_1] |0 \rangle
\label{bas1}
\eeq
are linearly independent many-body states. Thus, it turns out that $\hat{\rho}_{\tau} $ describes the states in the effective four-dimensional Hilbert space spanned by $| \psi^{\beta} \rangle$. The reduced density matrix \eq{red_matr} is characterized by four eigenvalues $\lambda_i$, and the entanglement entropy of the interval $\tau$ with the rest of the pulse is then given by
\beq
\mathcal{S} = - \sum_{i=1}^4 \lambda_i \ln \lambda_i .
\eeq
The basis \eq{bas1} is, however, not orthonormal, and the corresponding Gram matrix $\langle \psi^{\gamma'} | \psi^{\gamma} \rangle$ differs from the identity. Our central observation is that its components coincide with the two-point functions \eq{G_def0}, $\langle \psi^{\gamma'} | \psi^{\gamma} \rangle =\bar{\mathcal{G}}_{\gamma' \gamma} (\tau)$. Therefore, the eigenvalues $\lambda_i$ coincide with the eigenvalues of the $4 \times 4$ matrix $\rho (T) \bar{\mathcal{G}} (\tau)$.
\begin{figure}[t]
\includegraphics[width=0.45\textwidth]{s_entangle1a.eps}
\caption{(Color online) Entanglement entropy $\mathcal{S}$ as a function of the subsystem size $\tau$ for $T \to \infty$ (black upper curve) and $T=0$ (blue lower curve); the detuning $\delta=0$ and the Rabi frequency $\Omega_{r}=4 \Gamma$ are the same for both curves. The horizontal lines indicate the limiting values $\ln 4$ (upper line) and $\ln 2$ (lower line). }
\label{fig:sent1}
\end{figure}
In the {\it bulk} regime $T \to \infty$ we find that one of the eigenvalues of the matrix $\rho (\infty) \bar{\mathcal{G}} (\tau) $ equals
\beq
\lambda_4 (\tau) = - \frac{|\lambda |^4 R (\tau)}{4 \Gamma (\Gamma +|\lambda |^2)} - |\lambda |^2 \frac{ M (\tau)-1}{2 (\Gamma +|\lambda |^2)}.
\eeq
At small $\tau$ we have
\beq
\lambda_1 (\tau) & \approx & 1 - \frac{|\lambda |^4 \Gamma \tau}{(\Gamma +|\lambda |^2)^2}, \\
\lambda_2 (\tau) & \approx & \frac{|\lambda |^4 \Gamma \tau}{(\Gamma +|\lambda |^2)^2}, \\
\lambda_3 (\tau) & \sim & O (\tau^2), \\
\lambda_4 (\tau) & \sim & O (\tau^3),
\eeq
which leads us to the following behavior of the entanglement entropy
\beq
\mathcal{S}_{\infty} \approx - \frac{|\lambda |^4}{(\Gamma +|\lambda |^2)^2} (\Gamma \tau) \ln (\Gamma \tau).
\label{s0_bulk}
\eeq
In the limit of large $\tau \to \infty$ we have
\beq
\lambda_{1,2} = \frac14 \left( 1 \pm \sigma \right)^2 , \quad \lambda_3 = \lambda_4 = \frac{1-\sigma^2}{4},
\eeq
where
\beq
\sigma = \frac{\sqrt{\Gamma (\Gamma + 2 |\lambda |^2)}}{\Gamma + |\lambda |^2} < 1,
\eeq
leading us to the expression
\beq
\mathcal{S}_{\infty}^{\infty} &=& \lim_{\tau \to \infty} \mathcal{S}_{\infty} \nonumber \\
&=& - (1 + \sigma ) \ln \frac{1 + \sigma}{2} - (1 - \sigma) \ln \frac{1 - \sigma}{2} .
\label{s_bulk_stat}
\eeq
For the weak drive $|\lambda|^2 \ll \Gamma$ ($\sigma \to 1$), $\mathcal{S}_{\infty}^{\infty}$ vanishes. It means that in the absence of inelastic processes there are no correlations, and therefore there is no entanglement. For the strong drive $|\lambda|^2 \gg \Gamma$ ($\sigma \to 0$) we obtain $\lambda_i \approx \frac14$, and $\mathcal{S}_{\infty}^{\infty}$ approaches its maximal upper bound $\ln 4$.
\begin{figure}[t]
\includegraphics[width=0.45\textwidth]{s_entangle2a.eps}
\caption{(Color online) The same quantities as in Fig.~\ref{fig:sent1} for the different Rabi frequency $\Omega_{r}=10 \Gamma$.} \label{fig:sent2}
\end{figure}
In the {\it boundary} regime $T=0$ the rank of $\rho (0)$ reduces by two (because of $R (0)=C(0) =0$), and we have only two nonzero eigenvalues
\beq
\lambda_{1,2} (\tau) = \frac12 \left( 1 \pm \sqrt{\left(1- \frac{|\lambda |^2}{\Gamma} R (\tau) \right)^2 - 2 \frac{|\lambda|^2}{\Gamma} |C (\tau)|^2} \right). \nonumber \\
\label{lam12}
\eeq
At small $\tau$ they behave like
\beq
\lambda_1 \approx 1-\frac{(\Omega_r \tau)^2 }{2}, \quad \lambda_2 \approx \frac{(\Omega_r \tau)^2 }{2},
\eeq
yielding
\beq
\mathcal{S}_{0} \approx - (\Omega_r \tau)^2 \ln (\Omega_r \tau).
\label{s0_bound}
\eeq
At large $\tau$ the eigenvalues \eq{lam12} saturate at the values
\beq
\lambda_{1,2} = \frac{1 \pm \sigma}{2},
\eeq
and therefore
\beq
\mathcal{S}_{0}^{\infty} &=& \lim_{\tau \to \infty} \mathcal{S}_{0} \nonumber \\
&=& - \frac{1 + \sigma}{2} \ln \frac{1 + \sigma}{2} - \frac{1 - \sigma}{2} \ln \frac{1 - \sigma}{2} .
\label{s_bound_stat}
\eeq
It is remarkable that
\beq
\mathcal{S}_{0}^{\infty} = \frac12 \mathcal{S}_{\infty}^{\infty},
\label{sbb}
\eeq
which means that the subsystem lying deep in the bulk of the scattered pulse is twice stronger entangled with the rest system than the subsystem at the forefront of the pulse. The existence of the finite values \eq{s_bulk_stat} and \eq{s_bound_stat} in the large-$\tau$ limit tells us that the area law is asymptotically fulfilled for the large subsystem size. In our 1D geometry, the ``area'' of the subinterval consists either of two points in the bulk case or of a single point in the boundary case, and this difference in the ``area'' measure is accounted by the factor $1/2$ in \eq{sbb}.
At small $\tau$, the expression \eq{s0_bulk} and \eq{s0_bound} contain the logarithmic terms, which means that the entanglement entropy for the small subsystem size violates the volume ($\sim \tau$) law in our model.
In Fig.~\ref{fig:sent1} and \ref{fig:sent2} the $\tau$-dependence of the entanglement entropy for the bulk (black curve) and boundary (blue curve) cases using the values $\Omega_r =4 \Gamma$ (moderate drive) and $\Omega_r = 10 \Gamma$ (strong drive). The upper limits $\ln 4$ and $\ln 2$ are indicated by the horizontal lines. In both cases we set the detuning $\delta=0$ for simplicity.
We observe that the bulk entanglement entropy is the monotonously growing function of the subsystem size. In turn, the boundary entanglement entropy exhibits oscillatory behavior before it reaches the saturation value. For the strong driving field both entropies nearly reach the corresponding maximally allowed values $\ln 4$ and $\ln 2$ at large $\tau$.
Our numerical analysis also shows that the entanglement entropies $\mathcal{S} (\tau)$ for finite values of $T$ lie between the blue and the black curves (not shown in Figs.~\ref{fig:sent1} and \ref{fig:sent2}), though not always being bounded by them, but always being bounded by $\ln 4$ from above.
\section{Conclusion}
We have exactly computed the full counting statistics in the fundamental quantum
optical setup -- a finite size pulse of the coherent light propagating in the multi-mode waveguide and interacting
with the two-level system. These results provide a quantitative determination of many-body
correlation effects of photons mediated by their interaction with the emitter. Our analysis takes into account the spatial parameters of the incident pulse as well as the parameters of the detector -- the waiting time $T$ and the counting time $\tau$, in terms of which the FCS is defined and analyzed. We show that the three types of counting statistics for the reflected, transmitted, and chiral photons have qualitatively different behavior (sub-Poissonian, super-Poissonian, and Poissonian). We have analyzed the entanglement entropy of a spatial part of the scattered pulse with the rest of it in the chiral model, and observed the fulfillment of the area law for large subsystem size and the violation of the volume law for small subsystem size, as well as the oscillatory behavior of the entanglement entropy as a function of the subsystem size $\tau$ in the case of the short waiting time $T$.
We believe that the full characterization of properties of the scattered coherent light presented here will be requested in future theoretical and experimental studies of many-body effects in fundamental models of quantum nanophotonics and extended for more complicated systems.
\section*{Acknowledgements}
We benefited a lot from discussions with D. Baeriswyl, A. Fedorov, D. Ivanov, G. Johansson, A. Komnik, M. Laakso, G. Morigi, M. Ringel, and M. Wegewijs. Work of V. G. is part of the D-ITP consortium, a program of the Netherlands Organisation for Scientific Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW).
|
{
"timestamp": "2015-06-24T02:12:01",
"yymm": "1504",
"arxiv_id": "1504.03350",
"language": "en",
"url": "https://arxiv.org/abs/1504.03350"
}
|
\section{Introduction}
\setcounter{equation}{0}
It is well known that the simplest and most important quantum field theory model is the Abelian Higgs model \cite{JT,Ryder} which embodies an electromagnetic gauge field and
spontaneously broken symmetry and allows mass generation through the Higgs mechanism. In the temporal gauge, its static limit gives rise to the classical Ginzburg--Landau
theory for superconductivity \cite{GL} so that in its two dimensional setting mixed-state configurations known as the Abrikosov vortices \cite{Ab} can be rigorously constructed
\cite{Br,JT,T1,T2,WY}. Inspired by the gauged sigma model of Schroers \cite{Sch1,Sch2}, the classical Abelian Higgs model is extended in \cite{Y1,Y2} to allow the coexistence
of vortices and anti-vortices. This extended model is also shown to generate cosmic strings and anti-strings when gravitation is switched on by the Einstein equations which give
rise to curvature and mass concentrations essential for matter accretion in the early universe
\cite{Kibble1,Kibble2,Vi,VS,Yob1,Yob2}. In order to understand the topological contents of such an extended Abelian Higgs model, a reformulation of it is carried in
\cite{SSY} in the context of
a complex line bundle over a compact Riemann surface $S$ as in \cite{Br,Ga,N1,N2}. In a sharp and interesting contrast with the Abelian Higgs model where vortices are topologically
characterized by the first Chern class, the vortices and anti-vortices in the extended Abelian Higgs model \cite{Y1,Y2} are characterized jointly and elegantly \cite{SSY} by the first Chern class of the
line bundle and the Thom class \cite{DS} of the associated dual bundle. In the former case, there are only finitely many minimum energy values which can be attained due to the fact that
the total number of vortices is confined by the total area $|S|$ of the two-surface $S$ where vortices reside. In the latter case, however, the confinement is made instead to the difference of
the numbers of vortices and anti-vortices, but the minimum energy is proportional to the sum of these numbers. Hence the possible minimum energy values becomes an explicitly determined
infinite sequence as in the situation of vortices over a non-compact surface in the classical Abelian Higgs theory \cite{JT,T1,T2}.
In a recent interesting work of Tong and Wong \cite{TW}, a product Abelian gauge field theory is formulated to include magnetic impurities in the form of an extra gauge-matter sector. This gauge-matter
sector is not treated as a background source but as a fully coupled sector. In other words, this is a product Abelian gauge field theory with two complex Higgs fields. It is shown in \cite{TW}
that, like in the classical Abelian Higgs model, the new product model allows a BPS (after Bogomol'nyi \cite{Bo} and Prasad--Sommerfield \cite{PS}) reduction, hence a construction of magnetic
vortices as in \cite{JT}. The present paper aims to enrich our understanding of Abelian (magnetic) vortices by achieving two goals. The first is to extend the product Abelian gauge field theory of Tong and Wong \cite{TW} using the ideas in
\cite{Sch1,Sch2,Y1,Y2} into a new product field theory that allows the coexistence of two species of vortices and anti-vortices. The second is to establish an existence theorem for such
vortices of beautiful topological characteristics. For clarity and simplicity, the underlying domain for the vortices to live is assumed to be a compact Riemann surface, as in \cite{SSY}.
In order to put our study in an appropriate perspective, we shall first present a reformulation of the Tong--Wong model \cite{TW} in terms of
a complex line bundle over a compact Riemann surface. In such a context, we show that a Bradlow type bound or limit appears as in the
Abelian Higgs theory for the existence of multiple vortices \cite{Br,Ga,N1,N2,WY}, which is a preparation for our work regarding the extended model.
An outline of the rest of the paper is as follows.
In Section 2, we present the Tong--Wong theory and our extended product Abelian gauge field theory, in their static limits.
We describe in detail the field-theoretical properties of the extended theory and derive its BPS equations. We then state our main existence theorems for the existence of multiple vortices in the Tong--Wong theory and
for the coexistence of multiple vortices and anti-vortices, of two species. In Section 3,
we convert the BPS equations into systems of nonlinear elliptic equations, state the main existence theorems
in terms of these equations, and carry out some preliminary discussion. In Section 4, we establish the existence and uniqueness
theorem for the Tong--Wong multiple vortex solutions by calculus of variations.
In Section 5, we prove the existence theorem for the vortex and anti-vortex solutions
of our extended model by using a Leray--Schauder fixed-point theorem argument \cite{GT}
under a necessary and sufficient condition. In Section 6, we explicitly compute the (minimum) energy of a vortex and anti-vortex solution and show that such energy arises topologically and is proportional to the sum of vortex and anti-vortex numbers of two species.
In Section 7, we make some concluding remarks regarding coexisting vortices and anti-vortices in the extended model.
\section{Energy functionals, BPS reductions, and existence theorems}
\setcounter{equation}{0}
Let $L$ be complex Hermitian line bundle over a Riemann surface $S$. Use $q,p$ to denote two sections $L\to S$ and $Dq,Dp$ the connections induced from
the real-valued connection 1-forms $\hat{A},\tilde{A}$, respectively, so that
\be
Dq=\dd q-\ii (\hat{A}-\tilde{A})q,\quad Dp=\dd p-\ii \tilde{A} p.
\ee
Using $*$ to denote the usual Hodge dual operating on differential forms, the energy density of the
Tong--Wong model \cite{TW} for a product
Abelian Higgs theory implementing magnetic impurities may be rewritten as
\bea\label{x1.2}
{\cal E}&=&\frac12 *(\hat{F}\wedge *\hat{F})+\frac12*(\tilde{F}\wedge *\tilde{F})+*(Dq\wedge * \overline{Dq})+*(Dp\wedge * \overline{Dp})\nn\\
&&+\frac12(1-|q|^2)^2+\frac12([1-|q|^2]+[|p|^2-1])^2,
\eea
where $\hat{F}=\dd\hat{A},\tilde{F}=\dd\tilde{A}$ are curvature 2-forms, which recovers the classical Ginzburg--Landau model
\cite{GL} when impurities are switched off by setting
\be
\tilde{A}=0,\quad p=1.
\ee
Note also that there holds the identity
\be
Dq\wedge *\overline{Dq}+(*Dq)\wedge \overline{Dq}=(Dq\pm\ii *Dq)\wedge *\overline{(Dq\pm\ii * Dq)}
\pm\ii (Dq\wedge \overline{Dq}-(*Dq)\wedge(*\overline{Dq})).
\ee
Thus we get
\bea\label{x2.5}
{\cal E}&=&\frac12\left|\hat{F}\mp *(1-|q|^2)\right|^2+\frac12\left|\tilde{F}\pm(*[1-|q|^2]+*[|p|^2-1])\right|^2\nn\\
&&\pm *\hat{F}(1-|q|^2)\mp *\tilde{F}([1-|q|^2]+[|p|^2-1])\nn\\
&&+|Dq\pm\ii *Dq|^2+|Dp\pm\ii * Dp|^2\nn\\
&&\pm\frac{\ii}2 (Dq\wedge \overline{Dq}-(*Dq)\wedge(*\overline{Dq}))\pm\frac{\ii}2 (Dp\wedge \overline{Dp}-(*Dp)\wedge(*\overline{Dp})).
\eea
On the other hand, with the current densities
\be
J(q)=\frac{\ii}2(q\overline{Dq}-\overline{q} Dq),\quad J(p)=\frac{\ii}2(p\overline{Dp}-\overline{p} Dp),
\ee
we have
\bea
\dd J(q)&=&-(\hat{F}-\tilde{F})|q|^2+\frac{\ii}2(Dq\wedge\overline{Dq}-*Dq\wedge *\overline{Dq}),\label{x2.7}\\
\dd J(p)&=&-\tilde{F}|p|^2+\frac{\ii}2(Dp\wedge\overline{Dp}-*Dp\wedge *\overline{Dp}).\label{x2.8}
\eea
Inserting (\ref{x2.7}) and (\ref{x2.8}) into (\ref{x2.5}), we have
\bea\label{x2.9}
{\cal E}&=&\frac12\left|\hat{F}\mp *(1-|q|^2)\right|^2+\frac12\left|\tilde{F}\pm(*[1-|q|^2]+*[|p|^2-1])\right|^2\nn\\
&&+|Dq\pm\ii *Dq|^2+|Dp\pm\ii * Dp|^2\nn\\
&&\pm *(\hat{F}-\tilde{F})\pm *\tilde{F}\pm*\dd J(q)\pm*\dd J(p),
\eea
which leads to the topological energy lower bound
\be\label{x2.10}
E=\int_S {\cal E}*1\geq\left|\int_S \hat{F}\right|.
\ee
From the form of (\ref{x2.9}) it is clear that the lower bound in (\ref{x2.10}) is attained by the solutions of the BPS equations
\bea
\hat{F}&=&\pm *(1-|q|^2),\label{x2.11}\\
\tilde{F}&=&\mp *([1-|q|^2]+[|p|^2-1]),\label{x2.12}\\
Dq\pm \ii *Dq&=&0,\label{x2.13}\\
Dp\pm \ii *Dp&=&0,\label{x2.14}
\eea
as derived by Tong and Wong in \cite{TW}.
The equations of motion of (\ref{x1.2}) are
\bea
D*Dq&=&-(1-|q|^2)q-([1-|q|^2]+[|p|^2-1])q,\label{x2.15}\\
\dd*\hat{F}&=&\ii(\overline{q}Dq-q\overline{Dq}),\label{x2.16}\\
D*Dp&=&([1-|q|^2]+[|p|^2-1])p,\label{x2.17}\\
\dd *\tilde{F}&=&-\ii(\overline{q}Dq-q\overline{Dq})+\ii (\overline{p}Dp-p\overline{Dp}),\label{x2.18}
\eea
which contain (\ref{x2.11})--(\ref{x2.14}) as its first integral and may be viewed as
a reduced form of (\ref{x2.15})--(\ref{x2.18}). These reduced first-order equations are often referred to as the BPS equations after Bogomol'nyi \cite{Bo} and Prasad--Sommerfield \cite{PS} who pioneered the
idea of such reduction for the classical Yang--Mills--Higgs equations. When the upper sign is taken, the system is said to be self-dual; the lower, anti-self-dual.
It may also be checked that the self-dual and anti-self-dual cases are related to each other through the transformation
\be
\hat {A} \rightleftharpoons -\hat{A},\quad\tilde{A}\rightleftharpoons- \tilde{A},\quad q \rightleftharpoons\overline{q},\quad p\rightleftharpoons\overline{p}.
\ee
Hence, in the sequel, we will only consider the self-dual situation.
The structure of (\ref{x2.13}) and (\ref{x2.14}) indicates that the zeros of $q,p$ are isolated and of integer multiplicities which may be assumed to be
\be\label{xz}
{\cal Z}(q)=\{z_{1,1},\dots,z_{1,N_1}\},\quad {\cal Z}(p)=\{z_{2,1},\dots,z_{2,N_2}\},
\ee
where for convenience a zero of multiplicity $m$ is counted as $m$ zeros in the zero set. The quantities $\frac1{2\pi}\int (\hat{F}-\tilde{F})$ and $\frac1{2\pi}\int \tilde{F}$ are the first Chern numbers induced from the connections $\hat{A}-\tilde{A}$ and $\tilde{A}$ over
$L\to S$ which are determined by the numbers of zeros, $N_1$ and $N_2$, by the formulas
\bea
\frac1{2\pi}\int_S (\hat{F}-\tilde{F})&=& N_1,\label{221}\\
\frac1{2\pi}\int_S\tilde{F}&=& N_2,\label{222}
\eea
respectively.
Regarding the Tong--Wong BPS equations (\ref{x2.11})--(\ref{x2.14}), here is our existence theorem.
\begin{theorem}\label{thm2.1}
For the BPS system consisting of equations (\ref{x2.11})--(\ref{x2.14}) over a compact Riemann surface $S$ with canonical total area
$|S|$ governing two connection 1-forms $\hat{A},\tilde{A}$ and two cross sections $q,p$ with the prescribed sets of zeros
given in (\ref{xz}), there exists a solution to realize
these sets of zeros if and only if $N_1$ and $N_2$ satisfy the bound
\be\label{Bradlow}
N_1+2N_2<\frac{|S|}{2\pi}.
\ee
Such a solution carries a minimum
energy of the form
\be\label{xE}
E= 2\pi (N_1+N_2),
\ee
and is unique up to gauge transformations.
\end{theorem}
The condition stated in (\ref{Bradlow}) is analogous to the so-called Bradlow bound \cite{BM,Dem,Nasir} in
the classical Abelian Higgs model \cite{Br,Ga,WY} which was actually deduced earlier by Noguchi \cite{N1,N2}.
Next, following \cite{Y1,Y2} based on the idea of gauged sigma model, we show that we may extend the Tong--Wong model \cite{TW} to accommodate vortices and anti-vortices by considering the modified energy density
\bea
{\cal E}&=&\frac12 *(\hat{F}\wedge *\hat{F})+\frac12*(\tilde{F}\wedge *\tilde{F})+\frac4{(1+|q|^2)^2}*(Dq\wedge * \overline{Dq})+\frac4{(1+|p|^2)^2}*(Dp\wedge * \overline{Dp})\nn\\
&&+2\left(\frac{1-|q|^2}{1+|q|^2}\right)^2
+2\left(\frac{1-|q|^2}{1+|q|^2}+\frac{|p|^2-1}{1+|p|^2}\right)^2. \label{x1.4}
\eea
In fact, let $\phi$ and $\psi$ be two $S^2$-valued scalar fields. Fix ${\bf n}=(0,0,1)\in S^2$ and define the vacuum manifold of the model to be
\be
{\bf n}\cdot\phi=0,\quad {\bf n}\cdot\psi=0.
\ee
Then we modify (\ref{x1.2}) into the form
\be\label{xx1.2}
{\cal E}=\frac12 *(\hat{F}\wedge *\hat{F})+\frac12*(\tilde{F}\wedge *\tilde{F})+|D\phi|^2+|D\psi|^2+
2({\bf n}\cdot\phi)^2+2({\bf n}\cdot[\phi-\psi])^2,
\ee
where
\be
D\phi=\dd \phi-({\bf n}\times\phi)(\hat{A}-\tilde{A}),\quad D\psi=\dd \psi-({\bf n}\times\psi)\tilde{A}.
\ee
Use the stereographic projection from the south pole $-{\bf n}$ of $S^2$ to represent $\phi$ and $\psi$ by complex-valued functions $q$ and $p$,
respectively, so that
\be\label{2.29}
\phi=\left(\frac{2\Re\{q\}}{1+|q|^2},\frac{2\Im\{q\}}{1+|q|^2},\frac{1-|q|^2}{1+|q|^2}\right),\quad \psi=\left(\frac{2\Re\{p\}}{1+|p|^2},\frac{2\Im\{p\}}{1+|p|^2},\frac{1-|p|^2}{1+|p|^2}\right),
\ee
where $\Re\{c\}$ and $\Im\{c\}$ denote the real and imaginary parts of a complex number $c$. Inserting (\ref{2.29}) into (\ref{xx1.2}), we arrive at (\ref{x1.4}).
It is interesting to observe that (\ref{x1.2}) is recovered from (\ref{x1.4}) when taking the limit $|q|\to 1,|p|\to 1$ in the
denominators $1+|q|^2$ and $1+|p|^2$ of (\ref{x1.4}).
The Euler--Lagrange equations of the energy density are found to be
\bea
D*\left(\frac{Dq}{(1+|q|^2)^2}\right)&=&\frac1{(1+|q|^2)^3}(Dq\wedge *\overline{Dq})+2*\left(\frac{1-|q|^2}{(1+|q|^2)^3}\right)q\nn\\
&& +2*\left(\frac{1-|q|^2}{1+|q|^2}+\frac{|p|^2-1}{1+|p|^2}\right)\left(\frac{1-|q|^2}{(1+|q|^2)^2}\right)q,\label{x1.5}\\
\dd*\hat{F}&=&4\ii \frac{(\overline{q}Dq-q\overline{Dq})}{(1+|q|^2)^2},\label{x1.6}\\
D*\left(\frac{Dp}{(1+|p|^2)^2}\right)&=&\frac1{(1+|p|^2)^3}(Dp\wedge *\overline{Dp})\nn\\
&&+2*\left(\frac{1-|q|^2}{1+|q|^2}+\frac{|p|^2-1}{1+|p|^2}\right)\left(\frac{|p|^2-1}{(1+|p|^2)^2}\right)p,\label{x1.7}\\
\dd*\tilde{F}&=&-4\ii \frac{(\overline{q}Dq-q\overline{Dq})}{(1+|q|^2)^2}+4\ii \frac{(\overline{p}Dp-p\overline{Dp})}{(1+|p|^2)^2},\label{x1.8}
\eea
which appear rather complicated and intractable. In order to obtain interesting solutions of these equations, we follow \cite{TW,Y1} to pursue a BPS reduction.
Introduce the current densities
\be
J(q)=\frac{\ii}{1+|q|^2}(q\overline{Dq}-\overline{q} Dq),\quad J(p)=\frac{\ii}{1+|p|^2}(p\overline{Dp}-\overline{p} Dp).
\ee
Then we have
\bea
K(q)&=&\dd J(q)\nn\\
&=&-\frac{2 |q|^2}{1+|q|^2}(\hat{F}-\tilde{F})+\ii\left(\frac{Dq\wedge \overline{Dq}-*Dq\wedge *\overline{Dq}}{(1+|q|^2)^2}\right),\\
K(p)&=&\dd J(p)\nn\\
&=&-\frac{2|q|^2}{1+|q|^2}\tilde{F}+\ii\left(\frac{Dp\wedge \overline{Dp}-*Dp\wedge *\overline{Dp}}{(1+|p|^2)^2}\right).
\eea
So, with $|Dq|^2=*(Dq\wedge *\overline{Dq})$, etc, we arrive at the decomposition
\bea
{\cal E}&=&\frac12\left|\hat{F}\mp 2*\left(\frac{1-|q|^2}{1+|q|^2}\right)\right|^2+\frac12\left|\tilde{F}\pm \left(2*\left(\frac{1-|q|^2}{1+|q|^2}\right)
+2*\left(\frac{|p|^2-1}{1+|p|^2}\right)\right)\right|^2\nn\\
&&+\frac2{(1+|q|^2)^2}|Dq\pm\ii *Dq|^2+\frac2{(1+|p|^2)^2}|Dp\pm\ii *Dp|^2\nn\\
&&\pm 2*(\hat{F}-\tilde{F})\pm 2* K(q)\pm 2 *\tilde{F}\pm 2*K(p). \label{x1.13}
\eea
The quantities
$\frac1{4\pi}\int K(q)$ and $\frac1{4\pi}\int K(p)$ are the Thom classes over $L^*\to S$, respectively \cite{SSY}.
Thus, the sum
\be
\tau = 2\hat{F}+2K(q)+2K(p) \label{x1.14}
\ee
is a topological density which leads to the topological energy lower bound
\be
E=\int_M {\cal E}\,*1\geq\left|\int_M\tau\right|, \label{x1.15}
\ee
measuring the tension \cite{CY,E1,E2,E3,E4} of the vortex-lines, so that the lower bound is saturated when the quartet $(q,p,\hat{A},\tilde{A})$ satisfies the equations
\bea
Dq\pm\ii *Dq&=&0,\label{x1.17}\\
Dp\pm\ii *Dp&=&0,\label{x1.18}\\
\hat{F}&=&\pm 2*\left(\frac{1-|q|^2}{1+|q|^2}\right),\label{x1.19}\\
\tilde{F}&=&\mp \left(2*\left(\frac{1-|q|^2}{1+|q|^2}\right)+2*\left(\frac{|p|^2-1}{1+|p|^2}\right)\right).\label{x1.20}
\eea
It may directly be checked that (\ref{x1.17})--(\ref{x1.20}) imply (\ref{x1.5})--(\ref{x1.8}). In other words, (\ref{x1.17})--(\ref{x1.20}) may be regarded as a reduction of the system of equations
(\ref{x1.5})--(\ref{x1.8}).
From (\ref{x1.17}) and (\ref{x1.18}), we know \cite{JT,Y1,Y2} that the zeros and poles of the sections $q,p$ are
isolated and possess integer multiplicities. For simplicity, we may denote the sets of zeros and poles of $q,p$ by
\bea
&& {\cal Z}(q)=\{z_{1,1}',\dots,z_{1,N_1}'\},\quad {\cal P}(q)=\{z_{1,1}'',\dots,z_{1,P_1}''\},\label{Zq}\\
&& {\cal Z}(p)=\{z_{2,1}',\dots,z_{2,N_2}'\},\quad {\cal P}(p)=\{z_{2,1}'',\dots,z_{2,P_2}''\},\label{Zp}
\eea
respectively, so that the associated multiplicities of the zeros and poles are naturally counted by their repeated appearances
in the above collections of points.
If we interpret $*\hat{F}$ as a magnetic or vorticity field, (\ref{x1.19}) indicates that it attains its maximum $*\hat{F}=2$ at the zeros
and minimum $*\hat{F}=-2$ at the poles of $q$. Thus, the zeros and poles of $q$ may be viewed as centers of vortices and anti-vortices.
In other words, we may identify the zeros and poles of $q$ as the locations of vortices and anti-vortices generated from the connection 1-form
$\hat{A}$. Similarly, the zeros and poles of $p$ may be interpreted as vortices and anti-vortices generated from the connection 1-form
$\hat{A}+\tilde{A}$. Therefore, in what follows, the zeros and poles of $q,p$ are interchangeably
and generically referred to as the vortices and anti-vortices of a
solution configuration $(\hat{A},\tilde{A},q,p)$.
Here is our existence theorem for the BPS equations (\ref{x1.17})--(\ref{x1.20}).
\begin{theorem}\label{mainthm}
Consider the BPS system consisting of equations (\ref{x1.17})--(\ref{x1.20}) of the energy density (\ref{x1.4})
formulated over a complex Hermitian line bundle $L$ over a compact Riemann surface $S$ with canonical total area
$|S|$ governing two connection 1-forms $\hat{A},\tilde{A}$ and two cross sections $q,p$ and comprising a reduction of
the Euler--Lagrange equations (\ref{x1.5})--(\ref{x1.8}). For the prescribed sets of zeros and poles for the fields $q$ and $p$
given respectively in (\ref{Zq}) and (\ref{Zp}), the coupled equations (\ref{x1.17})--(\ref{x1.20}) have a solution to realize
these sets of zeros and poles, if and only if the inequalities
\bea
\left|N_1+N_2-(P_1+P_2)\right|&<&\frac{|S|}{\pi},\label{C1}\\
\left|N_1+2N_2-(P_1+2P_2)\right|&<&\frac{|S|}{\pi},\label{C2}
\eea
regarding the total numbers of zeros and poles are fulfilled simultaneously. Moreover, such a solution carries a minimum
energy of the form
\be\label{xE}
E= 4\pi (N_1+N_2+P_1+P_2),
\ee
which is seen to be stratified topologically by the Chern and Thom classes of the line bundle $L$ and its dual respectively.
In particular, in terms of energy, zeros (vortices) and poles (anti-vortices) of $q,p$ contribute equally.
\end{theorem}
It is interesting to note that the inequalities (\ref{C1}) and (\ref{C2}) imply that the differences of
vortices and anti-vortices must stay within suitable ranges to ensure the existence of a solution:
\ber
\Big|N_1-P_1\Big|&<& \frac{3|S|}{\pi}, \label{b3}\\
\Big|N_2-P_2\Big|&<& \frac{2|S|}{\pi}.\label{b4}
\eer
However, it may be checked that the conditions (\ref{b3}) and (\ref{b4}) do not lead to (\ref{C1}) and (\ref{C2}). The latter may be called
the difference of total numbers of vortices and anti-vortices and the difference of `weighted total numbers' of vortices and anti-vortices.
We note that (\ref{b3}) and (\ref{b4}) give the upper bounds of the total `magnetic fluxes'
\bea
\int_S (\hat{F}-\tilde{F})&=&2\pi(N_1-P_1),\\
\int_S \tilde{F}&=&2\pi (N_2-P_2),
\eea
generated by the `magnetic fields' $\hat{F}-\tilde{F}$
and $\tilde{F}$, respectively, which may be compared with (\ref{221}) and (\ref{222}) for the fluxes of the Tong--Wong model \cite{TW}.
See \S 6 for details of calculation.
\section{Governing elliptic equations and basic properties}
\setcounter{equation}{0}
\setcounter{theorem}{0}
To proceed, we set
\be\label{x3.1}
u=\ln|q|^2,\quad v=\ln|p|^2,
\ee
in (\ref{x2.11})--(\ref{x2.14}). Thus
by \cite{JT,Y1,Y2} we are led to the following equivalent governing elliptic equations
\bea
\Delta u&=&4(\re^u-1)-2(\re^v-1)+4\pi\sum_{z\in{\cal Z}(q)}\delta_z,\label{x3.2}\\
\Delta v&=&-2(\re^u-1)+2(\re^v-1)+4\pi\sum_{z\in{\cal Z}(p)}\delta_z,\label{x3.3}
\eea
where $\Delta$ is the Laplace--Beltrami operator on $(S, g)$ defined by
\be \Delta u=\frac{1}{\sqrt{g}}\partial_j(g^{jk}\sqrt{g}\partial_ku), \ee
and $\delta_{z}$ denotes the Dirac measure concentrated at the point $z\in S$ with respect to the Riemannian metric $g$ over $S$.
In what follows, we use $\dd\Omega_g$ to denote the canonical surface element and $|S|$ the associated total area of the Riemann surface $(S, g)$.
The results stated in Theorem \ref{thm2.1} are contained in the following theorem concerning
the coupled elliptic equations (\ref{x3.2}) and (\ref{x3.3}).
\begin{theorem}\label{thm3.1}
The system of equations consisting of (\ref{x3.2}) and (\ref{x3.3}) has a solution if only only if
\be N_1+2N_2<\frac{|S|}{2\pi}.\label{z0}\ee
Moreover,
if a solution exists, it must be unique and satisfies the quantization conditions
\bea
\int_S (1-\re^u)\,\ud \Omega_g&=&2\pi (N_1+N_2),\label{z01}\\
\int_S (1-\re^v)\,\ud \Omega_g&=&2\pi(N_1+2N_2).\label{z02}
\eea
\end{theorem}
Similarly, setting (\ref{x3.1}) in (\ref{x1.17})--(\ref{x1.20}), we obtain
\ber
\Delta u&=&\frac{8 (\re^u-1)}{\re^u+1}-\frac{4(\re^v-1)}{\re^v+1}+4\pi\sum\limits_{z\in{\cal Z}(q)}\delta_{z}-4\pi\sum\limits_{z\in{\cal P}(q)}\delta_{z}, \label{b1}\\
\Delta v&=&-\frac{4(\re^u-1)}{\re^u+1}+\frac{4(\re^v-1)}{\re^v+1}+4\pi\sum\limits_{z\in{\cal Z}(p)}\delta_{z}-4\pi\sum\limits_{z\in{\cal P}(p)}\delta_{z}. \label{b2}
\eer
Regarding the equivalently reduced equations (\ref{b1}) and (\ref{b2}) from (\ref{x1.17})--(\ref{x1.20}), we have
\begin{theorem}\label{thb1}
The coupled equations \eqref{b1} and \eqref{b2} admit a solution $(u,v)$ with
the prescribed sets $\mathcal{Z}(q), \mathcal{P}(q), \mathcal{Z}(p), \mathcal{P}(p)$ in $S$ specified in (\ref{Zq}) and (\ref{Zp})
if and only if the inequalities
(\ref{C1}) and (\ref{C2})
are satisfied simultaneously.
Moreover, for the solution to the equations \eqref{b1} and \eqref{b2} obtained above, there hold the quantized integrals
\ber
\int_S \frac{1-\re^u}{1+\re^u}\,\ud\Omega_g=\pi\left(N_1-P_1+N_2-P_2\right),\label{b4a}\\
\int_S \frac{1-\re^v}{1+\re^v}\,\ud\Omega_g=\pi\left(N_1-P_1+2(N_2-P_2)\right)\label{b4b}.
\eer
\end{theorem}
For convenience, we first need to take care of the Dirac distributions by subtracting suitable background functions. To do so,
we let $u_0^1, u_0^2,v_0^1, v_0^2$ be the normalized solutions of the equations that determine the source functions
arising from the sets ${\cal Z}(q),{\cal P}(q), {\cal Z}(p),{\cal P}(p)$, respectively. For instance, $u_0^1$ is the unique solution \cite{Aubin} to
\be
\Delta u_0^1=-\frac{4\pi N_1}{|S|}+4\pi\sum\limits_{z\in{\cal Z}(q)}\delta_z, \quad \int_Su_0^1\,\ud\Omega_g=0.\label{b5}
\ee
Set
$u=u_0^1+U, v=v_0^1+V.$
Then we can rewrite \eqref{x3.2} and \eqref{x3.3} as
\ber
\Delta U&=&4(\re^{u_0^1+U}-1)- 2(\re^{v_0^1+V}-1)+\frac{4\pi N_1}{|S|}, \label{z3}\\
\Delta V&=&-2(\re^{u_0^1+U}-1)+2(\re^{v_0^1+V}-1)+\frac{4\pi N_2}{|S|}.\label{z4}
\eer
We first show the necessity of the condition \eqref{z0}.
If there is a solution of \eqref{z3}--\eqref{z4}, integration of which over $S$ gives
\ber
\int_S\re^{u_0^1+U}\ud \Omega_g=|S|-2\pi(N_1+N_2)\equiv a_1>0,\label{zz1}\\
\int_S\re^{v_0^1+V}\ud \Omega_g=|S|-2\pi(N_1+2N_2)\equiv a_2>0.\label{zz2}
\eer
Then we see that the condition \eqref{z0} is necessary to ensure the existence of a solution to
the system \eqref{z3}--\eqref{z4}. The quantized integrals \eqref{z01}--\eqref{z02} follow from
\eqref{zz1}--\eqref{zz2}.
Let
$u=u_0^1-u_0^2+U, v=v_0^1-v_0^2+V.$
Then the equations \eqref{b1} and \eqref{b2} can be rewritten as
\ber
\Delta U&=&8 f(u_0^1, u_0^2, U)- 4f(v_0^1, v_0^2, V)+\frac{4\pi(N_1-P_1)}{|S|}, \label{b7}\\
\Delta V&=&-4 f(u_0^1, u_0^2, U)+4 f(v_0^1, v_0^2, V)+\frac{4\pi(N_2-P_2)}{|S|},\label{b8}
\eer
where and in what follows we use the notation
\ber
f(s^1, s^2, t)\equiv\frac{\re^{s^1-s^2+t}-1}{\re^{s^1-s^2+t}+1}=\frac{\re^{s^1+t}-\re^{s^2}}{\re^{s^1+t}+\re^{s^2}},
\quad s^1,s^2,t\in\bfR. \label{b9}
\eer
For fixed $s^1, s^2\in \mathbb{R}$, we have
\ber
0<\frac{\ud}{\ud t}f(s^1, s^2, t)=\frac{2\re^{s^1+t}\re^{s^2}}{(\re^{s^1+t}+\re^{s^2})^2}\le \frac12, \quad \forall\,t\in\mathbb{R}.\label{e43}
\eer
We now show that the condition consisting of \eqref{C1} and \eqref{C2} is necessary for the existence of solutions for \eqref{b7}--\eqref{b8}. In fact, integrating \eqref{b7}--\eqref{b8} over $S$, we find
\ber
\int_Sf(u_0^1, u_0^2, U)\,\ud\Omega_g&=&a|S|, \label{b10}\\
\int_Sf(v_0^1, v_0^2, V)\,\ud\Omega_g&=&b|S|, \label{b11}
\eer
where $a, b$ are constants defined by
\ber
a&\equiv&-\frac{\pi}{|S|}\left(N_1-P_1+N_2-P_2\right), \label{b12}\\
b&\equiv&-\frac{\pi}{|S|}\left(N_1-P_1+2(N_2-P_2)\right). \label{b13}
\eer
From \eqref{b10}--\eqref{b11} we see that the quantized integrals \eqref{b4a}--\eqref{b4b} hold.
On the other hand, noting
\be
-1< f(s^1, s^2, t)< 1 \quad \text{ for any}\quad s^1, s^2, t\in \mathbb{R}, \label{b13a}
\ee
we arrive at
\ber
|a|<1\quad \text{and}\quad |b|<1, \label{b14}
\eer
which is equivalent to \eqref{C1} and \eqref{C2}. Thus the inequalities \eqref{C1} and \eqref{C2} are necessary for a solution to exist.
\section{Proof of existence for the Tong--Wong system}
\setcounter{equation}{0}
In this section we establish Theorem \ref{thm2.1} or Theorem \ref{thm3.1} through a thorough study of the coupled
vortex equations (\ref{x3.2}) and (\ref{x3.3}). To this end, we recast the problem into a variational problem and apply a direct minimization approach recently developed in \cite{LiebYang}.
We have shown that the condition \eqref{z0} is necessary to the existence of a solution to \eqref{x3.2}--\eqref{x3.3} on $S$,
and in what follows we prove that it is also sufficient.
To formulate the problem into a variational structure, we set
\be
f=U,\quad h=U+2V,\quad \text{or}\quad U=f,\quad V=\frac{h-f}{2}.
\ee
The we rewrite the equations \eqref{z3}--\eqref{z4} equivalently as
\ber
\Delta f&=&4\left(\re^{u_0^1+f}-1\right)-2\left(\re^{v_0^1+\frac{h-f}{2}}-1\right)+\frac{4\pi N_1}{|S|}, \label{z5}\\
\Delta h&=&2\left(\re^{v_0^1+\frac{h-f}{2}}-1\right)+\frac{4\pi (N_1+2N_2)}{|S|}.\label{z6}
\eer
Then we directly check that the equations \eqref{z5}--\eqref{z6} are the Euler--Lagrange equations of the following functional
\ber
I(f, h)&=&\frac12\left(\|\nabla f\|_2^2+\|\nabla h\|_2^2\right)+4\int_S\left(\re^{u_0^1+f}-f+\re^{v_0^1+\frac{h-f}{2}}-\frac{h-f}{2}\right)\ud\Omega_g\nn\\
&& +\frac{4\pi N_1}{|S|}\int_S f\ud\Omega_g+\frac{4\pi (N_1+2N_2)}{|S|}\int_S h\ud\Omega_g. \label{z7}
\eer
Here and in what follows we use the following notation
\ber
\|\nabla w\|_2^2=\int_S|\nabla w|^2\ud \Omega_g\equiv \int_Sg^{jk}\partial_jw\partial_kw\ud \Omega_g.\label{zz}
\eer
We know that the Sobolev space $W^{1, 2}(S)$ (cf. \cite{Aubin}) can be decomposed as
$
W^{1, 2}(S)=\dot{W}^{1, 2}(S)\oplus\mathbb{R}$
where
\ber
\dot{W}^{1, 2}(S)\equiv \left\{w\in W^{1, 2}(S)\Big|\quad \int_S w\,\ud\Omega_g=0 \right\} \label{b18}
\eer
is a closed subspace of $W^{1, 2}(S)$.
To save notation, in the following of this paper we also use $W^{1, 2}(S), \dot{W}^{1, 2}(S)$ and $L^p(S)$ to denote the spaces of vector-valued functions.
For $f, h\in W^{1, 2}(S)$ we decompose them as
\be
f=f'+\overline{f}, \, h=h'+\overline{h}, \quad f', h'\in \dot{W}^{1,2}(S), \quad \overline{f}, \overline{h}\in \mathbb{R}.\label{zz*}
\ee
On the subspace $\dot{W}^{1,2}(S)$ there hold the Poincar\'{e} inequality
\ber
\int_S w^2\ud \Omega_g\equiv\|w\|_2^2\le C\|\nabla w\|_2^2,\quad w\in \dot{W}^{1,2}(S)\label{zp}
\eer
and the Moser--Trudinger inequality \cite{Aubin,font}
\be
\int_S \re^w\ud \Omega_g\le C \exp\left(\frac{1}{16\pi}\int_S|\nabla w|^2\ud x\right),\quad w\in \dot{W}^{1,2}(S), \label{z7'}
\ee
where $C$ is a generic positive constant.
We see from \eqref{z7'} that the functional $I$ defined by \eqref{z7} is a $C^1$-functional.
By the definition of $I$ and the decomposition \eqref{zz*} we have
\ber
I(f,h)&=&\frac12(\|\nabla f'\|_2^2+\|\nabla h'\|_2^2)+ 4\left(\re^{\overline{f}}\int_S\re^{u_0^1+f'}\ud\Omega_g-a_1\overline{f}\right)\nn\\
&&+ 4\left(\re^{ \frac{\overline{h}-\overline{f}}{2}}\int_S\re^{v_0^1+\frac{h'-f'}{2}}\ud\Omega_g- a_2\frac{[\overline{h}-\overline{f}]}{2}\right),\label{z9}
\eer
where $a_1,a_2$, defined by \eqref{zz1}--\eqref{zz2}, are positive, as ensured by \eqref{z0}.
Hence, by \eqref{z9} and the Jensen inequality, we obtain
\ber
&&I(f,h)-\frac12(\|\nabla f'\|_2^2+\|\nabla h'\|_2^2)\ge4 \left(|S|\re^{\overline{f}}-a_1\overline{f}
+ |S|\re^{ \frac{\overline{h}-\overline{f}}{2}}-a_2\frac{[\overline{h}-\overline{f}]}{2}\right). \label{z10}
\eer
From \eqref{z9}--\eqref{z10} we also see that
\ber
I(f, h)\ge 4\left(\ln\frac{|S|}{a_1}+\ln\frac{|S|}{a_2}\right),\label{z11}
\eer
which implies the functional $I$ is bounded from below and the minimization problem
\be
a_0\equiv\min \left\{I(f, h)\big|\, (f, h)\in W^{1, 2}(S)\right\}
\ee
is well-defined.
Let $\{(f_k, h_k)\}$ be a minimizing sequence. Noting that the function $m(t)=\alpha\re^t-\beta, \, \alpha,\beta>0$ satisfies $m(t)\to +\infty$ as
$t\to \pm\infty$, we see from \eqref{z10} that $\overline{f}_k,\frac{ \overline{h}_k-\overline{f}_k}{2}$ must be bounded for all $k$, which implies $\{(\overline{f}_k, \overline{h}_k)\}$ are bounded for all $k$. And \eqref{z10} also implies
$\{(\nabla f_k'\,,\nabla h_k')\}$ are bounded in $L^2(S)$ for all $k$. Then by the Poincar\'{e} inequality \eqref{zp}, we conclude that $\{(f_k', h_k')\}$ are bounded in $\dot{W}^{1, 2}(S)$, which
with the boundedness of $\{(\overline{f}_k, \overline{h}_k)\}$ imply that $\{(f_k, h_k)\}$ are bounded in $W^{1,2}(S)$ for all $k$. Hence, there exits a subsequence of
$\{(f_k, h_k)\}$, still denoted by $\{(f_k, h_k)\}$, such that $\{(f_k, h_k)\}$ converges weakly to some $(f_\infty, h_\infty)\in W^{1,2}(S)$.
It is easy to see that the functional $I$ is also weakly lower semi-continuous. Then the limit $(f_\infty, h_\infty)\in W^{1,2}(S)$ is a critical point of
$I$. Of course, it gives a solution for the system \eqref{z5}--\eqref{z6}, and hence for \eqref{z3}--\eqref{z4}. So the sufficiency of \eqref{z0} follows.
We directly see that the functional $I$ is strictly convex. Therefore the functional $I$ admits at most one critical point. That is to say, a solution of \eqref{z3}--\eqref{z4} must be unique.
Therefore we have completed the proof of Theorem \ref{thm3.1}.
\section{Proof of existence for the vortex and anti-vortex system}
\setcounter{equation}{0}
In this section, we prove that the condition comprised of \eqref{C1} and \eqref{C2} is also sufficient for the existence of a solution of
the coupled equations \eqref{b1} and \eqref{b2}. We will extend a fixed-point theorem argument used in \cite{yang1} when treating a single equation.
To do so, it is convenient to rewrite the equations \eqref{b7} and \eqref{b8} equivalently as
\ber
\Delta U&=&8\left(f(u_0^1, u_0^2, U)-a\right)-4\left(f(v_0^1, v_0^2, V)-b\right), \label{b15}\\
\Delta V&=&-4\left(f(u_0^1, u_0^2, U)-a\right)+4\left(f(v_0^1, v_0^2, V)-b\right), \label{b16}
\eer
where $a, b$ are defined by \eqref{b12}--\eqref{b13}.
We begin with the following lemma.
\begin{lemma}\label{lemb1}
For any $(U', V')\in \dot{W}^{1, 2}(S)$, there exists a unique pair $(c_1(U'), c_2(V'))\in \mathbb{R}^2$ such that
\ber
\int_Sf(u_0^1, u_0^2, U'+c_1(U'))\,\ud\Omega_g&=&a|S|,\label{b19}\\
\int_Sf(v_0^1, v_0^2, V'+c_2(V'))\,\ud\Omega_g&=&b|S|,\label{b20}
\eer
where $a, b$ are defined by \eqref{b12}--\eqref{b13}.
\end{lemma}
{\bf Proof.} Under the condition consisting of \eqref{C1} and \eqref{C2}, we easily see that
\ber
-1<a,b<1.\label{b21}
\eer
Noting the expression \eqref{b9}, for any $(U',V')\in \dot{W}^{1, 2}(S)$, we have
\ber
\int_Sf(u_0^1, u_0^2, U'+t)\ud\Omega_g, \int_Sf(v_0^1, v_0^2, V'+t)\ud\Omega_g\to |S|\quad \text{as}\quad t\to \infty\label{b22}
\eer
and
\ber
\int_Sf(u_0^1, u_0^2, U'+t)\ud\Omega_g, \int_Sf(v_0^1, v_0^2, V'+t)\ud\Omega_g\to-|S|\quad \text{as}\quad t\to-\infty.\label{b23}
\eer
Then, for any $(U', V')\in \dot{W}^{1, 2}(S)$, we conclude from \eqref{b21}, \eqref{b22} and \eqref{b23} that there exists a
point $(c_1(U'), c_2(V'))\in \mathbb{R}^2$ such that \eqref{b19} and \eqref{b20} hold.
The uniqueness of $(c_1(U'), c_2(V'))$ follows from the strict monotonicity of $ f(s^1, s^2,t)$ with respect to $t$ (see \eqref{e43}).
\begin{lemma}\label{lemb2}
For any $(U', V')\in \dot{W}^{1, 2}(S)$, let $(c_1(U'), c_2(V'))$ be defined in Lemma \ref{lemb1}. Then, the mapping
$(c_1(\cdot), c_2(\cdot)): \dot{W}^{1, 2}(S)\to \mathbb{R}^2$, is continuous with respect to the weak topology of $\dot{W}^{1, 2}(S)$.
\end{lemma}
{\bf Proof. } Take a weakly convergent sequence $\{(U_k', V_k')\}$ in $\dot{W}^{1, 2}(S)$ such that $(U_k', V_k')\to (U_0', V_0')$ weakly in $\dot{W}^{1, 2}(S)$.
Then we see that
\ber
(U_k', V_k')\to (U_0', V_0') \quad \text{strongly in}\quad L^p(S) \quad \text{for any}\quad p\ge1,\label{b26'}
\eer
by the compact embedding $W^{1, 2}(S)\hookrightarrow L^p(S)(p\ge1)$.
We aim to prove that $(c_1(U_k'), c_2(V_k'))\to (c_1(U_0'), c_2(V_0'))$ as $k\to \infty$.
Claim: The sequence $\{(c_1(U_k'), c_2(V_k'))\}$ is bounded.
To show this claim we first prove that $\{(c_1(U_k'), c_2(V_k'))\}$ is bounded from above. We argue by contradiction. Without loss of generality, assume $c_1(U_k')\to \infty$ as $k\to\infty$.
Noting \eqref{b26'} and using the Egorov theorem, we see that for any $\vep>0$, there is a large constant $K_\vep>0$ and a subset $S_\vep\subset S$ such that
\ber
|U_k'|\le K_\vep, \quad x\in S\setminus S_\vep, \quad |S_\vep|<\vep,\quad \forall\, k.\label{b23a}
\eer
Then by \eqref{b23a} and \eqref{b13a} we have
\ber
|a||S|&=&\left|\int_{S\setminus S_\vep}f(u_0^1, u_0^2, U_k'+c_1(U_k'))\ud\Omega_g+\int_{S_\vep}f(u_0^1, u_0^2, U_k'+c_1(U_k'))\ud\Omega_g\right|\nn\\
&\ge&\left|\int_{S\setminus S_\vep}f(u_0^1, u_0^2, U_k'+c_1(U_k'))\ud\Omega_g\right|-\left|\int_{S_\vep}f(u_0^1, u_0^2, U_k'+c_1(U_k'))\ud\Omega_g\right|\nn\\
&\ge&\int_{S\setminus S_\vep}f(u_0^1, u_0^2, c_1(U_k')-K_\vep)\ud\Omega_g-\vep.\label{b24}
\eer
Hence taking $k\to \infty$ in \eqref{b24} we get
\berr
|a||S|\ge |S\setminus S_\vep|-\vep\ge |S|-2\vep.
\eerr
Noting that $\vep$ is arbitrary, we obtain
\[|a|\ge1,\]
which contradicts the condition \eqref{C1} $(|a|<1)$. Hence the sequence $\{(c_1(U_k'), c_2(V_k'))\}$ is bounded from above.
Now we show that $\{(c_1(U_k'), c_2(V_k'))\}$ is also bounded from below. In fact, we may suppose $c_1(U_k')\to -\infty$ as $k\to\infty$.
Using \eqref{b23a} and \eqref{b13a}, we have
\ber
a|S|&=&\int_{S\setminus S_\vep}f(u_0^1, u_0^2, U_k'+c_1(U_k'))\ud\Omega_g+\int_{S_\vep}f(u_0^1, u_0^2, U_k'+c_1(U_k'))\ud\Omega_g\nn\\
&\le&\int_{S\setminus S_\vep}f(u_0^1, u_0^2, c_1(U_k')+K_\vep)\ud\Omega_g+\vep.\label{b24a}
\eer
Then letting $k\to \infty$ in \eqref{b24a}, we obtain
\berr
a|S|\le -|S\setminus S_\vep|+\vep\le -|S|+2\vep,
\eerr
which implies $a\le-1$ since $\vep>0$ is arbitrary. Hence we get a contradiction with the condition \eqref{C1} again. So the sequence $\{(c_1(U_k'), c_2(V_k'))\}$ is bounded from below.
Therefore the claim follows.
By the claim above, up to a subsequence, we may assume that
\ber
(c_1(U_k'), c_2(V_k'))\to (c_1', c_2') \quad \text{as}\quad k\to \infty \quad \text{for some}\quad (c_1', c_2')\in\mathbb{R}^2. \label{b24z}
\eer
Then, using \eqref{e43}, the Schwartz inequality, \eqref{b26'} and \eqref{b24z} we have
\ber
&&\left|\int_Sf(u_0^1, u_0^2, U_k'+c_1(U_k'))\,\ud\Omega_g-\int_Sf(u_0^1, u_0^2, U_0'+c_1')\,\ud\Omega_g\right|\nn\\
&&=\left|\int_Sf_t(u_0^1, u_0^2, \theta[U_k'+c_1(U_k')]+[1-\theta][U_0'+c_1'])(U_k'-U_0'+c_1(U_k')-c_1')\,\ud\Omega_g\right|
\nn\\
&&\le\frac12\int_S\left|U_k'-U_0'+c_1(U_k')-c_1'\right|\,\ud\Omega_g\nn\\
&&\le \frac12\left(|S|^{\frac12}\|U_k'-U_0'\|_2+|S||c_1(U_k')-c_1'|\right)\to 0 \quad \text{as}\quad k\to \infty, \label{nn1}
\eer
where $\theta\in(0,1)$. Noting \eqref{nn1} and
\be
\int_Sf(u_0^1, u_0^2, U_k'+c_1(U_k'))\,\ud\Omega_g=a|S|,
\ee
we have
\ber
\int_Sf(u_0^1, u_0^2, U_0'+c_1')\,\ud\Omega_g=a|S|.
\eer
Similarly, we get\be
\int_Sf(v_0^1, v_0^2, V_0'+c_2')\,\ud\Omega_g=b|S|.
\ee
Hence from Lemma \ref{lemb1} we see that $(c_1', c_2')=(c_1(U_0'), c_2(V_0'))$. Then Lemma \ref{lemb2} follows.
At this point we can define an operator
\berr
T:\dot{W}^{1, 2}(S)\to \dot{W}^{1, 2}(S)
\eerr
as follows. For $(U', V')\in \dot{W}^{1, 2}(S)$, let $(c_1(U'), c_2(V'))$ be defined by Lemma \ref{lemb1}.
Define $(\tilde{U}',\tilde{V}')=T(U',V')$ where $\tilde{U}'$ and $\tilde{V}'$ are the unique solutions of
\ber
\Delta \tilde{U}'&=& 8\left(f(u_0^1, u_0^2, U'+c_1(U'))-a\right)-4\left(f(v_0^1, v_0^2, V'+c_2(V'))-b\right), \label{b25}\\
\Delta \tilde{V}'&=&-4\left(f(u_0^1, u_0^2, U'+c_1(U'))-a\right)+4\left(f(v_0^1, v_0^2, V'+c_2(V'))-b\right), \label{b26}
\eer
respectively.
In fact, for any $(U', V')\in \dot{W}^{1, 2}(S)$, since the right-hand sides of \eqref{b25} and \eqref{b26} have zero averages, the
solutions $\tilde{U}'$ and $\tilde{V}'$ of \eqref{b25} and \eqref{b26}, respectively, are unique (cf. \cite{Aubin}).
Next we show that the operator $T$ admits a fixed point in $\dot{W}^{1, 2}(S)$. To this end, we first establish the following lemma.
\begin{lemma}\label{lemb3}
The above operator $T:\dot{W}^{1, 2}(S)\to \dot{W}^{1, 2}(S)$ is completely continuous.
\end{lemma}
{\bf Proof.} Assume $(U_k', V_k')\to (U_0', V_0')$ weakly in $\dot{W}^{1, 2}(S)$.
Hence by the compact embedding theorem we see that \eqref{b26'} holds.
Denote
\ber
(\tilde{U}'_k,\tilde{V}'_k)= T(U'_k, V'_k) \quad \text{and}\quad (\tilde{U}'_0,\tilde{V}'_0)= T(U'_0, V'_0).\label{b27}
\eer
Therefore we have
\ber
\Delta (\tilde{U}'_k-\tilde{U}'_0)&=&8\big(f(u_0^1, u_0^2, U'_k+c_1(U'_k))-f(u_0^1, u_0^2, U'_0+c_1(U'_0))\big)\nn\\
&&-4\big(f(v_0^1, v_0^2, V'_k+c_2(V'_k))-f(v_0^1, v_0^2, V'_0+c_2(V'_0))\big)\nn\\
&=&8f_t(u_0^1, u_0^2, \hat{U}'+\hat{c}_1)\big(U_k'-U_0'+c_1(U_k')-c_1(U_0')\big)\nn\\
&&-4f_t(v_0^1, v_0^2, \hat{V}'+\hat{c}_2)\big(V_k'-V_0'+c_2(V_k')-c_2(V_0')\big), \label{b28}\\[2mm]
\Delta (\tilde{V}'_k-\tilde{V}'_0)&=&-4\big(f(u_0^1, u_0^2, U'_k+c_1(U'_k))-f(u_0^1, u_0^2, U'_0+c_1(U'_0))\big)\nn\\
&&+4\big(f(v_0^1, v_0^2, V'_k+c_2(V'_k))-f(v_0^1, v_0^2, V'_0+c_2(V'_0))\big)\nn\\
&=&-4f_t(u_0^1, u_0^2, \hat{U}'+\hat{c}_1)\big(U_k'-U_0'+c_1(U_k')-c_1(U_0')\big)\nn\\
&&+4f_t(v_0^1, v_0^2, \hat{V}'+\hat{c}_2)\big(V_k'-V_0'+c_2(V_k')-c_2(V_0')\big),\label{b29}
\eer
where $\hat{U}'_k$ lies between $U'_k$ and $U'_0$, $\hat{V}'_k$ between $V'_k$ and $V'_0$, $\hat{c}_1$ between $c_1(U_k')$ and $c_1(U_0')$,
and $\hat{c}_2$ between $c_2(V_k')$ and $c_2(V_0')$.
Multiplying both sides of \eqref{b28} and \eqref{b29} by $\tilde{U}'_k-\tilde{U}'_0$ and $\tilde{V}'_k-\tilde{V}'_0$, respectively, and integrating by parts, we obtain
\ber
\|\nabla (\tilde{U}'_k-\tilde{U}'_0)\|_2^2&\le&\int_S \Big\{4\big(|U_k'-U_0'|+|c_1(U_k')-c_1(U_0')|\big) \nn\\
&&+2\big(|V_k'-V_0'|+|c_2(V_k')-c_2(V_0')|\big)\Big\}|\tilde{U}'_k-\tilde{U}'_0|\,\ud\Omega_g,\label{b30}\\
\|\nabla (\tilde{V}'_k-\tilde{V}'_0)\|_2^2&\le&2\int_S \Big(|U_k'-U_0'|+|c_1(U_k')-c_1(U_0')| \nn\\
&&+|V_k'-V_0'|+|c_2(V_k')-c_2(V_0')|\Big)|\tilde{V}'_k-\tilde{V}'_0|\,\ud\Omega_g,\label{b31}
\eer
where the property \eqref{e43} is used.
Combining \eqref{b30} with \eqref{b31}, and using the Poincar\'{e} inequality, we arrive at
\ber
\|\nabla (\tilde{U}'_k-\tilde{U}'_0)\|_2^2+\|\nabla (\tilde{V}'_k-\tilde{V}'_0)\|_2^2&\le& C\Big(\|U_k'-U_0'\|_2^2+\|V_k'-V_0'\|_2^2\nn\\
&&+|c_1(U_k')-c_1(U_0')|^2+|c_2(V_k')-c_2(V_0')|^2\Big)\label{b32}
\eer
for some $C>0$.
Then, from \eqref{b26'}, Lemma \ref{lemb2}, and \eqref{b32}, we see that
\[(\nabla \tilde{U}'_k,\nabla \tilde{V}'_k) \to (\nabla\tilde{U}'_0,\nabla\tilde{V}'_0) \quad \text{strongly in}\quad L^2(S) \quad \text{as}\quad k\to\infty,\]
which, with \eqref{b26'}, yields
\[(\tilde{U}'_k,\tilde{V}'_k) \to (\tilde{U}'_0,\tilde{V}'_0) \quad \text{strongly in}\quad \dot{W}^{1,2}(S) \quad \text{as}\quad k\to\infty.\]
Then the proof of Lemma \ref{lemb3} is complete.
Before applying the Leray--Schauder fixed-point theory, we need to estimate the solution of the fixed-point equation,
\ber
(U'_t, V'_t)&=&tT(U'_t, V'_t), \quad 0\le t\le 1.\label{b33}
\eer
\begin{lemma}\label{lemb4}
For any $(U'_t, V'_t)$ satisfying \eqref{b33}, there exists a constant $C>0$ independent of $t\in[0, 1]$ such that
\ber
\|U'_t\|_{\dot{W}^{1, 2}(S)}+ \|V'_t\|_{\dot{W}^{1, 2}(S)}\le C.\label{b34}
\eer
\end{lemma}
{\bf Proof.} From \eqref{b33} we have
\ber
\Delta U'_t&=&8t\left(f(u_0^1, u_0^2, U'_t+c_1(U'_t))-a\right)-4t\left(f(v_0^1, v_0^2, V'_t+c_2(V'_t))-b\right), \label{b35}\\
\Delta V'_t&=&-4t\left(f(u_0^1, u_0^2, U'_t+c_1(U'_t))-a\right)+4t\left(f(v_0^1, v_0^2, V'_t+c_2(V'_t))-b\right). \label{b36}
\eer
Multiplying both sides of \eqref{b35} and \eqref{b36} by $U_t'$ and $V_t'$, respectively, and integrating by parts, we see that
\berr
\|\nabla U_t'\|_2^2&\le&\int_S \Big(8|f(u_0^1, u_0^2, U'_t+c_1(U'_t))|+4|f(v_0^1, v_0^2, V'_t+c_2(V'_t))|\Big)|U'_t|\ud\Omega_g\nn\\
&\le& 12\int_S|U'_t|\ud\Omega_g,\\
\|\nabla V_t'\|_2^2&\le&\int_S \Big(4|f(u_0^1, u_0^2, U'_t+c_1(U'_t))|+4|f(v_0^1, v_0^2, V'_t+c_2(V'_t))|\Big)|V'_t|\ud\Omega_g\nn\\
&\le& 8\int_S|V'_t|\ud\Omega_g,
\eerr
where we have used \eqref{b13a}.
Then by the Poincar\'{e} inequality, we get the desired estimate \eqref{b34}.
Now using Lemmas \ref{lemb3}, \ref{lemb4}, and the Leray--Schauder fixed-point theorem (cf. \cite{GL}), we see that the operator $T$ admits a fixed point, say $(U', V')$, in $\dot{W}^{1, 2}(S)$.
Thus $(U'+c_1(U'), V'+c_2(V'))$ is a solution of \eqref{b15} and \eqref{b16}, i.e. a solution of \eqref{b7} and \eqref{b8}.
Hence we have completed the proof of Theorem \ref{thb1}.
\section{Explicit calculation of minimum energy}
\setcounter{equation}{0}
In this section we establish the minimum energy formula (\ref{xE}) and show how it is stratified topologically.
By the equations \eqref{x1.17}--\eqref{x1.20}, the fact $*1=\dd\Omega_g$, and \eqref{b4a}--\eqref{b4b}, we see that
\ber
\int_S(\hat{F}-\tilde{F})&=&4\int_S*\frac{1-\re^u}{\re^u+1}-2\int_S*\frac{1-\re^v}{\re^v+1}=2\pi(N_1-P_1), \label{c1}\\
\int_S\tilde{F}&=&-2\int_S*\frac{1-\re^u}{\re^u+1}+2\int_S*\frac{1-\re^v}{\re^v+1}=2\pi(N_2-P_2), \label{c2}
\eer
are valid, which give us
\ber
\int_S\hat{F}=2\pi(N_1-P_1+N_2-P_2). \label{c3}
\eer
To calculate the lower bound of the energy, we need to compute the fluxes contributed by the current densities
$K(q)$ and $K(p)$.
Take a coordinate chart $\{\mathcal{U}_j\}$ of $S$. Assume $z_{1,j}''\in \mathcal{U}_j$, $j=1,\dots, P_1$. In local
coordinates, we have
$D_iq=\partial_iq-\ri(\hat{A}_i-\tilde{A}_i)q$, $i=1, 2$ and the density $K(q)$ in $\mathcal{U}_j$ can be written as
\ber
K(q)=-\frac{2|q|^2}{1+|q|^2}(\hat{F}-\tilde{F})+\ri\frac{D_iq\overline{D_jq}-\overline{D_i}qD_jq}{(1+|q|^2)^2}\ud x^i\wedge\ud x^j. \label{c4}
\eer
Besides, in
$
K(q)=\ud J(q)$, we have
\ber
\quad J(q)=\frac{\ri}{1+|q|^2}(q\overline{D_iq}-\overline{q}D_iq)\ud x^i.\label{c5}
\eer
Then it follows from the Stokes formula that
\ber
\int_SK(q)=\int_S\ud J(q)=\sum\limits_{j=1}^{P_1}\lim\limits_{r\to0}\oint_{\partial B(z_{1,j}'', r)} J(q),\label{c6}
\eer
where $B(z, r)$ denotes a disc centered at $z$ with radius $r>0$ and all the line integrals are taken
counterclockwise.
Note that near $z_{1,j}''\in \mathcal{P}(q)$, the section $q$ has the representation
\ber
q(z)=z^{-1}h_j(z,\overline{z}), \quad z=x^1+\ri x^2, \quad x^1(z_{1,j}'')=x^2(z_{1,j}'')=0,\label{c7}
\eer
where $h_j$ is a non-vanishing function defined near $z_{1,j}''$.
From the equation \eqref{x1.17} we see that
\ber
\hat{A}_1-\tilde{A}_1=-2{\rm Re}(\ri\overline{\partial}\ln u), \quad \hat{A}_2-\tilde{A}_2=-2{\rm Im}(\ri\overline{\partial}\ln u),\label{c8}
\eer
which, with $u=\ln|q|^2$, implies
\ber
D_1q=(\partial+\overline{\partial})q+\left(\frac{\partial\overline{q}}{\overline{q}}-\frac{\overline{\partial}q}{q}\right)q=q\partial u, \label{c9}\\
D_2q=\ri(\partial-\overline{\partial})q+\ri\left(\frac{\overline{\partial}q}{q}+\frac{\partial\overline{q}}{\overline{q}}\right)q=\ri q\partial u.\label{c10}
\eer
Then, by \eqref{c6}, \eqref{c9}, and \eqref{c10}, we have
\ber
\oint_{\partial B(z_{1,j}'', r)} J(q)&=&\ri\oint_{\partial B(z_{1,j}'', r)} \frac{|q|^2}{1+|q|^2}(\overline{[\partial}-\partial]u\ud x^1-\ri[\overline{\partial}+\partial]u\ud x^2)\nn\\
&=&\oint_{\partial B(z_{1,j}'', r)} \frac{\re^u}{1+\re^u}(\partial_2u\ud x^1-\partial_1u \ud x^2).\label{c11}
\eer
Noting \eqref{c7}, near $z_{1,j}''\in \mathcal{P}(q)$, we see that
\ber
u=-2\ln|z|+w_j, \label{c12}
\eer
where $w_j$ is a smooth function. Thus we obtain
\ber
\lim\limits_{r\to0}\oint_{\partial B(z_{1,j}'', r)} J(q)=4\pi,\label{c13}
\eer
which, with \eqref{c6}, gives
\ber
\int_SK(q)=4\pi P_1.\label{c14}
\eer
Following a similar procedure, we have
\ber
\int_SK(p)=4\pi P_2.\label{c15}
\eer
As described in \cite{SSY}, the normalized integrals $\frac1{4\pi}\int K(q)$ and $\frac1{4\pi}\int K(p)$, counting the numbers $P_1,P_2$ of
anti-vortices of the two species, are the Thom classes of the dual bundle $L^*\to S$, of two respective classification (Chern) classes,
$\frac1{2\pi}\int(\hat{F}-\tilde{F})$ and $\frac1{2\pi}\int \tilde{F}$.
Hence, by \eqref{x1.13}--\eqref{x1.15}, \eqref{c3}, \eqref{c14}, and \eqref{c15}, we obtain the following topologically
stratified minimum energy
\ber
E=\int_S2([\hat{F}-\tilde{F}]+\tilde{F}+K(q)+K(p))=4\pi(N_1+P_1+N_2+P_2),
\eer
as stated in Theorem \ref{mainthm}.
\section{Conclusions and remarks}
\setcounter{equation}{0}
In this work we have extended the formalism of Tong and Wong \cite{TW} of a product Abelian Higgs theory describing a coupled vortex system with magnetic impurities to accommodate
coexisting vortices and anti-vortices of two species realized as topological solitons governed by a BPS system of equations. In additional to the usual first Chern classes suited over a
complex Hermitian line bundle, the presence of anti-vortices switches on the Thom classes over the dual bundle, as in \cite{SSY}. When the underlying Riemann surface $S$ where vortices
and anti-vortices reside is compact, we have established a theorem which spells out a necessary and sufficient condition,
consisting of two inequalities, (\ref{C1}) and (\ref{C2}), for prescribed $N_1, N_2$ vortices and $P_1,P_2$ anti-vortices,
of two respective species, to exist.
This necessary and sufficient condition contains a few special situations worthy of mentioning.
\begin{enumerate}
\item[(i)]
When $N_2=P_2=0$ (only vortices and anti-vortices of the first species are present), the condition becomes
\be
\left|N_1-P_1\right|<\frac{|S|}\pi.
\ee
\item[(ii)] When $N_1=P_1=0$ (only vortices and anti-vortices of the second species are present), the condition reads
\be
\left|N_2-P_2\right|<\frac{|S|}{2\pi}.
\ee
\item[(iii)] When $N_1=N_2=N$ and $P_1=P_2=P$ (there are equal numbers of vortices and anti-vortices, respectively, of two species), the condition is
\be
\left|N-P\right|<\frac{|S|}{3\pi}.
\ee
\end{enumerate}
In all these situations, the numbers of vortices and anti-vortices may be arbitrarily large, provided that the differences of these numbers are kept in suitable ranges as given.
Although the vortices and anti-vortices of the two species do not appear in the model in a symmetric manner as seen in the field-theoretical Lagrangian density and the governing
equations, they make equal contributions to the total topologically stratified minimum energy as stated in (\ref{xE}) of an elegant form.
Let ${\cal M}(N_1,P_1,N_2,P_2)$ denote the moduli space of solutions of the BPS equations (\ref{x1.17})--(\ref{x1.20}) with $N_1+N_2$ and
$P_1+P_2$ prescribed vortices and anti-vortices, of two respective species. Since these solutions depend on at least $2(N_1+N_2+P_1+P_2)$
continuous parameters which specify the locations of zeros and poles of the two sections $q,p$, respectively, we obtain the following
lower bound for the dimensionality of ${\cal M}(N_1,P_1,N_2,P_2)$:
\be \label{6.5}
\dim({\cal M}(N_1,P_1,N_2,P_2))\geq 2(N_1+N_2+P_1+P_2).
\ee
Since we have not established the uniqueness of a solution with $N_1+N_2$ and $P_1+P_2$ prescribed vortices and anti-vortices of the two
species yet, we do not know whether the inequality (\ref{6.5}) is actually an equality. In this regard, it will be interesting to carry out
an investigation along the (well-known classical) index theory work of Atiyah, Hitchin, and Singer \cite{AHS1,AHS2}
on the Yang--Mills instantons, of Weinberg \cite{Wein}
on the BPS system of the Abelian Higgs model, and of Lee \cite{Lee}
on supersymmetric domain walls, for our new system of equations, (\ref{x1.17})--(\ref{x1.20}).
In a sharp contrast, if we use ${\cal M}(N_1,N_2)$ to denote the moduli space of the solutions of the Tong--Wong equations
(\ref{x2.11})--(\ref{x2.14}) with $N_1$ and $N_2$ prescribed vortices, of two respective species, the established uniqueness
of the solutions indicates the result
\be
\dim({\cal M}(N_1,N_2))= 2(N_1+N_2).
\ee
See \cite{Moore} for some recent related work.
\subsection*{Acknowledgements}
Han was partially supported by National Natural Science Foundation of China under Grant 11201118 and the Key Foundation for Henan colleges under Grant 15A110013. Both authors were partially supported by National Natural Science Foundation of China under Grants 11471100 and 11471099.
\small{
|
{
"timestamp": "2015-07-16T02:09:08",
"yymm": "1504",
"arxiv_id": "1504.03053",
"language": "en",
"url": "https://arxiv.org/abs/1504.03053"
}
|
\section{Introduction and main observation}
Unless otherwise stated, all manifolds considered in this note are
smooth, closed, connected and oriented. All metrics mentioned in
this note refer to Riemannian metrics.
Atiyah and Hirzebruch proved in \cite{AH} that if a $4k$-dimensional
spin manifold $M$ admits a non-trivial (smooth) circle action
($S^1$-action), then the $\hat{A}$-genus of $M$ must vanish. Indeed,
they showed via the Atiyah-Bott-Singer fixed point formula that the
equivariant index $\text{spin}(g,M)\in\text{R}(S^1)$ of the Dirac
operator on $M$ vanishes identically. In particular, the
$\hat{A}$-genus $\hat{A}(M)=\text{spin}(1,M)$ must vanish. Note that
the $\hat{A}$-genus is a rational linear combination of Pontrjagin
numbers and $\hat{A}(\cdot)$ can be viewed as a ring homomorphism
$\hat{A}(\cdot):~\Omega^{\text{SO}}_{\ast}\otimes\mathbb{Q}\rightarrow\mathbb{Q}$,
where $\Omega^{\text{SO}}_{\ast}$ is the oriented cobordism ring
(\cite{Hi}). In addition to the above-mentioned main result, by
suitably choosing generators of
$\Omega^{\text{SO}}_{\ast}\otimes\mathbb{Q}$ and a clever
manipulation for them, Atiyah and Hirzebruch also proved in
\cite{AH} that the $\hat{A}$-genus can be characterized as the only
linear combination of Pontrjagin numbers that vanishes on all
$4k$-dimensional spin manifolds admitting a non-trivial circle
action. To be more precise, they showed that (\cite[\S 2.3]{AH}) a
rational linear combination of Pontrjagin numbers vanishes on all
spin manifolds equipped with a non-trivial smooth circle action if
and only if it is a multiple of the $\hat{A}$-genus.
Recall that on $4k$-dimensional spin manifolds the vanishing of the
$\hat{A}$-genus is also an obstruction to the existence of a
positive scalar curvature metric, which is a classical result of
Lichnerowicz (\cite{Li}) and whose argument is based on a
Bochner-type formula related to the Dirac operator and the
Atiyah-Singer index theorem (a detailed and excellent proof can be
found in \cite[Ch.5]{Wu}). Another breakthrough related to the
existence of a positive scalar curvature metric on spin manifolds
came from Gromov and Lawson. They proved in \cite[Theorem B]{GL}
that if a simply-connected spin manifold $X$ is spin cobordant to a
manifold equipped with a positive scalar curvature metric, then $X$
itself also carries
such a metric. As an application of this result, Gromov-Lawson obtained the same type result as that of
Atiyah-Hirzebruch in the language of cobordism theory (\cite[Corollary B]{GL}):
a rational linear combination of Pontrjagin numbers vanishes on all $4k$-dimensional spin manifolds carrying
a positive scalar curvature metric if and only if it is a multiple of the
$\hat{A}$-genus.
Another result which has a similar flavor to the above Atiyah-Hirzebruch and Gromov-Lawson's results
should also be mentioned. In \cite{Ko}, Kotschick showed that a rational linear combination of Pontrjagin numbers
which is bounded on all $4k$-dimensional manifolds with a nonnegative sectional curvature metric
if and only if it is a multiple of the signature. After \cite[Corollary 2]{Ko}, the author commented that his result
bears some resemblance to Gromov-Lawson's above result.
However, although Atiyah-Hirzebruch's result is much earlier, clearly neither \cite{GL} nor \cite{Ko} was aware of
it as they didn't cite \cite{AH}. In spirit, both Gromov-Lawson and Kotschick's proofs
are similar to that of Atiyah-Hirzebruch (see \cite[p. 432]{GL} and \cite[p. 139]{Ko}):
suitably choose generators for $\Omega^{\text{SO}}_{\ast}\otimes\mathbb{Q}$ sastisfying certain
restrictions and then apply a proportionality argument to lead to the desired
proof. This idea was further taken up by Kotschick et al to solve a
long-standing problem posed by Hirzebruch in 1954
(\cite{Ko2}, \cite{KS}).
The \emph{main purpose} of this short note is to show that, by using
the generators of $\Omega^{\text{SO}}_{\ast}\otimes\mathbb{Q}$
chosen in \cite{AH}, together with the celebrated Calabi-Yau
theorem, we can give a direct proof of Gromov-Lawson's this result,
which avoids using their main result \cite[Theorem]{GL}. Moreover,
because of the use of the Calabi-Yau theorem, our statement is
indeed more sharper and thus improves on their original result. We
now state our main observation in this note, whose proof will be
given in Section \ref{section2}.
Note that the Ricci curvature of a metric on a manifold is an
assignment of a quadratic form at the tangent space of any point of
this manifold. The Ricci curvature is said to be \emph{nonnegative
and nonzero at any point} if, at any point of this manifold, this
quadratic form is nonnegative definite but nonzero. With this notion
understood, our main observation is the following
\begin{theorem}\label{maintheorem}
If a rational linear combination of Pontrjagin numbers vanishes on
all $4k$-dimensional simply-connected spin manifolds equipped with a
metric whose Ricci curvature is nonnegative and nonzero at any
point, then it must be a multiple of the $\hat{A}$-genus.
\end{theorem}
Clearly the conclusion in Theorem \ref{maintheorem} remains true if
we weaken the restrictions imposed on the manifolds in Theorem
\ref{maintheorem}. Recall that the scalar curvature is nothing but
the trace of the real symmetric matrix which represents the
quadratic form of the Ricci curvature under some orthonormal basis.
So the condition that the Ricci curvature be nonnegative and nonzero
at any point means that the eigenvalues of this matrix are all
nonnegative and at least one of them is positive. In particular the
trace is positive. This means the condition of the Ricci curvature
required in Theorem \ref{maintheorem} implies that the scalar
curvature of this metric is positive. Therefore, Theorem
\ref{maintheorem}, together with the Lichnerowicz vanishing theorem,
yields the following corollary.
\begin{corollary}(Gromov-Lawson, \cite[Corollary B]{AH})\label{maincoro}
A rational linear combination of Pontrjagin numbers vanishes on all
$4k$-dimensional spin manifolds equipped with a positive scalar
curvature metric if and only if it is a multiple of the
$\hat{A}$-genus.
\end{corollary}
\section{Proof of Theorem \ref{maintheorem}}\label{section2}
Let $\mathbb{H}P^k$ denote the quaternionic projective space whose
real dimension is $4k$. It is well-known that $\mathbb{H}P^k$ admits
a metric with positive sectional curvature and thus it has positive
Ricci curvature. Let $V_2(4)$ be the smooth hypersurface of degree
$4$ in the $3$-dimensional complex projective space $\mathbb{C}P^3$,
which is a compact K\"{a}hler surface and whose first Chern class
$c_1(V_2(4))=0$ in $H^2(V_2(4);\mathbb{Z})$ (Kummer surface). For
simplicity we denote by $N^1:=V_2(4)$ and $N^k:=\mathbb{H}P^k$
($k\geq 2$).
For our later use, we record some facts related to $N^k$ in the
following lemma. Although some of them have been sketchily explained
in \cite[\S 2.3]{AH}, for the reader's convenience, we would like to
either point out a reference or indicate its proof if some
non-standard fact is stated/recorded.
\begin{lemma}\label{lemma}
~\begin{enumerate}
\item
$$\Omega^{\text{SO}}_{\ast}\otimes\mathbb{Q}=
\mathbb{Q}[N^1,N^2,N^3,\ldots,].$$ In other words,
$\{N^k\}^{\infty}_{k=1}$ is a basis sequence for the oriented
cobordism graded ring tensored with $\mathbb{Q}$. This means, if we
denote by $\Omega^k$ the restriction of $\Omega^{\text{SO}}_{\ast}$
to the $4k$-dimensional manifolds, then the vector space
$\Omega^k\otimes\mathbb{Q}$ has a basis $\{\prod_{i=1}^tN^{k_i}\}$,
where $(k_1,\ldots,k_t)$ runs over all the partitions of weight $k$
$(=k_1+k_2+\cdots +k_t).$
\item
All $N^k$ and any of their finitely many product
$\prod_{i=1}^tN^{k_i}$ are simply-connected and spin.
\item
$\hat{A}(N^1)=2$ and $\hat{A}(N^k)=0$ with $k\geq 2$.
\item
Suppose we have a sequence of $t$ positive-dimensional Riemannian
manifolds $(M_i,g_i)$ $(1\leq i\leq t)$ such that any Ricci
curvature $\text{Ric}(g_i)$ is either positive or identically zero
and at least one of these $\text{Ric}(g_i)$ is positive. Then the
Ricci curvature $\text{Ric}(\prod_{i=1}^tg_i)$ of the Riemannian
product manifold $(\prod_{i=1}^tM_i,\prod_{i=1}^tg_i)$ is
nonnegative and nonzero at any point.
\item
The product manifold $\prod_{i=1}^tN^{k_i}$ admits a Riemanian
metric whose Ricci curvature is nonnegative and nonzero at any point
if at least one of these $k_i$ is larger than $1$.
\end{enumerate}
\end{lemma}
\begin{proof}~
\begin{enumerate}
\item
This was observed in \cite[\S 2.3]{AH}. Indeed, there exists a
simple procedure, which is due to Thom, to detect whether or not a
given sequence of $4k$-dimensional manifolds $(k=1,2,\ldots)$ is a
basis sequence (cf. \cite[p. 79]{Hi} or \cite[p. 43]{HBJ}).
\item
Simply-connectedness is clear. A manifold is spin if and only if its
second Stiefel-Whitney class $\omega_2=0$. $N^k$ $(k\geq 2)$ are
spin as $H^2(N^k;\mathbb{Z}_2)=0$. $N^1$ is spin as $c_1(N^1)=0,$
whose module $2$ reduction is exactly $\omega_2(N^1)$. The fact that
the product $M_1\times M_2$ of two (orientable) spin manifolds $M_1$
and $M_2$ is still spin follows from the Whitney direct sum formula
for
$\omega_2(M_1\times M_2)$ and the fact that the first
Stiefel-Whitney class is zero if and only if the manifold is
orientable (\cite{MS}).
\item
$\hat{A}(N^k)=0$ for $k\geq 2$ follow form the main result in
\cite{AH} as these $N^k$ admit nontrivial circle actions.
$\hat{A}(N^1)=2$ is quite well-known. In fact there is a closed
formula to calculate the $\hat{A}$-genus of general complete
intersections in the complex projective spaces (\cite{Br}).
\item
This statement follows from the simple fact that the quadric form of
the Ricci curvature $\text{Ric}(g_1\times g_2)$ of the product
metric $g_1\times g_2$ is exactly the direct sum of those of
$\text{Ric}(g_1)$ and $\text{Ric}(g_2)$.
\item
As we have mentioned that on each $N^k$ ($k\geq 2$) one has a
positive Ricci curvature metric. Since $c_1(N^1)=0$, the celebrated
Calabi-Yau theorem, which refers to S.-T. Yau's solution to the
famous Calabi conjecture (\cite{yau}), tells us that $N^1$ carries a
K\"{a}hler (hence Riemannian) metric whose Ricci curvature is
identically zero. Hence this statement follows from $(4)$. Note that
it is \emph{this} place where we apply the remarkable Calabi-Yau
theorem!
\end{enumerate}
\end{proof}
Now we are in a position to prove Theorem \ref{maintheorem}.
\begin{proof}
Fix a positive integer $k$.
Let $\hat{A}_k$ be the restriction of $\hat{A}$ to
$\Omega^k\otimes\mathbb{Q}$. Namely, $\hat{A}_k$ is a linear
homomorphism:
$$\hat{A}_k(\cdot):~\Omega^k\otimes\mathbb{Q}\rightarrow\mathbb{Q}.$$
Let $q$ be a rational linear combination of Pontrjagin numbers on
$4k$-dimensional manifolds. Since Pontrjagin numbers are cobordism
invariants, $q$ can also be viewed as a linear homomorphism
$$q:~\Omega^k\otimes\mathbb{Q}\rightarrow\mathbb{Q}.$$
In view of these, it suffices to
show that, if $q$ vanishes on all spin manifolds which admits a
metric whose Ricci curvature is nonnegative and nonzero at any
point, then $q$ is a multiple of $\hat{A}_k$.
If $k=1$, the conclusion is clear as there has only one Pontrjagin
number $p_1$ and $\hat{A}_1=\frac{1}{24}p_1$. We now suppose that
$k\geq 2$. By $(1)$ of Lemma \ref{lemma} we know that the vector
space $\Omega^k\otimes\mathbb{Q}$ has a basis
$\{\prod_{i=1}^tN^{k_i}\}$, where $(k_1,\ldots,k_t)$ runs over all
the partitions of weight $k$. $(5)$ of Lemma \ref{lemma} tells us
that $q$ vanishes on all these basis elements with only one possible
exception $(N^1)^k$. The same holds for $\hat{A}_k$ by $(3)$ of
Lemma \ref{lemma}. This implies
$$q-\frac{q\big((N^1)^k\big)}{2^k}\hat{A}_k$$
vanishes on all these
basis elements and thus
$$q=\frac{q\big((N^1)^k\big)}{2^k}\hat{A}_k~\text{
on}~\Omega^k\otimes\mathbb{Q},$$
which gives the desired proof.
\end{proof}
|
{
"timestamp": "2015-04-14T02:11:35",
"yymm": "1504",
"arxiv_id": "1504.03078",
"language": "en",
"url": "https://arxiv.org/abs/1504.03078"
}
|
\section{Introduction}The QCD sum-rule approach
\cite{svz,aliev,rubinstein}, based on the application of Wilson's
operator product expansion (OPE) to the properties of individual
hadrons, has been extensively used for predicting heavy-meson
decay constants. An~important finding of these analyses was the
strong sensitivity of the decay constants to the values of the
input OPE parameters and to the prescription of fixing the
effective continuum threshold \cite{lms_1}. The latter governs the
accuracy of the quark--hadron duality approximation and, to a
large extent, determines the extracted value of the decay
constant. Even~if~the parameters of the truncated OPE are known
with arbitrarily high precision, the decay constants may be
predicted with only limited accuracy, which we refer to as their
systematic uncertainty. In a series of papers \cite{lms_new}, we
have~formulated a new algorithm for fixing the effective threshold
within Borel QCD sum rules and for obtaining reliable
estimates~for the systematic uncertainties. This procedure opened
the possibility to provide predictions for the decay
constants~with a controlled accuracy \cite{lms_fp,lms_fD}.
Here, we study the decay constants of the vector beauty mesons
$f_{B^*}$ and $f_{B^*_s}$. As is already known from the analysis
of the decay constants of the pseudoscalar mesons $B$ and $B_s$
\cite{lms_fp}, the OPE uncertainties in the obtained
predictions~are rather large. The same occurs also for the $B^*$
and $B_s^*$ mesons. However, the OPE uncertainties to a great
extent cancel out in the ratios of the decay constants of vector
and pseudoscalar beauty mesons. An important result reported here
is that the systematic uncertainties of the decay constants are
rather small and well under control. Therefore, these ratios are
predicted with a very good accuracy. It should be taken into
account that we address a rather subtle effect at a few-percent
level; a priori, it is not clear whether QCD sum rules are, in
principle, capable to provide~theoretical predictions at this
level of accuracy. Obviously, the control over the systematics is
becoming crucial.
The ratio of the decay constants of vector over pseudoscalar heavy
mesons is an interesting quantity: it is known to be unity in the
heavy-quark limit and to approach this limit from below because of
the radiative corrections \cite{neubert}. For beauty mesons, the
few existing sum-rule analyses (which, however, could not gain
good control over the systematic uncertainties) reported
$f_{B^*}/f_B$ slightly above unity \cite{narison2014,kh}.
Constituent-quark models typically also yield $f_{B^*}/f_B>1$
\cite{faustov}. A similar conclusion has been reached by
interpolation of the lattice data from the charm-quark mass region
to the beauty-quark mass \cite{becirevic2014}.
The first indication that this ratio for beauty mesons is below
unity was given in our papers \cite{lms2014}. Recently, HPQCD
\cite{hpqcd2015} also reported an accurate value of
$f_{B^*}/f_B<1$, in excellent agreement with the results of
\cite{lms2014}. The analysis of \cite{lms2014}, although
conclusively indicating $f_{B^*}/f_B<1$, observed an unpleasant
dependence of the extracted decay constants of the vector beauty
mesons on the renormalization scale $\mu$ chosen for the
evaluation of the vector correlation function. This analysis
solves the problem of the sensitivity to the choice of the scale
$\mu$ by improving the extraction procedures for the decay
constants and arrives at new predictions stable with respect to
the choice of $\mu$. Our detailed results read
\begin{eqnarray}
f_{B^*}/f_B=0.944\pm0.011_{\rm OPE}\pm0.018_{\rm syst},\qquad
f_{B_s^*}/f_{B_s}=0.947\pm0.023_{\rm OPE}\pm0.020_{\rm syst},
\end{eqnarray}
in more than excellent agreement with the latest results from
lattice QCD \cite{hpqcd2015}. Let us emphasize once more that~the
OPE uncertainties cancel to a large extent in the above ratios.
Thus, decisive for obtaining an accurate sum-rule~result is our
capability to control the systematic uncertainties of the QCD
sum-rule method.
\section{QCD vector correlator and sum rule for vector-meson decay
constant $f_V$}The decay constants of ground-state vector mesons
may be extracted by analyzing the two-point correlation function
\begin{eqnarray}
\label{1.1}i\int d^4x\,e^{ipx}
\langle0|T\!\left(j_\mu(x)j^\dagger_\nu(0)\right)|0\rangle=
\left(-g_{\mu\nu}+\frac{p_\mu p_\nu}{p^2}\right)\Pi(p^2)
+\frac{p_\mu p_\nu}{p^2}\Pi_L(p^2)
\end{eqnarray}
of the heavy--light vector currents for a heavy quark $Q$ of mass
$m_Q$ and a light quark $q$ of mass $m$,
\begin{eqnarray}
j_\mu(x)=\bar q(x)\gamma_\mu Q(x),
\end{eqnarray}
or, more precisely, the Borel transform of its transverse
structure $\Pi(p^2)$ to the Borel variable $\tau$, $\Pi(\tau)$.
Equating~$\Pi(\tau)$~as calculated within QCD and the expression
obtained by insertion of a complete set of hadron states yields
the sum~rule
\begin{eqnarray}
\label{pitau}\Pi(\tau)=f^2_{V}M_V^2e^{-M_V^2\tau}
+\int\limits_{s_{\rm phys}}^{\infty}ds\,e^{-s\tau}\rho_{\rm
hadr}(s)=\int\limits^\infty_{(m_Q+m)^2}ds\,e^{-s\tau}\rho_{\rm
pert}(s,\mu)+\Pi_{\rm power}(\tau,\mu).
\end{eqnarray}
Here, $M_V$ labels the mass, $f_V$ the decay constant, and
$\varepsilon_\mu(p)$ the polarization vector of the vector meson
$V$ under~study:
\begin{eqnarray}
\label{decay_constant}\langle0|\bar q\gamma_\mu Q|V(p)\rangle
=f_VM_{V}\varepsilon_\mu(p).
\end{eqnarray}
For the correlator (\ref{1.1}), $s_{\rm phys}=(M_{P}+M_\pi)^2$ is
the physical continuum threshold, wherein $M_{P}$ denotes the mass
of~the lightest pseudoscalar meson containing $Q$. For large
values of $\tau$, the ground state dominates the
correlator~and~thus its properties may be extracted from the
correlation function (\ref{1.1}).
In perturbation theory, the correlation function is found as an
expansion in powers of the strong coupling ``constant''
$\alpha_{\rm s}(\mu)$. The best known three-loop perturbative
spectral density has been calculated in \cite{chetyrkin} in terms
of the pole mass~of the heavy quark $Q$ (that is, in the present
case, $M_b$) and for a massless second quark [hereafter, we use
the abbreviation $a(\nu)=\alpha_{\rm s}(\nu)/\pi$, where
$\alpha_{\rm s}(\nu)$ is the running coupling at renormalization
scale $\nu$ in the $\overline{\rm MS}$ scheme]:
\begin{eqnarray}
\label{rhopert}\rho_{\rm pert}(s)=\rho^{(0)}(s,M_b)
+a(\nu)\rho^{(1)}(s,M_b)+a^2(\nu)\rho^{(2)}(s,M_b,\mu)+\cdots.
\end{eqnarray}
For both quarks having nonzero masses, the two-loop spectral
density in terms of their pole masses was
obtained~in~\cite{rubinstein}.
The power corrections are also separately scale-independent; their
explicit expressions can be found in \cite{jamin,kh}. For
instance, for pseudoscalar ($P$) and vector ($V$) currents the
quark-condensate contributions may be written in the~form
\begin{eqnarray}
\label{pipower}\Pi^P_{\rm power}(\tau)
&=&-\overline{m}_b(\nu)\langle\bar qq(\nu)\rangle M_b^2
\left[\exp(-M_b^2\tau)\left(1+\frac{3}{2}C_F a\right)-\frac{3}{2}
C_Fa\,\Gamma(0,M_b^2\tau)\right],\\ \Pi^V_{\rm power}(\tau)
&=&-\overline{m}_b(\nu)\langle\bar qq(\nu)\rangle
\left[\exp(-M_b^2\tau)\left(1+\frac{1}{2}C_F a\right)
+\frac{1}{2}C_F a\, M_b^2\tau \,\Gamma(-1,M_b^2\tau)\right],
\end{eqnarray}
where $\overline{m}_b(\nu)$ is the $b$-quark $\overline{\rm MS}$
mass at renormalization scale $\nu$,
$\overline{m}_b(\nu)\langle\bar qq(\nu)\rangle$ is a
scale-independent combination,~and $\Gamma(n,z)$ is the incomplete
gamma function \cite{AS}.
However, even if the lowest-order contributions to the
perturbative expansion and the vacuum condensates of lowest
dimensions are known to good accuracy, a truncated OPE does not
allow one to calculate the correlator for sufficiently large
$\tau$, such that the continuum states give a negligible
contribution to $\Pi(\tau)$ in the corresponding range of $\tau$.
In~order to get rid of the continuum contribution, the concept of
duality is invoked: Perturbative-QCD spectral density $\rho_{\rm
pert}(s)$ and hadron spectral density $\rho_{\rm hadr}(s)$
resemble each other at large values of $s$; thus, for values of
the integration lower limit $\bar s$ chosen sufficiently large,
that is to say, (far) above the resonance region, one arrives at
the duality relation
\begin{eqnarray}
\label{duality1} \int\limits_{\bar s}^{\infty}ds\,e^{-s
\tau}\rho_{\rm hadr}(s)=\int\limits_{\bar s}^{\infty}ds\,e^{-s
\tau}\rho_{\rm pert}(s).
\end{eqnarray}
Now, in order to express the hadron continuum contribution in
terms of the perturbative contribution, the relation
(\ref{duality1}) should be extended down to the hadronic or
physical threshold $s_{\rm phys}$. However, since the spectral
densities~$\rho_{\rm pert}(s)$~and $\rho_{\rm hadr}(s)$ obviously
differ in the region near $s_{\rm phys}$, one can reasonably only
expect to obtain a relationship of the form
\begin{eqnarray}
\label{duality}
\int\limits_{s_{\rm phys}}^{\infty} ds\, e^{-s
\tau} \rho_{\rm hadr}(s) = \int\limits_{s_{\rm
eff}(\tau)}^{\infty} ds\, e^{-s \tau} \rho_{\rm pert}(s),
\end{eqnarray}
where the effective threshold $s_{\rm eff}(\tau)$ is clearly
different from the physical threshold $s_{\rm phys}$, $s_{\rm
eff}(\tau)\ne s_{\rm phys}$, and, moreover, must be a function of
the Borel parameter $\tau$ \cite{lms_1,lms_new}. By virtue of
(\ref{duality}), we may hence rewrite the QCD sum rule
(\ref{pitau})~as
\begin{eqnarray}
\label{sr} f_V^2 M_V^2 e^{-M_V^2\tau}= \int\limits^{s_{\rm
eff}(\tau)}_{(m_Q+m)^2} ds\, e^{-s\tau}\rho_{\rm pert}(s,\mu) +
\Pi_{\rm power}(\tau,\mu) \equiv \Pi_{\rm dual}(\tau,s_{\rm eff}(\tau)).
\end{eqnarray}
We refer to the right-hand side of this relation as the {\it dual
correlator\/}, and to the masses and decay constants extracted
from this expression as the corresponding \emph{dual\/}
quantities. In addition to $\rho_{\rm pert}(s,\mu)$ and $\Pi_{\rm
power}(\tau,\mu)$, the extraction~of $f_V$ requires, as further
input, a criterion that fixes the functional behaviour of the
effective continuum threshold~$s_{\rm eff}(\tau)$.
We shall demonstrate that QCD sum rules allow a very satisfactory
extraction of the vector-meson decay constants, with an accuracy
that is certainly competitive to that found within the framework
of lattice QCD.
\section{OPE and choice of renormalization scheme and scale for
heavy-quark mass}The starting point of our discussion is the OPE
for the correlator (\ref{1.1}). The three-loop perturbative
spectral density $\rho_{\rm pert}(s,M)$ was calculated in
\cite{chetyrkin} in terms of the pole mass~of the heavy quark. A
nice feature of the pole-mass OPE is that each of the known
perturbative contributions to the dual correlator is positive.
Unfortunately, the pole-mass~OPE does not provide a visible
hierarchy of the perturbative contributions to the extracted
predictions, which raises doubts whether the
$O(\alpha_s^2)$-truncated pole-mass OPE is indeed a good starting
point for a reliable analysis of decay constants.
A well-known remedy is to reorganize the perturbative expansion in
terms of the $b$-quark running $\overline{\rm MS}$ mass
$\overline{m}_b(\mu)$, related (in the notations of \cite{jamin})
to the corresponding pole mass $M_b$ by
\begin{eqnarray}
\label{a2} M_b=
\overline{m}_b(\mu)/\left(1+a(\mu)r^{(1)}_m+a^2(\mu)r_m^{(2)}\right)
+O(a^3).
\end{eqnarray}
The spectral densities in the $\overline{\rm MS}$ scheme are found
by expanding the pole-mass spectral densities in powers
of~$a(\mu)$ and omitting terms of order $O(a^3)$ and higher;
starting at order $O(a)$, they contain two parts: the ``genuine''
part~from \cite{chetyrkin} and the part induced by the lower
perturbative orders when expanding the pole mass in terms of the
running~mass. By this, however, due to the truncation of the
perturbative series, one gets an explicit (unphysical) dependence
of~the dual correlator and of the extracted decay constant on the
scale $\mu$. In principle, any scale should be equivalently~good.
In practice, however, the distinctness of the hierarchy of the
perturbative contributions to the dual correlator depends on the
precise choice of the scale. This opens a possibility of choosing
the scale $\mu$ such that the hierarchy of the new perturbative
expansion is improved.
Figures \ref{Plot:1a} and \ref{Plot:1b} depict the
dual decay constants of the $B^*$ and $B$ mesons, respectively.
For the~$b$-quark $\overline{\rm MS}$ mass, we use the value
determined in \cite{ourmb} by matching our QCD sum-rule results
for $f_B$ to those of lattice~QCD:\footnote{As shown in
\cite{ourmb}, the PGD average
$\overline{m}_b(\overline{m}_b)=(4.18\pm0.030)$ GeV \cite{pdg}
(see also \cite{erler} for a recent overview of the $b$-quark mass
results)~leads to a considerably larger value of $f_B$,
incompatible with the latest lattice-QCD results. However, the
precise value of $\overline{m}_b(\overline{m}_b)$ has~negligible
impact on the ratio of the decay constants of vector and
pseudoscalar mesons.}
\begin{eqnarray}
\label{ourmb}\overline{m}_b(\overline{m}_b)=(4.247\pm0.034)\ {\rm GeV}.
\end{eqnarray}
The numerical values adopted for other relevant OPE parameters are
\cite{jamin,ourmb,lms_fD,ms2014}
\begin{eqnarray}
\label{Table:1}
&& m(2\;{\rm GeV})=(3.42\pm0.09)\;{\rm MeV}, \quad m_s(2\;{\rm GeV})=(93.8\pm2.4)\;{\rm MeV},\quad \alpha_{\rm s}(M_Z)=0.1184\pm0.0020, \nonumber\\
&&\left\langle\frac{\alpha_{\rm s}}{\pi}GG\right\rangle=(0.024\pm0.012)\;{\rm GeV}^4,\quad
\langle\bar qq\rangle(2\;{\rm GeV})=-[(267\pm17)\;{\rm MeV}]^3,\quad \frac{\langle\bar ss\rangle(2\;{\rm
GeV})}{\langle\bar qq\rangle(2\;{\rm GeV})}=0.8\pm0.3.
\end{eqnarray}
The purpose of Figs.~\ref{Plot:1a} and \ref{Plot:1b} is the illustration of the main features of the dual correlators
(\ref{sr}), therefore the QCD sum-rule estimates shown here are obtained for
a $\tau$-independent effective threshold: $s_{\rm eff}=\mbox{const}$.
The numerical value of the latter is, in each
case, found by requiring maximal stability of the extracted decay
constant in the Borel~window. We emphasize that our results for the decay constants reported in the next
Sections are obtained using the $\tau$-dependent effective thresholds.
\begin{figure}[!t]
\begin{tabular}{cc}
\includegraphics[width=7.5581cm]{1.Pi_Bv_dual_pole.eps} &
\includegraphics[width=7.5581cm]{1.Pi_Bv_dual_mu_2.5.eps} \\
\includegraphics[width=7.5581cm]{1.Pi_Bv_dual_mu_3.eps} &
\includegraphics[width=7.5581cm]{1.Pi_Bv_dual_mu_5.eps}
\end{tabular}
\caption{\label{Plot:1a}QCD sum-rule estimates of the $B^*$-meson
decay constant using the pole-mass OPE (a) and the running-mass
OPE at the renormalization scales $\mu=2.5$ GeV (b), $\mu=3$ GeV
(c), and $\mu=5$ GeV (d). The running-mass OPE for
$\overline{m}_b(\overline{m}_b)=4.247$~GeV is shown. The pole-mass
OPE employs the corresponding two-loop pole mass $M_b=4.87$ GeV.
For each case, separately, a constant effective continuum
threshold~$s_{\rm eff}$ is determined by requiring maximal
stability of the predicted decay constant in a Borel window of the
maximal width $0.05\le\tau\ ({\rm GeV}^{-2})\le0.15$. Bold lines
(lilac)---total findings, solid lines
(black)---$O(1)$~contributions; dashed lines
(red)---$O(\alpha_{\rm s})$ contributions; dotted lines
(blue)---$O(\alpha_{\rm s}^2)$ contributions; dot-dashed lines
(green)---power contributions.}
\end{figure}
\begin{figure}[!ht]
\begin{tabular}{cc}
\includegraphics[width=7.5581cm]{2.Pi_B_dual_pole.eps} &
\includegraphics[width=7.5581cm]{2.Pi_B_dual_mu_2.5.eps} \\
\includegraphics[width=7.5581cm]{2.Pi_B_dual_mu_3.eps} &
\includegraphics[width=7.5581cm]{2.Pi_B_dual_mu_5.eps}
\end{tabular}
\caption{\label{Plot:1b} Same as Figure \ref{Plot:1a} but for the
$B$ meson.}
\end{figure}
From Figs.~\ref{Plot:1a} and \ref{Plot:1b}, we conclude that the
$O(\alpha_s^2)$-truncated pole-mass OPE exhibits no hierarchy of
the perturbative expansion and better should not be used.
Unfortunately, the hierarchy of the running-mass OPE is also not
guaranteed automatically and depends strongly on the scale $\mu$.
Let us define a scale $\hat\mu$ by demanding
$M_b=\overline{m}_b(\hat\mu)$. From the $O(a^2)$ relation between
$\overline{\rm MS}$ and pole mass,~we find $\hat\mu\approx2.23$
GeV. At this scale, the perturbative hierarchy of the
$\overline{\rm MS}$ expansion is worse than that of the pole-mass
expansion because the $O(1)$ spectral densities coincide, whereas
the $O(\alpha_s)$ spectral density in the $\overline{\rm MS}$
scheme receives a positive contribution compared to the pole-mass
scheme. For lower scales $\mu<\hat\mu$, the hierarchy of the
$\overline{\rm MS}$ expansion gets worse with decreasing $\mu$.
For higher scales $\mu>\hat\mu$, first the hierarchy of the
$\overline{\rm MS}$-expansion improves with rising $\mu$
(Figs.~\ref{Plot:1a} and \ref{Plot:1b}). However, as the scale
$\mu$ becomes sufficiently larger than $\hat\mu$, the ``induced''
contributions, which mainly reflect the bad-behaved expansion of
the pole mass in terms of the running mass, start to dominate over
the ``genuine'' contributions. This is evident in
Figs.~\ref{Plot:1a} and \ref{Plot:1b}: at $\mu=5$ GeV, the $O(1)$
contribution to the dual correlator rises~steeply with $\tau$,
whereas the $O(a)$ contribution becomes negative in order to
compensate the rising $O(1)$ contribution. Finally, for large
values of $\mu$ we mainly observe a compensation between the
``induced'' contributions. We may expect in this case the accuracy
of the expansion to deteriorate.
Figures \ref{Plot:1a} and \ref{Plot:1b} also reveal an essential
difference between pseudoscalar and vector correlators: at the
same scale $\mu$, the good reproduction of the observed mass of
the vector meson requires lower values of $\tau$ compared to its
pseudoscalar partner. This implies that the Borel window for the
vector correlator should be chosen at lower values of $\tau$ than
the corresponding window for the pseudoscalar correlator.
Moreover, for $\mu\gtrsim5$--6 GeV the vector-meson mass cannot be
reproduced in a reasonably broad $\tau$ window and so the QCD sum
rule cannot predict the vector-meson decay~constant.
For the present analysis, we thus choose as range of scales
$\mu=3$--5 GeV: On the one hand, in this range we observe a
reasonable hierarchy of the perturbative contributions to the
correlator. On the other hand, we shall see that for~this range of
scales one can find sufficiently broad $\tau$ windows where the
decay constants may be reliably extracted by our algorithm. For
the vector mesons, the upper bound of this window depends on
$\mu$.
\section{Extraction of the beauty-meson decay constants from our
QCD sum rule}In order to extract the decay constants, we first
have to find a $\tau$ window such that the OPE provides a
sufficiently accurate description~of the exact correlator (i.e.,
all higher-order radiative and power corrections are under
control). Next, we must determine the $\tau$ dependence of the
effective threshold $s_{\rm eff}(\tau)$. The appropriate algorithm
was developed and verified within quantum-mechanical potential
models \cite{lms_new} and shown to work successfully for the decay
constants~of heavy pseudoscalar mesons \cite{lms_fp}. We introduce
a {\em dual invariant mass\/} $M_{\rm dual}$ and a {\em dual decay
constant\/} $f_{\rm dual}$ by defining
\begin{eqnarray}
\label{mdual}M_{\rm dual}^2(\tau)\equiv-\frac{d}{d\tau}\log
\Pi_{\rm dual}(\tau, s_{\rm eff}(\tau)),\qquad\label{fdual}f_{\rm
dual}^2(\tau)\equiv M_V^{-2}e^{M_V^2\tau}\Pi_{\rm dual}(\tau,
s_{\rm eff}(\tau)).
\end{eqnarray}
For a properly constructed $\Pi_{\rm dual}(\tau, s_{\rm eff}(\tau))$,
the dual mass coincides with the actual ground-state
mass $M_V$. Therefore, any deviation of the dual mass from $M_V$
is an indication of the contamination of the dual correlator by
excited states.
For any trial function for the effective threshold, we derive a
variational solution by minimizing the difference~between the dual
mass (\ref{mdual}) and the actual (i.e., experimentally measured)
mass in the Borel window. This variational solution provides the
decay constant then via (\ref{fdual}). We consider a set of
polynomial Ans\"atze for the effective threshold,~viz.,
\begin{eqnarray}
\label{zeff} s^{(n)}_{\rm eff}(\tau)=
\sum\limits_{j=0}^{n}s_j^{(n)}\tau^{j},
\end{eqnarray}
and fix the coefficients $s_j^{(n)}$ (the knowledge of which then
allows us to compute the decay constant $f_V$) by minimizing
\begin{eqnarray}
\label{chisq} \chi^2 \equiv \frac{1}{N} \sum_{i = 1}^{N} \left[
M^2_{\rm dual}(\tau_i) - M_V^2 \right]^2
\end{eqnarray}
over the Borel window. Still, different Ans\"atze for $s_{\rm
eff}(\tau)$ yield different sum-rule predictions for the decay
constants.
Careful studies of quantum-mechanical potential models indicate
that it suffices to allow for polynomials up to~third order: In
this case, the band delimited by the predictions arising from
linear, quadratic, and cubic Ans\"atze for $s_{\rm eff}(\tau)$
encompasses the true value of the decay constant. Even a good
knowledge of the truncated OPE does not allow~us~to determine the
decay constant precisely but it enables us to provide a range of
values containing the true value of~this decay constant. The width
of this range may then be regarded as the {\em systematic error\/}
related to the principally~limited accuracy of QCD sum rules.
Presently, we are not aware of any other possibility to acquire a
more reliable estimate~for the systematic error. Noteworthy,
considering a $\tau$-independent threshold would not allow us to
probe the accuracy of the obtained estimate for $f_V$.
On top of the systematic error comes the {\em OPE-related error\/}
of the decay constant: the OPE parameters are known only with some
errors, which induce a corresponding error of $f_V$. We determine
this OPE-related (statistical) error by averaging the results for
the decay constant assuming for the OPE parameters Gaussian
distributions with the central values and standard deviations
quoted in (\ref{Table:1}) and a flat distribution over the scale
$\mu$ in the range $3<\mu\;({\rm GeV})<5$.
\subsection{\boldmath Decay constant of the $B^*$ meson}
\subsubsection{Choice of renormalization scale}
In principle, the decay constant should be independent of the
scale $\mu$ at which the correlation function is evaluated. In
practice, however, due to the truncations of the perturbative
expansion and the series of power corrections, and~the neccessity
to isolate the ground-state contribution from the hadron continuum
states, a reliable extraction of the~decay constant may be
performed in only a limited range of the scale $\mu$. For the
vector beauty meson, the suitable range of $\mu$ is found to be
$\mu=3$--5 GeV: For $\mu\le3$ GeV, the perturbative expansion for
the vector correlator does not exhibit a satisfactory perturbative
convergence and therefore gives no reason to believe that the
unknown higher-order radiative corrections both in the
perturbative part of the correlation function and in the radiative
corrections to the condensates are negligible. At higher scales
$\mu\ge5$ GeV, the $B^*$ mass cannot be reproduced with the
required accuracy, signalling that there the contamination of the
excited states cannot be cleaned out.
\subsubsection{Choice of Borel-parameter window}
We require that the $B^*$--$B$ mass splitting and the masses of
$B^*$ and $B$ mesons are reproduced, separately, with an accuracy
not worse than 5 MeV for any $\tau$ value within the selected
ranges. As follows from the properties of the~dual correlators,
this requirement provides two constraints on the choice of the
$\tau$ window for $B^*$:\begin{enumerate}\item The $\tau$ window
for $B^*$ should be chosen at lower values of $\tau$ compared to
the $B$-meson case.\item The precise choice of the $\tau$-window
for $B^*$ should correlate with the scale $\mu$ at which the
correlator is evaluated.\end{enumerate}To satisfy the above
criteria for $B^*$, we set the lower boundary at $\tau_{\rm
min}\;({\rm GeV}^{-2})=0.01$ and choose a $\mu$-dependent~upper
boundary of the form $\tau_{\rm max}\;({\rm GeV}^{-2})=0.31-0.05\,
\mu\;({\rm GeV})$, which choice enables us to extract $f_{B^*}$
with a systematic uncertainty not worse than 5 MeV and strongly
diminishes the unphysical scale dependence of the decay
constant~$f_{B^*}$.
\begin{figure}[ht]
\begin{tabular}{c}
\includegraphics[width=7.5cm]{3.Bv_Mdual_mu_4.eps}
\includegraphics[width=7.5cm]{3.Bv_fdual_mu_4.eps}\\
(a) \hspace{7cm} (b)\\
\includegraphics[width=7.5cm]{3.Bv_seff_mu_4.eps}\\
(c)
\end{tabular}
\caption{\label{Plot:fBv} Dependence on the Borel parameter $\tau$
of the dual mass (a) and the dual decay constant (b) of the $B^*$
meson, obtained~by adopting different Ans\"atze (\ref{zeff}) for
the effective threshold $s_{\rm eff}(\tau)$ and fixing these
thresholds by minimizing (\ref{chisq}); the results are presented
for the central values of all OPE parameters. (c) The
$\tau$-dependent effective thresholds as obtained by our
algorithm. The integer $n=0,1,2,3$ is the degree of the polynomial
in our Ansatz (\ref{zeff}) for $s_{\rm eff}(\tau)$: dotted lines
(red)---$n=0$; solid lines (green)---$n=1$; dashed lines
(blue)---$n=2$; dot-dashed lines (black)---$n=3$.}
\end{figure}
Figure~\ref{Plot:fBv} shows the application of our procedure for
fixing the effective threshold and extracting the resulting
$f_{B^*}$. The dependence of our QCD sum-rule result on the
relevant OPE parameters, i.e., the $b$-quark mass
$m_b\equiv\overline{m}_b(\overline{m}_b)$, the quark condensate
$\langle\bar qq\rangle\equiv\langle\bar qq(2\;{\rm GeV})\rangle$
and the gluon condensate $\langle aGG\rangle$, proves to be well
described by a linear~relation:
\begin{align}
\label{fB*}&f_{B^*}^{\rm dual}(m_b,\langle \bar qq\rangle,\langle
aGG\rangle)=(181.8\pm4_{\rm syst})
\left(1-\frac{11}{181.8}\delta_{m_b}\right)
\left(1+\frac{7}{181.8}\delta_{\langle qq\rangle}\right)
\left(1-\frac{1}{181.8}\delta_{\langle aGG\rangle}\right)
\mbox{MeV},
\end{align}
with
\begin{align}
\label{deltas}
&\delta_{m_b}=\frac{m_b-\mbox{4.247\;GeV}}{\mbox{0.034\;GeV}},\qquad
\delta_{\langle qq\rangle}=\frac{|\langle\bar
qq\rangle|^{1/3}-\mbox{0.267\;GeV}}{\mbox{0.017\;GeV}},\qquad
\delta_{\langle aGG\rangle}=\frac{\langle aGG\rangle-0.024\;{\rm GeV}^4}{0.012\;{\rm GeV}^4}.
\end{align}
The above parameters $\delta$ take values between $-1$ and $+1$
when the corresponding OPE quantity varies in its
$1\sigma$~interval. Varying all other OPE parameters in their
$1\sigma$ ranges leads to an effect on $f_{B^*}$ of less than 1
MeV and is not shown~here. Trusting in our experience from exactly
solvable examples, we assume that the systematic uncertainty
interval contains the true value of the decay constant and that
inside this interval the true decay-constant value has a flat
distribution.
As evident from Fig.~\ref{Plot:fBv}(a), using a constant threshold
leads to a contamination of the dual correlator by excited~states
(beyond the acceptable level), while this contamination is
strongly reduced for $n>0$: the values of the decay constant in
Fig.~\ref{Plot:fBv}(b) resulting for $n>0$ are nicely grouped
together, whereas the $n=0$ prediction emerges some 10 MeV~below.
A particularly convincing feature of the presented extraction
procedure is the insensitivity of the extracted value~of $f_{B^*}$
(as well as that of $f_B$) to scale variations in the interval
$\mu=3$--5 GeV (Fig.~\ref{Plot:fV_vs_mu}), achieved by demanding
an accurate reproduction of the $B^*$ mass in the full $\tau$
window, which requires a specific choice of the $\tau$ window
correlated with the scale $\mu$ at which the correlator is
evaluated. Such choice of the $\tau$ window allows us to keep the
systematic uncertainty, estimated by the half width of the band
encompassing the results for the linear, quadratic, and cubic
thresholds, at a level below 4 MeV in the full $\mu$ range.
Therefore, (\ref{fB*}) describes well the result for any $\mu$
from the range $\mu=3$--5~GeV.
\begin{figure}[ht]
\begin{tabular}{cc}
\includegraphics[width=8.2cm]{4.fdual_B+Bv_vs_mu.eps}&
\includegraphics[width=8.2cm]{4.fdual_Bs+Bsv_vs_mu.eps}\\(a)&(b)
\end{tabular}
\caption{\label{Plot:fV_vs_mu}
Renormalization-scale dependence of
the predicted decay constants: (a) $f^{\rm dual}_B(\mu)$ and
$f^{\rm dual}_{B^*}(\mu)$, (b) $f^{\rm dual}_{B_s}(\mu)$ and
$f^{\rm dual}_{B^*_s}(\mu)$. For each decay constant, we depict
the $\mu$-related uncertainty, i.e., the standard deviation
calculated assuming a flat $\mu$ distribution in the range
$\mu=3$--5 GeV. Dotted lines (red)---vector beauty mesons; solid
lines (blue)---pseudoscalar beauty mesons.}
\end{figure}
Assuming Gaussian distributions for all OPE parameters collected
in (\ref{Table:1}), we get the distribution of $f_{B^*}$
depicted~in Fig.~\ref{Plot:bootstrap}. For the average and the
standard deviation of the $B^*$-meson decay constant, we obtain
\begin{eqnarray}
\label{fDv_constant} f_{B^*}=\left(181.8\pm13.1_{\rm OPE}\pm4_{\rm
syst}\right)\mbox{MeV}.
\end{eqnarray}
The OPE uncertainty is composed as follows: 11 MeV are due to the
variation of $m_b$ and 6 MeV arise from the quark condensate. The
uncertainties of all other OPE parameters contribute less than 1
MeV to the OPE uncertainty~of~$f_{B*}$.
\begin{figure}[ht]\begin{tabular}{ccc}
\includegraphics[width=8.5cm]{5.fBstarfB.eps}&
\includegraphics[width=8.5cm]{5.fBsstarfBs.eps}
\end{tabular}
\caption{\label{Plot:bootstrap}Distributions of the ratios
$f_{B^*}/f_B$ and $f_{B_s^*}/f_{B_s}$ of beauty-meson decay
constants, obtained by generating 1000 bootstrap events. For both
ratios, their final distributions possess Gaussian-like shapes,
with the standard deviations quoted in the plots.}
\end{figure}
The corresponding QCD sum-rule outcome for the $B$-meson decay
constant $f_B$ from our earlier investigation \cite{ourmb} reads
\begin{eqnarray}
\label{fB}f_B^{\rm dual}(m_b,\langle\bar qq\rangle,\langle
aGG\rangle)=(192.6\pm3_{\rm syst})
\left(1-\frac{12.6}{192.6}\delta_{m_b}\right)
\left(1+\frac{6.8}{192.6}\delta_{\langle qq\rangle}\right)
\left(1+\frac{1}{192.6}\delta_{\langle aGG\rangle}\right)
\mbox{MeV}.
\end{eqnarray}
As is obvious from (\ref{fB*}) and (\ref{fB}), the OPE
uncertainties cancel out, to a great extent, in the ratio, which,
consequently, can be predicted with a rather high accuracy:
\begin{eqnarray}
\label{fBvs/fBs}f_{B^*}/f_B=0.944\pm0.011_{\rm OPE}\pm0.018_{\rm syst}.
\end{eqnarray}
The main contribution to the OPE error in the ratio arises from
the gluon condensate, which enters with different~sign in the
pseudoscalar and the vector correlator (in detail:
$\pm0.01_{\langle aGG\rangle}\pm0.005_{m_b}\pm0.001_{\langle
qq\rangle}$). The total uncertainty~of~the ratio is dominated by
the \emph{systematic\/} uncertainties of the decay constants.
Figure~\ref{Plot:bootstrap} shows the distribution of the ratio as
obtained by a bootstrap analysis.
\subsection{\boldmath Decay constant of the $B^*_s$ meson}
For $B^*_s$, we choose the same Borel-parameter window as for
$B^*$ and again require that the deviation of the dual~mass from
the known $B_s^*$ mass does not exceed 10 MeV in the full $\tau$
window. Our findings for the $B^*_s$-meson decay constant may be
cast in the form
\begin{align}
\label{fbsstar}
&f_{B^*_s}^{\rm dual}(\mu=\overline{\mu},m_b,\langle\bar
ss\rangle,\langle aGG\rangle)=(213.6\pm6)
\left(1-\frac{13.2}{213.6}\delta_{m_b}\right)
\left(1+\frac{11.8}{213.6}\delta_{\langle ss\rangle}\right)
\left(1-\frac{1}{213.6}\delta_{\langle aGG\rangle}\right)
\mbox{MeV},
\end{align}
where $\overline{\mu}$ is defined in (\ref{fbsstarmu}) and
\vspace{-0.5cm}
\begin{eqnarray}
\delta_{\langle ss\rangle}=\frac{|\langle\bar
ss\rangle|^{1/3}-\mbox{0.248\;GeV}}{\mbox{0.033\;GeV}}.
\end{eqnarray}
Unfortunately, the sensitivity of $f_{B_s^*}$ to the choice of the
scale $\mu$ at which the vector correlator is evaluated turns out
to be rather pronounced. This dependence on the choice of $\mu$
may be parametrized by a series in powers of
$\log(\mu/\overline{\mu})$:
\begin{equation}
\label{fbsstarmu}
f_{B_s^*}^{\rm dual}(\mu)=213.6\;\mbox{MeV}
\left[1-0.12\log(\mu/\overline{\mu})+0.11\log^2(\mu/\overline{\mu})
+0.43\log^3(\mu/\overline{\mu})\right],
\qquad\overline{\mu}=3.86\;\mbox{GeV}.
\end{equation}
Averaging over the OPE parameters (using Gaussian distributions of
all OPE parameters except for $\mu$, for which~a~flat distribution
in the range $\mu=3$--5 GeV is assumed) yields
\begin{eqnarray}
\label{fbsstarfinal} f_{B_s^*}=(213.6\pm18.2_{\rm OPE}\pm6_{\rm
syst})\;\mbox{MeV},
\end{eqnarray}
with the following main contributions to the OPE error: 11.5 MeV
from the $s$-quark condensate and 14.1 MeV~from~$m_b$; an
uncertainty of 3.2 MeV arises from the $\mu$ dependence of
$f_{B^*_s}$.
For the pseudoscalar $B_s$ meson, our corresponding estimates read
\begin{eqnarray}
\label{fbs}
f_{B_s}^{\rm dual}(m_b,\langle\bar ss\rangle,\langle aGG\rangle)=
(225.6\pm3_{\rm syst})
\left(1-\frac{14.1}{225.6}\delta_{m_b}\right)
\left(1+\frac{11.5}{225.6}\delta_{\langle ss\rangle}\right)
\left(1+\frac{1}{225.6}\delta_{\langle aGG\rangle}\right)
\mbox{MeV}.
\end{eqnarray}
As seen in Fig.~\ref{Plot:fV_vs_mu}, the sensitivity of $f_{B_s}$
to the choice of $\mu$ is negligible. The total OPE uncertainty of
$f_{B_s}$ is rather~large:
\begin{eqnarray}
\label{fbsfinal}f_{B_s}=(225.6\pm18.3_{\rm OPE}\pm3_{\rm
syst})\;\mbox{MeV}.
\end{eqnarray}
The decomposition of the OPE error reads: 11.5 MeV are due to the
error of $s$-quark condensate and 14.1 MeV due to the error of
$\overline{m}_b(\overline{m}_b)$, the uncertainties of the other
OPE parameters contribute at the level of 1 MeV.
Similar to the $f_{B^*}/f_B$ case, except for the gluon-condensate
contribution the OPE uncertainties cancel, to a great extent,~in
the ratio of the decay constants, which may thus be predicted
rather accurately:
\begin{eqnarray}
\label{bsstarbsfinal} f_{B_s^*}/f_{B_s}=0.947\pm0.023_{\rm
OPE}\pm0.020_{\rm syst}.
\end{eqnarray}
The OPE uncertainty in the ratio is dominated by the sensitivity
of $f_{B_s^*}$ to the choice of the scale $\mu$. The (obligatory)
bootstrap analysis gives for the ratio $f_{B_s^*}/f_{B_s}$ the
nearly Gaussian distribution shown in Fig.~\ref{Plot:bootstrap}.
\section{Summary and conclusions}
Exploiting the tools offered by QCD sum rules, we analyzed in
great detail the decay constants of the beauty vector mesons,
paying special attention to the uncertainties arising in our
predictions for the decay constants: the OPE~error, related to the
precision with which the QCD parameters are known, and the
systematic error, intrinsic to the QCD sum-rule approach as a
whole, reflecting the limited accuracy of the extraction
procedure. Our findings are as follows:
\begin{itemize}
\item[(i)]
As was already noted in the case of heavy pseudoscalar mesons
\cite{lms_fD}, also for the vector correlator the perturbative
expansion in terms of the heavy-quark pole-mass does not exhibit
good convergence. Reorganizing the OPE in terms of the
corresponding running mass allows us to choose a range of scales
for which, upon evaluation of the correlator, the perturbative
hierarchy becomes explicit. For scales $\mu\le2.5$--3 GeV, also
the running-mass OPE does not exhibit any hierarchy of
perturbative contributions; at too large scales $\mu\gtrsim5$--6
GeV, we observe a strong cancellation between the large positive
zero-order and the large negative first-order contributions,~thus
signalling that the accuracy of the OPE may deteriorate. There is,
however, a sizeable interval of scales, $3\le\mu\;({\rm
GeV})\le5$, where the $O(a^2)$-truncated OPE provides a good
description of the dual correlation function.
\item[(ii)]
Requiring the known value of the meson mass to be well reproduced
in a relatively broad $\tau$ window leads, in~the case of the
vector mesons, to some correlation between the scale $\mu$ at
which the correlator is evaluated and the upper boundary of the
$\tau$ window: for $\mu\gtrsim5$ GeV, the Borel window for the
vector correlator shrinks and thus no meaningful extraction of the
decay constants of $B^*$ and $B_s^*$ from sum rules is possible.
The observed correlation between the parameters of the Borel
window and the value of $\mu$ strongly reduces the (unphysical)
$\mu$ dependence of the extracted beauty-meson decay constants.
\item[(iii)]
The $\tau$-dependence of the effective threshold and the details
of the algorithm for fixing this quantity are crucial for
obtaining realistic estimates of the systematic uncertainty of the
extracted decay constant. For the analysis~of the ratios of the
decay constants of vector to pseudoscalar beauty mesons, where the
mass splitting between~the vector and the pseudoscalar partners
amounts to some $45$ MeV only, the stringent requirement to
reproduce~this splitting and the individual masses of vector and
pseudoscalar beauty mesons with an accuracy not worse than 5 MeV
in the full $\tau$ range is crucial for obtaining the low
systematic uncertainty of the extracted decay constants.
\item[(iv)]
The decay constants of pseudoscalar and vector beauty mesons
exhibit a strong dependence on the precise value of
$\overline{m}_b(\overline{m}_b)$. Therefore, the $B_{(s)}$ and
$B^*_{(s)}$ decay constants suffer from large OPE uncertainties.
The~systematic uncertainties of the extracted decay constants are
of the level of a few MeV and remain under good control.
\item[(v)]
The ratios $f_{B^*}/f_B$ and $f_{B_s^*}/f_{B_s}$ can be predicted
with very good accuracy because of large cancellations between the
OPE uncertainties in the ratios and a good control over the
systematic uncertainties of the decay constants. Our final results
read
\begin{eqnarray*}
f_{B^*}/f_{B}=0.944\pm0.021,\qquad
f_{B_s^*}/f_{B_s}=0.947\pm0.030,
\end{eqnarray*}
where the error given is the total uncertainty, including the
systematic and the OPE uncertainty. The resulting distributions
are close to normal distributions (Fig.~\ref{Plot:bootstrap}),
thus the quoted errors are Gaussian standard deviations.
\item[(vi)]
Our results are in excellent agreement with and have a precision
comparable to the recent lattice QCD values~\cite{hpqcd2015}
\begin{eqnarray*}
f_{B^*}/f_{B}=0.941\pm0.026,\qquad
f_{B_s^*}/f_{B_s}=0.953\pm0.023.
\end{eqnarray*}
\end{itemize}
\vspace{3ex}\noindent{\bf Acknowledgements.} The authors thank
B.~Blossier, O.~P\`ene, H.~Sazdjian, and B.~Stech for interesting
discussions. D.~M.~is grateful to the Alexander von
Humboldt-Stiftung (Germany) for supporting his stay in Heidelberg,
where part of this work was done. S.~S.~thanks MIUR (Italy) for
partial support under contract No.~PRIN 2010--2011.
|
{
"timestamp": "2015-04-14T02:09:42",
"yymm": "1504",
"arxiv_id": "1504.03017",
"language": "en",
"url": "https://arxiv.org/abs/1504.03017"
}
|
\section{Introduction}
Blazars, the subclass of active galactic nuclei (AGN) showing jets almost aligned with the observer's line of sight \citep{BR78,UP95}, offer an invaluable laboratory of physics. Their spectral energy distribution (SED) shows a characteristic double-hump shape and is usually well modeled as due to synchrotron and inverse Compton radiation \citep{Gh98}. Relativistic Doppler boosting of the observed emission is likely involved in the large-amplitude variability observed essentially at all frequencies \citep[e.g.][]{Gh93}.
Although the interpretative scenario seems to be well established, many open problems are still present. The availability of continuously improving multi-wavelength (MW) data has revealed the need for more sophisticated approaches, with models assuming that the observed emission originates in multiple zones with typically independent physical parameters \citep[e.g.][]{Ale12}. The possibility of inhomogeneity in the emitting region, mimicked by multi-zone models, is definitely plausible. However, this immediately introduces a strong degeneracy in the already large parameter space, in turn requiring additional information to disentangle the various possible components in the observed emission.
The dominance of non-thermal emission processes (e.g. synchrotron radiation, etc.) in the blazar emission suggests that a wealth of information might come from polarimetric studies \citep[][to mention some of the most recent papers]{Lar13,Sor13,Sas14,Sor14,Zha14,Ito15}. In the optical, the detection of polarized emission was considered the smoking-gun signature for synchrotron emission from a non-thermal distribution of electrons\citep{AS80}. In general, the addition of polarimetric data to the modeling of blazar photometric/spectral information has widely shown its potential to derive information about, e.g., the magnetic field state \citep[e.g.][]{Lyu05,Mar14}, or to drive the modeling of different SED components \citep{Bar14}.
A relatively less explored regime is that of short timescale polarimetry \citep{Tom01a,Tom01b,And05,Sas08,Cha12,Ito13}. Short timescale photometry, on the contrary, is indeed a common practice in the field and has revealed to be a powerful diagnostic technique \citep{Mon06,Ran10,Dan13,Zha13,San14}, also in the very-high energy regime \citep[e.g.][]{Aha07,Alb07,Abd10,Fos13}.
In this paper we present and discuss well-sampled observations of two blazars: \object{BL\,Lacertae} (hereinafter \object{BL\,Lac}) and \object{PKS\,1424+240}. The observations were carried out with the optical polarimeter PAOLO\footnote{\url{http://www.tng.iac.es/instruments/lrs/paolo.html}} equipping the 3.6\,m INAF / Telescopio Nazionale Galileo (TNG) at the Canary Island of La Palma. The relatively large collective area of the TNG enabled us to explore time scales as short as several tens of seconds in both photometry and polarimetry.
The paper is organized as follows: observations are described in Sect.\,\ref{sec:data}. In Sect.\,\ref{sec:resdis} results of the analyses and a general discussion are presented, and conclusions are drawn in Sect.\,\ref{sec:conc}.
\section{Observations}
\label{sec:data}
PAOLO is an optical polarimeter integrated in the Naysmith focus instrument DOLORES\footnote{\url{http://www.tng.iac.es/instruments/lrs/}} at the TNG. The observations presented here were part of the commissioning and scientific activities of the instrument.
\object{BL\,Lac} is the prototype of the class of BL\,Lac objects, and is located at a redshift $z=0.069$ \citep{MH97}. The host is a fairly bright and massive elliptical galaxy \citep{Sca00,Hyv07}. Due to its relative proximity it is one of the most widely studied objects of the class. \object{PKS\,1424+240} is also a BL\,Lac object and its redshift is still uncertain. \citet{Fur13} report a lower limit at $z \gtrsim 0.6$, which can make it one of most luminous objects in its class. Its host galaxy was possibly detected by \citet{MR10} at typically a few percent of the nuclear emission, although \citet{Sca00} reported much fainter limits.
\object{BL\,Lac} was observed for about 8 hours during the night of 2012 September 1 -- 2. The observations consisted of short integrations of about 20-40\,s each with the $r$ filter, interrupted every $\sim 45$\,min to observe polarized and unpolarized polarimetric standard stars (\object{BD+28d4211}, \object{W2149+021}, \object{HD\,204827}) for a total of more than 300 data points. The data reduction is carried out following standard procedures and aperture photometry is performed using custom tools\footnote{\url{https://pypi.python.org/pypi/SRPAstro.FITS/}}. Photometric calibration was secured by comparison with isolated unsaturated stars in the field with magnitudes derived by the APASS catalogue\footnote{\url{http://www.aavso.org/apass}}. Photometric and polarimetric light curves are shown in Fig.\,\ref{fig:bllac}.
\begin{figure}
\centering
\resizebox{\hsize}{!}{\includegraphics{bllac.pdf}}
\caption{PAOLO observations of BL\,Lac. In the top panel we show the magnitude of the source (AB mag) not corrected for Galactic reddening and for the host galaxy brightness. In the middle panel we show the polarization degree and in the bottom panel the position angle. }
\label{fig:bllac}
\end{figure}
\object{PKS\,1424+240} was observed for about 5 hours during the night of 2014 June 1 -- 2. The observations consisted of short integrations of 1-2\,min each with the $r$ filter interrupted at the beginning and at the end of the sequence to observe an unpolarized polarimetric standard star (GD\,319) for a total of more than 100 data points. Reduction and calibration were carried out as for \object{BL\,Lac.} Photometric and polarimetric light curves are shown in Fig.\,\ref{fig:pks}.
\begin{figure}
\centering
\resizebox{\hsize}{!}{\includegraphics{pks.pdf}}
\caption{PAOLO observations of PKS\,1424+240. The top panel shows the magnitude of the source (AB mag) versus time not corrected for Galactic reddening and for the host galaxy brightness. The middle panel shows the polarization degree and the bottom panel the position angle versus time.}
\label{fig:pks}
\end{figure}
The removal of the few percent instrumental polarization typical of Nasmyth focus instruments \citep{Tin07,Wit11,Cov14} can be carried out rather efficiently and with PAOLO we can estimate \citep{Cov14} a residual r.m.s. of the order of $\sim 0.2$\% or better. If the observations cover a limited range in hour angles the correction is generally more accurate. This is a systematic uncertainty superposed onto our observations and it is already included in the reported errors for our data. The results reported here supersede the preliminary ones shown in \citet{Cov14}.
Where required, $\chi^2$ minimization is performed by using the downhill (Nelder-Mead) simplex algorithm as coded in the {\tt python}\footnote{\url{http://www.python.org}} {\tt scipy.optimize}\footnote{\url{http://www.scipy.org/SciPyPackages/Optimize}} library, v.\,0.14.0. The error search is carried out following \citet{Cas76}. Throughout this paper the reported uncertainties are at $1\sigma$.
Distances are computed assuming a $\Lambda$CDM-universe with $\Omega_\Lambda = 0.73, \Omega_{\rm m} = 0.27,$ and H$_0 = 71$\,km\,s$^{-1}$\,Mpc$^{-1}$ \citep{Kom11}. Magnitudes are in the AB system. Flux densities are computed following \citet{Fuk96}. The raw and reduced data discussed here are available from the authors upon request.
\section{Results and discussion}
\label{sec:resdis}
\object{BL\,Lac} and \object{PKS\,1424+240} are sources belonging to the same class and, during our observations, also showed a comparable brightness. This is already a remarkable finding since the latter is more than one order of magnitude farther away than the former. \object{PKS\,1424+240} is therefore intrinsically about 100 times more luminous in the optical than \object{BL\,Lac} in the considered period. The host galaxy of \object{BL\,Lac} was measured at $R \sim 15.5$ \citep{Sca00}, roughly 30\% of the source luminosity during our observations. The source showed intense short-term variability, as expected for a blazar which has previously been found to be strongly variable at any time-scale \citep{Rai13}. On the contrary, \object{PKS\,1424+240} was remarkably stable during the observations with slow (hours) variations at most at a few percent level. This behavior is rather unexpected although this source presented a less intense variability (at least compared to \object{BL\,Lac}) during long-term monitoring campaigns \citep[e.g.][]{Arc14,Ale14} and in particular close to our observation epoch\footnote{\url{http://users.utu.fi/kani/1m/PG\_1424+240.html}}.
\subsection{Analysis of flux variability}
\label{sec:var}
\begin{figure}
\centering
\resizebox{\hsize}{!}{\includegraphics{bllaclc.pdf}}
\caption{BL\,Lac light-curve after subtraction of the host galaxy contribution and correction for Galactic extinction. A few episodes of rapid variability are labelled (see Table\,\ref{tab:bllacrapvar}) and fits based on Eq.\,(\ref{exppeak}) are also shown (blue solid line).}
\label{fig:bllaclc}
\end{figure}
The rapid variability observed in \object{BL\,Lac}, although in most cases of rather low level in absolute terms ($\sim 5-10$\%), is characterized by a fair number of well sampled rise/decay phases \citep[see also][for a similar behavior in S5\,0716+714]{Mon06}. Following \citet{Dan13}, we modeled these episodes with a sum of exponentials after having converted the light-curves to flux densities. The rationale is based on the idea that the derived time-scales, $\tau$, can give constraints on the size of the emitting regions. In addition, the time-scales of the decay phases, if the emission is due to synchrotron radiation, can allow us to derive inferences about the cooling times of the accelerated electrons and, in turn, the magnetic fields.
The adopted empirical functional form \citep{Dan13} is:
\begin{equation}
\label{exppeak}
f_{\rm i}(t)=\frac{2F_{\rm i}}{exp\left( \frac{t_{\rm i}-t}{\tau_{\rm {r, i}}}\right) +exp\left( \frac{t-t_{\rm i}}{\tau_{\rm {d, i}}}\right) },
\end{equation}
where $ F_{\rm i} $ is the flare normalization, $ \tau_{\rm {r, i}} $ and $ \tau_{\rm {d, i}} $ are, respectively, the flux rise and decay time-scales, and $t_{\rm i}$ is the time of the pulse maximum. The inverse of Eq.\,\ref{exppeak}, $1/f_{\rm i}(t)$, is used when the light-curve shows a decay followed by a rise, and $t_{\rm i}$ corresponds in this case to the pulse minimum.
The dense sampling of our light-curve allowed us to derive four events with well constrained time-scales (Table\,\ref{tab:bllacrapvar} and Fig.\,\ref{fig:bllaclc}). In all cases the time-scales for rise or decay phases are approximately in the range 2-15\,min, considering the uncertainties.
Variability time-scales as short as a few minutes have already been singled out for BL\,Lac objects, mainly at high energies \citep[e.g.][]{Aha07,Alb07,Arl13} or X-rays \citep[e.g.][]{WW95}, where the flux variation is a large fraction of the total. In the optical the percentage amplitudes of flux variations are typically lower, possibly due to the superposition of several emission episodes with largely different time-scales \citep[e.g.][]{Cha11,Dan13,San14} originating from different emitting regions. Therefore, strictly speaking, constraints derived by the light-curve analysis hold only for a portion of the emitting region of the order of the ratio of the flux variability to the total flux.
\begin{table}
\caption{Parameters of rapid flares during our BL\,Lac monitoring. The epochs are relative to 00:00 UT on 2012 September 2. $F_i$ is the amplitude of the variability episode. $1\sigma$ errors are computed with two parameters of interest \citep{Cas76}.}
\label{tab:bllacrapvar}
\centering
\begin{tabular}{lrccl}
\hline \hline
Event & Epoch & $F_i$ & $\tau$ & notes \\
& (hours) & (mJy) & (min) & \\
\hline
A & $-2.7$ & $0.48_{-0.05}^{+0.11}$ & $3.3^{+1.2}_{-0.6}$ & decay \\
B & $-1.3$ & $0.27_{-0.15}^{+2.80}$ & $2.5^{+17.1}_{-1.3}$ & rise \\
C & $0.7$ & $0.13_{-0.04}^{+1.87}$ & $3.6^{+6.4}_{-1.9}$ & rise \\
D & $2.3$ & $0.06^{+1.94}_{-0.02}$ & $2.4^{+9.1}_{-1.5}$ & decay \\
\hline
\end{tabular}
\end{table}
The size of the emitting region can be constrained as:
\begin{equation}
R \lesssim \frac{\delta c \tau}{1+z},
\end{equation}
where $z$ is the source redshift, $c$ the speed of light, and $\delta$ is the relativistic Doppler factor of the emitting region. Assuming a reference time scale of $\sim 5$\,min we get $R \lesssim 3\times10^{-5} \times \frac{\delta}{10}\,{\rm pc} \sim 10^{14} \times \frac{\delta}{10}$\,cm. The rapid variability identified here amounts to only a few percent of the total emitted flux from \object{BL\,Lac}.
Under the hypothesis that the (variable) emission is due to synchrotron and Compton processes, the cooling time-scale can limit the time-scale of a decay phase as:
\begin{equation}
\label{eq:cool}
\tau_{\rm d} \gtrsim t_{\rm cool} = \frac{3m_{e}c(1+z)}{4\sigma_{\rm T} \delta u^{'}_{0}\gamma_{e}}\,{\rm s},
\end{equation}
where $m_{\rm e}$ is the electron mass, $\sigma_{\rm T}$ the Thomson cross-section, $u^{'}_{0}=u^{'}_{B}+u^{'}_{\rm rad}=(1+q)B^{'2}/8\pi$ the co-moving energy density of the magnetic field (determining the synchrotron cooling rate) plus the radiation field (determining the inverse-Compton cooling rate), $q= u^{'}_{\rm rad}/u^{'}_{B}$ the Compton dominance parameter, typically of order of unity for BL\,Lacs \citep{Tav10}, and $\gamma_{e}$ is the characteristic random Lorentz factor of electrons producing the emission.
The peak frequency of the synchrotron emission is at
\begin{equation}
\label{eq:syn}
\nu_{\rm syn}=\frac{0.274 \delta e \gamma_{e}^{2}B^{'}}{(1+z)m_{e}c}\,{\rm Hz},
\end{equation}
where $e$ is the electron charge.
Finally, substituting $\gamma_e$ in Eq(s).\,\ref{eq:cool} and \ref{eq:syn}, the co-moving magnetic field can be constrained as:
\begin{eqnarray}
B^{'} \gtrsim [\pi m_e c (1+z) e / \sigma_{\rm T}^{2}]^{1/3} \nu_{\rm syn}^{-1/3} t_{\rm cool}^{-2/3} \delta^{-1/3} \sim \\ \nonumber
\sim 4 \times 10^7 (1+z)^{1/3} \nu_{\rm syn}^{-1/3} t_{\rm cool}^{-2/3} \delta^{-1/3}\,{\rm G}.
\end{eqnarray}
\object{BL\,Lac} is an intensively monitored object. \citet{Rai13} reported on a comprehensive study of its long-term behavior, including the epoch of our observations. From that data set the position of the synchrotron peak frequency can be inferred to be close to $\nu_{\rm syn} \sim 5 \times 10^{14}$\,Hz, and therefore, again assuming a reference time scale for decay of $\sim 5$\,min, and considering that the cooling time should be shorter than this, we get $B' \gtrsim 6 \times (\frac{\delta}{10})^{-1/3}$\,G.
The SED of \object{BL\,Lac} and that of a number of sources of the same class were studied in \citet{Tav10} based on observations carried out in 2008. A single zone model allowed the authors to estimate an average magnetic field $B \sim 1.5$\,G, a Doppler factor $\delta \sim 15$, and radius of the emitting region $R \sim 7 \times 10^{-4}$\,pc. Compared to the results from our analysis, based however on observations carried out in 2012, the emitting region of \object{BL\,Lac} turns out to be, as expected, a small fraction of that responsible for the whole emission and the magnetic field is locally higher but still close to the one zone model inference.
A similar analysis for \object{PKS\,1424+240} is not possible due to the very low level of variability shown during our observations. A fit with a constant is indeed perfectly acceptable although during the first $\sim 30$\,min of observations the source was slightly brighter by $\sim 0.01-0.02$\,mag.
The length of our monitoring does not allow us to derive general conclusions, although the difference in the observed flux variability between the two objects is remarkable. \object{PKS\,1424+240} is actually at a higher redshift compared to \object{BL\,Lac} ($z \sim 0.6$ vs. $z = 0.069$). Time dilation will lead to a reduction of any intrinsic variability for the former source by a factor of about 1.5 with respect to the latter. In addition, based on the SEDs shown in \citet{Tav10} and \cite{Ale14}, the optical band is at a higher frequency than the synchrotron peak for \object{BL\,Lac}, and at a lower frequency (or close to) for \object{PKS\,1424+240}. As widely discussed in \citet{Kir98}, under the assumption that magnetic fields in the emitting region are constant, flux and spectral variability depend on the observed frequency. If electrons with a given energy, corresponding to photons at a given frequency, cool more slowly than they are accelerated, variability is smoothed out, as it might be the case for \object{PKS\,1424+240}. Variability is expected to be particularly important close to frequencies emitted by the highest-energy electrons, where both radiative cooling and acceleration have similar timescales. Different short-term variability behaviors for sources with the synchrotron peak at lower or higher frequencies than the observed band were indeed already singled out \citep{HW96,HW98,Rom02,Hov14}.
In the literature it is also customary to look for the total flux doubling/halving times \citep[e.g.][]{Sba11,Imp11,Fos13}. The small amplitude of the variability we observed does not allow us to derive strong constraints, since this would always require large extrapolation. However, the shortest time-scales we could detect are of the order of less than four hours for \object{BL\,Lac}, consistent with the values found in other blazars, which is consistent with the idea that the the whole emitting region is much larger than the regions responsible for the rapid variability.
A variability analysis can be carried out for the polarimetric light curves too. The results show variability timescales at the same level as the total flux curves, although with larger uncertainties. Rapid time variability on minute to hours time scales for the polarized flux was singled out in other blazars, as for instance AO\,0235+164 \citep{Hag08}, S5\,0716+714 \citep{Sas08}, CGRaBS\,J0211+1051 \citep{Cha12} or CTA\,102 \citep{Ito13}. Intranight variability for a set of radio-quiet and radio-loud AGN was studied by \citet{Vil09}.
\subsection{Polarimetry}
Blazar emission is known to be characterized by some degree of polarization that is often variable, both in intensity and direction, on various time-scales \citep[see][for a recent review about optical observation of BL\,Lacs]{Fal14}. Occasionally, some degree of correlation or anticorrelation between the total and polarized flux is observed \citep[e.g.][]{Hag08, Rai12,Sor13,Gau14}, while often no clear relation is singled out. The complexity of the observed behaviors likely implies that, even when a single zone modeling can satisfactorily describe the broad-band SEDs, more emission components are actually active. It was proposed \citep[e.g.][]{Bar10,Sak13} that a globally weakly polarized fraction of the optical flux is generated in a relatively stable jet component, while most of the shorter term variability, both in total and polarized flux, originates from the development and propagation of shocks in the jet.
\object{BL\,Lac} and \object{PKS\,1424+240} show rather different behaviors in the linear polarimetry as well. The degree of polarization of \object{BL\,Lac} starts at about 11\% and decreases slowly for a few hours to about 9\%; then, for the remaining three hours of our monitoring, it decreases more quickly to about 6\%. The position angle increases rather quickly after the first hour, from about $14^\circ$ to $23^\circ$; then it remains stable for a couple of hours and then increases again to about $30^\circ$. Superposed on these general trends there is considerable short-term variability above the observational errors. \object{PKS\,1424+240}, on the contrary, shows a fairly constant polarization degree at about 4\% and a position angle close to $127^\circ$, with some variability only at the beginning of our monitoring. These behaviors are in general agreement with the results reported by \citet{And05} studying intra-night polarization variability for a set of BL\,Lac objects.
The \object{PKS\,1424+240} jet was likely in a low activity state, although it was not in its historical minimum (see Sect.\,\ref{sec:resdis}). This is also confirmed by the publicly available information and data at other wavelengths, such as high-energy gamma rays provided by the {\it Fermi}/LAT Collaboration\footnote{{\tt http://fermisky.blogspot.it/2014\_06\_01\_archive.html}}, and soft X-rays available from the {\it Swift}/XRT monitoring program\footnote{{\tt http://www.swift.psu.edu/monitoring/source.php?source=PKS1424+240}}. \citet{Ale14} reported a higher polarization degree, $7-9$\%, in 2011, when the source was brighter than during our monitoring. Lower polarization degrees, $4.4-4.9$\%, were reported by \citet{Mea90} in 1988, when the source was instead fainter. The position angle was about $113-119^\circ$, similar to that observed during our monitoring. The latter is also consistent with the direction of the jet as measured by VLBA radio observations at 2\,cm (Lister et al. 2013). The kinematics of the most robust radio component showed a position angle of $141^{\circ}$ with a velocity vector direction of $108^{\circ}$, i.e. with a very small offset \citep[$33^{\circ}\pm20^{\circ}$,][]{Lis13}. VLBA observations in the framework of the MOJAVE Project\footnote{{\tt http://www.physics.purdue.edu/astro/MOJAVE/sourcepages/1424+240.shtml}} \citep{Lis09} showed a decreasing trend in the polarization degree from 5\% in 2011 to 2.8\% in 2013, with a roughly stable position angle ($126^{\circ}-154^{\circ}$), consistent with our results in the optical. In general, looking at historical data, the polarization degree of \object{PKS\,1424+240} seems to be almost constant ($\sim 4$\%) below a given optical flux (likely $\lesssim 9.0$\,mJy, based on the refereed studies). The optical position angle seems to be quite stable and aligned with the kinematic direction of the radio jet and with the radio polarization position angle. This behavior might suggest some kind of ``magnetic switch'' (i.e. a threshold effect) in the jet activity \citep[e.g.][]{PuCo90, Mei97, Mei99}.
Neglecting the short-term variability of \object{BL\,Lac}, the total rotation of the position angle, taking the minimum at approximately $\sim -2$\,hours (see Fig.\,\ref{fig:bllac}) and the value at the end of our monitoring, amounts to about $15^\circ$, i.e. $2 - 2.5^\circ$/hour ($45-60^\circ$/day). Rapid position angle rotations of this magnitude are not unusual for blazars in general, and for \object{BL\,Lac} specifically \citep[e.g.][]{All81,Sil93,Mar08}. The observation of relatively stable and long-lasting rotational trends (days to months) suggested that the polarized emission could be generated in a jet with helical magnetic fields or crossed by transverse shock waves, or in a rather stable jet with an additional linearly rotating component \citep{Rai13}.
\begin{figure*}
\centering
\resizebox{\hsize}{!}{\includegraphics{bllacpol.pdf}} \\
\resizebox{\hsize}{!}{\includegraphics{bllacpolqu.pdf}}
\caption{({\it upper left}) BL\,Lac (host galaxy subtracted) flux density vs. linear polarization \citep[host galaxy corrected, assuming unpolarized emission, e.g.][]{Cov03}. At least three different regimes are singled out: at early time the flux changes rapidly with a slowly varying and rather high polarization (brown, circles), then an intermediate phase with chaotic flux and polarization variations (green, stars), and finally a sharp decrease in polarization with almost constant flux (blue, squares). Times in the legend are in hours (see Fig.\,\ref{fig:bllac}). ({\it upper right}) BL\,Lac position angle vs. linear polarization. Same symbols of the upper left panel. The position angle tends to increase when the linear polarization decreases. The trend becomes very clear at the end of our observation. ({\it bottom}) Flux density vs. Stokes parameters $Q$ and $U$. Periods with approximately linear dependence between polarization and total flux are also singled out. Same symbols as in the upper left panel.}
\label{fig:bllacpol}
\end{figure*}
Our well-sampled monitoring observations allow us to disentangle different behaviors even during the relatively short-duration coverage of \object{BL\,Lac} (Fig.\,\ref{fig:bllacpol}, upper left plot). At the beginning of our monitoring period, we see a rapid flux decrease with polarization slowly decreasing. After that, the source enters a phase characterized by rapid small-scale variability both in the total and polarized flux. Finally, the flux begins to increase regularly by a small amount and the polarization decreases abruptly down to the lowest observed level. The relation between polarization and position angle (Fig.\,\ref{fig:bllacpol}, upper right plot) shows the already mentioned rotation of the position angle with the decrease of the linear polarization. However, again superposed on this general trend there is considerable variability \citep[see also][for a similar analysis]{Hag08}.
In \citet{Rai13} the long-term (years) flux light curve was modeled assuming the flux variation to be (mainly) due to Doppler factor variations with a nearly constant Lorentz factor, i.e. due to small line of sight angle variations. We applied the same technique for our rapid monitoring. Knowing the viewing angle required to model the flux variations, it is then possible to predict the expected polarization in different scenarios. In the case of helical magnetic fields, following \citet{Lyu05}, we can derive a polarized flux fraction at 9-10\%, roughly in agreement with our observations. However, a detailed agreement, explaining the short-time variability for both the total and polarized flux, is not possible. Alternatively, we may consider transverse shock wave models \citep{Hug85}, with which again rough agreement for the polarization degree is reached, but no detailed agreement is possible.
A geometric model for the flux variation is therefore unable to simultaneously interpret the total flux and polarization behavior at the time resolution discussed here.
As already introduced in Sect.\,\ref{sec:var}, a possible interpretation of both total and polarized flux curves can be derived if it is assumed that the observed emission is due to a constant (within the time-scale of our monitoring) component with some degree of polarization and one \citep[or many,][]{Bri85} rapidly varying emission component(s) with different polarization degree and position angle \citep[see also][]{Sas08,Sak13}. The idea is rather simple; using the first three Stokes parameters the observed polarization can be described as:
\begin{equation}
S = \left \{
\begin{aligned}
I_{\rm obs} & = & I_{\rm const} & + & I_{\rm var} \\
Q_{\rm obs} I_{\rm obs} & = & Q_{\rm const} I_{\rm const} & + & Q_{\rm var} I_{\rm var} \\
U_{\rm obs} I_{\rm obs} & = & U_{\rm const} I_{\rm const} & + & U_{\rm var} I_{\rm var}
\end{aligned}
\right .
\label{eq:stokes}
\end{equation}
where the suffixes "obs", "const" and "var" refer to the observed (total), constant and variable quantities. The redundancy in Eq.\,\ref{eq:stokes} can be reduced following various possible assumptions, often depending on the availability of multi-wavelength datasets or long-term monitoring \citep[see, e.g.][]{Hol84,Qui93,Bri96,Bar10}. \citet{Hag02} assumed, based on their long-term polarimetric monitoring, that the stable component in \object{BL\,Lac} could be characterized by $P \sim 9.2$\% and $\theta \sim 24^\circ$.
As discussed in \citet{Hag99} and \citet{Hag08}, if a linear relation between polarized and total flux is singled out, this can allow one to estimate the polarization degree and position angle of the variable component. A linear relation between the Stokes parameters and the total flux implies that polarization degree and position angle are essentially constant \citep{Hag99} and their values can be derived as the slopes of the linear relations. At the beginning of our monitoring we can identify a sufficiently long and well defined linear relation between the Stokes parameters $Q$ and $U$ and the total flux (see Fig\,\ref{fig:bllacpol}, bottom panel). As already mentioned, we find considerable variability superposed on the linear trend. Neglecting the shorter term variability, we can roughly estimate $P_{\rm var} \sim 22$\% and $\theta \sim 34^\circ$. The constant component turns out to be remarkably consistent with the one identified by \citet{Hag02} at a flux level $\sim 9.5$\,mJy.
At any rate, the increasing complexity singled out by long- and short-term monitoring requires new theoretical frameworks for a proper interpretation. \citet{Mar14}, for instance, proposed a scenario in which a turbulent plasma is flowing at relativistic speeds crossing a standing conical shock. In this model, total and polarized flux variations are due to a continuous noise process rather than by specific events such as explosive energy injection at the basis of the jet. The superposition of ordered and turbulent magnetic field components can easily explain random fluctuations superposed on a more stable pattern, without requiring a direct correlation between total and polarized flux. As discussed in \citet{Mar14}, simulations based on this scenario can also give continuous and relatively smooth position angle changes as observed during our monitoring of \object{BL\,Lac}.
\begin{table}
\caption{Parameters of interest for the model based on the scenario extensively discussed in \citet{Zha14} and \citet{Zha15}. Angles are in the co-moving frame.}
\label{tab:bllacsim}
\centering
\begin{tabular}{lc}
\hline \hline
Parameter & Value \\
\hline
Bulk Lorentz factor & 15 \\
Length of the disturbance ($L$) & $3.8\times10^{14}$\,cm \\
Radius of the disturbance ($A$) & $4.0\times10^{15}$\,cm \\
Orientation of the line of sight & $90^\circ$ \\
Helical magnetic field strength & 2.5\,G \\
Helical pitch angle & $47^\circ$ \\
Electron density & $4.5\times10^{2}$\,cm$^{-3}$ \\
\hline
\end{tabular}
\end{table}
\citet{Zha14} presented a detailed analysis of a shock-in-jet model assuming a helical magnetic field throughout the jet. They considered several different mechanisms for which a relativistic shock propagating through a jet may produce a synchrotron (and high-energy) flare. They find that, together with a correlation between synchrotron and synchrotron self-Compton flaring, substantial variability in the polarization degree and position angle, including large position angle rotations, are possible. This scenario assumes a cylindrical geometry for the emitting region moving along the jet, which is pervaded by a helical magnetic field and a turbulent component. On its trajectory, it encounters a flat stationary disturbance, which could be a shock. This shock region does not occupy the entire layer of the emitting region, but only a part of it. In the comoving frame of the emitting region, this shock will travel through the emitting region, and temporarily enhance the particle acceleration, resulting in a small flare. After the shock moves out, the particle distribution will revert to its initial condition due to cooling and escape.
The 3DPol (3D Multi-Zone Synchrotron Polarization) code presented in \citet{Zha14} and MCFP (Monte Carlo/Fokker-Planck) code presented in \citet{Che12} realizes the above model. As elaborated in \citet{Zha15}, since the shock is relatively weak and localized, the enhanced acceleration will lead to a small time-symmetric perturbation in the polarization signatures. Some of the key parameters for the model are reported in Table\,\ref{tab:bllacsim}, and the fits to the polarization degree and position angle light curves are shown in Fig. \ref{fig:bllacmod}. Near the end of the observation, the polarization degree experienced a sudden drop, while the position angle continued to evolve in a time-symmetric pattern. Therefore an increase in the turbulent contribution is necessary although, due to the lack of a multi-wavelength SED, it cannot be well constrained. The total flux, given the very low variability amplitude observed during our observations, was set at a constant level of about 12\,mJy (see Fig.\,\ref{fig:bllaclc}). Nevertheless, rapid polarimetry clearly reveals its diagnostic power, showing need of inhomogeneity and turbulence in the emitting region.
\begin{figure}
\centering
\resizebox{\hsize}{!}{\includegraphics{bllacmod.pdf}}
\caption{Fit of the \citet{Zha14,Zha15} model scenario (green solid line) to the \object{BL\,Lac} linear polarization \citep[host galaxy corrected, assuming unpolarized emission, e.g.][]{Cov03} and position angle data. The general behavior is fairly well described by the model, although for the polarization the addition of weakly constrained turbulence in the magnetic field is required.}
\label{fig:bllacmod}
\end{figure}
\section{Conclusions}
\label{sec:conc}
In this work we are presenting results from rapid time-resolved observations in the $r$ band for two blazars: \object{BL\,Lac} and \object{PKS\,1424+240}. The observations were carried out at the 3.6\,m TNG and allowed us to measure linear polarimetry and photometry almost continuously for several hours for both sources. In practice, long-term monitoring observations of relatively bright blazars can only be achieved with dedicated small-size telescopes; however the richness of information obtainable with a rather large facility as the TNG allows us to study regimes which were in the past only partially explored.
\object{BL\,Lac} and \object{PKS\,1424+240} show remarkably different variability levels, with the former characterized by intense variability at a few percent level, while the latter was almost constant for the whole duration of our observations. The shortest well constrained variability time scales for \object{BL\,Lac} are as short as a few minutes, allowing us to derive constraints on the physical size and magnetic fields of the source regions responsible for the variability.
The variability time-scales for the polarization of \object{BL\,Lac} are compatible with those derived for the total flux, while \object{PKS\,1424+240} shows an almost constant behavior also in the polarization. The position angle of \object{BL\,Lacertae} rotates quasi-monotonically during our observations, and an analysis of the total vs. polarized flux shows that different regimes are present even at the shortest time-scales.
Different recipes to interpret the polarimetric observations are considered. In general, with the simplest geometrical models, only the average level of polarization can be correctly predicted. More complex scenarios involving some turbulence in the magnetic fields are required, and promising results are derived by a numerical analysis carried out following the framework described in \citet{Zha14,Zha15}, which requires some symmetry in the emitting region, as shown by the time-symmetric position angle profile. The time-asymmetric polarization profile, and its decrease during the second part of the event, which is accompanied by a few small flares, can be described by adding some turbulent magnetic field structure to the model.
\begin{acknowledgements}
This work has been supported by ASI grant I/004/11/0. HZ is supported by the LANL/LDRD program and by DoE/Office of Fusion Energy Science through CMSO. Simulations were conducted on LANL's Institutional Computing machines. The work of MB is supported by the Department and Technology and the National Research Foundation of South Africa through the South African Research Chair Initiative (SARChI)\footnote{Any opinion, finding, and
conclusion or recommendation expressed in this material is that of the authors and the NRF does not accept any
liability in this regard.}. We also thank the anonymous referee for her/his competent comments that greatly enhanced the quality of the paper.
\end{acknowledgements}
|
{
"timestamp": "2015-04-14T02:09:44",
"yymm": "1504",
"arxiv_id": "1504.03020",
"language": "en",
"url": "https://arxiv.org/abs/1504.03020"
}
|
\section{Interim Pairwise (In)Stability} \label{appendix:insurance}
The following example shows that Theorem \ref{thm:interimpairwise} depends on our stochastic dominance notion of a blocking pair; if agents compare lotteries by computing expected utilities, then pairs of agents might benefit from circumventing a mechanism that always produces a stable match.
\begin{example}
There are three agents on each side. Men $m_2$ and $m_3$ are known to rank women in the order $w_1, w_2, w_3$; $m_1$ has this preference with probability $1-p$, and with probability $p$ ranks $w_3$ first. Symmetrically, women $w_2$ and $w_3$ are known to rank men in the order $m_1, m_2, m_3$; $w_1$ has this preference with probability $1-p$, and with probability $p$ ranks $m_3$ first.
\end{example}
For any realization, there is a unique stable match; note that when $m_1$ ranks $w_3$ first and $w_1$ ranks $m_1$ first and $m_3$ last, this match gives $w_2$ her least-preferred partner, $m_3$. Under a stable matching mechanism, both $m_2$ and $w_2$ get their first choice with probability $p(1-p)$, their second choice with probability $(1-p)^2+p^2$, and their third choice with probability $p(1-p)$. So long as their utility from their second choice is above their average utility from a lottery over their first and third choices, $m_2$ and $w_2$ prefer matching with one another to the outcome of the stable matching.
\section{Perfectly Correlated Preferences} \label{appendix:example}
Theorem \ref{thm:exante} demonstrates that a stable matching mechanism may be blocked ex-ante by a coalition when preferences are drawn independently and uniformly at random.
The following example considers an opposite extreme extreme, where one side has identical preferences ex-post. It demonstrates that even in this case, it may be possible for a coalition to profitably deviate ex-ante from a mechanism that always selects the unique stable matching.
In this appendix, we use the language of ``schools" and ``students," and assume that schools all rank students according to a common test.
\begin{example}
Each student has one of four possible preference profiles, drawn independently: \\
\centerline{$\begin{array}{l l r}
A , B , C & w.p. \,\, &(1-\delta)/2 \\
A , C , B & w.p. \,\, & \delta/2 \\
B , A , C & w.p. \,\, & (1-\delta)/2 \\
B , C , A & w.p. \,\, &\delta/2 \\ \\
\end{array}$}
Schools have aligned preferences ex-post. The possibilities are the following: \\
\centerline{$\begin{array}{l l r}
1 , 2 , 3 & w.p. \,\, & (1-\epsilon)/2 \\
1 , 3 , 2 & w.p. \,\, & \epsilon/2 \\
2 , 1 , 3 & w.p. \,\, & (1-\epsilon)/2 \\
2 , 3 , 1 & w.p. \,\, & \epsilon/2 \\ \\
\end{array}$}
\end{example}
{\onehalfspacing
If all agents participate in an assortative match, schools $A$ and $B$ get their first, second, and third choices with probabilities $(\frac{1}{2}, \frac{1-\delta}{2}, \frac{\delta}{2})$ respectively. Students $1$ and $2$ get their first, second, and third choices with probabilities $\left( \frac{3}{4},\frac{1}{4},0\right) - \frac{\epsilon}{8}\left( 2 - \delta, 2 - 5 \delta + 3 \delta^2, -4 + 6 \delta - 3 \delta^2 \right)$.
If only $(A,B,1,2)$ participate in an assortative match, then the associated match probabilities for schools $A$ and $B$ are $(\frac{1}{2}, \frac{1-\epsilon}{2},\frac{\epsilon}{2})$, and for students $1$ and $2$ are $\left( \frac{3}{4},\frac{1}{4},0\right) - \frac{\delta}{4} \left(0,1,-1 \right)$.
All four of $A, B, 1, 2$ prefer the latter option if $\epsilon < \delta < 2 \epsilon (1- \frac{3}{2} \delta + \frac{3}{4} \delta^2)$.
}
\subsection{Ex-post Stability}
\input{expost.tex}
\subsection{Interim Stability}
\input{interim.tex}
\subsection{Ex-ante Stability}
\input{exante.tex}
\section{Introduction} \vspace{-.1 in}
\input{intro.tex} \vspace{-.1 in}
\section{Related Work} \label{sec:related} \vspace{-.1 in}
\input{related} \vspace{-.1 in}
\section{Model and Notation} \vspace{-.1 in}
\input{model.tex} \vspace{-.1 in}
\section{Results} \vspace{-.1 in}
\input{results.tex} \vspace{-.1 in}
\section{Discussion} \vspace{-.1 in}
\input{discussion.tex}
\newpage
\bibliographystyle{apalike}
\begin{spacing}{0.8}
|
{
"timestamp": "2015-04-14T02:16:36",
"yymm": "1504",
"arxiv_id": "1504.03257",
"language": "en",
"url": "https://arxiv.org/abs/1504.03257"
}
|
\section{Introduction}
Nonlinear systems modeling inverse problems are typically ill-posed, in the sense that
their solutions do not depend continuously on the data and their data are
affected by noise \cite{dsk, kns,vogel_book}. In this work
we focus on the stable approximation of a solution
of these problems. Procedures in the classes of Levenberg-Marquardt and trust-region
methods are discussed, and a suitable version of
trust-region algorithm is shown to have regularizing properties both theoretically and numerically.
The underlying motivation for our study is twofold: most of the practical methods in literature
have been designed for well-posed systems, see e.g., \cite{cgt, nw}, and thus are
unsuited in the context of inverse problems; adaptation of existing procedures for handling ill-posed
problems, carried out in the seminal papers \cite{gpp,hanke, hanke1,k,vogel,wy}, deserves further theoretical and numerical insights.
Let
\begin{equation}\label{prob}
F(x)=y,
\end{equation}
with $F:\mathbb{R}^n\rightarrow \mathbb{R}^n$ continuously differentiable, be obtained from
the discretization of a problem modeling an inverse problem.
It is realistic to have noisy data $y^\delta$ at disposal, satisfying
\begin{equation}\label{yd}
||y-y^\delta||_2\le \delta,
\end{equation}
for some positive $\delta$.
Thus, in practice it is necessary to solve a problem of the form
\beqn{prob_noise}
F(x)=y^\delta,
\end{equation}
and, due to ill-posedeness, possible solutions
may be arbitrarily far from those of the original problem.
In \cite{hanke,hanke1}, Hanke supposed that
an initial guess, close enough to some solution $x^\dagger$ of \req{prob}, is available.
Then, he proposed a {\em regularizing} Levenberg-Marquardt procedure which is able to compute
a stable approximation $x_{k_*}^\delta$ to $x^\dagger$ or to some other solution of
the unperturbed problem (\ref{prob}) close to $x^\dagger$.
This task is achieved through a nonlinear stepsize control in the Levenberg-Marquardt procedure
and the discrepancy principle as the stopping criterion, so that
the iterative process is stopped at the iteration $k_*$ satisfying
\begin{equation}\label{discrepance}
\|y^\delta-F(x_{k_*}^\delta)\|_2\le \tau \delta <\|y^\delta-F(x_{k}^\delta)\|_2, \;\;\;0\le k< k_*,
\end{equation}
with $\tau>1$ appropriately chosen \cite{mor}.
Remarkably $x_{k_*}^\delta$ converges to a solution of (\ref{prob}) as $\delta$ tends to zero.
Further regularizing iterative methods have been proposed, including first-order methods and Newton-type methods.
Analogously to the Levenberg-Marquardt procedure proposed by Hanke, instead of promoting convergence to a solution of
(\ref{prob_noise}), they form approximations
of increasing accuracy to some solution of the unperturbed problem (\ref{prob})
until the discrepancy principle (\ref{discrepance}) is met. We refer to \cite{dsk, kns} for the description and analysis of such methods.
The above mentioned regularizing Levenberg-Marquardt method belongs to the unifying framework of
nonlinear stepsize control algorithms for unconstrained optimization developed by Toint \cite{t}
and including trust-region methods \cite{cgt}.
Therefore, elaborating on original ideas by Hanke, we introduce and analyze a
regularizing variant of the trust-region method.
The main feature of our variant is the rule for selecting the trust-region radius which guarantees two properties.
First, it guarantees the same regularizing properties as the method by Hanke.
Second, as for standard trust-region procedure, it enforces a monotonic decrease of
the value of the function
\begin{equation}\label{mq}
\Phi(x)=\frac{1}{2}\| y^\delta-F(x)\|_2^2,
\end{equation}
at the iterates $x_k^{\delta}$.
Convergence properties are enhanced with respect to
the regularizing Levenberg-Marquardt procedure in the following respects.
With exact data, if there exists an accumulation point of the iterates
which solves \req{prob}, then any accumulation point of the sequence solves \req{prob}.
With noisy data, the methods have the potential to satisfy the discrepancy
principle \req{discrepance}.
As for standard trust-region methods, these properties can be enhanced independently of the closeness of the initial guess
to a solution of (\ref{prob}).
Our contribution covers theoretical and practical aspects of the method proposed.
From a theoretical point of view, we propose the use of a trust-region radius
converging to zero as $\delta $ tends to zero. Trust-region methods with this distinguishing feature
have been proposed in several papers, see \cite{fan1, fan2,fan3, zw}, but none of such works
was either devised for ill-posed problems or applied to them; thus,
our study offers new insights on the potential of this choice for the trust-region radius.
Moreover, we have made a first attempt toward global convergent methods for
ill-posed problems; to our knowledge, this topic has been considered only
in a multilevel approach proposed by Kaltenbacher \cite{k}.
Finally, local convergence analysis has been carried out without the assumption (commonly made in literature) on
the boundness of the inverse of the Jacobian $J$ of $F$, since it may not be fulfilled in the situation of ill-posedeness.
Taking into account that the standard convergence analysis of trust-region methods always requires
the invertibility of $J$, our results represent a progress in the theoretical investigation of convergence.
Concerning numerical aspects,
we discuss an implementation of the regularizing trust-region method,
and test its ability to approximate a solution of \req{prob} in presence of noise. Comparison
with a standard trust-region scheme highlights the impact of the proposed trust-region radius choice on regularization.
The paper is organized as follows. In \S \ref{RLM} we describe the main features of
the regularizing Levenberg-Marquardt method proposed by Hanke.
In \S \ref{defTR} we introduce our regularizing version of the trust-region methods
and in \S \ref{local} we study the local convergence properties. A comparative numerical analysis
of all the procedures studied is done in \S \ref{exps}.
{\bf Notations.} We indicate the iterates of the procedures analyzed as $x_k^{\delta}$; if the data are exact, $x_k$
may be used in alternative to $x_k^{\delta}$.
By $x_0^\delta=x_0$ we denote an initial guess which may incorporate a-priori knowledge of an exact solution.
The symbol $\|\cdot\|$ indicates the Euclidean norm. The Jacobian matrix of $F$ is denoted as $J$.
\section{Regularizing Levenberg-Marquardt method for ill-posed problems}\label{RLM}
We describe the regularizing version of the
Levenberg-Marquardt method proposed in \cite{hanke}
for solving (\ref{prob_noise}),
and analyze some issues for its practical implementation.
At $k$-th iteration of the Levenberg-Marquardt,
given $x_k^{\delta}\in \mathbb{R}^n$ and $\lambda_k>0$, let
\begin{equation}\label{lm}
m_k^{\mathrm{LM}}(p) =\frac{1}{2}\|F(x_k^{\delta})-y^{\delta}+J(x_k^{\delta}) p\|^2+\frac{1}{2}\lambda_k \|p\|^2,
\end{equation}
be a quadratic model around $x_k^{\delta} $ for the function $\Phi$ in (\ref{mq}), see \cite{l,m}.
The step $p_k$ taken minimizes $m_k^{\mathrm{LM}}$, and $x_{k+1}^\delta=x_k^{\delta}+p_k$.
We observe that, if $p(\lambda)$ is the solution of
\begin{equation}\label{system_1}
(B_k+\lambda I)p(\lambda) =-g_k,
\end{equation}
with $B_k=J(x_k^{\delta})^TJ(x_k^{\delta})$ and $g_k=J(x_k^{\delta})^T(F(x_k^{\delta})-y^{\delta})$, then $p_k=p(\lambda_k)$.
If problem \req{prob_noise} is ill-posed,
and the scalars $\lambda_k$ are limited to promote convergence of procedure, see
\cite{m}, then the solution of \req{prob} may be significantly misinterpreted
\cite{g,kns, vogel_book}. The regularizing Levenberg-Marquardt method \cite{hanke}
attempts to approximate solutions of \req{prob} by choosing
$\lambda_k$ as the solution $\lambda_k^q$ of the nonlinear scalar equation
\begin{equation} \label{seculare_q}
\|F(x_k^{\delta})-y^{\delta} +J(x_k^{\delta}) p(\lambda)\|= q \|F(x_k^{\delta})-y^{\delta}\|,
\end{equation}
for some fixed $q\in (0, 1) $.
Under suitable assumptions discussed below, $\lambda_k^q$ is uniquely
determined from (\ref{seculare_q}).
As for (\ref{seculare_q}),
it is useful to establish relations between
$\lambda$, $\|p(\lambda)\|$ and $\|F(x_k^{\delta})-y^{\delta}+ J(x_k^{\delta}) p(\lambda) \|$.
\vskip 5pt
\begin{lemma}\label{Sigma_decomp} \cite[Lemma 4.2]{bcgmt}
Suppose $\|g_k\|\neq 0$ and let $p(\lambda)$ be the minimum norm solution of
\req{system_1} with $\lambda \geq 0$.
Suppose furthermore that $J(x_k^{\delta})$ is of rank $\ell$ and its singular-value
decomposition is given by $U_k \Sigma_k V_k^T$ where
$\Sigma_k$ is the diagonal matrix with entries $\varsigma_1,\ldots,\varsigma_{n}$ on the diagonal.
Then, denoting $r=(r_1,r_2,\dots,r_n)^T= U_k^T(F(x_k^{\delta})-y^{\delta})$, we have that
\begin{eqnarray}
& & \|p(\lambda)\|^2
= \sum_{i=1}^\ell \frac{\varsigma_i^2 r_i^2}{(\varsigma_i^2 + \lambda)^2} ,
\label{Sigmanorms1} \\
& & \|F(x_k^{\delta})-y^{\delta} +J(x_k^{\delta}) p(\lambda) \|^2
= \sum_{i=1}^\ell \frac{\lambda^2 r_i^2}{(\varsigma_i^2 + \lambda)^2}
+\sum_{i=\ell+1}^n r_i^2.\label{Sigmanorms2}
\end{eqnarray}
\end{lemma}
\vskip 5pt \noindent
Using this result, the solution of (\ref{seculare_q}) is characterized as follows.
\vskip 5pt
\begin{lemma}\label{sol_cond_q}
Suppose $\|g_k\|\neq 0$. Let $p(\lambda)$ be the minimum norm solution of
\req{system_1} with $\lambda \geq 0$, $\mathcal{R}(J(x_k^{\delta}))^\bot$ be
the orthogonal complement of the range $\mathcal{R}(J(x_k^{\delta}))$ of $J(x_k^{\delta})$,
and $P_k^\delta$ be the orthogonal projector onto $\mathcal{R}(J(x_k^{\delta}))^\bot$.
Then
\begin{description}
\item{(i)} Equation \req{seculare_q} is not solvable if $\|P_k^\delta (F(x_k^{\delta})-y^{\delta})\| > q\|F(x_k^{\delta})-y^{\delta}\|$.
\item{(ii)} If
\begin{equation}\label{gamma_libro}
\|F(x_k^{\delta})-y^{\delta}+J(x_k^{\delta})(x^\dagger-x_k^{\delta})\|\le \frac{q}{\theta_k}\|F(x_k^{\delta})-y^{\delta}\|,
\end{equation}
for some $\theta_k>1$, then equation \req{seculare_q} has a unique solution
$ \lambda_k^q $ such that
\begin{equation}\label{bound_lambda}
\lambda_{k}^{q}\in \left( 0, \frac{q}{1-q}\|B_k\| \right].
\end{equation}
\end{description}
\end{lemma}
\begin{proof}
$(i)$ Equation (\ref{Sigmanorms2}) implies
\begin{eqnarray*}
& & \lim_{\lambda \rightarrow 0} \|F(x_k^{\delta})-y^{\delta} +J(x_k^{\delta}) p(\lambda) \|=\|P_k^\delta (F(x_k^{\delta})-y^{\delta})\| , \\%\label{res_lim_0}\\
& & \lim_{\lambda \rightarrow \infty} \|F(x_k^{\delta})-y^{\delta} +J(x_k^{\delta}) p(\lambda) \|=\|F(x_k^{\delta})-y^{\delta}\|
\end{eqnarray*}
Thus, since $\|F(x_k^{\delta})-y^{\delta} +J(x_k^{\delta}) p(\lambda) \|$ is
monotonically increasing as a function of $\lambda$,
we conclude that \req{seculare_q} does not admit solution if
$\|P_k^\delta (F(x_k^{\delta})-y^{\delta})\| > q\|F(x_k^{\delta})-y^{\delta}\|$.
$(ii)$ Trivially $\|P_k^\delta (F(x_k^{\delta})-y^{\delta})\|\le \|F(x_k^{\delta})-y^{\delta}+J(x_k^{\delta})(x-x_k^{\delta})\| $, for any $x$.
Hence, for the monotonicity of $\|F(x_k^{\delta})-y^{\delta} +J(x_k^{\delta}) p(\lambda) \|$,
if \req{gamma_libro} holds, then equation \req{seculare_q} admits a solution which
is positive and unique. Finally,
$$
F(x_k^{\delta})-y^{\delta} +J(x_k^{\delta}) p(\lambda)=\lambda(J(x_k^{\delta})\Jdk^T+\lambda I)^{-1}(F(x_k^{\delta})-y^{\delta}),
$$
see e.g., \cite[Proposition 2.1]{hanke}, and by \req{seculare_q}
\begin{eqnarray*}
q \|F(x_k^{\delta})-y^{\delta}\
&=& \lambda_{k}^{q}\|(J(x_k^{\delta})\Jdk^T+\lambda_{k}^{q} I)^{-1}(F(x_k^{\delta})-y^{\delta})\|\\
&\ge& \frac{\lambda_{k}^{q} }{\|J(x_k^{\delta})\|^2+\lambda_{k}^{q}} \|F(x_k^{\delta})-y^{\delta}\|,
\end{eqnarray*}
which yields \req{bound_lambda}.
\end{proof}
\vskip 5pt
In \cite{hanke}, the analysis of resulting Levenberg-Marquardt method was made under
the subsequent assumptions on the solvability of the problem \req{prob},
the Taylor remainder of $F$, and the vicinity of
the initial guess $x_0$ to some
solution $x^\dagger$ of \req{prob}.
\begin{ipotesi}\label{ALM}
Given an initial guess $x_0$, there exist positive $\rho$ and $c$ such that
system (\ref{prob}) is solvable in $ B_{\rho} (x_0)$, and
\begin{equation} \label{condfond}
\quad \|F(x)-F(\tilde x)-J(x)(x-\tilde x)\|\le c \|x-\tilde x\|\, \|F(x)-F(\tilde x)\|, \;\;\; x,\tilde x \in B_{2\rho} (x_0).
\end{equation}
\end{ipotesi}
\begin{ipotesi} \label{A5} Let $x_0$, $c$ and $\rho$ as in Assumption \ref{ALM},
$x^\dagger$ be a solution of \req{prob} and $x_0$ satisfy
\begin{eqnarray}
\|x_0-x^\dagger\|&<& \min\left\{\frac{q}{c}, \rho\right \}, \hspace*{43pt}\mbox{if}\;\;\;\;\delta=0,\label{loc1}\\
\|x_0-x^\dagger\|& <& \min\left\{\frac{q\tau-1}{c(1+\tau)}, \rho \right\}, \;\;\;\;\mbox{if}\;\;\;\;\delta>0 \label{loc2}
\end{eqnarray}
where $\tau>1/q$.
\end{ipotesi}
\vskip5pt
From Assumption \ref{ALM} it follows that
inequality (\ref{gamma_libro}) is satisfied for any $x_k^\delta$ belonging to $ B_{2\rho} (x_0)$
and consequently there exists a solution to (\ref{seculare_q}),
see \cite[Theorems 2.2, 2.3]{hanke}.
Under Assumptions \ref{ALM} and \ref{A5}, the approximations $x_{k^*}^\delta$
generated by the Levenberg-Marquardt method satisfy (\ref{discrepance}) and converge
to a solution of (\ref{prob}) as $\delta$ tends to zero.
\vskip 5pt
\begin{theorem}\label{conv_lm}
Let Assumptions \ref{ALM} and \ref{A5} hold and $ x_k^{\delta} $ be the
Levenberg-Marquardt iterates determined by using \req{seculare_q}.
For noisy data, suppose
$k<k_*$ where $k_*$ is defined in \req{discrepance}.
Then, any iterate $x_k^\delta$ belongs to $B_{2\rho}(x_0)$.
With exact data, the sequence $\{x_k\}$ converges to a solution of \req{prob}.
With noisy data, the stopping criterion \req{discrepance} is satisfied after a finite number $k_*$ of iterations
and $\{x_{k^*}^\delta\}$ converges to a solution of \req{prob} as $\delta$ goes to zero.
\end{theorem}
\begin{proof}
See \cite{hanke}, Theorem 2.2 and Theorem 2.3.
\end{proof}
\vskip 5pt
Let us focus on a specific issue concerning the implementation of the method which,
to our knowledge, has not been addressed either in \cite{hanke} or in related papers.
The numerical solution of (\ref{seculare_q}) requires the application of a root-finder method
and Newton method is the most efficient procedure, though in general it requires
the knowledge of an accurate approximation to the solution. On the other hand,
nonlinear equations which are monotone and convex (or concave)
on some interval containing the root are particularly suited to an application of Newton method,
see e.g. \cite[Theorem 4.8]{henrici}. Equation (\ref{seculare_q}) does not have such properties
but it can be replaced by an equivalent equation with strictly decreasing and concave function in $[\lambda_k^q, \infty)$;
thus, Newton method applied to the reformulated equation converges globally to
$\lambda_k^q$ whenever the initial guess overestimates such a root.
\vskip 5pt
\begin{lemma}\label{sec_cond_q}
Suppose $\|F(x_k^{\delta})-y^{\delta}\|\neq 0$, and that (\ref{seculare_q}) has positive solution $\lambda_k^q$. Let
\begin{equation}\label{sec_new}
\psi(\lambda)=\frac{\lambda}{\|F(x_k^{\delta})-y^{\delta} +J(x_k^{\delta}) p(\lambda) \|}-\frac{\lambda}{q\|F(x_k^{\delta})-y^{\delta}\|} =0.
\end{equation}
Then, Newton method applied to \req{sec_new}
converges monotonically and globally to the root $\lambda_k^q$ of (\ref{seculare_q})
for any initial guess in the interval $[\lambda_k^q, \infty)$.
\end{lemma}
\begin{proof}
Trivially, solving (\ref{seculare_q}) is equivalent to finding the positive root of the equation
(\ref{sec_new}). We now show that $\psi(\lambda)$ is strictly decreasing in $[\lambda_k^q, \infty)$ and concave on $(0, \infty)$. By (\ref{Sigmanorms2}),
\begin{equation}\label{rapl}
\frac{\lambda}{\|F(x_k^{\delta})-y^{\delta} +J(x_k^{\delta}) p(\lambda) \|}= \left (\, \sqrt{\sum_{i=1}^l \left(\frac{r_i}{\zeta_i^2+\lambda}\right)^2+
\sum_{i=l+1}^n \left(\frac{r_i}{\lambda}\right)^2} \, \right)^{-1},
\end{equation}
and this function is concave on $(0, \infty)$, cfr. \cite[Lemma 2.1]{cgt_ima}. Thus, $\psi$ is concave on $(0, \infty)$
and trivially $\psi'(\lambda)$ is strictly decreasing.
Now we show that $\psi'(\lambda_k^q)$ is negative; thus, using the monotonicity of $\psi'(\lambda)$, we
get that $\psi(\lambda)$ is strictly decreasing in $[\lambda_k^q, \infty)$. Differentiation of $\psi(\lambda)$ and
(\ref{seculare_q}) give
\begin{eqnarray*}
\psi'(\lambda_k^q) &=& \frac{(\lambda_k^q)^3}{\|F(x_k^{\delta})-y^{\delta} +J(x_k^{\delta}) p(\lambda_k^q) \|^3}
\left(\sum_{i=1}^l \frac{r_i^2}{(\zeta_i^2+\lambda_k^q)^3}+ \sum_{i=l+1}^n \frac{r_i^2}{(\lambda_k^q)^3} \right)
\, -\, \frac{1}{q\|F(x_k^{\delta})-y^{\delta}\|}
\\ &=& \frac{(\lambda_k^q)^2}{\|F(x_k^{\delta})-y^{\delta} +J(x_k^{\delta}) p(\lambda_k^q) \|^3}
\left (\, \sum_{i=1}^l \frac{r_i^2\lambda_k^q}{(\zeta_i^2+\lambda_k^q)^3}+ \sum_{i=l+1}^n \left(\frac{r_i}{\lambda_k^q} \right)^2
-\frac{\|F(x_k^{\delta})-y^{\delta} +J(x_k^{\delta}) p(\lambda_k^q) \|^2}{(\lambda_k^q)^2}\, \right).
\end{eqnarray*}
Moreover, using (\ref{rapl}), it holds
\begin{eqnarray*}
\psi'(\lambda_k^q) &=& \frac{(\lambda_k^q)^2}{\|F(x_k^{\delta})-y^{\delta} +J(x_k^{\delta}) p(\lambda_k^q) \|^3}
\left (\, \sum_{i=1}^l \frac{r_i^2\lambda_k^q}{(\zeta_i^2+\lambda_k^q)^3
- \sum_{i=1}^l \left(\frac{r_i}{\zeta_i^2+\lambda_k^q}\right)^2 \, \right) \\
&=& -
\frac{(\lambda_k^q)^2}{\|F(x_k^{\delta})-y^{\delta} +J(x_k^{\delta}) p(\lambda_k^q) \|^3}
\sum_{i=1}^l \frac{r_i^2\zeta_i^2}{(\zeta_i^2+\lambda_k^q)^3}, \\
\end{eqnarray*}
i.e. $\psi'(\lambda_k^q)$ is negative.
The claimed convergence of Newton method
follows from results on univariate concave functions given in \cite[Theorem 4.8]{henrici}.
\end{proof}
\vskip 5pt
For the practical evaluation of $\psi(\lambda)$ and $\psi'(\lambda)$ we refer to \cite{cgt,more}.
Since (\ref{seculare_q}) may have not solution,
Hanke observed that it may be replaced by
\begin{equation}\label{RQ}
\|F(x_k^{\delta})-y^\delta +J(x_k^{\delta}) p_k\| \ge q \|F(x_k^{\delta})-y^{\delta}\|,
\end{equation}
later denoted as the {\em q-condition}, \cite[Remark p. 6]{hanke} but
this criterion was not analyzed or employed in numerical computation.
Since our aim is to tune $\lambda_k$ in view of global convergence, while
preserving regularizing properties, in the next section we allow more flexibility in
its selection and design a trust-region method based on condition (\ref{RQ}).
\section{A regularizing trust-region method}\label{defTR}
Trust-region methods are globally convergent approaches where
the stepsize between two successive iterates is determined via a
nonlinear control mechanism \cite{cgt}.
At a generic iteration $k$ of a trust-region method, the step
$p_k$ solves
\begin{equation}\label{TR}
\begin{array}{l}
\displaystyle \min_p m_k^{\mathrm{TR}}(p)=\frac{1}{2}\|F(x_k^{\delta})-y^{\delta}+J(x_k^{\delta}) p\|^2,\\
\mbox { s.t. } \|p\|\le \Delta_k,
\end{array}
\end{equation}
where $\Delta_k$ is a given positive trust-region radius.
If $\|g_k\|\neq 0$ then $p_k$ solves \req{TR} if and only if
it satisfies \req{system_1} for some nonnegative $\lambda_k$ such that
\begin{equation}\label{sol_tr}
\lambda_k(\|p_k\|-\Delta_k)=0.
\end{equation}
Therefore, whenever the minimum norm solution $p^+$ of
$$
B_kp=-g_k,
$$
satisfies $\|p^+\|\le \Delta_k$, then the scalar
$\lambda_k$ is null and $p_k=p(0)$ solves \req{TR}.
Otherwise, the step $p_k=p(\lambda_k)$ is a Levenberg-Marquardt step.
If $\|p_k\|=\Delta_k$, then the trust-region is said to be active.
Starting from an arbitrary initial guess,
trust-region methods generate a sequence of iterates such
that the value of $\Phi$ in (\ref{mq}) is monotonically decreasing and
this feature is enforced by an adaptive choice of the radius $\Delta_k$.
Specifically, let $p_k$ be the trust-region step and
\begin{equation}\label{rho_tr}
\rho_k=\frac{ared(p_k)}{pred(p_k)},
\end{equation}
be the ratio between the achieved $ared(p_k)$ and predicted $pred(p_k$) reductions given by
\begin{eqnarray}
& & ared(p_k)= \Phi(x_k^{\delta})-\Phi(x_k^{\delta}+p_k), \label{ared}\\
& & pred(p_k)= \Phi(x_k^{\delta})-m_k^{\mathrm{TR}}(p_k) . \label{pred}
\end{eqnarray}
Then, the trust region radius is reduced if $\rho_k$ is below some small positive
threshold; otherwise it is left unchanged or enlarged \cite{cgt}.
Since trust-region steps and Levenberg-Marquardt steps have the same
form \req{system_1}, trust-region and Levenberg-Marquardt methods
fall into a single unifying framework which can be
used for their description and theoretical analysis
\cite{cgt_ima, more, t}.
Due to such a strict connection, we elaborate on the regularizing Levenberg-Marquardt
described in the previous section,
and introduce a regularizing variant
of trust-region methods for solving ill-posed problems.
The standard trust-region strategy is modified so that the nonlinear stepsize
control enforces both the monotonic reduction of $\Phi$ and the $q$-condition (\ref{RQ}).
To this end, we first characterize the parameters $\lambda$ such that $p(\lambda)$ satisfies (\ref{RQ}).
\vskip 5pt
\begin{lemma}\label{sol_q}
Assume $\|g_k\|\neq 0$. Let $p(\lambda)$ be the minimum norm solution of
\req{system_1} with $\lambda \geq 0$ and $P_k^\delta$ be the orthogonal projector onto $\mathcal{R}(J(x_k^{\delta}))^\bot$.
Then, equation \req{RQ} is satisfied for any $\lambda\ge 0$
whenever
\begin{equation}\label{cond_lambda}
\|P_k^\delta (F(x_k^{\delta})-y^{\delta})\| \ge q\|F(x_k^{\delta})-y^{\delta}\|.
\end{equation}
Otherwise, it
is satisfied for any $\lambda\ge \lambda_k^q$ where $\lambda_k^q$ satisfies \req{bound_lambda}.
\end{lemma}
\begin{proof}
The claims easily follow from Lemma \ref{sol_cond_q}.
\end{proof}
\vskip 5pt
Now we are ready to characterize the size of the trust-region radius guaranteeing (\ref{RQ}).
\vskip 5pt
\begin{lemma}\label{delta_qcond}
Let $p_k$ solve the trust-region problem (\ref{TR}). If
\begin{equation}\label{radius}
\Delta_k\le \frac{1-q}{\|B_k\|}\|g_k\|,
\end{equation}
then $p_k$ satisfies the $q$-condition (\ref{RQ}).
\end{lemma}
\begin{proof}
By Lemma \ref{sol_q} we know that
the $q$-condition is satisfied either for $\lambda\ge 0$, or
for any $\lambda\ge \lambda_k^q$.
In the former case, the claim trivially holds. In the latter case, by
\req{system_1} it follows
$$
\|p(\lambda_k^q)\| \ge \frac{\|g_k\|}{\|B_k+\lambda_k^qI\|},
$$
and by \req{bound_lambda} it holds
$$
\|B_k+\lambda_k^qI\| \le \frac{\|B_k\|}{1-q}.
$$
By construction $\|p_k\|\le \Delta_k$, and if (\ref{radius}) holds then we obtain
$$
\|p_k\|=\|p(\lambda_k)\|\le \frac{1-q}{\|B_k\|}\|g_k\| \le \frac{\|g_k\|}{\|B_k+\lambda_k^qI\|} \le
\|p(\lambda_k^q)\|.
$$
Since $\|p(\lambda)\|$ is monotonically decreasing, it follows
$\lambda_k\ge\lambda_k^q$ and condition \req{RQ} is satisfied.
\end{proof}
\vskip 5pt
We stress that the bound \req{radius} provides a practical rule for choosing the trust-region radius that guarantees the satisfaction of
the q-condition \req{RQ}. Conversely, in papers \cite{wy} and \cite{zw}, where trust-region for ill-posed problems are studied,
such a condition is respectively assumed to be satisfied and explicitely enforced rejecting the step whenever it does not hold.
The result in Lemma \ref{delta_qcond} suggests the trust-region iteration described in Algorithm \ref{algoTR}.
We distinguish between the parameters
needed in the case of exact-data and the parameters required with noisy-data.
\algo{TRalgo}{$k$th iteration of the regularizing Trust-Region method for problem \req{prob_noise}}{
Given $x_k^{\delta}$, $\eta\in (0,1)$, $\gamma\in (0,1)$, $0<C_{\min}<C_{\max}$.
\vskip 1pt
Exact-data: given $\delta=0$, $q\in (0,1)$.
\vskip 1pt
Noisy-data: given $\delta>0$, $\tau>1$, $q>1/\tau$.
\\[1ex]
\begin{description}
\item[1.] Compute $B_k=J(x_k^{\delta})^TJ(x_k^{\delta})$ and $g_k=J(x_k^{\delta})^T(F(x_k^{\delta})-y^{\delta})$.
\item[2.] Choose $\displaystyle\Delta_k\in \left.\left [C_{\min}\|g_k\|,
\min\left\{ C_{\max}, \frac{1-q}{\|B_k\|} \right\}\|g_k\|\right]\right.$.
\item[3.] Repeat\\
3.1 Compute the solution $p_k$ of the trust-region problem (\ref{TR}).
\\
3.2 Compute $\rho_k$ given in \req{rho_tr}--\req{pred}.
\\
3.3 If $\rho_k< \eta$, then set $\Delta_{k}=\gamma \Delta_k$.\\
\hspace*{-25pt} Until $\rho_k\ge \eta$.
\item [4.] Set $\xk+1d=x_k^{\delta}+p_k$.
\end{description}
}\label{algoTR}
Due to well know properties of trust-region methods,
Algorithms \ref{algoTR} is well-defined, i.e. if $\|g_k\|\ne 0$
the step $p_k$ is found within a finite number of attempts, provided that the following Assumption is met \cite{cgt}.
\vskip 5pt
\begin{ipotesi}\label{A2}
There exists a positive constant $\kappa_J$ such that
$$
\|J(x)\|\le \kappa_J,
$$
for any $x$ belonging to the level set
${\cal L}=\{ x\in \mathbb{R}^n \, \mbox{ s.t. }\, \Phi(x)\le \Phi(x_0)\}$.
\end{ipotesi}
\vskip 5pt
Global convergence of the trust-region method is stated in the following theorem;
we refer to \cite[Theorem 11.9]{nw} for the proof.
\begin{theorem}
Suppose that Assumption \ref{A2} holds and $J$ is Lipschitz continuous on $\mathbb{R}^n$.
Then, the sequence $\{x_k^{\delta}\}$ generated by Algorithm \ref{algoTR} satisfies
\begin{equation}\label{conv_grad}
\lim_{k\rightarrow \infty} \nabla\Phi(x_k^{\delta})=\lim_{k\rightarrow \infty}\|J(x_k^{\delta})^T(F(x_k^{\delta})-y^{\delta})\|=0.
\end{equation}
\end{theorem}
We observe that assumption on Lipschitz continuity of $J$ is made also in the paper \cite{k}.
By construction, the sequence $ \{\|F(x_k^{\delta})-y^{\delta}\|\}$ is monotonically decreasing
and bounded below by zero; hence it is convergent.
Equation \req{conv_grad} implies that any accumulation point of the sequence $\{x_k^{\delta}\}$
is a stationary point of $\Phi$.
As for exact data, we conclude that if there exists an accumulation point of $\{x_k\}$ solving \req{prob},
then any accumulation point of the sequence solves \req{prob}.
In the case of noisy data, if the value of $\Phi$ at some accumulation point of $\{x_k^{\delta}\}$
is below the scalar $\tau \delta$, then there exists an iterate $x_{k _*}^\delta$
such that the discrepancy principle is met.
It remains to show the behaviour of the sequences generated
by Algorithm \ref{algoTR} when, for some $k$,
$x_k^{\delta}$ is sufficiently close to a solution $x^\dagger$ of \req{prob}.
For instance,
this occurs with exact data when
the accumulation points of $\{x_k\}$ solve \req{prob}
and $k$ is sufficiently large. In the next section we show that
the trust-region method described in Algorithm \ref{algoTR} shares
the same local regularizing properties as the regularizing Levenberg-Marquardt method.
\section{Local behaviour of the trust-region method}\label{local}
We analyze the local properties of the trust-region method
under the same assumptions made for the Levenberg-Marquardt method.
Hence, we suppose that there exists an iteration index $\bar k$ such that the iterate $x_{\bar k}^\delta$ satisfies the following assumptions
that are the counterpart of Assumptions \ref{ALM} and \ref{A5} for the Levenberg-Marquardt method.
\begin{ipotesi}\label{ATR}
Suppose that for some iteration index $\bar k$
there exist positive $\rho$ and $c$ such that
system (\ref{prob}) is solvable in $ B_{\rho} (x_{\bar k}^\delta)$, and
\begin{equation} \label{condfond_TR}
\quad \|F(x)-F(\tilde x)-J(x)(x-\tilde x)\|\le c \|x-\tilde x\|\, \|F(x)-F(\tilde x)\|, \;\;\; x,\tilde x \in B_{2\rho} (x_{\bar k}^\delta).
\end{equation}
Moreover, letting $x^\dagger$ be a solution of \req{prob}, and $\tau>1/q$ if the data are noisy,
suppose that $x_{\bar k}^\delta$ satisfies
\begin{eqnarray}
\|x_{\bar k}-x^\dagger\|&<& \min\left\{\frac{q}{c}, \rho\right \}, \hspace*{43pt}\mbox{if}\;\;\;\;\delta=0,\label{locTR1}\\
\|x_{\bar k}^\delta-x^\dagger\|& <& \min\left\{\frac{q\tau-1}{c(1+\tau)}, \rho \right\}, \;\;\;\;\mbox{if}\;\;\;\;\delta>0\label{locTR2}.
\end{eqnarray}
\end{ipotesi}
To our knowledge, except for paper \cite{wy,zw},
local convergence properties of trust-region strategies
have been analyzed under assumptions
which involve the inverse of $J$ and its upper bound in a neighbourhood of a solution,
and thus are stronger than \req{condfond_TR}.
The following theorems show the local behaviour of the regularizing trust-region method.
We prove that locally the trust-region is active, the iterates $x_k^\delta$ with $k>\bar k$ remain into the ball $B_{\rho} (x_{\bar k}^\delta)$ and the resulting algorithm is regularizing. We remark that in standard trust-region methods,
the trust-region becomes eventually inactive. On the other hand, regularization requires strictly positive scalars $\lambda_k$,
and consequently an active trust-region in all iterations.
First we focus on the noise-free case and we show that the error
$\| x_{k}- x^\dagger\|$ decreases in a monotonic way for $k\ge\bar k$, and the sequence
$\{x_{k}\}$ converges to a solution of (\ref{prob}).
\vskip 5pt
\begin{theorem}\label{lambdaTR_bound}
Suppose that Assumptions \ref{A2} and \ref{ATR} hold and $\delta=0$. Then, Algorithm \ref{algoTR} generates a sequence
$\{x_{k}\}$ such that, for $k>\bar k$,
\begin{description}
\item{(i)} the trust-region is active, i.e. $\lambda_k>0$ and
$x_k$ belongs to $B_{\rho}(x_{\bar k})$;
\item{(ii)} $\| x_{k+1}- x^\dagger\|<\| x_{k}- x^\dagger\|$;
\item{(iii)} there exists a constant $\bar \lambda > 0$ such that $ \lambda_k\le \bar \lambda$.
\end{description}
Moreover,
\begin{description}
\item{(iv)} the sequence $\{x_k\}$ converges to a solution of \req{prob}.
\end{description}
\end{theorem}
\begin{proof}
{\it (i)-(ii)} The scalar $\lambda_{\bar k}$ in Algorithm \ref{algoTR} is such that
$\lambda_{\bar k}\ge \lambda_{\bar k}^q$.
From Assumption \ref{ATR}, condition
\req{gamma_libro} is satisfied at
$k=\bar k$ with $\theta_{\bar k}= \displaystyle \frac{q}{c\|x_{\bar k}-x^\dagger\|}$,
and consequently by Lemma \ref{sol_cond_q} $\lambda_{\bar k}^q$ is strictly positive.
Hence, the trust-region is active.
Further, by a straightforward adaptation of \cite[Proposition 4.1]{kns}\footnote{cfr. equation (4.6) in \cite[Proposition 4.1]{kns}}
it follows
$$
\|x_{\bar k}-x^\dagger\|^2-\|x_{\bar k+1}-x^\dagger\|^2\ge \frac{2(\theta_{\bar k}-1)}{\theta_{\bar k} \lambda_{\bar k}}\|F(x_{\bar k})-y+J(x_{\bar k}) p_{\bar k}\|^2,
$$
and this implies
$\|x_{\bar k+1}-x^\dagger\|<\|x_{\bar k}-x^\dagger\|$ as (\ref{locTR1}) guarantees $\theta_{\bar k}>1$. Consequently $x_{\bar k+1}\in B_{\rho} (x_{\bar k})$.
Repeating the above arguments, by induction we can prove that condition \req{gamma_libro} holds for $k\ge \bar k$, $\lambda_k>0$, and the following inequality holds
\begin{equation}\label{monoton_1_new}
\qquad \
\|x_k-x^\dagger\|^2-\|x_{k+1}-x^\dagger\|^2\ge \frac{2(\theta_k-1)}{\theta_k \lambda_k}\|F(x_{k})-y+J(x_{ k}) p_k\|^2,
\end{equation}
with
$$\theta_k=\displaystyle\frac{q}{c\|x^\dagger-x_k\|}>1.$$
Thus, by induction, the sequence $\{\|x_k-x^\dagger\|\}_{k=\bar k}^\infty$ is monotonic decreasing and the sequence $\{\theta_k\}_{k=\bar k}^\infty$ is monotonic increasing.
{\it (iii)} Since the trust-region is active, by \req{system_1}
\begin{equation}\label{bound_supl}
\Delta_k=\|p_k\|=\|(B_k+\lambda_kI)^{-1}g_k\|\le \frac{\|g_k\|}{\lambda_k}.
\end{equation}
Thus our claim follows if $\Delta_k/\|g_k\|$ is larger than a suitable threshold, independent from $k$.
Let us provide such a bound by estimating the value of $\Delta_k$
which guarantees condition $\rho_k\ge\eta$.
If this condition is fulfilled for the value of $\Delta_k$
fixed in Step 2 of Algorithm \ref{algoTR}, then $\Delta_k/\|g_k\|\ge C_{\min}$;
otherwise, the trust-region radius is progressively reduced, and we provide a bound
for the value of $\Delta_k$ at termination of Step 3 of Algorithm \ref{algoTR}
in the case where $\Phi(x_k+p_k)>m_k^{\mathrm{TR}}(p_k)$. This occurrence represents the
most adverse case; in fact if $\Phi(x_k+p_k)\lem_k^{\mathrm{TR}}(p_k)$
then $\rho_k\ge 1>\eta$ and the repeat loop terminate for a trust-readius greater than
or equal to the one estimated below.
Trivially,
$$
1-\rho_k = \frac{\Phi(x_k+p_k)-m_k^{\mathrm{TR}}(p_k)}{\Phi(x_k)-m_k^{\mathrm{TR}}(p_k)},
$$
and
\begin{eqnarray*}
\Phi(x_k+p_k)-m_k^{\mathrm{TR}}(p_k)&\le & \frac{1}{2}\|F(x_k+p_k)-F(x_k)-J(x_k) p_k\|^2 \\
&&+\|F(x_k+p_k)-F(x_k)-J(x_k) p_k\|\|F(x_k)-y+J(x_k) p_k\| .
\end{eqnarray*}
By \req{condfond_TR} and the mean value \cite[Theorem 11.1]{nw}, it holds
\begin{equation}\label{ineq_p2} \ \ \
\|F(x_k+p_k)-F(x_k)-J(x_k) p_k\|\le c \|p_k\| \|F(x_k+p_k)-F(x_k)\|\le c\kappa_J \|p_k\|^2.
\end{equation}
Consequently, as $\Delta_k\le C_{\max}\|g_k\|$,
$$
\Phi(x_k+p_k)-m_k^{\mathrm{TR}}(p_k) \le \frac{1}{2} c\kappa_J \Delta_k^2\|F(x_{0})-y\| (c\kappa_J^3C_{\max}^2\|F(x_{0})-y\|+2).
$$
Theorem 6.3.1 in \cite{cgt} shows that
$$
\Phi(x_k) -m_k^{\mathrm{TR}}(p_k)
\geq \frac{1}{2}\|g_k\|
\min \left\{ \Delta_k,\frac{\|g_k\|}{\|B_k\|} \right\}.
$$
Then,
$$
\Phi(x_k)-m_k^{\mathrm{TR}}(p_k)\geq \frac{1}{2}\Delta_k \|g_k\|,
$$
whenever $\Delta_k\le \displaystyle \frac{\|g_k\|}{\kappa_J^2}$
and this implies
$$
1-\rho_k\le \frac{c\kappa_J\Delta_k \|F(x_{0})-y\| (c \kappa_J^3C_{\max}^2\|F(x_{0})-y\|+2)}{ \|g_k\|}.
$$
Namely, termination of the repeat loop occurs with
$$
\Delta_k\le \|g_k\| \omega,\nonumber
$$
and
\begin{equation}\label{omega}
\omega=\min \left\{\frac{1}{\kappa_J^2},\frac{1-\eta}{c\kappa_J\|F(x_{0})-y\| (c \kappa_J^3C_{\max}^2\|F(x_{0})-y\|+2)} \right\}.
\end{equation}
Taking into account Step 2 and the updating rule at Step 3.3, we can conclude that,
at termination of Step 3, the trust-region radius $\Delta_k$ satisfies
$$
\Delta_k\ge \min\left \{C_{\min} , \, \gamma \omega
\right \}\|g_k\| .
$$
Finally, by \req{bound_supl} $\lambda_k\le \bar \lambda$ as
\begin{equation} \label{bound_lambda_1}
\lambda_k\le \frac{\|g_k\|}{\Delta_k} \le \max \left \{ \frac{1}{ \gamma \omega},\, \frac{1}{C_{\min}}
\right \}.
\end{equation}
{\it (iv)} Since both the function $(\theta-1)/\theta$ and the sequence $\{\theta_k\}_{k=\bar k}^\infty$
are monotonic increasing, it follows
$(\theta_k-1)/\theta_k>(\theta_{\bar k}-1)/\theta_{\bar k}$ and by \req{monoton_1_new}
\begin{equation}\label{monoton_3}
\|x_k-x^\dagger\|^2-\|x_{k+1}-x^\dagger\|^2\ge\frac{2(\theta_{\bar k}-1)}{\theta_{\bar k} \lambda_k}\|F(x_k)-y+J(x_k) p_k\|^2.
\end{equation}
Therefore, following the lines of the proof of Theorem 4.2 in \cite{kns},
we can conclude that $\{x_k\}$ is a Cauchy sequence, i.e., it is convergent.
Finally, by \req{monoton_3}, $\lambda_k\le \bar \lambda$
and \req{RQ}
$$
\|x_k-x^\dagger\|^2-\|x_{k+1}-x^\dagger\|^2\ge\frac{2(\theta_{\bar k}-1)q^2}{\theta_{\bar k} \bar \lambda} \|F(x_k)-y\|^2,
$$
Hence $ \|F(x_k)-y\|$ tends to zero and the limit of $x_k$ has to be a solution of (\ref{prob}).
\end{proof}
A similar result can be given for the noisy case. In the following theorem we prove that for $\bar k<k<k_*$, where
$k_*$ is defined in \req{discrepance}, the trust region is active and therefore $\lambda_k>0$.
Moreover, the stopping criterion
\req{discrepance} is satisfied after a finite number of iterations and the method is regularizing as the error decreases monotonically and
the sequence $\{x_{k_*}^\delta\}$ converges to a solution of \req{prob} whenever $\delta$ goes to zero.
\begin{theorem}\label{lambdaTR_bound_noise}
Suppose that Assumptions \ref{A2} and \ref{ATR} hold and $\delta>0$. Moreover,
suppose $\tau>1/q$ and $\bar k<k_*$, where $\tau$ and $k_*$ are defined in \req{discrepance}.
Then, Algorithm \ref{algoTR} generates a sequence $x_k^{\delta}$ such that, for $\bar k\le k<k_*$,
\begin{description}
\item{(i)} the trust-region is active, i.e. $\lambda_k>0$ and
$x_k^{\delta}$ belongs to $B_{\rho}(x_{\bar k}^\delta)$;
\item{(ii)} $\| x_{k+1}^\delta- x^\dagger\|<\| x_k^{\delta}- x^\dagger\|$;
\item{(iii)} there exists a constant $\bar \lambda > 0$ such that $ \lambda_k\le \bar \lambda$.
\end{description}
Moreover,
\begin{description}
\item{(iv)} the stopping criterion
\req{discrepance} is satisfied after a finite number $k_*$ of iterations and
the sequence $\{x_{k_*}^\delta\}$ converges to a solution of \req{prob} whenever $\delta$ goes to zero.
\end{description}
\end{theorem}
\begin{proof}
{\it (i)-(ii)} By \req{condfond_TR} and \req{yd} we get
\begin{eqnarray*}
\|y^\delta- F(x_{\bar k}^\delta)-J(x_{\bar k}^\delta)(x^\dagger-x_{\bar k}^\delta)\|&\le& \delta+
\|y -F(x_{\bar k}^\delta)-J(x_{\bar k}^\delta)(x^\dagger-x_{\bar k}^\delta)\| \\
& \le& \delta+c\|x^\dagger-x_{\bar k}^\delta\|\, \|y-F(x_{\bar k}^\delta)\| \\
&\le & (1+ c\|x^\dagger-x_{\bar k}^\delta\|)\delta+ c\|x^\dagger-x_{\bar k}^\delta\|\, \|y^\delta-F(x_{\bar k}^\delta)\| .
\end{eqnarray*}
Then, at iteration $\bar k$, condition \req{discrepance} gives
\begin{eqnarray*}
\|y^\delta- F(x_{\bar k}^\delta)-J(x_{\bar k}^\delta)(x^\dagger-x_{\bar k}^\delta)\|& \le &
\left( \frac{1+ c\|x^\dagger-x_{\bar k}^\delta\|}{\tau}+ c\|x^\dagger-x_{\bar k}^\delta\| \right) \|y^\delta-F(x_{\bar k}^\delta)\| ,
\end{eqnarray*}
and (\ref{locTR2}) yields \req{gamma_libro} with
$
\theta_k=\displaystyle \frac{q\tau}{1+c(1+\tau)\|x^\dagger-x_{\bar k}^\delta\|}>1.
$
Then, Lemma \ref{sol_cond_q} yields $\lambda_{\bar k}^q>0$ and therefore $\lambda_{\bar k}\ge \lambda_{\bar k}^q$ is strictly positive.
Further, by a straightforward adaptation of \cite[Proposition 4.1]{kns}, it follows
$$
\|x_{\bar k}^\delta-x^\dagger\|^2-\|x_{\bar k+1}^\delta-x^\dagger\|^2\ge \frac{2(\theta_{\bar k}-1)}{\theta_{\bar k} \lambda_{\bar k}}\|F(x_{\bar k}^\delta)-y^\delta+J(x_{\bar k}^\delta) p_{\bar k}\|^2,
$$
and this implies
$\|x_{\bar k+1}^\delta-x^\dagger\|<\|x_{\bar k}^\delta-x^\dagger\|$ and consequently $x_{\bar k+1}^\delta\in B_{\rho} (x_{\bar k}^\delta)$.
Repeating the above arguments, by induction, we can prove that, for $ \bar k<k<k_*$, condition \req{gamma_libro} holds,
$\lambda_k>0$, and
\begin{equation}\label{monoton_1_new_noise}\qquad \
\|x_k^\delta-x^\dagger\|^2-\|x_{k+1}^\delta-x^\dagger\|^2\ge \frac{2(\theta_k-1)}{\theta_k \lambda_k}\|F(x_{k}^\delta)-y^\delta+J(x_{ k}^\delta) p_k\|^2,
\end{equation}
with
$$\theta_k=\displaystyle \frac{q\tau}{1+c(1+\tau)\|x^\dagger-x_{ k}^\delta\|}.
$$
Thus $\|x_{k+1}^\delta-x^\dagger\|<\|x_k^\delta-x^\dagger\|$
and $\theta_{k+1}>\theta_k$ for $\bar k\le k<k_*$.
{\it ({iii})} Proceeding as in the proof of point {\it (iii)} of Theorem \ref{lambdaTR_bound}, just replacing $x_k$ with $x_k^\delta$, we get that for $ \bar k<k<k_*$,
$\lambda_k<\bar \lambda$ with
$$
\bar \lambda\le \max \left \{ \frac{1}{ \gamma \omega },\, \frac{1}{C_{\min}}
\right \}.
$$
where $ \omega$ is obtained replacing $y$ with $y^{\delta}$ in (\ref{omega}).
{\it (iv)} Summing up from $\bar k$ to $k_*-1$, by (\ref{RQ}) and (\ref{monoton_1_new_noise}) it follows
$$
(k_*-\bar k)\tau^2 \delta^2 \le \sum_{k=\bar k}^{k_*-1} \|F(x_k^{\delta})-y^{\delta}\|^2\le\frac{\theta_{\bar k} \bar \lambda} {2(\theta_{\bar k}-1)q^2}\|x_{\bar k}^\delta-x^{\dagger}\|^2.
$$
Thus, $k_*$ is finite for $\delta>0$, and convergence of $x^\delta_{k_*}$ to a solution of (\ref{prob}) as $\delta$ goes to 0 is shown in \cite[Theorem 2.3]{hanke}.
\end{proof}
\section{Numerical results} \label{exps}
In this section we report on the performance of the regularizing
trust-region method
and make comparisons with the regularizing Levenberg-Marquardt method and
a standard version of the trust-region method.
The test problems are ill-posed and with noisy data, and arise from the discretization of
nonlinear Fredholm integral equations of the first kind
\begin{equation}\label{int_eq}
\int _0^1 k(t,s,x(s))ds=y(t), \quad t\in [0,1].
\end{equation}
The integral equations considered model inverse problems from groundwater hydrology and geophysics.
Their kernel is of the form
\begin{equation}\label{kernel1}
k(t,s,x(s))=\mbox{log}\left(\frac{(t-s)^2+H^2}{(t-s)^2+(H-x(s))^2}\right),
\end{equation}
see \cite[\S 3]{vogel}, or
\begin{equation}\label{kernel2}
k(t,s,x(s))=\frac{1}{\sqrt{1+(t-s)^2+x(s)^2}},
\end{equation}
see \cite[\S 6]{k}.
The interval $[0,1]$ was discretized with $n=64$ equidistant grid
points $t_i=(i-1) h$, $h=1/(n-1)$, $i=1, \ldots, n$.
Function $x(s)$ was approximated from the $n$-dimensional subspace of $H_0^1(0,1)$
spanned by standard piecewise linear functions. Specifically, we let
$s_j=(j-1) h$, $h=1/(n-1)$, $j=1, \ldots, n$, and looked for an approximation
$\hat x(s)=\sum_{j=1}^n \hat x_j \phi_j(s)$ where
$$
\phi_1(s)=
\left\{
\begin{array}{ll}
\frac{s_2-s}{h} & \mbox{if} \quad s_1\le s\le s_2 \\
0 & \mbox{otherwise}
\end{array}
\right .,
\qquad
\phi_n(s)=
\left\{
\begin{array}{ll}
\frac{s-s_{n-1}}{h} & \mbox{if} \quad s_{n-1}\le s\le s_n \\
0 & \mbox{otherwise}
\end{array}
\right . ,
$$
and
$$
\phi_j(s)=
\left\{
\begin{array}{ll}
\frac{s-s_{j-1}}{h} & \mbox{if} \quad s_{j-1}\le s\le s_j,\\
\frac{s_{j+1}-s}{h} & \mbox{if} \quad s_{j}\le s\le s_{j+1},\\
0 & \mbox{otherwise}
\end{array}
\right . \qquad j=2,\ldots n-1.
$$
Finally, the integrals $\int _0^1 k(t_i,s,\hat x(s))ds$, $1\le i\le n$, were approximated
by the composite trapezoidal rule on the points $s_j$, $1\le j\le n$.
The resulting discrete problems are square nonlinear systems \req{prob} with unknown
$x=(\hat x_1, \ldots, \hat x_n)^T$. We observe that $\hat x(s_j)=\hat x_j$;
thus, the $j$-th component of $x$ approximates a solution of \req{int_eq} at $s_j$.
Two problems with kernel \req{kernel1} and two problems with kernel \req{kernel2} were considered
and built so that solutions (later denoted as true solutions) are known.
Concerning kernel \req{kernel1}, the first problem is given in \cite[p. 46]{vogel}; it
admits as true continuous solutions the functions
$x_{true}(s)=c_1e^{d_1(s+p_1)^2}+c_2e^{d_2(s-p_2)^2}+c_3+c_4$ and
$x_{true}(s)=2H-c_1e^{d_1(s+p_1)^2}-c_2e^{d_2(s-p_2)^2}-c_3-c_4$
where $H=0.2, c_1=-0.1, c_2=-0.075, d_1=-40, d_2=-60, p_1=0.4, p_2=0.67, c_3$ and $c_4$
are chosen such that $x_{true}(0)=x_{true}(1)=0$. The second problem was given in \cite[p. 835]{wy}
and it has true continuous solutions $x_{true}(s)=1.3s(1-s)+0.2$ and $x_{true}(s)=1.3s(s-1)$.
The third and fourth problems have kernel \req{kernel2};
the former has solutions $x_{true}(s)=1$ and $x_{true}(s)=-1,\;s\in [0,1]$,
see \cite[p. 660]{k}, while the latter has the discontinuous functions
\begin{equation}
x_{true}(s) = \begin{cases} 1 & \text{ if } 0 \leq s \leq \frac{1}{2} \\
0 & \text{ if } \frac{1}{2}< s \leq 1 \end{cases},\quad
x_{true}(s) = \begin{cases} - 1 & \text{ if } 0 \leq s \leq \frac{1}{2} \\
0 & \text{ if } \frac{1}{2}< s \leq 1 \end{cases}
\end{equation}
as the true solutions, \cite[p. 662]{k}.
The nonlinear systems arising from the discretizations of the four problems
are denoted as {\tt P1}, {\tt P2}, {\tt P3} and {\tt P4} respectively, while $x^\dagger\in \mathbb{R}^n$
denotes a solution of the discretized problems.
Given the error level $\delta$, the exact data $y$ was perturbed by
normally distributed values with mean $0$ and variance $\delta$ using
the {\sc Matlab} function {\tt randn}.
All procedures were implemented in {\sc Matlab} and run using {\sc Matlab 2014}b on an Intel Core(TM) i7-4510U
2.6 GHz, 8 GB RAM; the machine precision is $\epsilon_m\approx 2\cdot 10^{-16}$.
The Jacobian of the nonlinear function $F$ was computed by finite differences.
The parameter $q$ used in \req{seculare_q} and in \req{RQ}
was set equal to $1.1/\tau$ and the discrepancy principle \req{discrepance}
with $\tau=1.5$ was used as the stopping criterion. A maximum number of 300 iterations was allowed and a failure
was declared when this limit was exceeded.
In the implementation of the regularizing trust-region method,
Step 3 in Algorithm \ref{algoTR} was performed setting $\eta=\frac{1}{4}$, $\gamma= \frac{1}{6}$.
Then, in Step 2 the trust-region radius was updated
as follows
\begin{eqnarray}
& \Delta_0 =\mu_0\|F_0\|, & \ \mu_0=10^{-1},\label{tr_algo1}\\
& \Delta_{k+1} =\mu_{k+1}\vert\vert F(x_{k+1}^\delta)\vert\vert , \qquad & \ \mu_{k+1} = \begin{cases} \displaystyle \frac{1}{6}\mu_k & \text{ if } q_k<q, \label{tr_algo2}\\
2\mu_k & \text{ if } q_k>\nu q, \end{cases},
\end{eqnarray}
with $q_k=\displaystyle \frac{\vert\vert J(x_k^{\delta})p_k+F(x_k^{\delta})\vert\vert}{\vert\vert F(x_k^{\delta})\vert\vert }$, and $\nu=1.1$.
The maximum and minimum values for $\Delta_k$ were set to
$\Delta_{\max}=10^4$ and $\Delta_{\min}=10^{-12}$.
This updating strategy turned out to be efficient
in practice and was based on the following considerations.
Clearly, $\Delta_k$ is cheaper to compute than the upper bound in (\ref{radius}) and
preserves the property of converging to
zero as $\delta$ tends to zero and a solution of problem (\ref{prob_noise}) is approached. Further,
$\Delta_k$ is adjusted taking into account the $q$-condition and by monitoring the value $q_k$;
therefore, if the $q$-condition was not satisfied at the last computed iterate $x_k^{\delta}$,
it is reasonable to take a smaller
radius than in the case where the $q$-condition was fulfilled.
The computation of the parameter $\lambda_k$ was performed applying Newton method to the equation
\begin{equation}\label{sec_TR}
\psi(\lambda)=\frac{1}{\|p(\lambda)\|}-\frac{1}{\Delta_k}=0,
\end{equation}
and each Newton iteration requires the Cholesky factorization of a shifted
matrix of the form $B_k+\lambda I$ \cite{cgt}.
Typically high accuracy in the solution of
the above scalar equations is not needed \cite{bcgmt, cgt} and this
fact was experimentally verified also for our test problems.
Hence, after extensive numerical
experience, we decided to terminated the Newton process
as soon as $|\Delta_k-\|p(\lambda)\||\le 10^{-2} \Delta_k$.
In our implementation of the standard trust-region method, we chose
the trust-region radius accordingly to technicalities well-known in literatures, see
e.g. \cite[\S 6.1]{cgt} and \cite[\S 11.2]{nw}. In particular, we set $\Delta_0=1$,
$$
\Delta_{k+1}=\left \{
\begin{array}{lll}
&\displaystyle \frac{\|p_k\|}{4}, & \quad \mbox{ if } \rho_k<\frac{1}{4},\\
&\Delta_k, & \quad \mbox{ if } \frac{1}{4}\le \rho_k\le\frac{3}{4},\\
& \min\{2\Delta_k,\Delta_{\max}\},& \quad \mbox{otherwise},
\end{array}
\right.
$$
with $\Delta_{\max}=10^4$ and chose $\Delta_{\min}=10^{-12}$
as the minimum values for $\Delta_k$.
Finally the Levenberg-Marquardt approach was implemented
imposing condition (\ref{seculare_q}) and solving
\req{sec_new} by Newton method. In order to find an accurate solution for (\ref{seculare_q})
it was necessary to use a tighter tolerance, equal to $10^{-5}$, than that used in the trust-region algorithm.
Our experiments were made varying the noise level $\delta$ on the data $y^{\delta}$.
Tables \ref{table1} and \ref{table2} display the results obtained by the regularizing
trust-region algorithm with noise $\delta=10^{-4}$ and $\delta= 10^{-2}$ respectively.
Runs for four different initial guesses $ x_0 $ are reported in the tables.
For problems P1 and P2 the initial guesses are $x_0=0e,-0.5e,-e,-2e$ and $x_0=0e,0.5e,e,2e$ respectively, where
$e$ denotes the vector $e=(1,\ldots,1)^T$.
For problem P3 the initial guess was chosen as the vector $x_0(\alpha)$ with $j$-th component given
by $(x_0(\alpha))_j=g_{\alpha}(s_j)$ for $j=1,\dots,n$, where $g_\alpha(s)=(-4\alpha+4)s^2+(4\alpha-4)s+1$,
and $s_j$ being the grid points in $[0,1]$. We have used the following values of $\alpha$,
$\alpha=1.25,1.5,1.75,2$.
For problem P4 the initial guess $x_0(\beta, \chi)$ has components
$(x_0(\beta,\chi))_j=g_{\beta,\chi}(s_j)$ for $j=1,\dots,n$ with $g_{\beta,\chi}=\beta-\chi s$ and $(\beta,\chi)=(1,1),(0.5,0),(1.5,1),(1.5,0)$.
In the tables we report: the initial guesses (for increasing distance from the true solutions) the number of iterations
{\tt it} performed; the final norm of function $F$;
the number of function evaluations {\tt nf} performed; the rounded
average number {\tt cf} of Cholesky factorizations per iteration.
To assess the quality of the results obtained, we measured the distance
between the final iterate $x_{k^*}^\delta$ and the true solution approached; in particular
${\tt e_I}=\max_{2\le j\le n-1} |x_{true}(s_j)-(x_{k^*}^\delta)_j|$ is the maximum absolute value of the difference between the
components associated to internal points $s_j\in(0,1)$, while
${\tt e_T}=\max_{1\le j\le n} |x_{true}(s_j)-(x_{k^*}^\delta)_j|$ is the maximum absolute value of the difference between the
components associated to points $s_j$ including the end-points of the interval $[0,1]$.
The symbol $``*''$ indicates that either the procedure failed to satisfy the discrepancy principle within the
prescribed maximum number of iteration, or the final $x_{k^*}^\delta$ was not an approximation of one of the
true solutions described above.
\begin{center}
\begin{table
{\small
\begin{tabular}{lr|cccccc|ll}
\hline\hline
\mbox{ Problem} & & \multicolumn{6}{c|}{RTR} & \multicolumn{2}{c}{RLM} \\
& $x_0$ & {\tt it} & ${\tt \vert\vert F \vert\vert}$ & {\tt nf} & {\tt cf} & ${\tt e_I}$ & ${\tt e_T}$ & ${\tt e_I}$& ${\tt e_T}$\\ \hline\hline
{\tt P1} & $0\,e$ & \z43 & 1.3e$-$4 & \z44 & 5 & 5.5e$-3$ & 5.5e$-3$ & 4.5e$-3$ & 4.5e$-3$ \\
& $-0.5\, e$ & \phantom{0} 63 & 1.2e$-$4 & \z71 & 5 & 3.2e$-2$ & 7.9e$-2$& 3.0e$-2$ & 7.1e$-2$ \\
& $-1\, e$ & \phantom{0} 82 & 1.4e$-$4 & \z94 & 4 & 3.4e$-2$ & 8.4e$-2$ & 4.0e$-2$ & 7.2e$-2$\\
& $-2\, e$ & 115 & 1.5e$-$4 & 138 & 4 & 3.4e$-2$ & 8.6e$-2$ & 2.9e$-2$ & 6.1e$-2$\\
\hline
{\tt P2} &$0\, e$ & \z54 & 1.2e$-$4 & \z55 & 5 & 7.4e$-3$ & 7.4e$-3$ & * & * \\
& $0.5\,e$ & \z56 & 1.4e$-$4 & \z59 & 5 & 1.1e$-2$ & 1.3e$-2$ & * & * \\
&$1\, e$ & \z73 & 1.4e$-$4 & \z84 & 4 & 1.0e$-2$ & 1.3e$-2$ & 7.3e$-3$ & 8.3e$-3$ \\
&$2\, e$ & 118 & 1.4e$-$4 & 138 & 4 & 9.3e$-3$ & 1.1e$-2$ & 4.8e$-3$ & 4.8e$-3$ \\
\hline
\hline
{\tt P3}&$x_0(1.25)$ & \z35 & 1.4e$-$4 & \z36 & 3 & 1.2e$-2$ & 1.2e$-2$ & 3.1e$-3$ & 3.1e$-3$ \\
&$x_0(1.5) $ & \z43 & 1.4e$-$4 & \z44 & 3 & 5.1e$-2$ & 5.1e$-2$ & 6.2e$-2$ & 6.2e$-$2\\
&$x_0(1.75) $ & \z45 & 1.3e$-$4 & \z46 & 3 & 3.2e$-1$ & 3.2e$-1$ & 3.1e$-1$ & 3.1e$-$1 \\
& $x_0(2)$ & \z65 & 1.4e$-$4 & \phantom{0} 71 & 3 & 4.6e$-1$ & 4.6e$-1$ & 3.8e$-1$ & 3.8e$-1$ \\
\hline
{\tt P4}&$x_0(1,1)$ &\z68 & 1.5e$-$4 &\z82 & 3 & 4.8e$-1$ & 4.8e$-1$ & * & * \\
&$x_0(0.5,0)$ & \z64 & 1.5e$-$4 & \z75 & 3 & 4.9e$-1$ & 4.9e$-1$ & 4.7e$-1$ & 4.7e$-1$ \\
&$x_0(1.5,1)$ & \z69 & 1.5e$-$4 & \z78 & 3 & 5.1e$-1$ & 5.1e$-1$ & 4.8e$-1$ & 4.8e$-1$ \\
&$x_0(1.5,0)$ & \z68 & 1.5e$-$4 & \z78 & 4 & 5.2e$-1$ & 7.1e$-1$ & 5.1e$-1$ & 6.3e$-1$ \\ \hline\hline
\end{tabular}
}
\caption{Results obtained by the regularizing trust-region method and the regularizing
Levenberg-Marquardt method with noise $\delta=10^{-4}$ and varying initial guesses.}\label{table1}
\end{table}
\end{center}
\begin{center}
\begin{table
{\small
\begin{tabular}{lr|cccccc|ll}
\hline\hline
\mbox{Problem} & & \multicolumn{6}{c|}{RTR} & \multicolumn{2}{c}{RLM} \\
& $x_0$ & {\tt it} & {\tt $\vert\vert F \vert\vert$} & nf & {\tt cf} & {\tt $e_I$} & {\tt $e_T$} & {\tt $e_I$}& {\tt $e_T$}\\ \hline\hline
{\tt P1} & 0$\,e$ & \z20 & 1.5e$-$2 & {\z21} & {6} & 1.9e$-2$ & 1.9e$-2$ & 1.8e$-2$ & 1.8e$-2$ \\
&$-0.5\, e$ & \phantom{0} 29 & { 1.0e$-$2}& {\phantom{0} 30} & {6} & 2.2e$-2$ & 3.1e$-1$ & 2.1e$-2$ & 3.1e$-1$\\
&$-1\, e$ & {\z35} & { 1.4e$-$2} & {\phantom{0} 36} & {5} & 3.6e$-2$ & 6.1e$-1$ & 3.3e$-2$ & 6.1e$-1$ \\
&$-2\, e$ & {\phantom{0} 40} & { 1.3e$-$2} & {\phantom{0} 41} & {5} & 4.9e$-2$ & 1.2e$+$0 & 4.5e$-2$ & 1.2e$+$0 \\
\hline
{\tt P2} & $0\,e$ & {\phantom{0} 30} & { 1.4e$-$2} & {\phantom{0} 31} &{5} & 6.9e$-3$& 1.3e$-2$ & * & * \\
&$0.5\, e$ & {\phantom{0} 25} & { 1.4e$-$2} & {\phantom{0} 26} & {5} & 1.7e$-2$& 2.1e$-1$ & * & * \\
&$1\, e$ & {\phantom{0} 29} & { 1.4e$-$2} & {\phantom{0} 30} & {5} & 3.8e$-2$& 5.4e$-1$ & 1.3e$-1$ & 5.2e$-1$ \\
&$2\, e$ & {\phantom{0} 37} & { 1.4e$-$2} & {\phantom{0} 39} & {5} & 5.5e$-2$ & 1.2e$+$0 & 2.2e$-1$ & 1.1e$+$0 \\
\hline
{\tt P3}&$x_0(1.25)$ &{\phantom{0} 15} & {1.2e$-$2} &{\phantom{0} 16} & {4} & 1.5e$-1$ & 1.5e$-1$ & 1.5e$-1$ & 1.5e$-1$ \\
&$x_0(1.5)$ & {\phantom{0} 17} & { 1.4e$-$2} & {\phantom{0} 18} & {4} & 3.2e$-1$ & 3.2e$-1$ & 3.2e$-1$ & 3.2e$-1$\\
&$x_0(1.75)$ & {\phantom{0} 19} & { 1.4e$-$2} & {\phantom{0} 20} & {4} & 5.0e$-1$& 5.0e$-1$ & 5.1e$-1$ & 5.1e$-1$ \\
&$x_0(2)$ & {\phantom{0} 22} & { 1.5e$-$2} & {\phantom{0} 23} & {4} & 6.9e$-1$ & 6.9e$-1$ & 7.0e$-1$ & 7.0e$-1$ \\
\hline
{\tt P4}&$x_0(1,1)$&{\phantom{0} 17} & { 1.4e$-$2} &{\phantom{0} 18} & {5} & 5.7e$-1$ & 5.7e$-1$ & 5.4e$-1$ & 5.4e$-1$ \\
&$x_0(0.5,0)$ & {\z20} & { 1.3e$-$2} & {\phantom{0} 21} & {4}& 5.5e$-1$ & 5.5e$-1$ & * & * \\
&$x_0(1.5,1)$ & {\phantom{0} 22} & { 1.4e$-$2} & {\phantom{0} 23} & {4} & 5.1e$-1$ & 5.1e$-$1 & 5.0e$-1$ & 5.0e$-1$ \\
& $x_0(1.5,0)$& {\phantom{0} 26} & { 1.5e$-$2} & {\phantom{0} 27} & {4} & 5.2e$-1$ & 8.8e$-1$ & * & * \\ \hline\hline
\end{tabular}
}
\caption{Results obtained by the regularizing trust-region method and the regularizing
Levenberg-Marquardt method with noise $\delta=10^{-2}$ and varying initial guesses.}\label{table2}
\end{table}
\end{center}
Tables \ref{table1} and \ref{table2} show that the regularizing trust-region method solves all
the tests. By Step 3 of our Algorithm \ref{algoTR}, the difference between the
number of function evaluations and the number of trust-region iterations,
if greater than one, indicates the number of trial iterates that were rejected
because a sufficient reduction on $\Phi$ was not achieved.
We observe that in 20 out of 32 runs, all the iterates generated were accepted;
this occurrence seems to indicate that the trust-region updating rule works well in practice.
Further insight on the trust-region updating rule (\ref{tr_algo1})-(\ref{tr_algo2})
can be gained analyzing the regularizing properties
of the implemented trust-region strategy. First, we verified numerically that,
though not explicitly enforced, the $q$-condition is satisfied in most of the iterations.
As an illustrative example, we consider problem P2 with
$\delta=10^{-4}$ and $x_0=0e$ and, in the left plot in Figure \ref{fig:figure9},
we display the values
$q_k=\frac{\vert\vert J(x_k^{\delta})p_k+F(x_k^{\delta})\vert\vert}{\vert\vert F(x_k^{\delta})\vert\vert }$
at the trust-region iterations, marked by an asterisk, and the value
$q=1.1/\tau\approx0.733$ fixed in our experiments, depicted by a solid line.
We observe that, even if we have not imposed the $q$-condition, it is satisfied
at most of the iterations.
The plot on the right of Figure \ref{fig:figure9} shows
a monotone decay of the error between $x_k^{\delta}$ and $x^\dagger$ through the iterations,
which results to be in accordance with the theoretical results in Theorem \ref{lambdaTR_bound_noise}.
The regularizing properties of the implemented trust-region scheme are also shown in
Figure \ref{fig:figure10} where, for each test problem we plot the error
$\vert\vert x_{k^*}^\delta-x^{\dagger}\vert\vert$ for decreasing noise levels;
it is evident that, in accordance with theory,
the error decays as the noise level decreases.
\begin{figure}[h]
\centering
\includegraphics[width=2.4in,height=2in]{figure2/figure9.eps}
\includegraphics[width=2.4in,height=2in]{figure2/errore_monotono.eps}
\caption{Regularizing trust-region applied to P2, $x_0=0e$, $\delta=10^{-4}$: values $q_k=\frac{\vert\vert J(x_k^{\delta})p_k+F(x_k^{\delta})\vert\vert}{\vert\vert F(x_k^{\delta})\vert\vert }$ (marked by an asterisk) and value of $q=1.1/\tau$ (solid line) versus the iterations (on the left); semilog plot of the error $\vert\vert x_k^{\delta}-x^{\dagger}\vert\vert$ versus the iterations (on the right).}
\label{fig:figure9}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=2.4in,height=2in]{figure2/rumore_lungo_P1.eps}
\includegraphics[width=2.4in,height=2in]{figure2/rumore_lungo_P2.eps}
\includegraphics[width=2.4in,height=2in]{figure2/rumore_lungo_P3.eps}
\includegraphics[width=2.4in,height=2in]{figure2/rumore_lungo_P4.eps}
\caption{Regularizing trust-region applied to P1, $x_0=0e$ (top left), P2, $x_0=0e$ (top right), P3,
$x_0=x_0(\alpha)=x_0(1.25)$ (lower left) and to P4, $x_0=x_0(\beta, \chi)=x_0(0.5,0)$ (lower right):
log plot of the error $\vert\vert x_{k^*}^{\delta}-x^{\dagger}\vert\vert$
versus the noise $\delta$.}
\label{fig:figure10}
\end{figure}
Let now compare the regularizing trust-region and Levenberg-Marquardt procedures.
On runs successful for both methods, the two methods provide solutions
of similar accuracy and such an accuracy increases with the vicinity
of the initial guess to the true solution; as an example Figure \ref{fig:figure20} shows the solutions computed by
the two methods for problems P1 and P3 for $\delta=10^{-2}$. On the other hand,
for large noise $\delta$ and initial guesses farther from the true solution, for both methods the accuracy at the endpoints of the interval
$[0,1]$ may deteriorate; for this occurrence we refer to Table \ref{table2} and runs on
problems P1 and P2.
Concerning failures, in 7 runs out of 32 the Levenberg-Marquardt algorithm does not act as a regularizing method as the generated sequence
approaches a solution of the noisy problem.
In Figure \ref{fig:figure4} we illustrate two unsuccessful runs of the Levenberg-Marquardt method;
approximated solution computed by the regularizing trust-region and Levenberg-Marquardt procedures are shown for
runs on problems P2 and P4.
\begin{figure}[h]
\centering
\includegraphics[width=2.4in,height=2in]{figure2/figure21.eps}
\includegraphics[width=2.4in,height=2in]{figure2/figure20.eps}
\includegraphics[width=2.4in,height=2in]{figure2/P3_delta2_0.eps}
\includegraphics[width=2.4in,height=2in]{figure2/P3_rumore2_0_LM.eps}
\caption{Regularizing trust-region (left) and regularizing Levenberg-Marquardt (right), true solution (solid line) and approximate solutions (dotted line). Upper part: P1, $\delta=10^{-2}$, $x_0=0e$; lower part: P3, $\delta=10^{-2}$, $x_0=x_0(\alpha)=x_0(1.25)$.}
\label{fig:figure20}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=2.4in,height=2in]{figure2/figure5.eps}
\includegraphics[width=2.4in,height=2in]{figure2/figure4.eps}
\includegraphics[width=2.4in,height=2in]{figure2/P4_TR_bene.eps}
\includegraphics[width=2.4in,height=2in]{figure2/P4_LM_male.eps}
\caption{True solution (solid line) and approximate solutions (dotted line) computed by the regularizing trust-region method (on the left) and the regularizing Levenberg-Marquardt method (on the right). Upper part: problem P2, $\delta=10^{-2}, x_0=0e$;
lower part: problem P4, $\delta=10^{-2}, x_0=x_0(\beta, \chi)=x_0(0.5,0)$.
}
\label{fig:figure4}
\end{figure}
The overall experience on the Levenberg-Marquardt algorithm seems to indicate that
the use of the $q$-condition is more flexible than condition (\ref{seculare_q})
and provides stronger regularizing properties. In order to
support this claim, in Figure \ref{fig:LM_q} we report
four solution approximations computed by the Levenberg-Marquardt algorithm
for varying values of $q$, i.e. $q=0.67,0.70,0.73,0.87$. It is evident that
the method is highly sensitive to the choice of the parameter $q$ and the quality of the solution approximation does not steadily improves as $q$ increases.
\begin{figure}[h]
\centering
\includegraphics[width=2.4in,height=2in]{figure2/LM_101_new.eps}
\includegraphics[width=2.4in,height=2in]{figure2/LM_105_new.eps}
\includegraphics[width=2.4in,height=2in]{figure2/LM_11_new.eps}
\includegraphics[width=2.4in,height=2in]{figure2/LM_13_new.eps}
\caption{Problem P4, $\delta=10^{-2}$, $x_0=x_0(\beta,\chi)=x_0(1.5,0)$:
approximate solution computed by the regularizing Levenberg-Marquardt
method for values of $q=0.67,\, 0.70,\, 0.73,\, 0.87$. }
\label{fig:LM_q}
\end{figure}
\begin{figure}
\subfigure[]{\includegraphics[width=2.4in,height=2in]{figure2/figure6.eps}}
\subfigure[]{\includegraphics[width=2.4in,height=2in]{figure2/figure_tr_classica.eps}}
\subfigure[]{\includegraphics[width=2.4in,height=2in]{figure2/figure8_1.eps}}
\subfigure[]{\includegraphics[width=2.4in,height=2in]{figure2/figure8.eps}}
\subfigure[]{\includegraphics[width=2.4in,height=2in]{figure2/figure_tr_reg3.eps}}
\subfigure[]{\includegraphics[width=2.4in,height=2in]{figure2/figure_tr_classica3.eps}}
\subfigure[]{\includegraphics[width=2.4in,height=2in]{figure2/figure_tr_reg4.eps}}
\subfigure[]{\includegraphics[width=2.4in,height=2in]{figure2/figure_tr_classica4.eps}}
\caption{True solution (solid line) and approximate solutions (dotted line) computed by the regularizing trust-region method (on the left) and the standard trust-region method (on the right). (a)-(b) problem P1, $\delta=10^{-2}, x_0=0e$; (c)-(d) problem P2, $\delta=10^{-2}, x_0=0e$; (e)-(f) problem P3, $\delta=10^{-2}$, $x_0=x_0(1.25)$; (g)-(h) problem P4, $\delta=10^{-2}$, $x_0=x_0(0.5,0)$.}
\label{fig:figure8}
\end{figure}
We conclude this section considering the standard trust-region strategy.
It is well known that the standard updating rule promotes
the use of inactive trust-regions, at least in the late stage of the procedure.
Clearly, this can adversely affect the solution of our test problems
and our experiments confirmed this fact. In particular, for $\delta=10^{-2}$ and
problems P1 and P2, the sequences computed by the standard trust-region method approach solutions of the noisy problem. The same behaviour occurs in most of the runs with P1 and P2 and noise level $\delta=10^{-4}$. Conversely, the approximations provided by the regularizing trust-region procedure are accurate approximations of true solutions in all the tests.
Moreover, the approximations computed by the standard trust-region applied to problems P3 and P4 are less accurate than those computed by the regularizing trust-region although they do not show the strong oscillatory behavior arising in problems P1 and P2. In problem P4, this behavior is evident when the second, third and fourth starting guesses are used, while the approximation computed starting from the first initial guess is as accurate as the one computed by the regularizing trust-region. This good result of the standard trust-region on problem P4 with $x_0=x_0(1,1)$ is due to the fact that the trust-region is active in all iterations and therefore a regularizing behaviour is implicitely provided. As an example in Figure \ref{fig:figure8} we compare some solution approximations computed by the regularizing trust-region (left) and by the standard trust-region (right) with $\delta=10^{-2}$ applied to problem P1 (figures (a)-(b)), P2 (figures (c)-(d)), P3 (figures (e)-(f)) and P4 (figures (g)-(h)).
\section{Conclusions}
We have presented a trust-region method for nonlinear ill-posed systems, possibly with noisy data, where the regularizing behaviour is guaranteed by a suitable choice of the trust-region radius.
The proposed approach shares the same local convergence properties as the regularizing Levenberg-Marquardt method
proposed by Hanke in \cite{hanke} but it is more likely to satisfy th discrepancy principle irrespective of the closeness of the initial guess to
a solution of (\ref{prob}).
The numerical experience presented confirms the effectiveness of the
trust-region radius adopted and the regularizing properties of the resulting trust-region method. It also enlights that the new approach is less sensitive than the regularizing Levenberg-Marquardt method to the choice of the parameter $q$ in (\ref{seculare_q}) and (\ref{RQ}). Finally, as expected the numerical results show that the solution of the noisy problems may be misinterpreted by the standard trust-region method.
|
{
"timestamp": "2015-04-15T02:06:01",
"yymm": "1504",
"arxiv_id": "1504.03442",
"language": "en",
"url": "https://arxiv.org/abs/1504.03442"
}
|
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